Frequency- and transverse wave-vector-dependent spin Hall conductivity in two-dimensional electron gas with disorder

TL;DR

This paper investigates the wave number and frequency dependence of spin Hall conductivity in a disordered 2D electron gas with Rashba spin-orbit interaction, revealing resonances and spin sign changes linked to spatial electric field variations.

Contribution

It provides a detailed analysis of wave vector and frequency-dependent spin Hall conductivity, including the effects of disorder and different spin current definitions, highlighting resonance phenomena and spin sign reversal.

Findings

Spin Hall conductivity is zero at zero wave number and frequency.

Resonance occurs when the wavelength of electric field variation matches the spin precession length.

The out-of-plane spin component flips sign with electric field direction change.

Abstract

We determine wave number $q$ and frequency $\omega$ dependent spin Hall conductivity $\sigma_{yx}^s(q, \omega)$ for a disordered two dimensional electron system with Rashba spin orbit interaction when $\q$ is {\it transverse} to the electric field. Both the conventional definition of spin current and its new definition which takes care of the conservation of spins, have been considered. The spin Hall conductivitivities for both of these definitions are qualitatively similar. $\sigma_{yx}^s(q, \omega)$ is zero at $q=0, \omega =0$ and is maximum at $q=0$ and at small but finite $\omega$ whose value depends on different parameters of the system. Interestingly for $\omega \to 0$, $\sigma_{yx}^s(q)$ resonates when $\Lambda \simeq L_{so}$ which are the wavelength $(\Lambda = 2\pi/q)$ of the electric field's spatial variation and the length for one cycle of spin precession respectively. The sign of the out-of-plane component of the electrons' spin flips when the sign of electric field changes due to its spatial variation along transverse direction. It changes the mode of spin precession from clockwise to anti-clockwise or {\it vice versa} and consequently a finite spin Hall current flows in the bulk of the system.