On the exciton binding energy in a quantum well

TL;DR

This paper models the exciton binding energy in a one-dimensional quantum well, deriving bounds and analyzing how it varies with well width, including effects of material mismatches, providing more accurate insights than previous studies.

Contribution

It introduces a refined model with bounds for exciton binding energy in quantum wells, considering finite potential and material mismatches, improving accuracy over prior work.

Findings

Binding energy varies from 1 Ry to 4 Ry with width in infinite potential wells.

For finite potential, the energy approaches 1 Ry at small and large widths, with a peak at intermediate sizes.

The peak in binding energy persists in specific GaAs-based quantum wells despite mismatches.

Abstract

We consider a model describing the one-dimensional confinement of an exciton in a symmetrical, rectangular quantum-well structure and derive upper and lower bounds for the binding energy $E_b$ of the exciton. Based on these bounds, we study the dependence of $E_b$ on the width of the confining potential with a higher accuracy than previous reports. For an infinitely deep potential the binding energy varies as expected from $1 Ry$ at large widths to $4 Ry$ at small widths. For a finite potential, but without consideration of a mass mismatch or a dielectric mismatch, we substantiate earlier results that the binding energy approaches the value $1 Ry$ for both small and large widths, having a characteristic peak for some intermediate size of the slab. Taking the mismatch into account, this result will in general no longer be true. For the specific case of a $Ga_{1-x}Al_{x}As/GaAs/Ga_{1-x}Al_{x}As$ quantum-well structure, however, and in contrast to previous findings, the peak structure is shown to survive.