Resonant tunneling time delay and quantum well sheet density

Abstract
Time delays, dwell times, resonant state lifetimes, and electron sheet densities associated with tunneling through quantum well structures are analyzed from a scattering theory (S-matrix) viewpoint. Some of the results differ from intuitively motivated expressions which have appeared in the resonant tunneling literature. It is shown that the sheet density is given by a formula similar to the Tsu–Esaki formula [Appl. Phys. Lett. 22, 562 (1973)] for current density. Sheet density and dwell time are related to Hermitian matrices which are expressed in terms of the S-matrix and in terms of partial widths associated with resonant states.