Three energy bands as a function of crystal momentum k. The gaps between them are band gaps — forbidden energies.
The Core Idea
Electrons in a crystal don't have a continuous range of allowed energies — they have bands: intervals of energy where states exist, separated by gaps where no states are allowed. This band structure determines whether a material is a conductor, semiconductor, or insulator. It's the single most important concept in solid-state physics.
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.
Weak potential — bands are nearly free-electron parabolas that barely split at the zone boundaries.
Tuning the potential
The crystal potential V₀ controls the band gap width. Drag the potential slider to see:
- V₀ → 0: Free electrons. Bands overlap, no gaps — the "empty lattice" limit.
- V₀ → large: Deep potential wells, tightly bound electrons. Wide gaps, narrow bands — the tight-binding limit.
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.
Strong potential — wide band gaps and narrow bands. Electrons are nearly localised to individual lattice sites.
The Brillouin zone
Band structure is plotted within the first Brillouin zone — the range k ∈ [−π/a, π/a]. Beyond this, the bands repeat (translational symmetry of the reciprocal lattice). The zone boundaries (k = ±π/a) are where the most interesting physics happens: Bragg reflection opens the gaps.
Further reading
- Kittel, Introduction to Solid State Physics — the standard undergraduate text.
- Electronic band structure (Wikipedia)
- Ashcroft & Mermin, Solid State Physics — more rigorous treatment.