Appendix H: Tunneling Through Potential Barriers

Abstract

The tunneling phenomenon was briefly discussed in Chapter 2 for a single potential barrier, a single potential well, and a double barrier structure. This effect was first reported by Esaki in a narrow germanium p-n junction (Esaki).At present, there are many devices based on the tunneling effect such as resonant tunneling diodes, point contact diodes, Schottky diodes, bipolar transistors, and field-effect transistors. One important aspect of the tunneling effect is that the tunneling time of carriers is proportional to the function exp(−2ρL), where ρ is the decaying wave vector inside the barrier and L is the width of the potential barrier. The wave functions of the tunneling carriers are characterized as propagating waves in the wells and evanescent waves inside the barriers. In Chapter 2, we have shown the form of the transmission coefficient for a particle tunneling through a rectangular barrier. When the product of the decaying wave vector and the width of the barrier is much larger than unity, the transmission coefficient is approximated by Equation 2.94. For a potential barrier with an arbitrary shape depicted in Fig. H1 (solid line), the exact derivation of the transmission coefficient becomes more complicated. It can be obtained with the help of approximation methods. For example, the Wentzel-Kramers-Brillouin (WKB) method becomes very handy in obtaining an approximate form of the tunneling probability. The method of obtaining an approximate expression for the transmission coefficient in case of an arbitrary potential barrier is shown in Fig. H1. As opposed to the rectangular potential barriers shown as the dashed line in the figure, one can consider the barrier as being composed of small rectangular segments as shown in the figure. The form of the transmission coefficient for each segment is identical to the expression presented in Equation 2.94. The total transmission coefficient Introduction to Nanomaterials and Devices, First Edition. Omar Manasreh.