Exciton-phonon interaction

even in such cases, when the conditions of applicability of perturbation theory are satisfied, i.e. when the matrix elements (/)2/ (9)( ) / i=- f], are small compared to the quantities 7 /) — E g), the contribution of multimolecular states, in general, is important. 

In Figs 3.1 and 3.2 we display the results of calculations of the hypochromatic effect (i.e. the quantity of the relative change of oscillator strength) for the lower excited state, which occurs when molecules aggregate to the crystal, as a function of the angle 9a between the dipole moment of the transition to the state A with the polymer axis. In Fig. 3.1 the calculations were performed using the values 

In the previous sections we assumed that the crystal molecules rest at their equilibrium positions at the lattice sites. Such an assumption makes it impossible to discuss several properties of excitons as, for example, the absorption lineshape and others, which result from the exciton-phonon interaction. 

With the aim of obtaining an operator, describing this interaction, we assume that the crystal molecules are displaced from their equilibrium positions and that their orientations also differ from that in equilibrium. 

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The commonly used scheme of energy relaxation in RGS includes some stages (Fig.2d, solid arrows). Primary excitation by VUV photons or low energy electrons creates electron-hole pairs. Secondary electrons are scattered inelastically and create free excitons, which are self-trapped into atomic or molecular type centers due to strong exciton-phonon interaction. 

Using the R dependence of the D s in (2.20), we may write for the hamiltonian of the linear exciton-phonon interaction... 

Using the exciton-wave representation, the exciton-phonon interaction hamiltonians (2.22), (2.24) fuse to a single expression ... 

This equation is an approximation in the sense that high-order diagrams are missing. The first neglected diagrams, in fourth order in the exciton-phonon interaction, are of the form... 

If the exciton-phonon interaction Hep is strong compared to the emission probability, high-order terms in Hep contribute to P (2.131), providing strong luminescence at the expense of the one-phonon (Raman) process. In contrast, if the emission probability dominates the phonon creation probability, the peak (2.133) dominates the secondary emission at the expense of the luminescence.77 Examples of this competition will be discussed for the surface-state secondary emission, where the picosecond emission of the surface states, and its possible modulation, allow very illustrating insights into the competition of the various channels modulated by static or thermal disorder, or by interface effects. 

Isotopically Mixed Crystals Excitons in LiH Crystals Exciton-Phonon Interaction Isotopic Effect in the Emission Spectrum of Polaritons Isotopic Disordering of Crystal Lattices Future Developments and Applications Conclusions... 

Falini, G., Albeck, S., Weiner, S. and Addadi, L. (1996). Control of aragonite or calcite polymorphism by mollusk shell macromolecules. Science, 271, 67-9. [19] Fanconi, B. M., Gerhold, G. A. and Simpson, W. T. (1969). Influence of exciton phonon interaction on metallic reflection from molecular crystals. Mol. Cryst. Liq. Cryst., 6, 41-81. [231]... 

Here at the expansion we have put nk = 1, and xk is taken in the step form Xk = x if ko < k < fcmax and yk = 0 for all other k that is, we believe that the exciton-phonon interaction is effective only in the region between phonon wavenumbers k0 = 50 cm-1 and fcmaT = 200 cm-1. Note that we have also made an extrapolation of the function Kk, expression (359), from large values of k to small values. 

The exciton-phonon interaction operator in the form (3.155) was used by several authors (see, for example, (16)—(18)). 

The coefficients F in (3.155) can be expressed as first derivatives of the quantities Vf and M/m with respect to the displacements of molecules from equilibrium. The operator (3.155), obtained by this approximation, was used in Ref. (18) by analysis of the conditions of applicability of a weak exciton-phonon interaction when the operator (3.155) can be considered as a small perturbation. 

At the same time Frenkel (see for this argument also the monograph by Davydov (19)) discussed another limiting case, which is that of a strong exciton-phonon interaction. In the case of a strong exciton-phonon interaction we assume that the time of the excitation transfer from one molecule to an adjacent... 

The states (3.162) were obtained under the assumption that the resonant inter-molecular interaction is small. Here we notice that attempts to improve the above method of computation of crystal states in the case of a strong exciton-phonon interaction has been made in a series of works (see (23)-(25)). 

Finally we remark that within the same excitonic band, if its width is sufficiently large, one can find both states, for which the exciton-phonon interaction is strong, and states where it is weak so that these states can be computed by means of perturbation theory (for a more detailed discussion of this problem see Ref. (18)). Only in the limit of very narrow excitonic bands do the excitonic states show the character of localized excitons, on which we have concentrated our attention. In all references which we have mentioned above the variational method was used, which gives only the lowest states in the excitonic band. 

As shown above, in such cases we can deal with free excitons, not connected with a local crystal deformation. Since all the states were obtained by the variational method, the physically realizable states are those which minimize the crystal energy. As shown in Ref. (23), in one-dimensional crystals with strong resonant intermolecular interaction the exciton-phonon interaction leads to a situation when the minimal energy of a crystal with exciton corresponds to a state with local deformation. 

An example is pyrene, which is a crystal with a rather strong exciton-phonon interaction and a barrier height U 262 cm-1. It was demonstrated that upon photogeneration of excitons in this crystal, even at low temperature, the process of self-trapping not always requires relaxation to the bottom of the free-exciton band (with k = 0 black arrows in Fig. 3.3), but sometimes takes place directly... 

Another interesting situation arises in the ST of excitons in which the electron and hole are spatially separated and are localized on different filaments (polymers or quantum wires) or different planes (or quantum wells). In such structures the electron-hole Coulomb interaction changes when these filaments or planes are deformed. As a result a strong exciton-phonon interaction may exist, even if the individual quasiparticles (electron and hole) have very small interaction with the phonons. The theory of ST of this type of excitations may be found in Ref. (44). 

It follows from the above discussion that considering excitons and intramolecular phonons as independent particles is approximate. It is clear, in particular, that a sufficiently strong exciton-phonon interaction can create propagating bound states, when electronic and vibronic excitations are centered on the same molecule. These states correspond to the previously discussed weak resonant interaction case. But the existence of such states in the vibronic spectrum does not exclude the existence of free excitons and intramolecular phonons states. Both types of states can usually coexist in the vibronic spectrum, in analogy to the case of two interacting particles, where continuum states, corresponding to free particles, coexist with bound states. 

Including in (3.181) only terms linear with respect to the operators a, a, as representative of the exciton-phonon interaction, Merrifield (52) has discussed the process of dressing of excitons into a coat of virtual optical phonons. Although Merrifield (52), by means of the variational method, has applied his approach to a one-dimensional chain, his results are qualitatively valid in the general case. In particular, he has shown that by increasing the exciton-phonon interaction the excitonic bandwidth decreases. It decreases even for the lowest vibronic state, where real phonons are absent. 

The discussion, initiated by Simpson and Peterson (50), Merrifield (52), and Suna (53) has been continued by Rashba (54) (see also the paper by Philpott (55)). Rashba assumed that the main mechanism of the exciton-phonon interaction is the changing A of the frequency of intramolecular vibrations, when the molecule is excited. 

The biexciton states described above have energies outside the free-exciton two-particle continuum and can decay only due to exciton-photon or exciton-phonon interactions. However, if the biexciton energy lies in the continuum, such a biexciton can decay spontaneously into states with two free (unbound) excitons. These quasibound states display themselves as resonant peaks in the two-particle continuum and can be observed in linear and nonlinear optical processes if these resonances are sufficiently narrow, e.g. if they are situated in the region of low density of two-particle states. 

In the above discussion we have not considered dissipative processes. To include these processes we can, in the framework of the above applied perturbation theory, aside from the perturbation operator U, include a time-independent operator which induces transitions between states tE o- In this case the tensor ej, ij is again defined by the expression (7.52), where in the resonant denominators the energy Huj must be replaced by a complex quantity tuv + ih ylu , k), 7 = 7 + iy", with I7I -C uj. Knowledge of the function 7(0 , k) is important, for example, in the analysis of a lineshape. Below we take into account the exciton-phonon interaction and, for simplicity, consider only the first order of perturbation theory. 

Let us assume that the light frequency uj is in the vicinity of a well-isolated dipole-allowed exciton resonance. In this case the operator of the exciton-phonon interaction, linear with respect to the operator of displacements of molecules from their equilibrium positions, has the form (3.155). It can be shown that in the first order of perturbation theory the real part of 7 uj, k) is given by the following formula (we assume here that the crystal temperature T = 0) ... 

The exciton-phonon interaction and the role of surface defects... 

We discussed in Ch. 3 the role of exciton-phonon interaction in the bulk. Below we stress some effects important also for surface excitons. 

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