We have studied the problem of the penetration of a uniform external electric field at the surface of a metal or degenerate semiconductor within the random-phase approximation. In its most general form, this problem involves the solution of a complex integrodifferential equation. This problem has been studied most recently by Newns using Fourier analysis and numerical inversion of an infinite-dimensional response matrix. We have found that using a parametric representation of the potential, $V\left(x\right)=V_0{(1-\alpha )}^{-1\mathrm{}}(e^{-\lambda x}-\alpha e^{-2\mathrm{}\lambda x})$, in the integral equation led to simple transcendental equations for determining the parameters. The potential which resulted was tested for full self-consistency by using it as a source for a single iteration of the full integral equation. A comparison between the first iteration and results obtained by Newns (for the case $r_s=2\mathrm{}$) showed that they agreed within a few percent.

• Received 29 March 1971

DOI:https://doi.org/10.1103/PhysRevB.4.1246