Introduction to Nanomaterials and Devices

Appendix D

Wentzel-Kramers-Brillouin (WKB) Approximation

The potential barriers and wells considered thus far are geometrically simple. If the barrier height is an arbitrary function of the position, the solution of Schrödinger equation becomes very complicated. A simple example in which the barrier height is a function of the distance is the triangular potential well, which is usually encountered at the semiconductor–heterojunction interfaces. This problem is discussed briefly in Chapter 2, in which the solution is expressed in Airy functions. Another example is the simple harmonic oscillator in which the potential is parabolic in distance. The solution of this problem is expressed in terms of Hermite polynomials. For an arbitrary potential barrier as shown in Fig. D.1, we follow Merzbacher treatment, where he considered the Wentzel-Kramers-Brillouin (WKB) approximation. Let us consider Fig. D.1a, in which we show an arbitrary spatially varying potential. The position a is called the turning point at which the wave function changes from propagating to decaying. The propagation vectors of both waves are given by

D.1

and

D.2

Figure D.1 (a) Variation of potential barrier as a function of the distance showing the corresponding energy levels. (b) An arbitrary potential well used for ...