Polariton spectra

LiI03 is a uniaxial hexagonal crystal (factor group C6). Vibrations of species A, E, and E2 are allowed in the Raman effect, but only A and E, are infrared-active, therefore polariton dispersion is expected for the transverse phonons of these two species. The phonon and polariton spectra were investigated by Claus 26>27> and Otaguro et al. 28>29). Here we want to show two series of spectra recorded by Claus. 

Experimental investigations of the effects of strong anharmonicity in phonon and polariton spectra are reported in many papers. In order to give some idea of the state-of-the-art concerning experimental research along these lines, as well as in the study of crystals, we shall make certain comments. 

A review is given of first principles methods for the calculation of exciton and polariton spectra in polymer crystals. Correlation effects are treated using the polaron model, the interaction of electron and hole pola-rons is calculated with the help of the resolvent operator formalism. The important role of charge-transfer components in the singlet and triplet exciton and polariton spectra of polydiacetylenes is discussed. 

Experimentally observed polariton [spectra in polymer crystals have mostly been analyzed in terms of phenomenological models, treating the solid as a continuum (ISIS). Microscopic treatments for molecular crystals have been proposed within the framework of the Frenkel-exciton model (19-22) using semiempirical Hamiltonians to represent the matter field. 

Chichibu, S.F., Uedono, A., Tsukazaki, A., Onuma, T., Zamfirescu, M., Ohtomo, A., Kavokin, A., Cantwell, G., Litton, C.W., Sota, T. and Kawasaki. M. (2005) Exciton-polariton spectra and limiting factors for the room-temperature photoluminescence efficiency in ZnO. Semiconduttor Science and Technology, 20, S67. 

The spectrum of polaritons can be found by means of Maxwell s macroscopic equations (see Ch. 4), provided that the dielectric tensor of the medium (44) is assumed to be known. Without going into details, we emphasize here that always a gap appears in the polariton spectrum (here we ignore spatial dispersion) in the region of the fundamental dipole-active vibration (C-phonon, exciton, etc.). At present, there is a sufficiently detailed theory for RSL by phonon-polaritons, taking many phonon bands into consideration. With this theory the RSL cross-section can be calculated for various scattering angles provided that the dielectric tensor of the crystal is known, as well as the dependence of the polarizability of the crystal on the displacement of the lattice sites and the electric field generated by this displacement (45). 

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. 

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). 

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)). 

The numerical simulations performed in (32), (33) for one-dimensional microcavity containing organic material with diagonal disorder qualitatively confirm the conclusions made above on the structure of the polaritons spectrum in an organic microcavity. Future simulations for realistic 2D organic microcavities would be very topical. 

The spectrum of the excitations is shown in Fig. 10.5 for 2 A = 80 meV. The dashed lines show the uncoupled molecular excitons and photons, and the solid lines show the coherent part of the spectrum with well-defined wavevector. The crosses show the end-points of the spectrum of excitations for which q is a good quantum number. The spectrum of incoherent (weakly coupled to light) states is shown by a broadened line centered at the energy Eq. It follows from the expression for the dielectric tensor that this spectrum is the same as the spectrum of out-of-cavity organics. The spectrum of absorption as well as the dielectric tensor depend on temperature. This means that in the calculation of the temperature dependence of the polariton spectrum we have to use the temperature dependence of the resonance frequency Eo as well as the temperature dependence of 7 determining the width of the absorption maximum. However, the spectrum of emission of local states which pump polariton states can be different from the spectrum of absorption. The Stokes shift in many cases... 

Puthoff et al. 77) found two polaritons associated with TO phonons of species Aj with tuning ranges from 630 to 500 cm 1 and 250 to nearly 0 cm-1. The Raman scattering at these extraordinary polaritons was used to study 36) the directional dispersion of the phonons in LiNb03. Recently Winter and Claus 78> also investigated the polaritons associated with the E phonons at 582 and 154 cm-1 by both photographic and photoelectric methods. These branches were important for the assignment of the phonon spectrum 39>. 

Surface electromagnetic waves or surface polaritons have recently received considerable attention. One of the results has been a number of review articles1, and thus no attempt is made here to present a comprehensive review. These review articles have been concerned with the surface waves, per se, and our interest is in the use of surface electromagnetic waves to determine the vibrational or electronic spectrum of molecules at a surface or interface. Only methods using optical excitation of surface electromagnetic waves will be considered. Such methods have been the only ones used for the studies of interest here. 

This split-off discrete state rejoins, for cK co0, the exciton energy ha>0 it behaves qualitatively in the same way as the lower branch of the 3D polariton.33 35 For this reason we call it the 2D polariton. It is the projection of the exciton K> on this 2D polariton (radiatively stable) that constitutes (1) the finite limit value of the curves AK t) for t- oo (Fig. 3.8), and (2) the weight of the discrete peak in the spectrum PK((o) (Fig. 3.9). The transition, in the 2D polariton branch, between the photon and the pure exciton characters occurs around the value K0 = co0/c in an area of width AK = r0/c (with ro = 15cm 1). Thus, the 2D polariton may be considered as a photon mode trapped in the 2D lattice, where it acquires its own dispersion.115,116,126 Therefore, the 2D polaritons cannot be excited by free photons, but they may be coupled to evanescent waves, by ATR for example.115,116... 

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... 

First of all we have to mention that the above described situation of resonance is not related to any quantum effects. Moreover, the role of the transverse electromagnetic field in crystal oscillations in the infrared part of the spectrum was discussed by means of the classical dynamics of crystal lattices a long time ago by Born and Ewald (2) (see also (3) and (4)), and later by a semiphenomenological approach in (5), (6). It is evident, however, that a quantum theory of polaritons in the region of electronic transitions can also be important particularly for the discussion of quantum effects. 

In (4.25), according to its derivation, the vector k is real. Consequently the eqn (4.25) gives a spectrum of polaritons consisting, in general, of allowed and forbidden energy bands.36 We can also proceed in a different way and express the refraction index n(w, s) defined by the relations... 

It follows from the above relation that the retarded interaction is important only in the vicinity of wavevectors k y/eoQ/c, i.e. in that part of the spectrum, where the frequencies of the Coulomb excitons are near to those of the transverse photons. When the retardation is ignored, the branches of the Coulomb excitons and the transverse photons intersect (Fig. 4.1a). This intersection is removed when the retardation is taken into account (Fig. 4.1b). In a similar way the dependence w(k) for polaritons can be found for crystals with different symmetries. 

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. 

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)). 

When the unit cell of an organic crystal contains two or more molecules, the spectrum of the cavity polaritons strongly depends on the relation between (i) the detuning u = coc — u>ly (ii) the energy of the Rabi splittings W1 and W2, and... 

Typical examples of the dispersion curves are shown in Fig. 10.2 for 99 = 7t/4, small positive detuning u = ivc — d>i = 35 meV, and different relations between Wy, W2 and Ac = 2 — uq. It is clear that when one of the coupling parameters IF is small, the spectrum consists of a doublet of polariton branches and of two branches which are close to the bare cavity photon and the bare exciton (Fig. 10.2a). When the Davydov splitting is small and W W2, then the pairs of the dispersion curves almost overlap (Fig. 10.2b). The electric fields in the overlapping curves are (with accuracy up to terms of the order of q2 IQ2) perpendicular to each other (see the discussion in Subsection 10.2.4). When all the parameters are of the same order, the spectrum consists of four well-pronounced polaritonic branches (Fig. 10.2c). 

From this macroscopic consideration it is seen that the states for which the wavevector is not a good quantum number do not form in a certain vicinity of q = 0 for both branches, and for q > q lx for the lower branch. In other words, the states with the well-defined wavevector exist in the intermediate region of the wavevectors only q n < q < qitlx for the lower branch, and q > q Pn for the upper branch. However, in contrast to the case of vanishing q, one can say that for q A> 1 the coherent polaritonic states do not form at all. The excited states from this part of the spectrum are not resonant with the cavity photon, and as a result no hybridization happens. Instead, these excited states are similar to the localized excited states in a non-cavity material, i.e. they are to be treated just as incoherent excited states. 

FlG. 10.5. The dispersion curves of the coherent polaritonic states (solid lines) and of uncoupled cavity photons and the molecular excitation (dashed lines). The crosses show the end-points of the part of the spectrum with well-defined wavevector. On the right, the broadened line of the molecular resonance is shown. The inset shows the excitonic weights (10.26) for upper ( cix ) 2) and lower ( cix ) 2) polaritonic branches. Reprinted with permission from Agranovich et al. (15). Copyright 2003, American Physical Society. 

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