Diamond, dispersion curve

Fig 1. (a) Phase (b) group velocity dispersion curves for aluminium. Circles show minimum dispersion points. Diamonds show excitation positions for transducer designed for X/d = 2.4 where d is the plate thickness. 

Diamond has been studied by coherent INS [16]. Fig. 11.7 shows the dispersion curves calculated by periodic DFT these are in excellent with the experimental data. Fig 11.8 shows a comparison of the INS spectrum derived from the dispersion curves and the TFXA experimental spectrum. Examination of the atomic displacements shows that the spectrum can be approximately described as C-C stretching modes above 1000 cm and deformation modes below 1000 cm. The agreement is good except for the features at 154 and 331 cm. These are not a failure of the calculation but are a consequence of the use of graphite for the analysing crystal ( 3.4.2.3) and will be discussed in 11.2.2. 

Graphite has been extensively studied by coherent INS [17,18], HREELS [19,20] and inffa and Raman spectroscopy [21]. Calculated dispersion curves are shown in Fig. 11.9 and the derived INS spectrum and the experimental spectrum in Fig. 11.10. As with diamond the... 

Fig. 11.7 Dispersion curves of diamond along conventional crystallographic directions as calculated by periodic DFT.

Fig. 11.8 INS spectrum of diamond derived from the dispersion curves of Fig. 11.7 (dashed line) compared with the experimental spectrum (solid line).

The study of the one-phonon density of states of crystals has shown the existence of singularities corresponding to critical points (CPs) located within or at the surface of the BZs along particular directions (the BZs for diamond and sphalerite structures are the same as the one shown in Fig. B2 of Appendix B). They arise from the topology of the ujt (q) dispersion curves, where the index t refers to a given phonon branch. It can be shown that the density of vibrational state g (uj) can be written as ... 

This analysis is based on topological considerations, but other CPs can emerge depending on the actual shape of the dispersion curves in the BZ [50]. Sometimes, the T point is included in the CPs, but when this is done, this can be only for the optical branches. The degeneracy of the TO and LO branches at the T point for the diamond structure. A similar topological degeneracy of the LA and LO branches at the X point (noted L(X)) also exists for this structure. Dispersion curves of phonons in diamond are shown in Fig. 3.1. The curves for silicon and germanium are qualitatively similar. 

Fig. 3.1. Phonon dispersion curves of diamond along the main symmetry directions calculated from a Born-von Karman model fitted to neutron scattering experimental data (after [50]). The frequencies are expressed in wavenumber v = uj/2itc. Along the [110] directions (E), the modes are neither purely longitudinal nor transverse, and three branches exist for each category. Copyright 1992 by the American Physical Society...

Subsequently, a peak in the RSL spectra, similar to the one observed by Krishnan, was not found in some crystals, such as silicon and germanium, which have the same type of structure as diamond and have even stronger anharmonicity than diamond. This encouraged Tubino and Birman (33) to improve the accuracy of the calculations of the structure of the phonon bands in crystals with a diamond-type structure. It was shown as a result of comprehensive investigations that the dispersion curve of the above-mentioned optical phonon in diamond has its highest maximum not at k = 0, but at k 0. The result of these calculations indicates that the peak experimentally observed in the RSL spectra of diamond falls within the region of the two-phonon continuum. It cannot correspond to a biphonon and is most likely related to features of the density of two-particle (dissociated) states. 

Fig. 6.5. Dispersion curves giving frequency v in terms of wave vector k in the [too] and [in] directions for a crystal with the diamond structure (L — longitudinal, T — transverse, A = acoustic, O = optical).

Fig. 4.1-18 Diamond. Phonon dispersion curves. Symbols, experimental data from neutron scattering solid curves, shell model calculation [1.16]...

Fig. 4.1-110 InP. Phonon dispersion curves. Experimental neutron data diamonds) [1.99] and Raman data (triangles) [1.100] and ah initio pseudopotential calculations (solid curves) [1.101]. From [1.101]...

Figure 3-6. Example of lattice dynamical calculations on ir-bonded tridimenional crystals with short range interactions. Dispersion curves for cubic diamond along the F — (K) —> X symmetry direction. Experimental points from neutron-scattering experiments dispersion curves from least squares frequency fitting of a six parameters short range valence force field (from [60]). The Raman active phonon is the triply degenerate state indicated with F j near 1300 cm Notice that at k 0 the degeneracy at F is removed because of the lowering of the symmetry throughout the whole BZ. Notice also the three acoustic branches for which v 0 at k F.

Complete dispersion curves along symmetry directions in the Brillouin zone are obtained from calculated force constants. Calculations of enharmonic terms and phonon-phonon interaction matrix elements are also presented. In Sec. IIIC, results for solid-solid phase transitions are presented. The stability of group IV covalent materials under pressure is discussed. Also presented is a calculation on the temperature- and pressure-induced crystal phase transitions in Be. In Sec. IV, we discuss the application of pseudopotential calculations to surface studies. Silicon and diamond surfaces will be used as the prototypes for the covalent semiconductor and insulator cases while surfaces of niobium and palladium will serve as representatives of the transition metal cases. In Sec. V, the validity of the local density approximation is examined. The results of a nonlocal density functional calculation for Si and... 

Fig. lr.1-111 InAs. Phonon dispersion curves (leftpanel) and density of states (rightpanel) [1.82]. Experimental neutron data (circles, T = 300K) [1.102], thennal-diffuse-scatteiing X-ray data (diamonds, r = 80K) [1.102], Raman data (squares, T = 330K) [1.103], and ab initio calculations (solidcurves [1.82]). From [1.82]... 

Fig. 82. InP(llO) (1x1) H. Surface phonon dispersion curve predicted for the hydrogenated surface by ab-initio linear response formalism [95Fril]. The shaded area represents the projection of the bulk bands on the surface Brillouin zone. The solid lines are the surface modes. The dotted curves indicate the dispersion of two surface-localised gap modes of clean InP(llO). The diamonds denote the frequency of these modes for a bare unrelaxed surface.

The phonon dispersion curves of all covalent semiconductors with diamond or sphalerite structure show one characteristic feature from which only diamond itself is an exception the TA phonon branches have very low frequencies and are very flat away from the zone center, although the corresponding shear moduli (slopes of w(q) at q = 0) have rather high values. Examples of dispersion curves for Ge and GaAs are shown in Figs.4.13,14. This behaviour is most easily understood with the bond charge model which has been developed for the lattice dynamics of covalent materials. 

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