Spin states of Dirac equation and Rashba spin-orbit interaction

TL;DR

This paper investigates the spin states of electrons described by the Dirac equation, revealing three independent spin states with distinct energy spectra and clarifying the conditions under which Rashba spin-orbit interaction occurs.

Contribution

It introduces three sets of eigenfunctions for the Dirac equation, each corresponding to different spin operators, and analyzes their associated energy spectra and spin-orbit interactions.

Findings

Only one spin state exhibits standard Rashba interaction.

Another spin state shows no Rashba interaction.

The third state results in an anisotropic band spectrum.

Abstract

The problem of the spin states corresponding to the solutions of Dirac equation is studied. In particular, the three sets of the eigenfunctions of Dirac equation are obtained. In each set the wavefunction is at the same time the eigenfunction of one of the three spin operators, which do not commute with each other, but do commute with the Dirac Hamiltonian. This means that the eigenfunctions of Dirac equation describe three independent spin states. The energy spectrum is calculated for each of these sets for the case of quasi-two-dimensional electrons in a quantum well. It is shown that the standard Rashba spin-orbit interaction takes place in one of such states only. In another one this interaction is not formed at all, and for the third one it leads to the band spectrum which is anisotropic in the plane domain of the propagation of a free electron and is different from the isotropic spectrum of Rashba type.