Vibrational mode longitudinal optical

Figure 2.17 The phonon dispersion relations for (a) GaN and (b) Si. TA, LA, LO, and TO refer to transverse acoustic, longitudinal acoustic, longitudinal optical and transverse optical phonons, respectively. Each of these represents a particular vibrational mode. Longitudinal modes run along bonds as in Figure 2.16, while for transverse modes the vibration velocity is perpendicular to the bonds. There are two transverse modes because there are two axes perpendicular to a bond direction. Figures after Levinshtein, Rumyantsev, Sergey, and Shur, Reference [5], p. 27 and 184, respectively. This material is used by permission of John Wiley Sons Inc.

Figure 8.11 (a) Dispersion curve for CuCl(s) along [110] of the cubic unit cell, (b) Density of vibrational modes [3], Here L, T, A and O denote longitudinal, transverse, acoustic and optic. Reproduced by permission of B. Hennion and The Institute of Physics. 

In three dimensions, transverse and longitudinal optic and acoustic modes result. The dispersion curve for CuCl along [100] of the cubic unit cell [3] is shown in Figure 8.11(a) as an example. The number of discrete modes with frequencies in a defined interval can be displayed as a function of the frequency. This gives what is termed the density of vibrational modes or the vibrational density of states (DoS). The vibrational DoS of CuCl is given in Figure 8.11(b). 

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). 

For the particular case of longitudinal optical modes, we found in Eq. (9-27) the electrostatic electron-phonon interaction, which turns out to be the dominant interaction with these modes in polar crystals. Interaction with transverse optical modes is much weaker. There is also an electrostatic interaction with acoustic modes -both longitudinal and transverse which may be calculated in terms of the polarization generated through the piezoelectric effect. (The piezoelectric electron phonon interaction was first treated by Meijer and Polder, 1953, and subsequently, it was treated more completely by Harrison, 1956). Clearly this interaction potential is proportional to the strain that is due to the vibration, and it also contains a factor of l/k obtained by using the Poisson equation to go from polarizations to potentials. The piezoelectric contribution to the coupling tends to be dominated by other contributions to the electron -phonon interaction in semiconductors at ordinary temperatures but, as we shall see, these other contribu-... 

In the solid state, the polar phonons (those that are IR active) split into two components, the transverse optical mode (TO) and the longitudinal optical mode (LO). This TO/LO splitting occurs because the electric field associated with the transverse wave = 0 while that associated with the longitudinal wave is 0. The coupling of these modes with the electric fields associated with the vibration gives rise to Vlo > Vto- This factor is relevant in relation to the shape and interpretation of the IR spectra of solid materials and will be further considered below. 

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). 

Here is the high-frequency dielectric constant, the static dielectric constant, and Lo the frequency of the longitudinal optical vibration mode. The values of P range from about 3 (GaP, ZnS, Csl, Nal), via 4(La202S), 5.6 (Y3AI5O12), to 7 (CaW04, YVO4). 

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]. 

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... 

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). 

Lattice vibrations are also classified as optical branch and acoustical branch modes or as transverse optical (TO) and longitudinal optical (LO) modes. These are not important to us. Lattice vibrations disappear if the crystal is destroyed by any means—e.g., by melting or solution. They are a cooperative phenomenon of a highly ordered system. 

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