Theory of conserved spin current and its application to a two-dimensional hole gas

We present a detailed microscopic theory of the conserved spin current which is introduced by us [Phys. Rev. Lett. 96, 076604 (2006)] and satisfies the spin continuity equation even for spin-orbit coupled systems. The spin-transport coefficients sigma(s)(mu nu) as a response to the electric field are shown to consist of two parts, i.e., the conventional part sigma(s0)(mu nu) and the spin-torque-dipole correction sigma(s tau)(mu nu). As one key result, an Onsager relation between sigma(s)(mu nu) and other kinds of transport coefficients is shown. The expression for sigma(s)(mu nu) in terms of single-particle Bloch states is derived, by use of which we study the conserved spin-Hall conductivity in the two-dimensional hole gas modeled by a combined Luttinger and space-inversion asymmetric Rashba spin-orbit coupling. It is shown that the two components in spin-Hall conductivity usually have the opposite contributions. While in the absence of Rashba spin splitting the spin-Hall transport is dominated by the conventional contribution, the presence of Rashba spin splitting stirs up a large enhancement of the spin-torque-dipole correction, leading to an overall sign change for the total spin-Hall conductivity. Furthermore, an approximate two-band calculation and the subsequent comparison with the exact four-band results are given, which reveals that the coupling between the heavy-hole and light-hole bands should be taken into account for strong Rashba spin splitting.