Energy spectrum, the spin polarization, and the optical selection rules of the Kronig-Penney superlattice model with spin-orbit coupling

TL;DR

This paper analytically solves the energy spectrum and spin polarization of a Kronig-Penney superlattice with spin-orbit coupling, revealing new band gaps and optical selection rules relevant for spintronic and optoelectronic applications.

Contribution

It provides exact solutions for spin-orbital bands and optical transition rules in a modified Kronig-Penney model with spin-orbit coupling and Zeeman field, which was not previously available.

Findings

Exact analytical relations between energy and wavevector are derived.

Large spin-orbital band gaps are identified inside the Brillouin zone.

Optical transition rates are nonzero at band edges, indicating allowed optical transitions.

Abstract

The Kronig-Penney model, an exactly solvable one-dimensional model of crystal in solid physics, shows how the allowed and forbidden bands are formed in solids. In this paper, we study this model in the presence of both strong spin-orbit coupling and the Zeeman field. We analytically obtain four transcendental equations that represent an implicit relation between the energy and the Bloch wavevector. Solving these four transcendental equations, we obtain the spin-orbital bands exactly. In addition to the usual band gap opened at the boundary of the Brillouin zone, a much larger spin-orbital band gap is also opened at some special sites inside the Brillouin zone. The $x$-component of the spin-polarization vector is an even function of the Bloch wavevector, while the $z$-component of the spin-polarization vector is an odd function of the Bloch wavevector. At the band edges, the optical transition rates between adjacent bands are nonzero.