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 wave vector. 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 wave vector, while the $z$ component of the spin-polarization vector is an odd function of the Bloch wave vector. At the band edges, the optical transition rates between adjacent bands are nonzero.

• Received 26 September 2017
• Revised 20 January 2018

DOI:https://doi.org/10.1103/PhysRevB.97.085430