Tight-Binding Model

hopping integrals · bandwidth · van Hove singularities

t = 1.00 eV
ε₀ = 0.00 eV
N = 20
Physics: Tight-binding 1D chain: E(k) = ε₀ − 2t·cos(ka), bandwidth W = 4t. Effective mass at band bottom: m* = ℏ²/(2ta²), negative at band top. Density of states: g(E) = N/(π·√(4t²−(E−ε₀)²)) — diverges (van Hove singularity) at band edges E = ε₀ ± 2t. 2D square lattice: logarithmic van Hove singularity at E = ε₀. Graphene: two triangular sublattices, bands E(k) = ±t|f(k)|, touch linearly at K/K' (Dirac cone). Bandwidth: cuprates ≈ 0.3 eV vs free electron bandwidth ≈ 10 eV — strong correlations in narrow bands.