Real-space approach to the multi-component envelope function problem in semiconductor heterostructures

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 4×4 Luttinger Hamiltonian and indicate how larger k⋅p Hamiltonians can be implemented. To demonstrate the flexibility of the method, we show the calculated hole subbands and envelope functions in a GaAs-AlxGa1-xAs quantum well with a magnetic field parallel to the interfaces and with an applied bias along the growth direction.