Nonsymmetrized Hamiltonian for semiconducting nanostructures in a magnetic field

Abstract

In the framework of the Burt-Foreman theory a nonsymmetrized eight-band effective-mass Hamiltonian is derived for nanostructures in the presence of a magnetic field. The Hamiltonian is tested for the case of a cylindrical quantum dot with parabolic in-plane confinement potential in a perpendicular magnetic field. We compare the results of our nonsymmetrized model to the single-band and conventional multiband calculations, where ad hoc symmetrization is used. The model is tested on $\mathrm{Ga}\mathrm{As}∕{\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}\mathrm{As}$, $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{As}$, and $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dots, where strain is not included in the model in order to resolve the influence of the boundary on the electronic structure. In structures with a large difference of Luttinger parameters between the constituent materials, such as $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dots, the conventional multiband models lead to unphysical high magnetic-field solutions that are substantially different from those obtained from the nonsymmetrized Hamiltonian and single-band model for the ground state. A similar behavior is observed for the case of $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum wells, where energy levels as a function of ${k}_{t}$ are analyzed. This discrepancy is attributed to an overestimation of band mixing in conventional models because of the inappropriate treatment of the boundary.