Biphonon states

In connection with the afore-said, for further analysis of the biphonon states we shall make use of a Hamiltonian of the form 

To obtain Hamiltonian (6.11) we proceeded above from the model of a molecular crystal. Actually, its range of application includes nonmolecular crystals as well, provided we are concerned with optical phonons whose bandwidth is much narrower than the phonon frequency. In these spectral regions the vibrations of the atoms inside the unit cell are similar to intramolecular vibrations in molecular crystals, since the comparatively narrow phonon bandwidth is indicative of the weakness of the interaction between the vibrations of atoms located in different unit cells. 

The wavefunction of states of the crystal with two vibration quanta, of interest to us, can be written as 

Before solving this system of equations we note that biphonon states, like other crystal states, transform according to the irreducible representations of the 

To find the biphonon energy levels, we introduce the two-particle Green s function in the zeroth approximation G°E (n, m , ), n n, a rn m, fj and so on, satisfying the equation 

---

Since for odd states ip(n, n) = 0, the Hamiltonian (6.28) cannot lead to the occurrence of odd biphonon states. This conclusion follows directly from the fact that the result of the action of the anharmonicity operator in Hamiltonian (6.28) on the odd wavefunctions of the biphonon is equal to zero. For example,... 

If we include in the Hamiltonian (6.28) not only the intramolecular anhar-monicity (proportional to A) and the anharmonicity due to the intermolecular interaction WA, but also the part of the intermolecular anharmonicity that is of the form Lnmlnm, where Lnm is the matrix element of the operator of intermolecular interaction between molecules n and m, corresponding to an interchange of quanta, the odd biphonon states can also separate from the band of... 

So far the odd biphonon states have not been discovered experimentally. In molecular crystals their contribution to the absorption spectrum or Raman scattering spectrum should be relatively small because in the odd states (6.29) there are no configurations in which both quanta sit on one and the same molecule. 

If, however, there is no biphonon, no gap appears in the polariton spectrum outside the region of two-particle states. Consequently, the experimental observation of such a gap is, at the same time, experimental proof of the existence of the biphonon state (see Section 6.7). 

A discussion of the more general situation with r 0 is more cumbersome and is not given here. We only point out that the appearance of biphonon states leads to a new resonance of k(E) when Fermi resonance is taken into account. But the appearence of quasibiphonon states can substantially alter the density of states inside the band of two-particle states. 

Let us assume that the anharmonicity constant A is large compared to the width T) = 2T (where T is the half-width) of the phonon band. In this case, as was shown in Subsection 6.2.2, the width of the biphonon band is of the order of T /A, i.e. small compared to the width T of the band of optical phonons. Important here is, however, the comparison of the width of the biexciton band with isotope shift. Indeed, in the limiting case of strong anharmonicity, the biphonon energy is E ss 2hui — 2A and the biphonon state 2) is just the coherent superposition of the states of two-fold excited molecules. It is clear then that an elementary generalization of an equation of the type (6.89) can be used to find biphonon local states. Specifically, the equation for the frequency of a local biphonon io, i.e. the localized state split off the biphonon zone, can be written as follows ... 

The bound state of two phonons is usually called a biphonon. Quite a comprehensive theory of biphonons has been developed and, what is of prime significance, convincing evidence has been obtained of their existence in various kinds of crystals. 

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 should be noted, to begin with, that biphonons are quite analogous to bound states of two magnons (4)-(6). 

In the papers by Van Kranendonk (7), (8) only the bound states of two different quasiparticles were considered under the condition that the motion of one of them can be ignored in a first approximation (the Van Kranendonk model, see Subsection 6.2.3). This made the Van Kranendonk model inapplicable for analysis of the biphonon spectrum in the frequency region of overtones, as well... 

A generalization of biphonon theory beyond the Van Kranendonk model was made later (14)—(17). Subsequently, the effect of biphonons on polariton dispersion in the spectral region of two-particle states was investigated in a number of papers (18)—(22), and the contribution of biphonons to the nonlinear polarizability of a crystal was discussed in (23)-(25). Problems of the theory of local and quasilocal biphonons in disordered media were discussed in a number of papers (14), (26) -(28). The influence of anharmonicity in crystals on the spectra of inelastically scattered neutrons was considered by Krauzman et al. (29), Prevot et al. (30), and in Ref. (31). 

Subsequently, a peak in the RSL spectra, similar to the one observed by Krishnan, was not found in some crystals, such as silicon and germanium, which have the same type of structure as diamond and have even stronger anharmonicity than diamond. This encouraged Tubino and Birman (33) to improve the accuracy of the calculations of the structure of the phonon bands in crystals with a diamond-type structure. It was shown as a result of comprehensive investigations that the dispersion curve of the above-mentioned optical phonon in diamond has its highest maximum not at k = 0, but at k 0. The result of these calculations indicates that the peak experimentally observed in the RSL spectra of diamond falls within the region of the two-phonon continuum. It cannot correspond to a biphonon and is most likely related to features of the density of two-particle (dissociated) states. 

A), biphonons have an extremely simple structure they go over into the states of molecules excited to the second vibrational level. Here the spectrum of the crystal in the frequency region being considered consists of two lines, with the corresponding crystal energies E = 2HQ — 2A (both quanta sit on the molecule and the state of the crystal is A-fold degenerate, where N is the number of molecules in the crystal) and E = 2Ml (the quanta sit on different molecules and the state of the crystal is [N(N — l)/2]-fold degenerate). 

If, on the contrary, anharmonicity is weak ( A particle states. But inside the band of two-particle states, as shown by Pitaevsky (38), only weakly bound states of biphonons are formed (even for the smallest value of A it is necessary, of course, that the value of the binding energy of the biphonon be greater than the width 6 of the phonon level regarding the feasibility of observing the states discussed by Pitaevsky(38), see below). 

If a unit cell of the crystal contains several molecules, then, as has already been noted, the shape of the spectrum of two-particle states becomes more complicated even when anharmonicity is ignored. As concerns the number of biphonon bands, it is equal, under conditions of strong anharmonicity A Vnm ) for nondegenerate vibrational transitions, to the number a of molecules in the unit cell. 

Let us now consider what is the analog of a Fermi resonance in a molecule when we consider the crystals. In going over from an isolated molecule to a crystal, the branches of optical phonons appear. In the region of overtone and sum frequencies, several bands of many-particle states arise and, if anharmonicity is sufficiently strong, bands of states with quasiparticles bound to one another (for instance, biphonons) will also appear. Thus, in crystals a large number of... 

This equation, a generalization of eqn (6.27), enables one to calculate the energy of biphonons, taking into account the Fermi resonance of the two-particle 13-phonon states with the band of C-phonons. A comparison of relations (6.54) and (6.27) indicates that taking the Fermi resonance into account leads to a renormalization of the anharmonicity constant... 

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. 

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)). 

FlG. 6.8. Dependence of the dielectric function on the frequency in the region of overtone frequencies E b and E b are the energies of the longitudinal and transverse biphonons, respectively m n and emax are the minimum and maximum energy values in the band of two-particle states. 

In the preceding section it was shown that the formation of bound states of phonons leads to the appearance of a new type of resonance of the dielectric tensor ij(co). It is clear, of course (23), that the nonlinear polarizabilities should have analogous resonances, and this also concerns, besides biphonons, other types of bound states of quasiparticles, such as biexcitons, electron-exciton complexes, etc. 

Note that for H(uj + uj ) = Eb it is necessary to take the damping of the biphonon into account in the expression for A(w + u/,0). In this case, along with the real part of the tensor Xije u>,uj), an imaginary part is also present. This corresponds, as is well known, to the occurrence of two-photon absorption that is accompanied, in the given case, by the excitation of a biphonon. As applied to excitons this question has been discussed by Hanamura (51) within the framework of a somewhat different approach. We refer to it here (see also Fly-tzanis (52)) because both the generation of a second harmonic and two-photon absorption are processes that are completely described by the nonlinear polarizability of the crystal found above with bound states taken into account. Actually, these processes can be investigated by a single method. 

Since the states of the biphonon, like those of the phonon, are characterized by only a value of the wavevector, an analysis of eqn (6.90) is analogous to that of eqn (6.89). On the basis of the result of such an analysis, which we have already used for phonons, it can be contended that the level of a local biphonon is formed if... 

---