Introduction

The intense research on magnetoelectric and multiferroic compounds in recent years1,2,3,4,5 has revealed a plethora of novel optical phenomena specific to these materials2,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37. Most of these optical effects can be rooted back to the simultaneous breaking of the time-reversal and the spatial inversion invariance in these compounds due to their coexisting magnetic and electric orderings2,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,25,26,30,33,34,37,38. Perhaps the most exotic optical effect recognized so far in magnetoelectric media is the non-reciprocal directional dichroism6,7,8,10,11,12,13,15,16,17,18,20,21,22,23,25,26,27,29,30,32,33,34,35 when counter-propagating beams with the same initial polarization are transmitted differently through the medium. In terms of light intensity, it is manifested in the so-called directional dichroism39,40, when the magnitude of light absorption is different for the two beams with opposite (±k) propagation vectors. This effect is of relativistic origin and is usually considered to be weak6,7,8,41,42. However, in some compounds, the so-called one-way transparency, i.e., the maximal directional anisotropy, has been achieved in resonance with magnetoelectric excitations17,20,22,23,29,32.

The symmetry-breaking via applied electric and magnetic fields is well-known to have profound effects on the light propagation in solids. In fact, apart from the first experimental demonstration of directional dichroism in the exciton resonances of a polar crystal in magnetic field43, early realizations of the effect were achieved on paraelectric and paramagnetic systems subject to external electric and magnetic fields6,42. As pointed out by Rikken and coworkers, in materials exposed to perpendicular static electric E and magnetic B fields, the refractive index has a term proportional to k ⋅ (E × B), hence, its magnitude is different for light propagation along ±k6. This polarization-independent effect was argued to be of relativistic origin, inherent to every material, since it can be traced back to the usual magnetic linear birefringence/dichroism, also called Cotton–Mouton or Voigt effect, via a Lorentz boost. The magnetic linear birefringence/dichroism, which is quadratic in B, describes the difference in the real/imaginary part of the refractive index for light polarization along and perpendicular to an externally applied magnetic field. In a frame moving with a velocity of v = c ⋅ E × B/B2 with respect to the original frame, there is a static electric field emerging and the optical anisotropy originally ∝ B2 is transformed to the directional optical anisotropy ∝k ⋅ (E × B).

Even without considering the microscopic origin, one can generally argue on a symmetry basis that such a triple-product term can be present in the refractive index. All fundamental interactions, except for the weak interaction, obey separately the time reversal, the spatial inversion, and the charge conjugation symmetries. Therefore, if any of these three operations are simultaneously applied to the light field and the material, the measured refractive index should stay invariant. In fact, this triple-product form fulfills this condition, since either of these three operations reverses two vectors of the triple product: Time reversal switches the sign of k and M, inversion reverses k and P, and charge conjugation reverses P and M. Concerning spatial symmetries, the triple product is an invariant scalar not only with respect to inversion but to all symmetry operations in O(3), as expected for light–matter interaction.

In this first proof-of-concept, experimental studies carried out on paraelectric and or paramagnetic materials, the magnitude of the directional dichroism was found to be of the order of a few percent at most6,42. Since externally induced and spontaneous built-in fields are equivalent in terms of symmetry, directional dichroism of the form δn ∝ k ⋅ (P × M) was expected to emerge in multiferroic compounds with finite crossed polarization (P) and magnetization (M). In addition to materials with finite P and M, directional optical anisotropy can produce contrast between antiferromagnetic domains39, as demonstrated in non-centrosymmetric (antipolar) antiferromagnetic crystals, where the reversal of k is equivalent to the reversal of the antiferromagnetic Néel vector27 or to the inversion of the quadrupole moment of the domain33.

While the trilinear form of the directional anisotropy has not been experimentally demonstrated to the full extent in multiferroics at terahertz (THz) frequencies, the effect was found to be highly amplified by the built-in symmetry-breaking fields, in some cases leading to one-way transparency11,20,22,23,29. In an attempt to prove the non-reciprocal nature of light propagation, parts of the trilinear expression were verified recently in various multiferroic or magnetoelectric crystals, by showing that δn changes sign upon the flipping of M6,7,8,11,21,22,29,42, k23, M or P12,14,15, M or k20. The only study, investigating the effect of the one-by-one reversal of all the three vectors and demonstrating the triple-product form of the optical magnetoelectric effect on a purely experimental basis, was carried out on the type-II multiferroic Ba2Mg2Fe12O22 in the microwave (GHz) regime using coplanar waveguides 26. In our study, we demonstrate that the triple-product form of the optical magnetoelectric effect also holds when freely propagating electromagnetic waves in the THz regime are brought into interaction with a multiferroic material.

For this purpose, we chose a type-I multiferroic, more specifically the polar easy-axis antiferromagnet Co2Mo3O844, as a benchmark material to systematically test the triple-product form, k ⋅ (P × M), of the directional optical anisotropy. Co2Mo3O8 belongs to the family of transition-metal molybdenum oxides M2Mo3O8 (M = Mn, Fe, Co) with a hexagonal structure in the polar space group P63mc45,46,47,48,49,50,51 (see Fig. 1 for the crystal structure). The Co2+ ions, located in both octahedrally and tetrahedrally coordinated sites, are responsible for the magnetism, as the Mo4+ ions from non-magnetic trimers46,52,53. Due to its polar crystal structure, Co2Mo3O8 has a pyroelectric polarization P along the c-axis and below TN = 40 K it undergoes a transition to a four-sublattice easy-axis collinear antiferromagnetic state with the Néel vector L pointing along the c-axis [see Fig. 1a]44,47,50,51. Similar to other members of the M2Mo3O8 family53,54,55, the polarization of Co2Mo3O8 is also affected by the magnetic ordering, implying a remarkable magnetoelectric coupling44. When the magnetic field is applied perpendicular to the c axis, a sizable magnetization can be induced via the canting of the spins away from the c axis, as sketched in Fig. 1(b). This state with perpendicular P and M is expected to exhibit directional anisotropy12, with different refractive indices for beams traveling along the ± k direction, as depicted in Fig. 1c.

Fig. 1: Lattice structure and directional dichroism.
figure 1

a Crystallographic structure of Co2Mo3O8, with ferroelectric polarization P along the c-axis. The magnetic Co2+ ions are located in both tetrahedral (blue) and octahedral (green) oxygen coordination. The antiferromagnetic spin arrangement is indicated by the arrows. b A magnetization M is induced for magnetic field H⊥c by a canting of the spins. (c) Illustration of the triple product k ⋅ (P × M) configurations leading to directional dichroism.

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Similar to the cases of the antiferromagnetic Fe2Mo3O824, and ferrimagnetic Zn-doped Fe2Mo3O825,56 and Mn2Mo3O857, in the low-energy range (<16 meV) we observe two strong magnetic excitations in zero fields both in the terahertz absorption and the inelastic neutron scattering (INS) data. In contrast to the sister compounds, both of these modes in Co2Mo3O8 are doubly degenerate and show a V-shape splitting in magnetic fields applied along the c-axis, as expected for a four-sublattice easy-axis collinear antiferromagnet. When Co2Mo3O8 is magnetized perpendicular to the c-axis, i.e., perpendicular to its pyroelectric polarization, these modes exhibit strong directional dichroism for light beams propagating perpendicular to both P and M. We systematically demonstrate via the sequential change of k, P, and M that the observed directional dichroism corresponds to a trilinear term in the refractive index, δn ∝ k ⋅ (P × M). Besides these dominant features, additional weaker excitations are resolved in the antiferromagnetic state by THz transmission spectroscopy.

Results and discussion

Dispersion of the magnetic excitations

According to linear spin-wave theory, in a simple collinear antiferromagnet, the number of magnon modes at each wavevector is expected to be equal to the number of magnetic sublattices. Given that Co2Mo3O8 realizes a four sublattice collinear antiferromagnetic order below TN, four modes are expected, which form two doublets in zero field.

For the first time in the M2Mo3O8 crystal family (where M stands for a metal ion), the magnon dispersion of Co2Mo3O8 was measured in a zero magnetic field. The results along the (h, 0, 0) and (−1, 0, l) directions, obtained by energy scans in INS, are shown in Fig. 2a, b, respectively. As expected for a four-sublattice antiferromagnetic order, below TN in zero field two intense modes are observed for both directions, which are located around 5 and 10.5 meV in the zone center. Along the (h, 0, 0)-line, both branches show a clear dispersion, while no dispersion can be resolved along the (−1, 0, l)-line. This indicates that the antiferromagnetic exchange between Co sites in the same honeycomb layer is the dominant magnetic interaction and interlayer exchange coupling is much weaker, as also found for Mn2Mo3O857.

Fig. 2: Comparison of magnon dispersion and THz spectra.
figure 2

a, b Dispersion of magnon modes along the (h, 0, 0)/(−1, 0, l) direction, as determined from inelastic neutron scattering. Symbols indicate the mode energies for the particular energy scans. The color coding represents the normalized neutron count. c Absorption spectrum with absorption coefficient α measured at 2 K in zero magnetic field for light polarization Eω∥c & Hω∥a. For direct comparison of the zone-center energies, as obtained by the two techniques, all frames share a common vertical scale, with energy/frequency units displayed on the left/right axis.

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Figure 2 provides a direct comparison between the energy of the two magnon modes resolved by INS and the energy of the modes observed in the THz absorption spectra in the zone center for light polarization Eω∥c & Hω∥a. The two modes found in the inelastic neutron data show up as the strongest features in the absorption spectrum, centered at 41 cm−1 ≈ 5.1 meV and 85 cm−1 ≈ 10.6 meV. However, a closer inspection of the THz spectra reveals additional weaker modes at 71 and 119 cm−1. Please note that the considerably lower energy resolution of the neutron scattering experiment does not allow making a statement on the sharp and weak mode at 71 cm−1, while the energy range of the other mode is uncovered by the current neutron study. Concerning the origin of the additional modes observed in the THz measurements, these may be identified either with low-lying transitions of primarily orbital character or with spin-stretching modes present in anisotropic spin systems with S > 1/258.

Field evolution and selection rules of the modes

The magnetic field dependence of the absorption spectrum is shown in Fig. 3a for Eω∥c & Hω∥a in static magnetic fields applied parallel (red curves) and perpendicular (blue curves) to the c axis. For H∥c, all modes except the one at 71 cm−1 show a V-shape splitting with an increasing magnetic field. From the shift of the mode frequencies, which is linearly proportional to the strength of the magnetic field, the effective g-factors of the different modes are determined and found to vary over an unusually wide range from 0 to 5.6 (see Table 1). In contrast, for magnetic fields applied perpendicular to the c axis, no splitting of the modes could be observed.

Fig. 3: THz absorption spectra in zero and finite magnetic fields.
figure 3

a Absorption coefficient, α, spectra measured with light polarization Eω∥c & Hω∥a at 2 K in magnetic fields H∥c (red curves) and H∥a (blue curves) up to 7 T. Spectra are shifted vertically by an offset in proportion to the magnetic field (150 cm−1/T) for clarity. b–d Zero-field absorption coefficient spectra measured with the three orthogonal light polarizations at 2 K. Modes are labeled in the same way as in Table 1.

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Table 1 Excitations observed in Co2Mo3O8 below TN.
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These observations imply that the ordered moments on all sublattices, which are co-aligned with the c-axis in zero field, acquire a canting along the magnetic field, when it is applied perpendicular to the c-axis. This leads to a uniform magnetization perpendicular to the built-in polarization, as schematically shown in Fig. 1b. In addition to measurements performed with polarization Eω∥c & Hω∥a, the absorption spectra were also studied in the orthogonal polarization configuration in zero field, as shown in Fig. 3b–d. Note that besides sharp resonances the absorption spectra for Eω∥a exhibit a gradual increase toward larger wavenumbers, which we assign to the low-energy tail of the lowest-lying optical phonon. A similar feature has been reported in this frequency range for the isostructural compound Fe1.86Zn0.14Mo3O8 and modeled accordingly56 using the phonon eigenfrequency of approx. 130 cm−1 observed for pure Fe2Mo3O859,60. The spectra in Fig. 3b, c reveal an additional resonance, mode E, at 80 cm−1, the strength of which is independent of the orientation of Hω, but the mode is only active for Eω∥a. Such an only electric-dipole active mode is not expected to exhibit directional dichroism17. In contrast, the directional dichroism observed for the other four resonances demonstrates unambiguously that ME1 − ME4 are magnetoelectric excitations, which are both electric- and magnetic-dipole active. The selection rules as well as the g-factors of the modes for H∥c are listed in Table 1.

Directional dichroism

According to linear-response theory, the directional dichroism at long wavelengths, i.e., in the THz or far-infrared range covered here, originates from the dynamic magnetoelectric effect18. In the present case, Co2Mo3O8 has a spontaneous polarization P∥c and by applying the external magnetic field H⊥c, we magnetize the material perpendicular to the c axis, M∥a. This reduces the \(6^{\prime} m^{\prime} m\) hexagonal magnetic point symmetry of the antiferromagnetic state54 to \(2^{\prime} m^{\prime} m\) orthorhombic symmetry and activates the time-reversal odd \({\chi }_{ac}^{\prime}\) and \({\chi }_{ca}^{\prime}\) components in the magnetoelectric tensor12. In this case, the relation δn ∝ k ⋅ (P × M) predicts directional dichroism for light beams propagating along and opposite to the b axis, being perpendicular to both the a and c axes. Correspondingly, the solution of the Maxwell equations yields four different values for the refractive index n, two for each of the two orthogonal linear polarization configurations12:

$${n}_{ac}^{\pm {k}_{b}}\approx \sqrt{{\epsilon }_{aa}^{\prime}{\mu }_{cc}^{\prime}}\mp {\chi }_{ca}^{\prime},$$
(1)
$${n}_{ca}^{\pm {k}_{b}}\approx \sqrt{{\epsilon }_{cc}^{\prime}{\mu }_{aa}^{\prime}}\pm {\chi }_{ac}^{\prime}.$$
(2)

Here, \({n}_{ac}^{\pm {k}_{b}}\) and \({n}_{ca}^{\pm {k}_{b}}\) denote the refractive indices for light polarizations Eω∥a & Hω∥c and Eω∥c & Hω∥a, respectively. States propagating in opposite directions are indicated by ±kb. \({\epsilon }_{ij}^{\prime}\) and \({\mu }_{ij}^{\prime}\) are time-reversal invariant components of the electric permittivity and magnetic permeability tensors, respectively. In the following, we discuss the directional dichroism in these two configurations and sequentially reverse k, P, and M, in order to prove the trilinear form of the optical magnetoelectric effect.

Figure 4a, b shows the absorption spectra at 2 K for the two polarization configurations measured in positive (red) and negative (blue) magnetic fields H∥a. For the polarization Eω∥a & Hω∥c, directional dichroism shows up for the ME1 mode at 41 cm−1 at finite magnetic fields. The absorption strength varies linearly with the magnetic field (or the induced transverse magnetization), i.e., it is enhanced and reduced for positive and negative magnetic fields, respectively. Notably, in −7 T the absorption of this resonance is almost fully suppressed and one-way transparency is nearly achieved, as will be directly evidenced in the following by the reversal of the propagation direction.

Fig. 4: Directional dichroism via inversion of the magnetic field.
figure 4

Magnetic field dependence (H∥a) of the absorption spectra at 2 K for polarization a Hω∥c & Eω∥a and b Hω∥a & Eω∥c. Red and blue curves correspond to positive and negative magnetic fields, respectively. c Magnetic field dependence of the peak value of the imaginary part of \({\chi }_{{{{{{\rm{me}}}}}}}^{\prime}\) for the modes ME1 and ME3. Note that the maximum absorption peak for the latter is not fully resolved due to its high absorption strength, as indicated by dashed lines.

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For the orthogonal light polarization Eω∥c & Hω∥a the absorption strength of the ME1 mode at 41 cm−1 is independent of the magnetic field, while for the modes ME2, ME3, ME4 at 71, 85, and 119 cm−1, respectively, the absorption coefficient shows a field dependence. However, in this case, it is reduced for positive and enhanced for negative fields. According to Eqs. (1) and (2), this leads to the same sign of \({\chi }_{ca}^{\prime}\) and \({\chi }_{ac}^{\prime}\), generating the directional dichroism in the first and the second polarization configuration, respectively.

The spectra of the magnetoelectric coefficients can be calculated from the absorption difference between oppositely magnetized states of the material according to

$${{{{{\rm{Im}}}}}}\{\chi ^{\prime} (\omega )\}\,=\frac{c}{4\omega }{{\Delta }}{\alpha }_{M}(\omega ) \,= \, \frac{c}{4\omega }\left\{\alpha (\omega )[+{{{{{\boldsymbol{k}}}}}},+{{{{{\boldsymbol{P}}}}}},+{{{{{\boldsymbol{M}}}}}}]\right.\\ \, -\left.\alpha (\omega )[+{{{{{\boldsymbol{k}}}}}},+{{{{{\boldsymbol{P}}}}}},-{{{{{\boldsymbol{M}}}}}}]\right\}.$$
(3)

The imaginary part of the dimensionless magnetoelectric coefficient spectrum \({\chi }_{ca}^{\prime}\) has a well-resolved maximum at 41 cm−1, which follows a linear field dependence, as shown in Fig. 4c. For the mode at 85 cm−1, the absorption peak is not properly resolved due to the high absorption, thus the foot of this absorption peak was fitted by a Lorentzian for both field directions to estimate the value of \({\chi }_{ac}^{\prime}\) at the resonance. The corresponding values of \({{{{{\rm{Im}}}}}}\{{\chi }_{ac}^{\prime}\}\), also plotted in Fig. 4c, follow a linear field dependence similar to \({{{{{\rm{Im}}}}}}\{{\chi }_{ca}^{\prime}\}\).

In order to prove that the term in the refractive index is also an odd function of P as well as k, we study the absorption coefficient upon the reversal of either the polarization of the material or the light propagation direction. Since the material is pyroelectric, the polarization P cannot be reversed by applying an electric field, instead of by rotating the sample by 180° about the a-axis. The reversal of the propagation direction k is conveniently done, without any rearrangement of the rest of the optical path, by exchanging THz emitter and receiver in our fiber-coupled Teraflash device. Both of these operations can be performed independently of each other and of the sign change of the magnetic field.

The three absorption difference spectra ΔαM(ω), ΔαP(ω) and Δαk(ω) are calculated based on experiments carried out at 2 K with H∥a and μ0∣H∣ = 7 T according to

$${{\Delta }}{\alpha }_{M}(\omega )=\alpha (\omega )[+{{{{{\boldsymbol{k}}}}}},+{{{{{\boldsymbol{P}}}}}},+{{{{{\boldsymbol{M}}}}}}]-\alpha (\omega )[+{{{{{\boldsymbol{k}}}}}},+{{{{{\boldsymbol{P}}}}}},-{{{{{\boldsymbol{M}}}}}}],$$
(4)
$${{\Delta }}{\alpha }_{P}(\omega )=\alpha (\omega )[+{{{{{\boldsymbol{k}}}}}},+{{{{{\boldsymbol{P}}}}}},+{{{{{\boldsymbol{M}}}}}}]-\alpha (\omega )[+{{{{{\boldsymbol{k}}}}}},-{{{{{\boldsymbol{P}}}}}},+{{{{{\boldsymbol{M}}}}}}],$$
(5)
$${{\Delta }}{\alpha }_{k}(\omega )=\alpha (\omega )[+{{{{{\boldsymbol{k}}}}}},+{{{{{\boldsymbol{P}}}}}},+{{{{{\boldsymbol{M}}}}}}]-\alpha (\omega )[-{{{{{\boldsymbol{k}}}}}},+{{{{{\boldsymbol{P}}}}}},+{{{{{\boldsymbol{M}}}}}}].$$
(6)

The corresponding spectra are shown in Fig. 5 for both polarization directions. The inset in the middle of the figure is a graphical illustration of how the three absorption differences are calculated according to the three equations above.

Fig. 5: Absorption difference spectra due to directional dichroism.
figure 5

a Absorption spectra measured at 2 K and zero magnetic field for polarization Hω∥c & Eω∥a. b Absorption difference spectra ΔαH, Δαk, and ΔαP for the same polarization in ∣H∣ = 7 T for H∥a. For clarity, curves are shifted vertically by constant offsets of 100 cm−1. c, d Corresponding set of data for the perpendicular polarization direction Hω∥a & Eω∥c. Dashed lines across the ME3 mode at 85 cm−1 indicate that due to the high absorption strength the peak maximum could not be fully resolved. The arrows represent the orientations of k, P, and M, following the color scheme of Fig. 1c.

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The three absorption difference spectra ΔαM, ΔαP, and Δαk for all magnetoelectric modes show very good agreement and nicely coincide for modes ME1 and ME3, as expected if the refractive index contains a term δn ∝ k ⋅ (P × M). The observation that the agreement for mode ME4 is not as good as for the other modes is likely explained by the vicinity of this mode to the high-frequency cutoff of our experimental setup, which leads to a lower signal-to-noise ratio and stability in that spectral range and, thus, a poorer reproducibility in the ±k or ±P experiments. Remarkably, for the polarization Eω∥a & Hω∥c the absorption difference in 7 T, displayed in Fig. 5b, is larger than the total absorption in zero field [Fig. 5a] for the ME1 mode. In the Eω∥c & Hω∥a polarization configuration, the absorption difference in 7 T, plotted in Fig. 5d, for ME2, ME3, and ME4 is close to or larger than the absorption in zero field [Fig. 5c]. Specific to ME3, the absorption peak is too strong to be fully resolved in 7 T. By fitting the flanks of Δα spectra with a Lorentzian line (not shown here) yields the maximum absorption difference Δα ≈ 200 cm−1, a value nearly twice as large as the absorption in zero field. We want to point out that out of the four magnetoelectric resonances one-way transparency17 is most closely realized for mode ME1, where the resonant absorption is almost completely suppressed in −7 T, as seen in Fig. 4a.

To summarize, by independent reversal of magnetization, polarization, and light propagation, all three possibilities of realizing directional dichroism have been demonstrated for spin-wave excitations in the THz or far-infrared spectral range. Our THz spectroscopy results, performed on the collinear antiferromagnet Co2Mo3O8 with orthogonal magnetization and polarization, directly confirm the trilinear-product form of the optical magnetoelectric effect k ⋅ (P × M). So far this product form has been confirmed on a purely experimental basis only for the propagation of GHz frequency waveguide modes in Ba2Mg2Fe12O2226. This compound is type-II multiferroic with a transverse conical structure, which has a long (~20 nm) periodicity61 and soft collective spin excitations in the sub-20 GHz range. In contrast, Co2Mo3O8 is a type-I multiferroic with a simple collinear Néel order and considerably harder antiferromagnetic magnons located in the THz regime. The distinct static and dynamic properties of these two compounds and the different spectral ranges investigated in the two studies imply that non-reciprocal light propagation in multiferroics with orthogonal P and M is generally governed by the optical magnetoelectric effect of trilinear form. The simple optical scheme and the well-defined geometry used in the present study, i.e., freely propagating light waves transmitted through plane-parallel samples, allow for precise quantification of the non-reciprocal effect. On this basis, we conclude that some of the spin-wave excitations in Co2Mo3O8 exhibit strong directional dichroism, with one of the excitations being very close to the one-way-transparency limit.

Methods

Synthesis

Single crystals were grown by the chemical transport reaction method at temperatures between 900 and 950 °C using anhydrous TeCl4 as the source of the transport agent. X-ray analysis of the crushed single crystals revealed a single phase composition with a hexagonal symmetry and space group P63mc. The room-temperature lattice constants a = b = 5.7677(1) Å and c = 9.9097(2) Å are close to the data reported in literature47,51. Single crystals were characterized by magnetometry and specific heat measurements, which confirmed the onset of long-range antiferromagnetic order at TN = 40 K.

THz spectroscopy

Temperature and magnetic field-dependent time-domain THz spectroscopy measurements were performed on plane parallel ab- and ac-cut single crystals of Co2Mo3O8 in transmission configuration. For the optical measurements, a Toptica TeraFlash time-domain THz spectrometer was used in combination with a superconducting magnet, which allows for measurements at temperatures down to 2 K and in magnetic fields up to ±7 T.

Inelastic neutron scattering

INS was measured on two co-aligned crystals with a total weight of 800 mg using the thermal neutron triple-axis spectrometer EIGER at the SINQ, Paul Scherrer Institut, Switzerland. The use of a double-focusing PG(002) monochromator and analyzer gave the energy resolution 0.7 meV at the elastic line. The final wave vector kf = 2.66 Å−1 was filtered by a PG filter. The sample was mounted in a ILL cryostat with the ac-plane horizontal, which gave access to excitations along the principal directions (h, 0, 0) and (−1, 0, l).