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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
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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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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One example: YMnO3
MnO5
Hexagonal : P63 cm
a
b
Mn3+ S=2 cY3+
ferroelectric
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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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Pc
T
5.5μC/cm2
900K
c
Experimental difficulty
C=ε0 εS/t
P=II(t)dt
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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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L = Σ
Si exp(2iπ
Qri )Order parameter
Neutron scattering
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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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Pc
T
5.5μC/cm2
900K
cTN
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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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Pailhes et al., 2009
They don’t vary in the same way.
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After Pailhes et al. Hybrid modes
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questions
• YMnO3 is ferromagnetic (?) below TN !– This was already published by Bertaut
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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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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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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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Cancel by symmetry
After I. A. Sergienko and E. Dagotto
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On the contrary, the Dzyaloshinskii-Moriya interaction— i.e., anisotropic exchangeinteraction Sn x Sn+1 — changes its sign under inversion.
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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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Sn
Sn+1
Sn x Sn+1
Sn
Sn+1
Sn x Sn+1
Effect of inversion
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• The problem is the symmetry• The solution is the symmetry• The method in Landau theory
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YMnO3 symmetry
• Non ferroelectric P63/mmc (194)
• ferroelectric P63cm (185).
M=0
Mc can be non zero
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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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YMnO3 symmetry
Non ferroferro
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• Symmetry analysis shows that the experimental observation was the only possible one.
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Symmetry restrictions
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• 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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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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GMR
R
M
I
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R
M
I
Write multiferro
P
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One historical example: Boracites
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Ni3 B7 O13 I
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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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Tenurite CuO
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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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Eu0.75 Y0.25 MnO3
H=0
H
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CuCrO2
Complexincommensuratestructure
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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FeVO4
6 Fe3+ 5/2 in a triclinic structure 1
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FeVO4
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FeCuO2
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A ferroic material
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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T>Tc
T<Tc
MHMbMaFF FMFM −++= 42
420
a is linear in T-Tc
M2 = -a/b H
T
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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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• 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
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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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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• 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)
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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Pz 4mm dimension 1Pxy 2mm dimension 2
Subgroups of 4/mmm
• Two possibilities:
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Secondary order parameterLet us call e the strain tensor
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Magnetic energyExample 4 atoms in Pca21
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• 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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• One example: in a mirror
Real vector Axial vector S x S vector
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In addition
• Incommensurate modulations suppressesSymmetry elements.
I have no time to explain details
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MnWO4ferroelectric
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MnWO4sensitive to magnetic field
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AF1, AF2, AF3
Collinear 1/4,,1/2,1/2
-0.241,1/2,0.457
P along a
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• The symmetry analysis was made by P. Toledano, and we find all the observed phases as possible sub groups
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Pr1/2 Ca1/2 MnO3CE type
Ferromagnetic coupling
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No centrosymmetry
From Khomskii et al.
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No ferroelectricity
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
YMnO3 – Anomaly in ε(T)
221
HgL γαχ
++=
2
2
211)0,0(),0()(
αααεεε gL
gLLHLHT −≈−
+===−==Δ
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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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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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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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 (%
)
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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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
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
models in magnetism timisoara 81
MnWO4A nice example
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P.G. Radaelli and L.C. Chapon, PRB, 76054428(2007)
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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.
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