Linear and nonlinear spin current response of anisotropic spin-orbit coupled systems

We calculate the linear and the second harmonic (SH) spin current response of two anisotropic systems with spin-orbit (SO) interaction. General expressions of wide applicability for the these response functions are first derived for a generic two-band Hamiltonian. The first system is a two-dimensional (2D) electron gas in the presence of Rashba andk-linear Dresselhaus SO couplings. The calculations show how narrow or wide the response spectra can be, what is their overall shape and size, and frequency shiftings, depending on which crystal orientation is selected. The quantitative knowing of this makes possible a comparative study for several orientations, which would allow to select a spectrum with particular characteristic. We find that vanishing linear and second order response tensors are achievable under SU(2) symmetry conditions, characterized by a collinear SO vector field. Additional conditions under which specific tensor components vanish are possible, without having such collinearity. Thus, a proper choice of the growth direction and SO strengths allows to select the polarization of the linear and SH spin currents according to the direction of flowing. The second system is an anisotropic 2D free electron gas with anisotropic Rashba interaction, which has been employed to study the optical conductivity of 2D puckered structures with anisotropic energy bands. The presence of mass anisotropy and an energy gap open several distinct scenarios for the allowed optical interband transitions, which manifest in the linear and SH response contrastingly. The linear response displays only out-of-plane spin polarized currents, while the SH spin currents flow with spin orientation lying parallel to the plane of the system strictly. The models illustrate the possibility of the nonlinear spin Hall effect in systems with SO interaction, under the presence or absence of time-reversal symmetry. The results suggest different ways to manipulate the linear and nonlinear optical generation of spin currents which could find spintronic applications.