Band theory of solids, a brief introduction

To understand why semiconductors conduct the way they do, it helps to start from how electron energies are distributed in a single, isolated atom. An isolated atom has its nucleus at the centre, surrounded by electrons occupying a number of discrete, well-separated energy levels (Fig. 14.2a).

When a large number of such atoms are packed together to form a solid, the outermost electronic energy levels are no longer the private property of a single atom -- they become shared across all the atoms in the solid. This sharing, and the resulting formation of energy BANDS, can be understood by considering solid sodium as a concrete example. An isolated sodium atom (atomic number 11) has the electronic configuration $1s^2\,2s^2\,2p^6\,3s^1$; its outermost 3s level can hold one more electron but is only half filled.

When solid sodium forms, its atoms interact through their electrons, and the Pauli exclusion principle -- no two electrons can share an identical set of quantum numbers, i.e. no two electrons of the same spin can occupy the same energy level -- governs how the energy levels fill. Each energy level can therefore hold at most two electrons (one spin-up, one spin-down), giving two available states per level. Bringing just two sodium atoms close enough that their outer 3s electrons could belong to either atom forces the single 3s level to split into two closely-spaced sub-levels, so the exclusion principle is not violated (Fig. 14.2b). In actual solid sodium, where atoms sit only about 2-3 angstrom apart, this same splitting happens for every atom simultaneously and for every energy level -- and since the number of atoms in a macroscopic piece of solid sodium is of the order of Avogadro's number, each atomic level splits into that same enormous number of sub-levels. These sub-levels are so closely spaced that the resulting set of levels appears essentially continuous; this continuum of very closely-spaced energy levels is called an ENERGY BAND (Fig. 14.2c), and the individual bands are named after the atomic level they arose from -- the 1s band, 2s band, 2p band, and so on. Outer, more strongly-interacting electrons produce more strongly broadened (wider) bands than the more weakly-affected inner core electrons.

For sodium, the topmost OCCUPIED energy level of the free atom is the 3s level; the corresponding band in the solid is therefore called the VALENCE BAND -- the topmost occupied energy band. Because the atomic 3s level itself was only half filled, the valence band of solid sodium is also only half filled (Fig. 14.3). When electrons in this valence band gain enough energy, they move up into the next available energy level, called the CONDUCTION level; the band formed from these levels is the CONDUCTION BAND. In sodium, the valence band and the conduction band actually OVERLAP -- there is no gap between them at all.

In a semiconductor or an insulator, by contrast, there IS a gap between the top of the valence band and the bottom of the conduction band, called the ENERGY GAP or BAND GAP (Fig. 14.4). It is worth remembering that a band diagram like this represents only the ENERGY of electrons in the solid -- it says nothing about the physical, spatial arrangement of atoms.

At absolute zero, every energy level in every band up to and including the valence band is completely filled in a semiconductor. At some finite temperature T, a few electrons gain thermal energy of order $kT$ (k being the Boltzmann constant). Electrons in bands BELOW the valence band cannot move up, since those higher bands are already fully occupied by other electrons -- only electrons already in the valence band (the topmost occupied band) can be excited across the gap into the empty conduction band, and only if the thermal energy they gain exceeds the band gap. (Electrons can also gain energy from an applied external electric field, though such energy is comparatively small, so typically only electrons already at the very top of the occupied levels can use it to take part in conduction. In sodium's case, because its valence band is only half-filled, electrons there can gain even this small field-energy and move to a slightly higher, still partly-empty level within the same band.) …