Band theory of solids

A single isolated atom has sharp, discrete allowed energy levels for its electrons, with well-defined gaps between them, exactly as quantum mechanics predicts for a confined system. Bring a large number of identical atoms close together into a crystal, and this picture changes dramatically. Neighbouring atoms' outer electron wavefunctions begin to overlap, and by the Pauli exclusion principle, no two electrons in the combined system can share exactly the same quantum state. What was a single sharp energy level in the isolated atom splits, in the crystal, into an enormous number of extremely closely spaced sub-levels — one for each atom's contribution to that shared level — and with a macroscopic number of atoms (on the order of 10²³ per cubic centimetre), these sub-levels merge into what is, for all practical purposes, a continuous allowed energy band.

Not every energy is allowed, however. Between one allowed band and the next lies a forbidden band gap, an energy range that a crystal electron simply cannot have, a direct consequence of the crystal lattice's periodicity acting on the electron wavefunctions. The resulting picture of alternating allowed bands and forbidden gaps is band theory, and it explains far more about a solid's electronic behaviour than atomic energy levels ever could on their own.

Whether a material conducts electricity, semiconducts, or insulates is determined entirely by how these bands are filled and how large the gaps between them are. In a conductor such as a metal, the highest occupied band (the valence band) is only partially filled, or overlaps with the next band up, so electrons near the top of the filled levels can move into neighbouring, only slightly higher-energy states under an applied electric field with almost no resistance — this is what makes metals conduct so well. In an insulator, the valence band is completely filled and separated from the next empty band (the conduction band) by a large band gap — several electronvolts — that ordinary thermal energy at room temperature cannot bridge, so essentially no electrons are available to carry current. A semiconductor has exactly the same qualitative band structure as an insulator — a filled valence band separated from an empty conduction band by a gap — but with a much smaller gap, typically around one electronvolt, small enough that a modest number of electrons are thermally excited across it at room temperature, giving semiconductors their intermediate, temperature- and impurity-sensitive conductivity.