A solvable model for excitonic complexes in one dimension

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.