Real-space approach to the multicomponent-envelope-function problem in semiconductor heterostructures.

Abstract

We describe a real-space numerical method for the solution of the multicomponent-envelope-function problem in semiconductor heterostructures. The method, based on a shooting technique, provides, with a very modest computational effort, an exact solution for arbitrarily shaped one-dimensional confining potentials, including high in-plane magnetic fields. Boundary conditions at interfaces are automatically taken into account. We apply our method to the 4\ifmmode\times\else\texttimes\fi{}4 Luttinger Hamiltonian and indicate how larger k\ensuremath{\cdot}p Hamiltonians can be implemented. To demonstrate the flexibility of the method, we show the calculated hole subbands and envelope functions in a GaAs-${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As quantum well with a magnetic field parallel to the interfaces and with an applied bias along the growth direction.