Phonon scattering processes

In order to provide a more quantitative picture of phonon scattering we consider the scattering of an electron in a plane wave state from a solid in the presence of phonons. We will denote the incident wave-vector of the electron hy k and the scattered wave-vector by k. The matrix element M k for the scattering process will be given by the expectation value of the scattering potential, that is, the potential of all the ions in the crystal, Vcriit), between initial and hnal states  

We are now interested in determining the behavior of this matrix element when, due to thermal motion, the ions are not at their ideal crystalline positions. Obviously, only the structure factor 5(q) is affected by this departure of the ions from their ideal positions, so we examine its behavior. The thermal motion leads to deviations from the ideal crystal positions, which we denote by s j. With these deviations, the structure factor takes the form 

For simplicity, we will assume from now on that there is only one atom in each unit cell, which allows us to eliminate the summation over the index j. We can express the deviations of the ions from their crystalline positions S in terms of the amplitudes of phonons and the corresponding eigenvectors e , with / denoting the phonon branch  

We will take the deviations from ideal positions to be small quantities and the corresponding phonon amplitudes to be small. To simplify the calculations, we define the vectors as 

Keeping only terms up to second order in in the last expression gives the following result  

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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]. 

Fig. 3.7. Polarized micro-Raman spectra of a (0001) ZnO bulk sample (a) and a (0001) ZnO thin film (d 1,970 nm) on (0001) sapphire (b). The vertical dotted and dashed lines mark ZnO and sapphire (S) phonon modes, respectively. MP denotes modes due to multi-phonon scattering processes in ZnO. Excitation with Ar+-laser line A = 514.5 nm and laser power P < 40 mW. Reprinted with permission from [38]...

At high temperature, TTF TCNQ is metallic, with a(T) oc T-2 3 since TTF TCNQ has a fairly high coefficient of thermal expansion, a more meaningful quantity to consider is the conductivity at constant volume phonon scattering processes are dominant. A CDW starts at about 160K on the TCNQ stacks at 54 K, CDW s on different TCNQ chains couple at 49 K a CDW starts on the TTF stacks, and by 38 K a full Peierls transition is seen. At TP the TTF molecules slip by only about 0.034 A along their long molecular axis. 

The nonadiabatic (nonsecular) contributions T, and T34 to the coherence decay are caused by inelastic 7 ,-type processes. Equation (41b) shows that these inelastic scattering processes are induced by anharmonic-ity (k ) in the ground state and a combination of anharmonicity and electron-phonon coupling (Vg ) in the excited state. Here describes the decay (creation) of the pseudolocalized phonon into (from) two band phonons. The relevant part of is in (39) the last term, which describes in the excited state the exchange of a pseudolocalized phonon with a band phonon. At low temperature (/c7 phonon scattering processes in the ground and excited state. 

The mean free path A may be determined by many different scattering mechanisms but the dominant one at temperatures not too close to o °K is phonon-phonon scattering, the coupling taking place through the anharmonicity of the lattice vibrations. There are two possible types of phonon-phonon scattering processes normal processes in which total phonon wave vector is conserved, and umklapp processes in which the total wave vector after collision differs from that before collision by a vector of the reciprocal lattice. Since normal processes do not affect the total phonon momentum or energy, they do not contribute to thermal resistance and only umklapp processes need be considered. For an umklapp process to occur between two phonons of wave vectors q and q we must have a relation of the form... 

Fig. 9.9 Electron-phonon scattering in a one-dimensional (1-d) and in a two-dimensional (2-d) metallic system. Since the Fermi energy is always large compared to the phonon energies, the single-phonon scattering processes are limited to the immediate neighbourhood of the Fermi surface. In the 1-d system, the Fermi surface consists of only two points and the conduction electrons can be scattered only from kp to -kp. In the (isotropic)...

Fig. 2. Schemalic shape of the temperature dependence of for solids. The temperature regions are marked, where various phonon. scattering processes play an appreciable role. 0 is the Debye temperature.

Fig. 136, The temperature dependence of the lattice thermal resistance, Wl= I/k, in LaTei,33, LaTe, 47 and LaTej 33. Solid lines are the result of calculation, points of experiment. The phonon scattering processes considered in the calculations are (1) phonons by phonons, (2) phonons by phonons plus phonons by electrons, (3) phonons by phonons plus phonons by defects, (4) phonons by phonons plus phonons by electrons plus phonons by defects.

In harmonic crystals, the vibrations are phonons, i.e. plane waves involving the motion of all atoms in the sample. Phonons are described by a frequency illuminated volume seen by the detector, and fields irradiated by aU polarizable units interfere. As a consequence, one phonon scattering processes (first-order Raman scattering) are subjected to the selection rules ... 

For an arbitrary state of the system, we are not interested in individual phonon scattering processes but rather in the thermal average over such events. We are also typically interested in the absolute value squared of the scattering matrix element, which enters in the expression for the cross-section, the latter being the experimentally measurable quantity. When we average over the sums that appear in the absolute value squared of the structure factor, the contribution that survives is... 

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