Generalization of the WKB Theory for Tunneling between Two Different One-Dimensional Periodic Potentials

A general analysis of tunneling in a one-dimensional heterojunction is given. The analysis is given in terms of a generalization of the WKB theory specifically developed to apply to potentials which, as $x$ tends to $\pm \infty$, tend to periodic functions of position, rather than to constants. The theory is particularly suited for narrow junctions. The tunneling probability is shown to factor into a bulk factor which is proportional to the product of the group velocities in the periodic potentials on both sides of the junction, and into a barrier factor. The latter depends primarily on solutions of the barrier Hamiltonian, and only for a thick junction may it be approximated by a simple exponential.