Spin Hall effect in two-dimensional $p$-type semiconductors in a magnetic field

TL;DR

This paper investigates the spin Hall effect in two-dimensional p-type semiconductors under a magnetic field, revealing that conventional and effective spin Hall conductivities vanish in low magnetic fields, indicating a singular limit.

Contribution

It provides a detailed calculation of spin Hall conductivity in p-type 2D semiconductors with magnetic fields, highlighting the cancellation effects and the singular behavior at low magnetic fields.

Findings

Universal intrinsic spin Hall conductivity approaches 9e/8π at low Fermi energies.

Intraband contributions cancel the intrinsic spin Hall effect, resulting in zero conventional conductivity.

Effective spin conductivity is proportional to magnetization and cancels out at small spin-orbit coupling.

Abstract

We calculate the spin Hall conductivity driven by Rashba spin-orbit interaction in $p$-type two-dimensional semiconductors in the presence of a perpendicular magnetic field. For a highly confined quantum well, the system is described by a $k$-cubic Rashba term for two-dimensional heavy holes. The eigenstates of the system can be described by Landau spinor states. First we consider the conventional spin Hall conductivity. The contribution of the interband transitions to the Kubo-Greenwood formula gives the density dependent intrinsic spin Hall conductivity, which approaches its universal value $\sigma^z_{xy}=9e/8\pi$ for weak spin-orbit coupling and low Fermi energies, in agreement with previous work. However two intraband contribution terms cancel this effect leading to zero conventional spin Hall conductivity. Adding the torque dipole contribution to the definition of spin current, we also study the {\it effective} spin conductivity. This is shown to be proportional to the total magnetization plus surface terms which exactly cancel it for small spin-orbit coupling. If in low magnetic field the intraband transitions evolve to vertex corrections, the fact that both {effective} and conventional spin Hall conductivities vanish is unexpected. This suggests that the zero magnetic field limit of the model is singular.