Applications and limitations of two important numerical methods for the computation of transmission coefficients

Abstract

As a consequence of the ever-shrinking sizes of nanoelectronic devices, hitherto neglected quantum effects, such as tunneling, are becoming important for device characterization. The study of electron reflection and transmission probabilities at potential barriers is one of the important areas of active research in this field. Analytic solutions for the quantum-mechanical transmission coefficient through a potential energy profile of arbitrary shape do not exist. One conceivable method for finding the transmission coefficient through such a potential involves transfer matrices. This technique is numerically limited, unfortunately, and fails to provide adequate results for potentials of interest in the development of practical nanoelectronic devices. However, within its capabilities, the transfer matrix method is a useful reference to which other results may be compared. Another method, utilizing backward recurrence, has been proposed as a numerically stable alternative for calculating the transmission coefficient through such potentials. This second method has yet to be widely applied. This thesis investigates the capabilities and limitations of each method, with an emphasis on their scope of applicability. Extensive programming, in the C language, has been done to examine the two methods. Output from these programs has been analyzed, and the backward-recurrence method has been shown to have wider applicability, and to be faster and much more numerically stable