Bulk phonon dispersion

Fig. 17. Bulk phonon dispersion of longitudinal modes in Cu and Ni crystals, along the (110) and (100) directions. After Ref. 37.)...

Figure 7. Bulk phonon dispersion curves for KBr and RbCl in their <100> and <111> high-symmetry directions. Both crystals have fee lattices and rocksalt structures. Note that the transverse branches, labeled TA (transverse acoustic) and TO (transverse optical), are doubly degenerate in these directions. (Adapted from Fig. 3 of Ref. 32.)...

The dynamical model employed in the theoretical calculations is the Shell Model, of which there are several variants [6, 9]. It is designed to approximate the physical situation of the ions in the crystalline environment more realistically than does the Bom-von Karman treatment with harmonic force constants between neighboring atoms as discussed in Section II, but its handling of the forces is not so very different. The Shell Model was developed for these materials to account for the bulk phonon dispersion that was measured by neutron scattering experiments as well as for their dielectric... 

The first factor in the parenthesis is associated with m and the second with the short-range interactions. Since the ratio between p and Rq is generally of the order of 0.1, the largest contribution to the cohesion energy is m, which justifies a posteriori the hard-sphere model. Despite its simplicity, Born s model has been used with success in many different instances for example, it has helped in the interpretation of many bulk phonon dispersion curves (Bilz and Kress, 1979). 

Fig. 53a, b. Ru(OOOl) (1x1) H. (a) Surface phonon dispersion [97Bra]. The lines are the result of a LDM calculation with bulk force constants, (b) Surface phonon dispersion curve in presence of the (1x1) H overlayer as determined by HATOF. The lines are the result for surface phonon and resonances of a LDM fitting the bulk phonon dispersion [97Bra]. LR denotes the longitudinal resonance. The SBZ is reported in the inset in (a). 

Fig, 26. Experimental dispersion curve of the Kr monolayer and measured line width broadening As of the Kr creation phonon peaks. The solid line in the dispersion plot is the clean Pt(lll) Rayleigh phonon dispersion curve and the dashed line the longitudinal phonon bulk band edge of the Pt(l 11) substrate, both in the r Mn azimuth which is coincident with the r Kk, azimuth. 

In this section, we compare BTE and MD predictions of bulk thermal conductivity of silicon to the available experimental data [74] over a temperature range of 500K to lOOOK (Fig. 4). The BTE predictions are obtained from the full phonon dispersion model described in section 2.3. This full phonon dispersion model involves an adjustable parameter, the Gruneisen constant (y), which is set... 

SOI and strained silicon transistors are comparable to or smaller than the phonon s mean free path (which, for silicon, has been estimated as 300 nm at 300K) [53], In this limit, the film surfaces alter the phonon dispersion relations [76], and the phonon-surface scattering may become the predominant scattering mechanism [3, 53], Since phonons are the main carriers of thermal energy in silicon, these effects alter the thermal conductivity, which differs from that of bulk silicon [10, 36, 77], Measurements of the thermal conductivities of silicon films of thicknesses down to 74 nm found a reduction of 50% with respect to the bulk value at 300K [53], This reduction depends on the temperature and the thickness of the film [3, 53],... 

Coherent Inelastic Scattering.—Inelastic neutron collisions with the solid can excite phonon modes (collective vibrations) and if the coherently scattered component can be detected variation with direction within the solid, i.e. the phonon dispersion curve, can be determined. This technique is well established for bulk solids and has been used recently to examine the properties of small particles (carbon black). 

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.)...

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 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).

For Nal, the large group of points in the center of the bulk band in FX near the F point are probably due to phonon-assisted bound state resonances which were also found for NaCl and for LiF [58, 61, 63]. In the case of NaCl, the bound state energies had been determined by other scattering experiments [75, 76] so that the peaks in the TOF spectra due to bulk phonon resonances could be reliably removed from the phonon dispersion diagram in Fig. 24. For Nal the values of the bound states still need to be established. 

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.)...

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