Unified treatment of bound, scattering, and resonant states in one-dimensional semiconductor nanostructures

Abstract

An exact and unified method is developed for finding a complete solution of the one-dimensional Schrödinger equation at any complex energy and for an arbitrary potential profile. This includes obtaining the binding energies, resonance energies and widths, transmission and reflection amplitudes, as well as the corresponding wave functions. In addition to finding the total widths of resonances, a simple but exact procedure is proposed for calculating their partial widths that determine relative probabilities of resonance decaying into (or exiting from) the left and right channels. The method is based on a direct calculation of the Jost matrix together with the Jost solutions of the Schrödinger equation. A combination of the variable-constant method with the complex coordinate rotation is used to replace this equation with an equivalent system of linear first-order differential equations whose solutions, taken at long distances from the interaction region, form the Jost matrix. The effectiveness and accuracy of the method are demonstrated by several numerical examples where the motion of particles through quantum-well semiconductor heterostructures as well as in a potential with infinite tails is considered.