The lack of both time-reversal and spatial inversion symmetry in polar magnets is a prerequisite for the occurrence of optical magnetoelectric effects such as nonreciprocal directional dichroism with the potential for the realization of optical diodes. In particular, antiferromagnetic materials with magnetic excitations in the THz range such as Fe2Mo3O8 are promising candidates for next-generation spintronic applications. In a combined experimental and theoretical effort we investigated the THz excitations of the polar honeycomb antiferromagnet Fe2Mo3O8 in external magnetic fields and their nonreciprocal directional dichroism, together with the temperature dependence of the electronic transitions in the mid- and near-infrared frequency range. Using an advanced single-ion approach for the Fe ions, we are able to describe optical excitations from the THz to the near-infrared frequency range quantitatively and successfully model the observed nonreciprocal directional dichroism in the THz regime.

An effective operator of the interaction of the orbital moment of d- electrons with the magnetic field is derived by combining the method of secondary quantization with the technique of irreducible tensor operators. It is found that in addition to renormalization of the matrix elements of the total orbital momentum L there are new terms in the Hamiltonian of the interaction with the magnetic field. The effects are numerically calculated by the example of the ground term of Fe 2+ ions in Fe 2 Mo 2 O 8 . Additional magnetic dipole transitions with triangle M = +/- 3 and triangle M = +/- 2 are allowed when the magnetic field is directed along the c - axis of the crystal and in the perpendicular orientation, respectively.

The problem of renormalization of the spin–orbit interaction operator of electrons of partially filled nf-shells due to exchange-covalent bonds with surrounding ligands has been solved. It is found that along with the change of the standard spin–orbit interaction parameter, new energy operators are generated, which can be interpreted as a spin-dependent crystal field operator. Simple formulas are obtained that allows to calculate its parameters via covalence parameters and overlap integrals. Numerical evaluations have been performed for multiplets Tb^3 + (f^8 )^7F_6 and Er^3 + (f^11 )^4I_15/2 in fluorides with cubic symmetry. Calculated parameters were found to be of the order of 10

The parameters of even and odd crystal fields acting on iron ions in tetrahedral and octahedral positions of the Fe2Mo3O8 crystal are calculated. The obtained energy level schemes of the lowest multiplets are discussed in the context of the available experimental data. By comparing the calculated intensities of magnetic and electric dipole transitions with experimental data, the parameters of the effective Hamiltonian of the interaction of 3d electrons with an electric field are refined. It is found that the main absorption lines at T < TN in the region of terahertz frequencies are due to excitations of iron ions and are not associated with collective oscillations of magnetic moments. The splitting of absorption lines upon application of an external magnetic field is a consequence of the difference in the orientations of the magnetic sublattices relative to the crystallographic axes.

The energy level schema of the ground term of the nickel ion in NiCr2O4 was calculated. The parameters of the interaction with the electric field were determined, and the distribution pattern of the electric dipole moments over different positions of nickel in the unit cell was calculated. The model of the NiCr2O4 magnetoelectric structure at T < Tc was constructed taking into account the data on neutron scattering and the results of the electric polarization measurements. The origin of the magnetodielectric effect was attributed to the peculiarities of the ground state of the nickel ion.

The effective Hamiltonians of the fine structure of Fe 2+ ( 5 D) terms are derived using the methods of operator perturbation theory taking into account covalent bonds between iron and oxygen ions. The energy operators of coupling of magnetic and electric dipole moments with the electric field are obtained for states of multiplets Fe 2+ ( 5 E) and Fe 2+ ( 5 T 2 ) with minimal sets of parameters. The energy spectra of low-lying states, the magnetic anisotropy parameters, and the distribution of quadrupole and induced electric dipole moments over all positions in the unit cell are calculated. The results of calculations are compared with available experimental data.

The change in the absorption spectra due to reversal of the direction of light propagation (nonreciprocity of absorption) is a consequence of a simultaneous violation of both time-reversal and spatial-inversion symmetries. Here, we report on a high-resolution spectroscopic study of absorption nonreciprocity in the noncentrosymmetric multiferroic CuB2O4 below the antiferromagnetic transition temperature TN = 21 K in the commensurate phase in magnetic fields up to 0.5 T. The study was performed in a broad spectral region covering several exciton transitions, which all are followed by an anomalously rich structure due to the multiple exciton-magnon-phonon satellites. Two components were resolved for the spectral line near 1.4 eV corresponding to the exciton transition between the ground and the first excited state. A quantitative theory of the optical absorption and nonreciprocity at this line was developed. The theory takes into account the interference between the electric and magnetic dipole contributions to the absorption and gives an adequate explanation of the relevant effects.

The parameters of the interaction of 3d electron states with an electromagnetic wave, as well as the probabilities of magnetic and electric dipole transitions between states of the ground term of an iron ion, which is split by the crystal field, exchange interaction, and spin–orbit coupling, have been calculated. The dependences of the absorption lines on the magnitude and direction of the applied magnetic field have been determined. It has been shown that the optical diode effect discovered in [Sh. Yu, B. Gao, J. W. Kim, S.-W. Cheong, M. K. L. Man, J. Madeo, K. M. Dani, and D. Talbayev, Phys. Rev. Lett. 120, 037601 (2018)] can be explained by the interference of magnetic and electric dipole transitions.

In the last decade, Fe2Mo3O8 was recognized for a giant magnetoelectric effect, the origin of which is still not clear. In the present paper, we contribute to the microscopic theory of the magnetoelectric coupling in this compound. Using crystal field theory and the molecular field approximation, we calculated the low-lying energy spectrum for iron ions and their interaction with electric and magnetic fields. Classical ionic contribution to the electric polarization related to the ionic shifts is also estimated. It is found that the electronic and ionic contributions to the electric polarization are comparable and these mechanisms support each other at T<TN. The suggested electronic mechanism provides insight into the nature of huge jumps in polarization upon phase transitions from paramagnetic (PM) to antiferromagnetic (AFM) and then to ferrimagnetic (FRM) states under an applied external magnetic field as well as the large differential magnetoelectric coefficient.

We propose a microscopic theory of the dynamic magnetoelectric coupling in CuB2O4. The energy levels, the wave functions of Cu2+ (3d9) in an antiferromagnetically ordered sublattice of copper ions, and the probability of magnetic and electric dipole transitions are calculated and invariant components of the magnetoelectric coupling tensor are determined for various magnetic field directions. The results of microscopic and group-theoretical calculations are compared with available experimental data on the absorption and photoluminescence spectra. The photoluminescence intensity diagrams are calculated for different directions of the radiation wavevector.

A FeCr 2 O 4 single crystal with the spinel structure is grown by the zone melting method with optical heating. The critical temperatures of establishing the orbital ordering T OO = 138 K, the formation of the collinear ferrimagnetic state T N = 65 K, and the formation of the spiral modulation of the magnetic structure T s = 38 K are determined from the temperature dependences of the heat capacity and magnetic susceptibility. An anomaly of the susceptibility at T ~ 21 K, below which the hysteresis curves become butterfly-like is probably caused by a change in the magnetic anisotropy. It is established that the magnetic resonance spectrum in the microwave X-band (~9.4 GHz) at T = 30 K changes under the reversal of the direction of the external magnetic field (nonreciprocity phenomenon). Magnetic and induced electric dipole transitions are considered theoretically. The detected nonreciprocity is explained by the interference of these transitions.

The mechanism of the magnetoelectric coupling of spins of Ni2+ and Cu2+ ions with the applied electric field in (Ni, Cu)B2O4, caused by the coupling of 3d electrons with the electric field and by the spin–orbit coupling, is analyzed. It is shown that the ordering of spins in the ab crystallographic plane of the crystal induces the electric polarization along the c axis of the crystal, and this polarization is mainly related to nickel ions. To obtain the electric polarization in the CuB2O4 antiferromagnet, it turns out to be necessary to apply the magnetic field in the ab plane.

Within the framework of the quantum mechanical approach, the microscopic theory for magnetoelectric coupling and spin-canting in the spin ordered FeCr2O4 is discussed. We describe two possible mechanisms of magnetoelectric coupling caused by the combined action of the odd crystal field from the tetrahedral environment of the iron ions, the spin-orbit interaction and the exchange field acting on the 3d iron electrons from the chromium spins. We also consider sources of spin-canting. The first one is DM interaction which induces an angle-canting about 2 degrees, the second is a new one due to the common action of the Jahn-Teller and spin-orbit couplings, which yields up to the 50 degrees