Nonparabolicity effects in a quantum well: Sublevel shift, parallel mass, and Landau levels

The influence of nonparabolicity on the subband structure in a quantum well is analyzed. Starting from an accurate expression for the bulk conduction-band structure expanded up to fourth order in k, we determine both the shift of the confinement energies and the energy dispersion parallel to the layers E${(\mathrm{k}}_{?}$). The resulting eigenvalue equations are of the same form as in the parabolic case, but somewhat more complicated. The anisotropy of the bulk conduction band is included, and it is found to have a larger effect in quantum wells than in the bulk. The results can be expressed in terms of the perpendicular mass, which is relevant for the determination of confinement energies, and the parallel mass, which gives the curvature of E${(\mathrm{k}}_{?}$) at the bottom of a subband. We derive approximate expressions for these masses in the form of explicit functions of the confinement energy, which is experimentally accessible. The enhancement of the parallel mass relative to the bulk mass is found to be 2–3 times stronger than that of the perpendicular mass. It is shown that the boundary conditions need to be modified in the nonparabolic case. The nonintuitive result is that the confinement energy for the ground state usually is increased relative to a similar calculation in the parabolic approximation. We include the effect of a perpendicular magnetic field and derive an analytic expression for the Landau levels. The cyclotron mass is found to increase with magnetic field and approach the parallel mass in the limit of small magnetic fields. The parallel mass is also relevant for transport parallel to the layers, density of states, and exciton properties. The agreement with experiment is encouraging. Previous theoretical approaches are critically reviewed and the differences and similarities with this work are pointed out.