Question

Explain the energy bands in solids.

Hint:

Discrete atomic levels split and merge into valence and conduction bands separated by a forbidden gap Eg; the size of Eg (0, about 1 eV, or large) classifies metals, semiconductors and insulators.

Answer 1

Step 1: From single atom to crystal.
Picture bringing \(N\) identical atoms from far apart to the tight spacing of a crystal. When they are far away each has the same set of discrete levels. As they approach, the wavefunctions of the outer electrons overlap, and to keep every electron in a distinct quantum state (Pauli principle) each single level fans out into \(N\) finely separated levels. With \(N \sim 10^{23}\) the spread is effectively continuous, giving a band of allowed energies.

Step 2: The two bands that matter.
Only the highest occupied band (the valence band) and the next empty band (the conduction band) decide electrical behaviour. Conduction requires empty nearby states for electrons to move into; a completely full band cannot carry net current.

Step 3: Read the gap \(E_g\).
The energy width separating these two bands, the forbidden gap \(E_g\), is the key. It is the energy an electron must gain to jump from the filled valence band into the empty conduction band.

Step 4: Three outcomes.
If the bands overlap (\(E_g = 0\)) plenty of free states are always available and the solid is a metallic conductor. If \(E_g\) is large (several eV) thermal energy at room temperature (\(\sim 0.025\) eV) cannot lift electrons across, so the solid is an insulator. If \(E_g\) is only about 1 eV, a small but useful number of electrons are thermally promoted, making the solid a semiconductor whose conductivity rises with temperature.
\[\boxed{\text{Overlap} \to \text{metal};\ \ E_g\sim1\,\text{eV} \to \text{semiconductor};\ \ E_g\ \text{large} \to \text{insulator}}\]