Electrostatics of two-dimensional structures: exact solutions and approximate methods

TL;DR

This paper provides exact analytical solutions and approximate methods for electrostatic problems in two-dimensional electron structures, aiding understanding of their shape, size, and density profiles under various geometries.

Contribution

It introduces exact solutions for specific geometries and proposes an approximate method validated against numerical simulations for complex cases.

Findings

Exact solutions for elongated metallic islands and antidots.

An asymptotic formula for constriction boundary shape.

Approximate method verified by numerical simulations.

Abstract

We consider a set of electrostatic problems relevant for determining the real-space structure and the ground-state energy of a two-dimensional electron liquid subject to smooth external potentials. Three fundamental geometries are investigated: an elongated metallic island, an antidot, and a constriction. In the first two cases complete closed-form analytical solutions are obtained, despite the absence of rotational or translational symmetries. These solutions govern the shape and size of large quantum dots, and also the size of the depletion regions and the density profiles around isolated antidots. For the constriction, an exact asymptotical formula for boundary shape is derived and arguments are given in favor of its universality. For the cases where the full analytical solution cannot be obtained, an approximate method is proposed as an alternative. Its accuracy is verified against numerical simulations in a periodic (checkerboard) geometry.