Empirical two-band model for quantum wells and superlattices in an electric field.

Abstract

This paper considers in detail the empirical two-band model formulated by Nelson et al. [Phys. Rev. B 35, 7770 (1987)] for the electronic states of semiconductor quantum wells and superlattices. The model is also extended to the case where the structure of interest is in an external potential. It is shown that one can define probability and probability current densities such that a continuity equation is satisfied and that solutions corresponding to different energies are orthogonal. Expressions are derived for the oscillator strengths of interband transitions and of intersubband transitions within the conduction band. The pair of coupled first-order differential equations resulting from the model can be recast into a single, second-order Schr\"odinger equation with an energy- and position-dependent effective mass. For a uniform electric field, it is shown that analytic solutions to this equation can be obtained with an error of order (\ensuremath{\gamma}F${)}^{2}$, where \ensuremath{\gamma} is the nonparabolicity parameter and F is the electric field. For a 200-\AA{} rectangular GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As quantum well, results are presented for electric-field-dependent conduction-subband energies, envelope functions, interband oscillator strengths, and tunneling resonance widths. These results are compared with the corresponding results obtained by a direct numerical integration of the two-band-model Schr\"odinger equation and with results obtained using the single-band envelope-function approximation.