Studies of Vibrational Surface Modes. II. Monatomic fcc Crystals

Abstract

We report the first thorough study of vibrational surface modes in realistic crystal models. This study was based on calculations for monatomic fcc crystals with (111), (100), and (110) surfaces. The most important general result is the following: In addition to the class of surface-mode branches that persist into the long-wavelength limit, studied extensively by previous workers, there is a second class of surface modes that exist only at relatively small wavelengths (of the order of an atomic spacing). The existence of such modes was pointed out in two earlier publications, but a detailed examination of the properties of these modes is presented in the present paper. For the (111) surface, there are five distinct surface-mode branches; for the (100) surface there are apparently at least 19; and for the (110) surface there are ten. A number of series of mixed (or pseudosurface) modes have also been identified. Many of the surface modes are primarily localized in the second layer beneath the surface or some deeper layer, rather than the surface layer itself. At some symmetry points, modes are obtained in which a single layer vibrates almost independently. Several cases have been found in which surface-mode branches attempt to cross each other and as a result exhibit hybridization. The surface modes and mixed modes have been studied throughout the interior of the two-dimensional Brillouin zone, as well as along the symmetry lines. The dispersion curves are shown, graphs are given for the attenuation of many of the modes with distance from the surface, and the polarizations of the modes are described. The behavior of the modes with respect to uniform changes in the density and changes in the surface force constants has been investigated: In all cases the Gr\"uneisen parameters for surface modes and bulk modes are approximately the same for uniform changes in density. If the surface force constants are increased by neglecting the relaxation of the surface particles, the surface-mode frequencies are increased, as one expects. In some cases, these frequencies are raised into the bulk continua and the modes become delocalized. This fact implies that the surface relaxation is important, and the qualitative features of the surface-mode spectrum are sensitive to changes in the surface force constants. Although the calculated surface-mode spectra are rather complicated, all of the surface modes and mixed modes can be explained in terms of a simple phenomenological model in which these modes are regarded as "peeling off" from the bulk bands in a systematic fashion: A mode which is primarily localized in the first layer peels off first from a given bulk band, then (if the perturbation due to the surface is strong enough) a second mode which is primarily localized in the second layer peels off, etc., with the $n$th-layer mode having the same character in the $n$th layer as the first-layer mode has in the first layer.