Side-jumps in the spin-Hall effect: construction of the Boltzmann collision integral

TL;DR

This paper systematically derives the side-jump contribution to the spin-Hall effect by constructing the collision integral in the Boltzmann equation, highlighting the role of off-diagonal density matrix elements and velocity corrections.

Contribution

It provides a detailed derivation of the collision integral accounting for side-jump effects in systems without band structure spin-orbit interactions, clarifying their impact on the spin-Hall current.

Findings

Two corrections to the collision integral sum to produce the side-jump contribution.

The spin-orbit-induced velocity correction does not contribute to the steady-state spin-Hall current.

The derived framework clarifies the microscopic origin of side-jump effects in spin transport.

Abstract

We present a systematic derivation of the side-jump contribution to the spin-Hall current in systems without band structure spin-orbit interactions, focusing on the construction of the collision integral for the Boltzmann equation. Starting from the quantum Liouville equation for the density operator we derive an equation describing the dynamics of the density matrix in the first Born approximation and to first order in the driving electric field. Elastic scattering requires conservation of the total energy, including the spin-orbit interaction energy with the electric field: this results in a first correction to the customary collision integral found in the Born approximation. A second correction is due to the change in the carrier position during collisions. It stems from the part of the density matrix off-diagonal in wave vector. The two corrections to the collision integral add up and are responsible for the total side-jump contribution to the spin-Hall current. The spin-orbit-induced correction to the velocity operator also contains terms diagonal and off-diagonal in momentum space, which together involve the total force acting on the system. This force is explicitly shown to vanish (on the average) in the steady state: thus the total contribution to the spin-Hall current due to the additional terms in the velocity operator is zero.