Frenkel exciton radius

These two types of exciton are schematically illustrated in Figure 4.13. The Mott-Wannier excitons have a large radius in comparison to the interatomic distances (Figure 4.13(a)) and so they correspond to delocalized states. These excitons can move freely throughout the crystal. On the other hand, the Frenkel excitons are localized in the vicinity of an atomic site, and have a much smaller radius than the Mott-Wannier excitons. We will now describe the main characteristics of these two types of exciton separately. 

In tightly bound (Frenkel) excitons, the observed peaks do not respond to the hy-drogenic equation (4.39), because the excitation is localized in the close proximity of a single atom. Thus, the exciton radius is comparable to the interatomic spacing and, consequently, we cannot consider a continuous medium with a relative dielectric constant as we did in the case of Mott-Wannier excitons. 

A Mott-Wannier exciton is a neutral quasi-particle, consisting of an excited bound-state electron and its associated "Coulomb hole" in a high-dielectric constant solid, that can also travel throughout the lattice without transporting net charge since the exciton radius is several lattice constants, its binding energy is as low as 0.01 eV it thus tends to be more "delocalized" than the Frenkel exciton. 

The electron and the hole in the crystal attract themselves and can create a bound state. Obviously, the Frenkel exciton corresponds to the situation when the electron and the hole in a bound state are localized in the same lattice cell (the same molecule). Therefore the Frenkel excitons are also called small-radius excitons. When the radius of the electron-hole bound state is much larger than the lattice constant, the corresponding quasiparticle is called a Wannier-Mott exciton, or a large-radius exciton. Let us consider the latter in more detail. 

Below we assume that superlattice layers 1 or 2 can be of different natures and can be, for example, organic or inorganic. In the vicinity of excitonic resonances even at 12 3> / ( is the Bohr radius of an exciton, of the order of the lattice constant in the case of Frenkel excitons) the nonlocality of the dielectric permeability can be taken into account. Consider, for instance, the frequency... 

If the energies of Frenkel and Wannier-Mott excitons are in resonance the size of the hybrid state is comparable with that for Wannier-Mott excitons, i.e. it is much larger than the radius of Frenkel excitons. In this case we can expect that the saturation concentration of excitons in hybrid structure will be of the same order as in a semiconductor quantum wire. Outside the resonance range, the coupling is governed by the parameter T2/(Ep — Ew) and is rather small. The condition of resonance is rather strict for the considered range of parameters and requires a careful choice of materials for both wires. And, naturally, the exciton linewidths should be small compared to 2Y. For these parameters these linewidths have to be smaller than the resonant splitting 2T 11 meV of the hybrid excitations. 

This relationship for Frenkel excitons was derived in (14) it can be seen from its derivation that it is independent of the model and, therefore, is valid also for ground state large-radius excitons as well as for electrons and holes in semiconductors. 

Energy transfer can also occur by excitons. An exciton is an excited state of the crystal lattice in which an electron and a hole are bound and can propagate through the lattice [14], They can be divided into two classes, viz. Frenkel and Wannter excitons. In the former the electron-hole separation is of the order of an atomic radius, i.e. it can be considered as a localized excitation. In the latter this separation is large in comparison with the lattice constant. Consequently, its binding energy is much smaller than that of the Frenkel exciton. 

One can see from Table 8.1 that the Frenkel exciton (n = 0) provides only about 30% of the electron-hole binding energy (the gap obtained in the QP framework is 3.7 eV for PTS and 3.2 eV for TCDU). The exciton must be localized with a radius of 25-30 A, n = 5 (also found by analyzing the excitonic wave function see Figure 3 of Suhai ), to obtain acceptably convergent excitation energies. Further, one can see that the correlation corrections reduce the excitation energies for both polymers by approximately 2.1 eV. Finally, one should point out that the exci-... 

Optical excitations in molecular crystals are well known as Frenkel excitons and the detailed descriptions have been derived by Davydov [4] and Craig and Walmsley [5]. Molecular excitons resemble very much the optical properties of the isolated molecules, since the exciton is confined on one molecule and only the weak interaction with the surrounding molecules leads to the formation of a collective excitation. This is contrary to the large radius Mott-Wannier excitons in conventional semiconductors, where the electron and the hole are typically loosely bound with... 

In heterostructures both the CB electrons and the VB holes will experience extra potential energies such as band offsets. For low-energy Frenkel excitons in a QD, the electrons and holes are confined within the same QD volume (type-I exciton) so that we will have a common potential energy V( R - a ) that confines the exciton, where R is the center of mass of the exciton (see O Eq. 23.19), a denotes the center of the QD. For conunon semiconductor QDs, the radius of the QD, Rqo, is in the order of the exciton Bohr radius Br so that one may neglect the free motion of the center of mass of the electron-hole pair in the QD, that is, K = 0 in O Eq. 23.20. For the ground state of the exciton we can approximate V( B-a ) = 0 when li-a < Rqd and y( U-a ) = oo elsewhere so that the wave function corresponding to the motion of the center of mass becomes (Gasiorowicz 1996)... 

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