Spin-Hall effect theory: new analytical solutions of the Pauli equation in a quantum dot

TL;DR

This paper derives new analytical solutions to the Pauli equation incorporating Rashba and Dresselhaus interactions, revealing their distinct roles in the spin-Hall effect within semiconductor quantum dots under electric fields.

Contribution

It provides the first analytical solutions for the Pauli equation with these spin-orbit interactions in quantum dots, clarifying their individual effects on the spin-Hall phenomenon.

Findings

Dresselhaus term causes spin-Hall effect when spin depends on y-coordinate.

Rashba interaction induces spin-Hall effect when spin depends on z-coordinate.

Spin density varies along the transverse coordinate depending on the dominant interaction.

Abstract

In this work, we present the analytical solution of the effective mass Pauli equation, with Rashba and linear Dresselhaus interactions, for an electron gas moving through a semiconductor quantum dot under a longitudinal electric field, which is defined along the $x$-direction. We study the relative influence of the Rashba and Dresselhaus terms on the spin-Hall effect for the first propagating and edge channels, by analyzing the mixing between spin-up and -down states and the zero-field spin splitting along the transverse directions. When the spin rotation depends only on the $y$-coordinate, the spin orientation and the spin density vary along this transverse coordinate and, in this case, we show that the spin-Hall effect is only due to the Dresselhaus term, for depolarized electrons. On the other hand, if the spin rotation depends on the $z$-coordinate, the spin-Hall effect is provoked only by the Rashba interaction.