The Mathematics
In a periodic potential V(x) = V(x + a), the electron wavefunctions satisfy Bloch's theorem:
ψₙₖ(x) = e^(ikx) · uₙₖ(x)
where uₙₖ(x) has the same periodicity as the lattice. The energy Eₙ(k) forms continuous functions of k within each band n: but different bands are separated by gaps.
For a 1D crystal with potential V(x) = V₀ cos(2πx/a), we solve the Schrödinger equation using a plane-wave expansion. The Hamiltonian in the plane-wave basis is:
H_{mn} = (ℏ²(k + mG)²/2m)δ_{mn} + (V₀/2)(δ_{m,n±1})
where G = 2π/a is the reciprocal lattice vector. Diagonalise this matrix for each k, and you get the band structure.
Band gaps and material properties
The Fermi level, the highest occupied energy at zero temperature, determines everything:
- Conductor: Fermi level cuts through a band. Electrons at the Fermi surface can accelerate → current flows.
- Insulator: Fermi level sits in a large band gap (> 4 eV). Electrons can't reach the next band → no conduction.
- Semiconductor: Fermi level sits in a small band gap (0.1–2 eV). Thermal excitation can promote electrons → modest conduction that increases with temperature.
Further reading
- Kittel, Introduction to Solid State Physics — the standard undergraduate text.
- Electronic band structure (Wikipedia)
- Ashcroft & Mermin, Solid State Physics — more rigorous treatment.