Surface phonon dispersion

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

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. 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. ... 

Surface phonon dispersion of the unreconstructed surface, lie theoretic dispersion curves are from Ref. 96 the data ( ) are due to Ernst et... 

C. Oshima, T, Aizawa, R. Souda, Y. Ishizawa Y. Sumiyoshi (1988). Solid State Commun., 65, 1601-1604. Surface phonon-dispersion curves of graphite(OOOl) over the entire energy region. 

Theoretical calculations of surface phonon dispersion have been carried out in two ways. One method is to use a Green s function technique which treats the surface as a perturbation of the bulk periodicity in the z-direction [34, 35]. The other is a slab dynamics calculation in which the crystal is represented by a slab of typically 15-30 layers thick, and periodic boundary conditions are employed to treat interactions outside the unit cell as the equations of motion for each atom are solved [28, 33, 35, 37]. In the latter both the bulk and the surface modes are found and the surface localized modes are identified by the decay of the vibrational amplitudes into the bulk in the former the surface modes can be obtained directly. When the frequency of a surface mode lies within a bulk band of the same symmetry, then hybridization can take place. In this event the mode can no longer be regarded as strictly surface localized and is referred to as a surface resonance [24]. Figure 8, adapted from Benedek and Toennies [24], shows how the bulk and surface modes develop as more and more layers are taken in a slab dynamics calculation. 

Figure 8. The evolution of surface phonon dispersion curves for a monatomic fee (111) surface in slab dynamics calculations as a function of the number of layers in the slab. The surface localized modes, marked by arrows in the last panel (iV = 15), lie below the bulk bands Mross the entire surface Brillouin zone and appear between the bands in the small gap near K in the TK region and in the larger gap in the MK region. (Reproduced from Fig. 1 of Ref. 24, with permission of Elsevier Science Publishers.)...

The incident helium beam energies in this instrument currently can be varied from about 15 to 60 meV by suitably cooling the nozzle source. These values correspond to incident wavevectors of — 5 to 11 A , or de Broglie wavelengths (27t/A ,) of about 0.6 to 1.3 A, and provide a good match with the energies and momenta of the surface phonons for determining the surface phonon dispersion. 

Figure 16. Surface phonon dispersion curves for LiF(OOl). The calculated bulk bands are indicated by the vertical-striped regions. The surface localized modes are shown by heavy solid lines, whereas the resonances lying within bulk bands are given by thinner solid lines. The mode label S refers to the Rayleigh wave, to the longitudinal resonance, Sg to the crossing resonance, and S2, S3, and S4 to optical modes. (Reproduced from Fig. 2 of Ref. 58, with permission of Elsevier Science Publishers.)...

Figure 17. Surface phonon dispersion for KBrfOOl). The data are compared to a Green s function calculation used to determine the bulk bands (shown by the shaded regions with polarizations perpendicular or parallel to the surface as indicated in the figure) and the surface localized modes (shown as solid lines). The predominant polarizations of the modes are indicated by perpendicular and parallel symbols, and the labels of the modes follow the notation in Fig. 16. Note that modes Sy and S5 are polarized shear horizontal and cannot be observed in this scattering arrangement. The data plotted as triangles are obtained from weaker peaks in the TOF spectra than the points represented by open circles. (Reproduced from Fig. 8 of Ref. 49, with permission.)...

Figure 18. Surface phonon dispersion for RbCl(OOl). The shaded regions correspond to the surface projected density of states, with the darker shades representing higher state densities. The Rayleigh wave, crossing resonance, and optical mode are indicated by RW, CR, and 2, respectively. (Reproduced from Ref. 119.)...

Figure 19. Comparison of the surface phonon dispersion of KBifOOl) and RbCl(OOl) in the FM region of the SBZ. The open triangles and open circles are for KBr and the closed triangles for RbCl. The shaded regions are adapted from the calculation in Fig. 17 for KBr. (Reproduced from Fig. 6 of Ref. 66, with permission.)...

Figure 20. Surface phonon dispersion for Rbl(OOl). The upper panel shows a comparison of the HAS data with a slab dynamics calculation for the unrelaxed surface, while the lower panel is a comparison of the same data with a similar calculation for a relaxed surface. The sagittal plane and shear horizontal modes are labeled by SP and SH, respectively, and the superscripts indicate which ion (Rb or T) is predominantly involved in the motion of the mode. The other labels follow the notation of Figs. 16 and 17. (Reproduced from Fig. 3 of Ref. 68, with permission.)...

Figure 21. Surface phonon dispersion for NaF(001). The bulk bands are indicated by shaded regions and the surface localized modes by heavy solid lines, as determined by a Green s function calculation. The mode labels follow the notation of Fig. 16. The z and x designations indicate the predominant mode polarization, perpendicular and parallel, respectively. (Reproduced from Fig. 4 of Ref. 70, with permission).

Figure 25. Surface phonon dispersion for CsF(OOl). The solid curve is a sine function which has been drawn to fit the data corresponding to the RW. The dashed horizontal lines in the < 110> direction (panel b) are estimates of the lower and upper limits of the expected bulk band gap. The four points near the upper limit lie in the energy region expected for the gap mode, based on energies for the corresponding gap mode, S4, in the mirror compound Nal, as in Fig. 23. (Reproduced from Figure 2 of Ref. 77, with permission of Elsevier Science Fhiblishers.)...

The effects of relaxation on the calculated surface phonon dispersion in Rbl have apparently been verified, particularly by the observation of a surface optical mode which lies above the bulk phonon optical bands. Except for the mysterious acoustic band mode in Rbl, the Shell model calculations have generally been quite accurate in predicting surface vibrational mode energies in both high-symmetry directions of the alkali halide (001) surfaces. 

Figure 26. Surface phonon dispersion for NiO(001). The HAS data (solid points) and EELS data (open squares) are compared with a slab dynamics calculation. The bulk bands are shown as the shaded regions, and the surface localized modes are indicated by solid lines and labeled as in Fig. 16. (This figure has been reproduced from Fig. 5 of Ref. 79, with permission.)...

Figure 31. Surface phonon dispersion for Cu(lll). The open circles are from HAS experiments, and the open triangles are from EELS experiments. The surface modes shown as solid lines and bulk band boundaries are based on a simple force constant model. The X and Y designations indicate the polarizations of the corresponding modes as identified in the reduced zone diagram in the inset. (Reproduced from Fig. 3 in Ref. 99, with permission.)...

Figure 32. Surface phonon dispersion for Nb(OOl). The data are the solid points which were taken at 900 K. Panels a and b correspond to slab dynamics calculations with two different force constant models the calculation in panel b uses the force constants from the bulk phonon fits. The solid lines represent the surface phonons and resonances polarized mainly longitudinally (or parallel), the lines with long dashes represent phonons polarized mainly perpendicularly, and those with short dashes are shear horizontal. (Reproduced from Fig. 6 of Ref. 107, with permission.)...

Figure 33. Surface phonon dispersion for W(OOl) in the FM portion of the SBZ showing the measured Rayleigh wave (R) and longitudinal (L) modes. The data in the upper panel were obtained at 1200 K, while in the lower panel the data shown by open circles were obtained at 500 K and those represented by closed circles were obtained at 300 K. The edges of the transverse acoustic (TA) and longitudinal acoustic (LA) bulk bands are given by the hatched lines. The vertical lines in the lower panel denote the widths in the energy transfer distributions of these points. (Reproduced from Figs. 10 and 13 of Ref. 110, with permission.)...

Figure 34. Surface phonon dispersion for 2H-TaSe2. The HAS data are shown as solid circles except for weak points which appear in the TOP spectra as hybridized longitudinal modes that are shown as crosses. All the data were obtained at 60 K, well into the low-temperature phase. The calculated striped and shaded regions, corresponding to transverse and longitudinal polarizations respectively, are the slab-adapted bulk phonon bands, while the solid line is a calculation for the Rayleigh wave based on the Dispersive Linear Chain Model (shown schematically in Fig. 35). The open circles at g = 0 are from Raman scattering experiments. (This figure has been corrected from Fig. 23 in Ref. 54.)...

Figure 36. Surface phonon dispersion of KBr films deposited onto NaCl(OOl) for two, three, four, and seven monolayers (ML). The solid squares correspond to strong peaks in the TOP spectra, whereas the open squares are calculated from weaker ones. The solid line is a sine curve matched to the RW phonon frequency at the M point for cleaved KBr(OOl), as in Fig. 17. Note that the frequencies at V do not go to zero. Also note that for the 4ML data (panel c), there appear to be a few good points (solid squares) which lie below the RW frequencies. (Reproduced from J. Duan, Ph.D. Dissertation, Florida State University, 1992.)...

Figure 42. Surface phonon dispersion curves for Ar/Ag(l 11) for films of one monolayer (MONO), two monolayers (BI), three monolayers (TRI), and 25 monolayers (BULK). The progression shows the increase in dispersion as the film s surface becomes increasingly like the surface on a terminated bulk. (Reproduced from Fig. 3 of Ref. 127, with permission.)...

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