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


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.


One example: YMnO3

MnO5

Hexagonal : P63 cm

a

b

Mn3+ S=2 cY3+

ferroelectric


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.


Pc

T

5.5μC/cm2

900K

c

Experimental difficulty

C=ε0 εS/t

P=II(t)dt


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


L = Σ

Si exp(2iπ

Qri )Order parameter

Neutron scattering


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


Pc

T

5.5μC/cm2

900K

cTN


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


Pailhes et al., 2009

They don’t vary in the same way.


After Pailhes et al. Hybrid modes


questions

• YMnO3 is ferromagnetic (?) below TN !– This was already published by Bertaut


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


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.


Superexchange explanation?

• Does superexchange enough to understand the coupling? – No, because of the symmetry. If you add the

three contributions, they cancel by symmetry.


Cancel by symmetry

After I. A. Sergienko and E. Dagotto


On the contrary, the Dzyaloshinskii-Moriya interaction— i.e., anisotropic exchangeinteraction Sn x Sn+1 — changes its sign under inversion.


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.



Sn

Sn+1

Sn x Sn+1

Sn

Sn+1

Sn x Sn+1

Effect of inversion


• The problem is the symmetry• The solution is the symmetry• The method in Landau theory


YMnO3 symmetry

• Non ferroelectric P63/mmc (194)

• ferroelectric P63cm (185).

M=0

Mc can be non zero


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


YMnO3 symmetry

Non ferroferro


• Symmetry analysis shows that the experimental observation was the only possible one.



Symmetry restrictions


• This is very limited• Solution: incommensurability

– An incommensurate modulation of the magnetism with a ferromagnetic component suppresses the corresponding symmetry elements



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.


GMR

R

M

I


R

M

I

Write multiferro

P


One historical example: Boracites


Ni3 B7 O13 I


Other materials• Structure: perovskite: BiFeO3 PrMnO3

• Structure: hexagonal: MMnO3 M=Y, Ho, etc…• Boracites• Spiral magnetic order: TbMnO3 MnWO4

• Fe Langasites.



Tenurite CuO


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


Eu0.75 Y0.25 MnO3

H=0

H


CuCrO2

Complexincommensuratestructure


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 +−−=


FeVO4

6 Fe3+ 5/2 in a triclinic structure 1


FeVO4


FeCuO2


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


T>Tc

T<Tc


420

a is linear in T-Tc

M2 = -a/b H

T


Interactions and symmetries

• This example is too simple: the symmetry is hidden and the role of the interactions is not clear


• 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


420

PEPPFF FEFE −++= 42

420

βα

Ferromagnet

Tc

P

Température


33

2210

+Q

-QPr

Ferroelectric

-Q

+QPr


A little more about Landau• Paraelectric I 4/mmm to

ferroelectric II at Tc.• F is formed by

successive invariants

From P. Toledano


• 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)


• Then, Pz2 = -a/b• Pz is the order parameter.

Tc

P

Température


33

2210

+Q

-QPr

Ferroelectric


Pz 4mm dimension 1Pxy 2mm dimension 2

Subgroups of 4/mmm

• Two possibilities:


Secondary order parameterLet us call e the strain tensor


Magnetic energyExample 4 atoms in Pca21


• This is rather complex, because spins don’t transform with the same symmetry operations than the “real” vectors,

• S x S is also different.


• One example: in a mirror

Real vector Axial vector S x S vector


In addition

• Incommensurate modulations suppressesSymmetry elements.

I have no time to explain details


MnWO4ferroelectric


MnWO4sensitive to magnetic field


AF1, AF2, AF3

Collinear 1/4,,1/2,1/2

-0.241,1/2,0.457

P along a


• The symmetry analysis was made by P. Toledano, and we find all the observed phases as possible sub groups



Pr1/2 Ca1/2 MnO3CE type

Ferromagnetic coupling


No centrosymmetry

From Khomskii et al.




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


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


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


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


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


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


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


MnWO4A nice example


P.G. Radaelli and L.C. Chapon, PRB, 76054428(2007)




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

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