A Microscopic Model of Non - Reciprocal Optical Effects in Cr

We develop a microscopic model that explains non-reciprocal optical effects in centrosymmetric Cr2O3. It is shown that light can couple directly to the antiferromagnetic order parameter. This coupling is mediated by the spinorbit interaction and involves an interplay between the breaking of inversion symmetry due to the antiferromagnetic order parameter and the trigonal field contribution to the ligand field at the Cr3+ ion. 42.65.-k,78.20.-e,78.20.Ls Typeset using REVTEX 1 The study of the interaction of light with magnetic substances has a long history. A classic example is the Faraday effect in ferromagnets where light couples directly to the ferromagnetic order parameter. Since the pioneering work of Argyres [1], it is known that electromagnetic radiation couples to the internal molecular field in a ferromagnet (which in turn is proportional to the order parameter) through the spin-orbit interaction. Such a coupling would of course be absent in antiferromagnets, where the internal molecular field is zero. In the absence of such a direct coupling between light and the antiferromagnetic order parameter, antiferromagnetic ordering could so far be probed only indirectly, for instance, by Raman scattering of magnetic excitations [2]. The discovery [3,4] of non-reciprocal optical effects (i.e., not invariant under time reversal) [5] below the Néel temperature, TN , in optical experiments on Cr2O3 has therefore been considered a breakthrough in the study of antiferromagnetic ordering by light for it is only this class of experiments which can distinguish between two magnetic states that are related to each other by the time-reversal operation. Fiebig et al. [4] have found that antiferromagnetic domains could be observed directly by non-reciprocal second harmonic generation (SHG), leading to the first photographs ever of antiferromagnetic domains [6,7]. These experiments show that light can indeed couple directly to the antiferromagnetic order parameter. Though such a coupling was anticipated earlier from symmetry considerations [8], no microscopic mechanism has been presented so far. In this Letter, we present a microscopic mechanism that explains all non-reciprocal optical effects in Cr2O3. While the spin-orbit interaction is, of course, essential in coupling the charge with the spin degrees of freedom it does not suffice for the generation of nonreciprocal effects. We find that non-reciprocal effects arise from an interplay between the breaking of crystal inversion symmetry by the antiferromagnetic order parameter and the trigonal distortion of the ligand field at the Cr ion. This effect, in addition to the spin-orbit interaction, leads to a coupling of the antiferromagnetic order parameter with light. We present furthermore a simple cluster model, containing the full crystal symmetry of Cr2O3, which allows, for the first time, the orders of magnitude of all matrix elements 2 contributing to the non-reciprocal phenomena in Cr2O3 to be predicted. We apply the microscopic model to the observed phenomenon of SHG [4] and explain how antiferromagnetic domains can be distinguished experimentally. We also apply our model to another nonreciprocal effect seen experimentally in Cr2O3 viz., gyrotropic birefringence [3] and solve the long-standing question regarding its order of magnitude. As an introduction to SHG in Cr2O3, we discuss the macroscopic theory [4] in brief. Above TN (≃ 307K), Cr2O3 crystallizes in the centrosymmetric point group D3d. The four Cr ions in the unit cell occupy equivalent c positions along the C3 (optic) axis. Since this structure has a center of inversion, parity considerations allow only magnetic dipole transitions related to the existence of an axial tensor of odd rank. Below TN , time reversal symmetry (R) is broken and in this case, since Neumann’s principle [9] cannot be applied to non-static phenomena, only symmetry operations of the crystal that do not include R may be used to classify the allowed tensors for the susceptibilities. For Cr2O3, the remaining invariant subgroup is D3. New tensors are allowed in this point group, for instance, a polar tensor of odd rank, that allow electric dipole transitions in SHG. From Maxwell’s equations, one can derive the expression for SHG by considering the contributions to (non-linear) magnetization M = γ : E E and polarization P = χ : E E. The source term S(r, t) in the wave equation [∇ × (∇×) + (1/c) ∂/∂t]E(r, t) = −S(r, t) can be written in a dipole expansion as [10], S(r, t) = μo ( ∂P(r, t) ∂t2 +∇× ∂M(r, t) ∂t + ... )