Energy Bands in Solids

Copper conducts, quartz doesn’t, and silicon sits in between, about 10¹¹ times worse than copper and 10¹³ times better than quartz. Odder still, silicon conducts better when it is warmed. Energy bands explain all three.

Three kinds of solid#

In a metal, a field E drives a current density j = σE, with conductivity σ. Heating a metal lowers σ: its free electrons collide more often and gain less drift speed between collisions, so its resistivity ρ = 1/σ rises. An insulator carries almost no current unless the field is enormous. A semiconductor, such as silicon or germanium, conducts weakly, and its conductivity rises with temperature.

From levels to bands#

An isolated atom has sharp energy levels (1s, 2s, 2p, 3s…), each holding a fixed number of electrons by the Pauli principle. Take sodium: 11 electrons as 1s² 2s² 2p⁶ 3s¹. The 3s level has room for two electrons but holds one.

Now gather N sodium atoms, far apart, as in a vapour. There are 2N identical 1s states (all filled), 2N 2s states, 6N 2p states, and 2N 3s states of which only N are filled. Bring the atoms close, as in a solid, and the outer 3s electrons of neighbours start to interact. The 2N states no longer share one energy: some move up, some down, and the sharp level spreads into a band of very closely spaced levels. Inner levels spread far less, because inner electrons barely feel the neighbours.

Who can take energy?

At 0 K electrons fill the bands from the bottom up. At temperature T, collisions trade energies of order kT (about 0.026 eV at 300 K). An electron can only accept a small amount of energy, from heat or from a battery’s field, if there is an empty state just above it. In a completely full band there isn’t, so those electrons can neither absorb heat nor carry current. Only electrons in a partly empty band conduct.

The four band pictures#

The full band is called the valence band, the empty one above it the conduction band. In a semiconductor, thermal collisions lift a few electrons across the small gap. Those electrons have the whole empty conduction band to move in, and the vacancies they leave let valence electrons shuffle too. There are few such electrons, so the conductivity is small, but it grows quickly as the temperature rises.