A comparison using a delta-function model of envelope function approximations for quantum wells

This paper compares four versions of the effective-mass approximation to the exact atomic wave function, as determined from a one-dimensional model potential, in predicting the energies of the bound states in a single quantum well such as those formed in compound semiconductor heterostructures. The model potential consists of delta-function barriers between atoms with a ‘‘well’’ in the strength of these delta functions. The model yields simple, closed-form solutions for the bound state energy. The corresponding effective-mass model that uses the calculated dispersion relations and continuity of the wave function derivatives at the well boundaries predicts bound state energies indistinguishable from those predicted by the exact solution. The parabolic-band model overestimates the lowest energy level of a 0.4-eV-deep, 50-Å-wide well by about 15%. Continuity of 1/m* times the wave functions underestimates the energies by about the same amount.