Dielectric continuum theory of the electronic structure of interfaces

General expressions for the density-density and potential-potential response functions, ground-state (i.e., surface) energy, and one-electron optical potential are derived for a model of planar interfaces between two media, each of which is described by a local frequency-dependent dielectric function. These expressions are utilized to evaluate the surface energies characteristic of interfaces between semiconductors (insulators) described by the uniform dielectic function ${\epsilon }_S\left(\omega \right)=1+{\omega }_p^2{({\Delta }^2-{\omega }^2-\frac{i\omega }{\tau })}^{-1\mathrm{}}$ metals described by ${\epsilon }_M\left(\omega \right)=1-\frac{{\omega }_p^2}{\omega (\omega +\frac{i}{\tau })}$, and the vacuum ${\epsilon }_V\left(\omega \right)\equiv 1$. Plasmon damping (i.e., nonzero ${\tau }^{-1\mathrm{}}$) is shown to limit the range of nonlocality of the one-electron optical potential to $\lambda \sim {\left(\frac{2\hslash \tau }{m}\right)}^{\frac{1}{2}}$ and to decrease the surface energy. The surface energy of semiconductor interfaces is found to diminish monotonically with increases in the band-gap parameter $E_g=\hslash \Delta$. The conventional expressions for the surface energy of metals as a function of their density, $n=\frac{m{\omega }_p^2}{4\pi e^2}$, are recovered in the $\tau \to \infty$ limit, although errors in some previous derivations of these expressions are displayed. Finally, the structure and limitations of local models of surface properties are examined critically, and the well-known hydrodynamic and step-density random-phase-approximation models of metal-vacuum interfaces are shown to be elementary consequences of classical electrostatics in the limit that ${\epsilon }_M\left(\omega \right)=1-\frac{{\omega }_p^2}{{\omega }^2}$.