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Band Gap

A region of forbidden energy levels between the valence and conduction bands, where electrons cannot exist in the solid state.
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The statement of the theorem

A band gap is a range of forbidden energies ΔE=(Egap,min,Egap,max)\Delta E = (E_{gap, min}, E_{gap, max}) such that the time-independent Schrödinger equation, (−ℏ22m∇2+V(r)−E)ψ=0\left(-\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}) - E\right) \psi = 0, possesses no solution ψ\psi satisfying the boundary conditions of the crystal lattice for any crystal momentum k\mathbf{k}. Formally, the gap exists when the determinant condition derived from the periodic potential yields no real solution for the wave vector β\beta for a given energy EE. The gap width is ΔE=Egap,max−Egap,min\Delta E = E_{gap, max} - E_{gap, min}.