Nodal Hydrogenic Wave Functions of Impurities on Bounded-Electron-Gas Surfaces

Recent suggestions by Levine for exactly treating hydrogenic impurities on idealized semiconductor surfaces are extended to the case of another exactly soluble model, a hydrogenic impurity on the boundary of a semi-infinite, dense electron gas. The surface selection rule $l+m=\mathrm{odd}$ is found to apply to this hypothetical surface. This rule is equivalent to the requirement that impurity wave functions have a nodal plane on the boundary. Unlike the dielectric case, the allowed wave functions of the present hypothetical model extend into the vacuum half-space but not into the electron-gas half-space. A surface dipole is thus formed which is directed into the surface, thus increasing the surface barrier for electron emission.