Phonons in Metal Films and Fast-Electron Energy Loss in the Phonon Spectral Range

Abstract

The phonons in a metal film and their excitation by fast-charged particles normally transmitted through the film are studied within the framework of a classical dielectric theory. The bulk phonons are described by means of the jellium model which treats the ions as a continuum plasma and the conduction electrons in the Thomas-Fermi screening approximation. The dispersion relation of the surface modes is obtained from the matching boundary conditions which must be satisfied by the solutions of Maxwell's equations. We find two kinds of modes connected with surface effects, which are distinguished by the nature of the fields which they induce inside the film. Firstly, there is a true-surface-mode branch which exists only at short wave-lengths. Secondly, we find a family of modes which carry oscillatory fields in the direction perpendicular to the film and which are called surface-bulk modes because of their mixed character. The energy loss experienced by fast electrons through the excitation of phonons is divided as usual into a bulk energy loss and various surface energy losses, all of which are studied analytically in detail. The effective bulk-loss intensity in a film remains significantly larger than the surface losses, even for very thin films. This indicates that the bulk phonons interact almost as strongly with electrons in thin films as in thick films, which contrasts with the case of plasmons where the effective bulk energy loss vanishes for sufficiently thin films. The predicted bulk energy loss due to phonons should be observable in sufficiently thick metal films with available experimental techniques. On the other hand, since the surface losses would be much enhanced for specularly reflected electrons at grazing-angle incidence they might become observable under such conditions.