### A.2 Wenzel-Kramers-Brillouin Method

The WKB method [102] is an approximative semiclassical approach to compute the
stationary solution of the Schrödinger equation without struggling with the
difficulties of a second order differential equation. Taking the time-independent,
one-dimensional Schrödinger equation

as a starting point and inserting the ansatz leads to Substituting the action by its expansion in power series of one obtains for terms of the order and for terms of the order . Integrating (A.20), one obtains which yields where the integration spans from to an arbitrary point . is also referred
to as the classical turning point, where the particle energy equals the potential
energy . Note that close to this point, the WKB approximation breaks down
and the expression for the wavefunction diverges since in the denominator
approaches zero. As a result, the wavefunction left and right to this point cannot be
adjusted, which is the case at the discontinuity of the semiconductor-dielectric
interface for instance. One way to overcome this problem is to apply Langer’s
procedure [102], which is not presented here. The above formula also applies to
classical forbidden regions where the particle energy lies below the potential
barrier .