In previous chapters we examined the behavior of electrons in solids when they are in their ground state, or when they are excited within the single-particle picture, from a state of energy below to one above the Fermi level. In all these discussions we did not explicitly consider the motion of electrons. In the present chapter, we shall study the dynamics of electrons in solids, that is, how they move as a function of time under the influence of external fields.

Up to now we have focused mainly on the behavior of electrons in a solid, given the fixed positions of all the ions. This is justified in terms of the Born–Oppenheimer approximation, and allowed us to study a number of interesting phenomena, including the interaction of solids with light. We next move on to a detailed account of what happens when the ions in a solid move away from their positions at equilibrium, that is, the lowest-energy structure of the entire system. We will again employ the Born–Oppenheimer approximation which is valid for most situations since the typical time scale for motion of ions, ${10}^{-15}$ sec (femtoseconds), is much slower than the relaxation time of electrons, ${10}^{-18}$ sec (attoseconds).

In thewe explored the structure of solids from a general viewpoint, emphasizing the crystalline nature of most common materials. We also described two simple, yet powerful, ways of understanding how sharing of electrons between the atoms that constitute the solid can explain the nature of bonds responsible for the cohesion and stability of the solid: these were the free-electron model, appropriate for metallic bonding, and the formation of bonding and anti-bonding combinations of atomic orbitals that can capture the essence of covalent bonding. Other types of bonding that we encounter in solids can be thought of as combinations of these two simple concepts. In this chapter we will explore the constraints that the periodic arrangement of atoms in a crystal impose on the behavior of electrons. The essence of these constraints is captured by Bloch’s theorem, a cornerstone of the mathematical description of electronic behavior in crystals.

In thewe considered the quantized vibrations of ions in a crystalline solid, the phonons, as independent “particles.” We used this concept to investigate some of the properties of solids related to ionic vibrations, like the specific heat and the thermal expansion coefficient. In this chapter we look at the interaction of phonons with other quasiparticles, like electrons, and other particles, like neutrons, as well as with photons, the quanta of the electromagnetic field.

Our goal in this chapter is to examine the electronic properties of several representative solids. To obtain the electronic band structure of these solids, we employ the density functional theory approach discussed in , which we solve numerically using the methods discussed in , , and . We will also rely heavily on ideas from the tight-binding approximation, which will help us understand the physics in a more transparent way.