LA-phonon

Fig. 3. Dispersion curves of the long-wavelength optical phonons, photons, and polaritons in the centre of BZ 1. In order to demonstrate the connection with the dispersion effects in the region 107 < k < 10 cm-1, the branches of an LO and an LA phonon in this region have been added in a different linear scale. The figures correspond to a cubic lattice with two atoms in the unit cell 3S)...

The electron-phonon coupling term is constructed so as to account for the modulation of the band gap - stemming from the diagonal single-particle matrix elements of Eq. (4) - as a function of the (la) phonon modes belonging to the oIh phonon branch and Zth site [29],... 

This band called iTOLA because it is attributed to a combination of two intra-valley phonons the first from the in-plane transverse optical branch (iTO) and the second phonon from the longitudinal acoustic (LA) branch, iTO-t-LA, where the acoustic LA phonon is responsible for the large dispersion that is observed experimentally [69]. 

Doublets of folded longitudinal acoustic (LA) phonons due to the superlattice periodicity [143] can also be seen in the Raman spectra of the SLs (indicated by arrows in Fig. 21.2). The positions of the doubled peaks agree well with the first doublet frequencies calculated within the elastic continuum model [144]. The observation of the LA phonon folding suggests that these superlattices possess the requisite structural quality for acoustic Bragg mirrors and cavities to be used for potential coherent phonon generation applications [145-147]. 

In binary crystal with large differences between the masses of the two atoms (e.g. in GaP or InP), the frequencies of the LA phonons at the BZ boundaries is significantly smaller than that of the TO phonons, and this difference is usually referred to as the phonon gap. 

The very same charge deformability of the mixed-valence Sm ion due to 4f" - 4f"5d excitations used for the description of the Raman intensities in fig. 37 has been used to describe the phonon anomalies (Bilz et al. 1979). Therefore we can conclude that the dominant F scattering intensities of Sm 25S near 250 cm" and 85 cm , respectively, arise mainly from the LO and LA phonon anomalies in the [111] direction, emphasizing scattering from L-point phonons. The available data on the LO(L) phonon frequencies of RS are depicted in fig. 38 as a function of the lattice parameter. The LO(L) phonons of intermediate-valence metallic SmS and Sm jY 25S lie between the divalent reference line given by YbS and EuS, and the trivalent reference line spanned by YS, GdS, PrS and LaS, thus exhibiting the behavior of an alloy of divalent and trivalent Sm ions. Figure 39 shows the bulk modulus of several RS compounds at room... 

Fig. 70. LA phonon dispersion curves for the [111] direction for metallic (VF) and semiconducting SmS (Mook et al. 1982).

Table II. Theoretical expressions for the oscillator strength of the r-L LA and TO phonons. For the LA phonon, scattering from 12 5 is included. 

Table V. Polarization dependent relative intensities of the LA phonon assisted r-L transition for X Ij [ill] and X COOl]... 

Except for the LA phonon there is little difference in the two works. 

For Ge reliable experimental data exists for Af A snd the polarization dependent relative intensities of the LA phonon in the strained crystal, as shown in Fig. 10. Reference 41 could not resolve the TO transition and thus there is no known value for the quantity Af pQ The lack of data for the TO phonon can be remedied by making use of the WMA spectra of Fig. 10 at zero stress. A line is found approximately 9 meV higher in energy than the strong LA peak and is a result of TO phonon assisted indirect transitions. This conclusion is based on the fact that at L the LA and TO phonons differ in energy by 9 meV. 

Fig. 10. Wavelength-modulated absorption spectra of the LA-phonon assisted indirect transition in Ge at 77K for X = 0 and 3.73 x 10 dyn-cm along [ lll for E 1 X and E X It. For the X = 0 spectra the TO-phonon assisted peak is also observed as marked.

As a further test of the calculation, we have evaluated the polarization dependent intensities of the LA phonon assisted transitions in the strained crystal for stresses along fOOll and [ill] This was accomplished through the use of the equations listed in Table V along with Eqs. (41) - (43). The results of this calculation along with the experimental values of Ref. 4 are presented in Table XI. The agreement between experiment and heor is quite good for both of the stresses for E H X and E 1 X. 

Although Herbert et al. have calculated a number of intervalley electron-phonon matrix elements there is overlap with our theoretical determinations in the case of T-L of Ge (LA-phonon). However, from Herbert s work it is not clear whether he has considered both intermediate states (i.e., I2 c 5 g). If we assume that he has taken into account only 12 q then there is fairly good agreement between his values and ours (see Table IX). 

Table XI. Theoretical and experimental values for the intensities of the LA-phonon assisted transition in Ge for stress, X, along fill and fool for the electric field vector, E, of the incident light polarized parallel and perpendicular to X.

Although in Ge a piezospectroscopy experiment has been performed on the LA phonon-assisted transition its interpretation is more complex due to the fact that there are two intermediate electron states. However, from this work as well as the WMA spectrum of the TO-phonon assisted transition at zero stress valuable information about Sg pjj and S -ph both these transitions was obtained. 

A related expression has been formulated by Klemens (1967) for the spontaneous decay of a LA phonon into two TA phonons. In this case it is found, again for T OK, that... 

It is important to note at this point that the LA phonon lifetime against spontaneous decay varies as oT, i.e, high frequency LA phonons are predicted to decay much more rapidly than low frequency phonons. In simple systems near T 0, the spontaneous three-phonon decay of LA phonons, described above, should dominate the observed temporal evolution of nonequilibrium phonon distributions (Orbach and Vredevoe, 1964). 

The above expressions describe the decay of nonequilibrium LA phonons in an ambient environment in which the temperature T 0K. 

The Raman results are beautifully corroborated by the direct inelastic neutron scattering by Alekseev et al. (1989) which are reproduced in fig. 23 together with LaBg as reference material. Indeed, the LO branch in the X direction is softened as a whole in SmB compared with LaB the same as the TO branch and the frequency at the T point agrees very well with the Raman measurement for both materials. Pronounced anomalies are also observed in SmBg contrary to LaBs for the LA branches in the [110] and [111] directions. These anomalies resemble the similar softening of LA phonons in the [111] direction in IV SmS and TmSe by Mook et al. (1978) and Mook and Holtzberg (1981). 

Fig, 74. (a) LA phonon dispersion in the [111] direction for intermediate-valent TmSe. An electronic plasmon mode is shown as the dashed horizontal line. Mixed electronic plasmon-phonon modes are shown with dotted lines. The heavy dashed line is the phonon mode without interaction, (b) Phonon density of states with (solid curve) and without (dashed curve) plasmon interaction. (After Treindl and Wachter 1980.)... 

Fig. 57. The evolution of LA phonon peak position as a function of the temperature and the unnormalized phonon energy (Liu 1989b). It gives qualitatively the shape of the LA phonon dispersion curve near the mode splitting frequency. All frequencies are in units of t].

---