Nonreciprocal directional dichroism at telecom wavelengths

Magnetoelectrics with ultra-low symmetry and spin-orbit coupling are well known to display a number of remarkable properties including nonreciprocal directional dichroism. As a polar and chiral magnet, Ni$_3$TeO$_6$ is predicted to host this effect in three fundamentally different configurations, although only two have been experimentally verified. Inspired by the opportunity to unravel the structure-property relations of such a unique light-matter interaction, we combined magneto-optical spectroscopy and first-principles calculations to reveal nonreciprocity in the toroidal geometry and compared our findings with the chiral configurations. We find that formation of Ni toroidal moments is responsible for the largest effects near 1.1 eV - a tendency that is captured by our microscopic model and computational implementation. At the same time, we demonstrate deterministic control of nonreciprocal directional dichroism in Ni$_3$TeO$_6$ across the entire telecom wavelength range. This discovery will accelerate the development of photonics applications that take advantage of unusual symmetry characteristics.


INTRODUCTION
Here, we focus on magneto-optical spectroscopy and first-principles-based simulations of nonreciprocity in the toroidal configuration in order to complete this fascinating series and, at the same time, unravel the structure-property relationships relevant to this unique lightmatter interaction. For instance, we find that the largest contrast is supported by the 3 A 2g → 3 T 2g on-site excitations near 1.1 eV due to the creation of Ni 2+ toroidal moments in this relatively narrow energy range -quite different from the magnetochrial and transverse magnetochiral mechanisms that provide broader but more modest effects over the color band range. The phonon side bands that ride on top of these d-to-d excitations are quantitatively assigned to particular phonon modes and shown to display nonreciprocal effects as well.
In a significant conceptual advance, we demonstrate dichroic contrast across the telecom range (i.e. optical fiber communication wavelengths). Remarkably, the telecom wavelength range dovetails perfectly with the strongest nonreciprocal response of Ni 3 TeO 6 in the toroidal geometry. This establishes a potentially significant application for nonreciprocal materials -beyond ferrites and the microwave regime -in which optical circulators or directional amplifiers can operate in the telecom range with low loss and without complicated sample fabrication. [34][35][36][37] These findings also open the door to the use of toroidal magnetoelectrics for optical signal processing and communication. 38

RESULTS AND DISCUSSION
Crystal structure, properties, and symmetry analysis The crystal structure of Ni 3 TeO 6 is both polar and chiral as expected for an R3 space group. 39 Chains of distorted NiO 6 and TeO 6 octahedra lie along the c direction, and the three different Ni centers are distinguished by their local environments. 40 In this work we denote the threefold axis of Ni 3 TeO 6 as the 'chiral axis' of the compound. 40,41 Although Ni 3 TeO 6 hosts interlocking ferroelectric and chiral domain patterns, 40,42 crystals can be polished to reveal a single chiral domain. 20 Below T N = 53 K, the system displays a collinear antiferromagnetic ground state. For H c, there is a spin-flop transition at 9 T leading to a conical spin and a metamagnetic transition at 52 T -both of which are accompanied by magnetoelastic effects. [43][44][45][46] There are no magnetically-driven phase transitions when field is applied perpendicular to the c-axis. 44 The optical properties of Ni 3 TeO 6 consist of several different color bands comprised of on-site Ni 2+ d-to-d excitations along with a charge gap near 2.6 eV. 47 These features are sensitive to the spin-flop and metamagnetic transitions, and because spin-orbit coupling endows the excitations with magnetoelectric character, these phases support nonreciprocal directional dichroism. 20 Here, the excited Ni state (with a hole in the t 2g orbital) provides spin-orbit coupling on the order of 40 meV and assures that the matrix elements containing polarization and magnetization are non-zero. 20 Significantly, the broad band nonreciprocal directional dichroism in the magnetochiral configuration of Ni 3 TeO 6 can be modeled using a quantitative first-principles-derived formalism. 20 Figure 1 summarizes our overall approach and the symmetry conditions that are important in this work. We begin by calculating absorption difference spectra as ∆α = α(H) -α(0 T) for the different measurement configurations. Nonreciprocity is determined from various differences in the ∆α spectra, for instance ∆α N DD (+k, ±H) = ∆α(+k, +H) -∆α(+k, -H) [ Fig. 1(a)]. Notice that when both k and H are reversed, α N DD (±k, ±H) = ∆α(+k, +H) -∆α(-k, -H) vanishes [ Fig. 1(b)]. Nonreciprocity can also be defined in terms of counter-propagating beams, ∆α N DD (±k, +H), as shown in Fig. 2.
The top panels in Fig. 2 summarize the three different measurement configurations predicted within the framework of symmetry operational similarity. 22,23 In the toroidal configuration [ Fig. 2(a)], electric polarization along c, combined with H perpendicular to c, breaks two-fold rotation, inversion, mirror, and time-reversal symmetries, so it has symmetry operational similarity with the wave vector of light k along the third direction. Thus, light propagation parallel to the toroidal moment can exhibit nonreciprocal directional dichroism.
The two other configurations of interest involve chirality (rather than electric polarization) which breaks inversion and mirror symmetries. The magnetochiral geometry is obtained when the light propagation direction and applied field are parallel to the chiral axis [ Fig.   2(b)]. The transverse magnetochiral configuration, on the other hand, requires both the light propagation direction and magnetic field to be perpendicular to the chiral axis [ Fig. 2(c)]. Note that we denote the C 3 axis in Ni 3 TeO 6 as the chiral axis; previous studies have argued that nonreciprocity in chiral systems can be easily understood considering chirality as a vector-like quantity. 22,41 Nonreciprocal effect in different measurement configurations Figure 2 summarizes nonreciprocal directional dichroism of Ni 3 TeO 6 at full field in the toroidal, magnetochiral, and transverse magnetochiral configurations. Overall, this behavior is a consequence of low symmetry, structural and magnetic chirality, and the presence of spinorbit coupling, although the specific appearance of ∆α NDD depends upon the measurement configuration as well. Focusing first on Ni 3 TeO 6 in the toroidal geometry [ Fig. 2(a)], we find a strong dichroic response in the vicinity of the 1.1 eV color band. This feature is assigned as a superposition of Ni 2+ on-site d-to-d excitations ( 3 A 2g → 3 T 2g ) emanating from the three different local environments of the Ni centers. 47 At 60 T, the largest field-induced changes are centered on the 1.1 eV color band and are on the order of 160 cm −1 . This corresponds to between zero and approximately 45% contrast, depending on the value of the absolute absorption. The dichroic contrast in the vicinity of the 3 A 2g → 1 E g and 3 A 2g → 3 T 1g color bands is significantly smaller, and that at higher energy near the 3 A g → 1 T 2g excitation is zero within our sensitivity. In the toroidal configuration, the overall size of ∆α NDD and its energy distribution is quite different than what is observed in the magnetochiral and transverse magnetochiral geometries. 20 For instance, while all configurations exhibit dichroic contrast near the 1.1 eV color band, that in the toroidal configuration is by far the largest. Field dependence of the dichroic contrast in the toroidal geometry Figure 3(a) displays nonreciprocal directional dichroism of Ni 3 TeO 6 as a function of magnetic field in the toroidal congifuration. As discussed previously, the toroidal geometry involves placing the electric polarization direction mutually orthogonal to both the light propagation and magnetic field directions. In addition to large positive and negative lobes in the dichroic contrast, there is a great deal of fine structure below 0.9 eV. These features can be assigned to phonon sidebands as discussed below. Examination reveals that α NDD appears at the lowest fields and grows systematically. This is because field-induced canting of Ni moments, which gives rise to net ferromagnetic moments and enables nonreciprocal behavior, can occur even at the smallest fields. That nonreciprocal directional dichroism can be seen at low fields is useful for a number of applications including optical isolators and rectifiers, high-fidelity holograms, and potentially in the telecom sector. In any case, we can quantify this trend by integrating α NDD over an appropriate energy window and plotting the result as a function of applied field [inset, Fig. 3(a)]. The shape reveals how the spins align in the direction of the applied field. The lack of sharp jumps or cusps is consistent with the absence of field-induced magnetic transitions for H ⊥ c. 44 No hysteresis is observed.
In addition to switching the applied field direction, we investigated symmetry effects 41 in Comparison between the experimental nonreciprocal directional dichroism in different geometries with the simulated versions provides insight into the nature of the contrast in different energy ranges. Figure 3(b) displays ∆α NDD of Ni 3 TeO 6 calculated by first-principlesbased methods in the toroidal configuration. 20 The simulated spectrum compares reasonably well with the experimental result, except for the phonon sideband effects around 0.8 eV which are not included in the model, capturing the maximum near 0.9 eV, the minimum near 1.03 eV as well as the lack of contrast at higher energies. Note that our simulation is based upon a local ionic picture and includes only intra-Ni-ionic excitations, 20 not the entire part of the structural (and magnetic) chirality of Ni 3 TeO 6 . The structural chirality of Ni 3 TeO 6 manifests (i) at the local level with a chiral crystal-field environment at each Ni site as well as (ii) in the global arrangement of Ni sites. Our theory includes the former but not the latter. We refer to these as local and global chirality, respectively. 48 Based on the results from our ionic picture-based simulation that nicely capture the dichroic response near 1 eV, we speculate that excitations near 1 eV are relevant to the local chiral component, namely the formation of the Ni ionic toroidal moments perpendicular to the magnetic field and bulk electric polarization (T ≡ P×M), which are coupled with the propagation of light and induce nonreciprocal directional dichroism. 41 On the other hand, nonreciprocity at higher energies, which is sizable only in the magnetochiral and transverse magnetochiral cases [ Fig. 2(b,c)], originates primarily from global chirality and is difficult to capture within our atomistic simulation framework. Since our simulated nonreciprocal directional dichroism is given as a THz range has recently been reported in Ni 3 TeO 6 , 11 possibly originating from the effects mentioned above. Below, we demonstrate that the phonon sidebands in Ni 3 TeO 6 not only display nonreciprocal directional dichroism [ Fig. 3(d)] but that this contrast occurs along with that related to the low-energy Ni 2+ crystal field excitations across the telecom range.

Nonreciprocal directional dichroism at telecom wavelengths
We therefore return to the spectra in Fig. 3(c) Supplementary Fig. 7]. In addition to demonstrating that nonreciprocal directional dichroism can be positioned within useful telecom windows, this work opens the door to the development of high efficiency / low dissipation optical diodes and rectifiers from crystalline materials. We anticipate that linear or circular polarizers would amplify the size of this effect in Ni 3 TeO 6 , but the ability to achieve polarization-independent signal is one of the beauties of magnetochiral materials.

Structure-property relations for photonics applications
Ni 3 TeO 6 is a superb platform for fundamental studies of nonreciprocity because it hosts this peculiar property across a wide range of excitations and in three different measurement configurations. To our knowledge, there are no other nonreciprocal materials that have been studied in so many different geometries. In this work, we focus on the Ni 2+ d-tod excitations and associated phonon sidebands in the near infrared and optical range in the toroidal configuration, unraveling structure-property relationships via comparison with contrast in the magnetochiral and transverse magnetochiral geometries. We find that polarity allows the creation of Ni toroidal moments, which results in large contrast near the color band at 1.1 eV. The chiral geometries, on the other hand, yield smaller contrast over a much broader energy range. Our modeling of the size and shape of the spectral response supports this picture of the importance of polarity vs. chirality in the creation of large vs. broadband contrast. These findings enhance our ability to design and deliver complex materials properties on demand. At the same time, the discovery of nonreciprocal effects across the full telecom wavelength range presents a number of exciting opportunities. While tunability under magnetic field is established in this work, the degree to which nonreciprocity can be controlled by other external stimuli or chemical substitution is unexplored. That said, the discovery of nonreciprocity in the telecom region is a significant conceptual advance that has the potential to jump-start the use of dichroic contrast in photonics applicationsparticularly in the area of high-efficiency optical diodes and rectifiers.

METHODS
Crystal growth and orientation: High quality single crystals of Ni 3 TeO 6 were grown by chemical vapour transport methods as described previously. 43 The crystals were polished to expose either the ab-plane or the c-axis and to control optical density. After polishing, the sample thicknesses were on the order of 30 µm. Optical microscope images and optical rotation data confirm that the crystals have a single chiral domain. 20 The crystals were coated with transparent epoxy to stabilize the structure during the field pulses.
Optical spectroscopy: Polarized optical transmittance was measured as a function of energy and temperature using a series of spectrometers as described previously (0.78 -2.5 eV; 4.2 -300 K). 45 Absorption was calculated as α(E) = 1 d ln(T (E)), where d is the sample thickness and T (E) is the measured transmittance.
Magneto-optical spectroscopy: Magneto-optical work was performed in the toroidal geometry in a capacitor-driven 65 T pulsed magnet at the National High Magnetic Field Laboratory in Los Alamos, NM. We employed the standard transmission probe fitted with a specially-designed Voigt end-piece for this work. These measurements covered the 0.75 -2.6 eV range with 2.4 meV resolution and were carried out at 4 K. Broadband light from a tungsten lamp was coupled to optical fibers and focused onto the sample for transmittance experiments. A collection fiber brought the light from the top of the probe to the grating spectrometer, where both CCD and InGaAs detectors were employed as appropriate. The spectra were taken in four different measurement configurations: (±k, ±H). Each run was carried out sequentially and consistently, starting with one k direction (and pulsing to obtained both ±H) and then switching to the other k direction by swapping the fibers (again pulsing both up and down). We calculated the absorption differences as: ∆α = α(H = ± 60 T) -α(H = 0 T). As an example, nonreciprocal directional dichroism is calculated as: ∆α NDD (+k, ± H) = ∆α(+k, +H) -∆α(+k, -H). Nonreciprocity can also be defined in terms of counter-propagating beams as ∆α N DD (±k, +H). Traditional smoothing techniques were employed in the CCD detector regime.

DATA AVAILABILITY
Data are available from the corresponding author upon reasonable request.

CODE AVAILABILITY
The codes implementing the calculations of this study are available from the corresponding author upon request. effects and applied it to Ni 3 TeO 6 . KP, HSK, and JLM wrote the manuscript. All authors commented on the text.

COMPETING INTERESTS
The authors declare no competing interests      context. Note that we used cubic term symbols here ( 3 A 2g , 3 T 2g , 1 E g , 3 T 1g , 1 T 2g ) to denote Ni 2+ atomic multiplets because the deviation from cubic symmetry by trigonal distortion at the Ni sites is small (≈0.1 eV in terms of crystal fields). a b -p l a n e ( a ) 1 5 6 5 -1 6 2 5 n m C -B a n d S -B a n d E -B a n d O -B a n d 1 2 6 0 -1 3 6 0 n m 1 3 6 0 -1 4 6 0 n m 1 4 6 0 -1 5 3 0 n m 1 5 3 0 -1 5 6 5 n m L -B a n d The U-band (1625-1675 nm) is not shown.