Excitons Mott-Wannier

This type of exciton is envisaged by Wannier [125] and Mott [132a], thus known commonly as a Wannier-Mott exciton. In this exciton, the electron and the hole revolve around each other resembling the simple (Bohr) structure of the hydrogen atom. The energies (45) and inter-carrier distances... 

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. 

The zero value of the energy (1.5) coincides with the bottom of the conduction band and its absolute value for k = 0 is equal to the binding energy of the electron and the hole in the exciton. The relatively simple relation between the Wannier-Mott exciton energy does not hold when the electron-hole interaction is treated more accurately (see (8)), and becomes a relation of a more general type. 

The theory of Wannier-Mott excitons and, in particular, the limits of validity of eqn (1.3), can be found, for instance, in the textbook by Knox (8) and the review article by Haken (19). 

As we have seen, the Frenkel exciton and the Wannier-Mott exciton correspond to two limiting situations related to the electron-hole binding process. In the first case one visualizes the electrons and the holes as localized on a given molecule, and their interaction with electrons of other molecules plays a secondary role. In this case the wavefunctions of exciton can be constructed from the wavefunctions of isolated molecules. In the case of Wannier-Mott excitons the mean electron-hole distance is much greater than the crystal lattice constant. It is clear that in this case the interaction energy of electrons and holes strongly depends on the properties of the medium. As can be seen from eqn (1.3), the electron-hole interaction in this simple model is determined by the dielectric permittivity of the medium. 

A similar situation appears in the theory of Wannier-Mott excitons (see, for example, (17), 4), where the interaction between the electron and the hole is given by —e2/er only at large enough distances. 

The appearance of an enhanced radiative width and renormalization of the exciton dispersion are the main effects arising in one- and two-dimensional structures under the influence of retardation (21). Qualitatively these effects are valid for Frenkel as well as for Wannier-Mott excitons. In contrast to 3D structure where in the exciton-photon interaction all three components of the momentum have to be conserved and as a result a picture with 3D polaritons arises, for structures of lower dimensionality only the in-plane momentum for 2D structures is conserved and only one component for the ID structure. An exciton in both cases is coupled to a continuum of photon states. There is no possibility of reversible strong... 

The picosecond time-scale observed in these experiments was the first example of superradiance of two-dimensional Frenkel excitons. Relative quantum yield measurements of the photoluminescence from bulk and the photoluminescence from the outermost monolayer indicate that the decay of excitons in the monolayer is purely radiative with a very small contribution from relaxation to the bulk. Later the same phenomenon for a 2D Wannier-Mott exciton in a semiconductor quantum well was observed by Deveaud et al. (4). 

Hybrid 2D Frenkel Wannier Mott excitons at the interface of organic and inorganic quantum wells. Strong coupling regime... 

FlG. 13.1. (a) The physical configuration under study (6) the Frenkel and Wannier-Mott excitons in nanostructure (from (13)). 

D translational invariance of the system, we classify the excitons by their inplane wavevector k. Supposing that for some bands of Frenkel excitons in the OQW and Wannier-Mott excitons in the IQW the energy separation is much less than the distance to other exciton bands we take into account only the hybridization between these two bands. We choose as a basis set the pure Frenkel and Wannier states, i.e. the state (denoted by F,k)) when the OQW is excited, while the IQW is in its ground state, and vice versa (denoted by W, k)), their energies being Ep(k) and W(k). We seek the new hybrid states in the form... 

It follows from these relations that in the vicinity of resonance the exciton in a hybrid structure, roughly speaking, lives half-time as a Frenkel exciton and half-time as a Wannier-Mott exciton. Thus, in a hybrid structure we can expect a strong exciton-light interaction typical for Frenkel excitons as well as strong resonance optical nonlinearity typical for Wannier-Mott excitons. 

To evaluate the matrix element F(k) determining the resonance interaction between Frenkel and Wannier-Mott excitons we write down the interaction Hamiltonian as... 

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. 

To summarize, we have demonstrated the possibility of strong resonance hybridization of ID Frenkel and Wannier-Mott excitons in parallel organic and semiconductor wires. Like the 2D case, the new states possess the properties of both types of excitons. They have a relatively large size (along the wires) like Wannier-Mott excitons, but they also have a large transition dipole moment which is typical for Frenkel excitons. Thus, one may expect the same as for 2D structures (see Fig. 13.1b) the strong nonlinear optical effects in such structures. 

On the hybridization of zero-dimensional Frenkel and Wannier Mott excitons... 

In the previous sections we have considered the hybridization of Frenkel and Wannier-Mott excitons in two-dimensional (quantum wells) and one-dimensional (quantum wires) geometries. For the sake of completeness, in this subsection we shall briefly and qualitatively discuss the zero-dimensional (0D) case that corresponds to a quantum dot geometry. We have in mind a configuration where a semiconductor QD is located near a small size organic cluster or is just covered by a thin shell of an organic material. 

Interesting results of the studies of the strong coupling regime of Wannier-Mott excitons in a quantum dot lattice embedded in organic medium and in dendrites and also unusual nonlinear properties of such structures can be found in the articles by Birman and coworkers (33)-(37). 

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