Exact eigenfunctions of a two-band semiconductor in a uniform electric field

In this paper we give exact wavefunctions in the envelope-function approximation for a model system consisting of a symmetric two-band semiconductor in a uniform electric field. The method of solution proceeds by analogy with the relativistic electron (Dirac equation). The exact solutions, which are Weber functions, are characterized with respect to their limiting forms for narrow bandgaps, wide bandgaps and low fields. Results of this analysis include re-derivation of the interband tunnelling probability and a corrected plane-wave approximation for the wavefunctions. A simple expression for the current density is derived from the effective Hamiltonian and its eigenfunctions. The heterostructure transport problem is formulated, leading to an analytical expression for the transmission through an arbitrary piecewise-linear potential. As an application of the results, the quantum-confined Stark effect is computed and compared with that obtained in the one-band model. The interband coupling further lowers the conduction subbands as the field is increased.