Phonons macroscopic modes

Summary. Coherent optical phonons are the lattice atoms vibrating in phase with each other over a macroscopic spatial region. With sub-10 fs laser pulses, one can impulsively excite the coherent phonons of a frequency up to 50THz, and detect them optically as a periodic modulation of electric susceptibility. The generation and relaxation processes depend critically on the coupling of the phonon mode to photoexcited electrons. Real-time observation of coherent phonons can thus offer crucial insight into the dynamic nature of the coupling, especially in extremely nonequilibrium conditions under intense photoexcitation. 

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). 

The surface Fuchs-Kliewer modes, like the Rayleigh modes, should be regarded as macroscopic vibrations, and may be predicted from the bulk elastic or dielectric properties of the solid with the imposition of a surface boundary condition. Their projection deep into the bulk makes them insensitive to changes in local surface structure, or the adsorption of molecules at the surface. True localised surface modes are those which depend on details of the lattice dynamics of near surface ions which may be modified by surface reconstruction, relaxation or adsorbate bonding at the surface. Relatively little has been reported on the measurement of such phonon modes, although they have been the subject of lattice dynamical calculations [61-67],... 

Of central importance for understanding the fundamental properties of ferroelec-trics is dynamics of the crystal lattice, which is closely related to the phenomenon of ferroelectricity [1]. The soft-mode theory of displacive ferroelectrics [65] has established the relationship between the polar optical vibrational modes and the spontaneous polarization. The lowest-frequency transverse optical phonon, called the soft mode, involves the same atomic displacements as those responsible for the appearance of spontaneous polarization, and the soft mode instability at Curie temperature causes the ferroelectric phase transition. The soft-mode behavior is also related to such properties of ferroelectric materials as high dielectric constant, large piezoelectric coefficients, and dielectric nonlinearity, which are extremely important for technological applications. The Lyddane-Sachs-Teller (LST) relation connects the macroscopic dielectric constants of a material with its microscopic properties - optical phonon frequencies ... 

The conventional macroscopic Fourier conduction model violates this non-local feature of microscale heat transfer, and alternative approaches are necessary for analysis. The most suitable model to date is the concept of phonon. The thermal energy in a uniform solid material can be jntetpreied as the vibrations of a regular lattice of closely bound atoms inside. These atoms exhibit collective modes of sound waves (phonons) wliich transports energy at tlie speed of sound in a material. Following quantum mechanical principles, phonons exhibit paiticle-like properties of bosons with zero spin (wave-particle duality). Phonons play an important role in many of the physical properties of solids, such as the thermal and the electrical conductivities. In insulating solids, phonons are also (he primary mechanism by which heal conduction takes place. 

The plasma frequency corresponds to an oscillation as a whole of the electronic charge density with respect to the fixed ionic charge. By analogy with the phonon excitation, the corresponding excitation is called plasmon and it can be considered as the quantization of classical plasma oscillation. The plasmon oscillation is longitudinal with respect to its propagation and is comparable to the TO phonon mode. The macroscopic electric field associated... 

In a heavily N-doped 6H sample (6xlOl9cm 3) Klein et al [34] observed an asymmetric broadening and a shift of the A,(LO) phonon which were attributed to the overdamped coupling between LO phonon and plasmon modes [35]. The interaction between these two excitations occurs via their macroscopic electric fields when the frequency of oscillation of a free-carrier plasma is close to that of the LO phonon. The dependence of the LO phonon-overdamped plasmon coupled modes on carrier concentration was reported by Yugami et al [36] in 3C-SiC films, where the carrier concentrations varied from 6.9 x 1016 to 2xl0,scm 3. They verified that the carrier concentrations obtained from RS were in fairly good agreement with the Hall measurement values, and that the Faust-Henry coefficient [35] for the 3C-SiC (C = + 0.35) was close to the value reported for 6H-SiC (C = + 0.39) [34]. 

The L0(F) mode cannot be treated in the same manner as the T0(F) frozen phonon, because of the presence of a macroscopic electric field. The method proposed in Ref.6 is the ab initio evaluation of effective charges that determine the LO-TO splitting and will be discussed in detail in Section 6. 

Since the number of phonon modes is very large in a macroscopic crystal, the sums in eq. (36.19) are in reality integrals over a continuum of vibronic states. The integral in eq. (36.17) can therefore be evaluated by the method of steepest descent. 

The Ai and Ei modes are both IR and Raman active, the twofold 2 ( 2° and are only Raman active, and the Bi modes and are silent. Owing to the macroscopic electric field associated with the relative atomic displacement of the longitudinal phonons, the polar Ai and Ei modes are split into longitudinal optical (LO) and transverse optical (TO) components. This electric field serves to stiffen the force constants of the phonon and thereby rises the LO frequency over that of the respective TO mode. The comparatively large mode polarity in GaN is a consequence of the strong ionic character of the bonds between group III and N atoms. 

Fuchs and Kliewer (1965) have predicted the existence of macroscopic surface optic modes in ionic crystals. We give here a simplified derivation of their result, based on the formalism of the dielectric constant. In the phonon frequency range, the bulk dielectric constant e( )) approximately varies with co as ... 

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