Longitudinal optic

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). 

Bulk silicon is a semiconductor with an indirect band structure, as schematically shown in Fig. 7.12 c. The top of the VB is located at the center of the Brillouin zone, while the CB has six minima at the equivalent (100) directions. The only allowed optical transition is a vertical transition of a photon with a subsequent electron-phonon scattering process which is needed to conserve the crystal momentum, as indicated by arrows in Fig. 7.12 c. The relevant phonon modes include transverse optical phonons (TO 56 meV), longitudinal optical phonons (LO 53.5 meV) and transverse acoustic phonons (TA 18.7 meV). At very low temperature a splitting (2.5 meV) of the main free exciton line in TO and LO replicas can be observed [Kol5]. 

Barker, A. S., 1973. Infrared absorption of localized longitudinal-optical phonons, Phys. Rev., B7, 2507-2520. 

A. Long-Wave Optical Phonons and Polaritons 1. Longitudinal Optical and Acoustical Phonons... 

We note that ionic crystals may have dielectric functions satisfying Eq. (4) for frequencies between their transverse and longitudinal optic phonon frequencies. SEW on such crystals are often called surface phonons or surface polaritons and the frequency range is the far IR. 

Ionic crystals also support SEW, but again no data exists where they have been used as substrates for attached molecule studies. That such studies may be feasible is illustrated in Fig. 21, which shows measured and calculated propagation distances for SrTi03 in the far infrared.— Again, these measurements were made with a molecular laser as a source. Unfortunately, for many crystals the frequency region over which SEW exist is very narrow (between the transverse and longitudinal optic phonon frequencies), and propagation distances are very short. However, ferroelectrics (and near-ferroelectrics like SrTiC ) may prove useful substrates for SEW spectroscopy. 

To identify c-BN, the characteristic transverse optical mode (TO) at 1065 cm-1 and longitudinal optical mode (LO) at 1340 cm-1 have been described [56]. When investigating commercial c-BN, commonly only one IR-peak between 1050 and 1100 cm-1 is observed. 

This scattering process is due to the interaction of electrons with the electric field induced by the lattice vibration polarization (polar longitudinal-optical phonons) occurring in polar semiconductors with partially ionic bonding. According to Devlin [55], the optical Hall mobility can be calculated by... 

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],... 

The vibration spectruin of GaAs, calculated by using the C, and C l values of Tabic 8-4 (TO = transverse optical LO = longitudinal optical LA = longitudinal acoustical TA = transverse acoustical). Experimental points arc from Dolling and Waugh (1965). 

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-... 

Typical luminescence spectra of as-grown crystals show ultra-violet emission due to free excitons (including their longitudinal optical phonons-EO replicas) as well as t5 ical bands of green and yellow-orange luminescence (the temperature is 77-80 K). When the temperature is 4.2 K, the luminescence of bound excitons (including their LO - replicas) dominates. ... 

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). 

Single crystal studies of solid hydrates are scarce. There are two experimental procedures possible (i) transmission spectra of thin crystal plates (see, for example. Refs. 16, 17) and (ii) reflection spectra of crystal faces . Using polarized infrared radiation, the species (symmetry) and other directional features of the water bands can be determined. In the case of reflection measurements, the true transverse and longitudinal optic phonon frequencies can be additionally computed by means of Kramers-Kronig analyses and oscillator fit methods, respectively. Both experimental techniques, however, are relatively difficult because of the lack of suitable monocrystals, the requirement of preparing sufficiently thin, i.e., <0.1 mm, crystal plates (except for studying overtone bands, see Sect. 4.2.6), and the efflorescence or absorption of water at the crystal surfaces. In favorable cases, thin sheets of orientated powdery material can be obtained . ... 

Bryk, T., and Mryglod, I. Longitudinal optic-like excitations in binary liquid mixtures. J. Phys. Cond. Matt., 2002, 14, No. 25, p. L445-451. 

---