Fermi resonance with polaritons

The functions l I (h) and 2(E) in the presence of a Fermi resonance min and max are the boundaries of the band of two-particle dissociated states. The open and filled circles represent the biphonon and quasibiphonon, respectively, e min- 

Relations (6.50), (6.52), and (6.53) are required to find the density of states in the presence of Fermi resonance, and this can be done in a way similar to the case of T = 0. We shall not give the corresponding calculations here, but shall turn to a discussion of a more general situation arising in the case of Fermi resonance with a polariton. 

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As is well known, the selection rules allow RSL by polaritons only in crystals without a center of inversion. This is precisely the kind of crystal in which Fermi resonance with polaritons (to be discussed below) was found to be the physical phenomenon in which the special features of the biphonon spectrum were most evident. 

In the subsequent sections of this chapter we discuss the fundamentals of biphonon theory, consider the special features of the Fermi resonance, including Fermi resonance with polaritons, and also analyze the data obtained in the infrared (IR) absorption and RSL spectra (see also the review (18)). 

Fermi resonance with polaritons 6.4.1 Microscopic theory... 

Note that the microscopic theory of Fermi resonance with polaritons, developed above, cannot be directly applied to cubic crystals, because triply degenerate states correspond to dipole-active transitions in such crystals (for the corresponding generalization of the theory, see (41)). However, as was mentioned previously, the polariton spectrum can also be found within the framework of macroscopic electrodynamics, which requires that the dielectric tensor of the crystal be known. The results of a proper analysis, as could be expected, are equivalent to those obtained in microscopic theory. We shall use the macroscopic theory in the following in application to cubic crystals. Using this approach we shall show additionally how the longitudinal and surface biphonons can also be found (see also (15)). 

Experimental investigations of biphonons and Fermi resonance with polariton... 

Surface biphonons could be investigated, for example, by the attenuated total reflection (ATR) method. In contrast to RSL by polaritons, this method is effective, as is well known, both for crystals with and without inversion center. In this sense, it is a more universal method. In conclusion we point out that in degenerate semiconductors Fermi resonance with plasmons (47) is also possible along with Fermi resonance with phonons and polaritons. The spectrum of plasmophonons has been measured in many semiconductors by the RSL method (see, e.g. Mooradian and McWhorter (48)). 

FlG. 6.15. Measured polariton dephasing rate 2/T2 ps 1 vs polariton frequency, under the conditions of Fermi resonance with the 2rq band (shaded region), at a crystal temperature 78 K. The full line is calculated (see text). 

Besides the region of basic (fundamental) frequencies of lattice vibrations, the polariton (light) branch in crystals also intersects the region of two-particle, three-particle, etc. states. Resonance with these states influences the dispersion law of the polariton and the result of this influence can be expediently investigated by the observation of the spectra of RSL by polaritons. What actually occurs here is a resonance, similar to the Fermi resonance, since one of the normal waves in the crystal (the polariton) resonates with states that are analogous to overtones or to combination tones of intramolecular vibrations. 

The most useful experimental data were obtained in investigations of the Fermi resonance of the polariton with two-particle states. This led to the discovery of biphonons in many crystals. Before beginning a discussion of the results obtained in these investigattions, we have several comments to make on the development of this research from the historical point of view. 

It is an essential fact that the above-mentioned gaps in the polariton spectrum, if they arise, as well as the corresponding interaction between the photon and phonon, are nonzero within the framework of linear theory and, in general, do not require that anharmonicity be taken into account. Therefore, it makes sense to denote as a polariton Fermi resonance only such situations where vibrations of overtone or combination tone frequencies resonate with the polariton. We now turn our attention to an analysis of such rather complex situations, requiring that multiparticle excited states of the crystal be taken into consideration. Shown schematically in Fig. 6.6 is a typical polariton spectrum, as well as a band of two-particle states of B phonons. If, under the effect of anharmonicity, biphonons with energy E = E are formed, these states also resonate with the polariton, influencing its spectrum. 

We direct our attention, first of all, to an analysis of the dispersion law for polaritons in the region of the Fermi resonance. For this purpose, by analogy with... 

FlG. 6.14. The polariton dispersion curve of ammonium chloride in the vicinity of the 2 4 two-phonon quasicontinuum (shaded region) obtained by near forward Raman scattering (filled circles). The dashed curve is the calculated dispersion curve in the absence of polariton Fermi resonance. Reprinted with permission from Mitin et al. (71). Copyright (1975), American Physical Society. 

Fermi resonance of vibrational excitons with polaritons. In this section we discuss the same effect only for molecular vibrations in multilayers. Below we present a few preliminary remarks which can be useful in the following discussion. 

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