  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.
