Surface Modes of Vibration and Optical Properties of an Ionic - Crystal Slab

A formalism is developed for calculating the normal modes of vibration of an ionic-crystal slab having the NaCl structure. The slab consists of a finite number of (100) planes oriented normal to the $z$ axis and is of infinite extent in the $x$ and $y$ directions. The derivation of the dynamical matrix takes into account the free surfaces of the slab as well as the effect of retardation of the Coulomb interaction. Retardation is easily excluded from the theory by letting the speed of light $c$ become infinite in the Coulomb contributions to the dynamical matrix. The unretarded normal-mode frequencies and eigenvectors are calculated for a seven-layer slab for values of the wave vector chosen along the $x$ axis and ranging from zero out to the boundary of the two-dimensional first Brillouin zone. The properties of the optical surface modes are discussed in detail. If an electromagnetic wave is incident upon the slab, the optical properties of the slab can be found by calculating the total electric field which exists in the regions of space on either side of the slab as a superposition of the incident field and the induced field arising from the ionic motion within the slab. For both $P$ and $S$ polarizations of the incident field, peaks in the infrared absorption are found to occur at frequencies corresponding to certain optical modes of the slab. The calculated results offer a qualitative explanation of the transmittance of a thin LiF film as experimentally observed by Berreman.