Spin-current quantization in a quantum point contact with spin-orbit interaction

We develop a realistic and analytically tractable model to describe the spin current which arises in a quantum point contact (QPC) with spin-orbit interaction (SOI) upon a small voltage is applied. In the model, the QPC is considered as a saddle point of two-dimensional potential landscape. The SOI acts within a finite region and is absent deep in the reservoirs. The SOI strength is not supposed to be strong. It is shown that the spin polarization appears in the third order of the perturbation theory as a result of definite combinations of electron transitions. They include two intersubband transitions to nearest subbands and one intrasubband transition. The spin current is proportional to the cube of the SOI strength and strongly depends on geometric parameters of the saddle point. The spin is polarized in the plane of the QPC and directed normally to the electron current if the SOI is of Rashba type. As a function of the saddle-point potential (i.e., the height of the QPC barrier), the spin conductance and especially the spin polarization have characteristic features (specifically, peaks) correlated with the charge conductance quantization steps. The peak shape depends on the length of the region where the SOI acts. In QPCs with sharp potential landscape, this picture is distorted by interference processes.