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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 Summary Introduction and definitions

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Introduction to the physics

  • f 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

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

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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: magnetoélectric Landau 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 c Y3+

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

50 100 150 200 4.4 4.6 4.8 5.0 5.2 5.4

χ (10

  • 3 emu/mol)

T (K)

2 4 6 8 10 12 14 0.00 0.05 0.10 0.15 from T=10K to T=100K

M(μB/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

20 40 60 80 100 120 16.5 17.0 17.5 18.0

ε

T(K)

YMnO3 - ε(T)

2

L − ∝ ε

ε = 1/ ε0 dP/dE dielectric constant

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models in magnetism timisoara 10

Pc T 5.5μC/cm2 900K c TN

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models in magnetism timisoara 11

20 40 60 80 100 0.000410 0.000415 0.000420 0.000425 0.000430

M(emu) T(K)

Small ferromagnetic component along c induced by the ferroelectric component

L order parameter P non zero everywhere, secondary M 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 superexchange changes 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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On the contrary, the Dzyaloshinskii-Moriya interaction— i.e., anisotropic exchange interaction 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 ferro ferro

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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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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 7.74 Co3V2O8

T(K)

ε

Kagome staircase - Co3 V2 O8

4 6 8 10 12 14 0.0 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=1 Co3 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

Complex incommensurate structure

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models in magnetism timisoara 43

CuCrO2

  • 10
  • 8
  • 6
  • 4
  • 2

2 4 6 8 10

  • 1

1 2 3 4

  • 10
  • 8
  • 6
  • 4
  • 2

2 4 6 8 10 13 14 15 16 17 18 19 20 21

(c)

20K Transversal magnetostriction, ΔL/L*10

6

H(T)

(b)

Polarization, P(μC/m

2)

H(T) Time(Sec)

Bohdan Kundys, Maria Poienar, Antoine Maignan, Christine Martin, Charles Simon

( )

gLP EP P L F F

AFM

+ − + =

2

α

b dH T T a L

N

2 / ) ) ( (

2 2

+ − − =

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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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SLIDE 48

Free energy from “Landau”

Tc M Température

MH M c M c M c M c F F − + + + + + = ....

4 4 3 3 2 2 1

MH M b M a F F

FM FM

− + + =

4 2

4 2 PE P P F F

FE FE

− + + =

4 2

4 2 β α

Ferromagnet

Tc P Température

PE P c P c P c P c F F − + + + + + = ....

4 4 3 3 2 2 1

+Q

  • Q

P r

Ferroelectric

  • Q

+Q

P r

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models in magnetism timisoara 49

T>Tc T<Tc

MH M b M a F F

FM FM

− + + =

4 2

4 2

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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SLIDE 52

Free energy from “Landau”

Tc M Température

MH M c M c M c M c F F − + + + + + = ....

4 4 3 3 2 2 1

MH M b M a F F

FM FM

− + + =

4 2

4 2 PE P P F F

FE FE

− + + =

4 2

4 2 β α

Ferromagnet

Tc P Température

PE P c P c P c P c F F − + + + + + = ....

4 4 3 3 2 2 1

+Q

  • Q

P r

Ferroelectric

  • Q

+Q

P r

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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

PE P c P c P c P c F F − + + + + + = ....

4 4 3 3 2 2 1

+Q

  • Q

P r

Ferroelectric

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models in magnetism timisoara 56

Pz 4mm dimension 1 Pxy 2mm dimension 2 Subgroups of 4/mmm

  • Two possibilities:
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models in magnetism timisoara 57

Secondary order parameter

Let us call e the strain tensor

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models in magnetism timisoara 58

Magnetic energy

Example 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

  • perations 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 suppresses

Symmetry elements. I have no time to explain details

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models in magnetism timisoara 62

MnWO4

ferroelectric

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models in magnetism timisoara 63

MnWO4

sensitive 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 MnO3 CE 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

2 3 2 2 2 1

) ( H c T L c T c + − + = ε ε

= + + + =

coupl FE AFM

F F F F F

Free energy : Minimization :

2 2

= + + − ⇒ = ∂ ∂ PH gPL E P P F γ α

E H gL P

2 2

1 γ α + + =

+ + + + =

2 2 4 2

4 2 H cL L b L a F EP P H cL L b L a F − + + + + = 2 4 2

2 2 2 4 2

α

2 2 2 2 2 2 2 4 2

2 2 2 4 2 H P L P g EP P H cL L b L a F γ α + + − + + + + =

∼20 ∼1 ∼10-4

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YMnO3 – Anomaly in ε(T)

2 2

1 H gL γ α χ + + =

2 2 2

1 1 ) , ( ) , ( ) ( α α α ε ε ε gL gL L H L H T − ≈ − + = = = − = = Δ

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models in magnetism timisoara 74

YMnO3 – ε(H)

E r B r

0.00 0.02 0.04

T=90K T=80K T=70K T=60K T=50K T=40K T=30K T=20K T=10K

0.00 0.02 0.04

  • 10
  • 5

5 10 0.00 0.02 0.04

μ0H(T)

  • 10
  • 5

5 10

ΔεH/ε0

(%)

μ0H(T)

  • 10
  • 5

5 10 15

μ0H(T)

20 40 60 80 1 2 3

coeffient me x10

12(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 2 2 2

1 1 ) , ( ) , ( ) ( α γ α γ α ε ε ε H gL H gL L H L H H − ≈ + − + + = = = − = Δ

2 2

1 H gL γ α χ + + =

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − ≈ Δ α α γ ε

2 2 2

2 1 ) ( gL H H

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + =

2 2 2 2

L L L L

fluctuations~χL

20 40 60 80 1 2 3

coeffient me x10

12(T

  • 2)

T(K)

γ

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models in magnetism timisoara 76

20 40 60 80 1 2 3

coeffient me x10

12(T

  • 2)

T(K)

YMnO3 constante diélectrique

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + − − + =

N

T T H L g T c λ α γ α ε ε 1

2 2 2 2 2 1

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models in magnetism timisoara 77

CuCrO2

  • 10
  • 8
  • 6
  • 4
  • 2

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)

12 24 36 48 4.0x10

  • 5

4.2x10

  • 5

4.4x10

  • 5

1 2 3 4 5

TN χ (emu.g

  • 1)

T(K)

  • Δε'/ε'H=0 (%)
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Co3 V2 O8

  • 10
  • 5

5 10

  • 6
  • 3

3 6

M (μB/f.u.)

μ0H(T)

  • 6
  • 3

3 6

T=50K T=20K

  • 6
  • 3

3 6

T=7K

  • 10
  • 5

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

5 10 0.3 0.4 0.5

dM/dH (μB/T·f.u.)

μ0H(T)

0.5 1.0

T=50K T=20K

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

  • ctahedra S= 0

Ferromagnet intrachain interac. Triangular ising lattice Antiferromagnetic interchain (TN =24K)

1 2 3 4 5 6 1 2 3 4 5 T=10K

M (μB/f.u.)

μ0H(T)

2 4 6 8 10 1 2 3 4 5 T=2K

M (μB/f.u.)

μ0H(T)

ΔH=3.6T ΔH=1.2T

R-3cm

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models in magnetism timisoara 80

Ca3 Co2 O6

1 2 3 4 5 6 1 2 3 4 5

M (μB/f.u.)

μ0H(T)

1 2 3 4 5 6

  • 1

ΔεH/εsat (%)

μ0H(T)

T=10K

1 2 3 4 5 6 0.0 0.5 1.0 1.5

χ(μB/T·f.u.)

μ0H(T)

Δε∼-χ

No polarization

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models in magnetism timisoara 81

MnWO4

A 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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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.