Theoretical description of cylindrical nano-structures, including pores in semiconductors

Abstract

Cylindrical and prismatic nano-structures (nano-wires and pores) with circular and hexagonal cross-section are studied using mirror boundary conditions to solve the Schrödinger equation in effective mass approximation. In comparison with “quantum billiard” problem, the solution using mirror boundary conditions allows to obtain the results in a much simpler way. It is possible to use even and odd mirror boundary conditions depending on the sign of the wave-function equated in the original and the image points, respectively. The even mirror boundary conditions provides non-vanishing wave-function at the boundary, corresponding to weak confinement allowing quantum tunneling. The odd mirror boundary conditions set wave-function to zero at the boundary, corresponding to a strong confinement. We report on spatial distributions of probability density in cross-section of a cylindrical and prismatic nano-structures, presenting the formulas for the energy of the corresponding quantum states.