k⋅ptheory of semiconductor superlattice electronic structure in an applied magnetic field

We present a k⋅p theory of semiconductor superlattices in an applied magnetic field. We consider superlattices with a [001] growth axis and the magnetic field along the growth axis. A single-basis set for the constituent materials is provided by a zone-center pseudopotential calculation with a reference Hamiltonian. The ${\Gamma }_{15}$ valence and ${\Gamma }_1$ conduction states are coupled with a spinor and treated explicitly. Nearby energy states are treated in Löwdin perturbation theory with the k⋅p operator and the difference between the material pseudopotential and the reference pseudopotential as the perturbation. The calculation is carried out consistently to first order in wave functions and second order in energies. Magnetic, exchange (in semimagnetic materials), spin-orbit, and strain (in strained-layer superlattices) interactions are included between the explicitly included states. When inversion-asymmetry and warping terms are dropped in the Hamiltonian, a Landau index becomes a good quantum number. Bloch and evanescent states are computed for a fixed Landau index in each material. Interface matching of the constituent-material bulk eigenfunctions is accomplished with use of results derived for the normal component of the current density operator. The Landau indices are not mixed by the interface matching. Superlattice translational symmetry is used to derive an eigenvalue equation for the superlattice wave vectors and eigenfunctions. The numerical implementation of the formal results is described and used to investigate a nonmagnetic superlattice ${\mathrm{Ga}}_{0.47}$ ${\mathrm{In}}_{0.53}$As/${\mathrm{Al}}_0$.${48\mathrm{I}\mathrm{n}}_{0.52}$As and a semimagnetic superlattice ${\mathrm{Hg}}_{0.95}$ ${\mathrm{Mn}}_{0.05}$Te/${\mathrm{Cd}}_{0.78}$ ${\mathrm{Mn}}_{0.22}$Te.