Size quantization of an exciton: A toy model of the “dead layer”

Abstract

Size-quantization levels of an exciton in large nanocrystals is studied theoretically. For the nanocrystal size, L, much bigger than the Bohr radius, aB, the level positions do not depend on aB. The correction to the levels in a small parameter aB∕L depends on the reflection phase of the exciton from the boundary. Calculation of this phase constitutes a three-body problem: electron, hole, and the boundary. This calculation can be performed analytically in the limit when the hole is much heavier than the electron. Physically, a slow motion of the hole towards the boundary takes place in the effective potential created by the fast motion of the electron orbiting the hole and touching the boundary. As a result, the hole is reflected before reaching the boundary. The distance of the closest approach of the hole to the boundary (the dead layer) exceeds aBparametrically.