Dielectric Response of a Semi-Infinite Degenerate Electron Gas

The dielectric response at frequency $\ensuremath{\omega}$ (neglecting retardation) of an idealized metal with two plane parallel surfaces is calculated in the Hartree approximation, with special emphasis on the limit of infinite thickness. It is assumed that the unperturbed system may be regarded as consisting of free particles confined between plane parallel boundaries. The validity of this approximation is discussed. The response is expressed in terms of the perturbing source potential in Fourier representation, and involves the inverse of an infinite matrix E; E is calculated in the limit of infinite interfacial separation at $\ensuremath{\omega}=0$ and at $\ensuremath{\omega}\ensuremath{\ne}0$ for one-dimensional potentials. A dielectric function ${\ensuremath{\epsilon}}_{\mathrm{Q}}(\ensuremath{\omega})$, where Q is a wave vector parallel to the surface, is introduced, which is inversely proportional to the sum of all elements in ${\mathbf{E}}^{\ensuremath{-}1}$. The surface-plasmon dispersion relation is given implicitly by ${\ensuremath{\epsilon}}_{\mathrm{Q}}(\ensuremath{\omega})+1=0$. The classical image theorem for a semi-infinite dielectric medium is obeyed at a given Q and $\ensuremath{\omega}$ if ${\ensuremath{\epsilon}}_{\mathrm{Q}}(\ensuremath{\omega})$ replaces the classical dielectric constant. The Fermi-Thomas approximation, the lowest-order correction to it, and the classical (high-frequency) approximation are derived. In agreement with earlier work, the corrections to the classical approximation, which relate to the damping and dispersion of surface plasmons, are found to be linear in the wave vector. Numerical results are obtained for a uniform static electric field normal to the surface. The calculated screening length $d$ is well approximated by $d\ensuremath{\simeq}{\ensuremath{\lambda}}^{\ensuremath{-}1}+\frac{1}{8}{\ensuremath{\lambda}}_{F}$, where ${\ensuremath{\lambda}}^{\ensuremath{-}1}$ is the Fermi-Thomas screening length and ${\ensuremath{\lambda}}_{F}$ the Fermi wavelength, the second term representing the lowest quantum correction; this result is 2-3 times ${\ensuremath{\lambda}}^{\ensuremath{-}1}$ in the metallic density range, owing almost entirely to the low electron density in the surface region. The results are compared with experiment. The long-range Friedel oscillations are discussed in an appendix.