Shift spin photocurrents in two-dimensional systems

The generation of nonlinear spin photocurrents by circularly polarized light in two-dimensional systems is theoretically investigated by calculating the shift spin conductivities. In time-reversal symmetric systems, shift spin photocurrent can be generated under the irradiation of circularly polarized light, while the shift charge photoccurrent is forbidden by symmetry. We show that the $k$-cubic Rashba-Dresselhaus system, the $k$-cubic wurtzite system, and Dirac surface states can support the shift spin photocurrent. By symmetry analysis, it is found that, in the Rashba type spin-orbit coupled systems, mirror symmetry requires that the spin polarization and the moving direction of the spin photocurrent be parallel, which we name longitudinal shift spin photocurrent. The Dirac surface states with warping term exhibit mirror symmetry, similar to the Rashba type system, and support longitudinal shift spin photocurrent. In contrast, in the Dresselhaus type spin-orbit coupled systems, the parity-mirror symmetry requires that the spin polarization and the moving direction of the spin photocurrent be perpendicular, which we dub transverse shift spin photocurrent. Furthermore, we find that the shift spin photocurrent always vanishes in any $k$-linear spin-orbit coupled system unless the Zeeman coupling ${\ensuremath{\mu}}_{z}$ is turned on. We find that the splitting of degenerate energy bands due to Zeeman coupling ${\ensuremath{\mu}}_{z}$ causes the van Hove singularity. The resulting shift spin conductivity has a significant peak at optical frequency $\ensuremath{\omega}=2{\ensuremath{\mu}}_{z}/\ensuremath{\hbar}$.