3 - Quantum theory of heterostructures

Summary

… I think I can safely say that nobody understands quantum mechanics.

The Character of Physical Law, Richard Feynman

Introduction

Modern technology has made possible the growth of thin crystalline epitaxial layers of different semiconductors. These epitaxial layers can be as small as a few lattice parameters. For small heterostructures (100 Å or less) a quantum treatment of heterostructures becomes necessary. In this chapter we will attempt to build a quantum picture of heterostructures. Note that the conventional quantum picture of a semiconductor crystal cannot be applied to rapidly varying semiconductor heterostructures since crystals are defined as periodic structures extending up to infinity. New theoretical techniques are thus required to describe these ‘spatially-varying crystals’.

Our quantum picture will be based upon an envelope model. An envelope model focuses on calculating the relative distribution of the wave-function from atomic cell to atomic cell rather than on the detailed distribution of the electron wave-function in each atomic cell.

The particular envelope picture we shall use is the so-called generalized Wannier picture. The generalized Wannier picture is capable of handling both the concept of band structure and the concept of its variation in space in a rigorous fashion. This model will therefore permit us to understand the impact of the interface upon the band structure in a heterojunction.

Other envelope pictures have been developed such as the Ben Daniel Duke Hamiltonian (effective-mass model [10], see also [11]), the k · p envelope model ([11]), and the tight-binding model ([8]).