The Kronig-Penney model is a 1D square-well approximation for a per...

The Kronig-Penney model is a 1D square-well approximation for a periodic array of charged nuclei, as shown below. d(x) d, - E X - -C +b +a Period =a In terms of the parameters a2 = 2mE/ h2 and B2 = - the Schrödinger equation can be written as dty + d2 = 0 05x5b dx² dty -B2=0 = -Cdx2 where E is the electron wavefunction energy and we assume E<©0. From Bloch's theorem, 4(x) = u(x)etter where u(x) is a periodic function with period a: u(+++ a) = u(x) . In terms of the parameters a2 = 2mE/n² and B2 = - the Schrödinger equation can be written as dty + ² 4 = 0 05x5b dx² d 4 B24 = 0 -Cdx2 where E is the electron wavefunction energy and we assume E<©0. From Bloch's theorem, 4 (x) = u(x)eth where u(x) is a periodic function with period a: u(x+a)=1(x). = a) Both 4 and dy dx must be continuous in order for the Schrödinger equation to have finite energies. Find general forms for 4 in the barrier and the well and also find four equations from boundary conditions that solve for the undetermined coefficients. b) Solve this system of four equations (possibly by setting a determinant to zero) to find an equation of the form f(b,c,a,B) = cos(ak). c) Plot f(E) versus E for representative values a = 1nm, b = 0.9nm, and Do =0.5, 5, 15, and 50eV (1 eV = 1.602x10-19 J). Label the allowed ranges of energy and the forbidden ranges (gaps). d) Plot the dispersion relation, i.e. the electron energy values as a function of k, for the representative values of part (c). On each of these plots, also draw the free-electron E VS. k. e) Redo part (b) if Do is very small. This is the case for most metals, such as alkali metals where electrons are very weakly bound. What are the allowed energies for a given k? What does this mean physically for the electron motion? Does this help explain the plot in (d) for small DO? f) Redo part (b) if Do is very large. This is the case for solids in which electrons are tightly bound to ion cores. What are the allowed energies for a given k? What does this mean physically for the electron motion? Does this help explain the plot in (d) for large Do?