General solution of the Dirac equation for quasi-two-dimensional electrons

TL;DR

This paper derives a comprehensive analytical solution to the Dirac equation for electrons in quasi-two-dimensional asymmetric quantum wells, revealing spin-dependent energy spectra and enabling advanced spin control in spintronics.

Contribution

It presents the first general analytical solution for the Dirac equation in this context, incorporating free parameters for continuous transformation of solutions and detailed cases for different quantum well configurations.

Findings

Exact energy spectrum depends on electron spin polarization.

Analytical expressions for effective mass and Rashba coefficients.

Conditions for specific spin states are established.

Abstract

The general solution of the Dirac equation for quasi-two-dimensional electrons confined in an asymmetric quantum well, is found. The energy spectrum of such a system is exactly calculated using special unitary transformation and shown to depend on the electron spin polarization. The general solution, being the only one, contains free parameters, whose variation continuously transforms one known particular solution into another. As an example, two different cases are considered in detailL: electron in a deep and in a strongly asymmetric shallow quantum well. The effective mass renormalized by relativistic corrections and Bychkov-Rashba coefficients are analytically obtained for both cases. The general solution allows - independently on the existence of the spin invariants - to establish conditions at which a specific (accompanied or non-accompanied by Rashba splitting) spin state can be realized. In principle, this opens new possibilities of the spin degree of freedom control in spintronics via synthesis of heteroctructures of the desirable properties.