The Stationary Phase Method for a Wave Packet in a Semiconductor Layered System. The applicability of the method

TL;DR

This paper applies the stationary phase method to analyze phase times of wave packets in layered semiconductor systems, including multiple barriers, using transfer matrix formalism to determine the method's applicability under various conditions.

Contribution

It extends the stationary phase method to complex layered semiconductor structures and derives conditions for its valid application based on phase derivatives.

Findings

Derived an expression for the transmitted wave phase depending on phase derivatives

Established criteria for the applicability of the stationary phase method in layered systems

Illustrated the approach with multiple barrier system analysis

Abstract

Using the formal analysis made by Bohm in his book, {\em "Quantum theory"}, Dover Publications Inc. 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 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 DBRT system), but one can assume that this stratum can be comprise of any number or any kind of semiconductor layers.