Excitons large radius

PbS has attracted much attention due to its special direct band gap energy (0.4 eV) and a relatively large exciton Bohr radius (18 nm) and their nanoclusters have potential applications in electroluminescent devices such as light-emitting diodes. PbS nanocrystals with rod like structures with diameters of 20-60 nm and lengths of 1-2 pm have been obtained using the sonochemical method and by using PEG-6000 [66]. Addition of PEG and the time of sonication have been found to play a key role in the formation of these rods. 

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. 

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. 

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. 

We must now explain, at least qualitatively, the existence of the critical radius (equation given in Ref. 11). In an infinite semiconductor medium a large-radius exciton appears as a result of the Coulombic attraction Feh( e, between an electron and a hole. The Hamiltonian of an exciton moving within a small SNc contains not only the Coulombic attraction but also the terms Tee (c,... 

For simplicity, however, we prefer to denote all excitons formed from bound states of conduction band electrons and valence band holes as Mott-Wannier excitons, recognizing that this term includes both small and large radius excitons. We call this limit the weak-coupling limit, as the starting point in the construction of the exciton basis is the noninteracting band limit. As we will see, a real space description of a Mott-Wannier exciton is of a hole in a valence band Wannier orbital bound to an electron in a conduction band Wannier orbital. 

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... 

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