Hopping rates

A celebrated derivation of the temperature dependence of the mobility within the hopping model was made by Miller and Abrahams [22], They first evaluated the hopping rate y j, that is the probability that an electron at site i jumps to site j. Their evaluation was made in the case of a lightly doped semiconductor at a very low temperature. The localized states are shallow impurity levels their energy stands in a narrow range, so that even at low temperatures, an electron at one site can easily find a phonon to jump to the nearest site. The hopping rate is given by 

Another famous hopping model is Mott s variable range hopping [23], in which it is assumed that the localized sites are spread over the entire gap. At low temperatures, the probability to find a phonon of sufficient energy to induce a jump to the nearest neighbor is low, and hops over larger distances may be more favorable. In that case, the conductivity is given by 

More recently, D. Emin [24] developed an altemative analysis of activated hopping by introducing the concept of coincidence. The turmeling of an electron from one site to the next occurs when the energy state of the second site coincides with that of the first one. Such a coincidence is insured by the thermal deformations of the lattice. By comparing the lifetime of such a coincidence and the electron transit time, one can identify two classes of hopping processes. If the coincidence time is much larger than the transit time, the jump is adiabatic the electron has time to follow the lattice deformations. In the reverse case, the jump is non-adiabatic. 

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Figure 4. (A) Cyclic voltammograms over a range of scan rates for a redox polymer (poly-[Fe 5-amino-1,10-phenanthrotme)3]3+/>)91 and (B) p-doping and undoping of a conducting polymer (polypyrrole) (B). [(A) Reprinted from X. Ren and P. O. Pickup, Strong dependence of the election hopping rate in poly-tris(5-amino-1,10-phenan-throline)iron(HI/II) on the nature of the counter-anion J. Electroanal. Chem. 365, 289-292,1994, with kind permission from Elsevier Sciences S.A.]...

Relation to Landau-Zener Surface Hopping Rates... 

Table 5.3 Effect of cation and solvent on hopping rate in durosemiquinone...

The same control mechanism can be put to work for the doubly layered electrophores [14]. From esr measurements it appears that a change only of the ion pairing brings about a different hopping rate and creates a different spin-density distribution within the timescale of the experiment. 

Fig. 6. The hop rate of muonium in KC1 as a function of temperature. The crossover from stochastic to quantum diffusion occurs at about 70 K, as evidence by a dramatic increase in the hop rate at lower temperatures. From Kiefl et al. (1989a).

Eq 5 reveals at a glance the two major components of the desired rate constant the primary electron-hopping rate constant, ke (r), which involves quantum and statistical mechanics associated with both heavy atom and electronic motion, on... 

In activated ion hopping processes such as occur in solid electrolytes, there is an inverse correlation between the magnitude of the activation energy and the frequency of successful ion hops this leads us to the concept of ion hopping rates. 

There are two times to consider, therefore, in order to characterise ionic conduction. One is the actual time, tj, taken to jump between sites this is of the order of 10 -10 s and is largely independent of the material. The other time is the site residence time, which is the time (on average) between successful hops. The site residence times can vary enormously, from nanoseconds in the good solid electrolytes to geological times in the ionic insulators. Ion hopping rates, cOp, are defined as the inverse of the site residence times, i.e. 

The ion hopping rate is an apparently simple parameter with a clear physical significance. It is the number of hops per second that an ion makes, on average. As an example of the use of hopping rates, measurements on Na )3-alumina indicate that many, if not all the Na" ions can move and at rates that vary enormously with temperature, from, for example, 10 jumps per second at liquid nitrogen temperatures to 10 ° jumps per second at room temperature. Mobilities of ions may be calculated from Eqn (2.1) provided the number of carriers is known, but it is not possible to measure ion mobilities directly. 

We earlier defined the hopping rate parameter, cOp. It corresponds approximately to the frequency at which the conductivity dispersion commences, arrowed in Fig. 2.8. 

In another model-based study Gillbro et al. [192] have used exciton-exciton annhilation to determine average EET hopping rates. In this technique the rate of exciton-exciton annhilation depends critically on the domain size and the pairwise EET rate [193, 194]. Average pairwise rates of 2 x 10 s -10 s were calculated for LHCII having EET domain sizes in the range of 300-1000 sites. 

Figure 4 Left Distribution of hopping times for an adatom at a solid-liquid interface at 600 K for conventional molecular dynamics and for superstate parallel-replica dynamics. Right Temperature dependence of the hopping rate for an adatom at a dry interface and at a wet interface as obtained using superstate parallel-replica dynamics.

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