Abstract
We apply a microscopic theory of polarization and magnetization to crystalline insulators at zero temperature and consider the orbital electronic contribution of the linear response to spatially varying, time-dependent electromagnetic fields. The charge and current density expectation values generally depend on both the microscopic polarization and magnetization fields, and on the microscopic free charge and current densities. But contributions from the latter vanish in linear response for the class of insulators we consider. Thus we need only consider the former, which can be decomposed into "site" polarization and magnetization fields, from which "site multipole moments" can be constructed. Macroscopic polarization and magnetization fields follow, and we identify the relevant contributions to them; for electromagnetic fields varying little over a lattice constant these are the electric and magnetic dipole moments per unit volume, and the electric quadrupole moment per unit volume. A description of optical activity and related magneto-optical phenomena follows from the response of these macroscopic quantities to the electromagnetic field and, while in this paper we work within the independent particle and frozen-ion approximations, both optical rotary dispersion and circular dichroism can be described with this strategy. Earlier expressions describing the magnetoelectric effect are recovered as the zero frequency limit of our more general equations. Since our site quantities are introduced with the use of Wannier functions, the site multipole moments and their macroscopic analogs are generally gauge dependent. However, the resulting macroscopic charge and current densities, together with the optical effects to which they lead, are gauge invariant, as would be physically expected.
Highlights
In a material that is optically active the plane of polarization of light rotates as the light propagates through the medium; the rotation is associated with a difference in the phase velocities of right- and left-handed circularly polarized light
While exponentially localized Wannier functions (ELWFs) would be a natural choice for the original Wannier functions, we show that whatever choice is made the resulting electronic charge and current densities predicted are gauge invariant [21], as would be physically expected; the expressions we extract for σ il (ω) and σ il j (ω) are gauge invariant
In retaining terms that are at most linear in q, we extract tensors σ il (ω) and σ il j (ω) that describe phenomena involving the rotation of the plane of polarization of light as it propagates through a medium; the former contributes through its antisymmetric part only when timereversal symmetry is broken in the unperturbed system, and can be considered as describing an “internal” Faraday effect, while the latter contributes more generally and describes optical activity
Summary
In a material that is optically active the plane of polarization of light rotates as the light propagates through the medium; the rotation is associated with a difference in the phase velocities of right- and left-handed circularly polarized light. The artificial divergences that can plague standard minimal coupling calculations are absent In this approach the site contributions to the electronic component of the microscopic polarization and magnetization fields, and to their multipole moments, depend on a modified form of the Wannier functions resulting from a generalized Peierls substitution [18]. In the absence of both time-reversal and spatial inversion symmetry, a magnetic field can still induce a polarization and an electric field can still induce a magnetization This phenomenon is called the magnetoelectric effect [22]; in an earlier work [19] we used our approach to derive the so-called orbital magnetoelectric polarizability (OMP) tensor that describes the magnetoelectric effect in the limit of fixed ion cores and with the neglect of spin contributions, and found agreement with earlier studies based on the “modern theory of polarization and magnetization” [23,24].
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