Coulomb matrix elements of bilayers of confined charge carriers with arbitrary spatial separation

TL;DR

This paper presents an efficient analytical method to compute Coulomb matrix elements for 2D confined charge carriers with arbitrary separation, aiding theoretical modeling of layered quantum systems.

Contribution

It introduces a new analytical expression and computational approach for Coulomb matrix elements in bilayer systems with arbitrary separation distances.

Findings

Method is efficient and accurate.

Provides functional dependence on separation distance.

Facilitates diagonalization of electron-hole pairs in quantum dots.

Abstract

We describe a practical procedure to calculate the Coulomb matrix elements of 2D spatially separated and confined charge carriers, which are needed for detailed theoretical descriptions of important condensed matter finite systems. We derive an analytical expression, for arbitrary separations, in terms of a single infinite series and apply a u-type Levin transform in order to accelerate the resulting infinite series. This procedure has proven to be efficient and accurate. Direct consequences concerning the functional dependence of the matrix elements on the separation distance, transition amplitudes and the diagonalization of a single electron-hole pair in vertically stacked parabolic quantum dots are presented.