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
Summary
Introduction and definitions
The example of YMnO3
Origin of the coupling term Dzyaloshinskii-Moriya
Importance of symmetry
Applications
Some examples
Landau theory and symmetries
The example of MnWO4
Examples are taken in work of Natalia Bellido, Damien
Saurel, Kiran Singh and Bohdan Kundys
models in magnetism timisoara
2
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: magnetoélectric
Landau in 1957
Dzyaloshinskii in 1959 predicts that Cr2O3
magnetoelectric
Astrov et al. 1960 E induces M, Folen, Rado Stalker 1961, B induces P.
models in magnetism timisoara
3
One example: YMnO3
Hexagonal : P63cm
ferroelectric
MnO5
b
Y3+
a
Mn3+ S=2
c
models in magnetism timisoara
4
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.
models in magnetism timisoara
5
Pc
5.5μC/cm2
c
900K
T
Experimental difficulty
P=II(t)dt
C=ε0εS/t
models in magnetism timisoara
6
5.4
0.15 from T=10K to T=100K
5.2
M(μB/fu)
-3
Mn3+
χ (10 emu/mol)
Antiferromagnetism
5.0
4.8
4.6
4.4
0
50
100
150
200
T (K)
0.10
0.05
0.00
0 2 4 6 8 10 12 14
μ Η(T)
L : alternate
magnetization
models in magnetism timisoara
7
Neutron scattering
Order parameter
L = Σ Si exp(2iπ Qri)
models in magnetism timisoara
8
YMnO3 - ε(T)
ε = 1/ ε0 dP/dE dielectric constant
18.0
ε
17.5
17.0
16.5
0
20
40
60
80
100
120
T(K)
ε ∝ −L
2
models in magnetism timisoara
9
Pc
5.5μC/cm2
c
TN
900K
T
models in magnetism timisoara
10
0.000430
Pc
5.5μC/cm2
M(emu)
0.000425
0.000420
0.000415
TN
0.000410
0
20
40
60
80
100
T(K)
T
Small ferromagnetic component along c
induced by the ferroelectric component
L order parameter
P non zero everywhere, secondary
M third order
models in magnetism timisoara
11
They don’t vary in the
same way.
Pailhes et al., 2009
models in magnetism timisoara
12
After Pailhes et al.
Hybrid modes
models in magnetism timisoara
13
questions
• YMnO3 is ferromagnetic (?) below TN!
– This was already published by Bertaut
models in magnetism timisoara
14
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
models in magnetism timisoara
15
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 superexchange
changes the energy and the polarization
4 It induces a ferromagnetic component.
models in magnetism timisoara
16
Superexchange explanation?
• Does superexchange enough to
understand the coupling?
– No, because of the symmetry. If you add the
three contributions, they cancel by symmetry.
models in magnetism timisoara
17
Cancel by symmetry
After I. A. Sergienko and E. Dagotto
models in magnetism timisoara
18
On the contrary, the Dzyaloshinskii-Moriya
interaction—
i.e., anisotropic exchange
interaction Sn x Sn+1— changes its sign under
inversion.
models in magnetism timisoara
19
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.
models in magnetism timisoara
20
models in magnetism timisoara
21
Sn x Sn+1
Sn+1
Sn x Sn+1
Sn+1
Sn
Sn
Effect of inversion
models in magnetism timisoara
22
• The problem is the symmetry
• The solution is the symmetry
• The method in Landau theory
models in magnetism timisoara
23
YMnO3 symmetry
• Non ferroelectric P63/mmc (194)
M=0
• ferroelectric P63cm (185).
Mc can be non zero
models in magnetism timisoara
24
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
models in magnetism timisoara
25
YMnO3 symmetry
Non ferro
ferro
models in magnetism timisoara
26
• Symmetry analysis shows that the
experimental observation was the only
possible one.
models in magnetism timisoara
27
models in magnetism timisoara
28
Symmetry restrictions
models in magnetism timisoara
29
• This is very limited
• Solution: incommensurability
– An incommensurate modulation of the
magnetism with a ferromagnetic component
suppresses the corresponding symmetry
elements
models in magnetism timisoara
30
models in magnetism timisoara
31
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.
models in magnetism timisoara
32
GMR
I
M
R
models in magnetism timisoara
33
Write multiferro
P
I
M
R
models in magnetism timisoara
34
One historical example: Boracites
models in magnetism timisoara
35
Ni3B7O13I
models in magnetism timisoara
36
Other materials
•
•
•
•
•
Structure: perovskite: BiFeO3 PrMnO3
Structure: hexagonal: MMnO3 M=Y, Ho, etc…
Boracites
Spiral magnetic order: TbMnO3 MnWO4
Fe Langasites.
models in magnetism timisoara
37
models in magnetism timisoara
38
Tenurite CuO
models in magnetism timisoara
39
Kagome staircase - Co3V2O8
Ni3V2O8 [1]: S=1
Co3V2O8 [1]: S=3/2
β-Cu3V2O8 [2]: S=1/2
0.6
δ=1/2
0.5
0.4
δ
Co3V2O8
7.74
δ=1/3
ε
0.3
7.73
0.2
0.1
0.0
δ=0
4
7.72
6
8
10
T(K)
12
14
4
6
models in magnetism timisoara
8
10
T(K)
12
14
40
Eu0.75Y0.25MnO3
H=0
H
models in magnetism timisoara
41
CuCrO2
Complex
incommensurate
structure
models in magnetism timisoara
42
CuCrO2
Time(Sec)
21
(b)
Polarization, P(μC/m2)
20
F = FAFM (L ) + αP 2 − EP + gLP
19
18
17
16
15
14
13
-10
-8
-6
-4
-2
0
2
4
6
8
10
6
8
10
Transversal magnetostriction, ΔL/L*106
H(T)
(c)
4
L2 = ( − a (T − TN ) + dH 2 ) / 2b
20K
3
2
1
0
-1
-10
-8
-6
-4
-2
0
2
4
H(T)
Bohdan Kundys, Maria Poienar, Antoine Maignan, Christine Martin, Charles Simon
models in magnetism timisoara
43
FeVO4
6 Fe3+ 5/2 in a triclinic structure 1
models in magnetism timisoara
44
FeVO4
models in magnetism timisoara
45
FeCuO2
models in magnetism timisoara
46
A ferroic material
models in magnetism timisoara
47
Free energy from “Landau”
Ferromagnet
M
F = F0 + c1M + c2 M 2 + c3 M 3 + c4 M 4 + .... − MH
Tc
FFM = FFM 0
a 2 b 4
+ M + M − MH
2
4
Température
Ferroelectric
P
F = F0 + c1 P + c2 P 2 + c3 P 3 + c4 P 4 + .... − PE
Tc
r
P
Température
+Q
-Q
r
P
-Q
+Q
FFE = FFE0 +
α
2
P +
2
β
4
P 4 − PE
T>Tc
FFM = FFM 0
a 2 b 4
+ M + M − MH
2
4
a is linear in T-Tc
T<Tc
M2 = -a/b
H
T
models in magnetism timisoara
49
Interactions and symmetries
• This example is too simple: the symmetry
is hidden and the role of the interactions is
not clear
models in magnetism timisoara
50
• 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
models in magnetism timisoara
51
Free energy from “Landau”
Ferromagnet
M
F = F0 + c1M + c2 M 2 + c3 M 3 + c4 M 4 + .... − MH
Tc
FFM = FFM 0
a 2 b 4
+ M + M − MH
2
4
Température
Ferroelectric
P
F = F0 + c1 P + c2 P 2 + c3 P 3 + c4 P 4 + .... − PE
Tc
r
P
Température
+Q
-Q
r
P
-Q
+Q
FFE = FFE0 +
α
2
P +
2
β
4
P 4 − PE
A little more about Landau
• Paraelectric I 4/mmm to
ferroelectric II at Tc.
• F is formed by
successive invariants
From P. Toledano
models in magnetism timisoara
53
• 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
54
• Then, Pz2 = -a/b
• Pz is the order parameter.
Ferroelectric
P
F = F0 + c1 P + c2 P 2 + c3 P 3 + c4 P 4 + .... − PE
Tc
r
P
Température
+Q
-Q
models in magnetism timisoara
55
• Two possibilities:
Pz 4mm dimension 1
Pxy 2mm dimension 2
Subgroups of 4/mmm
models in magnetism timisoara
56
Secondary order parameter
Let us call e the strain tensor
models in magnetism timisoara
57
Magnetic energy
Example 4 atoms in Pca21
models in magnetism timisoara
58
• This is rather complex, because spins
don’t transform with the same symmetry
operations than the “real” vectors,
• S x S is also different.
models in magnetism timisoara
59
• One example: in a mirror
Real vector
Axial vector
models in magnetism timisoara
S x S vector
60
In addition
• Incommensurate modulations suppresses
Symmetry elements.
I have no time to explain details
models in magnetism timisoara
61
MnWO4
ferroelectric
models in magnetism timisoara
62
MnWO4
sensitive to magnetic field
models in magnetism timisoara
63
AF1, AF2, AF3
P along a
-0.241,1/2,0.457
Collinear 1/4,,1/2,1/2
models in magnetism timisoara
64
• The symmetry analysis was made by P.
Toledano, and we find all the observed
phases as possible sub groups
models in magnetism timisoara
65
models in magnetism timisoara
66
Pr1/2Ca1/2MnO3
CE type
Ferromagnetic coupling
models in magnetism timisoara
67
No centrosymmetry
From Khomskii et al.
models in magnetism timisoara
68
models in magnetism timisoara
69
models in magnetism timisoara
70
No ferroelectricity
models in magnetism timisoara
71
YMnO3 - Landau
ε = ε 0 + c1T 2 − c2 L2 (T ) + c3 H 2
∼20
∼1
∼10-4
Free energy : F = F0 + FAFM + FFE + Fcoupl =
2
L2
L4
P
g
γ
= F0 + a + b + cL2 H 2 + α
− EP + P 2 L2 + P 2 H 2
2
4
2
2
2
Minimization :
∂F
= 0 ⇒ αP − E + gPL2 + γPH 2 = 0
∂P
1
P=
E
2
2
α + gL + γH
Electric
χ = ε-1
susceptibility
YMnO3 – Anomaly in ε(T)
1
χ=
α + gL2 + γH 2
1
1
gL2
Δε (T ) = ε ( H = 0, L) − ε ( H = 0, L = 0) =
− ≈− 2
2
α + gL α
α
YMnO3 – ε(H)
r
E
Δε~10-4
0.04
T=10K
T=20K
T=30K
T=50K
T=60K
0.02
0.00
r
B
ΔεH/ε00.04
(%)
T=40K
0.02
12
-2
coeffient me x10 (T )
0.00
0.04
3
T=80K
T=70K
T=90K
0.02
2
0.00
-10
1
0
0
20
40
T(K)
60
80
-5
0
5
μ0H(T)
10
-10
-5
0
5
μ0H(T)
10
-10
-5
0
5
μ0H(T)
10
15
Paramagnet
models in magnetism timisoara
74
YMnO3 –magnétodiélectric effect ε(H) in H2
1
χ=
α + gL2 + γH 2
Δε ( H ) = ε ( H , L) − ε ( H = 0, L) =
γH 2 ⎛⎜
gL2
Δε ( H ) ≈ − 2 1 − 2
α ⎜⎝
α
12
-2
coeffient me x10 (T )
1
1
γH 2
=
−
≈− 2
2
2
2
α + gL + γH α + gL
α
3
2
γ
1
0
L = L + ⎛⎜ L2 − L2 ⎞⎟
⎝
⎠
2
0
20
40
60
80
⎞
⎟
⎟
⎠
2
T(K)
models in magnetism timisoara
fluctuations~χL
75
YMnO3 constante diélectrique
γ
λ
2⎛
ε = ε 0 + c1T − 2 L − 2 H ⎜⎜1 +
α
α
⎝ T − TN
g
2
⎞
⎟⎟
⎠
12
-2
coeffient me x10 (T )
2
3
2
1
0
0
20
40
60
80
T(K)
models in magnetism timisoara
76
CuCrO2
TN
-5
4.4x10
0.5
5
27K
25K
4
Δε'/ε'H=0 (%)
10K
-1.5
-2.0
15K
-2.5
24K
-3.0
21K
-1
6K
-1.0
χ (emu.g )
-0.5
3
-5
4.2x10
2
-Δε'/ε'H=0 (%)
0.0
1
22K
-3.5
100kHz
23K
0
-5
4.0x10
-4.0
-10
-8
-6
-4
-2
0
2
4
6
8
10
0
12
24
36
48
T(K)
H(T)
models in magnetism timisoara
77
Co3V2O8
Δε∼χ
0.00
T=7K
-0.05
6
15
3
10
0
T=7K
-35
-0.10
0.00
ΔεΗ/ε0
(%)
T=20K
-0.05
-0.10
-0.15
T=7K
-60
6
dM/dH 1.03
M
(μB/T·f.u.) 0.50
(μB/f.u.) -3
T=20K
T=20K
-6
0.5
6
0.00
3
0.4
0
-0.05
-3
0.3
-6
T=50K
-0.10
-10
-5
0
5
μ0H(T)
10
T=50K
T=50K
-10
-10
-5
-5
00
55
H(T)
μμ00H(T)
10
10
Ca3Co2O6 – magnetization plateaux
R-3cm
Polyhèdra CoO6 :
triangular prism S=2
octahedra S= 0
Ferromagnet intrachain interac.
Triangular ising lattice
Antiferromagnetic interchain (TN=24K)
ΔH=3.6T
ΔH=1.2T
5
4
M (μB/f.u.)
M (μB/f.u.)
5
T=10K
3
2
1
0
0
1
2
3
4 5 6in
models
μ0H(T)
4
3
T=2K
2
1
0
0
2
4
magnetism timisoara
6
8
μ0H(T)
10
79
Ca3Co2O6
T=10K
χM
(μB(/T·f.u.)
μB/f.u.)
5
1.5
4
Δε∼-χ
1.03
2
0.5
1
ΔεH/εsat (%)
0.00
00
1
1
2
2
33
44
μμ0H(T)
H(T)
0
55
66
0
No polarization
-1
0
1
2
3
4
μ0H(T)
5
6
models in magnetism timisoara
80
MnWO4
A nice example
models in magnetism timisoara
81
P.G. Radaelli and L.C. Chapon, PRB, 76054428(2007)
models in magnetism timisoara
82
models in magnetism timisoara
83
models in magnetism timisoara
84
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.
models in magnetism timisoara
85