Finite-size effects in a two-dimensional electron gas with Rashba spin-orbit interaction

Within the Kubo formalism, we estimate the spin-Hall conductivity in a two-dimensional electron gas with Rashba spin-orbit interaction and study its variation as a function of disorder strength and system size. The numerical algorithm employed in the calculation is based on the direct numerical integration of the time-dependent Schrödinger equation in a spin-dependent variant of the particle-source method. We find that the spin-precession length $L_s$, controlled by the strength of the Rashba coupling, establishes the critical length scale that marks the significant reduction of the spin-Hall conductivity in bulk systems. In contrast, the electron mean free path, inversely proportional to the strength of disorder, appears to have only a minor effect.

Figure 1

Spin-Hall conductivity as a function of Fermi energy in units of $e∕8\pi$ for clean and disordered systems. System size is $30\times 30$. Spin-orbit interaction strength is fixed to $V_R=0.2t$. Average was done over 2000 random frequencies.

Figure 2

(Color online) Spin-Hall conductivity as a function of system size in units of $e∕8\pi$ represented on a logarithmic scale. Fermi energy is fixed to $E_F=-3.0t$ and the spin precession length is $L_s\simeq 16a_0$, corresponding to $V_R=0.2t$. Mean free path ranges from $l\simeq 45a_0$ for $W=0.6$ down to a few lattice constants when disorder increases. Averages are done over ${10}^3$ disorder samples and for each sample, 200 initial state vectors are considered.

Figure 3

Disorder dependence of the characteristic length scale $\xi$ for Fermi energy $E_F=-3.0t$ and spin-orbit interaction strength $V_R=0.2t$.