Asymptotic estimation of Hankel transforms and the screened Coulomb potential in trilayer structures.

Abstract

To find an asymptotic expression for the Hankel transform we develop a convenient method which recalls that of Lighthill for Fourier integrals. Using the proposed method together with the model based on the assumption of a specular reflection of electrons, we examine long-range behavior of the screened Coulomb potential in a dielectric-metal-dielectric structure. We find that the oscillatory part of this potential falls off like ${\mathrm{\ensuremath{\rho}}}^{\mathrm{\ensuremath{-}}5/2}$, where \ensuremath{\rho} is the distance in the direction perpendicular to the superlattice axis. This is in contrast to the nonoscillatory part of the screened potential which decays as ${\mathrm{\ensuremath{\rho}}}^{\mathrm{\ensuremath{-}}3}$. We give an example where the asymptotic estimation of the potential is confirmed by computer calculations.