Multiferroic and magnetoelectric materials

Maxim Mostovoy

University of Groningen Zernike Institute for Advanced Materials

ICMR Summer School on

Santa Barbara 2008 Outline

• Ferroelectric properties of magnetic defects

• and magnetoelectric effect

• Electromagnons Electrostatics of magnetic defects

Easy plane spins: M = M [e1 cos ϕ + e2 sin ϕ]

2 Polarization: Pa = −γ χe M ε 3ab ∂bϕ

Total polarization of :

2 ∫ dx Py = γ χe M [ϕ(+ ∞)−ϕ(− ∞)]

2 (2) Charge density: ρ = −div P = 2πγ χeM Γδ (x⊥ ) 1 Vortex charge: Q ∝ Γ = ∫ dx⋅∇ϕ winding number 2π C Polarization of domain walls

Bloch wall P = 0 e3 || xˆ

X

P || [e × xˆ] Néel wall 3 e3 ⊥ xˆ

X

e3 Magnetic vortex in magnetic field

Q H

pseudoscalar moment P P = 0 A ∝ ∑rα ⋅Sα α Magnetic vortex in magnetic field

H

toroidal moment P = 0 T ∝ ∑rα ×Sα P α Array of magnetic vortices is magnetoelectric

H

P Outline

• Ferroelectric properties of magnetic defects

• Toroidal moment and magnetoelectric effect

• Electromagnons Multipole expansion

Average vector potential A(R)

j(r) 1 j(r) A()R = d 3r c ∫ r − R

Magnetic dipole moment

1 1 A()1 = −M×∇ M= d3r r× j R 2c ∫ Quadrupole and toroidal moments Quadrupole moment 1 A )2( = −ε Q ∂ ∂ i ijk kl j l R 1 Q = d3r ([][]r× j r + r× j r ) ij 6c ∫ i j j i Toroidal moment (not in textbooks) 1 A )2( = ∇()T⋅∇ + 4π Tδ ()R R 1 T = d3r r×[]r× j 6c ∫ Meaning of toroidal moment

T = µ r ×S 1 3 B ∑ α α T = d r r ×[]r × j α 6c ∫

T T Spins interacting with inhomogeneous magnetic field H(r) E = −2µB ∑Sα ⋅H(rα ) α

Gradient expansion E )0( = −M ⋅H(0)

)1( E = −A∇ ⋅H − T⋅∇× H − Qij (∂i H j + ∂ j Hi ) 2 A = − µB ∑rα ⋅Sα 3 α Magnetoelectric properties of magnetic multipoles

Φme = −a E⋅H − t ⋅E× H − qij (Ei H j + E j H i )

Φ H = −A∇ ⋅H − T⋅∇× H − Qij (∂i H j + ∂ j H i )

The magnetic multipoles have the same symmetry as magnetoelectric

a∝ A t ∝ T qij ∝ Qij Magnetoelectric effect

Φ me = −λT⋅E× H

∂Φ P = − me = −T× H ∂E ∂Φ M = − me = +T×E ∂H Magnetoelectric effect in triangle

j = 0 j = π/2 pseudoscalar  cos ϕ sin ϕ toroidal   α =α0  moment moment −sin ϕ cos ϕ  j S1 S1

S2 S3 S2 S3 H H P P Triangular lattice

j = -2π/3

j = 0 j = +2 π/3

Cancellation of magnetoelectric effect Pseudoscalar Kagome

 1 0   α = α0    0 1 Toroidal Kagome

layered Kagomé lattice

 0 1   α = α0   −1 0 Magnetoelectric Coupling

−3 • KITPITE: α = 3 ⋅ 10 (Gaussian units)

LDA+U Cris Delaney &

−4 • Cr 2O3 α = 1 ⋅ 10 Outline

• Ferroelectric properties of magnetic defects

• Toroidal moment and magnetoelectric effect

• Electromagnons Phonon-magnon continuum

H me = −P∑λijk ui (S j ⋅Sk ) ijk Lorenzana & Sawatzky (1995)

magnon photon phonon magnon Suzuura et al (1996)

‘Charged magnons’ in NaV 2O5 Damascelli et al (1998) magnon photon magnon Photoexitation of magnons PL∂L

photon

(ω )0, Electro magnon (ω,Q) phonon λ ,0( Q) SDW

H. Katsura et al. (2006) Magnons in spiral state of RMnO 3

electromagnon Electromagnon mode c P P || c P || a

a b

bc–spiral E || a ab–spiral H > H H < H C C

Softening at H C ~ divergent εεεa Magnetic field dependence

T. Kimura et al (2005) N. Kida et al (2008) Eu 1-xYxMnO 3 ab spiral E || a

R. V. Aguilar et al (2007) Q || b RMnO 3 b Alternation of J E || a 1' along Q

-δJ -δJ

2 2' +δ J +δ J

E || a excites single magnon 1 a Q || b RMnO 3 b Alternation of J E || b 1' orthogonal to Q

+δ J -δJ

2 2' +δ J -δJ no coupling to single magnon 1 a Magnons in spiral state of RMnO 3

electro- magnon Bi-magnon continuum e || a

0.012

0.01

0.008 χ′′ 0.006 DOS

0.004

0.002

0 0 20 40 60 80 100 120 Photon frequency (cm -1 ) Conclusions

• Magnetic frustration gives rise to unusual spin orders that break inversion symmetry and give rise to multiferroic behavior and linear magnetoelectric effect