Coulombic excitons

In the first part of this introductory section, we summarize the main collective phenomena acquired by the dipolar exciton from the lattice-symmetry collectivization of molecular properties. The crystal is considered as an assembly of electrically neutral systems, the molecules, physically separated from each other and in electromagnetic interaction. This /V-body problem will be treated quantum-mechanically in the limit of low exciton densities. We redemonstrate the complete equivalence of this treatment with the theories of Lorentz and Ewald, as well as with the semiclassical approximation. In Section I.A, in a more compact but still gradual way, we establish the model of the rigid lattice of dipoles and the general theory of low-exciton-density systems in interaction with the radiation field. Coulombic excitons, photons,... 

Outside of a small region around the center of the Brillouin zone, (the optical region), the retarded interactions are very small. Thus the concept of coulombic exciton may be used, as well the important notions of mixure of molecular states by the crystal field and of Davydov splitting when the unit cell contains many dipoles. On the basis of coulombic excitons, we studied retarded effects in the optical region K 0, introducing the polariton, the mixed exciton-photon quasi-particle, and the transverse dielectric tensor. This allows a quantitative study of the polariton from the properties of the coulombic exciton. 

As illustrated in Fig. 2.8 of Section II, the general reflectivity lineshape shows (1) a sharp rise of the bulk 0-0 reflectivity (Section II.B.C) at E00, corresponding to the b coulombic exciton with a wave vector perpendicular to the (001) face (2) a dip, corresponding to the fission in the surface of a bulk polariton into one 46 -cm 1 phonon and one b exciton at E°° + 46 cm"1 (3) two vibrons E200 and E1 00 immersed in their two-particle-state continua with sharp low-energy thresholds. On this relatively smooth bulk reflectivity lineshape are superimposed sharp and narrow surface 0-0 transition structures whose observation requires the following ... 

In the intermediate domain of values for the parameters, an exact solution requires the specific inspection of each configuration of the system. It is obvious that such an exact theoretical analysis is impossible, and that it is necessary to dispose of credible procedures for numerical simulation as probes to test the validity of the various inevitable approximations. We summarize, in Section IV.B.l below, the mean-field theories currently used for random binary alloys, and we establish the formalism for them in order to discuss better approximations to the experimental observations. In Section IV.B.2, we apply these theories to the physical systems of our interest 2D excitons in layered crystals, with examples of triplet excitons in the well-known binary system of an isotopically mixed crystal of naphthalene, currently denoted as Nds-Nha. After discussing the drawbacks of treating short-range coulombic excitons in the mean-field scheme at all concentrations (in contrast with the retarded interactions discussed in Section IV.A, which are perfectly adapted to the mean-field treatment), we propose a theory for treating all concentrations, in the scheme of the molecular CPA (MCPA) method using a cell... 

The Hamiltonian (2.2) contains only the instantaneous Coulomb interaction between the carriers forming the crystal. Therefore, as in (12), the excitons corresponding to the operator (2.2) will be called Coulomb excitons. 

The long-wavelength field can be easily found if we take into account that in a medium without external charges the longitudinal component of the induction vector T> vanishes, and the macroscopic electric field is longitudinal, if the retardation, as assumed in the theory of Coulomb excitons, is not taken into account. From this considerations we obtain... 

This is, however, not the only reason to consider mechanical excitons because mechanical excitons can be used as states of zeroth approximation in calculation of the crystal dielectric tensor (see Sections 7.1, 7.3 and the monographs (12), (16) for this problem). In addition, as will be shown below, using the states of mechanical excitons and also the dielectric tensor, one can establish the energies of Coulomb excitons. Here we do not repeat the calculations of the dielectric tensor by making use of mechanical excitons states, since these calculations can be found in Ch. IV of the monograph (12). We will use only the results of these calculations. 

The frequencies of Coulomb excitons can be found from the dielectric tensor (see Section 7.4 and (12), 2). If, for a given direction, k mechanical excitons have only the vectors Po o which are not perpendicular to the direction of k, then the frequencies of corresponding Coulomb excitons for this direction of the wavevector can be found from the equations9... 

Making use of the above equation in the case of a simple model crystal, where the formula (2.45) is valid, we obtain for the Coulomb exciton energy E = frw the expression... 

Substituting the expression (2.52) into eqn (2.47) we obtain the frequencies of Coulomb excitons in terms of that of mechanical excitons... 

Equations (2.53) give, in particular, the Davydov splitting for Coulomb excitons... 

As we have shown above, the exciton energy depends not only on the characteristics of a single molecular term /, as it would follow from an elementary approach based on the Heitler-London approximation, but, in general, depends on all excited states of the molecule. This property is reflected by the fact that, as we have shown, the energies of Coulomb excitons can be expressed in terms of the crystal dielectric tensor, which includes contributions of all resonances. 

Thus, by making use of the formula (2.4) which gives the crystal wavefunction for the ground state, then using the formula (2.8), which for k = 0 gives the wavefunction of a Coulomb exciton with k = 0 in the excitonic band //, we obtain... 

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. 

The existence of this relation should be no surprise since, as we have demonstrated, the tensor ejj(u ,k) determines the frequencies of all normal electromagnetic waves in a condensed medium. But this relation, as we show below, permits a simplified consideration of some properties of Coulomb excitons including the dependence of Coulomb exciton energies on s for k —> 0, which by using microscopic theories is quite tedious. 

Substituting (4.77) into (4.76) we find for the Coulomb exciton energy the following expression... 

Note, however, that the dependence of the polariton energy on the wavevec-tor, which arose when only retardation is taken into account, is correct only if we can neglect the dependence of the energy of the Coulomb exciton Etl on k, arising from instantaneous Coulomb interaction. For example, if we apply this theory for 2D quantum well polaritons, the linear term in the dispersion of po-laritons will be cancelled because in this case the linear term as a function of the energy of the quantum well exciton on the wavevector has the same value with opposite sign. 

Linearity of the above relation allows one to find the tensor (01, k) by calculating the polarization P, induced in the crystal by the total transverse field E4-, neglecting the local counterpart of the transverse field which is very small.48 As unperturbed states in this case we have to use the Coulomb exciton states which are obtained taking full account of the Coulomb interaction between charges. Importantly in this case we can assume that charge transfer excitons are also taken into account. If the unperturbed states, obtained by taking into account the full Coulomb interaction, are known, considering the field E4- as a perturbation we can determine the polarization... 

The values a(w,k) and b(v, k) have resonances at frequencies corresponding to longitudinal and transverse polaritons. If one takes into account the dissipation, the imaginary parts of the polariton energies would appear in the denominators of these expressions. As longitudinal excitons do not interact with the transverse electric field, the resonances of a(v, k) coincide with the frequencies of longitudinal Coulomb excitons. 

As unperturbed states we use the states obtained by accounting for the Coulomb interaction (Coulomb excitons), and the transverse part of the macroscopic field in the medium, i.e. E (r,t) = —(l/c)d A/dt, where A(r,t) is the vector potential of the transverse field, is considered as the perturbing field. Denote by I no and En the wavefunctions and eigenenergies of the unperturbed states, and by = tE o + the corresponding wavefunctions when is taken into account. The operator of interaction of charges with the external field, in the linear approximation, is given as... 

It is evident that an explicit form of this dependence can be given only if we know explicitly the expressions F(k + q, k, qr) and the energies EM(k) of the Coulomb excitons. However, independently of the concrete form of those functions, it follows from (7.55) that Y(uj, k) vanishes for uj < E m/h, where... 

The detailed theory of the frequency dependence of 7(w) for the regions of exciton resonances has been developed by Toyozawa (14), (15). However, in this work the Coulomb excitons and transverse photons were used as zeroth-approximation states, so that the above discussed effect of the long-wavelength... 

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