Phonons of single quintuple films of Bi${}_2$Te${}_3$ and Bi${}_2$Se${}_3$ and corresponding bulk materials are calculated in detail by MedeA (a trademark of Materials Design) and Vienna ab initio simulation package (VASP). The calculated results with and without spin-orbit couplings are compared, and the important roles that the spin-orbit coupling plays in these materials are discussed. A symmetry breaking caused by the anharmonic potentials around Bi atoms in the single quintuple films is identified and discussed. The observed Raman intensity features in Bi${}_2$Te${}_3$ and Bi${}_2$Se${}_3$ quintuple films are explained.

Phonon modes in spherical silicon germanium alloy $\left(\text{SiGe}\right)$ nanocrystals containing up to 1147 atoms $\left(3.6\text{nm}\right)$ have been investigated as a function of the $\text{Si}$ concentration. Microscopic details of phonon modes, including phonon frequencies and vibrational amplitudes, phonon density-of-states are calculated directly from the dynamic matrices. In particular, the dependence of phonon frequency on the configuration (such as a different ratio of $\text{Si}$ to $\text{Ge}$ atoms), and location (surface or interior) of clusters of atoms in $\text{SiGe}$ alloy nanocrystals have been investigated. Low frequency surface phonons that are related to the spheroidal and torsional modes of a continuum sphere are identified and their frequency dependence on alloy concentration elucidated. The calculated results are compared with measured Raman spectra in bulk, thin films, and superlattices of $\text{SiGe}$ alloy reported in the literature. Insights into the behavior of Raman peaks usually identified as $\text{Ge}\text{−}\text{Ge}$, $\text{Si}\text{−}\text{Si}$, and $\text{Ge}\text{−}\text{Si}$ optical phonon modes are presented.

We have used a microscopic lattice dynamical model to study phonon modes in germanium (Ge) NC with size varying between 47 to 7289 atoms $\left(\mathrm{diameter}\sim 6.8\mathrm{nm}\right).\mathrm{}$ By separating these atoms into bulk and surface atoms we have found that surface modes can exist in Ge NC both at low frequencies $(<\mathrm{}50\mathrm{}{\mathrm{cm}}^{-1})$ and at high frequency $\left(\sim 260{\mathrm{cm}}^{-1}\right).\mathrm{}$ The latter mode is a resonant mode which occurs in the “pseudogap” between the acoustic and optical phonon branches in bulk Ge. From the low frequency surface modes we have been able to reconstruct the spheroidal and torsional Lamb modes which have been used to interpret experimental results. Finally, we found that the Lamb model starts to deviate from the lattice dynamical results for Ge NC with diameter $<4\mathrm{}\mathrm{nm}$ and breaks down for NC smaller than 3 nm.

We have studied the Raman intensities of low-frequency phonon modes in germanium $\left(\mathrm{Ge}\right)$ nanocrystals (NC) with varying sizes by using a microscopic valence force field model. The results are compared with the predictions of the continuum model of Lamb using a projection method. We found that the $l=0$ spheroidal Lamb modes are Raman active in the parallel polarization scattering geometry, while the $l=2$ spheroidal Lamb modes are active in the crossed polarization geometry. This result agrees with the group theory prediction that the torsional Lamb modes are not Raman active, but is in disagreement with the identification of torsional Lamb modes in the crossed polarization Raman spectra of NC suggested by many authors.