The Kronig-Penney model with the limit of a delta function potential

exercise 3_4341

The Kronig-Penney model was presented in lecture. The potential is composed of barriers of width b and height U 0 and wells of width a and zero height. Assume the first barrier for positive x is between a and a + b (i.e., with period a + b ) and that the energy is E < U 0, as shown below.

(a) Beginning with Schrodinger's equation, prove that inside the wells ψ 1 ( x ) = A e iαx + B e −iαx and inside the barriers it is ψ 2 ( x ) = C e βx + De −βx . What are α and β ?

(b) Write down four equations for the four coefficients (A, B, C, D), using continuity of ψ( x ) and its derivatives at x = 0 and periodicity of ψ( x ) and its derivatives at x = −b . Find the matrix Q such that Q [A, B, C, D] = 0.

(c) By setting the determinant of Q = 0 and taking the limit of the delta function potential, one can show that cos( ka ) = F ( αa ) where F ( αa ) º ( P / α a)sin( αa ) + cos( αa ) (You don't have to show this). The delta-function potential is defined such that b approaches zero and U 0 approaches infinity in such a way that P = β 2 ba /2. A plot of F ( αa ) versus αa is shown below as an example for P = 3 p /2. Note that the allowed values of the energy are given by the regions where F ( αa ) is between -1 and +1. These allowed regions lead to the E vs k energy dispersion shown below on the right with energy gaps at ka = n p when F ( αa ) is outside the allowed range (i.e., in the shaded region). For the delta-function potential and with P << 1, find at k = 0 the energy of the lowest energy band.

(d) For the same problem find the band gap at k = π/a.