Exciton binding energy in a quantum well

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 exciton Rydberg R at large widths to 4 R 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 R 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 ${\mathrm{Ga}}_{1-x}{\mathrm{Al}}_x{\mathrm{A}\mathrm{s}/\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{G}\mathrm{a}}_{1-x}{\mathrm{Al}}_x\mathrm{As}$ quantum-well structure, however, and in contrast to previous findings, the peak structure is shown to survive.