ABSTRACT
In the last chapter, we had reduced the many-body problem of interacting electrons and ions to a one-body problem using the DFT. This led to the KS equation
− ∇ +
=
v r r r e
eff i i i( ) ( ) ( )y ey
(4.1)
The knowledge of eigenvalues εi and eigenfunctions ψi is necessary to understand various properties of solids such as transport, optical, magnetic, and superconducting properties. In the next three chapters, we shall discuss how to solve Equation 4.1 for the periodic or crystalline solids. In this chapter, we shall discuss general consequences of a periodic potential on eigenfunctions and eigenvaules of Equation 4.1. We shall also discuss the effect of various symmetries such as inversion symmetry, time-reversal symmetry, and so on. In this chapter, we shall introduce the basic terminology used in discussing the electronic structure of solids. In Section 4.2, we start with the crystal potential and then in Section 4.3, we discuss the Bloch’s theorem. In Section 4.4, we introduce the concept of Brillouin zone. In Section 4.5, we discuss spin-orbit interaction and in Section 4.6, symmetry in crystals. In Section 4.7, inversion symmetry, time-reversal symmetry, and Kramers’ theorem are discussed. In Section 4.8, we introduce the concepts of band structure and Fermi surface. In Section 4.9, we define and develop the concepts of density of states, local density of states, and projected density of states, and in Section 4.10, we discuss the charge density. Finally, in Section 4.11, we briefly discuss the Brillouin zone integration. This chapter only covers periodic solids at absolute zero temperature. Calculations for nonperiodic solids are much more complicated and will be discussed in Chapters 8 and 11.
