Intrinsic second-order spin current

In recent years, the nonlinear Hall effect has attracted great attention with three different terms contributed by the Drude effect, Berry curvature dipole, and Berry connection polarizability. In this paper, we theoretically predict an intrinsic second-order spin current induced by spin-dependent Berry curvature polarizability based on time-independent perturbation theory. We show two other second-order spin conductivities contributed by the group velocity and spin-dependent Berry curvature dipole. A two-dimensional Rashba-Dresselhaus spin-orbit coupled system is studied as an example. and it is found that the intrinsic second-order contribution plays a major role when the chemical potential $\ensuremath{\mu}$ is greater than zero, while current mainly comes from the extrinsic terms when $\ensuremath{\mu}<0$. Our calculation provides a theoretical basis for understanding the nonlinear spin transport.