Multiferroic and magnetoelectric materialsLecturesOutlineElectric ↔ MagneticMultiferroicsTime-reversal symmetry breaking in magnetsInversion symmetry breaking in ferroelectricsNo chemistry betweenmagnetism and ferroelectricityLinear magnetoelectric effectAnomalies of magnetoelectric constant in boracitesOrthorombic RMnO3Dielectric constant anomalyat the transition to spiral statePolarization switching by magnetic fieldMagnetic control of dielectric propertiesGiant magnetocapacitance effectin DyMnO3Electric polarization reversalsin TbMn2O5CoCr2O4OutlineLinear magnetoelectric effectCr2O3Cr2O3Novel MultiferroicsBreaking of inversion symmetry by spin orderingInduced PolarizationSinusoidal SDWSpiral SDWBiFeO3Geometrical FrustrationCompeting interactionsMagnetic frustration in RMnO3Why TFE is lower than TM?Sinusoidal-helicoidal transitionDielectric constant anomalyat the transition to spiral stateMagnetic phase diagramsOutlineFerroelectricity induced by magnetostrictionRMn2O5Two-dimensional representation and induced polarizationHigher-order terms in effective spin HamiltonianEffective spin Hamiltonian(3d order)OutlinePolarization of domain wallsElectric charge of magnetic vortexElectrostatics of magnetic defectsMagnetic vortex in magnetic fieldMagnetic vortex in magnetic fieldArray of magnetic vorticesis magnetoelectricMagnetoelectric effectin spin triangleConclusionsMultiferroic and magnetoelectricmaterialsMaxim MostovoyUniversity of GroningenZernike Institute for Advanced MaterialsBoulder Summer SchoolJuly 2008Lectures• Spin-orbital exchange in Mott insulators9Multiferroics and magnetoelectricsOutline• Linear magnetoelectric effect, multiferroics• Phenomenological description• Microscopic mechanisms of magnetoelectriccoupling• OutlookElectric ↔ Magnetic• Duality of Maxwell equations• Aharonov-BohmAharonov-Casher• Thermodynamics of ferroelectrics and ferromagnets⎪⎪⎩⎪⎪⎨⎧∂∂+=×∇∂∂−=×∇tctcEHHE11⎪⎩⎪⎨⎧−→→EHHE()⎩⎨⎧=+⋅∇=×∇040PEEπ⎟⎟⎠⎞⎜⎜⎝⎛ΦΦ0Fe⎟⎟⎠⎞⎜⎜⎝⎛0ρρFµ⎪⎩⎪⎨⎧−+=Φ−+=ΦMHbMaMPEbPaPFMFE4242()⎩⎨⎧=+⋅∇=×∇040MHHπMultiferroics• Both ferroelectric and magnetic• Coupling between P and MPb(Fe2/3W1/3)O3BiFeO3Pb(Fe1/2Ta1/2)O3YMnO3G. A. SmolenskiiG.A. Smolenskii & I.E. Chupis, Sov. Phys. Usp. 25, 475 (1982)Time-reversal symmetry breaking in magnets0≠><S()()tt SS−=−Ferromagnets Antiferromagnets0≠M 0=MInversion symmetry breaking in ferroelectrics()()xPxP−=−BaTiO3Centrosymmetric+−+−+−−Noncentrosymmetric+−+−+−−PNo chemistry betweenmagnetism and ferroelectricityFM FEd0,s2d1,d2,d3…multiferroicsN. A. Hill, J. Phys. Chem. B 104, 6694 (2000)Linear magnetoelectric effectI. E. Dzyaloshinskii JETP 10 628 (1959), D. N. Astrov, JETP 11 708 (1960)Cr2O3+= EPeχHα=MHmχ+EαG.T. Rado PRL 13 335 (1964)Anomalies of magnetoelectricconstant in boracitesM = Co2+, Ni2+D. G. Sannikov, Ferroelectrics 219 177 (1998)X = I, Br, ClM3B7O13XOrthorombic RMnO3T. Kimura et al PRB 68,060403 (2003)Sinusoidal SDW Spiral SDWDielectric constant anomalyat the transition to spiral stateT. Kimura et al , Nature 426, 55 (2003)Polarization switching by magnetic fieldT. Kimura Annu. Rev. Mater. Res. 37 387(2007)Magnetic control of dielectric properties T. Kimura Annu. Rev. Mater. Res. 37 387(2007)T. Goto et al PRL 92, 257201(2004)Giant magnetocapacitance effectin DyMnO3Electric polarization reversalsin TbMn2O5N. Hur et al Nature 429, 392 (2004)CoCr2O4MP×is conservedY. Yamasaki et al, PRL 96, 207204 (2006)Outline• Linear magnetoelectric effect, multiferroics• Phenomenological description• Microscopic mechanisms of magnetoelectriccoupling• OutlookLinear magnetoelectric effectTime-reversal symmetry T (t Ø -t) and inversion I (x Ø - x) are brokenCr2O3jijiHPα=jjiiEMα=I.E. Dzyaloshinskii (1959), D.N. Astrov (1960)jiijHEα−=ΦmeIT symmetry (t Ø -t, x Ø - x) is conservedCr2O3space groupcR3⎟⎟⎠⎞⎜⎜⎝⎛yxHH⎟⎟⎠⎞⎜⎜⎝⎛yxEEzEzHI~x2z3⎟⎟⎠⎞⎜⎜⎝⎛−−1001⎟⎟⎠⎞⎜⎜⎝⎛−1001⎟⎟⎠⎞⎜⎜⎝⎛−−−=1331213/2πR3/2πR1−1−1+⎟⎟⎠⎞⎜⎜⎝⎛−−1001⎟⎟⎠⎞⎜⎜⎝⎛−10013/2πR1− 1−1+Symmetries of low-T phaseInversion combined with time reversalITI=~120o-rotation()yyxxzzHEHEHEF+−−=⊥αα||meInvariants:Cr2O3cR33zAFM order parameter TN= 306K4321MMMML−+−=0≠zL1symmetries of paramagnetic phase2xI 2x3zLz-+ +Ez- -+Hz+-+2I32x4zzzzzHEHEL||αλ=zL∝⊥αα,||Invariants:()yyxxzHEHEL+Mechanism of inversion symmetry breakingMaterialscovalent bonding between 3d0transition metal (Ti) and oxygen BaTiO3polarizability of 6s2lone pair BiMnO3, BiFeO3structural transition‘Geometric ferroelectrics’K2SeO4,Cs2CdI4h-RMnO3charge ordering‘Electronic ferroelectrics’LuFe2O4magnetic ordering‘Magnetic ferroelectrics’o-RMnO3, RMn2O5, CoCr2O4, MnWO4FerroelectricsS.-W. Cheong & M. M. Nature Materials 6, 13 (2007)ProperImproperNovel Multiferroicsmaterial TFE(K) TM(K)P(µC m-2)TbMnO328 41 600Ni3V2O86.3 9.1 100CuFeO211 14 300LiCu2O223 23 5MnWO48 13.5 60CoCr2O426 93 2TbMn2O538 43 400CuO230 230 100Breaking of inversion symmetry by spin orderingInversion I: (x,y,z) |(-x,-y,-z)Qe3Cycloidal spiralQe3Cycloidal spiralInduced PolarizationEnergy (cubic lattice)() ()[]MMMMPP⋅∇−∇⋅⋅−=λχePF22Induced electric polarization()()[]MMMMP⋅∇−∇⋅=eλχBary’akhtar et al, JETP Lett 37, 673(1983); Stefanovskii et al, Sov. J. Low Temp. Phys. 12, 478(1986), M.M. PRL 96, 067601(2006)Sinusoidal SDWQxsinAM=0=PQxcenter of inversionSpiral SDW()QxeQxeM sincos210+= M[]QeP ×∝3Qe3BiFeO3TFE= 1100 KTN= 640 KFerroelectricAntiferromagnetic()()LPLLLF ∂−∂+=λϕ2Periodic modulation of AFM ordering:Q∝λ PFree energyA.M. Kadomtseva et al. JETP Lett. 79, 571 (2004) Low-pitch spiralλ = 620 ÅGeometrical Frustration0<J0>′JCompeting interactions[]∑++⋅′+⋅=nnnnnJJE21SSSSFrustrated Heisenberg chain4JJ >′JJQ′=4cos[]∑++′+=nnnnnJJE21σσσσFrustrated Ising chain0>′J1±=nσ2JJ >′0<JMagnetic frustration in RMnO3JFMJAFMJFMMnbaFMAFMJJ2=κκ < 1 Ferromagnetic κ12cos =bQκ > 1 Incommensurate SDWWhy TFEis lower than TM?TbMnO3Ni3V2O8G. Lawes et al PRL 95, 087205 (2005)28K < T < 41KT < 28K6.3K < T < 9.1K3.9K < T < 6.3KM. Kenzelmann et al PRL 95, 087206 (2005)Sinusoidal-helicoidal transition () () ()MM222242222⎟⎟⎠⎞⎜⎜⎝⎛+++++=Φ QdxdcMbMaMaMazzyyxxmzxyxaaaa