The Crystal Lattice and Reciprocal Space

Abstract

In this chapter we start to discuss topics that are specific to crystalline solids and apply to the large majority of inorganic semiconductors. In particular we lay out the basis for the following chapters where we consider elementary excitations and so-called quasi-particles in semiconductors. These will be needed to describe and understand the linear optical properties. The properties of the quasi-particles themselves are governed by the underlying periodicity of the crystalline lattice. The latter is in particular reflected by the dispersion of the quasi-particles in reciprocal space.

Problems

9.1

Inspect or build some lattice and crystal models to become familiar with the topics presented in Sect. 9.1.

9.2

Show that the reciprocal lattice of a face-centered cubic (fcc) lattice is a body centered cubic (bcc) lattice and vice versa.

9.3

Calculate and draw the primitive unit cell and the Wigner–Seitz cell in real space and the first three Brillouin zones in reciprocal space for a simple cubic and a hexagonal two-dimensional lattice.

9.4

Inspect a model of a cubic crystal (e.g., zinc-blende). Find the non primitive cubic unit cell and the primitive one. Explain qualitatively that such a crystal should be optically isotropic for light propagating ($\varvec{k} \ne 0$!) e.g., in the directions (100) or (111) but not in (110).