Fuchs and Kliewer modes

Fuchs and Kliewer (1965) have predicted the existence of macroscopic surface optic modes in ionic crystals. We give here a simplified derivation of their result, based on the formalism of the dielectric constant. In the phonon frequency range, the bulk dielectric constant e( )) approximately varies with co as  

At a surface, new phonon modes are associated with the roots of the equation  

It is larger than coto and smaller than colo- Lucas and Vigneron (1984) and Lambin et al. (1985) have generalized this dielectric approach to anisotropic materials and to interfaces. 

More precisely, Fuchs and Kliewer have shown that there exist two modes ft)FK+ and ypK- whose frequencies are close to coto and colo at the zone centre, and which become degenerate as the product x a, where a is the layer thickness, increases. Their common value is intermediate between wto and colo- At the zone centre, the surface mode is uniform in the whole slab. In MgO, for example, it has a frequency close to 1.2 x 10 rad s  

It is difficult to build a universal model which reasonably describes the microscopic surface modes in real materials, because they are sensitive to the actual crystal structure and surface orientation. Nevertheless, in order to exemplify some of their characteristics, we now present a calculation of the vibrations in a semi-infinite linear chain of alternating anions and cations. The atoms, with respective masses Ma and Me (reduced mass p) are equidistant and assumed to be linked by springs of stiffness t. The displacements of the anions and cations, with respect to their equilibrium positions, in the cell n, are respectively denoted u and v . In the surface cell (n = 1), an anion is in contact with vacuum. The equations for motion in a cell n 1 are  

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Fig. 4.1. Phonon dispersion curves in MgO(lOO) (according to Chen et ai, 1977). Hatched zones are the projection of the bulk modes. Surface modes S are indexed by n (1 < n < 7) the Rayleigh mode is Si the Fuchs and Kliewer modes have a frequency close to 12x10 rad s S3 is an example of a microscopic mode.

The macroscopic modes have an attenuation length which varies as the inverse of the normal component of their wave vector. They penetrate deeply inside the crystal. On this length scale, the precise atomic structure is unimportant and the elasticity theory of continuous media or dielectric theories may be used. Depending upon whether they are acoustic or optic, one distinguishes the Rayleigh mode (1885) and the Fuchs and Kliewer modes (1965). 

As in the case of metals and semi-conductors, there exist specific surface excitations in insulating oxides. Three types of surface phonon modes may be distinguished the Rayleigh mode, the Fuchs and Kliewer modes and the microscopic surface modes. The first two modes have a long penetration length into the crystal. They are located below the bulk acoustic branches and in the optical modes, respectively. The latter are generally found in the gap of the bulk phonon spectrum. 

The peaks appear to be Lorentzian, and if we were to plot Im a60(x)) this would be even more obvious. That this is indeed so, to good approximation, was shown by Fuchs and Kliewer (1968), who discussed in detail the normal modes of an ionic sphere. 

FUCHS and KLIEWER [4.40] point out that these peaks in the Mie scattering efficiencies can be identified with the virtual modes of the sphere. 

At the time of a recent review [9], there remained very few examples of vibrational studies of adsorbate, or localised substrate modes, at metal oxide surfaces. By far the majority of studies concerned the characterisation by HREELS of phonon modes (such as Fuchs-Kliewer modes) pertaining to the properties of the bulk structure, rather than the surface, or to electronic transitions. Such studies have been excluded from this review in order to concentrate on the vibrational spectroscopy of surface vibrations on well-characterised metal oxide surfaces such as single crystals or epitaxially grown oxide films, for which there is now a substantial literature. Nevertheless, it is important to briefly describe the electronic and phonon properties of oxides in order to understand the constraints and difficulties in carrying out RAIRS and HREELS with sufficient sensitivity to observe adsorbate vibrations, and more localised substrate vibrational modes. 

The surface Fuchs-Kliewer modes, like the Rayleigh modes, should be regarded as macroscopic vibrations, and may be predicted from the bulk elastic or dielectric properties of the solid with the imposition of a surface boundary condition. Their projection deep into the bulk makes them insensitive to changes in local surface structure, or the adsorption of molecules at the surface. True localised surface modes are those which depend on details of the lattice dynamics of near surface ions which may be modified by surface reconstruction, relaxation or adsorbate bonding at the surface. Relatively little has been reported on the measurement of such phonon modes, although they have been the subject of lattice dynamical calculations [61-67],... 

Localised phonon excitations are in principle best studied by neutral-atom scattering, or off specular HREELS, in order to reduce the strong dipole excitation of Fuchs-Kliewer modes. Two off specular HREELS measurements on MgO(lOO) have been reported [25, 68], however there is some disagreement concerning the energy and assignment of the substrate derived loss peaks. Since the microscopic surface modes are expected to be sensitive to the surface structure, it has been suggested [9] that the differences may be associated with differences in surface preparation. 

Fig. 9. The HREELS spectrum of clean Cr2 03(0001) and Cr2 03(0001) [67]. The loss at 21.4meV in the spectrum of Cr2 O3(110) disappears on exposure to CO or O2, and is ascribed to a localised surface mode. Its isotopic shift of 2.4% is predicted theoretically, and is significantly smaller than that observed for the Fuchs-Kliewer modes of 4.5-6%.

In dielectric layers, where a surface phonon mode may occur, or in ionic crystals, multiple scattering from the surface phonon mode can result in Poisson replicas of the no-loss peak. These modes are referred to as Fuchs- Kliewer modes they are a general feature of HREELS spectra of ionic and polar materials, and metal oxides. Ordered overlays on surfaces can also exhibit collective modes, but at submonolayer coverages the HREELS loss peaks are due almost exclusively to single oscillations of the fundamentals. Substrate (silver) phonon modes are shown at 10 meV (83 cm ) in Figure 7. 

Cr2 03(0001) surface is characterised by losses at 21.4, 51.7, 78.6, 85.0 and 88.5meV, and combinations of these losses at higher energies. The latter four are identified as Fuchs-Kliewer phonon modes, and the intensity and energy of these modes are found to be uninfluenced by the adsorption of CO or O2 at 90K. 

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