multiferroic and magnetoelectric...
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Multiferroic and magnetoelectricmaterials
Maxim Mostovoy
University of GroningenZernike Institute
for Advanced Materials
Boulder Summer SchoolJuly 2008
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Lectures
• Spin-orbital exchange in Mott insulators
Multiferroics and magnetoelectrics
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Outline
• Linear magnetoelectric effect, multiferroics
• Phenomenological description
• Microscopic mechanisms of magnetoelectriccoupling
• Outlook
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Electric ↔ Magnetic• Duality of Maxwell
equations
• Aharonov-BohmAharonov-Casher
• Thermodynamics of ferroelectrics and ferromagnets
⎪⎪⎩
⎪⎪⎨
⎧
∂∂
+=×∇
∂∂
−=×∇
tc
tcEH
HE
1
1
⎪⎩
⎪⎨
⎧
−→
→
EH
HE
( )⎩⎨⎧
=+⋅∇=×∇
040
PEE
π
⎟⎟⎠
⎞⎜⎜⎝
⎛ΦΦ
0
F
e
⎟⎟⎠
⎞⎜⎜⎝
⎛
0ρρF
µ
⎪⎩
⎪⎨⎧
−+=Φ
−+=Φ
MHbMaM
PEbPaP
FM
FE42
42
( )⎩⎨⎧
=+⋅∇=×∇
040
MHH
π
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Multiferroics• Both ferroelectric and magnetic
• Coupling between P and M
Pb(Fe2/3W1/3)O3 BiFeO3
Pb(Fe1/2Ta1/2)O3
YMnO3
G. A. Smolenskii
G.A. Smolenskii & I.E. Chupis, Sov. Phys. Usp. 25, 475 (1982)
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Time-reversal symmetry breaking in magnets
0≠>< S
( ) ( )tt SS −=−
Ferromagnets Antiferromagnets
0≠M 0=M
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Inversion symmetry breaking in ferroelectrics
( ) ( )xPxP −=− BaTiO3Centrosymmetric
+ − + − + −−
Noncentrosymmetric
+ − + − + −−P
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No chemistry betweenmagnetism and ferroelectricity
FM FEd0,s2d1,d2,d3…
multiferroicsN. A. Hill, J. Phys. Chem. B 104, 6694 (2000)
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Linear magnetoelectric effectI. E. Dzyaloshinskii JETP 10 628 (1959), D. N. Astrov, JETP 11 708 (1960)Cr2O3
+= EP eχ Hα
=M Hmχ+Eα
G.T. Rado PRL 13 335 (1964)
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Anomalies of magnetoelectricconstant in boracites
M = Co2+, Ni2+
D. G. Sannikov, Ferroelectrics 219 177 (1998)
X = I, Br, ClM3B7O13X
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Orthorombic RMnO3
T. Kimura et al PRB 68,060403 (2003)
Sinusoidal SDW
Spiral SDW
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Dielectric constant anomalyat the transition to spiral state
T. Kimura et al , Nature 426, 55 (2003)
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Polarization switching by magnetic field
T. Kimura Annu. Rev. Mater. Res. 37 387(2007)
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Magnetic control of dielectric properties
T. Kimura Annu. Rev. Mater. Res. 37 387(2007)
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T. Goto et al PRL 92, 257201 (2004)
Giant magnetocapacitance effectin DyMnO3
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Electric polarization reversalsin TbMn2O5
N. Hur et al Nature 429, 392 (2004)
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CoCr2O4
MP× is conserved
Y. Yamasaki et al, PRL 96, 207204 (2006)
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Outline
• Linear magnetoelectric effect, multiferroics
• Phenomenological description
• Microscopic mechanisms of magnetoelectriccoupling
• Outlook
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Linear magnetoelectric effect
Time-reversal symmetry T (t Ø - t) and inversion I (x Ø - x) are broken
Cr2O3
jiji HP α=
jjii EM α=
I.E. Dzyaloshinskii (1959), D.N. Astrov (1960)
jiij HEα−=Φme
IT symmetry (t Ø - t, x Ø - x) is conserved
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Cr2O3space group
cR3
⎟⎟⎠
⎞⎜⎜⎝
⎛
y
x
HH
⎟⎟⎠
⎞⎜⎜⎝
⎛
y
x
EE
zE
zH
I~ x2 z3
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−10
01⎟⎟⎠
⎞⎜⎜⎝
⎛−1001
⎟⎟⎠
⎞⎜⎜⎝
⎛
−−−
=1331
21
3/2πR
3/2πR
1− 1− 1+
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−10
01⎟⎟⎠
⎞⎜⎜⎝
⎛−1001
3/2πR
1− 1− 1+
Symmetries of low-T phase
Inversion combined with time reversal
ITI =~
120o-rotation
( )yyxxzz HEHEHEF +−−= ⊥αα ||meInvariants:
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Cr2O3cR33z
AFM order parameter TN = 306K
4321 MMMML −+−= 0≠zL1 symmetries of paramagnetic phase
2x I 2x 3z
Lz - + +Ez - - +Hz + - +
2
I3
2x
4zzzzz HEHEL ||αλ =
zL∝⊥αα ,||Invariants: ( )yyxxz HEHEL +
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Mechanism of inversion symmetry breaking Materials
covalent bonding between 3d0
transition metal (Ti) and oxygen BaTiO3
polarizability of 6s2 lone pair BiMnO3, BiFeO3
structural transition
‘Geometric ferroelectrics’
K2SeO4, Cs2CdI4h-RMnO3
charge ordering
‘Electronic ferroelectrics’LuFe2O4
magnetic ordering
‘Magnetic ferroelectrics’o-RMnO3, RMn2O5, CoCr2O4, MnWO4
Ferroelectrics
S.-W. Cheong & M. M. Nature Materials 6, 13 (2007)
Pro
per
Impr
oper
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Novel Multiferroicsmaterial TFE (K) TM (K) P(µC m-2)
TbMnO3 28 41 600
Ni3V2O8 6.3 9.1 100
CuFeO2 11 14 300
LiCu2O2 23 23 5
MnWO4 8 13.5 60
CoCr2O4 26 93 2
TbMn2O5 38 43 400
CuO 230 230 100
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Breaking of inversion symmetry by spin ordering
Inversion I: (x,y,z) (-x,-y,-z)
Q
e3
Cycloidal spiralQ
e3
Cycloidalspiral
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Induced Polarization
Energy (cubic lattice)
( ) ( )[ ]MMMMPP⋅∇−∇⋅⋅−= λ
χ ePF
2
2
Induced electric polarization
( ) ( )[ ]MMMMP ⋅∇−∇⋅= eλχ
Bary’akhtar et al, JETP Lett 37, 673 (1983); Stefanovskii et al, Sov. J. Low Temp. Phys. 12, 478(1986), M.M. PRL 96, 067601 (2006)
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Sinusoidal SDW
QxsinAM =
0=P
Qx
center of inversion
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Spiral SDW( )QxeQxeM sincos 210 += M
[ ]QeP ×∝ 3
Q
e3
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BiFeO3
TFE = 1100 K
TN = 640 K
Ferroelectric
Antiferromagnetic
( ) ( ) LPLLLF ∂−∂+= λϕ 2
Periodic modulation of AFM ordering: Q ∝ λ P
Free energy
A.M. Kadomtseva et al. JETP Lett. 79, 571 (2004)
Low-pitch spiral λ = 620 Å
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Geometrical Frustration
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0<J0>′J
Competing interactions[ ]∑ ++ ⋅′+⋅=
nnnnn JJE 21 SSSSFrustrated Heisenberg chain
4J
J >′JJ
Q′
=4
cos
[ ]∑ ++ ′+=n
nnnn JJE 21 σσσσFrustrated Ising chain
0>′J 1±=nσ
2J
J >′
0<J
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Magnetic frustration in RMnO3
JFM
JAFM
JFM
Mnb
a
FM
AFM
JJ2
=κκ < 1 Ferromagnetic
κ1
2cos =bQ
κ > 1 Incommensurate SDW
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Why TFE is lower than TM?TbMnO3 Ni3V2O8
G. Lawes et al PRL 95, 087205 (2005)
28K < T < 41K
T < 28K
6.3K < T < 9.1K
3.9K < T < 6.3K
M. Kenzelmann et al PRL 95, 087206 (2005)
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Sinusoidal-helicoidal transition
( ) ( ) ( ) MM2
22
24222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛+++++=Φ Q
dxdcMbMaMaMa z
zy
yx
xm
zxyx aaaa <∆+=<
Ginzburg-Landau expansion
Anisotropy:
QxM x cosx̂M =1st transition: Sinusoidal SDW
P = 0 ( ) 0=−= SDWx TTa α
QxMQxM yx sinˆcosˆ yxM +=2nd transition: Helicoidal SDW
α∆
−=23
SDWSP TT3x
yaa = P || y
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Dielectric constant anomalyat the transition to spiral state
T. Kimura et al , Nature 426, 55 (2003)
⎪⎪⎩
⎪⎪⎨
⎧
<−
>−
=
SPSP
SPSP
yy
TTTT
A
TTTT
A
,21
,ε
TTMMP SPyxy −∝∝
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Q Q
e
eP
P
Polarization Flop in Eu1-xYxMnO3
a||H0H =
3e
Q3e
P
P
Q
Spin flops Polarization flops
QeP ×∝ 3
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Magnetic phase diagrams
x
y
zb
a
c
Pbnm
T. Kimura et alPRB 71,224425(2005) M.M. PRL 96,
067601 (2006)
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Outline
• Linear magnetoelectric effect, multiferroics
• Phenomenological description
• Microscopic mechanisms of magnetoelectriccoupling
• Outlook
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Effects of Dzyaloshinskii-Moriya interaction
[ ]2112 SSD ×⋅=DME
1212 r̂xD ×∝ λ1S 2S
−2O
x
r12
H. Katsura et al PRL 95 057205 (2005), Sergienko & Dagotto PRB 73 094434 (2006)
P
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Dzyaloshinskii-Moriya interaction
( )sl ⋅= λSOH( )
σεε
ψλψσψβ βα
βαβαα
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−⋅
+→ ∑sl
αβ
γ1
( )βα
βααγγβ
εελ
−⋅ 1
12
slUtt
S( )
Utt
S γαβγ
βα
αβ
εελ 12
1
−⋅sl
(1) (2)
βα γ
[ ] 21121112 ,212, ssssss ×∝⎥⎦
⎤⎢⎣⎡ +⋅=∝ SHexδ
∇×=ihrl
2
real wave functions
γααγ tt =αββα ll −=
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Moriya rules
012 =D1S 2S
1S 2S
1S 2S
Inbersion center mirror xz plane
xz
yx
mirror yz plane
yz∈12D
xz⊥12D
1S 2Su
r12
urD ×∝ 1212 ˆ
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Ferroelectricity induced by magnetostriction
Aq Bq P
43212
43211
SSSSLSSSSL
+−−=−−+=
( )22
21int LLP −−=Φ λ 21 LL ↔
I
22
21 LLP −∝
1 2 3 4
Ca3Co2-xMnxO6
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RMn2O5
PbMn4+Mn3+
b
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Two-dimensional representation and induced polarization
PMn4+Mn3+ Mn3+
1L
2L
A. B. Sushkov et al. J. Phys. Cond. Mat. (2008)
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1S 2S
O2-
O2-
1S 2S
P
Exchange striction
E = J (S1S2)
θ = 180o J > 0 θ θ = 90o J < 0
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0=P
Role of frustration
P
Néel ordering: Inversion symmetry not broken
ÆÆ∞∞ ordering: Inversion symmetry is broken
To induce P spin ordering must break inversion symmetry
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Higher-order terms in effective spin Hamiltonian
Hubbard model + coupling to external fieldsL.N. Bulaevskii, C.D. Batista, M. M., and D. Khomskii, arXiv:0709.0575
Effective spin Hamiltonian (2nd order)
∑ ⎟⎠⎞
⎜⎝⎛ −⋅=
jijiU
tH,
2)2(
eff 414 SS
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Effective spin Hamiltonian(3d order)
Interaction with magnetic field
scalar spin chiralityPersistent electric current
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Effective spin Hamiltonian(3d order)
Interaction with electric field
virtual statesSpin-induced charge
( )[ ]∑ ⋅−+⋅=⎟⎠⎞
⎜⎝⎛=
∂∂
=n
b
UteHQ 32321
3
1
)3(eff
1 28 SSSSSϕ
δ
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Polarization of electronic orbitals
Ground state O2-
1 2
Intermedite state O2-
P1 2
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Outline
• Linear magnetoelectric effect, multiferroics
• Phenomenological description
• Microscopic mechanisms of magnetoelectriccoupling
• Outlook
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[ ]xeP ˆ|| 3 ×
X
xe ˆ3 ⊥
e3
0=PX
xe ˆ||3Bloch wall
Polarization of domain walls
Néel wall
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Electric charge of magnetic vortex
Charge in the vortex core
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Electrostatics of magnetic defects[ ]ϕϕ sincos 21 eeM += MEasy plane spins:
ϕεχγ babeb MP ∂−= 2Polarization:
Total polarization of domain wall:
( ) ( )[ ]∞−−∞+=∫ ϕϕχγ 2MPdx ey
( )( )⊥Γ=−= xP 222div δχγπρ MeCharge density:
ϕπ
∇⋅=Γ∝ ∫C
dQ x21
Vortex charge: winding number
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Magnetic vortex in magnetic field
P
H
pseudoscalarmoment
∑ ⋅∝α
αα Sr AP = 0
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Magnetic vortex in magnetic field
P
H
P = 0
toroidalmoment
∑ ×∝α
αα SrT
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Array of magnetic vorticesis magnetoelectric
H
P
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Magnetoelectric effectin spin triangle
HS1
P
S2 S3
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KITPITE
layeredKagomé lattice
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=0110
0ααC. Delaney, M. M. and N. A. Spaldin, to be published
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Conclusions
• Magnetic frustration gives rise to unusual spin orders that break inversion symmetry and give rise to multiferroicbehavior and linear magnetoelectriceffect