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.