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Kronig-Penney Model

A simplified model using square potential barriers to approximate the periodic potential and calculate energy band structure, demonstrating the concept of band gaps.
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The statement of the theorem

Define the periodic potential V(x)V(x) as a sequence of rectangular barriers of width aa and height V0V_0, separated by wells of width bb, such that the period is L=a+bL = a+b. The transmission coefficient TT for an electron with energy EE across one period is given by the transfer matrix method. The condition for wave propagation (i.e., T=1T=1) leads to the transcendental equation:\ncos⁡(βL)=cos⁡(βb)cosh⁡(κa)−12γ(γsinh⁡(κa)sin⁡(βb))\cos(\beta L) = \cos(\beta b) \cosh(\kappa a) - \frac{1}{2 \gamma} \left( \gamma \sinh(\kappa a) \sin(\beta b) \right) \nwhere β=2mE/ℏ\beta = \sqrt{2mE}/\hbar, γ=V0/E−1\gamma = V_0/E - 1, and κ=2m(V0−E)/ℏ\kappa = \sqrt{2m(V_0 - E)}/\hbar. Allowed energies EE correspond to real values of βL\beta L, while forbidden energies correspond to complex βL\beta L.
Source: Wikipedia