Multiferroic and magnetoelectric materials Maxim Mostovoy University of Groningen Zernike Institute for Advanced Materials ICMR Summer School on Multiferroics Santa Barbara 2008 Outline • Ferroelectric properties of magnetic defects • Toroidal moment 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 domain wall: 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 3 Q = d r ([][]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 3 T= d r 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 Hi ) Φ H = −A∇ ⋅H − T⋅∇× H − Qij (∂i H j + ∂ j Hi ) The magnetic multipoles have the same symmetry as magnetoelectric tensor 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 spin 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 & Nicola Spaldin −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.