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

Size-quantization levels of an exciton in large nanocrystals is studied theoretically. For the nanocrystal size, $L$, much bigger than the Bohr radius, $a_B$, the level positions do not depend on $a_B$. The correction to the levels in a small parameter $a_B/L$ 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 $a_B$ parametrically.