Introduction to magnetismmagnetism.eu/esm/2009/slides/simon-slides-1.pdf · suppresses the corresponding symmetry elements. models in magnetism timisoara 31. ... Applications •

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

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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 −+++++= ....443

32

210

MHMbMaFF FMFM −++=42

420

PEPPFF FEFE −++=42

420βα

Ferromagnet

Tc

P

Température

PEPcPcPcPcFF −+++++= ....443

32

210

+Q

-QPr

Ferroelectric

-Q

+QPr

models in magnetism timisoara 49

T>Tc

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 −+++++= ....443

32

210

MHMbMaFF FMFM −++=42

420

PEPPFF FEFE −++=42

420βα

Ferromagnet

Tc

P

Température

PEPcPcPcPcFF −+++++= ....443

32

210

+Q

-QPr

Ferroelectric

-Q

+QPr

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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• 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 −+++++= ....443

32

210

+Q

-QPr

Ferroelectric

models in magnetism timisoara 56

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 α2222

222

42

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

x101

2 (T-2)

T(K) Paramagnet

Δε~10-4

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

x101

2 (T-2)

T(K)

γ

models in magnetism timisoara 76

0 20 40 60 800

1

2

3

coef

fient

me

x101

2 (T-2)

T(K)

YMnO3 constante diélectrique

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+−−+=NTT

HLgTc λαγ

αεε 122

22

210

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 (%

)

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

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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.

Introduction to the physics of multiferroics�SummaryWhat is a multiferroic?Diapositive numéro 4Why this exampleDiapositive numéro 6Diapositive numéro 7Diapositive numéro 8Diapositive numéro 9Diapositive numéro 10Diapositive numéro 11Diapositive numéro 12Diapositive numéro 13questionsquestionsOrigin of the coupling termSuperexchange explanation?Cancel by symmetryDiapositive numéro 19Dzyaloshinskii-Moriya interactionDiapositive numéro 21Diapositive numéro 22Diapositive numéro 23YMnO3 symmetryDiapositive numéro 25YMnO3 symmetryDiapositive numéro 27Diapositive numéro 28Diapositive numéro 29Diapositive numéro 30Diapositive numéro 31ApplicationsDiapositive numéro 33Diapositive numéro 34Diapositive numéro 35Diapositive numéro 36Other materialsDiapositive numéro 38Diapositive numéro 39Diapositive numéro 40Diapositive numéro 41CuCrO2CuCrO2FeVO4FeVO4FeCuO2A ferroic materialFree energy from “Landau”Diapositive numéro 49Interactions and symmetriesDiapositive numéro 51Free energy from “Landau”A little more about LandauDiapositive numéro 54Diapositive numéro 55Diapositive numéro 56Secondary order parameterMagnetic energyDiapositive numéro 59Diapositive numéro 60In additionMnWO4�ferroelectricMnWO4�sensitive to magnetic fieldAF1, AF2, AF3Diapositive numéro 65Diapositive numéro 66Diapositive numéro 67Diapositive numéro 68Diapositive numéro 69Diapositive numéro 70Diapositive numéro 71YMnO3 - LandauYMnO3 – Anomaly in (T)Diapositive numéro 74YMnO3 –magnétodiélectric effect (H) in H2YMnO3 constante diélectriqueCuCrO2Co3V2O8Diapositive numéro 79Diapositive numéro 80MnWO4�A nice example Diapositive numéro 82Diapositive numéro 83Diapositive numéro 84Conclusion