Exciton transverse

Bulk silicon is a semiconductor with an indirect band structure, as schematically shown in Fig. 7.12 c. The top of the VB is located at the center of the Brillouin zone, while the CB has six minima at the equivalent (100) directions. The only allowed optical transition is a vertical transition of a photon with a subsequent electron-phonon scattering process which is needed to conserve the crystal momentum, as indicated by arrows in Fig. 7.12 c. The relevant phonon modes include transverse optical phonons (TO 56 meV), longitudinal optical phonons (LO 53.5 meV) and transverse acoustic phonons (TA 18.7 meV). At very low temperature a splitting (2.5 meV) of the main free exciton line in TO and LO replicas can be observed [Kol5]. 

Nevertheless, certain collective excitations can occur in the condensed phase. These may be brought about by longitudinal coulombic interaction (plasmons in thin films) or by transverse interaction, as in the 7-eV excitation in condensed benzene, which is believed to be an exciton [12]. Special conditions must be satisfied by the real and imaginary parts of the dielectric function of the condensed phase for collective excitations to occur. After analyzing these factors, it has been concluded that in most ordinary liquids such as water, collective excitations would not result by interaction of fast charged particles [13,14]. 

This equation in o)2, which is exact if we consider the dependence K of oje and of de, determines all the eigenenergies of the exciton-photon coupled system. However, it is preferable to put it in the classical form, which privileges the photon subspace, by introducing the transverse dielectric tensor1213... 

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. 

FIGURE 6 Evolution of the intensity of the reflectance structures with in-plane orientation of the electric field (a) [10-10] orientation, (b) [-12-10] orientation [8], Note that the increase of Ais is accompanied by a decrease of Bu and makes it easier to detect A2S. Arrows indicate the average positions of transverse excitonic polaritons The eigenenergies, which are different for the two polarisations due to the spin exchange interaction, can be obtained from a lineshape fitting of the reflectivity spectra. 

In the present report, three subjects are to be reviewed (i) The triplet structure of excitons and the upper valence band (ii) the full set of the fundamental optical functions in the 0-30 eV energy range for polarizations E II c, E J. c, and their theoretical analysis and (iii) the main parameters of the elementary transverse and longitudinal transition components in the 0-30 eV energy range for polarizations E c, E L c, and their theoretical analysis. 

The analysis of the S2 and -Ims" spectra allowed to determine the energy values of transverse and longitudinal exciton transitions, and to evaluate their band areas and oscillator strengths (Tab. 1) the values off, given here, have been enlarged 10. ... 

In Table 2, the calculated values of energy maxima E, and band areas Si of transverse transitions were compared for the three variants of calculations of optical function sets. These variants differ comparatively little in energy El. The principal inconsistencies are observed in the values of Si. This is caused by the differences in the experimental R(E) spectra, i.e. by using of ZnO samples of different quality and by different registration techniques for polarized reflectivity spectra. The determined components of S2 and -Ims spectra are caused by direct interband transitions or metastable excitons, except for the most long-wavelength of them, which are associated with free excitons. The theoretical band calculations of ZnO ° strongly differ in the bands dispersion and positions. This makes it difficult to propose a... 

For the first time, the full sets of exciton optical fundamental functions of ZnO have been calculated at 1.6, 4.2, and 90 K. The most correct energy values of transverse and longitudinal exciton transitions of the three series, together with their areas and oscillator strengths, have been determined, as well as their characteristic features. 

If the retarded interaction is ignored and the operator Hmt is removed from the total Hamiltonian (4.2), then the operator H becomes a sum of two independent Hamiltonians, one of them (Hi) describing the crystal elementary excitations - those which occur when the retardation effects are ignored and the second (H2) giving the elementary excitations - transverse photons in vacuum. The presence of the operator H3 leads to the interaction between carriers with the transverse electromagnetic field. In the case of an atomic gas this interaction causes, in particular, the so-called radiative width of energetic levels of excited states. In the case of an infinite crystal possessing translational symmetry the radiative width of excitonic states vanishes.29... 

For the longitudinal excitons, p = py and T(j,k,/x) = 0 as follows from (4.11). In consequence, those excitons do not interact with the transverse photons and retardation effects for them are not essential. From (4.17) and (4.18) we obtain for the longitudinal excitons35... 

The index p labels the allowed energy bands. The number of values which p can achieve is given by a sum of the considered exciton bands + two j = 1,2- two branches of the transverse photons). 

The crystal possesses cubic symmetry. In this case exciton states, which correspond to nonvanishing values f1M(s), are either longitudinal exciton states (p = p,, P (k, p, ) s), or transverse exciton states, which for k —> 0 corresponds to the two-fold degenerate exciton band... 

In some cases it is useful to know the dependence of the exciton frequencies on the wavevector. The inverse dependence, i.e. the dependence of k on cj, is, as we have seen before, given by the relation k = wn(w, s)s/c. The calculation of the function w(k) can be carried out by using the previously obtained expressions for n(w, s). So, for example, for transverse waves in cubic crystals we obtain from (4.34) the following equation for w(k) ... 

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. 

If the retardation is ignored, then two coinciding branches of the transverse photons (polarization j = 1,2) and one exciton branch, polarized along the molecule transition dipole, are the lowest elementary excitations of the crystal. 

Fig. 4.3. The dispersion of polariton in cubic crystals. Nongyrotropic crystals (a) The dependences of exciton and photon energy on wavevector, the retardation neglected (b) the same but with retardation taken into account. The symbols and L indicate longitudinal and transverse polarization of excitons (c) retardation neglected but dependence of the exciton energy on the wavevector taken into account here and in (d), (e), and (f) only the lower branch of the polaritons shown (d) the retardation and dependence of exciton energy on wavevector are taken into account. Gyrotropic crystals (e) Dispersion of excitons in the cubic gyrotropic crystals if retardation is neglected (f) the same when retardation is also taken into account Aq denotes the position of the bottom of the polariton energy.

In order to discuss more correctly the question of radiative width we keep in mind the fact that the operator (4.88) provokes not only a radiative damping of exciton states, but also changes their dispersion rule. To obtain this dispersion, we add to (4.88) operators of the free exciton and transverse photon fields, as was done in Sections 4.1 and 4.2, and diagonalize the total Hamiltonian so obtained. Recall for comparison that in the case of an ideal 3D crystal after such diagonalization of the total Hamiltonian the radiative width of new excitations (polaritons) disappeared. We show below that in ID and 2D the results are completely different. 

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. 

The consideration of the strong dependence of dissipation of polaritons near exciton resonances will be performed below with the use of transverse dielectric tensor ej y(w, k). This tensor will be calculated assuming that the excitonic states, with complete account of the Coulomb interaction, are known. Since the derivation of the expression for k) is given in the monograph (3), see... 

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... 

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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