Nodal Hydrogenic Wave Functions of Donors on Semiconductor Surfaces

Abstract

Quantum-mechanical properties have been derived theoretically for an isolated donor atom located at the surface of any dielectric crystal. The idealized surface potential is chosen to be $V=\frac{\ensuremath{-}{e}^{2}}{\ensuremath{\kappa}r}$ inside the dielectric and $V=+\ensuremath{\infty}$ outside, where $e$ is the electronic charge, $r$ is the radius from the donor-ion core, and $\ensuremath{\kappa}$ is the dielectric constant of the crystal. $V$ is chosen to be $+\ensuremath{\infty}$ outside the crystal because the valence electron is energetically more stable inside the crystal because of its high electron affinity. By using this potential, not previously used, the allowed wave functions are shown to be those hydrogenic wave functions which possess planar nodes. They have (1) the selection rule $l+m=\mathrm{odd}$, (2) finite dipole moments, (3) $l$-shell degeneracy of $l$, (4) $n$-shell degeneracy of $n(n\ensuremath{-}1)$, and (5) unusual optical-polarization properties. The surfacedonor ground state consists of one lobe of a $2p$ wave function; its ionization energy is only \textonequarter{} that of a bulk donor; and its dipole moment is $9.54\ensuremath{\kappa}$ Debye units.