Understanding excitons using spherical geometry

TL;DR

This paper introduces a spherical geometry model for excitons that simplifies mathematical treatment while capturing essential physics, unifying different exciton types and enabling exact solutions for specific dielectric constants.

Contribution

The novel model uses a 3-sphere geometry to study excitons, providing exact wave functions for certain dielectric constants and bridging Frenkel and Wannier-Mott excitons.

Findings

Exact wave functions derived for specific dielectric constants

Unified treatment of Frenkel and Wannier-Mott excitons

Intermediate regime shows good agreement with exact results

Abstract

Using the spherical geometry, we introduce a novel model to study excitons confined in a three-dimensional space, which offers unparalleled mathematical simplicity while retaining much of the key physics. This new model consists of an exciton trapped on the 3-sphere (i.e. the surface of a four-dimensional ball), and provides a unified treatment of Frenkel and Wannier-Mott excitons. Moreover, we show that one can determine, for particular values of the dielectric constant $\epsilon$, the closed-form expression of the exact wave function. We use the exact wave function of the lowest bound state for $\epsilon=2$ to introduce an intermediate regime which gives satisfactory agreement with \alert{the} exact results for a wide range of $\epsilon$ values.