Introduction to the physics of multiferroics Charles Simon - PowerPoint PPT Presentation

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 YMnO 3 Origin of the coupling term Dzyaloshinskii-Moriya Importance of symmetry Applications Some examples Landau theory and symmetries The example of MnWO 4 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 Cr 2 O 3 magnetoelectric Astrov et al. 1960 E induces M, Folen, Rado Stalker 1961, B induces P. models in magnetism timisoara 3

One example: YMnO 3 Hexagonal : P6 3 cm ferroelectric MnO 5 b Y 3+ a Mn 3+ 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

5.5 μ C/cm 2 Pc c 900K T Experimental difficulty P=II(t)dt C= ε 0 ε S/t models in magnetism timisoara 6

Antiferromagnetism 5.4 -3 emu/mol) Mn 3+ 0.15 from T=10K to T=100K 5.2 M( μ B /fu) 5.0 0.10 4.8 0.05 4.6 χ (10 4.4 0 50 100 150 200 0.00 T (K) 0 2 4 6 8 10 12 14 μ Η (T) L : alternate magnetization models in magnetism timisoara 7

Neutron scattering L = Σ S i exp(2i π Qr i ) Order parameter models in magnetism timisoara 8

YMnO 3 - ε (T) ε = 1/ ε 0 dP/dE dielectric constant 18.0 17.5 ε 17.0 16.5 0 20 40 60 80 100 120 T(K) ε ∝ − 2 L models in magnetism timisoara 9

5.5 μ C/cm 2 Pc c T N 900K T models in magnetism timisoara 10

5.5 μ C/cm 2 0.000430 Pc 0.000425 M(emu) 0.000420 0.000415 T N 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 • YMnO 3 is ferromagnetic (?) below T N ! – This was already published by Bertaut models in magnetism timisoara 14

questions • YMnO 3 is ferromagnetic (?) below T N ! • 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 S n x S n +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

21 models in magnetism timisoara

Sn x Sn+1 Sn x Sn+1 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

YMnO 3 symmetry • Non ferroelectric P 63 /mmc (194) M=0 • ferroelectric P 63 cm (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

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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

31 models in magnetism timisoara

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

33 I R GMR M models in magnetism timisoara

Write multiferro P I M R models in magnetism timisoara 34

One historical example: Boracites models in magnetism timisoara 35

36 models in magnetism timisoara I O 13 B 7 Ni 3

Other materials • Structure: perovskite: BiFeO 3 PrMnO 3 • Structure: hexagonal: MMnO 3 M=Y, Ho, etc… • Boracites • Spiral magnetic order: TbMnO 3 MnWO 4 • Fe Langasites. models in magnetism timisoara 37

38 models in magnetism timisoara

39 Tenurite CuO models in magnetism timisoara

Kagome staircase - Co 3 V 2 O 8 Ni 3 V 2 O 8 [1]: S=1 Co 3 V 2 O 8 [1]: S=3/2 β -Cu 3 V 2 O 8 [2]: S=1/2 0.6 7.74 Co 3 V 2 O 8 δ =1/2 0.5 0.4 δ =1/3 ε δ 0.3 7.73 0.2 0.1 δ =0 7.72 0.0 4 6 8 10 12 14 4 6 8 10 12 14 T(K) T(K) models in magnetism timisoara 40

Eu 0.75 Y 0.25 MnO 3 H=0 H models in magnetism timisoara 41

CuCrO 2 Complex incommensurate structure models in magnetism timisoara 42

CuCrO 2 Time(Sec) 21 (b) 20 ( ) 2 ) 19 Polarization, P( μ C/m = + α − + 2 18 F F L P EP gLP 17 AFM 16 15 14 13 = − − + -10 -8 -6 -4 -2 0 2 4 6 8 10 2 2 L ( a ( T T ) dH ) / 2 b H(T) 6 Transversal magnetostriction, Δ L/L*10 N (c) 4 20K 3 2 1 0 -1 -10 -8 -6 -4 -2 0 2 4 6 8 10 H(T) Bohdan Kundys, Maria Poienar, Antoine Maignan, Christine Martin, Charles Simon models in magnetism timisoara 43

FeVO 4 6 Fe 3+ 5/2 in a triclinic structure 1 models in magnetism timisoara 44

45 models in magnetism timisoara FeVO 4

46 models in magnetism timisoara FeCuO 2

A ferroic material models in magnetism timisoara 47

Free energy from “Landau” Ferromagnet = + + + + + − 2 3 4 F F c M c M c M c M .... MH 0 1 2 3 4 M T c a b = + + − 2 4 F F M M MH FM FM 2 4 0 Température Ferroelectric = + + + + + − 2 3 4 F F c P c P c P c P .... PE 0 1 2 3 4 P T c α β +Q -Q r r = + + − 2 4 F F P P PE P P FE FE 2 4 0 -Q +Q Température

T>Tc a b = + + − 2 4 F F M M MH FM FM 0 2 4 a is linear in T-Tc T<Tc M 2 = -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 = + + + + + − 2 3 4 F F c M c M c M c M .... MH 0 1 2 3 4 M T c a b = + + − 2 4 F F M M MH FM FM 2 4 0 Température Ferroelectric = + + + + + − 2 3 4 F F c P c P c P c P .... PE 0 1 2 3 4 P T c α β +Q -Q r r = + + − 2 4 F F P P PE P P FE FE 2 4 0 -Q +Q Température

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 Px 2 +Py 2 , Pz 2 • Quartic invariants (Px 2 +Py 2 ) 2 , Pz 4 , Px 4 +Py 4 , (PxPy) 2 +a/2(Px 2 +Py 2 )+a’/2 Pz 2 +… • F=F 0 • Minimization of F with respect to Px,Py,Pz • a or a’ changes sign first (assume a, a’>0) models in magnetism timisoara 54