A solvable model for excitonic complexes in one dimension

TL;DR

This paper introduces an analytically solvable one-dimensional model for excitonic complexes, including charged and neutral excitons, providing exact energy spectra and mathematical insights into their properties in nanostructures.

Contribution

It presents a new extended Calogero model that allows exact analytical solutions for excitonic complexes in one-dimensional nanostructures, linking physics with special functions.

Findings

Analytical energy spectra for excitons and charged excitons in 1D.

Model related to Heun equation enabling exact solutions.

Provides mathematical and physical insights into excitonic complexes.

Abstract

It is known experimentally that stable few-body clusters containing negatively-charged electrons (e) and positively-charged holes (h) can exist in low-dimensional semiconductor nanostructures. In addition to the familiar exciton (e+h), three-body 'charged excitons' (2e+h and 2h+e) have also been observed. Much less is known about the properties of such charged excitons since three-body problems are generally very difficult to solve, even numerically. Here we introduce a simple model, which can be considered as an extended Calogero model, to calculate analytically the energy spectra for both a charged exciton and a neutral exciton in a one-dimensional nanostructure, such as a finite-length quantum wire. Apart from its physical motivation, the model is of mathematical interest in that it can be related to the Heun (or Heine) equation and, as shown explicitly, highly accurate, closed form solutions can be obtained.