Simultaneously Magnetic- and Electric-dipole Active Spin Excitations Govern the Static Magnetoelectric Effect in Multiferroic Materials
D. Szaller, S. Bordacs, V. Kocsis, U. Nagel, T. Room, I. Kezsmarki
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J a n Simultaneously Magnetic- and Electric-dipole Active Spin Excitations Govern the StaticMagnetoelectric E ff ect in Multiferroic Materials D´avid Szaller, S´andor Bord´acs, Vilmos Kocsis, Toomas R˜o˜om, Urmas Nagel, and Istv´an K´ezsm´arki Department of Physics, Budapest University of Technology and Economics, 1111 Budapest, Hungary Quantum-Phase Electronics Center, Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan National Institute of Chemical Physics and Biophysics, Akadeemia tee 23, 12618 Tallinn, Estonia Condensed Matter Research Group of the Hungarian Academy of Sciences, 1111 Budapest, Hungary (Dated: June 4, 2018)We derive a sum rule to demonstrate that the static magnetoelectric (ME) e ff ect is governed by optical tran-sitions that are simultaneously excited via the electric and magnetic components of light. By a systematicanalysis of magnetic point groups, we show that the ME sum rule is applicable to a broad variety of non-centrosymmetric magnets including ME multiferroic compounds. Due to the dynamical ME e ff ect, the opticalexcitations in these materials can exhibit directional dichroism, i.e. the absorption coe ffi cient can be di ff erentfor counter-propagating light beams. According to the ME sum rule, the magnitude of the linear ME e ff ect of amaterial is mainly determined by the directional dichroism of its low-energy optical excitations. Application ofthe sum rule to the multiferroic Ba CoGe O , Sr CoSi O and Ca CoSi O shows that in these compounds thestatic ME e ff ect is mostly governed by the directional dichroism of the spin-wave excitations in the GHz-THzspectral range. On this basis, we argue that the studies of directional dichroism and the application of ME sumrule can promote the synthesis of new materials with large static ME e ff ect. I. INTRODUCTION
Magnetoelectric (ME) multiferroics, where ferroelectric-ity coexists with (ferro)magnetism, represent the most exten-sively studied class of multiferroics.
A spectacular controlof the ferroelectric polarization by magnetic field and manip-ulation of the magnetic order via electric field can be real-ized in most of these materials as a direct consequence ofthe coupling between spins and local electric dipoles. Thiso ff ers a fundamentally new path for data storage by com-bining the best qualities of ferroelectric and magnetoresistivememories: fast low-power electrical write operation, and non-destructive non-volatile magnetic read operation. The e ffi -ciency of multiferroics in such memory applications dependson the strength of the magnetization-polarization coupling re-sponsible for the ME phenomena.The ME e ff ect has also been proposed to open new perspec-tives in photonics. The entanglement between spins and localpolarization governs not only the ground-state properties butalso the character of excited states. Consequently, the elec-tric component of light can induce precession of the spins andthe magnetic component of light can generate electric polar-ization waves. This is termed as the optical ME e ff ect andhas recently been observed for the spin excitations in severalmultiferroic compounds. As one of the most peculiar manifestations of the ME e ff ectin the optical regime, counter-propagating light beams can ex-perience di ff erent refractive indices in multiferroics. Strongdirectional dichroism, that is di ff erence in the absorption co-e ffi cient for light beams traveling in opposite directions, hasbeen reported for spin excitations in these materials and pro-posed as a new principle to design directional light switchesoperating in the GHz-THz region. Here, we show that optical studies of low energy magnonsand phonons in ME multiferroics, provide an e ffi cient tool tofurther elucidate microscopic mechanisms of multiferroicity. These studies can be particularly useful to promote the sys-tematic synthesis of new materials with large static ME ef-fect. We derive a relation, hereafter referred to as the MEsum rule , which shows the connection between the static MEe ff ect and the directional dichroism observed for low-energyexcitations. We specify the class of materials where thisME sum rule is directly applicable. Finally, we investigatethe consequences of the ME sum rule for three multiferroicmaterials, Ba CoGe O (BCGO), Sr CoSi O (SCSO) andCa CoSi O (CCSO). For this purpose, we compare their di-rectional dichroism spectra to the corresponding static ME co-e ffi cients reported in the literature. Absorption measure-ments used to determine the directional dichroism in the GHz-THz spectral range were performed in the present study andpartly reproduced from our former works.
The Kramers-Kronig relation, also known as the Hilberttransformation, connects the real ( ℜ ) and imaginary ( ℑ ) partsof a general frequency dependent response function (suscep-tibility), χ ( ω ), which corresponds to a linear and causal re-sponse function in the time domain: ℜ χ ( ω ) = π P Z ∞−∞ ℑ χ ( ω ′ ) ω ′ − ω d ω ′ , ℑ χ ( ω ) = − π P Z ∞−∞ ℜ χ ( ω ′ ) ω ′ − ω d ω ′ , where P stands for the Cauchy principal value integral. Inmany cases, either the real or the imaginary part of χ ( ω ) canbe determined experimentally and the Kramers-Kronig trans-formation is used to obtain the entire complex response func-tion. In the limit of ω =
0, these expressions are simplified tothe following form, which shows close similarity with sumrules: ℜ χ ( ω = ≡ χ (0) = π P Z ∞ ℑ χ ( ω ) ω d ω, (1) ℑ χ ( ω = ≡ = − π P Z ∞−∞ ℜ χ ( ω ) ω d ω. (2)Typeset by REVTEXEquation 1 shows that the static response of a system is fullydetermined by the corresponding dynamical susceptibility andthe frequency denominator on the right-hand side indicates thevital role in low-energy excitations to the static susceptibility.A common example is the dielectric permittivity of semi-conductors, which is usually larger for compounds withsmaller charge gap and can be considerably a ff ected by thecontributions from low-energy phonon modes. A particularlystrong enhancement is found in quantum paraelectrics due tothe presence of soft polar phonon modes. Besides low-energy or soft modes, in materials with ferroic orders, the acsusceptibility related to the domain dynamics can also influ-ence the static response.In multiferroic materials the coupling between the elec-tric polarization and the magnetization can be phenomeno-logically described by the magnetoelectric susceptibility ten-sors χ me ( ω ) and χ em ( ω ), where ∆ M ωγ = χ me γδ ( ω ) E ωδ is themagnetization generated by an oscillating electric field and ∆ P ωδ = χ em δγ ( ω ) H ωγ is the polarization induced by an oscillat-ing magnetic field, respectively. Here γ and δ stand for theCartesian coordinates and the two cross-coupling tensors areconnected by the { . . . } ′ time-reversal operation according to { χ me γδ ( ω ) } ′ = − χ em δγ ( ω ).In a broad class of materials lacking simultaneously spatialinversion and time reversal symmetries, including alsomultiferroic compounds, the time reversal odd part of the MEsusceptibility can induce a di ff erence in the complex refractiveindex of counter-propagating electromagnetic waves, N ± ( ω ) ≈ q ε δδ ( ω ) µ γγ ( ω ) ± h χ me γδ ( ω ) − { χ me γδ ( ω ) } ′ i . (3)Here N ± stands for the refractive indices of waves propagat-ing in opposite directions ( ± k ). The e δ and e γ unit vectors areparallel to the direction of the electric ( E ω ) and magnetic ( H ω )fields of light, respectively, while ε δδ ( ω ) and µ γγ ( ω ) are diag-onal components of the complex relative permittivity and per-meability tensors in the { e δ , e γ , e η k k } basis. From this point onwe restrict our study to those cases, when the solutions of theMaxwell equations are linearly polarized waves or the linearpolarization of the incident light is nearly preserved during thepropagation through the magnetoelectric medium. This con-dition needs to be satisfied to have direct comparison betweenthe static and optical ME data. The di ff erence in the imaginarypart of the N + and N − refractive indices gives rise to a dif-ference in the absorption coe ffi cients of counter-propagatingwaves, termed as directional dichroism: ∆ α ( ω ) = α + ( ω ) − α − ( ω ) = ω c ℑ ( χ me γδ ( ω ) − { χ me γδ ( ω ) } ′ ) , (4)where c is the speed of light in vacuum. II. RESULTSA. The ME sum rule
In several classes of non-centrosymmetric magnets χ me γδ ( ω )is antisymmetric with respect to the time reversal, as listed in Table I and discussed later in this article. In this case, thestatic ME properties and the optical directional dichroism aredescribed by the same element of the ME tensor, hence, Eqs. 1and 4 yield the following ME sum rule: χ me γδ (0) = c π P Z ∞ ∆ α ( ω ) ω d ω. (5)According to this sum rule the static ME e ff ect is mostly gov-erned by the directional dichroism of low-energy excitations,since the absorption di ff erence, ∆ α , is cut o ff by the ω de-nominator at higher frequencies. The ME sum rule in Eq. 5can also be derived using the Kubo formula as described inthe Appendix.Following Neumann’s principle, we specify the symme-try of those magnetic crystals for which χ me γδ ( ω ) changes signupon the time reversal. This o ff -diagonal ME tensor compo-nent is antisymmetric if and only if(1) all spatial symmetry operations of the magnetic pointgroup (MPG) transforming χ me γδ ( ω ) into − χ me γδ ( ω ) arecombined with the time reversal operation, and thereis at least one such symmetry operation present in theMPG and(2) none of the spatial symmetry operations that leave χ me γδ ( ω ) invariant are combined with time reversal and(3) symmetry elements connecting χ me γδ ( ω ) to χ me δγ ( ω )or −{ χ me δγ ( ω ) } ′ and symmetry elements transforming χ me γδ ( ω ) to − χ me δγ ( ω ) or { χ me δγ ( ω ) } ′ are not present in theMPG at the same time.When light propagates along the principal axis of the crys-tal labeled as the z axis, the o ff -diagonal tensor component χ mexy ( ω ) can generate directional dichroism, where x and y axisare perpendicular to the z axis. These three conditions are ful-filled for χ mexy ( ω ) if the MPG meets all of the following crite-ria: i) it contains either an n ′ z or 2 ′ x symmetry operation where n ∈ { , , , } , ii) the MPG does not have any element com-bined with time reversal besides the previous ones and 4 ′ z , andiii) the MPG does not contain any of the previously listed op-erations without a subsequent time reversal. Here subscriptsstand for the axis of the n -fold proper ( n ) or improper ( n ) rota-tions, primes ( ′ ) following spatial transformations indicate thetime reversal operation.For light propagation along the y axis perpendicular to theprincipal z axis, χ mexz ( ω ) can generate directional dichroism. Inthis case, the conditions (1)-(3) specifying the requirementsof χ mexz ( ω ) being antisymmetric with respect to the time re-versal are fulfilled if the MPG matches all of the followingcriteria: i) it contains at least one symmetry element from n ′ , ′ x , ′ y , ′ z , ′ z , ′ z o , ii) the MPG does not have any elementcombined with the time reversal besides the previous ones,and iii) the MPG does not contain 4 z , 4 z or any of the previ-ously listed operations without a subsequent time reversal.The MPGs fulfilling these requirements are listed in Ta-ble I together with example materials. For MPGs marked byasterisks, the refractive index of the corresponding materials TABLE I: Crystallographic magnetic point groups (MPG) hosting χ mexy and χ mexz ME tensor elements, which are antisymmetric with respect tothe time reversal, listed in the second and fifth columns, respectively. Here z denotes the principal symmetry axis and MPGs are labeled inthe international notation. The subscripts of the symmetry operations show the axes of the n proper and n improper rotations and the axesperpendicular to the m mirror planes. Subscript d denotes the diagonal direction between the x and y coordinate axes. Symmetry operationsmarked by prime ( ′ ) are combined with the time reversal. The χ mexy and χ mexz ME tensor elements correspond to light propagation along the z and y axes, respectively. For the MPGs marked with asterisks in the third and sixth columns, the solutions of the Maxwell equations inthe transverse-wave approximation are linearly polarized waves. The few remaining MPGs are chiral, hence, they show circular dichroism.Several example materials are given in the fourth and seventh columns, where H α –if specified– stands for an external magnetic field pointingto the α crystallographic direction. In these cases x and z are the actual high-symmetry axes, i.e. for H [100] , H [001] and H [110] the correspondingcoordinates are x , z and again x , respectively. In hexagonal manganites ScMnO and LuMnO , there are coexisting magnetic phases withsample dependent temperature ranges, thus, they are indicated in two lines of the table.Crystal χ mexy ( ω ) = −{ χ mexy ( ω ) } ′ Materials χ mexz ( ω ) = −{ χ mexz ( ω ) } ′ MaterialssystemTriclinic 1 ′ z * 1 ′ z *Monoclinic m ′ z * Ni B O I; BiTeI H [100]31 ′ z LiCoPO ; Cu OSeO H [110] ; Co TeO a z m ′ z * TbOOH; Ba Ni F ′ z m z * TbPO ; MnPS ; Co TeO
17 K < T <
21 K Rhombic m x m y m ′ z * LiNiPO b m x m ′ y ′ z * BCGO, SCSO,
CCSO, CuB O H [110] c ′ x y ′ z BCGO, SCSO,
CCSO, CuB O H [100] d Tetragonal 4 z m ′ z *2 ′ x m d ′ z *2 ′ x ′ d z Nd Si m x m d z m ′ z *Rhombohedral 3 ′ z * Cr O ′ z * Cr O m ′ y z * BiTeI H [001]31 ′ x z ′ x z m x ′ z * Gd Ti O m x ′ z * Gd Ti O m ′ y ′ z * Nb Mn O , Nb Co O Hexagonal 6 ′ z * 6 ′ z ScMnO , LuMnO z m ′ z * 6 ′ z m z * m x ′ y ′ z * 2 ′ x m ′ y z * Fe P ′ x ′ y z m x m ′ y ′ z * HoMnO ; YMnO , ErMnO , YbMnO e m x m y z m ′ z * 2 y ′ z m x m ′ y ′ z m z * a NdFe (BO ) H [010]42 b one-dimensional photonic crystal with four-layered unit cell c CdS, AlN, GaN, InN H [100] ; CaBaCo O ; GaFeO ; Co B O Br; KMnFeF d Cu OSeO H [100] ; [Ru(bpy) (ppy)][MnCr(ox)]; one-dimensional pho-tonic crystal with three-layered unit cell e ScMnO , LuMnO , TmMnO is described by Eq. 3. In all these MPGs, the solutions ofthe Maxwell-equations are linearly polarized waves. In someof these cases, when there is a finite magnetization perpen-dicular to the light propagation, the polarization can have asmall longitudinal component, which is neglected here. Thistransverse-wave approximation means that we neglect addi-tional terms in the refractive index, which are higher-orderproducts of tensor components like χ meyz ǫ zx /ǫ zz or χ mezy µ zx /µ zz for propagation along the z axis.In materials belonging to MPGs not marked by asterisk,natural and magnetic circular dichroism can appear, sincethese MPGs are all chiral and in some cases finite magne-tization is allowed parallel to the light propagation direction(Faraday configuration). However, for su ffi ciently thin sam-ples the linear polarization of the incident light is nearly pre- served even then. Thus, the index of refraction can be ap-proximated by Eq. 3 for all of the listed MPGs. This allowsa direct comparison between the static and dynamical ME ef-fects according to the ME sum rule in Eq. 5, since the staticmeasurements used to determine the o ff -diagonal ME tensorelements can be compared to the optical experiments with lin-early polarized light. In the second and third rows of Fig. 1 the2 ′ x y ′ z chiral state of BCGO and CCSO is studied in theFaraday configuration, where the material shows polarizationrotation. Nevertheless, the directional dichroism can be wellapproximated by Eq. 4. B. Application of the ME sum rule to multiferroic materials
In order to check the applicability of the ME sum rule,we compare the magnetic field dependence of the staticand optical ME e ff ects for three members of the multifer-roic melilite family, namely for Ba CoGe O , Ca CoSi O and Sr CoSi O . These compounds crystallize in the non-centrosymmetric tetragonal P42 m structure where Co + cations with S = / N ≈ The free rotation of the magnetizationwithin the tetragonal plane can already be realized by mod-erate fields of . As another consequence of the single-ion anisotropy, the magnetization is saturated at di ff erent mag-netic field values, H S atplane and H S ataxis , when the field is appliedwithin the easy plane and along the hard axis, respectively. Prior to saturation, the magnetization follows a nearly linearfield dependence due to the increasing canting of the sublat-tice moments for any direction of the magnetic field.The multiferroic character of these materials hasbeen intensively studied both theoretically, andexperimentally via their static ME properties.The strong optical ME e ff ect emerging at their spin-waveexcitations has also attracted much interest. Themagnetically induced ferroelectric polarization has beendescribed by the spin-dependent hybridization of the Co + d orbitals with the p orbitals of the surrounding oxygen ionsforming tetrahedral cages. When the magnetization isa linear function of the applied field, the direction of thesublattice magnetizations can be straightforwardly expressedas a function of the orientation and the magnitude of themagnetic field. Then, the components of the magneticallyinduced ferroelectric polarization are directly determinedfrom the orientation of the sublattice magnetizations withinthe spin-dependent hybridization model: P [100] = A plane H sin θ H S atplane − vut − H sin θ H S atplane vt − H cos θ H S ataxis H cos θ H S ataxis sin φ, (6) P [010] = A plane H sin θ H S atplane − vut − H sin θ H S atplane vt − H cos θ H S ataxis H cos θ H S ataxis cos φ, (7) P [001] = A axis H sin θ H S atplane − H sin θ H S atplane vut − H sin θ H S atplane − − H cos θ H S ataxis sin2 φ. (8)Here θ and φ are the polar and azimuthal angles of the mag-netic field relative to the [001] and [100] axes, respectively,and H is the magnitude of the field. A plane and A axis areconstants describing the strength of the magnetoelectric cou-pling. To make the formulas more compact, the tilting an-gle of the two inequivalent oxygen tetrahedra in the unit cellwas approximated by π/
4, which is close to the experimen-tal value of 48 ◦ for CCSO. For BCGO, the saturation fieldsare H S atplane ≈
16 T and H S ataxis ≈
36 T as found both in the static and optical experiments. By fitting the field dependence ofthe static polarization reproduced from Ref. 18,19 in Fig. 1(a)and (g), we obtain A plane = µ C / m and A axis = µ C / m for BCGO. Using these parameters, the field dependence ofevery component of the static χ em δγ = ∂ P δ /∂ H γ ME tensor canbe calculated for BCGO according to Eqs. 6-8.For these three compounds, several elements of the staticME tensor, which are used in the present study for comparisonwith the directional dichroism spectra, can be directly deter-mined from the measured field dependence of the ferroelec-tric polarization reported in the literature. Only in those cases when experimental curves are not available, the ME tensor el-ements are evaluated using the fitted parameters as describedabove.Fig. 1(a) displays the ferroelectric polarization inducedalong the [001] axis in BCGO by magnetic fields applied par-allel to the [110] direction, P [001] ( H [110] ), as reproduced fromRef. 18. The field dependence of the χ em [001] , [110] static ME ten-sor element for external fields along the [110] axis, given bythe derivative ∂ P [001] /∂ H [110] , is shown in Fig. 1(c). Via theME sum rule in Eq. 5, this element of the static ME tensoris related to the integral of the directional dichroism spectrumin the Voigt configuration, where the magnetic component oflight is parallel to the static magnetic field applied along the[110] direction and the electric component of light is parallelto the [001] axis. In this configuration, the directional dichro-ism spectra reported for BCGO by Ref. 10 correspond to thedi ff erence of the red and blue curves in Fig. 1(b), which arethe absorption spectra obtained for counter-propagating THzwaves. The comparison between the static and optical datausing Eq. 5 is shown Fig. 1(c). -100-75-50-250255075100 -90 0 90 180 270 -75-50-250255075100 0.25 0.50 0.75 1.00 02468101214 -75-50-250255075100 0.25 0.50 0.75 1.00 02468101214 BCGO: + / ; CCSO: 2 + / 2 ; pol.: E [001] H [010] ; static field: H [100] A b s o r p t i on c oe ff i c i en t Frequency [THz]
30 cm -1 Opt. exp.
Stat. mod.BCGO
CCSO
SCSO H [100] [T] BCGO: + / ; pol.: E [001] H [110] ; static field: H [110] A b s o r p t i on c oe ff i c i en t Frequency [THz]
30 cm -1 e m [ ][ ] e m [ ][ ] e m [ ][ ] e m [ ][ ] H [110] [T] BCGO Stat. exp.
Opt. exp. a b cj k lg h id e f H [T] H [T] HE H [110] [T] P [ ] [ C / m ] em [001][110] = P [001] / H [110] BCGO Exp. Model
BCGO Exp. Model em [001][010] = P [001] / H [010] P [ ] [ C / m ] [(cid:176)] H=1T in the (001) plane [110] [100] [ ] HE HE P [ ] [ C / m ] [(cid:176)] H=5 T in the (100) plane [001] [010] [001]BCGO Exp. Model em [100][001] = P [100] / H [001] H [T] BCGO: + / ; pol.: E [110] H [001] ; static field: H [110] A b s o r p t i on c oe ff i c i en t BCGO: + / ; CCSO: 2 + / 2 ; pol.: E [100] H [001] ; static field: H [010] 0 H [T]Frequency [THz]
30 cm -1 Opt. exp.
Stat. mod.BCGO
CCSO
SCSO H [010] [T] -90 0 90 180 270 E H [001] [110] [001]BCGO Model em [110][001] = P [110] / H [001] [(cid:176)], H=5 T in the (110) plane P [ ] [ C / m ] A b s o r p t i on c oe ff i c i en t Frequency [THz]
30 cm -1 H [110] [T] BCGO Opt. exp.
Stat. mod.
FIG. 1: (Color online) Comparison of the static and optical ME properties of multiferroic Ba CoGe O (BCGO), Ca CoSi O (CCSO) andSr CoSi O (SCSO) based on the ME sum rule in Eq. 5. Panel (a): Dependence of the ferroelectric polarization ( P ) on the magnitude of themagnetic field ( H ) in BCGO. Panels (d), (g) and (j): Dependence of P on the orientation of the field in BCGO. In these panels the solid lines areexperimental data reproduced from Ref. 18, while the dashed lines are calculated using the spin dependent p-d hybridization model accordingto Eqs. 6-8. The slopes of the green lines in the same panels are proportional to the corresponding elements of the ME tensor. Arrows labeledwith E ω and H ω show the electric and magnetic components of the absorbed light in the corresponding optical experiment, respectively. Panels(b), (e), (h) and (k): Field dependence of the magnon absorption spectra of BCGO in the GHz-THz range. The light polarizations indicatedin these panels correspond to the labels E ω and H ω shown in the panels of the first column. The spectra are shifted vertically proportionalto H . For BCGO, the spectra corresponding to counter-propagating light beams are plotted by red and blue lines, while for CCSO brownand dark green lines represent the two propagation directions. The absorption coe ffi cient of CCSO is multiplied by a factor of two for bettervisibility. The spectra in panels (b) and (k) are measured in the present study, while the data in panels (e) and (h) are reproduced from Ref. 11for BCGO and from Ref. 12 for CCSO, respectively. Panels (c), (f), (i) and (l): Magnetic field dependence of di ff erent components of the MEtensor. Symbols indicate the tensor elements calculated from the corresponding optical measurements using the ME sum rule; empty square,full diamond and empty triangle stand for BCGO, CCSO and SCSO, respectively. The field dependence of the static ME tensor componentsare plotted with solid, dashed and dotted lines for the three compounds in the same order. The points corresponding to the slope of the greenlines in the left panels are indicated by a green dot. The solid line in panel (c) is calculated directly from the measured polarization-magneticfield curve shown on panel (a), while the curves in panels (f), (i) and (l) are evaluated using Eqs. 6-8. Static experiments, optical measurementsand model calculations were carried out at T = T = T = The following part of Fig. 1 shows similar analysis for otherthree elements of the ME tensor in BCGO. In two cases, datafor SCSO and CCSO are also included. The dependence ofthe ferroelectric polarization on the orientation of a constantfield H is shown in panels (d), (g) and (j). The directionaldichroism spectra in these three cases are displayed in panels(e), (h) and (k), while the comparison between the static andoptical data is given in panels (f), (i) and (l).The P [100] ( θ ) curve in Fig. 1(g) is reproduced fromRef. 18, where θ is the angle of the magnetic field rela-tive to the [001] axis. Since the tilting of the magneticfield from the [010] direction by a small angle of δθ intro-duces a weak transversal field δ H = (0 , , H sin δθ ), for H k [010] χ em [100][001] = ∂ P [100] /∂ H [001] ≈ / H × ∂ P [100] /∂θ . The correspond-ing optical experiment can be realized in the Faraday config-uration, where H k [010], while the electric and magnetic com-ponents of light are parallel to the [100] and [001] axes, re-spectively. These THz absorption spectra are shown for thetwo opposite wave propagation directions in Fig. 1(h) as re-produced from Ref. 11 for BCGO and from Ref. 12 forCCSO.The P [001] ( φ ) curve in Fig. 1(d) is taken from Ref. 18and the P [110] ( θ ) curve in Fig. 1(j) is calculated usingEqs. 6-8. In the former and later cases, the elements ofthe static ME tensor are respectively obtained accordingto χ em [001][010] = P [001] /∂ H [010] ≈ / H × ∂ P [001] /∂φ for H k [100] and χ em [110][001] = P [110] /∂ H [001] ≈ / H × ∂ P [110] /∂θ for H k [110]. Thecorresponding THz absorption spectra are shown in panels (e)and (k), respectively. III. DISCUSSION
The comparison between the ME tensor elements calcu-lated from the static and optical data in the last column of Fig.1 supports the applicability of the ME sum rule in these multi-ferroic compounds. The magnitude and the field dependenceof the static and optical data in panels (f), (i) and (l) showquantitative agreement. Their di ff erence can be attributed tothe following factors: i) the directional dichroism measure-ments were performed at T = T ≤ ffi cients are largerby ∼ ff erent growths, iii) in Fig. 1(e) and (h)the polarization of light beams can change during the propa-gation through the samples due to natural and magnetic cir-cular dichroism, iv) the model used to calculate the field de-pendence of the static ME coe ffi cients is not accurate due tothe linear field dependence of the magnetization assumed hereto reduce the number of fitting parameters, and v) uncertaintyin the geometrical factors of samples used in the static andoptical experiments may also cause an error of typically ∼ .
075 THz in zero field. In the field region investigated here, its energy remains consid-erably smaller than those of the other magnon modes. Hence,it dominates the integral in the Eq. 5 sum rule due to the ω frequency denominator. This mode is not allowed in an easy-plane magnet if the magnetic component of light is parallel tothe static magnetic field as seen in Fig. 1(b). Correspondingly,in Fig. 1(c) the ME tensor element calculated from the direc-tional dichroism data is smaller than those for the transversespin excitations shown in panels (f), (i) and (l).Moreover, the ME tensor element calculated from the sumrule in Fig. 1(c) is one order of magnitude smaller than thevalue determined from the static measurement, though theyboth change sign in the same field region of µ H = − ff erence may come from directional dichro-ism exhibited by excitations out of range of our optical de-tection. Since all magnon modes expected in the micro-scopic spin model of BCGO are observed in the absorptionexperiments, we think that low-energy phonon modes canshow strong optical magnetoelectric e ff ect due to coupling tomagnon modes. Though directional dichroism has not beendirectly observed for phonon modes, recent optical studieson multiferroic Ba NbFe Si O reported about the magneto-electric nature of low-energy lattice vibrations. As anotherpossibility, spin excitations located out of our experimentalwindow and not captured by the spin-wave theory can alsocontribute to the directional dichroism spectrum.Besides the comparative analysis of static and optical MEdata carried out for the three compounds above to demonstratethe applicability of the ME sum rule, we also make predic-tions for the same and other multiferroic materials. Previousstudies report about magnetically induced ferroelectric polar-ization in the paramagnetic phase of BCGO and SCSO up to T =
300 K, while their magnetic ordering temperature is T N ≈ T N invarious configurations can survive up to room temperature.In the non-centrosymmetric soft magnet (Cu,Ni)B O theelectric control of the magnetization direction has beendemonstrated together with directional dichroism of near-infrared electronic excitations. Since the contribution fromthese d-d transitions to the ME sum rule is negligible due totheir high frequency and the ω denominator in Eq. 5, we ex-pect that directional dichroism should also be present for low-frequency magnon excitations in this material.The magnetic control of the ferroelectric polariza-tion and / or the electric control of the magnetizationhave been observed in a plethora of multiferroic mate-rials including perovskite manganites with cycloidal spinorder, the room temperature multiferroics BiFeO and Sr Co Fe O . Based on the ME sum rule, we pre-dict that these compounds can also show directional dichro-ism as already has been found for Eu . Y . MnO andGd . Tb . MnO in the spectral range of the magnon excita-tions.
IV. CONCLUSIONS
We derived a ME sum rule and discussed its validity fornon-centrosymmetric magnets. We showed that the ME sumrule can be used to predict the static ME properties based onthe directional dichroism spectra governed by the optical MEe ff ect and vica versa, whenever the ME susceptibility of amaterial is antisymmetric with respect to the time reversal.We verified this approach by a quantitative comparisonbetween static ME coe ffi cients and directional dichroismspectra experimentally determined for three multiferroiccompounds in the melilite family. In most cases we foundthat the dominant contribution to the ME sum rule comesfrom magnon excitations located in the GHz-THz region.Our approach is applicable to most of the magnetoelectricmultiferroics, where the magnetically induced electric polar-ization can be controlled by the magnitude or the direction ofexternal magnetic field.We thank Y. Tokura, H. Murakawa and K. Penc forvaluable discussions. This project was supported byHungarian Research Funds OTKA K108918, T ´AMOP-4.2.2.B-10 / / B-09 / / KMR-2010-0002, T ´AMOP 4.2.4. A / / Appendix: Derivation of the magnetoelectric sum rule from theKubo formula
The microscopic description of the linear response of aquantum system to external stimuli is given by the Kubo for-mula. For the frequency dependence of the ME susceptibilitytensor, the finite-temperature Kubo formula reads χ me γδ ( z ) = − ~ X m , n e − β ~ ω n − e − β ~ ω m P i e − β ~ ω i h n | M γ | m i h m | P δ | n i z − ω m + ω n , (A.1) where z = ω + i ε and ε −→ + . M γ and P δ are the magneticand electric dipole operators, respectively. | m i and | n i areeigenstates of the unperturbed system with energies of ~ ω m and ~ ω n , while β is the inverse temperature. In the zero-temperature limit the Boltzmann factors vanish except for the | i zero energy ground state: χ me γδ ( z ) = − ~ X m h | M γ | m i h m | P δ | i z − ω m − h | P δ | m i h m | M γ | i z + ω m ! . (A.2)If χ me γδ ( ω ) is antisymmetric with respect to the time reversal,the h | M γ | m i h m | P δ | i product of the transition matrix ele-ments of the magnetic and electric dipole operators is real.The imaginary part of the transition matrix element productvanishes since the magnetic dipole operator changes sign un-der time reversal operation, which also requires the conjuga-tion of the matrix elements due to the exchange of the initialand final states. The Kubo formula at zero temperature for thereal and imaginary part of the magnetoelectric susceptibilityyields: ℜ χ me γδ ( ω ) = − ~ P X m ω m h | M γ | m i h m | P δ | i ω − ω m , (A.3) ℑ χ me γδ ( ω > = π ~ X m h | M γ | m i h m | P δ | i δ ( ω − ω m ) . (A.4)These expressions can also be obtained by second order per-turbation theory. With ∆ α ( ω ) = ω c ℑ χ me γδ ( ω ) one can repro-duce Eq. 5: χ me γδ (0) = ~ P X m h | M γ | m i h m | P δ | i ω m = ~ P X m Z ∞ h | M γ | m i h m | P δ | i ω · δ ( ω − ω m )d ω = c π P Z ∞ ∆ α ( ω ) ω d ω. (A.5) A. J. Freeman and H. Schmid (eds.)
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