Potential landscapes and induced charges near metallic islands in three dimensions

TL;DR

This paper develops a 3D numerical method to calculate electrostatic potentials and induced charges around metallic islands, enabling detailed analysis of nanoscale systems with improved accuracy and efficiency.

Contribution

It generalizes a 2D relaxation algorithm to 3D for calculating electrostatic potentials and charges in systems with metallic islands, providing a self-consistent and efficient computational approach.

Findings

Successfully computes potential landscapes for nanoscale metallic systems.

Accurately determines all free and induced charges self-consistently.

Calculates the entire capacitance matrix for complex 3D geometries.

Abstract

We calculate electrostatic potential landscapes for an external probe charge in the presence of a set of metallic islands. Our numerical calculation in three dimensions (3D)uses an efficient grid relaxation technique. The well-known relaxation algorithm for solving the Poisson equation in two dimensions is generalized to 3D. In addition,all charges on the system, free as well as induced charges,are determined accurately and self-consistently to satisfy the desired boundary conditions. This allows the straightforward calculation of the potential on the outer boundary using the free space electrostatic Green's function,as well as the calculation of the entire capacitance matrix of the system. Physically interesting examples of nanoscale systems are presented and analyzed.