Longitudinal-optical phonon branches

Several theoretical and experimental studies assess the vibrational properties of the high-pressure phases of silicon. A group-theoretical analysis of lattice vibrations in the -tin structure has been made by Chen [98]. In the vicinity of the F point, the optical modes consist of one longitudinal optical (LO) branch and at higher frequencies of a doubly degenerate transverse optical (TO) branch, both of which are Raman active. Zone-center phonon frequencies of Si-11 have been calculated as a function of pressure using the ab initio pseudopotential method... 

Period of the chain is equal to a. Let us suppose the linear relationship between the interaction force between the nearest neighbors and atomic displacement. Every internal motion of the lattice could be represented by the superposition of the mutually orthogonal waves as follows from the lattice dynamic theoiy (see e.g. Bom and Huang 1954 Leibfried 1955). Aiy lattice wave could be described by the wave vector K from the first Brillouin zone in the reciprocal space. Dispersion curve co K) has two separated branches (for 2 atoms in the primitive unit), which could be characterized as acoustic and optic phonons. If we suppose also the transversal waves (along with longimdinal ones), we can get three acoustic and three optical phonon branches. There is always one longitudinal (LA or LO) and two mutually perpendicular transversal (TA or TO) phonons. 

Hereby, the branches with E - and / -symmetry are twofold degenerated. Both A - and / d-modes are polar, and split into transverse optical (TO) and longitudinal optical (LO) phonons with different frequencies wto and wlo, respectively, because of the macroscopic electric fields associated with the LO phonons. The short-range interatomic forces cause anisotropy, and A - and / d-modcs possess, therefore, different frequencies. The electrostatic forces dominate the anisotropy in the short-range forces in ZnO, such that the TO-LO splitting is larger than the A -E splitting. For the lattice vibrations with Ai- and F -symmetry, the atoms move parallel and perpendicular to the c-axis, respectively (Fig. 3.2). 

Indium nitride has twelve phonon modes at the zone centre (symmetry group Cev), three acoustic and nine optical with the acoustic branches essentially zero at k = 0. The infrared active modes are Ei(LO), Ei(TO), Ai(LO) and Ai(TO). A transverse optical mode has been identified at 478 cm 1 (59.3 meV) by reflectance [6] and 460 cm 1 (57.1 meV) by transmission [24], In both reports the location of a longitudinal optical mode is inferred from the Brout sum rule, giving respective values of 694 cm 1 (86.1 meV) and 719 cm 1 (89.2 meV). Raman scattering of single crystalline wurtzite InN reveals Ai(LO) and E22 peaks at 596 cm 1 and at 495 cm 1 respectively [25],... 

On the assumption that the electron correlations are relatively weak in the material, the model of Schulz [37], which takes into account the interaction between the conduction electrons and one acoustic phonon branch, has then been used to calculate the longitudinal phonon spectrum. In the model, two new optical modes are coming up at 2kF, corresponding to excitations in the phase and in the amplitude of the distortion. However, the agreement was still very poor in this case, the renormalization of the phase mode being considerably larger in the model than experimentally observed [34]. 

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]. 

Wurtzite ZnO structure with four atoms in the unit cell has a total of 12 phonon modes (one longitudinal acoustic (LA), two transverse acoustic (TA), three longitudinal optical (LO), and six transverse optical (TO) branches). The optical phonons at the r point of the Brillouin zone in their irreducible representation belong to Ai and Ei branches that are both Raman and infrared active, the two nonpolar 2 branches are only Raman active, and the Bi branches are inactive (silent modes). Furthermore, the Ai and Ei modes are each spht into LO and TO components with different frequencies. For the Ai and Ei mode lattice vibrations, the atoms move parallel and perpendicular to the c-axis, respectively. On the other hand, 2 modes are due to the vibration of only the Zn sublattice ( 2-low) or O sublattice ( 2-high). The expected Raman peaks for bulk ZnO are at 101 cm ( 2-low), 380 cm (Ai-TO), 407 cm ( i-TO), 437 cm ( 2-high), and 583 cm ( j-LO). 

In the case of strongly polar or concentrated species (which is typically the case for an oxide), the vibrational contribution to e(co) may become larger than Boo in the region of the resonance. The shapes of the functions -lm[g(< )] and lm[l/g(< )] then become different, the former exhibiting its maximum for (o=(Oo, whereas for the latter the maximum turns out to be shifted to a o=(coo+Ne lBoBcoR). If one considers the phonon modes in the infinite 3D material, the two modes o coq appear as the zero-wavevector limit of the transverse-optical (TO) and longitudinal-optical (LO) phonon branches, and for that reason are generally termed TO mode and LO mode [94]. 

Some interesting and important conclusions were drawn by separating the phonon spectrum in accordance with the polarization of the oscillations [15]. The whole spectrum was divided into six branches, each of which has an almost Gaussian form of the distribution curve g( ). For cubic crystals, these six branches consist of three acoustical branches (one branch of longitudinal and two branches of transverse waves) and three optical branches (one longitudinal and two transverse waves). The acoustical vibrations can be compared with the vibrations of atoms in a unit cell, and the optical vibrations with mutual oscillations of the sublattices in relation to one another. The curves of the density distribution of oscillations in each [Pg.180]

Fig. 9. Magnon dispersion for Tb in the ferromagnetic phase along the three principal axis directions. Symbols along the top of the diagram label the directions in the conventional notation shown in fig. 4. Dashed curves labelled PH are [longitudinal acoustic (LA), transverse acoustic (TA), and transverse optic (TO)] phonon branches which interact with the magnons. (After Mackintosh and Bjerrum-Moller 1972.)...

The temperature dependence of the phonons has also been measured in SmBe by Alekseev (1993) and by Alekseev et al. (1993b). They reported an unusual increase of the longitudinal acoustic branches, but a softening of the longitudinal optic branches with decreasing temperature. In addition, they found an unexpected peak at 20meV at high temperatures, and this was interpreted as a resonant gap mode. This mode apparently disappears at low temperature. 

It is evident Irom Fig. 10 that there are six distinct phonon modes in the monolayer graphite. The LO branch is a longitudinal optical mode. The LA branch is a longitudinal acousticlike mode. The ZO branch is a vertically vibrating transverse optical mode. The ZA branch is a vertically vibrating acoustic-like mode. The SHO branch is a shear horizontal optical mode. The SHA is a shear horizontal acoustic-like mode. The last two SH modes appear because of the lack of mirror symmetry in these experiments (39). 

In the spectra of CdSe, three modes at 258, 359 and 950 cm are clearly observed. In general, the LO and TO phonons are observed along with the surface modes in polar nanocrystals in resonance Raman spectra and/or surface enhanced Raman spectra [275]. However, LO and TO modes are observed simultaneously only in randomly oriented nanoparticles. Resonance Raman Spectra (RRS) of CdTe nanoparticles give band due to Longitudinal optical (LO) phonons at 170 cm (LO), 340 cm (2LO) and 510 cm (3LO) mode frequency is found to shift due to quantum confinement effect and confined phonons are observed using surface enhanced Raman spectroscopy [275]. Transverse optic (TO) phonon is reported at 145 cm and its position is invariant with decreasing particle size as the dispersion curve for TO phonon branch is almost fiat [275]. In CdSe nanoparticles, LO phonons are reported in the range 180- 200 cm, wheras in ZnSe at 140 cm [Ref. 275 and references therein]. Thus, the Raman spectra observed in the present work well identifies the phonons in these nanoparticles. 

Fig. A.5-22 BaTiOs. Phonon dispersion relation determined by neutron scattering along the [100] direction in the cubic phase, v is the phonon frequency. LA, longitudinal acoustic branch TA, transverse acoustic branch TO, transverse optical branch. The frequency of the TO branch is lower (softer) at 230 °C than at 430 " C, indicating mode softening...

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