Quantum size effect in semiconductor transport

A quantum transport theory for electric conductivity in semiconducting thin films is presented when the de Broglie wavelength of a thermal electron becomes comparable to the thickness of the film and size quantization is important. The theoretical results obtained are analyzed for the electron scattering by acoustic phonons and point defects represented by a $\delta$ -function potential. In the ultra-thin limit (UTL), the resistivity in the longitudinal configuration (when current flows parallel to the plane of the thin film) is found to be inversely proportional to the thickness of the film. In the transverse configuration (when current flows perpendicular to the plane of the thin film) the resistivity in UTL is inversely proportional to $d^3$ when ${3\epsilon }_0>>\frac{\hslash }{\tau }$, where $d$ is the thickness of the film, ${\epsilon }_0$ is the ground-state energy of the electron confined to a box of length $d$, and $\tau$ is the relaxation time. When ${3\epsilon }_0<<\frac{\hslash }{\tau }$, the resistivity is found to be inversely proportional to $d$.