Hydrodynamic Theory of Surface-Plasmon Dispersion

Abstract

The dispersion of surface-plasma oscillations is calculated analytically, using the Bloch hydrodynamic model for an electron gas and solving Maxwell's equations in the absence of retardation effects. This study is based on an approximation for a constitutive relation which provides a useful alternative to previous model descriptions of the surface. It is found that at long wavelengths the surface-plasmon dispersion relation includes a term which is linear in the momentum parallel to the surface and which is half as large as in the case of a study for specular reflection. In contrast to hydrodynamic treatments for specular reflection, the present one leads to an appreciable Landau damping of the surface-plasma oscillations. The magnitude of this Landau damping is roughly consistent with the results of a recent numerical study of plasmons for a more realistic model of the surface. Also discussed in some detail are the charge-density fluctuations associated with the surface plasmon, and a useful relation is established between the amplitudes of bulk- and surface-charge oscillations. In an appendix it is shown that when the imaginary part of the hydrodynamic dielectric function is neglected, one is led to surface modes which are quite different from the usual surface plasmon. Finally the results for surface plasmons are compared with those which are obtained for surface phonons in a metal film and the relationship between two different recent analyses of the surface-phonon problem is discussed in some detail.