Phonons, surface

So far we have discussed surface properties for the case where the atoms which constitute a crystal occupy their equilibrium positions. Taking into account of the kinetic energy of nuclei leads to the equation for their vibrational 

3) For the sake of simpKcity, we have neglected here the scattering of electrons. 

4) Only the real part of this expression has a physical sense. 

The essential features of surface vibrational dynamics can be analyzed, as it has been done for surface electronic states, by considering a ID semi-infinite chain of atoms. We assume for simplicity that the chain consists of two kinds of atoms with masses Mi and M2 (see Fig. 2.8) and only neighbouring atoms interact with each other through the elastic forces 

oc is the force constant and A2 is the difference between the displacements 2i and Z2n of two adjacent atoms from their equilibrium positions, where the subscript n specifies the number of the chain unit cell. Then Newton s second law reads 

A considerable number of experimental studies deal with the surface phonon dispersion of aUcah hahde (001) surfaces. All of these studies are based on inelastic HAS experiments. Historically, the first successful determination of a surface phonon dispersion curve up to the Brillouin zone boundary was made for the LiF(OOl) surface by Doak around the year 1980 [93]. Experimental data on this and other alkali halides, which were reported up to 1990, have been reviewed extensively in [74]. 

Another example is the case of RbBr(OOl), for which lattice dynamics calculations based on shell models predict the existence of a relaxation-induced surface-locaUzed SP mode peeled off the top of the optic bulk band [96]. This mode was not found in HAS experiments by Chern et al. [19], most likely because the surface relaxation of RbBr(OOl), which was predicted by the shell models to involve shifts of the ions of about 6% of the lattice constant, is much smaller, as shown by a recent LEED study [54]. 

As the experimentally determined optical surface phonons of BaFj] ) have about half the energies of corresponding bulk branches and could be well described with the bulk force constant between nearest neighbors and with the bulk bonding angles, surface relaxation was considered rather unlikely for BaFj] ). 

The preparation of low-index surfaces of alkali halides and alkaUne earth halides is straightforward for the surface with the smallest surface energy. By scratching 

Ca + sublattice as dark spots. Missing Ca + ions are tagged by white arrowheads, (b) 

The lattice dynamics is strongly perturbed by the presence of the surface. The elements of the force constant matrix connecting two atoms at or near the surface differ from those for the bulk. The number of neighbors is also different at the surface. Vibrational frequencies are generally expected to be lower than in the bulk. 

Phonon bands occur in the SBZ, similarly to the surface states discussed in Sect. 5.2.3. When the frequency of a surface mode corresponds to a gap in the bulk spectrum, the mode is localized at the surface and is called a surface phonon. If degeneracy with bulk modes exists, one speaks of surface resonances. Surface phonon modes are labeled Sj ( / = 1, 2, 3.), and surface resonances by Rj when strong mixing with bulk modes is present, the phonon is labeled MSj. The lowest mode that is desired from the (bulk) acoustic band is often called the Rayleigh mode, after Lord Rayleigh, who first predicted (in 1887) the existence of surface modes at lower frequencies than in the bulk. 

The experimental results have been obtained mainly through atom and electron scattering, HATOF and EELS. 

Surface phonon bands along symmetry lines of the SBZ are given for fee metals in Figs. 5.2-49-5.2-55 and in Table 5.2-20. In all figures the horizontal axis is the reduced wave vector, expressed as the ratio to its value at the zone boundary. Table 5.2-21 gives the surface Debye temperatures for some fee and bcc metals, as well as the amplitudes of thermal vibrations of atoms in the first layer p as compared with those of the bulk pb-In the harmonic approximation, the root mean square displacement of the atoms is proportional to the inverse of the Debye temperature. 

Surface phonon energies of fee metals. References to the original articles are given in the figure captions and/or 

We have seen in Chapter 2 that bond-breaking on a surface greatly modifies the atomic energy levels. The same is true for the atomic vibrations around the equilibrium positions. In this section, we will review the main experimental and theoretical results concerning surface phonons on oxide surfaces. 

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J. L. Erskine. CRC Crit. Rev. Solid State Mater. Sci. 13,311,1987. Recent review of scattering mechanisms, surface phonon properties, and improved instrumentation. 

In recent years there is a growing interest in the study of vibrational properties of both clean and adsorbate covered surfaces of metals. For several years two complementary experimental methods have been used to measure the dispersion relations of surface phonons on different crystal faces. These are the scattering of thermal helium beams" and the high-resolution electron-energy-loss-spectroscopy. ... 

W. Kress and F.W. de Wctte, Surface Phonons, Springer Series in Surface Science Vol. 21, p. 301,... 

Fuyukui, M., Watanabe, K. and Matsumoto, Y. (2006) Coherent surface phonon dynamics at K-coverd Pt(lll) surface investigated by time-resolved second harmonic generation. Phys. Rev. B, 74, 195412. 

However, others reached more ambiguous conclusions. Gates et al. developed a 2D model based on coupling NO vibration to surface phonons, but ignoring the possible role of electron-hole pairs, and successfully captured... 

By using a nonlinear optical process such as SHG, one can probe surface phonons and adsorbate-related vibrations exclusively [14,15,32,34]. Time-resolved SHG (TRSHG) detects the second harmonic (SH) of the probe beam as a function of time delay between pump and probe. The SH electric field is driven by the nonlinear polarization Pi 2w) at the surface, which... 

The SH intensity is proportional to P 2. Experimentally, the oscillatory part of the total SH is so small that one can ignore its second-order term. If coherent surface phonons are created by ISRS, the whole process including excitation and detection is the coherent time-domain analogue of stimulated hyper Raman scattering (y(4) process) [14]. The cross section of the SHG process is then proportional to the product of a Raman tensor in the pump transition and a hyper-Raman tensor dx k/dQn in the probe transition. 

Recent development of ultrashort intense laser pulses has enabled the observation of small-amplitude, high-frequency phonons in wide-gap materials. Typical examples include diamond (Sect. 2.5.1), GaN [72], ZnO [73,74], and TiC>2 [75,76]. Onishi and coworkers observed the bulk and surface phonon modes of TiC>2 at four different frequencies in their TRSHG measurements... 

This section deals with the dynamics of collective surface vibrational excitations, i.e. with surface phonons. A surface phonon is defined as a localized vibrational excitation of a semi-infinite crystal, with an amplitude which has wavelike characteristics parallel to the surface and decays exponentially into the bulk, perpendicular to the surface. This behavior is directly linked to the broken translational invariance at a surface, the translational symmetry being confined here to the directions parallel to the surface. 

Fig. 5. Schematics of the formation of the surface phonon dispersion of a (111) f.c.c. crystal.

The dispersion curves of surface phonons of short wavelength are calculated by lattice dynamical methods. First, the equations of motion of the lattice atoms are set up in terms of the potential energy of the lattice. We assume that thejxitential energy (p can be expressed as a function of the atomic positions 5( I y in the semi-infinite crystal. The location of the nth atom can be... 

Kinematics of surface phonon He spectroscopy. The thick lines correspond to scan curves of a 18 meV He beam. The thin lines display the Rayleigh phonon dispersion curve of Pt(lll) along the f M azimuth. 

The first successful measurement of surface phonons by means of inelastic He scattering was performed in Gottingen in 1980. By using a highly monochromatic He beam (Av/t 1%) Brusdeylins et al. were able to measure the dispersion of the Rayleigh wave of the LiF(001) crystal surfae. In earlier attempts the inelastic events could not be resolved satisfactorily due to the low beam monochromaticity. In Fig. 10a we show a typical TOF spectrum. 

Recently, we hav measured the surface phonon dispersion of Cu(l 10) along the rx, rF, and F5 azimuth of the surface Brillouin zone (Fig. 13) and analyzed the data with a lattice dynamical slab calculation. As an example we will discuss here the results along the TX-direction, i.e. the direction along the close-packed Cu atom rows. 

Fig. 15. Measured surface phonon disj persion of Cu(110) along the F X azimuth.

In addition to the acoustical modes and MSo, we observe in the first half of the Brillouin zone a weak optical mode MS7 at 19-20 me V. This particular mode has also been observed by Stroscio et with electron energy loss spectrocopy. According to Persson et the surface phonon density of states along the FX-direction is a region of depleted density of states, which they call pseudo band gap, inside which the resonance mode MS7 peals of. This behavior is explained in Fig. 16 (a) top view of a (110) surface (b) and (c) schematic plot of Ae structure of the layers in a plane normal to the (110) surface and containing the (110) and (100) directions, respectively. Along the (110) direction each bulk atom has six nearest neighbors in a lattice plane, while in the (100) direction it has only four. As exemplified in Fig. 17, where inelastic... 

Fig. 23. Calculated and measured surface phonon dispersion curves of the (111) surfaces of the noble metals Cu, Ag and Au. (After Ref. 45.)...

Detailed electronic energy-band calculations have revealed the existence of appropriate surface states near the Fermi energy, indicative of an electronically driven surface instability. Angle-resolved photoemission studies, however, showed that the Fermi surface is very curved and the nesting is far from perfect. Recently Wang and Weber have calculated the surface phonon dispersion curve of the unreconstructed clean W(100) surface based on the first principles energy-band calculations of Mattheis and Hamann. ... 

They found a whole bunch of soft phonons, which are primarily horizontally polarized, near the zone boundaries between M and X. The most unstable mode they observed is the Mj phonon, the displacement pattern of which is shown in Fig. 40 note the similarity between this pattern and the reconstruction model in Fig. 39. According to Wang and Weber, these soft phonons are caused by electron-phonon coupling between the surface phonon modes and the electronic 3 surface states at the Fermi surface. They attributed the predominant Ms phonon instability to an additional coupling between d(x — y ) and d(xy) orbitals of the Zj states. 

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