2D excitons

One can say that the obtained by us experimental results upon 2D exciton localization (taking place due to the growth of the crystal dielectric permeability anisotropy parameter) with o are very close to [27] where the behaviour of polaron excitons in parabolic quantum dots were considered and shown that the dot size decrease results in increasing the exciton binding energy. 

In this section we analyze the surface investigation of molecular crystals by the technique of UV spectroscopy, in the linear-response limit of Section I, which allows a selective and sharp definition of the surface excited states as 2D excitons confined in the first monolayer of intrinsic surfaces (surface and subsurfaces) of a molecular crystal of layered structure. The (001) face of the anthracene crystal is the typical sample investigated in this chapter. 

A theory of 2D excitons and polaritons is presented for this type of surfaces, with continuity conditions matching 2D states their 3D counterparts in the bulk substrate, investigated in Sections I and II. This leads to a satisfactory description of the excitations (polaritons, excitons, phonons) and their theoretical interactions in a general type of real finite crystals A crystal of layered structure (easy cleavage) with strong dipolar transitions (triplet states do not build up long-lived polaritons). 

These surface excitation phenomena are investigated in this section, on excitons in the intrinsic-surface-bulk system, and the next section, on disordered 2D excitons. 

The optical response of a monomolecular layer consists of scattered waves at the frequency of the incident wave. Since the surface model is a perfect infinite layer, the scattered waves are reflected and transmitted plane waves. In the case of a 3D crystal, we have defined (Section I.B.2) a dielectric permittivity tensor providing a complete description of the optical response of the 3D crystal. This approach, which embodies the concept of propagation of dressed photons in the 3D matter space, cannot be applied in the 2D matter system, since the photons continue propagating in the 3D space. Therefore, the problem of the 2D exciton must be tackled directly from the general theory of the matter-radiation interaction presented in Section I. 

This expression shows also a nonanalytic dispersion at K = 0, but which is continuous The nonanalyticity of the 2D exciton has regressed by one order compared to the 3D exciton cf. Section I.B.l. Furthermore, as we are interested in effects around K = 0 for the b component of the anthracene crystal, it is legitimate to replace the whole unit cell by a transition dipole... 

Figure 3.6. Isoenergetic contours in the middle of the Brillouin zone, for a 2D exciton with purely coulombic dispersion. For the large wave vectors, the parabolic dispersion in K2 prevails and the contours are quasi-circular, whereas in the vicinity of the middle of the zone, the coulombic dispersion in K cos2 0 tends to make K perpendicular to the transition dipole d. Compare these contours with the 3D exciton. Fig. 2.18.

Thus, flK(co) is imaginary for K < (o/c and real for K > cu/c these two cases correspond, respectively, to a radiatively unstable 2D exciton and to a radiatively stable 2D polariton. Solution of equations (1.3) and (3.12) provides the complete description of the 2D polariton dynamics. We analyze below the 2D polariton at different orders of exciton-photon coupling. 

For the radiative excitons, Im RK(a)0) gives the radiative half width of the 2D excitons. For K = 0, the value of this radiative half width is... 

Figure 3.8. The exciton decay in photon and polariton states The time evolution (in units of w0r) of a 2D exciton K created at r = 0 (Kid). This decay, illustrated for various wave vectors (in units of iu0/c), is purely exponential for K < to0/c, but exhibits very complex transient oscillatory behavior in the region K - oj0/c. For K > o>Jc the 2D exciton is radiatively stable.

Figure 3.10. Scheme of the 2D polaritons and radiatively very unstable 2D excitons in the coupled system of an exciton K and an effective photon continuum (a) The two subsystems are not coupled, (b) The coupled system with a discrete state split off below the continuum, called the 2D polariton excitonic solutions exist only in a small segment of the Brillouin zone,... 

The model of an isolated layer was refined by introducing substrate effects by coupling the surface 2D excitons to the bulk polaritons with coherent effects modulating the surface emission and incoherent k-dependent effects damping the surface reflectivity and emission, both effects being treated by a KK analysis of the bulk reflectivity. The excitation spectra of the surface emission allowed a detailed analysis of the intrasurface relaxation dominated by resonant Raman scattering, by vibron fission, and by nonlocal transfer of... 

The lifetimes of molecular fluorescence emissions are determined by the competition between radiative and nonradiative processes. If the radiative channel is dominant, as in the anthracene molecule, the fluorescence quantum yield is about unity-and the lifetime lies in the nanosecond range. In molecular assemblies, however, due to the cooperative emission of interacting molecules, much shorter lifetimes—in the picosecond or even in the femtosecond range—can theoretically be expected an upper limit has been calculated for 2D excitons [see (3.15) and Fig. 3.7] and for /V-multilayer systems with 100 > N > 2.78 The nonradiative molecular process is local, so unless fluorescence is in resonance by fission (Section II.C.2), its contribution to the lifetime of the molecular-assembly emission remains constant it is usually overwhelmed by the radiative process.118121 The phenomenon of collective spontaneous emission is often related to Dicke s model of superradiance,144 with the difference that only a very small density of excitation is involved. Direct measurement of such short radiative lifetimes of collective emissions, in the picosecond range, have recently been reported for two very different 2D systems ... 

We combine now the disordered part (4.3) with the retarded part of (4.1), which does not depend on the particular resonance energy of one domain or site, but on the transition dipole operators, and therefore is identical for our model to that of the perfect 2D lattice. Thus, we obtain the following effective hamiltonian for the disordered 2D 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... 

Now we turn to a discussion of the properties of excitons in layered molecular structures (1). First we consider the properties of excitations at the boundary of an anthracene crystal with the vacuum. Of course, this is a particular case of a boundary. However, this case has been investigated in many experiments and therefore can be considered as some kind of experimental confirmation of the approach we will use in our more general discussion. It can be considered now as well-established that the 2D exciton state - the lowest electronic excitation of the outermost monolayer of the anthracene crystal - is blue-shifted by 204 cm-1 with respect to the bottom of the exciton band in the bulk. Thus, the frequency of this electronic transition in the first monolayer lies between the bulk value of the exciton frequency and the frequency of excitation in an isolated molecule because the value of this molecular frequency in anthracene is blue-shifted by 2000 cm-1 with respect to the frequency of bulk excitation (see Fig. 9.1). 

For the sake of definiteness we assume that a state of SSSE is localized in the single crystallographic plane z = 0. Let this plane separate media with positive dielectric constants ei(w) (z > 0) and 2(10) (z < 0), respectively. We ignore an anisotropy in the plane z = 0 and assume that the dipole moment of the transition of the Coulomb 2D exciton is located in the plane z = 0. 

Now we turn to the situation when the QW width fluctuations, alloy disorder or impurities localize the 2D exciton (such a situation is more frequent for II-VI semiconductor quantum wells than for III—V ones). Then, the wavefunction of the center-of-mass exciton motion (ry) is no longer just a plane wave, and the corresponding polarization is given by... 

EXCITON-PHONON COUPLING OF LOCALIZED QUASI-2D EXCITONS IN SEMICONDUCTOR QUANTUM WELL HETEROSTRUCTURES... 

We calculate the lateral size dependence of Huang-Rhys factors for localized quasi-2D excitons interacting with phonons in semiconductor QW heterostructures. The Huang-Rhys factors increase with decreasing the localization area. This indicates an enhancement of exciton-phonon interactions with decreasing the localization area of quasi-2D excitons. 

In this paper, we present the lateral size dependence of Huang-Rhys factors for the QW heterostructures with the localized quasi-2D excitonic states. The Huang-Rhys factor is a quantity representing the coupling strength of a localized particle with phonons [4]. All exciton-phonon interaction mechanisms are analyzed. They are the polar optic (Froehlich) interaction, the optic deformation potential, the acoustic deformation potential, and the acoustic piezoelectric interaction. We would... 

We calculated the size dependence of the zero-temperature Huang-Rhys factors (4) using parameters of GaAs [7]. Fig. 1(a) shows the optic Huang-Rhys factors of 2D-excitons (flat dot) as a function of the QD lateral size. The Froehlich interaction is seen to dominate over the optic deformation potential interaction (caused by the heavy hole interaction in a p-like valence band). The total optic Huang-Rhys factor gradually increases with the decreasing QD lateral size. 

Figure 1. Optic (a) and acoustic (b) zero-temperature Huang-Rhys factors of localized 2D-excitons. 

The acoustic Huang-Rhys factor for localized 2D-excitons is shown in Fig. 1(b). The acoustic deformation potential interaction totally prevails over the acoustic peizoelectric interaction. The total acoustic Huang-Rhys factor is 1 and increases with the decreasing QD size. This indicates rather strong exciton-acoustic-phonon coupling which further enhances with the decrease of the QD lateral size. 

In conclusion, our calculations show an enhancement of the exciton-phonon interactions in quasi-2D QD-like islands in Q W heterostmctures. This entails the respective increase of the excitonic dephasing rate. Our conclusion is valid unless the quasi-2D exciton may be considered as weakly confined in the lateral plane. 

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