# Confirming the trilinear form of the optical magnetoelectric effect
in the polar honeycomb antiferromagnet Co2Mo3O8

S. Reschke  Experimental Physics V, Center for Electronic Correlations and
Magnetism, Institute for Physics, Augsburg University, D-86135 Augsburg,
Germany  TOPTICA Photonics AG, Lochhamer Schlag 19, 82166 Gräfelfing, Germany
D. G. Farkas  Department of Physics, Budapest University of Technology and
Economics, 1111 Budapest, Hungary  Condensed Matter Research Group of the
Hungarian Academy of Sciences, 1111 Budapest, Hungary  A. Strinić  S. Ghara
Experimental Physics V, Center for Electronic Correlations and Magnetism,
Institute for Physics, Augsburg University, D-86135 Augsburg, Germany  K.
Guratinder  O. Zaharko  Laboratory for Neutron Scattering and Imaging, Paul
Scherrer Institut, CH-5232 Villigen PSI, Switzerland  L. Prodan  V. Tsurkan
Experimental Physics V, Center for Electronic Correlations and Magnetism,
Institute for Physics, Augsburg University, D-86135 Augsburg, Germany
Institute of Applied Physics, MD-2028 Chişinău, Republic of Moldova  D.
Szaller  Institute of Solid State Physics, TU Wien, 1040 Vienna, Austria  S.
Bordács  Department of Physics, Budapest University of Technology and
Economics, 1111 Budapest, Hungary  J. Deisenhofer  I. Kézsmárki  Experimental
Physics V, Center for Electronic Correlations and Magnetism, Institute for
Physics, Augsburg University, D-86135 Augsburg, Germany

###### Abstract

Magnetoelectric phenomena are intimately linked to relativistic effects and
also require the material to break spatial inversion symmetry and time
reversal invariance. Magnetoelectric coupling can substantially affect light-
matter interaction and lead to non-reciprocal light propagation. Here, we
confirm on a fully experimental basis, without invoking either symmetry-based
or material-specific assumptions, that the optical magnetoelectric effect in
materials with non-parallel magnetization ($\bm{M}$) and electric polarization
($\bm{P}$) generates a trilinear term in the refractive index, $\delta
n\propto\bm{k}\cdot(\bm{P}\times\bm{M})$, where $\bm{k}$ is the propagation
vector of light. Its sharp magnetoelectric resonances, that are simultaneously
electric and magnetic dipole active excitations, make Co2Mo3O8 an ideal
compound to demonstrate this fundamental relation via independent variation of
$\bm{M}$, $\bm{P}$ and $\bm{k}$. Remarkably, the material shows almost perfect
one-way transparency in moderate magnetic fields at some of the
magnetoelectric resonances.

## Introduction

The intense research on magnetoelectric and multiferroic compounds in recent
years [1, 2, 3, 4, 5] has revealed a plethora of novel optical phenomena
specific to these materials [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 2, 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 orderings [7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 2, 38, 30, 33,
34, 37]. Perhaps the most exotic optical effect recognized so far in
magnetoelectric media is the non-reciprocal directional dichroism [6, 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 dichroism [39, 40], when the magnitude
of light absorption is different for the two beams with opposite
($\pm\mathbf{k}$) propagation vectors. This effect is of relativistic origin
and usually considered to be weak [41, 6, 7, 8, 42]. However, in some
compounds the so-called one-way transparency, i.e. the maximal directional
anisotropy, has been achieved in resonance with magnetoelectric excitations
[17, 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 field [43], early
realizations of the effect were achieved on paraelectric and paramagnetic
systems subject to external electric and magnetic fields [6, 42]. As pointed
out by Rikken and coworkers, in materials exposed to perpendicular static
electric $\bm{E}$ and magnetic $\bm{B}$ fields, the refractive index has a
term proportional to $\bm{k}\cdot(\bm{E}\times\bm{B})$, hence, its magnitude
is different for light propagation along $\pm\bm{k}$ [6]. 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, that is quadratic
in $B$, describes the difference in the real/imaginary part of the refractive
index for light polarizations along and perpendicular to an externally applied
magnetic field. In a frame moving with a velocity of
$\bm{v}=c\cdot\bm{E}\times\bm{B}/B^{2}$ with respect to the original frame,
there is a static electric field emerging and the optical anisotropy
originally $\propto B^{2}$ is transformed to the directional optical
anisotropy $\propto\bm{k}\cdot(\bm{E}\times\bm{B})$.

Even without considering the microscopic origin, one can generally argue on
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 reverse two
vectors of the triple product: Time reversal switches the sign of $\bm{k}$ and
$\bm{M}$, inversion reverses $\bm{k}$ and $\bm{P}$, and charge conjugation
reverses $\bm{P}$ and $\bm{M}$. 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 these 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 percents at most [6, 42].
Since externally induced and spontaneous built-in fields are equivalent in
terms of symmetry, directional dichroism of the form $\delta
n\propto\bm{k}\cdot(\bm{P}\times\bm{M})$ 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
domains [39], as demonstrated in non-centrosymmetric (antipolar)
antiferromagnetic crystals, where the reversal of $\bm{k}$ is equivalent to
the reversal of the antiferromagnetic Néel vector [27] or to the inversion of
the quadrupole moment of the domain [33].

While the trilinear form of the directional anisotropy has not been
experimentally demonstrated in full extent in multiferroics, the effect was
found to be highly amplified by the built-in symmetry breaking fields, in some
cases leading to one-way transparency [11, 20, 22, 23, 29]. Parts of the
trilinear expression were verified recently in various multiferroic or
magnetoelectric crystals, by showing that $\delta n$ changes sign upon the
flipping of $\bm{M}$ [6, 7, 8, 42, 11, 21, 22, 29], $\bm{k}$ [23], $\bm{M}$
and/or $\bm{P}$ [12, 14, 15], $\bm{M}$ and/or $\bm{k}$ [20]. The only study
investigating the effect of the one-by-one reversal of all the three vectors
was carried out on Ba2Mg2Fe12O22, where the triple-product form of the optical
magnetoelectric effect in the microwave region was rigorously demonstrated on
experimental basis [26].

Here, we choose the polar easy-axis antiferromagnet Co2Mo3O8 [44] as a
benchmark material to systematically test the triple-product form,
$\bm{k}\cdot(\bm{P}\times\bm{M})$, of the directional optical anisotropy.
Co2Mo3O8 belongs to the family of type-I multiferroic transition-metal
molybdenum oxides M2Mo3O8 (M = Mn, Fe, Co) with a hexagonal structure in the
polar space group $P6_{3}mc$ [45, 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 form non-magnetic trimers [52, 46, 53]. Due to its polar crystal
structure Co2Mo3O8 has a pyroelectric polarization $\bm{P}$ along the $c$-axis
and below $T_{\mathrm{N}}{}=40$ K it undergoes a transition to a four-
sublattice easy-axis collinear antiferromagnetic state with the Néel vector
$\bm{L}$ pointing along the $c$-axis [see Fig. 1(a)] [47, 50, 51, 44].
Similarly to other members of the M2Mo3O8 family [54, 53, 55], the
polarization of Co2Mo3O8 is also affected by the magnetic ordering, implying a
remarkable magnetoelectric coupling [44]. 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 $\bm{P}$ and $\bm{M}$ is expected to exhibit
directional anisotropy[12], with different refractive indices for beams
travelling along the $\pm\bm{k}$ direction, as depicted in 1(c).

Similarly to the cases of the antiferromagnetic Fe2Mo3O8 [24], and
ferrimagnetic Zn-doped Fe2Mo3O8 [25, 56] and Mn2Mo3O8 [57], in the low-energy
range ($<$16 meV) we observe two strong magnetic excitations in zero field
both in the terahertz absorption and the inelastic neutron scattering 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 a
strong directional dichroism for light beams propagating perpendicular to both
$\bm{P}$ and $\bm{M}$. 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, $\delta
n\propto\bm{k}\cdot(\bm{P}\times\bm{M})$. Besides these dominant features,
additional weaker excitations are resolved in the antiferromagnetic state by
terahertz (THz) spectroscopy.

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 $\bm{H}\perp c$ by a canting
of the spins. (c) Illustration of the triple product
$\bm{k}\cdot(\bm{P}\times\bm{M})$ configurations leading to directional
dichroism.

## Results

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 $T_{N}$, four modes are expected,
which form two doublets in zero field.

For the first time in the $M_{2}$Mo3O8 crystal family (where $M$ stands for a
metal ion), the magnon dispersion of Co2Mo3O8 was measured in zero magnetic
field. The results along the $(h,0,0)$ and $(-1,0,l)$ directions, obtained by
energy scans in inelastic neutron scattering, are shown in Fig. 2(a) and (b),
respectively. As expected for a four-sublattice antiferromagnetic order, below
$T_{\mathrm{N}}$ 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 inter-layer exchange coupling is much
weaker, as also found for Mn2Mo3O8[57].

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 colour coding
represents the normalized neutron count. (c) Absorption spectrum with
absorption coefficient $\alpha$ measured at 2 K in zero magnetic field for
light polarization $\bm{E}^{\omega}\parallel c$ & $\bm{H}^{\omega}\parallel
a$. 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.

Figure 2 provides a direct comparison between the energy of the two magnon
modes resolved by inelastic neutron scattering and the energy of the modes
observed in the THz absorption spectra in the zone center for light
polarization $\bm{E}^{\omega}\parallel c$ & $\bm{H}^{\omega}\parallel a$. The
two modes found in the inelastic neutron data show up as the strongest
features in the absorption spectrum, centered at 41 cm${}^{-1}\approx 5.1$ meV
and 85 cm${}^{-1}\approx 10.6$ meV. However, a closer inspection of the THz
spectra reveals additional weaker modes at 71 cm-1 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 $S>1/2$ [58].

Figure 3:  (a) Absorption coefficient, $\alpha$, spectra measured with light
polarization $\bm{E}^{\omega}\parallel c$ & $\bm{H}^{\omega}\parallel a$ at 2
K in magnetic fields $\bm{H}\parallel c$ (red curves) and $\bm{H}\parallel a$
(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 labelled in the same way as in Table 1.

Field evolution and selection rules of the modes
The magnetic field dependence of the absorption spectrum is shown in Fig. 3(a)
for $\bm{E}^{\omega}\parallel c$ & $\bm{H}^{\omega}\parallel a$ in static
magnetic fields applied parallel (red curves) and perpendicular (blue curves)
to the $c$ axis. For $\bm{H}\parallel c$, all modes except the one at 71 cm-1
show a V-shape splitting with 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. This observation implies
that the ordered moments have zero projections within the $ab$ plane in the
studied magnetic field range, again supporting the easy-axis character of the
antiferromagnetic ground state. In addition to measurements performed with
polarization $\bm{E}^{\omega}\parallel c$ & $\bm{H}^{\omega}\parallel a$, the
absorption spectra were also studied in the orthogonal polarization
configuration in zero field, as shown in Fig. 3(a). The spectra in Fig. 3(a)
reveal an additional resonance $E$ at 80 cm-1, the strength of which is
independent of the orientation of $\bm{H}^{\omega}$, but the mode is only
active for $\bm{E}^{\omega}\parallel a$. Such an only electric-dipole active
mode is not expected to exhibit directional dichroism [17]. In contrast, the
directional dichroism observed for the other four resonances demonstrates
unambiguously that $ME_{1}-ME_{4}$ are magnetoelectric excitations, which are
both electric- and magnetic-dipole active. The selection rules as well as the
$g$-factors of the modes for $\bm{H}\parallel c$ are listed in Table 1.

mode | $\bm{\omega_{0}}$ | $g$ | polarization | ME activity
---|---|---|---|---
| $[\mathrm{cm}^{-1}]$ |  |  $H^{\omega}_{b}$–$E^{\omega}_{a}$ |  $H^{\omega}_{c}$–$E^{\omega}_{a}$ |  $H^{\omega}_{a}$–$E^{\omega}_{c}$ |
$ME_{1}$ | 41 | 1.6 | $\checkmark$ | $\checkmark$ | $\checkmark$ |  $H^{\omega}_{c}$–$E^{\omega}_{a}$
$ME_{2}$ | 71 | 0.0 | $\checkmark$ | $\times$ | $\checkmark$ |  $H^{\omega}_{a}$–$E^{\omega}_{c}$
$E$ | 80 | 3.3 | $\checkmark$ | $\checkmark$ | $\times$ | —
$ME_{3}$ | 85 | 5.6 | $\checkmark$ | $\times$ | $\checkmark$ |  $H^{\omega}_{a}$–$E^{\omega}_{c}$
$ME_{4}$ | 119 | 3.7 | $\checkmark$ | n.r. | $\checkmark$ |  $H^{\omega}_{a}$–$E^{\omega}_{c}$
Table 1:  Excitations observed in Co2Mo3O8 below $T_{N}$. For each mode, we
list the resonance frequency $\omega_{0}$ and the effective $g$-factor for
$\bm{H}\parallel c$, and specify in which polarization configurations the mode
is observed/silent ($\checkmark$/$\times$) in zero field. Mode $ME_{4}$ at 119
cm-1 was not resolved (n.r.) in one polarization configuration because of a
low-lying phonon mode with high absorption in the corresponding spectral
range. The last column refers to the polarization configuration, where the
directional dichroism is observed. Figure 4:  Magnetic field dependence
($\bm{H}\parallel a$) of the absorption spectra at 2 K for polarization (a)
$\bm{H}^{\omega}\parallel c$ & $\bm{E}^{\omega}\parallel a$ and (b)
$\bm{H}^{\omega}\parallel a$ & $\bm{E}^{\omega}\parallel c$. 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
$\chi^{\prime}_{\mathrm{me}}$ for the modes $ME_{1}$ and $ME_{3}$. Note that
the maximum of the absorption peak for the latter is not fully resolved due to
its highly increased absorption strength of this mode, as indicated by dashed
lines.  Figure 5:  (a) Absorption spectra measured at 2 K and zero magnetic
field for polarization $\bm{H}^{\omega}\parallel c$ &
$\bm{E}^{\omega}\parallel a$. (b) Absorption difference spectra
$\Delta\alpha_{H}$, $\Delta\alpha_{k}$ and $\Delta\alpha_{P}$ for the same
polarization in $|H|=7$ T$\parallel a$. For clarity, curves are shifted
vertically by constant offsets of 100 cm-1. (c),(d) Same set of data for the
perpendicular polarization direction $\bm{H}^{\omega}\parallel a$ &
$\bm{E}^{\omega}\parallel c$ . Dashed lines across the 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 $\bm{k}$, $\bm{P}$
and $\bm{M}$ , following the color scheme of Fig. 1(c).

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 effect [18]. In the present case, Co2Mo3O8
has a spontaneous polarization $\bm{P}\parallel c$ and by applying the
external magnetic field $\bm{H}\perp c$, we magnetize the material
perpendicular to the $c$ axis, $\bm{M}\parallel a$. This reduces the
$6^{\prime}m^{\prime}m$ hexagonal magnetic point symmetry of the
antiferromagnetic state [54] to $2^{\prime}m^{\prime}m$ orthorhombic symmetry
and activates the time-reversal odd $\chi^{\prime}_{ac}$ and
$\chi^{\prime}_{ca}$ components in the magnetoelectric tensor [12]. In this
case, the relation $\delta n\propto\bm{k}\cdot(\bm{P}\times\bm{M})$ 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
configurations [12]:

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

Here $n^{\pm k_{b}}_{ac}$ and $n^{\pm k_{b}}_{ca}$ denote the refractive
indices for light polarizations $\bm{E}^{\omega}\parallel a$ &
$\bm{H}^{\omega}\parallel c$ and $\bm{E}^{\omega}\parallel c$ &
$\bm{H}^{\omega}\parallel a$, respectively. States propagating in opposite
directions are indicated by $\pm k_{b}$. $\epsilon^{\prime}_{ij}$ and
$\mu^{\prime}_{ij}$ 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 $\bm{k}$, $\bm{P}$ and $\bm{M}$, in order to prove
the trilinear form of the optical magnetoelectric effect.

Figures 4(a) and (b) show the absorption spectra at 2 K for the two
polarization configurations measured in positive (red) and negative (blue)
magnetic fields $\bm{H}\parallel a$. For the polarization
$\bm{E}^{\omega}\parallel a$ & $\bm{H}^{\omega}\parallel c$, directional
dichroism shows up for the $ME_{1}$ mode at 41 cm-1 at finite magnetic fields.
The absorption strength varies linearly with 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.

For the orthogonal light polarization $\bm{E}^{\omega}\parallel c$ &
$\bm{H}^{\omega}\parallel a$, the absorption strength of the $ME_{1}$ mode at
41 cm-1 is independent of the magnetic field, while for the modes $ME_{2}$,
$ME_{3}$, $ME_{4}$ 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 $\chi^{\prime}_{ca}$ and $\chi^{\prime}_{ac}$,
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

$\displaystyle\Im\\{\chi\prime(\omega)\\}=\frac{c}{4\omega}\Delta\alpha_{M}(\omega)$
$\displaystyle=\frac{c}{4\omega}\\{\alpha(\omega)[+\bm{k},+\bm{P},+\bm{M}]-\alpha(\omega)[+\bm{k},+\bm{P},-\bm{M}]\\}.$

The imaginary part of the dimensionless magnetoelectric coefficient spectrum
$\chi^{\prime}_{ca}$ has a well-resolved maximum at 41 cm-1, which follows a
linear field dependence, as in the inset of Fig. 4(a). 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 $\chi^{\prime}_{ac}$ at the
resonance. The corresponding values of $\Im\\{\chi^{\prime}_{ac}$}, also
plotted in the inset of Fig. 4(a), follow a linear field dependence similar to
$\Im\\{\chi^{\prime}_{ca}$}.

In order to prove that the term in the refractive index is also an odd
function of $\bm{P}$ as well as $\bm{k}$, 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
$\bm{P}$ cannot be reversed by applying an electric field, instead by rotating
the sample by 180∘ about the $a$ axis. The reversal of the propagation
direction $\bm{k}$ 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 $\Delta\alpha_{M}(\omega)$,
$\Delta\alpha_{P}(\omega)$ and $\Delta\alpha_{k}(\omega)$ are calculated based
on experiments carried out at 2 K with $\bm{H}\parallel a$ and $\mu_{0}|H|=7$
T according to

$\displaystyle\Delta\alpha_{M}(\omega)$ $\displaystyle=$
$\displaystyle\alpha(\omega)[+\bm{k},+\bm{P},+\bm{M}]-\alpha(\omega)[+\bm{k},+\bm{P},-\bm{M}],$
$\displaystyle\Delta\alpha_{P}(\omega)$ $\displaystyle=$
$\displaystyle\alpha(\omega)[+\bm{k},+\bm{P},+\bm{M}]-\alpha(\omega)[+\bm{k},-\bm{P},+\bm{M}],$
$\displaystyle\Delta\alpha_{k}(\omega)$ $\displaystyle=$
$\displaystyle\alpha(\omega)[+\bm{k},+\bm{P},+\bm{M}]-\alpha(\omega)[-\bm{k},+\bm{P},+\bm{M}].$

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.

The three absorption difference spectra $\Delta\alpha_{M}$, $\Delta\alpha_{P}$
and $\Delta\alpha_{k}$ nicely coincide, as expected if the refractive index
contains a term $\delta n\propto\bm{k}\cdot(\bm{P}\times\bm{M})$. Remarkably,
for the polarization $\bm{E}^{\omega}\parallel a$ & $\bm{H}^{\omega}\parallel
c$ the absorption difference in 7 T, displayed in Fig. 5(b) , is larger than
the total absorption in zero field [Fig. 5(a)] for the $ME_{1}$ mode. In the
$\bm{E}^{\omega}\parallel c$ & $\bm{H}^{\omega}\parallel a$ polarization
configuration, the absorption difference in 7 T, plotted in Fig. 5(d), for
$ME_{2}$, $ME_{3}$ and $ME_{4}$ is close to or larger than the absorption in
zero field [Fig. 5(c)]. Specific to $ME_{3}$, the absorption peak is too
strong to be fully resolved in 7 T. By fitting the flanks of $\Delta\alpha$
spectra with a Lorentzian line (not shown here) yields the maximum absorption
difference $\Delta\alpha\approx 200$ cm-1. This value is nearly twice as large
as the absorption in zero field, indicating that one-way transparency [17] is
realized for this mode.

## Conclusions

By independent reversal of magnetization, polarization and light propagation,
all three possibilities of realizing directional dichroism have been
demonstrated for the first time. Our THz spectroscopy results on Co2Mo3O8 with
orthogonal magnetization and polarization directly confirm the trilinear-
product form of the optical magnetoelectric effect
$\bm{k}\cdot(\bm{P}\times\bm{M})$, which was expected on theoretical grounds,
but not confirmed yet in magnetoelectric crystals on a purely experimental
basis. Some of the low-energy excitations of this compound, presumably spin-
waves, exhibit strong directional dichroism close to the one-way-transparency
limit.

## Methods

Synthesis
Single crystals were grown by the chemical transport reaction method at
temperatures between 900–950 ∘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 $P6_{3}mc$.
The room-temperature lattice constants $a=b=5.7677(1)$ Å and $c=9.9097(2)$ Å
are close to the data reported in literature [47, 51]. Single crystals were
characterized by magnetometry and specific heat measurements, which confirmed
the onset of long-range antiferromagnetic order at $T_{\mathrm{N}}{}=40$ 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 $\pm 7$ T.

Inelastic neutron scattering
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
$k_{\mathrm{f}}=2.66$ 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 $(-1,0,l)$.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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###### Acknowledgements.

J.D. acknowledges stimulating discussions with Prof. Jorge Stephany. This
research was partly funded by Deutsche Forschungsgemeinschaft DFG via the
Transregional Collaborative Research Center TRR 80 “From Electronic
correlations to functionality” (Augsburg, Munich, Stuttgart). The support via
the project ANCD 20.80009.5007.19 (Moldova) is also acknowledged. This
research was supported by the National Research, Development and Innovation
Office – NKFIH, FK 153003 and Bolyai 00318/20/11. D.S. acknowledges the
support of the Austrian Science Fund (FWF) [I 2816-N27, TAI 334-N] and that of
the Austrian Agency for International Cooperation in Education and Research
[WTZ HU 08/2020].

## Author Contributions

L.P. and V.T. synthetized and characterized the crystals; S.R., D.G.F., and
A.S. performed the THz measurements; S.R., J.D., and I.K. analyzed the THz
measurements; S.G., K.G., O.Z. performed and analyzed the neutron scattering
experiments. S.K, J.D, and I.K. wrote the manuscript with contributions from
D.S. and S.B.; I.K. planned the project.

## Competing Interests

The authors declare that there are no competing interests.