3. Band Structure In this chapter, we start our journey into the world of condensed matter physics. This is the study of the properties of "stuff". Here, our interest lies in a particular and familiar kind of stuff: solids. Solids are collections of tightly bound atoms. For most solids, these atoms arrange themselves in regular patterns on an underlying crystalline lattice. Some of the elec- trons of the atom then disassociate themselves from their parent atom and wander through the lattice environment. The properties of these electrons determine many of the properties of the solid, not least its ability to conduct electricity. One might imagine that the electrons in a solid move in a fairly random fashion, as they bounce from one lattice site to another, like a ball in a pinball machine. However, as we will see, this is not at all the case: the more fluid nature of quantum particles allows them to glide through a regular lattice, almost unimpeded, with a distorted energy spectrum the only memory of the underlying lattice. In this chapter, we will focus on understanding how the energy of an electron depends on its momentum when it moves in a lattice environment. The usual formula for kinetic energy, E = +mv2 = p2/2m, is one of the first things we learn in theoretical physics as children. As we will see, a lattice changes this in interesting ways, the consequences of which we will explore in chapter 4. 3.1 Electrons Moving in One Dimension We begin with some particularly simple toy models which capture much of the relevant physics. These toy models describe an electron moving in a one-dimensional lattice. We'll take what lessons we can from this before moving onto more realistic descriptions of electrons moving in higher dimensions. 3.1.1 The Tight-Binding Model The tight-binding model is a caricature of electron motion in solid in which space is made discrete. The electron can sit only on the locations of atoms in the solid and has some small probability to hop to a neighbouring site due to quantum tunnelling. To start with our "solid" consists of a one-dimensional lattice of atoms. This is described by N points arranged along a line, each separated by distance a. - 65 -
a Consider a single electron moving on this lattice. We will assume that the electron can only sit on a given lattice point; it's not allowed to roam between lattice points. This is supposed to mimic the idea that electrons are bound to the atoms in a lattice and goes by the name of the tight-binding approximation. (We'll see exactly what we're neglecting in this approximation later.) When the electron sits on the nth atom, we denote the quantum state as |n). These states are considered orthogonal to each other, so (n|m) =Inm Clearly the total Hilbert space has dimension N, and is spanned by |n) with n = 1, ... , N. What kind of Hamiltonian will govern the dynamics of this electron? If