Bulk phonon symmetry

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 situation at surfaces is more complicated, and richer in information. The altered chemical environment at the surface modifies the dynamics to give rise to new vibrational modes which have amplitudes that decay rapidly into the bulk and so are localized at the surface [33]. Hence, the displacements of the atoms at the surface are due both to surface phonons and to bulk phonons projected onto the surface. Since the crystalline symmetry at the surface is reduced from three dimensions to the two dimensions in the plane parallel to the surface, the wavevector characterizing the states becomes the two-dimensional vector Q = qy). (We follow the conventional notation using uppercase letters for surface projections of three-dimensional vectors and take the positive sense for the z-direction as outward normal to the surface.) Thus, for a given Q there is a whole band of bulk vibrational frequencies which appear at the surface, corresponding to all the bulk phonons with different values of (which effectively form a continuum) along with the isolated frequencies from the surface localized modes. 

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. 

Fig. 5.2-58 Surface phonon dispersion curves for Si(lll) 2x1 measured by HATOF. Energies at symmetry points X, 10.2 and 11.1 meV S, 10.5 and ll.bmeV. The flat phonon mode at 10.5 meV is associated with the 2 x 1 reconstruction. The surface mode couples with transverse bulk phonons near the center of the SBZ, giving rise to considerable broadening. The shaded area corresponds to the width ofthe 10.5 meV peak [2.90], The energy of the optical mode (not shown in the figure) is 56.0 meV [2.91]...

The surface phonon dispersion relation m(k ) is represented by a surface in 3D space. Usually one displays it by plotting the function m(k ) along certain symmetry lines of the 2D Brillouin zone together with the dispersion relation m(k, j ) for bulk phonons. Then the surface phonon dispersion is given by a line whereas the bulk phonon frequencies fill an area originating from different values of (Fig. 2.10). 

Surface modes can be clearly identified in the dispersion relations ft>(qn) when they appear in regions where no bulk bands appear. Similar to the identification of surface electronic states, the projected bulk modes form the bulk phonon bands in the surface Brillouin zone, as shown in Figure 9.46. In the bulk case, there are three acoustic phonon bands and 3(S-1) optical phonon bands, with S as the number of atoms in the primitive unit cell of the bulk crystal. Along high-symmetry directions in the bulk, such as the (100) or (111) directions in cubic crystals, the phonons can be classified either as transverse or longitudinal, depending on whether or not their displacements are perpendicular or parallel to the direction of the 3D wave vector. 

The consequences of this approximation are well known. While E s is good enough for calculating bulk moduli it will fail for deformations of the crystal that do not preserve symmetry. So it cannot be used to calculate, for example, shear elastic constants or phonons. The reason is simple. changes little if you rotate one atomic sphere... 

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. 

The calculated Rayleigh mode (SJ, the lowest lying phonon branch, is in good agreement with the experimental data of Harten et al. for all three metals. Due to symmetry selection rules the shear horizontal mode just below the transverse bulk band edge can not be observed by scattering methods. The mode denoted by Sg is the anomalous acoustic phonon branch discussed above. Jayanthi et al. ascribed this anomalous soft resonance to an increased Coulomb attraction at the surface, reducing the effective ion-ion repulsion of surface atoms. The Coulomb attraction term is similar for all three metals... 

The above derivation of the effective Hamiltonian is only complete when, for some reasons, the uniform strains of the crystal are not relevant. This is clearly the case for crystals with low concentration of Jahn-Teller impurities. Contrary to that, bulk deformations often arise in its low-symmetry structural phases of Jahn-Teller crystals [2,11]. The uniform strains describing the bulk deformations of the crystal cannot be reduced to a combination of phonon modes, as it was first pointed out by... 

Symmetry considerations under TERS conditions have recently been considered as a method for obtaining molecular orientations [93, 94]. Berweger and Raschke considered using the polar phonon mode selection rules for TERS to determine nanocrystallographic information fi om solids [93]. TERS enables control of experimental parameters required to extract information from the Raman tensor such as the polarization- and k-vector-dependent field enhancement relative to the surface. The selection mles were demonstrated for both near-field TERS and far-field scatting from bulk and nanocrystalline LiNb03. 

It has been shown that the binding energy of a Be acceptor can be varied from its bulk value of 28 meV to a maximum of around 55 meV in a narrow GaAs/AlAs quantum well. T he c orresponding s eparation o f t he i mpurities i nternal 1 s a nd 2 p levels is around 21 to 42 meV, which is equivalent to a wavelength of between 60 and 30 mm. Pump probe spectroscopy of the same samples has shown that the lifetime of the upper (2p) level varies from around 350 ps in bulk material to 80 ps in the narrowest quantum well. This variation in lifetime is thought to be due to non-radiative scattering due to zone-folded acoustic phonon modes, which arise from the symmetry of the multiple quantum well potentials. 

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. 

On the other hand, the two modes of CeBeij show a softening of about 2% with respect to the reference line. All other symmetry modes of CeBe j do not show any anomaly. This applies also to the bulk modulus of CeBcij, partly indicative of the behavior of the long-wavelength acoustic phonons as shown in fig. 36. As a function of Q/V the bulk modulus of the RBe j compounds follows a straight line with CeBe j right on it (Mock et al. 1985). However, an even stronger softening of the two F modes is found upon Ce dilution in... 

However, contrary to CeBCu this mode softening in Cei La Bei3 for 0.8 is also observed for all other symmetry modes with respect to the average behavior of the reference materials. The phonon softening in Cej La, Be,3 for 0.1 X 0.8, independent of the mode symmetry is also reflected by the behavior of the Debye temperature 0 (Besnus et a. 1983), which is displayed at the bottom of fig. 35. No temperature dependent phonon anomaly has been observed for the optical phonons of CeBcij, contrary to the anomalous softening of the bulk modulus upon cooling down below 350 K (Lenz et al. 1984). 

The selected high-resolution spectra and the simulated spectra (as shown in Fig. 23) did not directly reveal any reason why the appearance of PMI and PDI bulk spectra was so different. The calculations showed, however, that PMI has a significant static dipole moment in the So state (around 6 Debye units), which increases by 1 Debye unit upon excitation into the Si state. By symmetry, PDI has no dipole moment in either state. This pointed to strong linear electron-phonon coupling in the case of PMI, which was in line with the measured Debye-Waller factors for PMI ( d = 0.15) and PDI ( d = 0.4). 

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. 

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