4.1 Introduction

In the first two chapters of this book we learned about the way a particle moves in a potential. Because in quantum mechanics particles have a wavy character, this modifies how they scatter from a change in potential compared with the classical case. In Section 3.8 we calculated transmission and reflection of an unbound particle from a one-dimensional potential step of energy V ₀. The particle was incident from the left and impinged on the potential barrier with energy E > V ₀. Significant quantum mechanical reflection probability for the particle occurred because the change in particle velocity at the potential step was large. This result is in stark contrast to the predictions of classical mechanics in which the particle velocity changes but there is no reflection.

In Section 3.10 we applied our knowledge of electron scattering from a step potential to the design of a new type of transistor. The analytic expressions developed were very successful in focusing our attention on the concept of matching electron velocities as a means of reducing quantum mechanical reflection that can occur at a semiconductor heterointerface. In this particular case, it is obvious that we could benefit from a model that is capable of taking into account more details of the potential. Such a model would be a next step in developing an accurate picture of transistor operation over a wide range of voltage bias conditions. We need a method for finding solutions to complicated potential structures for which...