Spin–charge conversion and current vortex in spin–orbit coupled systems

Using response theory, we calculate the charge current vortex generated by spin pumping at a point-like contact in a system with Rashba spin–orbit coupling (SOC). We discuss the spatial profile of the current density for finite temperature and for the zero-temperature limit. The main observation is that the Rashba spin precession leads to a charge current that oscillates as a function of distance from the spin pumping source, which is confirmed by numerical simulations. In our calculations, we consider a Rashba model on a square lattice, for which we first review the basic properties related to charge and spin transport. In particular, we define the charge current and spin current operators for the tight-binding Hamiltonian as the currents coupled linearly with the U(1) and SU(2) gauge potentials, respectively. By analogy to the continuum model, the SOC Hamiltonian on the lattice is then introduced as the generator of the spin current.