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Introduction to the physics of multiferroics
Charles Simon Laboratoire CRISMAT, CNRS and ENSICAEN,
F14050 Caen.
“Models in magnetism: from basics aspects to practical use”Timisoara september 2009
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models in magnetism timisoara 2
SummaryIntroduction and definitionsThe example of YMnO3Origin of the coupling term Dzyaloshinskii-Moriya Importance of symmetryApplicationsSome examplesLandau theory and symmetriesThe example of MnWO4
Examples are taken in work of Natalia Bellido, Damien Saurel, Kiran Singh and Bohdan Kundys
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models in magnetism timisoara 3
What is a multiferroic?Definitions are various: For me in this lecture:
A ferromagnetic and ferroelectric compound. (spontaneous magnetization in zero field and spontaneous polarization in zero field)
It was predicted by P. Curie in 1894 “Les conditions de symétrie nous permettent d’imaginer qu’un corps se polarise magnétiquement lorsqu’on lui applique un champ électrique”
Debye in 1926: magnetoélectricLandau in 1957
Dzyaloshinskii in 1959 predicts that Cr2 O3 magnetoelectric
Astrov et al. 1960 E induces M, Folen, Rado Stalker 1961, B induces P.
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models in magnetism timisoara 4
One example: YMnO3
MnO5
Hexagonal : P63 cm
a
b
Mn3+ S=2 cY3+
ferroelectric
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models in magnetism timisoara 5
Why this example
• Because is it quite simple in symmetry and interactions
• However, this is rather complex, and if you find it difficult, this is normal, I find it complex.
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models in magnetism timisoara 6
Pc
T
5.5μC/cm2
900K
c
Experimental difficulty
C=ε0 εS/t
P=II(t)dt
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models in magnetism timisoara 7
0 50 100 150 2004.4
4.6
4.8
5.0
5.2
5.4
χ (1
0-3 e
mu/
mol
)
T (K) 0 2 4 6 8 10 12 140.00
0.05
0.10
0.15 from T=10K to T=100K
M(μ
B/fu
)
μ Η(T)
Antiferromagnetism
Mn3+
L : alternate magnetization
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models in magnetism timisoara 8
L = Σ
Si exp(2iπ
Qri )Order parameter
Neutron scattering
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models in magnetism timisoara 9
0 20 40 60 80 100 120
16.5
17.0
17.5
18.0
ε
T(K)
YMnO3 - ε(T)
2L−∝ε
ε
= 1/ ε0
dP/dE dielectric constant
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models in magnetism timisoara 10
Pc
T
5.5μC/cm2
900K
cTN
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models in magnetism timisoara 11
0 20 40 60 80 100
0.000410
0.000415
0.000420
0.000425
0.000430
M(e
mu)
T(K)
Small ferromagnetic component along c induced by the ferroelectric component
L order parameterP non zero everywhere, secondaryM third order
Pc
T
5.5μC/cm2
TN
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models in magnetism timisoara 12
Pailhes et al., 2009
They don’t vary in the same way.
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models in magnetism timisoara 13
After Pailhes et al. Hybrid modes
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models in magnetism timisoara 14
questions
• YMnO3 is ferromagnetic (?) below TN !– This was already published by Bertaut
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models in magnetism timisoara 15
questions
• YMnO3 is ferromagnetic (?) below TN !• What is the origin of the coupling? Why
there is an effect on polarization? – Two steps
• The microscopic coupling (exchange, LS coupling)• The long range ordering (symmetry)
– Both are difficult
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models in magnetism timisoara 16
Origin of the coupling term
1 Displacement of oxygen is responsible to the polarization
2 Origin of the antiferromagnetism?superexchange by oxygen
3 antiferromagnetism by superexchangechanges the energy and the polarization
4 It induces a ferromagnetic component.
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models in magnetism timisoara 17
Superexchange explanation?
• Does superexchange enough to understand the coupling? – No, because of the symmetry. If you add the
three contributions, they cancel by symmetry.
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models in magnetism timisoara 18
Cancel by symmetry
After I. A. Sergienko and E. Dagotto
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models in magnetism timisoara 19
On the contrary, the Dzyaloshinskii-Moriya interaction— i.e., anisotropic exchangeinteraction Sn x Sn+1 — changes its sign under inversion.
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models in magnetism timisoara 20
Dzyaloshinskii-Moriya interaction
• Of course, this expansion in term of LS coupling does not mean that this term in the dominant one, but an least, this is the first one you can think about.
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models in magnetism timisoara 21
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models in magnetism timisoara 22
Sn
Sn+1
Sn x Sn+1
Sn
Sn+1
Sn x Sn+1
Effect of inversion
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models in magnetism timisoara 23
• The problem is the symmetry• The solution is the symmetry• The method in Landau theory
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models in magnetism timisoara 24
YMnO3 symmetry
• Non ferroelectric P63/mmc (194)
• ferroelectric P63cm (185).
M=0
Mc can be non zero
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models in magnetism timisoara 25
1 identity
2 symmetry by a plane
No in plane components
3 rotation axis 2 with translation
C axis component possible
4 combinations of two
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models in magnetism timisoara 26
YMnO3 symmetry
Non ferroferro
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models in magnetism timisoara 27
• Symmetry analysis shows that the experimental observation was the only possible one.
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models in magnetism timisoara 28
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models in magnetism timisoara 29
Symmetry restrictions
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models in magnetism timisoara 30
• This is very limited• Solution: incommensurability
– An incommensurate modulation of the magnetism with a ferromagnetic component suppresses the corresponding symmetry elements
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models in magnetism timisoara 31
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models in magnetism timisoara 32
Applications• Magnetic memories that you can write with electric field• RAM (random acces memory) FRAM (ferroélectric, no battery), MRAM
(magnétic, no battery, difficult to write).• Multiferro: write with electric field, read with magnetic sensor.
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models in magnetism timisoara 33
GMR
R
M
I
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models in magnetism timisoara 34
R
M
I
Write multiferro
P
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models in magnetism timisoara 35
One historical example: Boracites
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models in magnetism timisoara 36
Ni3 B7 O13 I
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models in magnetism timisoara 37
Other materials• Structure: perovskite: BiFeO3 PrMnO3
• Structure: hexagonal: MMnO3 M=Y, Ho, etc…• Boracites• Spiral magnetic order: TbMnO3 MnWO4
• Fe Langasites.
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models in magnetism timisoara 38
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models in magnetism timisoara 39
Tenurite CuO
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models in magnetism timisoara 40
4 6 8 10 12 14
7.72
7.73
7.74Co3V2O8
T(K)
ε
Kagome staircase - Co3 V2 O8
4 6 8 10 12 140.0
0.1
0.2
0.3
0.4
0.5
0.6
δ=0
δ=1/3
δ=1/2
δ
T(K)
Ni3 V2 O8 [1]: S=1Co3 V2 O8 [1]: S=3/2 β-Cu3 V2 O8 [2]: S=1/2
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models in magnetism timisoara 41
Eu0.75 Y0.25 MnO3
H=0
H
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models in magnetism timisoara 42
CuCrO2
Complexincommensuratestructure
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models in magnetism timisoara 43
CuCrO2
-10 -8 -6 -4 -2 0 2 4 6 8 10
-1
0
1
2
3
4
-10 -8 -6 -4 -2 0 2 4 6 8 10
13
14
15
16
17
18
19
20
21
(c)20K
Tran
sver
sal m
agne
tost
rictio
n, Δ
L/L*
106
H(T)
(b)
Pol
ariz
atio
n, P
(μC
/m2 )
H(T)
Time(Sec)
Bohdan Kundys, Maria Poienar, Antoine Maignan, Christine Martin, Charles Simon
( ) gLPEPPLFF AFM +−+= 2α
bdHTTaL N 2/))(( 22 +−−=
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models in magnetism timisoara 44
FeVO4
6 Fe3+ 5/2 in a triclinic structure 1
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models in magnetism timisoara 45
FeVO4
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models in magnetism timisoara 46
FeCuO2
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models in magnetism timisoara 47
A ferroic material
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Free energy from “Landau”
Tc
M
Température
MHMcMcMcMcFF −+++++= ....44
33
2210
MHMbMaFF FMFM −++= 42
420
PEPPFF FEFE −++= 42
420
βα
Ferromagnet
Tc
P
Température
PEPcPcPcPcFF −+++++= ....44
33
2210
+Q
-QPr
Ferroelectric
-Q
+QPr
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models in magnetism timisoara 49
T>Tc
T<Tc
MHMbMaFF FMFM −++= 42
420
a is linear in T-Tc
M2 = -a/b H
T
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models in magnetism timisoara 50
Interactions and symmetries
• This example is too simple: the symmetry is hidden and the role of the interactions is not clear
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models in magnetism timisoara 51
• We have already discussed in this school the possible origins of ferromagnetism
• Let us discuss briefly the possible origin of ferroelectricity:– A shift of one of the atoms from the
symmetrical position due electron electron repulsion
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Free energy from “Landau”
Tc
M
Température
MHMcMcMcMcFF −+++++= ....44
33
2210
MHMbMaFF FMFM −++= 42
420
PEPPFF FEFE −++= 42
420
βα
Ferromagnet
Tc
P
Température
PEPcPcPcPcFF −+++++= ....44
33
2210
+Q
-QPr
Ferroelectric
-Q
+QPr
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models in magnetism timisoara 53
A little more about Landau• Paraelectric I 4/mmm to
ferroelectric II at Tc.• F is formed by
successive invariants
From P. Toledano
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models in magnetism timisoara 54
• Quadratic invariants Px2+Py2, Pz2
• Quartic invariants (Px2+Py2) 2, Pz4, Px4+Py4, (PxPy)2
• F=F0 +a/2(Px2+Py2)+a’/2 Pz2+…
• Minimization of F with respect to Px,Py,Pz• a or a’ changes sign first (assume a, a’>0)
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models in magnetism timisoara 55
• Then, Pz2 = -a/b• Pz is the order parameter.
Tc
P
Température
PEPcPcPcPcFF −+++++= ....44
33
2210
+Q
-QPr
Ferroelectric
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models in magnetism timisoara 56
Pz 4mm dimension 1Pxy 2mm dimension 2
Subgroups of 4/mmm
• Two possibilities:
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models in magnetism timisoara 57
Secondary order parameterLet us call e the strain tensor
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models in magnetism timisoara 58
Magnetic energyExample 4 atoms in Pca21
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models in magnetism timisoara 59
• This is rather complex, because spins don’t transform with the same symmetry operations than the “real” vectors,
• S x S is also different.
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models in magnetism timisoara 60
• One example: in a mirror
Real vector Axial vector S x S vector
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models in magnetism timisoara 61
In addition
• Incommensurate modulations suppressesSymmetry elements.
I have no time to explain details
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models in magnetism timisoara 62
MnWO4ferroelectric
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models in magnetism timisoara 63
MnWO4sensitive to magnetic field
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models in magnetism timisoara 64
AF1, AF2, AF3
Collinear 1/4,,1/2,1/2
-0.241,1/2,0.457
P along a
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models in magnetism timisoara 65
• The symmetry analysis was made by P. Toledano, and we find all the observed phases as possible sub groups
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models in magnetism timisoara 66
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models in magnetism timisoara 67
Pr1/2 Ca1/2 MnO3CE type
Ferromagnetic coupling
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models in magnetism timisoara 68
No centrosymmetry
From Khomskii et al.
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models in magnetism timisoara 69
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models in magnetism timisoara 70
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models in magnetism timisoara 71
No ferroelectricity
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Electric susceptibility
χ
= ε-1
YMnO3 - Landau
23
22
210 )( HcTLcTc +−+= εε
=+++= couplFEAFM FFFFF 0Free energy :
Minimization : 00 22 =++−⇒=∂∂ PHgPLEPPF γα
EHgL
P 221
γα ++=
++++= 2242
0 42HcLLbLaF EPPHcLLbLaF −++++=
242
222
42
0 α 22222
2242
0 22242HPLPgEPPHcLLbLaF γα ++−++++=
∼20 ∼1 ∼10-4
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YMnO3 – Anomaly in ε(T)
221
HgL γαχ
++=
2
2
211)0,0(),0()(
αααεεε gL
gLLHLHT −≈−
+===−==Δ
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models in magnetism timisoara 74
YMnO3 – ε(H)Er
Br
0.00
0.02
0.04
T=90KT=80KT=70K
T=60KT=50KT=40K
T=30KT=20K
T=10K
0.00
0.02
0.04
-10 -5 0 5 10
0.00
0.02
0.04
μ0H(T)-10 -5 0 5 10
ΔεH/ε0(%)
μ0H(T)-10 -5 0 5 10 15
μ0H(T)
0 20 40 60 800
1
2
3
coef
fient
me
x1012
(T-2)
T(K) Paramagnet
Δε~10-4
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models in magnetism timisoara 75
YMnO3 –magnétodiélectric effect ε(H) in H2
2
2
22211
),0(),()(
αγ
αγα
εεε
HgLHgL
LHLHH
−≈+
−++
=
==−=Δ
221
HgL γαχ
++=
⎟⎟⎠
⎞⎜⎜⎝
⎛−−≈Δ
ααγε
2
2
2
21)( gLHH
⎟⎠⎞⎜
⎝⎛ −+= 2222 LLLL
fluctuations~χL
0 20 40 60 800
1
2
3
coef
fient
me
x1012
(T-2)
T(K)
γ
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models in magnetism timisoara 76
0 20 40 60 800
1
2
3
coef
fient
me
x1012
(T-2)
T(K)
YMnO3 constante diélectrique
⎟⎟⎠
⎞⎜⎜⎝
⎛−
+−−+=NTT
HLgTc λαγ
αεε 12
22
22
10
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models in magnetism timisoara 77
CuCrO2
-10 -8 -6 -4 -2 0 2 4 6 8 10-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
22K
21K
23K
24K
15K
10K
6K
27K
25K
100kHz
Δε'/ε
' H=0
(%)
H(T)
0 12 24 36 484.0x10-5
4.2x10-5
4.4x10-5
0
1
2
3
4
5TN
χ (e
mu.
g-1)
T(K)
-Δε'/ε' H
=0 (%
)
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Co3 V2 O8
-10 -5 0 5 10-6
-3
0
3
6
M (μB/f.u.)
μ0H(T)
-6
-3
0
3
6
T=50K
T=20K
-6
-3
0
3
6
T=7K
-10 -5 0 5 10-0.10
-0.05
0.00
μ0H(T)
-0.15
-0.10
-0.05
0.00
-0.10
-0.05
0.00
ΔεΗ/ε0
(%)
-10 -5 0 5 10
0.3
0.4
0.5
dM/dH(μB/T·f.u.)
μ0H(T)
0.5
1.0
T=50K
T=20K
0
5
10
15
T=7K
T=7K
T=20K
T=50K
Δε∼χ
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models in magnetism timisoara 79
Ca3 Co2 O6 – magnetization plateaux
Polyhèdra CoO6 :
triangular prism S=2
octahedra S= 0
Ferromagnet intrachain interac.
Triangular ising lattice
Antiferromagnetic interchain (TN =24K)
0 1 2 3 4 5 60
1
2
3
4
5
T=10K
M (μ
B/f.u
.)
μ0H(T)0 2 4 6 8 10
0
1
2
3
4
5
T=2K
M (μ
B/f.u
.)
μ0H(T)
ΔH=3.6T ΔH=1.2T
R-3cm
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models in magnetism timisoara 80
Ca3 Co2 O6
0 1 2 3 4 5 60
1
2
3
4
5
M
(μB/
f.u.)
μ0H(T)
0 1 2 3 4 5 6
-1
0
Δε H
/εsa
t (%
)
μ0H(T)
T=10K
0 1 2 3 4 5 60.0
0.5
1.0
1.5
χ(μ B/
T·f.u
.)
μ0H(T)
Δε∼-χ
No polarization
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models in magnetism timisoara 81
MnWO4A nice example
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models in magnetism timisoara 82
P.G. Radaelli and L.C. Chapon, PRB, 76054428(2007)
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models in magnetism timisoara 83
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models in magnetism timisoara 84
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models in magnetism timisoara 85
Conclusion• Spin orbit coupling is necessary to create coupling between ferromagnetism
and ferroelectricity• Incommensurability is very useful to help with symmetry• There is no ab initio calculation of the intensity of the coupling• There is more to understand in the coupling terms• Magnetic group theory is needed.