Magnetoelectric Effects in Multiferroics

Magnetoelectric Effects in Multiferroics Th. Lottermoser, M. Fiebig, T. Arima, Y. Tokura PROMOX2, APRIL 2005 ERATO-SSS

Magnetoelectric Effects in Multiferroics Introduction: Magnetoelectric Effect & Multiferroics Multiferroic Manganites with Perovskite Structure Electric Polarization Control by Magnetic Field Multiferroic Manganites with Hexagonal Structure Magnetization Control by Electric Field Magnetoelectric Second Harmonic Generation (SHG) Domains and Domain Walls Superlattices as Artificial Multiferroics Magnetoelectric Effects in Multiferroics -1- ERATO-SSS

Linear Magnetoelectric Effect Polarization and magnetization of a medium: Covariant relativistic formulation: with: Relativistic equivalence of electric and magnetic fields requires "magneto-electric" cross-correlation (~ α) in matter: 1960: Theoretically not well understood Small effect (10-5 ) Limited choice of compounds 2000: 20 10 New theoretical concepts 0 Year "Gigantic" effects: induction of phase transitions New materials: multiferroics, composites, "magnetoelectricity on design" 1985 1990 1995 2000 2005 Magnetoelectric Effects in Multiferroics -2- ERATO-SSS Publications / year Publications on magnetoelectric 100 15 90 80 10 70 60 50 40 30 5 0 1960 1965 1970 1975

Multiferroic Compounds Compounds with simultaneous (anti-)ferromagnetic, ferroelectric, ferrotoroidic and/or ferroelastic ordering (Aizu 1969) Multiferroics Compounds with only simultaneous magnetic and electric ordering Magnetic ferroelectrics or ferroelectromagnets Multiferroic hysteresis in Ni 3 B 7 O 13 I (Ascher et al. 1966) 1958 Idea of new compounds with coexisting magnetic and electric ordering by Smolenskii and Ioffe 1966 First experimental proof of a multiferroic effect by Ascher et al. 1975 Suggestions for technical applications based on multicferroic properties by Wood and Austin... 2000 Why are there so few magnetic ferroelctrics? by Hill Magnetoelectric Effects in Multiferroics -3- ERATO-SSS

Why Multiferroics? The magnetoelectric coupling constant α ij is small (α ij2 < χ e ii χm jj ) Applying external magnetic/electric fields lead only to small ME effects Multiferroics with magnetic and electric ordering exhibit strong internal electromagnetic fields! ME contribution to free energy: H ME = α DB with D = ε 0 (E+P), B = µ 0 (H+M) Observation of giant ME effects Magnetic or electric phase transitions Control of magnetization/polarization by electric/magnetic field Other effects based on the coexisting orders in multiferroics: Interaction of magnetic and electric domains and domain walls Appereance of ME contributions to nonlinear optical signals Magnetoelectric Effects in Multiferroics -4- ERATO-SSS

Electric Polarization Control by Magnetic Field Magnetic control of ferroelectric polarization in TbMnO 3 : At 42 K incommensurate antiferromagnetic Mn 3+ ordering At 27 K incommensurate/commensurate lock-in transition of Mn 3+ spins magnetoeleastic displacements of Mn 3+ ions spatial variation of electric dipolmoments ferroelectric ordered phase Kimura et. al., Nature 426, 55 (2003) Applying of an external magnetic field leads to 90 rotation of polarization giant ME effect! Magnetoelectric Effects in Multiferroics -6- ERATO-SSS

Electric Polarization Control by Magnetic Field Hur et. al., Nature 429, 392 (2004) Electric polarization reversal and memory in in TbMn 2 O 5 induced by magnetic fields: Exchange interactions between Mn 3+, Mn 4+, Tb 3+ spins and lattice polarization lead to several phase transitions. Below 10 K: Magnetic ordring of Mn 3+ /Mn 4+ and Tb 3+ spins Ferroelectric polarization P = P 1 + P 2 (H) with P 1 antiparallel P 2 Modulation of P 2 (H) by external magnetic field leads to sign reversal of the overall polarization P Magnetoelectric Effects in Multiferroics -7- ERATO-SSS

Optical Second Harmonic Generation In general: Multipole expansion of source term S for SHG: Three nonlinear contributions: Electric dipole (ED): Magnetic dipole (MD: Electric quadrupole (EQ): SH source term S i (2ω) χ ijk E j (ω)e k (ω) Incident laser beam Nonlinear signal: electric, magnetic, i-type χ(i) c-type χ(c) Interference! SH intensity: I SH S(c) + S(i) 2 χ(c) + A e iψ χ(i) 2 I 2 (ω) = (χ 2 (c) + A 2 χ 2 (i) + 2A χ(c) χ(i) cos ψ) I 2 (ω) always > 0 interference term Amplitude A and phase ψ can be controlled in the experiment. Magnetoelectric Effects in Multiferroics -9- ERATO-SSS

SHG in a Ferroelectromagnetic Multiferroic Two-dimensional expansion of the Second Harmonic (SH) susceptibility χ for electric ( ) and magnetic ( ) order parameters P NL [ χˆ(0) + χˆ( ) + χˆ( ) + χˆ( ) +...] E( ω) E( ) ( 2ω) = ε0 ω χ(0): Paraelectric paramagnetic contribution always allowed χ(): (Anti)ferroelectric contribution allowed below χ( ): (Anti)ferromagnetic contribution the respective χ( ): Multiferroic contribution ordering temperature SHG allows simultaneous investigation of magnetic and electric structures Selective access to electric and magnetic sublattices Multiferroic contribution reveals the magneto-electric interaction between the sublattices Magnetoelectric Effects in Multiferroics -10- ERATO-SSS

PEL PM Crystallographic and Magnetic Structure of RMnO 3 Mn 3+ (z = 0) Mn 3+ (z=c/2) y R 3+ Mn 3+ O 2- z x FEL x AFM RMnO3 (R = Sc, Y, In, Ho, Er, Tm, Yb, Lu) : A highly correlated and ordered system! Ferroelectric phase transition: T C = 570-990 K Breaking of inversion symmetry I! Antiferromagnetic phase transition of the Mn 3+ sublattice: T N = 70-130 K Breaking of time-reversal symmetry T, but not of inversion symmetry I! Additional rare-earth order: T C 5 K for Ho, Er, Tm, Yb Magnetoelectric Effects in Multiferroics -11- ERATO-SSS

Magnetic Structure and SHG Selection Rules At least 8 different triangular inplane spin structures with different magnetic symmetries and different selection rules for SHG α structures: SHG for k z allowed α x (ϕ = 0 ): χ xxx = 0, χ yyy 0 α y (ϕ = 90 ): χ xxx 0, χ yyy = 0 α ρ (ϕ = 0-90 ): χ xxx sin ϕ, χ yyy cos ϕ β structures: SHG for k z not allowed β x, β y, β ρ : χ xxx = 0, χ yyy = 0 Determine β structure from α-β transition α x β y : χ xxx = 0, χ yyy cos ϕ α y β x : χ xxx sin ϕ, χ yyy = 0 α structures y x β structures Contrary to diffraction techniques: α and β models clearly distinguishable! Magnetoelectric Effects in Multiferroics -12- ERATO-SSS

SH spectrum and Magnetic Symmetry SH intensity 0 SH intensity YMnO 3 6 K HoMnO 3 6 K χ 2.0 2.2 2.4 2.6 2.8 50 K 3.0 xxx 2 ErMnO 3 6 K HoMnO 3 χ yyy 2 0 2.0 2.2 2.4 2.6 2.8 3.0 SH energy (ev) 2.0 2.2 2.4 2.6 2.8 3.0 SH energy (ev) The magnetic symmetry, not the R ion, determines the SH spectrum of RMnO 3 Magnetoelectric Effects in Multiferroics -13- ERATO-SSS

H/T Phase Diagram of Hexagonal RMnO 3 Temperature (K) 140 120 100 80 60 40 RMnO 3 (H = 0) Representation B (hyst.) A Repres. index 1 (none) 2 20 0 80 Sc In Lu Yb Tm Er Ho Y 70 60 HoMnO 3 B 2 Temperature (K) Temperature (K) 50 40 30 20 10 0 80 70 60 50 40 30 20 10 A 1 B 2 B 1 B 2 A 2 TmMnO 3 0 0 1 2 3 4 Magnetic field µ 0 H z (T) A 2 B 2 YbMnO 3 A 2 ErMnO 3 A 2 0 1 2 3 4 Magnetic field µ 0 H z (T) J. Appl. Phys. 83, 8194 (2003) α structures y x β structures Magnetoelectric Effects in Multiferroics -14- ERATO-SSS

Magnetoelectric 3d - 4f Superexchange in RMnO 3 k: R sites with 3 and 3m symmetries i k : all R ions at k sites (4+2) j: 6 Mn ions neighboring an R ion A: Mn-R exchange matrix (4 types) S: spins of Mn and R ions α model: β model: no change! lowers ground-state energy: Gigantic magnetoelectric effect which originates in 3d-4f superexchange; triggers hidden phase transition! Ferroelectric distortion modifies the superexchange:, Scales with order par. Substitution leads to: ME contribution Magnetoelectric Effects in Multiferroics -15- ERATO-SSS

Magnetization Control by Electric Field Electric field suppresses magnetic SHG and leads to additional field depended contribution to Faraday rotation Induction of magnetic phase transition! Nature 430, 541 (2004) ME contribution to free energy: H α zz P z S z Ho Ferroelectric poling and ferromagnetic ordering of Ho 3+ spins giant ME effect Magnetoelectric Effects in Multiferroics -16- ERATO-SSS

Magnetoelectric Second Harmonic Generation Source term S ED (0) S ED () S MD,EQ () S ED () Sublattice sym. P6 3 /mcm P6 3 cm P6 3 /mcm P6 3 cm SHG for k z SHG for k x = 0 = 0 = 0 0 0 = 0 0 SH intensity χ yyy k z 0 2.0 2.2 2.4 2.6 2.8 S H energy (ev) χ zyy χ yyz χ zzz χ yyy k x 10 2500 10 2.0 2.2 2.4 2.6 2.8 SH energy (ev) 1 Identical magnetic spectra for k z and k x indicate bilinear coupling to,. Unarbitrary evidence for the first observation of "magnetoelectric SHG" Nature 419, 818 (2002) Magnetoelectric Effects in Multiferroics -17- ERATO-SSS

Observation of Multiferroic Domains Independent ferroelectric ( ) and ferroelectromagnetic ( ) domain structures; antiferromagnetic domain structure ( ) is not! (a) (c) (b) Multiferroic domains": = +1 for = ±1, = ±1 = -1 for = ±1, = 1 Any reversal of the FEL order parameter is clamped to a reversal of the AFM order parameter 0.5 mm (d) + + + + FEL FEL & AFM AFM Origin: Piezomagnetic interaction between lattice distortions at the FEL domain wall and magnetization induced by the AFM domain wall decreases the free energy Nature 419, 818 (2002) Magnetoelectric Effects in Multiferroics -18- ERATO-SSS

Interaction of electric and magnetic domain walls M AFM wall 0 d 0 d m p AFM wall carries an intrinsic macroscopic mag-netization FEL wall induces strain due to switching of polarization σ H pm FEL wall piezomagnetic energy s s=0 s=d 0 Width of walls: AFM - O[10 3 ] unit cells: small in-plane anisotropy iii FEL O[10 0 ] unit cells: large uniaxial anisotropy Piezomagnetic contribution H pm = q ijk M i σ jk with σ P z higher-order magnetoelectric effect Generation of an antiferromagnetic wall clamped to a ferroelectric wall leads to reduction of free energy. Phys. Rev. Lett. 90, 177204 (2003) Magnetoelectric Effects in Multiferroics -19- ERATO-SSS s=d

Spin-Rotation Domains in HoMnO3 Observation of drastically increased dielectric constant during magnetic spin-reorientation phase transition by Lorenz et. al. (PRL 92, 87204 (2004)) But: ME not allowed by symmetry considerations! Inside AFM domain walls reduced local symmetry P2 due to uncompensated magnetic moment. P2 allows ME contribution Pz αzx,y Mx,y Local ME effect Magnetoelectric Effects in Multiferroics -20- ERATO-SSS

Superlattices as Artificial Multiferroics repeated 10 times Tricolor superlattice LaAlO 3 SrMnO 3 LMnO 3 substrate 2 u.c. 4 u.c. 6 u.c. 1. Asymmetric sequence of layers breaks inversion symmetry 2. Ferromagnetic ordering at SrMnO3/LaMn03 interface Polar ferromagnet Nonlinear magneto-optical Kerr rotation (NOMOKE) Kerr rotation angle φ ( ) Kerr rotation angle φ ( ) 18 16 14 12 10 8 6 4 2 0 LSAT External field: 0.35 T Kerr rotation Magnetization -2 6 Kerr rotation Magnetization 5 4 m STO P (2ω) 3 tanφ Re nm P (2ω )sinθ 2 1 External field: 0.35 T 0-1 0 50 100 150 200 250 300 Temperature (K) 1.0 0.8 0.6 0.4 0.2 0.0 0.5 0.4 0.3 0.2 0.1 0.0 Magnetization (µ Magnetization (µ B /Mn) B /Mn) Magnetoelectric Effects in Multiferroics -22- ERATO-SSS

Summary Coexistence of magnetic and electric ordering in multiferroics allow giant magnetoelectric effects: Electric/magnetic phase transitions Control of polarization/magnetization by external magnetic/electric fields New effects based on the specific nature of multiferroics: Interaction of magnetic and electric domains and domain walls Magnetolectric second harmonic generation Magnetoelectric Effects in Multiferroics -23- ERATO-SSS

Acknowledgment Germany: Th. Lonkai, D. Fröhlich, St. Leute, C. Degenhardt, T. Satoh Russia: R.V. Pisarev, V.V. Pavlov, A.V. Goltsev Japan: K. Kohn, Y. Tanabe, E. Hanamura, H. Yamada Magnetoelectric Effects in Multiferroics -24- ERATO-SSS