Analytically solvable model to the spin Hall effect with Rashba and Dresselhaus spin-orbit couplings

TL;DR

This paper introduces an analytical model for the spin Hall effect in 2D electron systems with Rashba and Dresselhaus spin-orbit couplings, revealing resonant behaviors and conductance features.

Contribution

It develops a solvable Hamiltonian approximation that captures the combined effects of Rashba and Dresselhaus couplings in magnetic fields, providing closed-form eigenstates and energies.

Findings

Eigenstates and energies match numerical results across coupling strengths.

Spin Hall conductance shows a resonant peak at specific magnetic fields.

Charge conductance exhibits quantized plateaus and jumps at resonance.

Abstract

When the Rashba and Dresslhaus spin-orbit coupling are both presented for a two-dimensional electron in a perpendicular magnetic field, a striking resemblance to anisotropic quantum Rabi model in quantum optics is found. We perform a generalized Rashba coupling approximation to obtain a solvable Hamiltonian by keeping the nearest-mixing terms of Laudau states, which is reformulated in the similar form to that with only Rashba coupling. Each Landau state becomes a new displaced-Fock state with a displacement shift instead of the original Harmonic oscillator Fock state, yielding eigenstates in closed form. Analytical energies are consistent with numerical ones in a wide range of coupling strength even for a strong Zeeman splitting. In the presence of an electric field, the spin conductance and the charge conductance obtained analytically are in good agreements with the numerical results. As the component of the Dresselhaus coupling increases, we find that the spin Hall conductance exhibits a pronounced resonant peak at a larger value of the inverse of the magnetic field. Meanwhile, the charge conductance exhibits a series of plateaus as well as a jump at the resonant magnetic field. Our method provides an easy-to-implement analytical treatment to two-dimensional electron gas systems with both types of spin-orbit couplings.