Microwave magnetochiral effect in Cu2OSeO3
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Microwave magnetochiral effect in Cu OSeO Masahito Mochizuki
1, 2 Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara, Kanagawa 229-8558, Japan PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan
We theoretically find that in a multiferroic chiral magnet Cu OSeO , resonant magnetic excita-tions are coupled to collective oscillation of electric polarization, and thereby attain simultaneousactivity to ac magnetic field and ac electric field. Because of interference between these magnetic andelectric activation processes, this material hosts gigantic magnetochiral dichroism on microwaves,that is, the directional dichroism at gigahertz frequencies in Faraday geometry. The absorptionintensity of microwave differs by as much as ∼
30% depending on whether its propagation directionis parallel or antiparallel to the external magnetic field.
PACS numbers: 76.50.+g,78.20.Ls,78.20.Bh,78.70.Gq
Collective excitations of spins in magnets, so-calledmagnons or spin waves, can be activated not only viaa direct process with ac magnetic field H ω coupled tomagnetizations but also via an electric excitation by acelectric field E ω coupled to charge degrees of freedom.When the magnon or spin-wave modes have simultane-ous activity to the H ω and E ω components of electro-magnetic waves, interference between the two activationprocesses, that is, the magnetically activating and theelectrically activating processes, gives rise to peculiar op-tical and/or microwave phenomena, so-called optical MEeffect. One of the most important examples is the direc-tional dichroism, that is, oppositely propagating electro-magnetic waves exhibit different absorptions.Multiferroic materials with concurrent magnetic andferroelectric orders [1–8] provide an opportunity to re-alize the electric-dipole active magnons (so-called elec-tromagnons) [9–13], and thus the optical ME effect viathe magnetoelectric coupling [14–16]. Indeed observa-tions of the directional dichroism have been reported forseveral multiferroic materials such as Ba CoGe O [17–19], R MnO ( R =rare-earth ions) [20, 21], andCuFe − x Ga x O [22], in which nontrivial spin orders in-duce the ferroelectric polarization via the relativisticspin-orbit interaction. In these materials, the opticalME effect is observed at the electromagnon resonancefrequencies in the terahertz (THz) regime.The directional dichroism is observed also at higherfrequencies, i.e., x-ray and visible-light regimes in severalpolar magnets, which is caused by electron transitionsamong the spin-orbit multiplets [23–29]. However, ob-servations of the effect at gigahertz (GHz) frequenciesare quite limited and the effect observed so far is verytiny whose difference in absorption intensity is only 2 . OSeO exhibits unprecedentedly large magnetochi-ral dichroism at GHz frequencies, that is, the microwavedirectional dichroism in Faraday geometry. In the pres-ence of H , the conical spin phase or the field-polarizedferrimagnetic phase emerges in the bulk samples depend-ing on the strength of H . When H is applied in a cer-tain direction, and a microwave is irradiated parallel orantiparallel to H , the absorption intensities for the oppo-sitely propagating microwaves differ by as much as 30%.Such a huge directional dichroism at microwave frequen-cies has never been observed in a single material. Thiseffect is traced back to the resonantly enhanced magne-toelectric coupling, and therefore essentially distinct inmicroscopic mechanism from the traditional microwavenon-reciprocal device based on microwave polarization,potentially leading to a unique microwave device.The crystal and magnetic structures of Cu OSeO con-sist of a network of tetrahedra composed of four Cu ( S =1/2) ions at their apexes as shown in Figs. 1(a)and (b). In each tetrahedron, three-up and one-downcollinear spin configuration is realized below T c ∼
58K [33, 34]. This four-spin assembly can be regarded asa magnetic unit, and is described by a classical magneti-zation vector m i whose norm m is unity. We employ aclassical Heisenberg model on a cubic lattice to describethe magnetism in a bulk specimen of Cu OSeO [35–37],which contains the ferromagnetic-exchange interactionand the Dzyaloshinskii-Moriya interaction among the ef-fective magnetizations m i and the Zeeman coupling to a bc (d) P = P || [001] M||H || Q H
691 Oe (c)
Conical Ferromagnetic (e) (g) M || H || [010] (f) (b) abc < > < > M || H || [110] 0-8-16816 4515 300-45 -15-30 P c ( µ C / m ) θ (deg) abc (h) ab θ (>0) M || H (a) P c =( λ /2 V ) sin2 θ FIG. 1: (color online). (a) Crystal structure of Cu OSeO .(b) Magnetic structure of Cu OSeO . (c) Phase diagramof the spin model given by Eq. (1) with J =1 meV and D/J =0.09. (d) Schematic figure of the conical spin struc-ture. (e) Symmetry axes in the chiral cubic P2 P k [001] under H k [110]. (g) Ab-sence of P under H k [010]. (h) Calculated net polarization P c in the field-polarized ferromagnetic state under H ⊥ c as afunction of the angle θ between H and the b axis. the external H . The Hamiltonian is given by, H = − J X m i · m j − D X i, ˆ γ m i × m i +ˆ γ · ˆ γ − gµ B µ H · X i m i , (1)where g =2, and ˆ γ runs over ˆ a , ˆ b , and ˆ c in the cu-bic setting. Here details of the exchange interactionswithin each tetrahedron are neglected and only the ef-fective interactions among these magnetic units m i aretaken into account. Such a coarse graining is justified asfar as low-energy excitations with long-wave-length mag-netic modulations are considered. We take J =1 meVand D/J =0.09 so as to reproduce the experimentally ob-served T c and periodicity ( ∼
50 nm) of the conical state.Figure 1(c) shows a phase diagram of this spin model at T =0 as a function of the magnetic field H , which exhibitsa phase transition between the conical phase and the fer-romagnetic phase at H =691 Oe, in agreement with theexperiments for the bulk specimen [38, 39]. Schematicfigure of the spin structure in the conical phase is shownin Fig. 1(d), in which both the propagation vector Q and the net magnetization M are parallel to H . Note thatthe spin textures considered here are slowly varying inspace, and thus the coupling to the background crystalstructure is significantly weak. This justifies our treat-ment with a spin model on the cubic lattice after thecoarse graining of magnetizations.The presence or absence of the ferroelectric polariza-tion P and, if any, its direction can be known from thesymmetry consideration [38, 40]. The crystal structure ofCu OSeO belongs to the chiral cubic P2 h i , andtwo-fold screw axis, 2 , along h i as shown in Fig. 1(e).This crystal symmetry is not polar, and thus there existsno spontaneous P . Although the conical and the ferro-magnetic spin states are not polar, either, combinationof the crystal and the magnetic symmetries renders thesystem polar, and allows the emergence of P . As shownin Fig. 1(f), the emergence of P k [001] perpendicular tothe net magnetization M ( k H ) is expected for the coni-cal and the ferromagnetic states formed under H k [110]since only the 2 axis along [001] remains as a symmetryaxis. On the other hand, the emergence of P is forbid-den under H k [010] since three 2 axes survive as shownin Fig. 1(g).Microscopically the local polarization p i at the i thtetrahedron is given using the magnetization components m ia , m ib , and m ic as, p i = ( p ia , p ib , p ic ) = λ ( m ib m ic , m ic m ia , m ia m ib ) . (2)The net magnetization M and the ferroelectric polariza-tion P are calculated by sums of the local contributionsas M = gµ B NV P Ni =1 m i and P = NV P Ni =1 p i , respectively,where the index i runs over the Cu-ion tetrahedra, N isthe number of the tetrahedra, and V (=1.76 × − m ) isthe volume per tetrahedron. The coupling constant λ isevaluated as λ =5 . × − µ Cm from the experimentaldata [31]. Figure 1(h) shows calculated net polarization P in the field-polarized ferromagnetic state when H isapplied within the c -plane as a function of the angle θ between H and the b axis (see the inset). We find thatthe positive (negative) P emerges along [001] when θ isnegative (positive).According to Fig. 1(h), one realizes that oscillation of M (∆ M ω k [100]) induces oscillation of P (∆ P ω k [001])when M k H k [010] [see Figs. 2(a)-(c)], and conversely∆ P ω k [001] induces ∆ M ω k [100] via the magnetoelectriccoupling. This means that both H ω k [100] and E ω k [001]components of microwave can activate the coupled oscil-lation of M and P (Note that the response time of M ( P ) against the change of P ( M ) is governed by the elec-tron transitions among the orbital multiplets, and thusis much shorter than the typical time scale of the oscil-lations). To see this, we calculate the following dynam-ical susceptibilities by numerically solving the Landau-Lifshitz-Gilbert (LLG) equation using the fourth-order H ω H ω E ω K ω H ω E ω additive subtractive sgn (Re k ω ) = +1 sgn (Re k ω ) = -1 k ω ||H ||[010], H ω ||[100], E ω ||[001] abc K ω ac b ac b H ω H ω E ω E ω H ω E ω E ω ∆ M ω ∆ P ω ∆ P ω abc H ω H ω MP =0 H ∆ M ω (a) (b) (c)(d) stronglyabsorbed weaklyabsorbed FIG. 2: (color online). (a)-(c) In the presence of net mag-netization M k H under H k [010], oscillating magnetizationcomponent ∆ M ω ( k [100]) is accompanied by the oscillatingpolarization component ∆ P ω ( k [001]). (d) Configuration ofmicrowave H ω and E ω components, for which the magne-tochiral dichroism occurs under H k [010]: k ω k± H , H ω k [100]and E ω k [001]. Runge-Kutta method: χ mm αβ ( ω ) = ∆ M ωα µ ∆ H ωβ magnetic susceptibility ,χ ee αβ ( ω ) = ∆ P ωα ǫ ∆ E ωβ dielectric susceptibility ,χ em αβ ( ω ) = ∆ P ωα √ ǫ µ ∆ H ωβ magnetoelectric susceptibility ,χ me αβ ( ω ) = r µ ǫ ∆ M ωα ∆ E ωβ electromagnetic susceptibility . Because of the symmetry, the relation χ em αβ ( ω ) = χ me βα ( ω )holds. The LLG equation is given by d m i dt = − m i × H eff i + α G m m i × d m i dt , (3)where α G (=0.04) is the Gilbert-damping coefficient. Theeffective field H eff i is calculated from the Hamiltonian H = H + H ′ ( t ) as H eff i = − ∂ H /∂ m i . The first term H is the model Hamiltonian given by Eq. (1). The pertur-bation term H ′ ( t ) represents a short rectangular pulse ofmagnetic field ∆ H ( t ) or electric field ∆ E ( t ), which aregiven, respectively, by, H ′ ( t ) = − gµ B µ X i ∆ H ( t ) · m i (4) Frequency (GHz) Im χ mmaa Im χ eecc Im χ meac =Im χ emca (×8)(×50) Im χ mmaa Im χ meca =Im χ emca (×8) Im χ eecc (×50) I m χ µ ν α β conical state ( H =518.4Oe) ferromag. state ( H =1382.4Oe) k ω || H ||[010]|| b , H ω ||[100]|| a , E ω ||[001]|| c FIG. 3: (color online). Imaginary parts of the calculateddynamical magnetic, dielectric, and magnetoelectric suscep-tibilities, Im χ mm aa , Im χ ee cc , and Im χ em ca , as functions of the fre-quency for the conical state at H =518.4 Oe and the ferro-magnetic state at H =1382.4 Oe when the static magneticfield H k [010]( k b ) is applied. and H ′ ( t ) = − X i ∆ E ( t ) · p i . (5)After applying the pulse at t =0, we trace time profilesof M ( t ) and P ( t ), and obtain their Fourier transforms∆ M ωα and ∆ P ωα . Dividing these quantities by Fouriercomponents of the field pulses, ∆ H ω and ∆ E ω , we ob-tain the susceptibilities. The calculations are performedusing a system of N =20 × ×
140 sites with the periodicboundary condition.In Fig 3, we display imaginary parts of the calculateddynamical magnetic, dielectric, and magnetoelectric sus-ceptibilities, i.e., Im χ mm aa , Im χ ee cc , and Im χ em ca , as functionsof the frequency for the conical state and the ferromag-netic state when the static magnetic field H k [010]( k b )is applied. These susceptibilities have resonant peaks inthe GHz regime in agreement with the microwave exper-iments [32, 41]. Moreover all of these dynamical suscep-tibilities in each phase have peaks at the same frequencyindicating that the resonant modes have simultaneousactivity to H ω k [100]( k a ) and E ω k [001]( k c ).This magnetoelectric activity causes the microwavemagnetochiral effect in Cu OSeO [see Fig. 2(d)]. For theelectromagnetic wave, the relation H ω k k ω × E ω holds,indicating that relative directions of the H ω and E ω components are determined by the propagation vector k ω , and their relationship should be reversed upon thesign reversal of k ω . When the lineally polarized elec-tromagnetic wave with H ω k a and E ω k c propagatesparallel (antiparallel) to H k b , that is, sgn[Re k ω ]=+1(sgn[Re k ω ]= −
1) with k ω = k ω ˆ b , the H ω and E ω com-ponents contribute in an additive (a subtractive) wayto excite the coupled oscillation of M and P , resultingin weaker (stronger) absorption of the electromagnetic α + , α − ( c m - ) g µ B µ H/J= ω (GHz) ∆ α ( c m - ) α + α − Frequency (GHz) ω (GHz) (a) Conical phase (b)
Ferromagnetic phase α + , α − ( c m - ) g µ B µ H/J=g µ B µ H/J= α + α − ∆ α ( c m - ) FIG. 4: (color online). Calculated frequency-dependence ofthe absorption coefficients, α + and α − , for microwaves withsgn(Re k ω )=+1 and sgn(Re k ω )= −
1, respectively, at severalvalues of H when K ω k H k [010], H ω k [100] and E ω k [001] inthe conical state (a) and the ferromagnetic state (b). F r equen cy ( G H z ) FMConical ∆ α / α - ( % ) Magnetic Field (Oe)
404 8 12 160
Magnetic Field (A/m)
FIG. 5: (color online). Calculated magnetic-field dependenceof resonance frequency and magnitude of the magnetochiraldichroism ∆ α/α − . wave.The expression of the complex refractive index N ( ω )is derived by solving the Fourier-formed Maxwell’s equa-tions as [17], N ( ω ) = cω k ω ∼ p [ ǫ cc ∞ + χ ee cc ( ω )][ µ aa ∞ + χ mm aa ( ω )]+ sgn (Re k ω )[ χ me ac ( ω ) + χ em ca ( ω )] / , (6)for k ω = k ω ˆ b k [010], H ω k [100] k a and E ω k [001] k c . The absorption coefficient α ( ω ) is related to N ( ω ) as, α ( ω ) = 2 ωκ ( ω ) c = 2 ωc Im N ( ω ) , (7)and thus attains the directional dependence via the signof Re k ω . Here κ ( ω ) = Im N ( ω ) is the extinction coeffi-cient.Figures 4(a) and (b) display calculated frequency-dependence of α + and α − in the conical phase and theferromagnetic phase, respectively, for several values of H , where α + and α − are the absorption coefficientsfor microwaves propagating parallel and antiparallel to H k [010], respectively. In the calculation, we take ǫ ∞ zz =8and µ ∞ zz =1 according to the experimental data [42, 43].We find that the microwave absorption is resonantly en-hanced at the eigen-frequency of the electromagnon exci-tation, and there exists significant difference between α + and α − .Calculated magnitude of the directional dichroism∆ α / α − where ∆ α = α + − α − is plotted in Fig. 5. Thisquantity is governed by the amplitude of ferroelectric po-larization P , which is nearly proportional to the squareof the net magnetization M . The dichroism increasesin the conical state with the growth of P and M as themagnetic field increases. In contrast, once the systementers the ferromagnetic state, the saturated M givesnearly constant directional dichroism although there stillexists slight field-dependence due to the field-dependentresonant frequency via ω in Eq. (7). The directionaldichroism is enhanced at the phase boundary betweenthe conical and the ferromagnetic phases, and its mag-nitude reaches as much as ∼ OSeO host gigantic microwave magnetochi-ral dichroism. This phenomenon results from interfer-ence between the magnetic and electric activation pro-cesses of electromagnons with GHz resonance frequen-cies. It has been demonstrated that long-period mag-netic structures in the chiral multiferroics without inver-sion symmetry can host gigantic dynamical magnetoelec-tric phenomena at GHz regime. In order to further en-hance the effect, search for novel chiral multiferroics withlarger P or stronger magnetoelectric coupling is needed.For example, chiral multiferroics based on the inverseDzyaloshinskii-Moriya mechanism [2] as an origin of its P is worth trying to search because this mechanism tendsto induce large P relative to the spin-dependent metal-ligand hybridization mechanism in Cu OSeO . 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