Electron transmission across an interface of different one-dimensional crystals.

Abstract

An exact solution is obtained for the quantum-mechanical reflection and transmission coefficients for the electronic motion across the boundary of two different one-dimensional crystals, each described by a Kronig-Penney model. The result is compared with those obtained in the envelope-function (plane-wave) approximations with different degrees of refinement. It turns out that, in general, neither the simplest such approximation, corresponding to a continuous envelope function \ensuremath{\Phi} and a discontinuous gradient \ensuremath{\Phi}' (with a single-parameter matching of the electron flux ${m}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\Phi}' across the interface), nor the refined procedure, originally proposed by Harrison (corresponding to a two-parameter matching of discontinuous \ensuremath{\Phi} and \ensuremath{\Phi}' subject to the flux-continuity condition), produce an adequate description. Only with a special assumption about the size of the crystal cell at the interface do these procedures provide a consistent approximation; for this case we have derived an explicit expression for the matching coefficients in terms of ``crystal'' structure parameters. For arbitrary sizes of the boundary cell, the exact solution for the reflection coefficient in the vicinity of the band edges can be modeled with a three-parameter envelope-function approximation.