Intrinsic electric polarization in spin-orbit coupled semiconductor heterostructures

We present Maxwell equations with source terms for the electromagnetic field interacting with a moving electron in a spin-orbit coupled semiconductor heterostructure. We start with the eight--band ${\bm k}{\bm p}$ model and derive the electric and magnetic polarization vectors using the Gordon--like decomposition method. Next, we present the ${\bm k}{\bm p}$ effective Lagrangian for the nonparabolic conduction band electrons interacting with electromagnetic field in semiconductor heterostructures with abrupt interfaces. This Lagrangian gives rise to the Maxwell equations with source terms and boundary conditions at heterointerfaces as well as equations for the electron envelope wave function in the external electromagnetic field together with appropriate boundary conditions. As an example, we consider spin--orbit effects caused by the structure inversion asymmetry for the conduction electron states. We compute the intrinsic contribution to the electric polarization of the steady state electron gas in asymmetric quantum well in equilibrium and in the spin Hall regime. We argue that this contribution, as well as the intrinsic spin Hall current, are not cancelled by the elastic scattering processes.