Coulombic excitonic interactions

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. 

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. 

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. 

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... 

In this subsection we introduce some notations for the bare excitations whose interaction leads to the formation of cavity polariton states. These bare excitations are cavity photons and Coulomb excitons. 

Besides Coulomb excitons, it is convenient to use also the so-called mechanical excitons. These are the excited states of the crystal found in the approximation that the contribution of the macroscopic electric field to the intermolecular interaction is omitted, and only the contribution of the short-range intermolecular interaction is taken into account. The energies of the mechanical excitons determine the poles of the dielectric tensor ... 

So far, we have considered the Bose-Einstein condensation of Coulomb excitons and thus we neglected the influence of the retardation. Such a consideration is correct only for excitons with small (or zero) oscillator strengths. Thus the above-described consideration on the Bose-Einstein condensation can be applied, for example, to the case of triplet excitons, or to singlet excitons for which the exciton-photon interaction energy is smaller than the exciton level width, caused by, for example, scattering by phonons. 

We present a detailed calculation of the transition temperature of a model, filamentary excitonic superconductor. The proposed structure consists of a linear chain of transition-metal atoms to which is complexed a ligand system of highly polarizable dye molecules. The model is discussed in the light of recent developments in our understanding of one-dimensional metals. We show that for the structure proposed, the momentum dependence of the exciton interaction results in the superconducting state being favoured over the Peierls state, and in vertex corrections to the electron-exciton interaction which are small. The calculation of the transition temperature is based on what we believe to be reasonable estimates of the strength of the excitonic interaction, Coulomb repulsion and band structure. 

As the number of eigenstates available for coherent coupling increases, the dynamical behavior of the system becomes considerably more complex, and issues such as Coulomb interactions become more important. For example, over how many wells can the wave packet survive, if the holes remain locked in place If the holes become mobile, how will that affect the wave packet and, correspondingly, its controllability The contribution of excitons to the experimental signal must also be included [34], as well as the effects of the superposition of hole states created during the excitation process. These questions are currently under active investigation. 

Exciton formation The holes and electrons must combine in the emitter region of the device via a Coulombic interaction to form excitons [67] (neutral excited species) other excited states are also possible such as excimer [68] or exciplex excited states [69,70]. 

The free valencies of a crystal can form pairs, each such pair wandering through the crystal as an entity until it breaks up. Such formations are well known in the theory of the solid state. A pair of opposite valencies in an ionic crystal (electron - - hole bound by Coulomb interaction) forms what is called a Mott exciton. A pair of like valencies (election + electron or hole + hole bound by exchange interactions) forms a so-called doublon. Such formations have recently been investigated, 12, IS). 

This exciton diffuses to the donor/acceptor interface via an energy-transfer mechanism (i.e., no net transport of mass or charge occurs). (3) Charge-transfer quenching of the exciton at the D/A interface produces a charge- transfer (CT) state, in the form of a coulombically interacting donor/acceptor complex (D A ). The nomenclature used to describe this species has been relatively imprecise, and has... 

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