Quantum states of electron in the image potential of the spherically deformed ultrathin nanostructure

Abstract

The quantum problem in the interaction of the electron with conducting ultrathin nanostructure is considered. The last one is represented by the model of the spherically deformable flat layer. The external part of the electron interaction with the surface is given by the electrostatic polarization image potential, and the internal part of this interaction is represented by the potential well with the given parameters — the depth and width. The quantum problem is reduced to the one-dimensional Schr¨odiger equation which is solved numerically by the finite difference method. In the case of the flat layer two types of the Rydberg states assigned to the parity number of the reflection symmetry are observed. The effects of the symmetry breaking are investigated under the spherical deformation of the nanostructure in dependence of the surface radius curvature. The conditions of the discrete states representation by the spectral expression for the hydrogen-like atom are investigated. Values of the quantum defect are calculated as functions of the thickness and curvature radius of the nanostructure. In the case of the flat geometry calculations for the structure with the graphene parameters are in well agreement with results of the theoretical works of the other authors. The number of the bounded states becomes finite under the spherical deformation of the nanostructure. At the same time the hydrogen-like property of the states is violated at the upper limit of the energy spectrum as well as at the lower one. Refs 15. Figs 5.