Abstract
The coupling between spins and electric dipoles governs magnetoelectric phenomena in multiferroics. The dynamical magnetoelectric effect, which is an inherent attribute of the spin excitations in multiferroics, drastically changes the optical properties of these compounds compared with conventional materials where lightmatter interaction is expressed only by the dielectric permittivity or magnetic permeability. Here we show via polarized terahertz spectroscopy studies on multiferroic Ca_{2}CoSi_{2}O_{7}, Sr_{2}CoSi_{2}O_{7} and Ba_{2}CoGe_{2}O_{7} that such magnetoeletric spin excitations exhibit quadrochroism, that is, they have different colours for all the four combinations of the two propagation directions (forward or backward) and the two orthogonal polarizations of a light beam. We demonstrate that oneway transparency can be realized for spinwave excitations with sufficiently strong optical magnetoelectric effect. Furthermore, the transparent and absorbing directions of light propagation can be reversed by external magnetic fields. This magnetically controlled opticaldiode function of magnetoelectric multiferroics may open a new horizon in photonics.
Introduction
Nature offers a plethora of dichroic materials whose colour is different for two specific polarizations of the transmitted light. Dichroism generally appears in media that are not isotropic and provides information about their symmetry. Linear dichroism (absorption difference for two orthogonal linear polarizations of a light beam) emerges in materials where the symmetry is lower than cubic, while circular dichroism (absorption difference for the two circular polarizations) is observed in materials with finite magnetization or chiral structure. In all these cases, the four transverse wave solutions of the Maxwell equations for a given axis of light propagation group into two pairs, where each pair contains two counterpropagating waves (±k) with the same absorption coefficient^{1,2}.
In materials with simultaneously broken spatial inversion and time reversal symmetry this twofold ±k directional degeneracy of the Maxwell equations can be lifted and each of the four waves may be absorbed with a different strength^{3,4,5,6,7}. We call this phenomenon quadrochroism and the corresponding materials quadrochroic or fourcoloured media. Following the early prediction of ±k directional anisotropy^{4,8,9} and experimental observation^{10,11} of weak directional anisotropy effects, recent studies on multiferroics found strong directional dichroism (difference in the absorption coefficient for counterpropagating light beams) in the GHzTHz^{6,12,13,14,15} and visible spectral range^{16,17,18} as a hallmark of quadrochroism.
Magnetoeletric multiferroic materials simultaneously exhibit two ferroic properties in a single phase, namely they are both ferromagnetic and ferroelectric^{19}. In a broader sense, ferroelectrics concurrently hosting ferrimagnetic or antiferromagnetic order are also referred to as magnetoelectric multiferroics. When the coexisting magnetic and electric order parameters are intimately coupled to produce the static magnetoelectric effect, electric polarization can be generated by a magnetic field and magnetization appears in response to an electric field^{20,21}.
In magnetoelectric multiferroics, the coupling between the electric and magnetic states also leads to the dynamical or optical magnetoelectric effect described by the response functions (ω) and (ω)^{4,5,6,12,13,22}. Consequently, oscillating magnetization and polarization (M^{ω} and P^{ω}) are induced by the electric and magnetic components of light (E^{ω} and H^{ω}), respectively, besides the conventional terms arising from the dielectric permittivity () and magnetic permeability () tensors: M^{ω}=[(ω)−1]H^{ω}+(ω)E^{ω} and P^{ω}=_{0}[(ω)−1]E^{ω}+(ω)H^{ω}.
The static magnetoelectric effect has been widely proposed to form the basis for a fundamentally new type of memory devices by realizing a fast lowpower electrical write operation and a nondestructive nonvolatile magnetic read operation^{23,24,25}. Here we show that its dynamical analogue, the optical magnetoelectric effect, may generate quadrochroism and give rise to new optical functionalities such as magnetically controllable oneway transparency. We argue that quadrochroism is an inherent property of magnetoelectric multiferroics.
Results
Quadrochroism induced by the optical magnetoelectric effect
Optical properties of magnetoelectric media can be described phenomenologically by solving the Maxwell equations with the generalized constitutive relations quoted above^{4}, which yields the following form of the refractive index for a given polarization^{12,26}:
The±signs correspond to wavevectors ±k, while i and j specify the directions of the electric field (E^{ω}∥e_{i}) and the magnetic field (H^{ω}∥h_{j}) of light, respectively. The unit vectors e_{i} and h_{j} together with k form a righthanded system. Using i, j subscripts, polarization dependence is also indicated for the refractive index and other scalar quantities such as the absorption and reflectivity coefficients. The second term in Equation 1 shows the key role of the optical magnetoelectric effect in generating quadrochroism by lifting the ±k degeneracy of the Maxwell equations. The derivation of Equation 1 for magnetoelectric materials of various symmetries is given in the Methods section.
The optical magnetoelectric effect is exclusively generated by such transitions where both the electric and magneticdipole moments induced by an absorbed photon with energy ħω are finite:
that is, the magnetic (m_{j}) and electric (p_{i}) dipole operators must simultaneously have nonvanishing matrix elements between the ground state 0 and the excited states n separated by ħω_{n0} energy. V is the volume of the system, δ_{n} is the inverse lifetime of the excited state n, _{0} and μ_{0} are respectively the permittivity and the permeability of the vacuum. (ω) and (ω) are the sum of terms with the real (ℜ) and the imaginary (ℑ) parts of the matrix element products, respectively, and do not correspond to the real and imaginary parts of the total (ω).
This classification is based on the time parity of the two parts, (ω) changes sign under time reversal and (ω) remains invariant, which plays an important role in the peculiar optical phenomena studied here. The timeodd (ω) is solely responsible for the ±k directional anisotropy, i.e., , as the two cross effects are related by the Kubo formula according to and . The static magnetoelectric effect is also governed by , since vanishes in the zerofrequency limit. Equation 2 provides a direct link between the static magnetoelectric effect and the directional anisotropy by pointing to the common microscopic origin of the two phenomena.
When the optical magnetoelectric effect is sufficiently strong, a fourcoloured material can become fully transparent for a given propagation direction, while it still absorbs light travelling in the opposite direction as depicted in Fig. 1a. Specific to an excitation with a given resonance energy ħω_{n0} and polarization (e_{i},h_{j}), we define the magnetoelectric ratio as . For a magnetoelectric resonance separated from other excitations, the absorption, given by the imaginary part of the refractive index, vanishes for either forward or backward propagation, that is, the resonance shows oneway transparency if
where c is the speed of light in vacuum, is the dielectric permittivity for the given light polarization due to optical transitions at higher frequencies. Note that oneway transparency is only possible when the interference between the magnetic and electricdipole matrix elements is perfectly constructive or destructive, that is, γ_{i,j} is purely real. Correspondingly, the ± signs refer to the two cases when the polarization and the magnetization simultaneously induced by the electromagnetic field oscillate with zero and π phase difference, respectively. Note that in CGS units Equation 3 has the form γ_{i,j}=±, hence, the magnetic and electricdipole matrix elements must be of the same order of magnitude to approach the oneway transparency. (For technical details see the Methods section.)
Although directional anisotropy is not manifested in the normalincidence reflectivity of a vacuummaterial interface, the oneway transparency of a bulk material can still be accompanied by a peculiar behaviour of the normalincidence reflectivity, namely the magnetoelectric resonance remains hidden in the reflectivity spectrum, R(ω). Purely electric and magneticdipole transitions appear in the reflectivity spectrum with opposite line shapes according to the Fresnel formula R_{i,j}= as schematically shown in Fig. 1b. Thus, the relative strength of magnetic and electric susceptibilities for a given transition determines the shape of R(ω) near the resonance. When the condition γ_{i,j}=c/ is fulfilled, the reflectivity spectrum is featureless at the resonance with the constant value of R_{i,j}= as if there were no resonance present in that frequency range.
Multiferroic character of Ca_{2}CoSi_{2}O_{7}
This compound has a noncentrosymmetric tetragonal structure^{27,28} where Co^{2+} cations with S=3/2 spin form squarelattice layers stacked along the tetragonal [001] axis. The static magnetic and magnetoelectric properties of Ca_{2}CoSi_{2}O_{7} (refs 29, 30), as shown in Fig. 2, resemble the properties of Ba_{2}CoGe_{2}O_{7} (refs 31, 32, 33, 34, 35) and Sr_{2}CoSi_{2}O_{7} (ref. 36). This material undergoes an antiferromagnetic ordering below T_{N}=5.7 K with a small ferromagnetic moment (M). In the ordered state, the spins likely form a coplanar structure within the tetragonal planes similarly to the case of Ba_{2}CoGe_{2}O_{7} (refs 37, 38). Below T_{N}, magnetic field applied along the [110] (or ) axis in the tetragonal plane induces ferroelectric polarization (P) along the [001] direction, which changes sign at B≈12 T accompanied by an anomaly in the magnetization. The rotation of B=5 T magnetic field within the tetragonal plane results in a nearly sinusoidal modulation of P with zero crossings for field pointing along the [100] and [010] axes. The variation of M is less than 5% implying that inplane magnetic anisotropies, due to for example, small orthorhombicity of the crystal structure, become negligible in this field range. On this basis, we expect that the magnetic symmetry of Ca_{2}CoSi_{2}O_{7} can be approximated by the mm′2′ (m′m2′) polar point group for B∥[110] (B∥) and by the 22′2′ (2′22′) chiral point group for B∥[100] (B∥[010]) similarly to Ba_{2}CoGe_{2}O_{7} (refs 12, 13, 35, 39) and Sr_{2}CoSi_{2}O_{7} (ref. 36).
Quadrochroism in Ca_{2}CoSi_{2}O_{7}, Sr_{2}CoSi_{2}O_{7} and Ba_{2}CoGe_{2}O_{7}
Former studies on Ba_{2}CoGe_{2}O_{7} have demonstrated the role of magnetoelectric phenomena in the THz optical properties of this material and reported about strong directional dichroism of magnon modes^{12,13,22,40,41}. Here, we investigate the fourcoloured nature of magnons and the possibility to observe oneway transparency in the THz frequency range (0.2−2 THz) on single crystals of Ca_{2}CoSi_{2}O_{7}, Sr_{2}CoSi_{2}O_{7} and Ba_{2}CoGe_{2}O_{7}. Since the (ω) magnetoelectric tensor component responsible for the directional anisotropy has odd time parity, reversing the direction of an applied static magnetic field is equivalent to the reversal of light propagation direction. Therefore, we fixed the propagation direction and recorded the absorption spectra (α) in magnetic fields parallel and antiparallel to the propagation direction (±B) for two orthogonal polarizations as shown in Fig. 3.
While the static magnetic and magnetoelectric properties of these three compounds show close similarities, magnon modes in Ca_{2}CoSi_{2}O_{7} differ from the magnons observed for the other two compounds in their location, strength and field dependence. Irrespective of these differences, most of the magnon modes show strong quadrochroic character in each compound, that is, the strength of light absorption is different for all the four cases: the two polarizations and the two (±k) propagation directions. For certain modes oneway transparency is closely realized as the absorption coefficient is nearly zero for light propagating along a given direction, while the absorption remains strong for counterpropagating light beams with the same polarization. Furthermore, in each compound the sign of directional dichroism is reversed for some of the resonances within the field range of B=12–20 T.
Next, we show that the directional dichroism observed in Fig. 3 is the consequence of the magnetically induced chirality as previously proposed for Ba_{2}CoGe_{2}O_{7}, where magnetic switching between the two chiral enantiomers (22′2′ and 2′22′) has been verified by the detection of natural circular dichroism for the magnon modes^{13}. For this purpose, we studied the effect of the magnetic field orientation on the absorption spectra of Ca_{2}CoSi_{2}O_{7}. The sign of the directional dichroism is reversed by changing the orientation of the sample from B∥[100] to B∥[010] via π/2 rotation of the crystal around the tetragonal axis (compare spectra in Fig. 4a,b). On the other hand, rotating the crystal by π around any of the [100], [010] and [001] axes during the measurement leaves the absorption spectra unchanged. These observations support our assignment that the symmetry of these materials corresponds to the 22′2′ and 2′22′ chiral point group for B∥[100] and B∥[010], respectively. Thus, the directional dichroism observed here in Faraday configuration (B∥k) is the manifestation of chirality.
As a more fundamental quantity, the optical magnetoelectric susceptibility spectra calculated from the absorption spectra of counterpropagating beams with a given polarization according to ℑ{(ω)}≈ are plotted in Fig. 4c. ℑ{} has opposite signs in the left and righthanded forms of the material, which are established by magnetic fields pointing parallel to the [100] and [010] crystallographic axes, respectively. A summary of the ℑ{(ω)} spectra obtained for the three materials is given in Fig. 5. In all the three cases, the static magnetic field was applied parallel to the [100] axis, and two independent components of the magnetoelectric tensor have been evaluated from the absorption spectra measured with the two orthogonal polarizations of light.
When the crystal is rotated by π/4 around the [001] axis, that is, the direction of the external magnetic field is changed from B∥[100] to B∥[110], the material is expected to become not chiral but polar. This was experimentally verified by the lack of ±k directional anisotropy in Faraday configuration as shown for Ca_{2}CoSi_{2}O_{7} and Sr_{2}CoSi_{2}O_{7} in the Supplementary Fig. 2. In this case, the spin excitations show no directional dichroism because the magnetoelectric ratio, γ_{i,j}, is either zero or infinity for each resonance when (E^{ω}∥[001], H^{ω}∥) and (E^{ω}∥, H^{ω}∥[001]). Hence, the modes loose their quadrochroic character and become ordinary dichroic transitions.
Oneway transparency of magnetoelectric excitations
As clear from Equation 3, oneway transparency in a finite spectral range is not prohibited by any fundamental law of nature. While the conventional magnon modes are only magneticdipole active, the spinwave excitations in multiferroics can have a magnetoelectric (both electric and magneticdipole) character^{6,12,13,14,15,22,26} and be even dominantly electricdipole active^{42,43,44,45,46,47,48,49}. Since the ratio of the matrix elements can be controlled via the spin system and is mainly determined by the lattice vibrations, oneway transparency can be achieved by synthesis of tailored material.
The magnetoelectric ratio for a separate transition can be calculated from the absorption measured for counterpropagating beams with a given polarization, and :
Here, the upper/lower sign corresponds to the case when the transition is located at the magnetic/electricdipole side of the border of oneway transparency, which can be determined from polarized reflectivity spectra as explained earlier or by the systematic polarization dependence of the absorption spectra. (See Methods section for the derivation of Equation 4.)
Figure 1 shows the magnetoelectric ratio, γ_{i,j}, for magnon modes with strong directional anisotropy in Ca_{2}CoSi_{2}O_{7}, Sr_{2}CoSi_{2}O_{7}, Ba_{2}CoGe_{2}O_{7} and in two other multiferroics Gd_{0.5}Tb_{0.5}MnO_{3} (ref. 15) and Eu_{0.55}Y_{0.45}MnO_{3} (ref. 14). Note that the absorption coefficient for counterpropagating light beams differs by more than one order of magnitude for modes located in the region of 0.5<γ_{i,j}<2. The selected modes are separated from other transitions, hence, their magnetoelectric ratio can be determined according to Equations 3 and 4. For Ca_{2}CoSi_{2}O_{7}, Sr_{2}CoSi_{2}O_{7} and Ba_{2}CoGe_{2}O_{7} the labelling of the modes follows the notation used in Fig. 3. GTMO #1 and EYMO #1 are the lowestenergy spincurrent driven electromagnon modes in Gd_{0.5}Tb_{0.5}MnO_{3} (ref. 15) and Eu_{0.55}Y_{0.45}MnO_{3} (ref. 14), respectively. GTMO #2 is the exchangestriction induced electromagnon^{42,44,46} and EYMO #2 is the conventional antiferromagnetic resonance^{50}, both located at ≈0.6–0.7 THz (the selection rules for these transitions are as specified in refs 14, 15). The magnetoelectric ratio can be unambiguously determined from the directional anisotropy data together with the systematic polarization dependence of the absorption spectra for the modes in Ba_{2}CoGe_{2}O_{7} (refs 12, 13, 40) and for the higherenergy modes in Gd_{0.5}Tb_{0.5}MnO_{3} (refs 15, 44, 46) and Eu_{0.55}Y_{0.45}MnO_{3} (refs 14, 50). The dominantly magnetic and electricdipole characters of modes BCGO #1 and BCGO #2, respectively, are in agreement with the theoretical predictions^{22,40}. Because of the close similarity between the magnon spectra of Sr_{2}CoSi_{2}O_{7} and Ba_{2}CoGe_{2}O_{7}, we use the same assignment for Sr_{2}CoSi_{2}O_{7}. The spin excitations in Ca_{2}CoSi_{2}O_{7} are tentatively assigned to the magneticdipole side of the γ_{i,j}=c/ boundary, while the spincurrent driven mode in Gd_{0.5}Tb_{0.5}MnO_{3} and Eu_{0.55}Y_{0.45}MnO_{3} is assumed to be at the electricdipole side. Note that the selected magnon modes in Sr_{2}CoSi_{2}O_{7}, Ca_{2}CoSi_{2}O_{7} and Ba_{2}CoGe_{2}O_{7} show nearly oneway transparency as also evident from Fig. 3.
DC magnetoelectric effect governed by directional dichroism
As already discussed in the context of Equation 2, directional optical anisotropy and static magnetoelectric effect share a common microscopic origin. Their relation can be quantitatively formulated by the KramersKronig transformation when taking the zerofrequency limit:
Hence, the static magnetoelectric coefficient is proportional to the integral of the directional dichroism over the whole frequency range. Due to the ω′^{2} denominator, the main contributions to the integral come from the lowestfrequency magnetoelectric excitations, for example, from magnon modes. We note that the second equality in Equation 5 holds only if (ω)≡0, that is, 0m_{j}nnp_{i}0 is real. This is indeed the case for multiferroic compounds belonging to many magnetic point groups such as 42′2′, 22′2′, 2′, mm′2′, mmm′. The connection between the static and optical magnetoelectric phenomena is also manifested in the present materials. The magnetically induced ferroelectric polarization together with the differential magnetoelectric effect (indicated for Ca_{2}CoSi_{2}O_{7} in the caption of Fig. 2b) change sign at B≈12 T, 16 T and 15 T in Ca_{2}CoSi_{2}O_{7}, Sr_{2}CoSi_{2}O_{7} and Ba_{2}CoGe_{2}O_{7}, respectively^{31,36}. Since directional dichroism is governed by the same differential magnetoelectric effect in the optical region, the sign of the integral in Equation 5 should simultaneously be reversed. Accordingly, in the same magnetic field region, we observed the sign reversal of the directional dichroism for most of the modes in each compound.
The optical magnetoelectric effect measures the interference between the electric polarization and the magnetization that are simultaneously induced in the light absorption processes. The interference term, that is, 0m_{j}nnp_{i}0 matrix element product, is sensitive to the relative orientation of the static electric polarization and the static magnetization in multiferroics. Indeed directional dichroism can emerge in multiferroics where the ferroelectric polarization is not parallel to the magnetization^{7,12,14,15}, in magnets with ferrotoroidic order^{7}, and in the present case of magnetically induced chirality. Therefore, the measurement of directional anisotropy may be applied for spatial mapping of multiferroic domain structures, a capability that has been considered to be the privilege of nonlinear optical techniques such as the optical second harmonic generation^{51,52,53,54}.
As a more fundamental issue, the optical magnetoelectric susceptibility determined from the directional dichroism spectrum can help to distinguish between possible microscopic mechanisms of magnetoelectric coupling in multiferroics. The directional dichroism observed for magnon modes in Ba_{2}CoGe_{2}O_{7} has been closely reproduced by the spindependent metalligand hybridization model^{13} previously proposed to describe the static magnetoelectric properties of this compound. In the spincycloidal state of perovskite manganites, directional dichroism is strong for the lowenergy mode induced by the spincurrent mechanism (GTMO #1 and EYMO #1 in Fig. 1), while it is weak for the higherenergy mode driven by magnetostriction (GTMO #2)^{14,15}. This is in agreement with the expected magnetoelectric nature of GTMO #1 and EYMO #1 magnon modes and the dominantly electricdipole character of GTMO #2 magnon mode^{6,14,15,42,43,45}.
Discussion
We have demonstrated that quadrochroism generated by the optical magnetoelectric effect in multiferroics is an inherent property of the spinwave excitations located in the THz spectral range. Depending on the ratio of lightinduced oscillating magnetization and electric polarization, these excitations can exhibit oneway transparency in specific light polarization. As another consequence of quadrochroism, directional dichroism is not cancelled but remains finite for unpolarized light. Moreover, an excitation showing oneway transparency remains hidden in the reflectivity. Recently, directional anisotropy has been predicted^{55} and also observed^{56} for the spin excitations of skyrmion crystals at GHz frequencies.
Besides spin excitations in the GHzTHz spectral range, the optical magnetoelectric effect can strongly influence the optical properties of multiferroics in the infraredvisible region. Crystal field transitions of the d or f shell electrons of magnetic ions show strong directional anisotropy as was observed in multiferroic CuB_{2}O_{4} (refs 16, 17) and (Cu,Ni)B_{2}O_{4} (ref. 18). Although the crystalfield excitations predominantly have an electricdipole character, strong spinorbit interaction can tune their magnetoelectric ratio closer to γ_{i,j}=c/.
In contrast to the common belief that lattice vibrations are purely electricdipole active transitions, the optical magnetoelectric effect can also emerge for phonon modes in multiferroics as has been reported recently for the magnetoelectric atomic rotations in Ba_{3}NbFe_{3}Si_{2}O_{14} (ref. 57). To complete the list of excitations, we recall that directional dichroism was first observed for excitonic transitions of the polar semiconductor CdS in external magnetic fields by Hopfield and Thomas^{58} in 1960.
Since the whole zoo of spin, orbital and lattice excitations can exhibit optical magnetoelectric effect in multiferroics, directional control of light in these materials may work over a broad spectrum of the electromagnetic radiation from the THz range to the visible region. On the other hand, the presence of static magnetization and electric polarization in multiferroics facilitates the switching between the transparent and absorbing directions of light propagation via static magnetic or electric fields. The oneway transparency observed here emerges for ‘conventional’ (zonecenter) magnon modes of multiferroic compounds with simple antiferromagnetic order, hence, such opticaldiode function is expected to be present in a broad variety of multiferroics.
Methods
Polarized THz absorption spectroscopy
For the study of polarized absorption in magnetic fields up to 30 T, we used Fourier transform spectroscopy. The measurement system for the B=0–12 T region is based on a MartinPuplett interferometer, a mercury arc lamp as a light source and a Si bolometer cooled down to 300 mK as a light intensity detector. This setup covers the spectral range 0.13–6 THz with a maximum resolution of 0.004 THz. The polarization of the beam incident to the sample is set by a wiregrid polarizer, while the detection side (light path from the sample till the detector) is insensitive to the polarization of light. Experiments up to 30 T were carried out in the High Field Magnet Laboratory in Nijmegen, where a Michaelson interferometer is used together with a 1.6 K Si bolometer providing a spectral coverage of 0.3–6 THz.
All the measurements were carried out in Faraday configuration, that is, by applying magnetic fields (anti)parallel to the direction of light propagation using oriented single crystal pieces with the typical thickness of 1 mm. The crystals were grown by a floatingzone method and were characterized by magnetization and ferroelectric polarization experiments before the optical study^{31}.
Formulae for quadrochroism
The quadrochroism of spin excitations located in the longwavelength region of light can be described by the Maxwell’s equations
by introducing the dynamical magnetoelectric effects into the constitutive relations:
The microscopic form of the dielectric permittivity () and magnetic permeability () is given by the Kubo formula:
Here, and are the sum of terms with the real (ℜ) parts of the matrix element products (also including δ_{ji} and δ_{ji}, respectively) while, and are the sum of terms with the imaginary (ℑ) parts of the matrix element products. In contrast to the parity of the magnetoelectric tensor introduced in Equation 2, () is invariant and () changes sign under time reversal. is the background dielectric constant from modes located at higher frequencies than the studied frequency window, while magnetic permeability contribution from higherfrequency excitations is neglected being usually much smaller than unity.
Among crystals exhibiting ±k directional anisotropy, the highest symmetry ones are the chiral materials with the 432 cubic point group when magnetization develops along one of their principal axes. In this case, their magnetic point symmetry is reduced to 42′2′ where ′ means the timereversal operation. The fourfold rotational symmetry is preserved around the direction of the magnetization chosen as the yaxis in the following. According to Neumann’s principle, for materials belonging to the point group the dynamical response tensors have the following form:
and the general relation yields in the present symmetry . By solving the Maxwell equations for propagation parallel and antiparallel to the magnetization direction (±k∥y), one obtains the following refractive indices for the left (l) and right (r) circularly polarized eigenmodes:
The reflectivity of an interface between the vacuum and a material with 42′2′ symmetry can be determined from the Maxwell equations in the usual way by using the boundary conditions. For normal incidence the components of the magnetoelectric tensor do not appear in the reflectivity and one can reproduce the general expression:
where Z_{l/r}=Z_{0} and Z_{0} are the surface impedance of the interface and the impedance of vacuum, respectively. Keeping only the leading term in the surface impedance, that is, Z=Z_{0} for polarization E^{ω}∥e_{i} and H^{ω}∥h_{j}, the expression for the normalincidence reflectivity given above is generally valid for materials belonging to other magnetic point groups as well.
When the symmetry is reduced to , as is the case in the magnetically induced chiral state of Ca_{2}CoSi_{2}O_{7} and Ba_{2}CoGe_{2}O_{7}, the equivalence of x and zaxis does not hold anymore. Consequently the form of the tensors changes and the eigenmodes become elliptically polarized. Since the (100) plane studied for these materials shows strong linear dichroism/birefringence because of their crystal structure (with the dielectric constant ^{∞}≈12 and 8 for polarizations along the optical axes [100] and [001], respectively, as determined from transmission data), the eigenmodes are expected to be nearly linearly polarized. The approximate form of the refractive index, when keeping only the timereversal odd components of and the timereversal even components in and (that is, neglecting polarization rotation):
In a broad class of orthorhombic multiferroics, the magnetization and ferroelectric polarization are perpendicular to each other. Choosing the magnetization and polarization parallel to the yaxis and zaxis, respectively, the magnetic point group is the . In this case, the form of and tensors is the same as in but the magnetoelectric tensors become different:
For propagation perpendicular both to the direction of the magnetization and the ferroelectric polarization (±k∥x), the refractive indices for the two linearly polarized eigenmodes are:
While in the lowestorder approximation the general formulae are and as given in Equation 1, quadrochroism can be generated by other higherorder terms, such as , that are odd both in time reversal and spatial inversion. Note that ′ and ′′ correspond to the part of being odd and even under time reversal, respectively, while the situation is reversed for and .
The criterion for oneway transparency
We calculate the absorption coefficient in the vicinity of a separate magnetoelectric resonance, that is, when the photon with the frequency ω=ω_{n0} resonantly excites the 0 → n transition. When using the Kubo formula for , and , the refractive index for a given polarization, , has the following form:
where the square root in the first term is expanded up to the second order. Then, the absorption coefficient vanishes for one direction, that is, either =0 or =0, if
where c is the speed of light in vacuum, is the dielectric permittivity for the given polarization due to other optical transitions and ξ=ℜ{γ_{i,j}}/γ_{i,j}. Note that oneway transparency is only possible when γ_{i,j} is purely real corresponding to ξ =+ 1 or −1 and the formula simplifies to γ_{i,j}=c/ as given in Equation 3. Note that γ_{i,j} is real when =0, which is the case for many multiferroic materials belonging to different magnetic point groups including 42′2′ (Nd_{5}Si_{4}) (ref. 59), 22′2′ (CuB_{2}O_{4}) (ref. 16), 2′ (LiCoPO_{4}) (ref. 60), mm′2′ (GaFeO_{3}) (ref. 61), mmm′ (LiNiPO_{4}) (ref. 62). When γ_{ij} is real, Equation 4 can be straightforwardly derived from Equation 21. When the condition of oneway transparency holds for a separate resonance in a given polarization , the surface impedance does not change near the resonance and has the frequency independent Z=1/ value. Thus, the reflectivity also coincides with the background value R=.
Additional information
How to cite this article: Kézsmárki, I. et al. Oneway transparency of fourcoloured spinwave excitations in multiferroic materials. Nat. Commun. 5:3203 doi: 10.1038/ncomms4203 (2014).
References
Barron, L. D. Molecular Light Scattering and Optical Activity Cambridge University Press (2004).
Azzam, R. M. A. & Bashara, N. M. Ellipsometry and polarized light NorthHolland: Amsterdam, (1979).
Hornreich, R. M. & Shtrikman, S. Theory of gyrotropic birefringence. Phys. Rev. 171, 1065–1074 (1968).
O'Dell, T. H. The Electrodynamics of Magnetoelectric Media NorthHolland: Amsterdam, (1970).
Arima, T. Magnetoelectric optics in noncentrosymmetric ferromagnets. J. Phys. Condens. Matter 20, 434211 (2008).
Cano, A. Theory of electromagnon resonances in the optical response of spiral magnets. Phys. Rev. B 80, 180416(R) (2009).
Szaller, D., Bordács, S. & Kézsmárki, I. Symmetry conditions for nonreciprocal light propagation in magnetic crystals. Phys. Rev. B 87, 014421 (2013).
Baranova, N. B. & Zeldovich, B. Y. a. Theory of a new linear magnetorefractive effect in liquids. Molec. Phys. 38, 1085–1098 (1979).
Barron, L. D. & Vrbancich, Magnetochiral birefringence and dichroism. Molec. Phys. 51, 715–730 (1984).
Rikken, G. L. J. A. & Raupach, E. Observation of magnetochiral dichroism. Nature 390, 493–494 (1997).
Rikken, G. L. J. A., Strohm, C. & Wyder, P. Observation of magnetoelectric directional anisotropy. Phys. Rev. Lett. 89, 133005 (2002).
Kézsmárki, I. et al. Enhanced directional dichroism of terahertz light in resonance with magnetic excitations of the multiferroic Ba2CoGe2O7 oxide compound. Phys. Rev. Lett. 106, 057403 (2011).
Bordács, S. et al. Chirality of matter shows up via spin excitations. Nat. Phys. 8, 734–738 (2012).
Takahashi, Y., Shimano, R., Kaneko, Y., Murakawa, H. & Tokura, Y. Magnetoelectric resonance with electromagnons in a perovskite helimagnet. Nat. Phys. 8, 121–125 (2012).
Takahashi, Y., Yamasaki, Y. & Tokura, Y. Terahertz magnetoelectric resonance enhanced by mutual coupling of electromagnons. Phys. Rev. Lett. 111, 037204 (2013).
Saito, M., Ishikawa, K., Taniguchi, K. & Arima, T. Magnetic control of crystal chirality and the existence of a large magnetooptical dichroism effect in CuB2O4 . Phys. Rev. Lett. 101, 117402 (2008).
Saito, M., Taniguchi, K. & Arima, T. Gigantic Optical Magnetoelectric Effect in CuB2O4 . J. Phys. Soc. Jpn 77, 013705 (2008).
Saito, M., Ishikawa, K., Konno, S., Taniguchi, K. & Arima, T. Periodic rotation of magnetization in a noncentrosymmetric soft magnet induced by an electric field. Nat. Mater. 8, 634–638 (2009).
Schmid, H. Multiferroic magnetoelectrics. Ferroelectrics 162, 317–338 (1994).
Fiebig, M. Revival of the magnetoelectric effect. J. Phys. D 38, R123–R152 (2005).
Wang, K. F., Liu, J.M. & Ren, Z. F. Multiferroicity: the coupling between magnetic and polarization orders. Adv. Phys. 58, 321–448 (2009).
Miyahara, S. & Furukawa, N. Theory of magnetoelectric resonance in twodimensional S=3/2 antiferromagnet Ba2CoGe2O7 via spindependent metalligand hybridization mechanism. J. Phys. Soc. Jpn 80, 073708 (2011).
Ramesh, R. & Spaldin, N. A. Multiferroics: progress and prospects in thin films. Nat. Mater. 6, 21–29 (2007).
Martin, L. W. et al. Advances in the growth and characterization of magnetic, ferroelectric, and multiferroic oxide thin films. Mater. Sci. Eng. R 68, 89–133 (2010).
Wu, S. M. et al. Full electric control of echange bias. Phys. Rev. Lett. 110, 067202 (2013).
Miyahara, S. & Furukawa, N. Nonreciprocal Directional dichroism and toroidalmagnons in helical magnets. J. Phys. Soc. Jpn 81, 023712 (2012).
Hagiya, K., Ohmasa, M. & Iishi, K. The modulated structure of synthetic coakermanite, Ca2CoSi2O7 . Acta Cryst. B 49, 172–179 (1993).
Jia, Z. H., Schaper, A. K., Massa, W., Treutmanna, W. & Ragera, H. Structure and phase transitions in Ca2CoSi2O7Ca2ZnSi2O7 solidsolution crystals. Acta Cryst. B 62, 547–555 (2006).
Akaki, M., Tozawa, J., Akahoshi, D. & Kuwahara, H. Gigantic magnetoelectric effect caused by magneticfieldinduced canted antiferromagneticparamagnetic transition in quasitwodimensional Ca2CoSi2O7 crystal. Appl. Phys. Lett. 94, 212904 (2009).
Akaki, M., Tozawa, J., Akahoshi, D. & Kuwahara, H. Magnetic and dielectric properties of A2CoSi2O7 (A=Ca, Sr, Ba) crystals. J. Phys. C 150, 042001 (2009).
Murakawa, H., Onose, Y., Miyahara, S., Furukawa, N. & Tokura, Y. Ferroelectricity induced by spindependent metalligand hybridization in Ba2CoGe2O7 . Phys. Rev. Lett. 103, 137202 (2010).
Yi, H. T., Choi, Y. J., Lee, S. & Cheong, S. W. Multiferroicity in the squarelattice antiferromagnet of Ba2CoGe2O7 . Appl. Phys. Lett. 92, 212904 (2008).
Romhányi, J., Lajkó, M. & Penc, K. Zero and finitetemperature mean field study of magnetic field induced electric polarization in Ba2CoGe2O7: effect of the antiferroelectric coupling. Phys. Rev. B 84, 224419 (2011).
Yamauchi, K., Barone, P. & Silvia Picozzi, S. Theoretical investigation of magnetoelectric effects in Ba2CoGe2O7 . Phys. Rev. B 84, 165137 (2011).
PerezMato, J. M. & Ribeiro, J. L. On the symmetry and the signature of atomic mechanisms in multiferroics: the example of Ba2CoGe2O7 . Acta Cryst. A 67, 264–268 (2011).
Akaki, M., Iwamoto, H., Kihara, T., Tokunaga, M. & Kuwahara, H. Multiferroic properties of an akermanite Sr2CoSi2O7 single crystal in high magnetic fields. Phys. Rev. B 86, 060413(R) (2012).
Zheludev, A. et al. Spin waves and the origin of commensurate magnetism in Ba2CoGe2O7 . Phys. Rev. B 68, 024428 (2003).
Hutanu, V. et al. Determination of the magnetic order and the crystal symmetry in the multiferroic ground state of Ba2CoGe2O7 . Phys. Rev. B 86, 104401 (2012).
Toledano, P., Khalyavin, D. D. & Chapon, L. C. Spontaneous toroidal moment and fieldinduced magnetotoroidic effects in Ba2CoGe2O7 . Phys. Rev. B 84, 094421 (2011).
Penc, K. et al. Spinstretching modes in anisotropic magnets: spinwave excitations in the multiferroic Ba2CoGe2O7 . Phys. Rev. Lett. 108, 257203 (2012).
Romhányi, J. & Penc, K. Multiboson spinwave theory for Ba2CoGe2O7: A spin3/2 easyplane Ńeel antiferromagnet with strong singleion anisotropy. Phys. Rev. B 86, 174428 (2012).
Pimenov, A. et al. Possible evidence for electromagnons in multiferroic manganites. Nat. Phys. 2, 97–100 (2006).
Sushkov, A. B., Mostovoy, M., Valdes Aguilar, R., Cheong, S.W. & Drew, D. Electromagnons in multiferroic RMn2O5 compounds and their microscopic origin. J. Phys. Condens. Matter 20, 434210 (2008).
Takahashi, Y. et al. evidence for an electricdipole active continuum band of spin excitations in multiferroic TbMnO3 . Phys. Rev. Lett. 101, 187201 (2008).
Valdes Aguilar, R. et al. Origin of electromagnon excitations in multiferroic RMnO3 . Phys. Rev. Lett. 102, 047203 (2009).
Kida, N. et al. Terahertz timedomain spectroscopy of electromagnons in multiferroic perovskite manganites. J. Opt. Soc. Am. B 26, A35–A51 (2009).
Mochizuki, M., Furukawa, N. & Nagaosa, N. Theory of electromagnons in the multiferroic mn perovskites: the vital role of higher harmonic components of the spiral spin order. Phys. Rev. Lett. 104, 177206 (2010).
Seki, S., Kida, N., Kumakura, S., Shimano, R. & Tokura, Y. Electromagnons in the spin collinear state of a triangular lattice antiferromagnet. Phys. Rev. Lett. 105, 097207 (2010).
Kida, N., Kumakura, S., Ishiwata, S., Taguchi, Y. & Tokura, Y. Gigantic terahertz magnetochromism via electromagnons in the hexaferrite magnet Ba2Mg2Fe12O22. Phys. Rev. B 83, 064422 (2011).
Takahashi, Y. et al. Farinfrared optical study of electromagnons and their coupling to optical phonons in Eu1−xYxMnO3 (x=0.1, 0.2, 0.3, 0.4, and 0.45). Phys. Rev. B 79, 214431 (2009).
Fiebig, M., Pavlov, V. V. & Pisarev, R. V. Secondharmonic generation as a tool for studying electronic and magnetic structures of crystals: review. J. Opt. Soc. Am. B 22, 96–118 (2005).
Van Aken, B. B., Rivera, J.P., Schmid, H. & Fiebig, M. Observation of ferrotoroidic domains. Nature 449, 702–705 (2007).
Ju, S. & Guo, G.Y. Colossal nonlinear optical magnetoelectric effects in multiferroic Bi2FeCrO6 . Appl. Phys. Lett. 92, 202504 (2008).
Ju, S. & Cai, T.Y. Ab initio study of ferroelectric and nonlinear optical performance in BiFeO3 ultrathin films. Appl. Phys. Lett. 95, 112506 (2009).
Mochizuki, M. & Seki, S. Magnetoelectric resonances and predicted microwave diode effect of the skyrmion crystal in a multiferroic chirallattice magnet. Phys. Rev. B 87, 134403 (2013).
Okamura, Y. et al. Microwave magnetoelectric effect via skyrmion resonance modes in a helimagnetic multiferroic. Nat. Commun. 4, 2391 (2013).
Chaix, L. et al. THz Magnetoelectric Atomic Rotations in the Chiral Compound Ba3NbFe3Si2O14 . Phys. Rev. Lett. bf 110, 157208 (2013).
Hopfield, J. J. & Thomas, D. G. Photon momentum effects in the magnetooptics of excitons. Phys. Rev. Lett. 4, 357–359 (1960).
Cadogan, J. M., Ryan, D. H., Altounian, Z., Wang, H. B. & Swainson, I. P. The magnetic structures of Nd5Si4 and Nd5Ge4 . J. Phys. Condens. Matter 14, 7191–7200 (2002).
Vaknin, D., Zarestky, J. L., Miller, L. L., Rivera, J.P. & Schmid, H. Weakly coupled antiferromagnetic planes in singlecrystal LiCoPO4 . Phys. Rev. B 65, 224414 (2002).
Abrahams, S. C. & Reddy, J. M. Magnetic, electric, and crystallographic properties of gallium iron oxide. Phys. Rev. Lett. 13, 688–690 (1964).
Santoro, R. P., Segal, D. J. & Newnham, R. E. Magnetic properties of LiCoPO4 and LiNiPO4 . J. Phys. Chem. Solids 27, 1192–1193 (1966).
Acknowledgements
We thank K. Penc and S. Miyahara for discussions. This work was supported by Hungarian Research Funds OTKA K108918, TÁMOP4.2.1/B09/1/KMR20100002, TÁMOP 4.2.4. A/211120120001 and TÁMOP4.2.2.B10/120100009, by KAKENHI, MEXT of Japan, by Funding Program for WorldLeading Innovation R&D on Science and Technology (FIRST program) on ‘Quantum Science on Strong Correlation’ and by a GrantinAid for Scientific Research under the Contract No. 24224009, by the Estonian Ministry of Education and Research under Grant SF0690029s09, and Estonian Science Foundation under Grants ETF8170 and ETF8703, by the bilateral program of the Estonian and Hungarian Academies of Sciences under the Contract No. SNK64/2013, by EuroMagNET II under the EU Contract No. 228043, and by HFMLRU/FOM, member of the European Magnetic Field Laboratory.
Author information
Authors and Affiliations
Contributions
D.S., S.B., V.K., I.K., T.R., U.N., H.E. performed the THz experiments. H.M., V.K., Y. Tokunaga, Y. Taguchi synthetized the samples and carried out static magnetoelectric measurements. All the authors contributed to the analysis and discussions of the results. I.K. wrote the manuscript and supervised the project.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Figures 12 (PDF 101 kb)
Rights and permissions
About this article
Cite this article
Kézsmárki, I., Szaller, D., Bordács, S. et al. Oneway transparency of fourcoloured spinwave excitations in multiferroic materials. Nat Commun 5, 3203 (2014). https://doi.org/10.1038/ncomms4203
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/ncomms4203
This article is cited by

Confirming the trilinear form of the optical magnetoelectric effect in the polar honeycomb antiferromagnet Co2Mo3O8
npj Quantum Materials (2022)

Nonreciprocal directional dichroism at telecom wavelengths
npj Quantum Materials (2022)

Electric Dipole Active Magnetic Resonance and Nonreciprocal Directional Dichroism in Magnetoelectric Multiferroic Materials in Terahertz and Millimeter Wave Regions
Applied Magnetic Resonance (2021)

Timedomain terahertz spectroscopy in high magnetic fields
Frontiers of Optoelectronics (2021)

Imaging switchable magnetoelectric quadrupole domains via nonreciprocal linear dichroism
Communications Materials (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.