# 9.5 Band theory of solids

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By the end of this section, you will be able to:
• Describe two main approaches to determining the energy levels of an electron in a crystal
• Explain the presence of energy bands and gaps in the energy structure of a crystal
• Explain why some materials are good conductors and others are good insulators
• Differentiate between an insulator and a semiconductor

The free electron model explains many important properties of conductors but is weak in at least two areas. First, it assumes a constant potential energy within the solid. (Recall that a constant potential energy is associated with no forces.) [link] compares the assumption of a constant potential energy (dotted line) with the periodic Coulomb potential, which drops as $-1\text{/}r$ at each lattice point, where r is the distance from the ion core (solid line). Second, the free electron model assumes an impenetrable barrier at the surface. This assumption is not valid, because under certain conditions, electrons can escape the surface—such as in the photoelectric effect. In addition to these assumptions, the free electron model does not explain the dramatic differences in electronic properties of conductors, semiconductors, and insulators. Therefore, a more complete model is needed. The periodic potential used to model electrons in a conductor. Each ion in the solid is the source of a Coulomb potential. Notice that the free electron model is productive because the average of this field is approximately constant.

We can produce an improved model by solving Schrödinger’s equation for the periodic potential shown in [link] . However, the solution requires technical mathematics far beyond our scope. We again seek a qualitative argument based on quantum mechanics to find a way forward.

We first review the argument used to explain the energy structure of a covalent bond. Consider two identical hydrogen atoms so far apart that there is no interaction whatsoever between them. Further suppose that the electron in each atom is in the same ground state: a 1 s electron with an energy of $-13.6\phantom{\rule{0.2em}{0ex}}\text{eV}$ (ignore spin). When the hydrogen atoms are brought closer together, the individual wave functions of the electrons overlap and, by the exclusion principle, can no longer be in the same quantum state, which splits the original equivalent energy levels into two different energy levels. The energies of these levels depend on the interatomic distance, $\alpha$ ( [link] ).

If four hydrogen atoms are brought together, four levels are formed from the four possible symmetries—a single sine wave “hump” in each well, alternating up and down, and so on. In the limit of a very large number N of atoms, we expect a spread of nearly continuous bands of electronic energy levels in a solid (see [link] (c)). Each of these bands is known as an energy band    . (The allowed states of energy and wave number are still technically quantized, but for large numbers of atoms, these states are so close together that they are consider to be continuous or “in the continuum.”)

Energy bands differ in the number of electrons they hold. In the 1 s and 2 s energy bands, each energy level holds up to two electrons (spin up and spin down), so this band has a maximum occupancy of 2 N electrons. In the 2 p energy band, each energy level holds up to six electrons, so this band has a maximum occupancy of 6 N electrons ( [link] ).

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