Three-phonon processes

Important contributions on the Raman side came after 1970 which could make use of the well estabhshed assignment of the fundamentals in interpreting the spectra [97-99, 134]. The most complete studies on two- and three-phonon processes have been carried out by Harvey and Butler (Raman, only gxg combinations assigned) [100] and by Eckert (Raman and IR) [109]. 

We first consider the vibrational relaxation that can be induced by aijQqiqj (three-phonon processes) or Qq qi (four-phonon processes). In the three-phonon processes there are two accepting modes, while in the four-phonon processes there are three accepting modes. To calculate the rate of vibrational relaxation, we use... 

Charged point defects on regular lattice positions can also contribute to additional losses the translation invariance, which forbids the interaction of electromagnetic waves with acoustic phonons, is perturbed due to charged defects at random positions. Such single-phonon processes are much more effective than the two- or three phonon processes discussed before, because the energy of the acoustic branches goes to zero at the T point of the Brillouin zone. Until now, only a classical approach to account for these losses exists, which has been... 

In the gas phase, the asymmetric CO stretch lifetime is 1.28 0.1 ns. The solvent can provide an alternative relaxation pathway that requires single phonon excitation (or phonon annihilation) (102) at 150 cm-1. Some support for this picture is provided by the results shown in Fig. 8. When Ar is the solvent at 3 mol/L, a single exponential decay is observed with a lifetime that is the same as the zero density lifetime, within experimental error. While Ar is effective at relaxing the low-frequency modes of W(CO)6, as discussed in conjunction with Fig. 8, it has no affect on the asymmetric CO stretch lifetime. The DOS of Ar cuts off at "-60 cm-1 (108). If the role of the solvent is to open a relaxation pathway involving intermolecular interactions that require the deposition of 150 cm-1 into the solvent, then in Ar the process would require the excitation of three phonons. A three-phonon process would be much less probable than single phonon processes that may occur in the polyatomic solvents. In this picture, the differences in the actual lifetimes measured in ethane, fluoroform, and CO2 (see Fig. 3) are attributed to differences in the phonon DOS at 150 cm-1 or to the magnitude of the coupling matrix elements. 

The first process shown above (Fj) is third order and by far the most commonly cited relaxation mechanism in pure crystals. It corresponds to a three-phonon process whereby the initial phonon either splits into two new phonons of lower energy (first term, down conversion) or interacts with a higher energy phonon (second term, up conversion). The down conversion process leads to a finite linewidth at 0 K, whereas the up conversion process gives zero contribution at low temperature. 

Similar relationships can be obtained for three-phonon processes. At low temperature, n (iv, T) is much smaller than unity and the above expression tends to unity for summation processes, and to zero for the difference processes, which are, therefore, not observed at low temperature. At higher temperature, the absorption intensity increases for both processes [45], at a difference with the one-phonon process, which is temperature-independent. 

We have mentioned the existence of CPs in the one-phonon density of states, but this can be measured only for compound crystals. The situation is different in multi-phonon absorption because the high-frequency phonons of the BZ boundary are mostly involved (note that in three-phonon processes, the q = 0 zone-centre phonons can also be involved without problem for momentum conservation). For the two-phonon absorption, the density of states is proportional to an integral similar to the one in expression 3.20, with wt replaced by the sum of the two pulsations and of the phonons of the combination. Besides the trivial case where = u> = 0, the condition Vq (wt (q) + wt (q)) = 0 is fulfilled when Vq (wt) = Vq(uv) = 0 or when Vq (wt) = — Vq (wt ). The observed two-phonon absorption is the sum of the contributions of the possible two-phonon processes. Figure 3.2 shows the RT absorption of silicon in the two- and three-phonon absorption region. In the two-phonon region, it is fitted with the two-phonon dispersion curves... 

All other terms that are not related to the above terms by permutation of the Indices are zero. As noted before, < gives the optical phonon frequency y gives the amplitudes for three-phonon processes and a and 3 give the amplitudes for the bare four-phonon processes. 

The transition probabilities for other three-phonon processes are readily derived, by reference to those for two-phonon and one-phonon processes. 

The difTeiential cross sections of uniaxially oriented polyethylene for the one-phonon,two-phonon and three-phonon processes were calculated by Kitagawa and Miyazawa (1969b), as shown in Fig. VIL3. In the region below 200 cm, the transverse one-phonon cross section is much larger than the longitudinal one-phonon cross section. On the other... 

The reflection spectrum measured at room temperature from 90 to 600 cm" has a maximum near 130 cm (R 85%) and a minimum near 185 cm (R 1%). These two extrema were attributed to the lattice vibrations cojo and (Olq, respectively. Axe [1], see p. 187. Absorption measurements on an EuSe film on LiF at 2 K gave o)to = 134.0 cm" Ikezawa, Suzuki [2]. Bulk samples have their characteristic reststrahlen band between (Ojo and colo, see Fig. 116 which shows reflection and transmission spectra of stoichiometric EuSe obtained by sublimation for the paramagnetic state at 300 K and for the antiferromagnetic NNSS state at 3 K. The numerous intrinsic structures in T(v) (and weaker in R(v)) observed beside the reststrahlen band were explained by one-, two-, and three-phonon processes consistent with measurements of second-order Raman scattering (cf. p. 248). Absorption spectra calculated from R and T at 300, 80, and 3 K (K ax S x10" cm" at v 130 cm ) are given in the paper, Mutzenich etal. [3]. The reststrahlen wavelength is calculated by — 76 im... 

Table 5.1. Schematic representation of the eight three-phonon processes described by H3 and listed in (5.151). Note that positive q s appear as convergent and negative q s as divergent vectors...

The region between 860 and 2400 cm is believed to be caused by multi-phonon processes in which one photon generates two or more phonons. The a values for the two-phonon processes are about 10 times as large as the one-phonon processes, and the three-phonon processes give a values about 10 times as large. 

The two phonon processes produce an absorption peak in YAG at 1450 cm" which is just about 2 j. The three-phonon processes produce the peak at 2120 cm", which is nearly equal to 35j of the AIO4 vibration, and it is an intrinsic feature of YAG and is not caused by trace impurities. The 2 3 and 3 3 peaks of Fe04 groups in YIG occur at 1200 and 1700 cm" (Cockayne, 1966 Wood and Remeika, 1967). 

In direct gap GaAs, an excited electron at the bottom of the conduction band can relax spontaneously back into a hole in the valence band by emitting a photon at the band gap energy. This electron-hole radiative recombination process can only occur in Si if momentum is conserved, i.e., the excited electron wave vector must be reduced to zero. This, in pure Si, occurs via the transfer of momentum to a phonon that is created with equal and opposite wave vector to that of the initial state in the conduction band. Such a three-body process is quite inefficient compared with direct gap recombination.1218 This is why Si is such a poor light emitter. 

With regards to the second feature of real crystals mentioned earlier, there are different types of anharmonicity-induced phonon-phonon scattering events that may occur. However, only those events that result in a total momentum change can produce resistance to the flow of heat. A special type, in which there is a net phonon momentum change (reversal), is the three-phonon scattering event called the Umklapp process. In this process, two phonons combine to give a third phonon propagating in the reverse direction. 

The spin-lattice relaxation is enabled via spin-orbital coupling involving a phonon process. Spin-lattice relaxation time (tsl) is temperature dependent. Generally speaking, tsl becomes smaller on increasing the temperature. One can distinguish three types of spin-lattice relaxation processes [103] ... 

Discussing the relaxation time origin we will also follow [2]. Three mechanisms of relaxation can be accounted here thermal activation over the potential barrier Vo (described with tt), tunnelling through the barrier accompanied by phonon emission (r,), and two phonon process analogous to Raman scattering (tr) ... 

V is the speed of sound, 6 is the Debye temperature, tc is the total phonon-scattering rate, w is the phonon-scattering rate due to three phonon normal processes. In this model, two additional scattering mechanisms of phonon (by point defects and by charge carriers) are considered. 

Note that the case for emission of a blue photon by Tm3+ is slightly different than that of Er ". A three-step process is indicated. This may consist of excitation of the Tm3+ ion and energy transfer from two excited Yb3+ ions, or excitation of the Tm3+ ion by exchange (sequential excitation) with three excited Yb3+ sites. Note that two relaxation steps (phonon emission to the host lattice) would be Involved. 

For three-phonon interactions one distinguishes two types of collisions normal processes (N processes), in which the total momentum is conserved and the direction of flow does not change (these processes lead to infinite thermal conductivity) and Umklapp processes (U processes), in which the sum of the wave vectors is not conserved and changes sharply, leading to a finite thermal resistivity of a crystal. In U processes the following conditions are fulfilled ... 

---