The stationary phase method for a wave packet in a semiconductor layered system. The applicability of the method

Abstract

Using the formal analysis made by Bohm [D. Bohm, Quantum Theory (Dover, New York, 1979)] to calculate approximately the phase time for a transmitted and the reflected wave packets through a potential barrier, we calculate the phase time for a semiconductor system formed by different mesoscopic layers. The transmitted and the reflected wave packets are analyzed and the applicability of this procedure, based on the stationary phase of a wave packet, is considered in different conditions. For the applicability of the stationary phase method an expression is obtained in the case of the transmitted wave depending only on the derivatives of the phase, up to third order. This condition indicates whether the parameters of the system allow us to define the wave packet by its leading term. The case of a multiple barrier systems is shown as an illustration of the results. This formalism includes the use of the transfer matrix to describe the central stratum, whether it is formed by one layer (the single barrier case), or two barriers and an inner well (the double barrier resonant tunneling semiconductor structure system), but one can assume that this stratum can be comprise of any number or any kind of semiconductor layers.