angular momentum space.pdf

Magnetic anisotropy in f-electron systems:A crystal-field perspectiveA crystal-field perspectiveNicola Magnani

(nicola magnani@kit edu)

INSTITUTE OF NANOTECHNOLOGY, KARLSRUHE INSTITUTE OF TECHNOLOGY

([email protected])

KIT University of the State of Baden-Wuerttemberg and National Research Center of the Helmholtz Association www.kit.edu

Outline

Part 1: An introduction to crystal field in f-electron systemsPart 1: An introduction to crystal field in f-electron systems

Part 2: Crystal-field anisotropy in lanthanide-based SMMs y py(selected examples)

Part 3: Crystal-field anisotropy in actinide-based SMMs (selected examples)

Institute of Nanotechnology (INT-KIT)2 Nicola Magnani Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013

Part 1

An introduction toAn introduction tocrystal field iny

f-electron systems


f-electron orbitals: the central field approximation2 2 2

1 1

2

N Ni

i i ji ij

p Ze eHm r r

22 N i rUZepHji ij

2 N e body)-(one

210

i

ii

i rUrm

H b d )(

1

1

jij

ij

rUreH

body)(o e

1N r R r Y body)-(two

1and both commute with i i H L l S s

01

,i i i ii n l i l m i i

ir R r Y


1i ii i

Spherical harmonics Ylm 1l = 0

l = 1

Y

Y

cos312

1

01

00

2

220

235

41

rrzY

l = 1

l = 2

iY expsin23

21

2

11

1

z

r

l = 3

Y

151

1cos3541

1

202

l = 4

iY

iY

2expsin215

41

expcossin215

21

222

12

l = 5

l = 6

242


l 6

m = 0 m = 1 m = 2 m = 3 m = 4 m = 5 m = 6

The spectra of rare-earth ionsl = 3 2 1 0 -1 -2 -3

4f 3 L 6 S 3/210

HH 2S+1L

4f 3: L = 6, S = 3/2

0H NH l4fn

2S+1L

HHH

i

iiiSO rH1

sl

2S 1

Jmax=L+S

SOHHH 10 Jmin=|L-S|

2S+1L

J i =|L-S|

. . . N 7or


Jmin |L S| 1 JJgJeff Jmax L+S

The crystal field: a simple point-charge example

A magnetic ion placed at the origin of the Cartesian coordinate system, with two equal li d ( f ff ti h Z ) l d t

(0,0,R)

r = (x,y,z)

ligands (of effective charge Ze) placed at the same distance along the z axis.

(0,0,-R)

2222

222

22

Rzyx

Ze

Rzyx

ZeeZVk k

k

rR

2222

2

//211

//211

RRRRRZe

yy

2222 //21//21 RrRzRrRzR)(23163355311)()!2(1 765432 xOaaaaaaan n


)(10242561281682

1)()!2(1 0 2

xOaaaaaaana n n

The crystal field: a simple point-charge example

5

4224

4

23

3

22

222 835303

235

231

//211

Rzzrr

Rzrz

Rrz

Rz

RRrRz

7

624426

6

4325

165105315231

8157063

822//21

Rrzrzrz

Rzrzrz

RRRRRRrRz

2r 4260235

2 YRr

045

4

32 Y

Rr0

67

6

132 Y

Rr

...

31

131

514 067

60

45

40

23

22

axial YRrY

RrY

RrZeV


3135 RRR

The crystal field: a simple point-charge example(0,0,R)

(-R 0 0)

(0 R 0)(0 R 0)

(-R,0,0)

(x,y,z)(0,R,0)(0,-R,0)

(R,0,0)

145

37 4

44

40

45

42

cubic YYYRrZeV

(0,0,-R)

...27

1323

143

46

46

067

6

5cub c

YYYRr

R


2132 6667R

Constraints on non-zero matrix elements

formthehaveelementsmatrixitson tocontributieachharmonics, spherical offunction a asrewritten been has Once CFH

ddsind2"""* ''' rrY mlnmlmlnformthehave elementsmatrix itson tocontributieach

ddid "*'* mmm YYYRR ddsind "*'""* '' mlmlmllnln YYYrRR therequireson gintegratin,2d Since "'

2"' e mmmmmmi

nonzero. be result to for the fulfilled be to"'condition

qgg0

mmm

mmm

."'"' andeven bemust "':sconstraintadditional in tworesultson gintegratin Similarly,

llllllll


.6 andeven bemust that means this)3"'( electrons For lllf

Constraints on non-zero matrix elements

...066060440402202axial i

ii

ii

i iYrAiYrAiYrAViii

145 4

44

40

44

4cubic i iYiYiYrAV

714

4406

4444cubic

ii

...27 4

64

60

66

6

i

i iYiYiYrA

( l ib i f d l l 4)(even less contributions for d electrons: l 4)

Now all thats left to do is to calculate these matrix elements!


Now all that s left to do is to calculate these matrix elements!

Angular momentum space: operator equivalents mlll

i

ml

mml

li OrYYr 1

ml

mll

lmlCF OrAH

~ml ,


Angular momentum space: operator equivalents



SLMLSM :statesset Basis

zyx LLLzyx ,,,, mlll

i

ml

mml

li OrYYr 1

2S+1L

y 133 e.g. 222022 LLLrzYr z 222222222 LLyxYYr

2S+1L

22 yx LLyxYYr JLSJM :statesset Basis

4fn

Jmax=L+S

2S+1L

. . . kk JL


Jmin=|L-S|


~~;,..., with states 12

ml

ml

mll

lmlCF

J

OBOrAH

JJMJ mllli

ml

mml

li OrYYr 1

2S+1L cases some(in degeneracy their lifts,,

mlll

mllllCF

2S+1Ld lilhb dd

nHamiltonia above theof validity The.partially)only

4fn model.ticelectrostathebeyondextends

2S+1L


2S+1LJMJs


Simple example: 4f 1 (Ce3+), 2F5/2, in a cubic CF 4040 21~5~ OOBOOBH 4606644044 215 OOBOOBHCF

1112/ 4444

JJJ MMMJJMJ

JJO

2/389852/19852/185

2/352/3)2/3)(2/5()2/7)(2/5(2/5111

2

334

JJJJ

JJ

JJJ

MMMJJMJ4~B

51242/389852/19852/185

44

2

JJ MOM

JJ



Simple example: 4f 1 (Ce3+), 2F5/2, in a cubic CF

MJ 5/2 3/2 1/2 -1/2 -3/2 -5/2

5/2 60 0 0 0 605 0

MJ 5/2 -3/2 -5/2 3/2 1/2 -1/2

5/2 60 605 0 0 0 0120

-240

3/2 0 -180 0 0 0 605-3/2 605 -180 0 0 0 0

240

1/2 0 0 120 0 0 0

-1/2 0 0 0 120 0 0

-5/2 0 0 60 605 0 0

3/2 0 0 605 -180 0 04~B

-3/2 605 0 0 0 -180 0

5/2 0 605 0 0 0 60

1/2 0 0 0 0 120 0

1/2 0 0 0 0 0 120


-5/2 0 605 0 0 0 60-1/2 0 0 0 0 0 120


Simple example: 4f 1 (Ce3+), 2F5/2, in a cubic CF

2/1

2F5/2

4~360B

6/2/32/55 4360B

6/2/52/35 Institute of Nanotechnology (INT-KIT)18 Nicola Magnani Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013

Crystal-field splitting: cerium(III)


Crystal-field splitting: praseodymium(III)

Kramers theorem: all the crystal-field states of an ion with dd b f l t h d


an odd number of electrons have even degeneracy.

Zero-field splitting in S-state ions

Ions with a L = 0 ground term should in principle have no spatial anisotropy. However, small mixing with higher-lying p py g g y gmultiplets can give rise to zero-field splitting.

mlmlZFS OBH ~ S ml ,

Assuming only rank-2 contributions: SS 2200 ~~ OBOBH

SSSSS

12

12

12

12

22

22

2222

~~~

OBOBOB

OBOBHZFS

cost

~13~

222

2222

202

yxz

SSEDS

SSBSSSB


cost. yxz SSEDS

Full configuration diagonalization

2S+1L

2S+1L4fn

2S+1L


2S+1LJ

LS versus jj coupling

J mixing can be taken into account perturbatively:


J mixing can be taken into account perturbatively:N. M. et al., PRB 71, 054405 (2005)

End of part 1 - Bibliography

D.J. Newman and B. Ng, Crystal Field Handbook (Cambridge University Press, 2000)

C. Goerller-Walrand and K. Binnemans, in: Handbook on the Physics and Chemistry of Rare Earths Vol. 23 (Elsevier, 1996)

SPECTRA h l l /d l d / t /i d ht l SPECTRA: chmwls.anl.gov/downloads/spectra/index.html

SMMS: cluster.univpm.it/_build/html/productsdir/smms/magnetism.html


Part 2

Crystal-field anisotropyCrystal field anisotropyin rare-earth based SMMs:

some examples


Origin of magnetic relaxation in rare-earth-based single-ion complexes

Direct process Orbach processAllowed if Jz is not a good quantum

E

n

e

r

g

y

E

n

e

r

g

y

g qnumber (like QTM)

Thermally activated

e

t

i

c

f

i

e

l

d

e

t

i

c

f

i

e

l

d

(slow at low T)

M

a

g

n

e

M

a

g

n

e


Origin of uniaxial anisotropy in SMMs

sandwich structures naturally accommodatenaturally accommodate oblate 4f orbitals


J.D. Rinehart and J.R. Long, Chem. Sci. 2 (2011) 2078

Crystal-field splitting in [Pc2R] complexes

1Ueff~260 cm-1~440 cm-1

Tb might be the best SMM, but Tm is much

Institute of Nanotechnology (INT-KIT)28

Tb might be the best SMM, but Tm is much better for studying the crystal field of the series!

Nicola Magnani Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013

N. Ishikawa et al., JACS 125 (2003) 8694

Pc2Tm: optical spectroscopy


N.M. et al, PRB 79 (2009) 104407


Pc2Tm: neutron spectroscopy


N.M. et al, PRB 79 (2009) 104407

Specific heat can help fixing the degeneracy

Doublet: GS 1ES 2ES 3ES

2 22

12expln

n nn

Bm

EgTkC


Pc2Tm: specific heat


N.M. et al, PRB 79 (2009) 104407

Crystal-field states assignment

- States nearest to the GS t h ll J tmust have small Jz to

reproduce low-temperature susceptibility

- States with Jz = 6 are much lower in energy than expected

- All excited levels detected below 2 meV must correspond to singly degenerate states;to singly-degenerate states; presence of a significant C4contribution is evident


N.M. et al, PRB 79 (2009) 104407

Crystal-field states assignment


Pc2Tm: field-dependent specific heat

Model AModel C


N.M. et al, PRB 79 (2009) 104407

Pc2Tm: magnetic susceptibility

Model B

Model A


N.M. et al, PRB 79 (2009) 104407

Pc2R: Revised crystal-field parameters

2,0

Ueff

6,0

4,4

4,0

The revised crystal-field model is perfectlycompatible with the relaxation properties.


N.M. et al, PRB 79 (2009) 104407

End of part 2 (almost) Conclusions

The CF parameters for Tm-DD complex were unambiguously determined by a variety of experimental techniques.y y p q

D4d terms are qualitatively in line with those previously determined (signs and orders of magnitude)

NOT quantitatively: A20 smaller by a factor 3!

C symmetry terms small (with respect to the axial potential) but not C4-symmetry terms small (with respect to the axial potential) but not negligible (clear effect on the energy spectra)

Lower-symmetry terms have a much smaller influence on the low- Lower-symmetry terms have a much smaller influence on the low-energy wavefunction, even if present (|A22r2| < 0.5 cm-1)


Addendum: Exploiting magnetically active radicals

(Rinehart et al., Nat. Chem. 2011, 3, 538; J. Am. Chem. Soc. 2011, 133, 14236)


(Lukens, N.M., and Booth, Inorg. Chem. 2012, 51, 10105)


How can the coupling be strengthened further?

I t f di t

*

Increase t: for direct spin exchange, this means increasing the overlapf the overlap.

Actinides 5f shell has a larger

t i th

f*

extension than rare earths 4fDecrease U,matching thematching the reduction potential of the ligand and metal.

A ti id h h

U2t

E2

TS Actinides have much richer chemistry than lanthanides (exp. more oxidation


more oxidation states)

(Lukens, N.M., and Booth, Inorg. Chem. 2012, 51, 10105)


How can the coupling be strengthened further?

58 59 60 61 62 63 64 65 66 67 68 69 70

Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm YbCe Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb[Xe]4f15d16s2 [Xe]4f25d16s2 [Xe]4f35d16s2 [Xe]4f45d16s2 [Xe]4f55d16s2 [Xe]4f65d16s2 [Xe]4f75d16s2 [Xe]4f85d16s2 [Xe]4f95d16s2 [Xe]4f105d16s2 [Xe]4f115d16s2 [Xe]4f125d16s2 [Xe]4f135d16s2

Lanthanides with accessible tetravalent states should be coupled with oxidizing radicals

Lanthanides with accessible divalent states should be coupled with

reducing radicalsg g

Actinides have a much richer chemistry!


Part 3

Anisotropy in actinide-basedAnisotropy in actinide based SMMs: some examplesp


Why actinides?


Why actinides?

Switching to actinide-based SMMs could combine the best of both worlds (3d and 4f) since 5f electrons can lead to theof both worlds (3d and 4f), since 5f electrons can lead to the simultaneous presence of strong ligand-field potential and magnetic superexchange coupling.

5New actinide-containing SMMs reported per year

2

3

4

0

1

2


2009 2010 2011 2012 2013


A paradigmatic case: UTp3

Ueff /100 !!!

= 267 cm-1

More complex relaxation processes than expected from the energy spectrum


J.D. Rinehart and J.R. Long, Dalton Trans. 41 (2012) 13572

Direct processes are enhanced by the non-axial part of the p y pligand-field potential.

Possible solution: ACTINOCENES (D8h)( 8h)

2

(Karraker et al., JACS 1970, 92, 4841)

PROBLEM: tetravalent U 5f 2


Butterfly magnetic hysteresis in neptunocene (5f3)

Jz = 5/2z

237NpI = 5/2


(N.M. et al., Angew. Chem. Int. Ed. 2011, 50, 1696)


Exchange coupling and slow relaxation in the heterovalent Np trimetallic [NpVIO2Cl2][NpVO2Cl(thf)3]2

140 K


(N.M. et al., Phys. Rev. Lett. 2010, 104, 197202)


Exchange coupling in U(V) complexes

3.36

(Arnold et al., Nat. Chem. 2012, 4, 221)


5f1 4fn complexes

Ln = Dy


(Arnold et al., JACS 2013, 135, 3841)


End of part 3 Conclusions and perspectives

Both the ligand-field anisotropy and the superexchange interaction in actinide-based magnetic molecules are generally stronger than in their lanthanide counterparts. This could make light actinides a better choice to design SMMs, despite their lower magnetic moments

b t f t t l f d t l i t ti hi h i i t but unfortunately, some fundamental interactions which give rise to fast relaxation processes (non-axial ligand field, hyperfine coupling, orbital moment reduction) seem to be enhanced too.

Challenge: exploit the richer actinide chemistry (more oxidation states, larger coordination sphere) to obtain a favorable balance.

Pu(V): strong exchange and large moment?


Acknowledgments

Annie Powell and her group (KIT)

Christos Apostolidis, Roberto Caciuffo and his group (JRC-ITU)

Wayne Lukens, Norman Edelstein, Corwin Booth, Richard Andersen (LBNL)

Clint Sharrad Richard Winpenny and his group (Uni Manchester)Clint Sharrad, Richard Winpenny and his group (Uni-Manchester)

Polly Arnold and her group (Uni-Edinburgh)

Marinella Mazzanti and her group (CEA-Grenoble)


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