Hopping recombination

The quasi-steady-state hopping recombination rate K(oo) = Kq is related to the coefficient i eff via equation (4.2.14) as in the diffusion-controlled case. As in equation (4.2.15), this Ifu is defined by the asymptotics of the solution, Y(r,oo) = y(r), as r —> oo. It is important, however, that R ff cannot generally be treated as the effective recombination radius. It holds provided that the hop length is much smaller than the distinctive scale ro of tunnelling recombination... 

Since typically ro 1 A, and A is restricted in a crystal or liquid by a distance of several A, the condition (4.3.7) can be violated which makes the diffusion description inadequate and an alternative hopping recombination formalism should be used. 

Note that the steady-state y(r) obtained from equation (4.3.9) as t — oo does not completely agree with the asymptotic behaviour of the quantity established earlier due to absence here of the term R ff/r. This term comes from the diffusion character of distant tunnelling recombination. The term Reff/r arises from the next after equation (4.3.8) term in the expansion of the integral (4.3.5) in parameter (R/X)2 deriving equation (4.3.10) the asymptotic value of y(r) is only used rather than its behaviour at large r. A comparison of hopping recombination with equation (4.2.18) yields... 

In solid state materials, single-step electron transport between dopant species is well known. For example, electron-hole recombination accounts for luminescence in some materials [H]. Multistep hopping is also well known. Models for single and multistep transport are enjoying renewed interest in tlie context of DNA electron transfer [12, 13, 14 and 15]. Indeed, tliere are strong links between tire ET literature and tire literature of hopping conductivity in polymers [16]. 

The diffusion of H and D atoms in the molecular crystals of hydrogen isotopes was explored with the EPR method. The atoms were generated by y-irradiation of crystals or by photolysis of a dopant. In the H2 crystals the initial concentration of the hydrogen atoms 4x 10 mol/cm is halved during 10 s at 4.2 K as well as at 1.9 K [Miyazaki et al. 1984 Itskovskii et al. 1986]. The bimolecular recombination (with rate constant /ch = 82cm mol s ) is limited by diffusion, where, because of the low concentration of H atoms, each encounter of the recombinating partners is preceded by 10 -10 hops between adjacent sites. 

Figure 3.32. Energy level scheme of the device in Figure 3.31. Photoinduced electron transfer takes place from the photoexcited ruthenium dye into the Ti02 conduction band. The recombination directly back to the dye has to be suppressed. Instead, the current is directed through the circuit to the counterelectrode and the hole conductor that brings the electrons back via hopping transport. HTM hole transport material.

A representative example for the information extracted from a TRMC experiment is the work of Prins et al. [141] on the electron and hole dynamics on isolated chains of solution-processable poly(thienylenevinylene) (PTV) derivatives in dilute solution. The mobility of both electrons and holes as well as the kinetics of their bimolecular recombination have been monitored by a 34-GHz microwave field. It was found that at room temperature both electrons and holes have high intrachain mobilities of fi = 0.23 0.04 cm A s and = 0.38 0.02 cm / V s V The electrons become trapped at defects or impurities within 4 ps while no trapping was observed for holes. The essential results are (1) that the trap-free mobilities of electrons and holes are comparable and (2) that the intra-chain hole mobility in PTV is about three orders of magnitude larger than the macroscopic hole mobility measured in PTV devices [142]. This proves that the mobilities inferred from ToF and FET experiments are limited by inter-chain hopping, in addition to possible trapping events. It also confirms the notion that there is no reason why electron and hole mobilities should be principally different. The fact... 

Fig. 3 Charge transfer in DNA hairpins after photoexcitation of stilbene linker (St) by a laser pulse [45]. A hole, first, undergoes a transition from photoexcited St to the adjacent GC pair as shown by the solid arrow. Then it can either hop to next GC pairs (dot-dashed arrow) or return to St with the subsequent electron-hole recombination (dotted arrow)...

Table 4 The ranges of temperature where tunneling, protonation, hopping, or recombination are dominant in irradiated DNA samples...

We mentioned the main models for generation, transfer, and recombination of the charge carriers in polymers. Very often, these models are interwoven. For example, the photogeneration can be considered in the frame of the exciton model and transport in the frame of the hopping one. The concrete nature of the impurity centers, deep and shallow traps, intermediate neutral and charged states are specific for different types of polymers. We will try to take into account these perculiarities for different classes of the macro-molecules materials in the next sections. 

All local concentrations C of particles entering the non-linear functions F in equation (2.1.40) are taken at the same space points, in other words, the chemical reaction is treated as a local one. Taking into account that for extended systems we shouldn t consider distances greater than the distinctive microscopic scale Ao, the choice of equation (2.1.40) means that inside infinitesimal volumes vo particles are well mixed and their reaction could be described by the phenomenological reaction rates earlier used for systems with complete reactant mixing. This means that Ao value must exceed such distinctive scales of the reaction as contact recombination radius, effective radius of a dynamical interaction and the particle hop length, which imposes quite natural limits on the choice of volumes v0 used for averaging. 

Defect diffusion traditionally is treated as a process in continuum medium. However, discreteness of the crystalline lattice becomes important in particular situations, e.g., when defect recombination occurs in several hops (nearest neighbour recombination) [3, 4] or even for nearest-site hops of defects if their recombination is controlled by the tunnelling whose probability greatly changes on a scale of lattice constant [45, 46],... 

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