klokishner sophia
TRANSCRIPT
(1) Institute of Applied Physics of the Academy of Sciences of Moldova
Academy str5 Chisinau MD 2028 Moldova
(2) Department of Chemistry Texas AampM University P O Box 30012College StationTX 77843-3012 USA
(3) Chemistry Department Ben-Gurion University of the Negev Beer-Sheva 84105 Israel
Origin of Magnetic Anisotropyin Single Molecule and Single Chain Magnets
Containing Ions with Unquenched Orbital Angular Momenta
S Klokishner (1) A Palii (1) SOstrovsky (1) OReu (1) PTregenna-Piggott K Dunbar (2) BTsukerblat(3)
OUTLINE
1 General introduction
2 Magnetic anisotropy in the ReII4MnII
4single molecule magnet quantum-spin and classical-spin approach
3 Experimental study and theoretical modeling of the CuII
2TbII2 single molecule
magnet
4 Basic ingredients of single-chain magnet behavior
5 Highly anisotropic Co(II)-based single-chain magnet
Antiferromagnetic coupling between
Mn3+ and Mn4+
Ground state S=10
SINGLE MOLECULAR MAGNETS- DISCOVERY OF THE PHENOMENON
R Sessoli D Gatteschi A Caneschi M A Novak Nature 1993365 141
4S(Mn4+)-
ferromagnetic
8S(Mn3+)-
ferromagnetic
Mn12O12 (CH3COO)16 (H2O)4] ndashmolecule Mn12-acetateeight Mn3+ ions (si =2)
and four Mn4+ ions (si =32)
Quenched orbital angular momenta for the manganese ions
Mn(III) ions large spin S=2 and single ion easy axis anisotropy (single-ion ZFS parameter D~-35 cm-1) due to the Jahn-Teller elongation of the coordination octahedron
Zero-field splitting is a second-order effect Weak anisotropy
Mn(IV) ions very small zero-field splitting parameter
Zero-field splitting DSMS2 with DSlt0 uniaxial magnetic anisotropy
Mechanism of the formation of spin reversal barrier in the Mn12 cluster
Barrier heightU =S2DS
E(MS)=DS(MS2 S(S+1)3)
DS is the effective (molecular) zero-field
splitting parameter for the ground S-multiplet
DS=-064 K S=10
Barrier height E =S2DSThe widely used design rule rdquoto increase Srdquo is not as efficient as promised
bull The zero-field splitting parameter DS for the ground state decreases with increasing S
bull The barrier height does not increase with S as S2 but as S0
(OWaldmann InorgChem 2007 46 10035)
bull The strategy tordquo increase the total spin S of the ground staterdquo by synthesis of big spin-clusters has not produced better SMMs yet
SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta
KR Dunbar et al AngewChemIntEd
2003 421523
)1(3 lmE
)0(23 lmA
)( 421
3 tTMn(III)
Mn(II)
CN
Cyano-bridged cluster [MnIII(CN)6]2[MnII(tmphen2]3
Incorporation of orbitally degenerate 3d ndashmetal ions
Strongly magnetically anisotropic ground E-state
The interplay between strong single ion anisotropy arising from the trigonal crystal field combined with SO interaction and antiferromagnetic Heisenberg-type exchangerarr
appreciable barrier for reversal of magnetization
SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta
Trigonally distorted molecular cube[MnCl]4[Re(triphos)(CN)3]4
JAmChemSoc 2004 126 15004-15005
Rendashlight blue Mnndashred C-grayNndashdark blue Clndashyellow P-purple
Replacing 3d transition metal ions with 4d or 5d ones
bullSpin-orbit interaction in 4d and 5d ions is at least one order of magnitude larger than in 3d ions
bull Much stronger exchange interaction than in the case of 3d-ions
bullThe barrier for reversal of magnetization can be significantly increased compared to clusters of 3d-ions
bullDespite the prospect of stronger magnetic interactions only few SMMs incorporating heavier transition metal ions are known
Mononuclear Lanthanide Single Molecule Magnets
Introduction of lanthanide metal ions
SMMs Based on Polyoxometalates encapsulating lanthanides Er HoDy
ECoronado et al JACS20081308874 InorgChem2009483467
Much stronger single-ion anisotropy and much slower relaxation as compared with those exhibiting by 3d 4d and 5d ions
Presence of an axial crystal field acting on the 4f-ion and stabilizing a Stark sublevel with a large absolute value of the total angular momentum projection |MJ| thus achieving an easy axis of magnetization
Lanthanide Double-Decker Complexes Functioning as SMMs DyHoTb NIshikawa et al JACS20031258694
3d-4f Single Molecule Magnets
Linear trinuclear heterobimetallic Co2Gd complex
VChandrasekhar et alInorgChem2007465140
Ground term of the Gd3+-ion ndash8S72
Spin -ion
Co ndashions unquenched orbital angular momenta
Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]
VChandrasekhar et alInorgChem2008474918
bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought
by the Dy3+ ion
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2
Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are
omitted
Temperature dependenceof the out-of-phase ac
susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and
[NiIILTbIII(hfac)2]2 compounds
[CuLTb(hfac)2]2
SOsa et alJACS2004126420
MOTIVATION OF THE WORK
Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters
Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems
Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster
2
1
5
73
4
6
8
Z
C3
J
J J
J
J J
J
J
J
J
J
J
Re(II) Mn(II)
Molecular cube compressedalong one of the trigonal axes
No zero field splitting for the Re(II) ion with spin frac12
Mn(II) ions do not carry anymagnetic anisotropy
The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways
J J - exchange integrals J ne J
Trigonally distorted mixed-ligand surrounding of the Re(II) ion
Re(II)
P
PP
C
CC
Local trigonal Z-axis
Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5
Htrig=(lz 23)2^
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
gt0
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
||
lt0
Relevant case
Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||
2E (ml=1)||
5+6mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )
Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0
bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12
bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z
Exchange Interaction for the Re(II)-Mn(II) Pair
Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators
J(Re-Mn)rarr exchange interaction parameter2E (ml=1)
||5+6
mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|
Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian
bull One can restrict the consideration of the exchange interaction by the ground Kramers space
Hamiltonian of exchange interaction
Local (red) and molecular (blue) Z-axes and the network of exchange pathways
for the Re4Mn4 ndashcluster
8ˆ6ˆ4ˆ
8ˆ5ˆ3ˆ
6ˆ5ˆ2ˆ2
8ˆ7ˆ6ˆ4ˆ
8ˆ6ˆ5ˆ1ˆ2ˆ
444
333
222
4444
1111
ZZZ
ZZZ
ZZZ
ZZZZ
ZZZZex
ss
ss
ssJ
sss
sssJH
Re(II) Mn(II)
2
1
5
73
4
6
8
Z1
C3(Z)
J
J J
J
J J
J
J
J
J
J
J
Z2
Z3
Z4
All operators are defined in the local frames
iiZ - is the z -component
5ˆ1Zs - is the z -component
of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc
of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame
0 50 100 150 200 250 30012
14
16
18
20
22
24
g|| Re = 4 gMn = 195
J = -35 cm-1
J = -105 cm-1
T c
m3 K
mol
-1
Temperature K
Comparison between theory and experiment
Temperature K
J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
3K4K
2KT
M
B
H THT
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
OUTLINE
1 General introduction
2 Magnetic anisotropy in the ReII4MnII
4single molecule magnet quantum-spin and classical-spin approach
3 Experimental study and theoretical modeling of the CuII
2TbII2 single molecule
magnet
4 Basic ingredients of single-chain magnet behavior
5 Highly anisotropic Co(II)-based single-chain magnet
Antiferromagnetic coupling between
Mn3+ and Mn4+
Ground state S=10
SINGLE MOLECULAR MAGNETS- DISCOVERY OF THE PHENOMENON
R Sessoli D Gatteschi A Caneschi M A Novak Nature 1993365 141
4S(Mn4+)-
ferromagnetic
8S(Mn3+)-
ferromagnetic
Mn12O12 (CH3COO)16 (H2O)4] ndashmolecule Mn12-acetateeight Mn3+ ions (si =2)
and four Mn4+ ions (si =32)
Quenched orbital angular momenta for the manganese ions
Mn(III) ions large spin S=2 and single ion easy axis anisotropy (single-ion ZFS parameter D~-35 cm-1) due to the Jahn-Teller elongation of the coordination octahedron
Zero-field splitting is a second-order effect Weak anisotropy
Mn(IV) ions very small zero-field splitting parameter
Zero-field splitting DSMS2 with DSlt0 uniaxial magnetic anisotropy
Mechanism of the formation of spin reversal barrier in the Mn12 cluster
Barrier heightU =S2DS
E(MS)=DS(MS2 S(S+1)3)
DS is the effective (molecular) zero-field
splitting parameter for the ground S-multiplet
DS=-064 K S=10
Barrier height E =S2DSThe widely used design rule rdquoto increase Srdquo is not as efficient as promised
bull The zero-field splitting parameter DS for the ground state decreases with increasing S
bull The barrier height does not increase with S as S2 but as S0
(OWaldmann InorgChem 2007 46 10035)
bull The strategy tordquo increase the total spin S of the ground staterdquo by synthesis of big spin-clusters has not produced better SMMs yet
SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta
KR Dunbar et al AngewChemIntEd
2003 421523
)1(3 lmE
)0(23 lmA
)( 421
3 tTMn(III)
Mn(II)
CN
Cyano-bridged cluster [MnIII(CN)6]2[MnII(tmphen2]3
Incorporation of orbitally degenerate 3d ndashmetal ions
Strongly magnetically anisotropic ground E-state
The interplay between strong single ion anisotropy arising from the trigonal crystal field combined with SO interaction and antiferromagnetic Heisenberg-type exchangerarr
appreciable barrier for reversal of magnetization
SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta
Trigonally distorted molecular cube[MnCl]4[Re(triphos)(CN)3]4
JAmChemSoc 2004 126 15004-15005
Rendashlight blue Mnndashred C-grayNndashdark blue Clndashyellow P-purple
Replacing 3d transition metal ions with 4d or 5d ones
bullSpin-orbit interaction in 4d and 5d ions is at least one order of magnitude larger than in 3d ions
bull Much stronger exchange interaction than in the case of 3d-ions
bullThe barrier for reversal of magnetization can be significantly increased compared to clusters of 3d-ions
bullDespite the prospect of stronger magnetic interactions only few SMMs incorporating heavier transition metal ions are known
Mononuclear Lanthanide Single Molecule Magnets
Introduction of lanthanide metal ions
SMMs Based on Polyoxometalates encapsulating lanthanides Er HoDy
ECoronado et al JACS20081308874 InorgChem2009483467
Much stronger single-ion anisotropy and much slower relaxation as compared with those exhibiting by 3d 4d and 5d ions
Presence of an axial crystal field acting on the 4f-ion and stabilizing a Stark sublevel with a large absolute value of the total angular momentum projection |MJ| thus achieving an easy axis of magnetization
Lanthanide Double-Decker Complexes Functioning as SMMs DyHoTb NIshikawa et al JACS20031258694
3d-4f Single Molecule Magnets
Linear trinuclear heterobimetallic Co2Gd complex
VChandrasekhar et alInorgChem2007465140
Ground term of the Gd3+-ion ndash8S72
Spin -ion
Co ndashions unquenched orbital angular momenta
Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]
VChandrasekhar et alInorgChem2008474918
bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought
by the Dy3+ ion
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2
Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are
omitted
Temperature dependenceof the out-of-phase ac
susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and
[NiIILTbIII(hfac)2]2 compounds
[CuLTb(hfac)2]2
SOsa et alJACS2004126420
MOTIVATION OF THE WORK
Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters
Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems
Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster
2
1
5
73
4
6
8
Z
C3
J
J J
J
J J
J
J
J
J
J
J
Re(II) Mn(II)
Molecular cube compressedalong one of the trigonal axes
No zero field splitting for the Re(II) ion with spin frac12
Mn(II) ions do not carry anymagnetic anisotropy
The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways
J J - exchange integrals J ne J
Trigonally distorted mixed-ligand surrounding of the Re(II) ion
Re(II)
P
PP
C
CC
Local trigonal Z-axis
Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5
Htrig=(lz 23)2^
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
gt0
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
||
lt0
Relevant case
Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||
2E (ml=1)||
5+6mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )
Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0
bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12
bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z
Exchange Interaction for the Re(II)-Mn(II) Pair
Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators
J(Re-Mn)rarr exchange interaction parameter2E (ml=1)
||5+6
mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|
Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian
bull One can restrict the consideration of the exchange interaction by the ground Kramers space
Hamiltonian of exchange interaction
Local (red) and molecular (blue) Z-axes and the network of exchange pathways
for the Re4Mn4 ndashcluster
8ˆ6ˆ4ˆ
8ˆ5ˆ3ˆ
6ˆ5ˆ2ˆ2
8ˆ7ˆ6ˆ4ˆ
8ˆ6ˆ5ˆ1ˆ2ˆ
444
333
222
4444
1111
ZZZ
ZZZ
ZZZ
ZZZZ
ZZZZex
ss
ss
ssJ
sss
sssJH
Re(II) Mn(II)
2
1
5
73
4
6
8
Z1
C3(Z)
J
J J
J
J J
J
J
J
J
J
J
Z2
Z3
Z4
All operators are defined in the local frames
iiZ - is the z -component
5ˆ1Zs - is the z -component
of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc
of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame
0 50 100 150 200 250 30012
14
16
18
20
22
24
g|| Re = 4 gMn = 195
J = -35 cm-1
J = -105 cm-1
T c
m3 K
mol
-1
Temperature K
Comparison between theory and experiment
Temperature K
J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
3K4K
2KT
M
B
H THT
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
Antiferromagnetic coupling between
Mn3+ and Mn4+
Ground state S=10
SINGLE MOLECULAR MAGNETS- DISCOVERY OF THE PHENOMENON
R Sessoli D Gatteschi A Caneschi M A Novak Nature 1993365 141
4S(Mn4+)-
ferromagnetic
8S(Mn3+)-
ferromagnetic
Mn12O12 (CH3COO)16 (H2O)4] ndashmolecule Mn12-acetateeight Mn3+ ions (si =2)
and four Mn4+ ions (si =32)
Quenched orbital angular momenta for the manganese ions
Mn(III) ions large spin S=2 and single ion easy axis anisotropy (single-ion ZFS parameter D~-35 cm-1) due to the Jahn-Teller elongation of the coordination octahedron
Zero-field splitting is a second-order effect Weak anisotropy
Mn(IV) ions very small zero-field splitting parameter
Zero-field splitting DSMS2 with DSlt0 uniaxial magnetic anisotropy
Mechanism of the formation of spin reversal barrier in the Mn12 cluster
Barrier heightU =S2DS
E(MS)=DS(MS2 S(S+1)3)
DS is the effective (molecular) zero-field
splitting parameter for the ground S-multiplet
DS=-064 K S=10
Barrier height E =S2DSThe widely used design rule rdquoto increase Srdquo is not as efficient as promised
bull The zero-field splitting parameter DS for the ground state decreases with increasing S
bull The barrier height does not increase with S as S2 but as S0
(OWaldmann InorgChem 2007 46 10035)
bull The strategy tordquo increase the total spin S of the ground staterdquo by synthesis of big spin-clusters has not produced better SMMs yet
SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta
KR Dunbar et al AngewChemIntEd
2003 421523
)1(3 lmE
)0(23 lmA
)( 421
3 tTMn(III)
Mn(II)
CN
Cyano-bridged cluster [MnIII(CN)6]2[MnII(tmphen2]3
Incorporation of orbitally degenerate 3d ndashmetal ions
Strongly magnetically anisotropic ground E-state
The interplay between strong single ion anisotropy arising from the trigonal crystal field combined with SO interaction and antiferromagnetic Heisenberg-type exchangerarr
appreciable barrier for reversal of magnetization
SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta
Trigonally distorted molecular cube[MnCl]4[Re(triphos)(CN)3]4
JAmChemSoc 2004 126 15004-15005
Rendashlight blue Mnndashred C-grayNndashdark blue Clndashyellow P-purple
Replacing 3d transition metal ions with 4d or 5d ones
bullSpin-orbit interaction in 4d and 5d ions is at least one order of magnitude larger than in 3d ions
bull Much stronger exchange interaction than in the case of 3d-ions
bullThe barrier for reversal of magnetization can be significantly increased compared to clusters of 3d-ions
bullDespite the prospect of stronger magnetic interactions only few SMMs incorporating heavier transition metal ions are known
Mononuclear Lanthanide Single Molecule Magnets
Introduction of lanthanide metal ions
SMMs Based on Polyoxometalates encapsulating lanthanides Er HoDy
ECoronado et al JACS20081308874 InorgChem2009483467
Much stronger single-ion anisotropy and much slower relaxation as compared with those exhibiting by 3d 4d and 5d ions
Presence of an axial crystal field acting on the 4f-ion and stabilizing a Stark sublevel with a large absolute value of the total angular momentum projection |MJ| thus achieving an easy axis of magnetization
Lanthanide Double-Decker Complexes Functioning as SMMs DyHoTb NIshikawa et al JACS20031258694
3d-4f Single Molecule Magnets
Linear trinuclear heterobimetallic Co2Gd complex
VChandrasekhar et alInorgChem2007465140
Ground term of the Gd3+-ion ndash8S72
Spin -ion
Co ndashions unquenched orbital angular momenta
Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]
VChandrasekhar et alInorgChem2008474918
bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought
by the Dy3+ ion
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2
Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are
omitted
Temperature dependenceof the out-of-phase ac
susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and
[NiIILTbIII(hfac)2]2 compounds
[CuLTb(hfac)2]2
SOsa et alJACS2004126420
MOTIVATION OF THE WORK
Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters
Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems
Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster
2
1
5
73
4
6
8
Z
C3
J
J J
J
J J
J
J
J
J
J
J
Re(II) Mn(II)
Molecular cube compressedalong one of the trigonal axes
No zero field splitting for the Re(II) ion with spin frac12
Mn(II) ions do not carry anymagnetic anisotropy
The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways
J J - exchange integrals J ne J
Trigonally distorted mixed-ligand surrounding of the Re(II) ion
Re(II)
P
PP
C
CC
Local trigonal Z-axis
Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5
Htrig=(lz 23)2^
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
gt0
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
||
lt0
Relevant case
Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||
2E (ml=1)||
5+6mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )
Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0
bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12
bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z
Exchange Interaction for the Re(II)-Mn(II) Pair
Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators
J(Re-Mn)rarr exchange interaction parameter2E (ml=1)
||5+6
mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|
Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian
bull One can restrict the consideration of the exchange interaction by the ground Kramers space
Hamiltonian of exchange interaction
Local (red) and molecular (blue) Z-axes and the network of exchange pathways
for the Re4Mn4 ndashcluster
8ˆ6ˆ4ˆ
8ˆ5ˆ3ˆ
6ˆ5ˆ2ˆ2
8ˆ7ˆ6ˆ4ˆ
8ˆ6ˆ5ˆ1ˆ2ˆ
444
333
222
4444
1111
ZZZ
ZZZ
ZZZ
ZZZZ
ZZZZex
ss
ss
ssJ
sss
sssJH
Re(II) Mn(II)
2
1
5
73
4
6
8
Z1
C3(Z)
J
J J
J
J J
J
J
J
J
J
J
Z2
Z3
Z4
All operators are defined in the local frames
iiZ - is the z -component
5ˆ1Zs - is the z -component
of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc
of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame
0 50 100 150 200 250 30012
14
16
18
20
22
24
g|| Re = 4 gMn = 195
J = -35 cm-1
J = -105 cm-1
T c
m3 K
mol
-1
Temperature K
Comparison between theory and experiment
Temperature K
J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
3K4K
2KT
M
B
H THT
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
Quenched orbital angular momenta for the manganese ions
Mn(III) ions large spin S=2 and single ion easy axis anisotropy (single-ion ZFS parameter D~-35 cm-1) due to the Jahn-Teller elongation of the coordination octahedron
Zero-field splitting is a second-order effect Weak anisotropy
Mn(IV) ions very small zero-field splitting parameter
Zero-field splitting DSMS2 with DSlt0 uniaxial magnetic anisotropy
Mechanism of the formation of spin reversal barrier in the Mn12 cluster
Barrier heightU =S2DS
E(MS)=DS(MS2 S(S+1)3)
DS is the effective (molecular) zero-field
splitting parameter for the ground S-multiplet
DS=-064 K S=10
Barrier height E =S2DSThe widely used design rule rdquoto increase Srdquo is not as efficient as promised
bull The zero-field splitting parameter DS for the ground state decreases with increasing S
bull The barrier height does not increase with S as S2 but as S0
(OWaldmann InorgChem 2007 46 10035)
bull The strategy tordquo increase the total spin S of the ground staterdquo by synthesis of big spin-clusters has not produced better SMMs yet
SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta
KR Dunbar et al AngewChemIntEd
2003 421523
)1(3 lmE
)0(23 lmA
)( 421
3 tTMn(III)
Mn(II)
CN
Cyano-bridged cluster [MnIII(CN)6]2[MnII(tmphen2]3
Incorporation of orbitally degenerate 3d ndashmetal ions
Strongly magnetically anisotropic ground E-state
The interplay between strong single ion anisotropy arising from the trigonal crystal field combined with SO interaction and antiferromagnetic Heisenberg-type exchangerarr
appreciable barrier for reversal of magnetization
SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta
Trigonally distorted molecular cube[MnCl]4[Re(triphos)(CN)3]4
JAmChemSoc 2004 126 15004-15005
Rendashlight blue Mnndashred C-grayNndashdark blue Clndashyellow P-purple
Replacing 3d transition metal ions with 4d or 5d ones
bullSpin-orbit interaction in 4d and 5d ions is at least one order of magnitude larger than in 3d ions
bull Much stronger exchange interaction than in the case of 3d-ions
bullThe barrier for reversal of magnetization can be significantly increased compared to clusters of 3d-ions
bullDespite the prospect of stronger magnetic interactions only few SMMs incorporating heavier transition metal ions are known
Mononuclear Lanthanide Single Molecule Magnets
Introduction of lanthanide metal ions
SMMs Based on Polyoxometalates encapsulating lanthanides Er HoDy
ECoronado et al JACS20081308874 InorgChem2009483467
Much stronger single-ion anisotropy and much slower relaxation as compared with those exhibiting by 3d 4d and 5d ions
Presence of an axial crystal field acting on the 4f-ion and stabilizing a Stark sublevel with a large absolute value of the total angular momentum projection |MJ| thus achieving an easy axis of magnetization
Lanthanide Double-Decker Complexes Functioning as SMMs DyHoTb NIshikawa et al JACS20031258694
3d-4f Single Molecule Magnets
Linear trinuclear heterobimetallic Co2Gd complex
VChandrasekhar et alInorgChem2007465140
Ground term of the Gd3+-ion ndash8S72
Spin -ion
Co ndashions unquenched orbital angular momenta
Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]
VChandrasekhar et alInorgChem2008474918
bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought
by the Dy3+ ion
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2
Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are
omitted
Temperature dependenceof the out-of-phase ac
susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and
[NiIILTbIII(hfac)2]2 compounds
[CuLTb(hfac)2]2
SOsa et alJACS2004126420
MOTIVATION OF THE WORK
Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters
Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems
Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster
2
1
5
73
4
6
8
Z
C3
J
J J
J
J J
J
J
J
J
J
J
Re(II) Mn(II)
Molecular cube compressedalong one of the trigonal axes
No zero field splitting for the Re(II) ion with spin frac12
Mn(II) ions do not carry anymagnetic anisotropy
The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways
J J - exchange integrals J ne J
Trigonally distorted mixed-ligand surrounding of the Re(II) ion
Re(II)
P
PP
C
CC
Local trigonal Z-axis
Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5
Htrig=(lz 23)2^
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
gt0
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
||
lt0
Relevant case
Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||
2E (ml=1)||
5+6mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )
Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0
bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12
bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z
Exchange Interaction for the Re(II)-Mn(II) Pair
Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators
J(Re-Mn)rarr exchange interaction parameter2E (ml=1)
||5+6
mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|
Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian
bull One can restrict the consideration of the exchange interaction by the ground Kramers space
Hamiltonian of exchange interaction
Local (red) and molecular (blue) Z-axes and the network of exchange pathways
for the Re4Mn4 ndashcluster
8ˆ6ˆ4ˆ
8ˆ5ˆ3ˆ
6ˆ5ˆ2ˆ2
8ˆ7ˆ6ˆ4ˆ
8ˆ6ˆ5ˆ1ˆ2ˆ
444
333
222
4444
1111
ZZZ
ZZZ
ZZZ
ZZZZ
ZZZZex
ss
ss
ssJ
sss
sssJH
Re(II) Mn(II)
2
1
5
73
4
6
8
Z1
C3(Z)
J
J J
J
J J
J
J
J
J
J
J
Z2
Z3
Z4
All operators are defined in the local frames
iiZ - is the z -component
5ˆ1Zs - is the z -component
of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc
of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame
0 50 100 150 200 250 30012
14
16
18
20
22
24
g|| Re = 4 gMn = 195
J = -35 cm-1
J = -105 cm-1
T c
m3 K
mol
-1
Temperature K
Comparison between theory and experiment
Temperature K
J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
3K4K
2KT
M
B
H THT
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
Barrier height E =S2DSThe widely used design rule rdquoto increase Srdquo is not as efficient as promised
bull The zero-field splitting parameter DS for the ground state decreases with increasing S
bull The barrier height does not increase with S as S2 but as S0
(OWaldmann InorgChem 2007 46 10035)
bull The strategy tordquo increase the total spin S of the ground staterdquo by synthesis of big spin-clusters has not produced better SMMs yet
SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta
KR Dunbar et al AngewChemIntEd
2003 421523
)1(3 lmE
)0(23 lmA
)( 421
3 tTMn(III)
Mn(II)
CN
Cyano-bridged cluster [MnIII(CN)6]2[MnII(tmphen2]3
Incorporation of orbitally degenerate 3d ndashmetal ions
Strongly magnetically anisotropic ground E-state
The interplay between strong single ion anisotropy arising from the trigonal crystal field combined with SO interaction and antiferromagnetic Heisenberg-type exchangerarr
appreciable barrier for reversal of magnetization
SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta
Trigonally distorted molecular cube[MnCl]4[Re(triphos)(CN)3]4
JAmChemSoc 2004 126 15004-15005
Rendashlight blue Mnndashred C-grayNndashdark blue Clndashyellow P-purple
Replacing 3d transition metal ions with 4d or 5d ones
bullSpin-orbit interaction in 4d and 5d ions is at least one order of magnitude larger than in 3d ions
bull Much stronger exchange interaction than in the case of 3d-ions
bullThe barrier for reversal of magnetization can be significantly increased compared to clusters of 3d-ions
bullDespite the prospect of stronger magnetic interactions only few SMMs incorporating heavier transition metal ions are known
Mononuclear Lanthanide Single Molecule Magnets
Introduction of lanthanide metal ions
SMMs Based on Polyoxometalates encapsulating lanthanides Er HoDy
ECoronado et al JACS20081308874 InorgChem2009483467
Much stronger single-ion anisotropy and much slower relaxation as compared with those exhibiting by 3d 4d and 5d ions
Presence of an axial crystal field acting on the 4f-ion and stabilizing a Stark sublevel with a large absolute value of the total angular momentum projection |MJ| thus achieving an easy axis of magnetization
Lanthanide Double-Decker Complexes Functioning as SMMs DyHoTb NIshikawa et al JACS20031258694
3d-4f Single Molecule Magnets
Linear trinuclear heterobimetallic Co2Gd complex
VChandrasekhar et alInorgChem2007465140
Ground term of the Gd3+-ion ndash8S72
Spin -ion
Co ndashions unquenched orbital angular momenta
Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]
VChandrasekhar et alInorgChem2008474918
bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought
by the Dy3+ ion
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2
Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are
omitted
Temperature dependenceof the out-of-phase ac
susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and
[NiIILTbIII(hfac)2]2 compounds
[CuLTb(hfac)2]2
SOsa et alJACS2004126420
MOTIVATION OF THE WORK
Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters
Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems
Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster
2
1
5
73
4
6
8
Z
C3
J
J J
J
J J
J
J
J
J
J
J
Re(II) Mn(II)
Molecular cube compressedalong one of the trigonal axes
No zero field splitting for the Re(II) ion with spin frac12
Mn(II) ions do not carry anymagnetic anisotropy
The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways
J J - exchange integrals J ne J
Trigonally distorted mixed-ligand surrounding of the Re(II) ion
Re(II)
P
PP
C
CC
Local trigonal Z-axis
Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5
Htrig=(lz 23)2^
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
gt0
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
||
lt0
Relevant case
Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||
2E (ml=1)||
5+6mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )
Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0
bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12
bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z
Exchange Interaction for the Re(II)-Mn(II) Pair
Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators
J(Re-Mn)rarr exchange interaction parameter2E (ml=1)
||5+6
mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|
Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian
bull One can restrict the consideration of the exchange interaction by the ground Kramers space
Hamiltonian of exchange interaction
Local (red) and molecular (blue) Z-axes and the network of exchange pathways
for the Re4Mn4 ndashcluster
8ˆ6ˆ4ˆ
8ˆ5ˆ3ˆ
6ˆ5ˆ2ˆ2
8ˆ7ˆ6ˆ4ˆ
8ˆ6ˆ5ˆ1ˆ2ˆ
444
333
222
4444
1111
ZZZ
ZZZ
ZZZ
ZZZZ
ZZZZex
ss
ss
ssJ
sss
sssJH
Re(II) Mn(II)
2
1
5
73
4
6
8
Z1
C3(Z)
J
J J
J
J J
J
J
J
J
J
J
Z2
Z3
Z4
All operators are defined in the local frames
iiZ - is the z -component
5ˆ1Zs - is the z -component
of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc
of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame
0 50 100 150 200 250 30012
14
16
18
20
22
24
g|| Re = 4 gMn = 195
J = -35 cm-1
J = -105 cm-1
T c
m3 K
mol
-1
Temperature K
Comparison between theory and experiment
Temperature K
J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
3K4K
2KT
M
B
H THT
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta
KR Dunbar et al AngewChemIntEd
2003 421523
)1(3 lmE
)0(23 lmA
)( 421
3 tTMn(III)
Mn(II)
CN
Cyano-bridged cluster [MnIII(CN)6]2[MnII(tmphen2]3
Incorporation of orbitally degenerate 3d ndashmetal ions
Strongly magnetically anisotropic ground E-state
The interplay between strong single ion anisotropy arising from the trigonal crystal field combined with SO interaction and antiferromagnetic Heisenberg-type exchangerarr
appreciable barrier for reversal of magnetization
SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta
Trigonally distorted molecular cube[MnCl]4[Re(triphos)(CN)3]4
JAmChemSoc 2004 126 15004-15005
Rendashlight blue Mnndashred C-grayNndashdark blue Clndashyellow P-purple
Replacing 3d transition metal ions with 4d or 5d ones
bullSpin-orbit interaction in 4d and 5d ions is at least one order of magnitude larger than in 3d ions
bull Much stronger exchange interaction than in the case of 3d-ions
bullThe barrier for reversal of magnetization can be significantly increased compared to clusters of 3d-ions
bullDespite the prospect of stronger magnetic interactions only few SMMs incorporating heavier transition metal ions are known
Mononuclear Lanthanide Single Molecule Magnets
Introduction of lanthanide metal ions
SMMs Based on Polyoxometalates encapsulating lanthanides Er HoDy
ECoronado et al JACS20081308874 InorgChem2009483467
Much stronger single-ion anisotropy and much slower relaxation as compared with those exhibiting by 3d 4d and 5d ions
Presence of an axial crystal field acting on the 4f-ion and stabilizing a Stark sublevel with a large absolute value of the total angular momentum projection |MJ| thus achieving an easy axis of magnetization
Lanthanide Double-Decker Complexes Functioning as SMMs DyHoTb NIshikawa et al JACS20031258694
3d-4f Single Molecule Magnets
Linear trinuclear heterobimetallic Co2Gd complex
VChandrasekhar et alInorgChem2007465140
Ground term of the Gd3+-ion ndash8S72
Spin -ion
Co ndashions unquenched orbital angular momenta
Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]
VChandrasekhar et alInorgChem2008474918
bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought
by the Dy3+ ion
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2
Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are
omitted
Temperature dependenceof the out-of-phase ac
susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and
[NiIILTbIII(hfac)2]2 compounds
[CuLTb(hfac)2]2
SOsa et alJACS2004126420
MOTIVATION OF THE WORK
Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters
Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems
Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster
2
1
5
73
4
6
8
Z
C3
J
J J
J
J J
J
J
J
J
J
J
Re(II) Mn(II)
Molecular cube compressedalong one of the trigonal axes
No zero field splitting for the Re(II) ion with spin frac12
Mn(II) ions do not carry anymagnetic anisotropy
The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways
J J - exchange integrals J ne J
Trigonally distorted mixed-ligand surrounding of the Re(II) ion
Re(II)
P
PP
C
CC
Local trigonal Z-axis
Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5
Htrig=(lz 23)2^
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
gt0
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
||
lt0
Relevant case
Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||
2E (ml=1)||
5+6mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )
Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0
bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12
bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z
Exchange Interaction for the Re(II)-Mn(II) Pair
Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators
J(Re-Mn)rarr exchange interaction parameter2E (ml=1)
||5+6
mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|
Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian
bull One can restrict the consideration of the exchange interaction by the ground Kramers space
Hamiltonian of exchange interaction
Local (red) and molecular (blue) Z-axes and the network of exchange pathways
for the Re4Mn4 ndashcluster
8ˆ6ˆ4ˆ
8ˆ5ˆ3ˆ
6ˆ5ˆ2ˆ2
8ˆ7ˆ6ˆ4ˆ
8ˆ6ˆ5ˆ1ˆ2ˆ
444
333
222
4444
1111
ZZZ
ZZZ
ZZZ
ZZZZ
ZZZZex
ss
ss
ssJ
sss
sssJH
Re(II) Mn(II)
2
1
5
73
4
6
8
Z1
C3(Z)
J
J J
J
J J
J
J
J
J
J
J
Z2
Z3
Z4
All operators are defined in the local frames
iiZ - is the z -component
5ˆ1Zs - is the z -component
of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc
of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame
0 50 100 150 200 250 30012
14
16
18
20
22
24
g|| Re = 4 gMn = 195
J = -35 cm-1
J = -105 cm-1
T c
m3 K
mol
-1
Temperature K
Comparison between theory and experiment
Temperature K
J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
3K4K
2KT
M
B
H THT
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta
Trigonally distorted molecular cube[MnCl]4[Re(triphos)(CN)3]4
JAmChemSoc 2004 126 15004-15005
Rendashlight blue Mnndashred C-grayNndashdark blue Clndashyellow P-purple
Replacing 3d transition metal ions with 4d or 5d ones
bullSpin-orbit interaction in 4d and 5d ions is at least one order of magnitude larger than in 3d ions
bull Much stronger exchange interaction than in the case of 3d-ions
bullThe barrier for reversal of magnetization can be significantly increased compared to clusters of 3d-ions
bullDespite the prospect of stronger magnetic interactions only few SMMs incorporating heavier transition metal ions are known
Mononuclear Lanthanide Single Molecule Magnets
Introduction of lanthanide metal ions
SMMs Based on Polyoxometalates encapsulating lanthanides Er HoDy
ECoronado et al JACS20081308874 InorgChem2009483467
Much stronger single-ion anisotropy and much slower relaxation as compared with those exhibiting by 3d 4d and 5d ions
Presence of an axial crystal field acting on the 4f-ion and stabilizing a Stark sublevel with a large absolute value of the total angular momentum projection |MJ| thus achieving an easy axis of magnetization
Lanthanide Double-Decker Complexes Functioning as SMMs DyHoTb NIshikawa et al JACS20031258694
3d-4f Single Molecule Magnets
Linear trinuclear heterobimetallic Co2Gd complex
VChandrasekhar et alInorgChem2007465140
Ground term of the Gd3+-ion ndash8S72
Spin -ion
Co ndashions unquenched orbital angular momenta
Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]
VChandrasekhar et alInorgChem2008474918
bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought
by the Dy3+ ion
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2
Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are
omitted
Temperature dependenceof the out-of-phase ac
susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and
[NiIILTbIII(hfac)2]2 compounds
[CuLTb(hfac)2]2
SOsa et alJACS2004126420
MOTIVATION OF THE WORK
Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters
Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems
Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster
2
1
5
73
4
6
8
Z
C3
J
J J
J
J J
J
J
J
J
J
J
Re(II) Mn(II)
Molecular cube compressedalong one of the trigonal axes
No zero field splitting for the Re(II) ion with spin frac12
Mn(II) ions do not carry anymagnetic anisotropy
The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways
J J - exchange integrals J ne J
Trigonally distorted mixed-ligand surrounding of the Re(II) ion
Re(II)
P
PP
C
CC
Local trigonal Z-axis
Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5
Htrig=(lz 23)2^
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
gt0
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
||
lt0
Relevant case
Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||
2E (ml=1)||
5+6mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )
Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0
bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12
bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z
Exchange Interaction for the Re(II)-Mn(II) Pair
Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators
J(Re-Mn)rarr exchange interaction parameter2E (ml=1)
||5+6
mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|
Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian
bull One can restrict the consideration of the exchange interaction by the ground Kramers space
Hamiltonian of exchange interaction
Local (red) and molecular (blue) Z-axes and the network of exchange pathways
for the Re4Mn4 ndashcluster
8ˆ6ˆ4ˆ
8ˆ5ˆ3ˆ
6ˆ5ˆ2ˆ2
8ˆ7ˆ6ˆ4ˆ
8ˆ6ˆ5ˆ1ˆ2ˆ
444
333
222
4444
1111
ZZZ
ZZZ
ZZZ
ZZZZ
ZZZZex
ss
ss
ssJ
sss
sssJH
Re(II) Mn(II)
2
1
5
73
4
6
8
Z1
C3(Z)
J
J J
J
J J
J
J
J
J
J
J
Z2
Z3
Z4
All operators are defined in the local frames
iiZ - is the z -component
5ˆ1Zs - is the z -component
of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc
of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame
0 50 100 150 200 250 30012
14
16
18
20
22
24
g|| Re = 4 gMn = 195
J = -35 cm-1
J = -105 cm-1
T c
m3 K
mol
-1
Temperature K
Comparison between theory and experiment
Temperature K
J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
3K4K
2KT
M
B
H THT
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
Mononuclear Lanthanide Single Molecule Magnets
Introduction of lanthanide metal ions
SMMs Based on Polyoxometalates encapsulating lanthanides Er HoDy
ECoronado et al JACS20081308874 InorgChem2009483467
Much stronger single-ion anisotropy and much slower relaxation as compared with those exhibiting by 3d 4d and 5d ions
Presence of an axial crystal field acting on the 4f-ion and stabilizing a Stark sublevel with a large absolute value of the total angular momentum projection |MJ| thus achieving an easy axis of magnetization
Lanthanide Double-Decker Complexes Functioning as SMMs DyHoTb NIshikawa et al JACS20031258694
3d-4f Single Molecule Magnets
Linear trinuclear heterobimetallic Co2Gd complex
VChandrasekhar et alInorgChem2007465140
Ground term of the Gd3+-ion ndash8S72
Spin -ion
Co ndashions unquenched orbital angular momenta
Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]
VChandrasekhar et alInorgChem2008474918
bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought
by the Dy3+ ion
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2
Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are
omitted
Temperature dependenceof the out-of-phase ac
susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and
[NiIILTbIII(hfac)2]2 compounds
[CuLTb(hfac)2]2
SOsa et alJACS2004126420
MOTIVATION OF THE WORK
Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters
Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems
Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster
2
1
5
73
4
6
8
Z
C3
J
J J
J
J J
J
J
J
J
J
J
Re(II) Mn(II)
Molecular cube compressedalong one of the trigonal axes
No zero field splitting for the Re(II) ion with spin frac12
Mn(II) ions do not carry anymagnetic anisotropy
The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways
J J - exchange integrals J ne J
Trigonally distorted mixed-ligand surrounding of the Re(II) ion
Re(II)
P
PP
C
CC
Local trigonal Z-axis
Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5
Htrig=(lz 23)2^
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
gt0
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
||
lt0
Relevant case
Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||
2E (ml=1)||
5+6mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )
Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0
bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12
bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z
Exchange Interaction for the Re(II)-Mn(II) Pair
Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators
J(Re-Mn)rarr exchange interaction parameter2E (ml=1)
||5+6
mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|
Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian
bull One can restrict the consideration of the exchange interaction by the ground Kramers space
Hamiltonian of exchange interaction
Local (red) and molecular (blue) Z-axes and the network of exchange pathways
for the Re4Mn4 ndashcluster
8ˆ6ˆ4ˆ
8ˆ5ˆ3ˆ
6ˆ5ˆ2ˆ2
8ˆ7ˆ6ˆ4ˆ
8ˆ6ˆ5ˆ1ˆ2ˆ
444
333
222
4444
1111
ZZZ
ZZZ
ZZZ
ZZZZ
ZZZZex
ss
ss
ssJ
sss
sssJH
Re(II) Mn(II)
2
1
5
73
4
6
8
Z1
C3(Z)
J
J J
J
J J
J
J
J
J
J
J
Z2
Z3
Z4
All operators are defined in the local frames
iiZ - is the z -component
5ˆ1Zs - is the z -component
of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc
of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame
0 50 100 150 200 250 30012
14
16
18
20
22
24
g|| Re = 4 gMn = 195
J = -35 cm-1
J = -105 cm-1
T c
m3 K
mol
-1
Temperature K
Comparison between theory and experiment
Temperature K
J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
3K4K
2KT
M
B
H THT
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
3d-4f Single Molecule Magnets
Linear trinuclear heterobimetallic Co2Gd complex
VChandrasekhar et alInorgChem2007465140
Ground term of the Gd3+-ion ndash8S72
Spin -ion
Co ndashions unquenched orbital angular momenta
Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]
VChandrasekhar et alInorgChem2008474918
bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought
by the Dy3+ ion
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2
Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are
omitted
Temperature dependenceof the out-of-phase ac
susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and
[NiIILTbIII(hfac)2]2 compounds
[CuLTb(hfac)2]2
SOsa et alJACS2004126420
MOTIVATION OF THE WORK
Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters
Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems
Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster
2
1
5
73
4
6
8
Z
C3
J
J J
J
J J
J
J
J
J
J
J
Re(II) Mn(II)
Molecular cube compressedalong one of the trigonal axes
No zero field splitting for the Re(II) ion with spin frac12
Mn(II) ions do not carry anymagnetic anisotropy
The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways
J J - exchange integrals J ne J
Trigonally distorted mixed-ligand surrounding of the Re(II) ion
Re(II)
P
PP
C
CC
Local trigonal Z-axis
Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5
Htrig=(lz 23)2^
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
gt0
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
||
lt0
Relevant case
Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||
2E (ml=1)||
5+6mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )
Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0
bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12
bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z
Exchange Interaction for the Re(II)-Mn(II) Pair
Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators
J(Re-Mn)rarr exchange interaction parameter2E (ml=1)
||5+6
mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|
Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian
bull One can restrict the consideration of the exchange interaction by the ground Kramers space
Hamiltonian of exchange interaction
Local (red) and molecular (blue) Z-axes and the network of exchange pathways
for the Re4Mn4 ndashcluster
8ˆ6ˆ4ˆ
8ˆ5ˆ3ˆ
6ˆ5ˆ2ˆ2
8ˆ7ˆ6ˆ4ˆ
8ˆ6ˆ5ˆ1ˆ2ˆ
444
333
222
4444
1111
ZZZ
ZZZ
ZZZ
ZZZZ
ZZZZex
ss
ss
ssJ
sss
sssJH
Re(II) Mn(II)
2
1
5
73
4
6
8
Z1
C3(Z)
J
J J
J
J J
J
J
J
J
J
J
Z2
Z3
Z4
All operators are defined in the local frames
iiZ - is the z -component
5ˆ1Zs - is the z -component
of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc
of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame
0 50 100 150 200 250 30012
14
16
18
20
22
24
g|| Re = 4 gMn = 195
J = -35 cm-1
J = -105 cm-1
T c
m3 K
mol
-1
Temperature K
Comparison between theory and experiment
Temperature K
J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
3K4K
2KT
M
B
H THT
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2
Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are
omitted
Temperature dependenceof the out-of-phase ac
susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and
[NiIILTbIII(hfac)2]2 compounds
[CuLTb(hfac)2]2
SOsa et alJACS2004126420
MOTIVATION OF THE WORK
Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters
Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems
Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster
2
1
5
73
4
6
8
Z
C3
J
J J
J
J J
J
J
J
J
J
J
Re(II) Mn(II)
Molecular cube compressedalong one of the trigonal axes
No zero field splitting for the Re(II) ion with spin frac12
Mn(II) ions do not carry anymagnetic anisotropy
The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways
J J - exchange integrals J ne J
Trigonally distorted mixed-ligand surrounding of the Re(II) ion
Re(II)
P
PP
C
CC
Local trigonal Z-axis
Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5
Htrig=(lz 23)2^
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
gt0
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
||
lt0
Relevant case
Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||
2E (ml=1)||
5+6mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )
Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0
bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12
bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z
Exchange Interaction for the Re(II)-Mn(II) Pair
Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators
J(Re-Mn)rarr exchange interaction parameter2E (ml=1)
||5+6
mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|
Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian
bull One can restrict the consideration of the exchange interaction by the ground Kramers space
Hamiltonian of exchange interaction
Local (red) and molecular (blue) Z-axes and the network of exchange pathways
for the Re4Mn4 ndashcluster
8ˆ6ˆ4ˆ
8ˆ5ˆ3ˆ
6ˆ5ˆ2ˆ2
8ˆ7ˆ6ˆ4ˆ
8ˆ6ˆ5ˆ1ˆ2ˆ
444
333
222
4444
1111
ZZZ
ZZZ
ZZZ
ZZZZ
ZZZZex
ss
ss
ssJ
sss
sssJH
Re(II) Mn(II)
2
1
5
73
4
6
8
Z1
C3(Z)
J
J J
J
J J
J
J
J
J
J
J
Z2
Z3
Z4
All operators are defined in the local frames
iiZ - is the z -component
5ˆ1Zs - is the z -component
of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc
of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame
0 50 100 150 200 250 30012
14
16
18
20
22
24
g|| Re = 4 gMn = 195
J = -35 cm-1
J = -105 cm-1
T c
m3 K
mol
-1
Temperature K
Comparison between theory and experiment
Temperature K
J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
3K4K
2KT
M
B
H THT
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
MOTIVATION OF THE WORK
Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters
Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems
Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster
2
1
5
73
4
6
8
Z
C3
J
J J
J
J J
J
J
J
J
J
J
Re(II) Mn(II)
Molecular cube compressedalong one of the trigonal axes
No zero field splitting for the Re(II) ion with spin frac12
Mn(II) ions do not carry anymagnetic anisotropy
The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways
J J - exchange integrals J ne J
Trigonally distorted mixed-ligand surrounding of the Re(II) ion
Re(II)
P
PP
C
CC
Local trigonal Z-axis
Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5
Htrig=(lz 23)2^
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
gt0
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
||
lt0
Relevant case
Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||
2E (ml=1)||
5+6mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )
Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0
bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12
bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z
Exchange Interaction for the Re(II)-Mn(II) Pair
Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators
J(Re-Mn)rarr exchange interaction parameter2E (ml=1)
||5+6
mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|
Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian
bull One can restrict the consideration of the exchange interaction by the ground Kramers space
Hamiltonian of exchange interaction
Local (red) and molecular (blue) Z-axes and the network of exchange pathways
for the Re4Mn4 ndashcluster
8ˆ6ˆ4ˆ
8ˆ5ˆ3ˆ
6ˆ5ˆ2ˆ2
8ˆ7ˆ6ˆ4ˆ
8ˆ6ˆ5ˆ1ˆ2ˆ
444
333
222
4444
1111
ZZZ
ZZZ
ZZZ
ZZZZ
ZZZZex
ss
ss
ssJ
sss
sssJH
Re(II) Mn(II)
2
1
5
73
4
6
8
Z1
C3(Z)
J
J J
J
J J
J
J
J
J
J
J
Z2
Z3
Z4
All operators are defined in the local frames
iiZ - is the z -component
5ˆ1Zs - is the z -component
of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc
of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame
0 50 100 150 200 250 30012
14
16
18
20
22
24
g|| Re = 4 gMn = 195
J = -35 cm-1
J = -105 cm-1
T c
m3 K
mol
-1
Temperature K
Comparison between theory and experiment
Temperature K
J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
3K4K
2KT
M
B
H THT
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster
2
1
5
73
4
6
8
Z
C3
J
J J
J
J J
J
J
J
J
J
J
Re(II) Mn(II)
Molecular cube compressedalong one of the trigonal axes
No zero field splitting for the Re(II) ion with spin frac12
Mn(II) ions do not carry anymagnetic anisotropy
The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways
J J - exchange integrals J ne J
Trigonally distorted mixed-ligand surrounding of the Re(II) ion
Re(II)
P
PP
C
CC
Local trigonal Z-axis
Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5
Htrig=(lz 23)2^
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
gt0
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
||
lt0
Relevant case
Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||
2E (ml=1)||
5+6mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )
Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0
bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12
bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z
Exchange Interaction for the Re(II)-Mn(II) Pair
Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators
J(Re-Mn)rarr exchange interaction parameter2E (ml=1)
||5+6
mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|
Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian
bull One can restrict the consideration of the exchange interaction by the ground Kramers space
Hamiltonian of exchange interaction
Local (red) and molecular (blue) Z-axes and the network of exchange pathways
for the Re4Mn4 ndashcluster
8ˆ6ˆ4ˆ
8ˆ5ˆ3ˆ
6ˆ5ˆ2ˆ2
8ˆ7ˆ6ˆ4ˆ
8ˆ6ˆ5ˆ1ˆ2ˆ
444
333
222
4444
1111
ZZZ
ZZZ
ZZZ
ZZZZ
ZZZZex
ss
ss
ssJ
sss
sssJH
Re(II) Mn(II)
2
1
5
73
4
6
8
Z1
C3(Z)
J
J J
J
J J
J
J
J
J
J
J
Z2
Z3
Z4
All operators are defined in the local frames
iiZ - is the z -component
5ˆ1Zs - is the z -component
of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc
of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame
0 50 100 150 200 250 30012
14
16
18
20
22
24
g|| Re = 4 gMn = 195
J = -35 cm-1
J = -105 cm-1
T c
m3 K
mol
-1
Temperature K
Comparison between theory and experiment
Temperature K
J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
3K4K
2KT
M
B
H THT
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
Trigonally distorted mixed-ligand surrounding of the Re(II) ion
Re(II)
P
PP
C
CC
Local trigonal Z-axis
Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5
Htrig=(lz 23)2^
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
gt0
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
||
lt0
Relevant case
Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||
2E (ml=1)||
5+6mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )
Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0
bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12
bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z
Exchange Interaction for the Re(II)-Mn(II) Pair
Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators
J(Re-Mn)rarr exchange interaction parameter2E (ml=1)
||5+6
mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|
Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian
bull One can restrict the consideration of the exchange interaction by the ground Kramers space
Hamiltonian of exchange interaction
Local (red) and molecular (blue) Z-axes and the network of exchange pathways
for the Re4Mn4 ndashcluster
8ˆ6ˆ4ˆ
8ˆ5ˆ3ˆ
6ˆ5ˆ2ˆ2
8ˆ7ˆ6ˆ4ˆ
8ˆ6ˆ5ˆ1ˆ2ˆ
444
333
222
4444
1111
ZZZ
ZZZ
ZZZ
ZZZZ
ZZZZex
ss
ss
ssJ
sss
sssJH
Re(II) Mn(II)
2
1
5
73
4
6
8
Z1
C3(Z)
J
J J
J
J J
J
J
J
J
J
J
Z2
Z3
Z4
All operators are defined in the local frames
iiZ - is the z -component
5ˆ1Zs - is the z -component
of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc
of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame
0 50 100 150 200 250 30012
14
16
18
20
22
24
g|| Re = 4 gMn = 195
J = -35 cm-1
J = -105 cm-1
T c
m3 K
mol
-1
Temperature K
Comparison between theory and experiment
Temperature K
J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
3K4K
2KT
M
B
H THT
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5
Htrig=(lz 23)2^
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
gt0
2T2(t2 )5
2A1 (ml=0)
2E (ml=1)
||
lt0
Relevant case
Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||
2E (ml=1)||
5+6mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )
Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0
bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12
bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z
Exchange Interaction for the Re(II)-Mn(II) Pair
Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators
J(Re-Mn)rarr exchange interaction parameter2E (ml=1)
||5+6
mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|
Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian
bull One can restrict the consideration of the exchange interaction by the ground Kramers space
Hamiltonian of exchange interaction
Local (red) and molecular (blue) Z-axes and the network of exchange pathways
for the Re4Mn4 ndashcluster
8ˆ6ˆ4ˆ
8ˆ5ˆ3ˆ
6ˆ5ˆ2ˆ2
8ˆ7ˆ6ˆ4ˆ
8ˆ6ˆ5ˆ1ˆ2ˆ
444
333
222
4444
1111
ZZZ
ZZZ
ZZZ
ZZZZ
ZZZZex
ss
ss
ssJ
sss
sssJH
Re(II) Mn(II)
2
1
5
73
4
6
8
Z1
C3(Z)
J
J J
J
J J
J
J
J
J
J
J
Z2
Z3
Z4
All operators are defined in the local frames
iiZ - is the z -component
5ˆ1Zs - is the z -component
of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc
of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame
0 50 100 150 200 250 30012
14
16
18
20
22
24
g|| Re = 4 gMn = 195
J = -35 cm-1
J = -105 cm-1
T c
m3 K
mol
-1
Temperature K
Comparison between theory and experiment
Temperature K
J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
3K4K
2KT
M
B
H THT
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12
bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z
Exchange Interaction for the Re(II)-Mn(II) Pair
Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators
J(Re-Mn)rarr exchange interaction parameter2E (ml=1)
||5+6
mj=32 (ml=1 ms=12 )
4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|
Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian
bull One can restrict the consideration of the exchange interaction by the ground Kramers space
Hamiltonian of exchange interaction
Local (red) and molecular (blue) Z-axes and the network of exchange pathways
for the Re4Mn4 ndashcluster
8ˆ6ˆ4ˆ
8ˆ5ˆ3ˆ
6ˆ5ˆ2ˆ2
8ˆ7ˆ6ˆ4ˆ
8ˆ6ˆ5ˆ1ˆ2ˆ
444
333
222
4444
1111
ZZZ
ZZZ
ZZZ
ZZZZ
ZZZZex
ss
ss
ssJ
sss
sssJH
Re(II) Mn(II)
2
1
5
73
4
6
8
Z1
C3(Z)
J
J J
J
J J
J
J
J
J
J
J
Z2
Z3
Z4
All operators are defined in the local frames
iiZ - is the z -component
5ˆ1Zs - is the z -component
of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc
of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame
0 50 100 150 200 250 30012
14
16
18
20
22
24
g|| Re = 4 gMn = 195
J = -35 cm-1
J = -105 cm-1
T c
m3 K
mol
-1
Temperature K
Comparison between theory and experiment
Temperature K
J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
3K4K
2KT
M
B
H THT
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
Hamiltonian of exchange interaction
Local (red) and molecular (blue) Z-axes and the network of exchange pathways
for the Re4Mn4 ndashcluster
8ˆ6ˆ4ˆ
8ˆ5ˆ3ˆ
6ˆ5ˆ2ˆ2
8ˆ7ˆ6ˆ4ˆ
8ˆ6ˆ5ˆ1ˆ2ˆ
444
333
222
4444
1111
ZZZ
ZZZ
ZZZ
ZZZZ
ZZZZex
ss
ss
ssJ
sss
sssJH
Re(II) Mn(II)
2
1
5
73
4
6
8
Z1
C3(Z)
J
J J
J
J J
J
J
J
J
J
J
Z2
Z3
Z4
All operators are defined in the local frames
iiZ - is the z -component
5ˆ1Zs - is the z -component
of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc
of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame
0 50 100 150 200 250 30012
14
16
18
20
22
24
g|| Re = 4 gMn = 195
J = -35 cm-1
J = -105 cm-1
T c
m3 K
mol
-1
Temperature K
Comparison between theory and experiment
Temperature K
J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
3K4K
2KT
M
B
H THT
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
0 50 100 150 200 250 30012
14
16
18
20
22
24
g|| Re = 4 gMn = 195
J = -35 cm-1
J = -105 cm-1
T c
m3 K
mol
-1
Temperature K
Comparison between theory and experiment
Temperature K
J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters
0 1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
3K4K
2KT
M
B
H THT
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
0 50 100 150 200 250 3000
10
20
30
40
50
T
||T
T c
m3 K
mol
-1
Temperature K
Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components
of the magnetic susceptibility calculated with the set of the best-fit
parameters
Temperature K
|| T
T
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10
0
10
20
30
40
50
60
70
80
90
Ene
rgy
cm
-1
MS
2) Low-lying energy levels as functions of MS (MS is the projection of the
total spin of four Mn(II) ions)calculated with the set of the best-
fit parameters
Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for
illustration but not in calculations )
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones
0 50 100 150 200 250 3000
10
20
30
40
50
60
T (quantum)
T (quantum-classical)
||T (quantum)
||T (quantum-classical)
T c
m3 K
mol
-1
Temperature K0 50 100 150 200 250 300
14
16
18
20
22
24Quantum-classical approach
Quantum approach
T c
m3 K
mol
-1
Temperature K
quantum-classical approach
quantum approach
quantum-classical approach
quantum approach
quantum-classical approachquantum approach
cos1 kkkZ ssks cossin1 kkkkX ssks
kkkkY sinsinssks 1
25ks
The pseudo-spins of the Re(II) ions are treated quantum-mechanically
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster
The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization
The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters
SUMMARY
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2Tb3+(1)
Tb3+(2)Cu2+(2)
Cu2+(1)
J1 J1
J2
J2
Exchange interactions in the [CuIILTbIII(hfac)2]2 complex
x
y
z
Z
N1
N2 O2
O3
O1
O4
O5
O6
O7
O8
XY
Cu
Tb
Cluster fragment
bullVery low local symmetry of Tb ions
bullIsolated tetranuclear molecules in the crystal
bullDifferent strength of exchange interaction for two types of Cu-Tb pairs
bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs
The [CuIILTbIII(hfac)2]2 Single Molecule Magnet
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
INTERACTIONS INCLUDED IN THE MODEL )2()1(
2(2)Tb1
(1)Tb21
(2)Tb2
(1)Tb1
(2)Tb
(1)Tb
CuCuZeZe
HHJJHHH sjsjsjsj
Crystal field and Zeeman interaction for
Tb-ions
Exchange interaction
between Tb and Cu ions
Zeeman interaction for
Cu-ions
Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion
iim
lmli
ml
i
opi
iicc YBWVV
pRrr )()( ecm
lpcm
lml
BBB
Exchange charges (Malkin BZ) pp
ml
p p
plecml C
RRSeB
)(5
2 2
)( pπlπpσσpsspl RSγGRSGRSGRS 222
00404 mpnmfRSsnmfRS pps
114 mpnmfRS p
S ndashoverlap integrals G-phenomenological parameters of the model
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
MAGNETIC SUSCEPTIBILITY OF
THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
xxT
yyT
zzT
T e
mu
K m
ol-1
T K
Temperature dependences of the XXT YYT and ZZT components for the
[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters
Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55T
em
u K
mol
-1
T KT as a function of temperature
for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2
clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53
solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2
Cu2Tb2Ni2Tb2
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster
Wave function of the th state of the cluster
ZJ JcZ
)2(2
Tb)1(1
TbZCu
ZZCu
ZZ sjsjJ
Expectation value of the cluster total angular momentum projection
ZZ JJ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Ground cluster state
13876102
ZJ JCZ
First excited state
0864202
ZJ JCZ
212166 )2()1()2()1( ZCu
ZCu
ZTb
ZTb ssjj
Probability of the one-phonon transitionground statelt-gtfirst excited state
02)2()1( excitedfirstHHground Tb
eLTbeL
Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster
calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-
1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the
expectation value of the operator Jz
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
jTb=6 sCu =12 Tb rarr7F6 J=13
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal
Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion
Tb-Cu pair
Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz
Tb +szCu of the pair
Cluster energy pattern indicative of a system with almost axial anisotropy
Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions
Stark structure of the ground 7F6multiplet of the TbIII ion in
[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled
Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53
J1=283 cm-1
0
100
200
300
400zj
plusmn 18plusmn 29plusmn 38
plusmn 01plusmn 03
plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17
plusmn 54plusmn 64
b)a)
Ene
rgy
cm
-1
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1
0
5
10
15
20
25
30
35
40
45
I
0
plusmn 107
plusmn 117plusmn 09plusmn 117
0
plusmn 127
Ene
rgy
cm
-1
Calculated energy level diagram for the Cu2Tb2 cluster
Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117
Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
SUMMARY
The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster
The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization
Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760
Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the
absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems
can be considered as a magnet
BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR
hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS
hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT
THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)
hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET
RJ Glauber JMathPhys 19634294
ji
ZZex jiJH 2
Z is the operator for the Z component of the spin or pseudospin J is the coupling constant
TkT
Bbexp)( 0
Barrier to reverse the magnetization direction energy loss in one spin flip-flop process
Jb 2
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA
Barrier height |DS|S2
CONVENTIONAL MODEL
J parameter of ferromagnetic exchange between the spin units DS single-unit zero-
field splitting parameter
i
ziSii
iF SDSSJH 212
MAGNETIC UNIT
Ener
gyEn
ergy
DSlt0
S ndash spin of the ground state
BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET
4JF S2 -4JF S2
|DS|S2 +8JFS2
Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in
[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2
Miysaka H et al Chem-Eur J 2005
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE
KRDunbar et al JApplPhys2005
0 50 100 150 200 250 300
05
10
15
20
25
30
35
0 10 20 30
05
10
15
20
25
30
35
Temperature K
MT
cm
3 Km
ol-1
Temperature K
MT
cm
3 Km
ol-1
Temperature dependence of T for the Co(H2L)(H2O) compound
evidencesa) unquenched orbital angular momentum
T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1
S=32b) antiferromagnetic exchange interaction
between Co ionsc) T maximum at 25 K rarr spin canting
(noncolinear spin structure)
ORTEP representation of the
Co(H2L)(H2O) unit
View of a 1D zigzag chain of Co(H2L)(H2O)
The cobalt octahedra and CPO3tetrahedra are shaded in green and
pink respectively
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
MOTIVATION OF THE WORK
Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta
PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the
Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E
4A2
4E
4T1 0
Zero-field splitting of the isolated trigonal 4A2 term
Providing gt 0 the barrier
for the reversal of magnetization
in a single magnetic unit cannot exist
MS=32
MS=12
2D4A2 Dgt04E
4A2
4T1
0
Orbital magneticcontribution isfully suppressed
Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
Wavefunction of the ground Kramers doublet of the
Co2+ion
Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting
Single-ion Hamiltonian (local frames)
2
2
XA ZA
XB
ZB
Z
X
A B
Local (XCZC C=AB) and molecular (XZ) coordinates
21|)21(252321231
14
jj
jgr
T
mjCjSl
-1500 -1000 -500 0 500 1000 1500
-1000
-500
0
500
1000
3212
12123252
|mj |
Ene
rgy
cm
-1
cm-1
Energy levels of the Co-ion in the axial surrounding
Zero-fieldspliitingparameterpositive
Spin-orbit interactionAxial crystal field
BAplpHpZCo sl23322
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
BABA
BABABA
XZZX
ZZXXYYex
ssss
ssssssJH
sin
cos2
mmsmm
mmsmm
slXsl
slXsl
A
A
0231231
0231231
Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian
BsAsBsAsBsAsJJH
ZZYYXX
BAex
22 ss
molecular frame
local frames of ions A and B
Ising pseudo-spin-12 Hamiltonian for the exchange
problem
BA ZZeffex JH 2
||
cos3cos92
JJJeff
components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt
23
23
23
ml=1 ms= 32
ml=1 ms= 12
ml=1 ms= 12
ml=1 ms= 324E (ml=1)
STRONG NEGATIVE TETRAGONALFIELD ON Co(II)
Spin-orbital splitting of the tetragonal 4E term
pp ZZ
pSO slEH 234
Axial form of the spin-orbit interaction
J |||| λκ
0 3|| ggg e
Principal values of the g-tensor for the ground Kramers doublet
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )
A
B
A
B
ZB
Z
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
A
B
A
B
ZB
Z
ZA
XA
ZA
XA
XB
X
Noncolinear spin structure of the chain and illustration
for a single spin flip-flop process
BZBZAZAZi
AZBZBZAZeff iigiiiiJH HH ||12
)1(2 cos2 21
i
AZ
BZ
BZ
AZeff iiiiJH
Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )
Ferromagnetic interaction
12sin2 22 iiiiJH A
XBX
BX
AX
ieff
Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|
Molecular frames
H=H1+H2+H3
H3 - antisymmetric Dzyaloshinsky-Moria exchange
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
0
5
10
15
20
25
1 10 100 1000Frequency (Hz)
(
emu
mol
)
22 K21 K20 K19 K18 K
0
1
2
3
4
5
6
7
1 10 100 1000Frequency (Hz)
(em
um
ol)
22 K21 K20 K19 K18 K
Frequency dependenceof the prime and primeprime
components of the ac magnetic susceptibility
measured at various temperatures
BARRIER FOR MAGNETIZATION REVERSAL
TEMPERATURE DEPENDENCE OF THE
RELAXATION TIME
044 046 048 050 052 054 056 0584
5
6
7
8
9
Ln(1
)
1T (K -1) 01 ln ln
TBkb
The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo
respectively The solid line corresponds to the best fit of the data to the Arrhenius
expression
b=2|Jeff |=194 cm-1
Jeff =-97 cm-1
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
MAGNETIC BEHAVIOR OF THE CO CHAIN
011
YYXX
XXZZ
ZZ XFZF ||H||HHHHH
Principal values of the magnetic susceptibility tensor
0 10 20 30 40 5000
05
10
15
20
25
30
e
mu
K m
ol-1
T K0 10 20 30 40 50
0
1
2
3
4
5
6
7
8
9
ZZT
XXT
e
mu
K m
ol-1
T K
Temperature dependence of T
circles ndashexperimental data solid line ndashtheoretical curve calculated
with =-180 cm-1=08Jeff=-97 cm-1=15 o
Nonzero diagonal components of the T tensor calculated using the
parameter values =-180 cm-1=08Jeff=-97cm-1=15o
3XXZZ
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH
MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form
BABABABABA XZZXZZXXYYex ssssssssssJH sincos2
Isotropic exchange (local frames)
PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused
by exchange and Zeeman interactions
UNPERTURBED HAMILTONIAN PERTURBATION
BAi
iZEex HHV
EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN
BApsllH ppZBAp p
))23()32(( 2
0
21effeffeff HHH
Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff
(2)first and second order terms with respect to V
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
EFFECTIVE HAMILTONIAN
BAppXpXpXpXpZpZBXAZBXAZ
BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff
HHgHgJ
JJJJH
|| )(2
2222
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
BAZXJ
BA XZJ
BAZZJ
BAYYJBAXXJ
-3000 -2000 -1000 0 1000-10
-8
-6
-4
-2
0
2
Exch
ange
para
met
ers
cm-1
cm-1
Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
-3000 -2000 -1000 0 1000
1
2
3
4
5
6
7
8
g fa
ctor
s
cm-1
||g
g
Principal values of the g-tensor as functions of
bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1
bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters
11 111579cos9 cmJcmJJ oeff
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
SUMMARY
The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]
Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment
The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian
The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital
angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the
anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From
this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic
anisotropy described by the zero-field splitting spin Hamiltonian