Eigenstates of Wannier excitons near a semiconductor surface

The exact energies and wave functions, in the effective-mass approximation, of an electron moving in the potential of a massive hole near a single rigid surface of a semiconductor are presented. The electron-hole interaction potential is taken to be the Coulombic $\frac{1}{\epsilon r}$ inside the semiconductor and $+\infty$ outside. The Schrödinger equation for this electron-hole interaction potential is separable in the prolate spheroidal coordinate system, and thus can be solved numerically to any desired accuracy rather easily. The ground state of the exciton is seen to change continously from a hydrogenic $1s$ state, when the hole is well inside the bulk, to a hydrogenic $2p_z$ state, when the hole is located right on the surface in agreement with the known results in these two limits. The variation of the energies and wave functions of the ground and the excited states as a function of the distance of the hole from the surface are discussed and compared with some previous calculations.