Енергетичний спектр електронів у електричному полі зарядженого квантового нанокільця

Abstract

Here a thin uniformly charged ring immersed into a semiconductor matrix has been considered. By solving the Poisson equation, the electrostatic potential is found. It turns out that the potential can be expressed in terms of complete elliptic integrals of the first kind. Behavior of the potential in the vicinity of singular points allows to introduce and justify some approximate expressions for the electron potential energy. Due to accepting these approximations it became possible to separate the radial and transverse movements of carriers. It has been shown that the transverse movement in applied here basic approximation is described in terms of harmonic oscillator functions with corresponding energy values. If the ring is positively charged, a potential well for electrons appears. In the immediate vicinity of the ring the potential energy is approximated by a linear dependence on the radial coordinate. Therefore the corresponding dynamics are described by Airy functions, which made it possible to establish the dispersion equation in its explicit form and to find approximate solutions for eugen values of energy.