Band Gap Opening: Nearly-Free-Electron Model

Periodic Potential

Potential strength V₀ 1.20 eV

Lattice constant a 4.0 Å

Zone Scheme

Electron Probe

Band 1 (lowest)

Crystal momentum k/(π/a) 0.00

Readout

E(k)–

Band gap E_g–

Group velocity v_g–

Effective mass m*/mₑ–

Band 1 Band 2 Band 3

How this works

A free electron in a crystal would have a simple parabolic energy E₀(k)=ħ²k²/2mₑ. But the ion lattice creates a periodic potential V(x)=V₀·cos(2πx/a), and wherever two plane waves k and k−2π/a become degenerate — at the Brillouin-zone boundary k=±π/a — that potential mixes them and splits their energies apart, opening a band gap. This simulator diagonalizes the real plane-wave Hamiltonian (a 9-wave basis) at every k to get the exact nearly-free-electron bands, no shortcuts.

H_nn = ħ²(k+nG)²/2mₑ
H_n,n±1 = V₀/2,  G = 2π/a
E(k) ± gap at k = ±π/a, ±2π/a, …

V₀ — raise it and watch every gap widen; at V₀=0 the bands collapse back onto the free-electron parabola.
Lattice constant a — shrinking a pushes the zone boundary ±π/a outward, squeezing the whole band picture.
Probe k — drag it to the top of band 1, right at the zone boundary: the band flattens, group velocity drops toward zero, and the effective mass swings negative — that's a hole.

Band 1 Band 2 Band 3 V(x) potential Zone boundary

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