Eur. Phys. J. B 22, 89-98 (2001)
https://doi.org/10.1007/PL00011138

Hydrogenic impurity ground state in quantum well: the envelope function revisited

¹ GPS (Laboratoire associé au Centre National de la Recherche Scientifique et aux Universités Paris 6 et Paris 7) , Université Denis Diderot and Université Pierre et Marie Curie, CNRS, Tour 23, 2, place Jussieu, 75251 Paris Cedex 05, France
² Laboratoire de Physique Statistiqueb, École Normale Supérieure, 24 Rue Lhomond, 75251 Paris Cedex 05, France

Corresponding author: ^a combescot@gps.jussieu.fr

Received: 11 December 2000
Published online: 15 July 2001

Abstract

We present a new variationnal method for calculating the ground state energy of an electron bound to an impurity located in a quantum well. This method relies on an envelope function which is determined exactly from a formal minimization procedure. The obtained energies are lower by as much as 10% than the ones found by the widely used free electron envelope function. Their large width limits are reached with exponentially small corrections as they should. We also find that, except for narrow wells, the shape of these exact envelope functions strongly depends on the impurity position, being consequently quite different from the usual free electron ones. In order to discuss the improvements brought by our new procedure in the most striking way, we have used a model semiconductor quantum well with infinite barrier height and simplified band structure. Extensions can be made to finite barrier and more realistic band structures, following the same technique.