Spin-Hall Conductivity in Electron-Phonon Coupled Systems

TL;DR

This paper investigates how electron-phonon interactions affect the ac spin-Hall conductivity in two-dimensional spin-orbit coupled systems, revealing model-dependent modifications to the conductivity's universality and resonance features.

Contribution

It derives the ac spin-Hall conductivity considering electron-phonon coupling and compares the effects in different Rashba models, highlighting the role of spin-vertex corrections.

Findings

Electron-phonon interactions break the universality of spin-Hall conductivity in the linear Rashba model.

In a generalized Rashba model, phonons do not alter low-frequency conductivity due to absence of vertex corrections.

Electron-phonon coupling renormalizes interband resonance in the linear Rashba case.

Abstract

We derive the ac spin-Hall conductivity $\sigma_{\rm sH}(\omega)$ of two-dimensional spin-orbit coupled systems interacting with dispersionless phonons of frequency $\omega_0$. For the linear Rashba model we show that the electron-phonon contribution to the spin-vertex corrections breaks the universality of $\sigma_{\rm sH}(\omega)$ at low-frequencies and provides a non-trivial renormalization of the interband resonance. On the contrary, in a generalized Rashba model for which the spin-vertex contributions are absent, the coupling to the phonons enters only through the self-energy, leaving the low frequency behavior of $\sigma_{\rm sH}(\omega)$ unaffected by the electron-phonon interaction.