Mathematical Analysis of a Nonparabolic Two-Band Schrödinger-Poisson Problem

Abstract

A mathematical model for the quantum transport of a two-band semiconductor that includes the self-consistent electrostatic potential is analyzed. Corrections beyond the usual effective mass approximation are considered. Transparent boundary conditions are derived for the multiband envelope Schrödinger model. The existence of a solution of the nonlinear system is proved by using an asymptotic procedure. Some numerical examples are included. They illustrate the behavior of the scattering and the resonant states.

Application of the multiband $kp$-models to quantum transport in quasi-periodic crystals

“…The Luttinger-Kohn model has been largely applied to the study of the electronic properties of solids and in particular of semiconductor heterostructures [5,6,17,19]. The reason for the success of this and other similar approaches (which are generally denoted as kp models) is that the expansion (8) factorizes the solution ψ as a product of rapidly varying function (the periodic Bloch functions u n 0, ) whose period scales with the interatomic distance and smooth functions (the envelope functions φ) that vary at the macroscopic scale.…”