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 ${\sigma }_{\mu \nu }^s$ as a response to the electric field are shown to consist of two parts, i.e., the conventional part ${\sigma }_{\mu \nu }^{s0}$ and the spin-torque-dipole correction ${\sigma }_{\mu \nu }^{s\tau }$. As one key result, an Onsager relation between ${\sigma }_{\mu \nu }^s$ and other kinds of transport coefficients is shown. The expression for ${\sigma }_{\mu \nu }^s$ 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.

Figure 1

(a) Approximate band structure of the GaAs 2DHG. (b) Heavy-hole Fermi wave vectors as a function Rashba coefficient $\alpha$ for $n_h=1.5\times {10}^{12}{\mathrm{cm}}^{-2}$.

Figure 2

(Color online) Conserved spin-Hall conductivity ${\sigma }_{yx}^s$ (black line) and its two components ${\sigma }_{yx}^{s0}$ (blue line) and ${\sigma }_{yx}^{s\tau }$ (red line) as a function of the heavy-hole Fermi wave vector for $\alpha =0$ and $n_h=1.5\times {10}^{12}{\mathrm{cm}}^{-2}$.

Figure 3

(Color online) (a) Conventional spin-Hall conductance component ${\sigma }_{yx}^{s0}$, (b) spin-torque-dipole component ${\sigma }_{yx}^{s\tau }$, and (c) the total spin-Hall conductance ${\sigma }_{yx}^s$ (in unit of $e∕4\pi$) as a function of Rashba coefficient $\alpha$ and hole density $n_h$ (contour plot).

Figure 4

(Color online) Conserved spin-Hall conductivity ${\sigma }_{yx}^s$ as a function of Rashba coefficients $\alpha$ for $n_h=1.5\times {10}^{12}{\mathrm{cm}}^{-2}$. The black line is the exact four-band result, while the red line gives the approximate two-band result for the heavy holes.