Exact energy-dispersion relations for N-well superlattice configurations.

Abstract

Exact energy-dispersion relations for N coupled quantum wells are obtained where N\ensuremath{\ge}1. Each such relation is a composite function of transcendental forms which in turn determine the eigenenergies of the system. This relation is explicitly given for the cases of N=2,3,4 for an even potential with arbitrary barrier and well widths. For the more-standard case of constant barrier and well widths, a band structure emerges with the number of bound states in the outermost band varying from 1 to N. For arbitrary N, results reduce to that of a well of width a or Na in the limits of infinite and zero barrier widths, respectively, where a is the fundamental well width. With variation in well parameters, the number of states at a given value of N varies from a total of one state to a band structure with N states per band. The manner in which the ground state of the configuration varies with N is found. Plots of the dispersion relations for arbitrary N reveal the manner in which these curves merge to the single-well result with increase in barrier width. All dispersion relations are found to be asymptotic, in the limit of large decay wave number, to the same asymptotes as for the single finite well, independent of f, where f represents the ratio of barrier to well widths. Zeros of the dispersion relation merge to zeros of a single well of width a, for large f, and to a well of width Na for small f.