The jellium model is commonly used in condensed matter physics to study the properties of a two-dimensional electron gas system. Within this approximation, one assumes that electrons move in the presence of a neutralizing background consisting of uniformly spread positive charge. When properties of bulk systems (of infinite size) are studied, shape of the jellium domain is irrelevant. However, the same cannot be said when one is dealing with finite systems of electrons confined in a finite two-dimensional region of space. In such a case, geometry and shape of the jellium background play a role on the overall properties of the system. In this work, we assume that the region where the electrons are confined is represented by a jellium background charge with an elliptical shape. It is shown that, in this case, the Coulomb self-energy of the elliptically shaped region can be exactly calculated in closed analytical form by using suitable mathematical transformations. The results obtained reveal the external influence of geometry/shape on the properties of two-dimensional systems of few electrons confined to a small finite region of space.

The two-dimensional electron gas (2DEG) model has received a great deal of attention in condensed matter physics due to the richness and complexity of the emerging phenomena associated with it. The variety of possible scenarios makes this model fascinating from a theoretical and experimental point of view. In particular, a 2DEG in a strong perpendicular magnetic field has come to the forefront of current research as a result of the discovery of the integer quantum Hall effect (IQHE) [

However, there have been recent developments in the field of nanotechnology that make possible the fabrication of finite systems of few electrons confined in a finite 2D domain [

We consider a uniformly charged 2D domain with elliptical shape (elliptical plate) with area

At this juncture, we write the expression in (

A uniformly charged ellipse turns into a uniformly charged disk with radius

To conclude, in this work we focused our attention on a finite 2D jellium model with an elliptical shape. We show that in such a case the Coulomb self-energy of the positive background can be analytically calculated in closed form by using simple mathematical transformations. The results derived can be used in systematic studies of the properties of finite systems of electrons embedded in a elliptically shaped background of uniform positive charge. One can also use the above formulas to derive exact expressions for systems of few

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported in part by US Army Research Office (ARO) Grant no. W911NF-13-1-0139 and National Science Foundation (NSF) Grants nos. DMR-1104795 and DMR-1410350.