Resonant tunneling time delay and quantum well sheet density

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.