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

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 spinorbital 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.

The impacts of the quantum-dot confining potential on the spin-orbit effect

“…Note that all of the four bulk wave functions belong to the '−' branch. In the coordinate region |x| < a, the eigenfunction reads [35] Ψ(x) = c 1 sin θ1…”

“…The bulk spectrum and the corresponding bulk wave functions in the energy region E b ≥ − 1 2 mα 2 − ∆ 2 2mα 2 can be found elsewhere [35]. The bulk spectrum of planewave solution reads [35]…”

“…Inside the well |x| < a, the eigenfunction Ψ(x) of Hamiltonian (1) can be expanded in terms of the four degenerate bulk wave functions [35]. However, outside the well |x| > a for the FSW (classical forbidden region), the electron must have a dissipative energy E b < − 1 2 mα 2 − ∆ 2 2mα 2 , otherwise, the bound state can not be formed.…”

“…The eigenfunction Ψ(x) can be written as a linear combination of these four degenerate bulk wave functions, i.e., two from the '+' branch and two from the '−' branch. In the coordinate region |x| < a, the eigenfunction reads [35] Ψ(x) = c 1 cos θ1…”

Searching strong `spin'-orbit coupled one-dimensional hole gas in strong magnetic fields

“…While the low-energy subband dispersions of the 1D hole gas in a cylindrical Ge nanowire are quite different [24]. The lowest two subband dispersions of the 1D hole gas anticross with each other at the center of the k z space [24,25], and the shape of the dispersion is very similar to that of a strong spin-orbit coupled 1D electron gas described by the Hamiltonian H c = p 2 /(2m * e ) + ασ z p + g e µ B Bσ x /2 [26,27]. However, there is an additional spin degeneracy in the hole subband dispersions [25].…”