Tunneling transfer

Besides its temperature dependence, hopping transport is also characterized by an electric field-dependent mobility. This dependence becomes measurable at high field (namely, for a field in excess of ca. 10d V/cm). Such a behavior was first reported in 1970 in polyvinylcarbazole (PVK) [48. The phenomenon was explained through a Poole-ITenkel mechanism [49], in which the Coulomb potential near a charged localized level is modified by the applied field in such a way that the tunnel transfer rale between sites increases. The general dependence of the mobility is then given by Eq. (14.69)... 

Equation (106) shows that the interaction of the proton with the motion of the center of mass, described by the terms proportional to fx, is formally of the same form as the interaction with the medium atoms, and the first three terms in the Hamiltonian in Eq. (106) are equivalent to addition of one more degree of freedom to the vibrational subsystem. Thus, this problem does not differ from that for the process of tunnel transfer of the particles stimulated by the vibrations which were discussed in Section IV. So we may use directly the expressions obtained previously with substitution of the appropriate parameters. 

Borgis, D. and Hynes, J. T. Dynamical theory of proton tunneling transfer rates in solution general formulation, Chem. Phys., 170(1993),315-346... 

Fig. 8-1. Potential energy barrier for tunneling transfer of electrons across an interface of metal electrode (a) cathodic electron transfer from an occupied level of metal electrode to a vacant level of l drated oxidant particles, (b) anodic electron transfer fiom an occupied level of hjrdrated reductant particles to a vacant level of metal electrode. M. = electrode surface OHP = outer Helmholtz plane cfuh = Fermi level of electnms in metal electrode. [From Gerischer, I960.]...

Fig. 8-38. Redox electron transfer at film-covered metal electrodes (a) transfer of redox electionB across a thin semiconductor film (direct tunneling transfer of electrons), (b) transfer of redox elections throu a thidc semiconductor film (indirect transfer of elections through electron levels in a thick film).

Figure 2.4 Carboxylic acid dimer in the potential energy minima with local vibrational states. OV represents the correlation time for a thermally activated proton transfer, and TU, the correlation time for tunneling transfer. (Reproduced with permission from ref. 29.)...

The basic idea of resonance tunneling relies on the reasonable assumption that there are impurity states in the oxide film (regarded as a semiconductor), the energy of which is in resonance with that of electrons in the metal on which the film has been formed. One considers the situation in terms of two coordinated tunnel transfers, one from the metal to the impurity state and then from the impurity state to an ion adsotbed at the oxide/solution interface. 

One of the interesting results of the work presented in ref. 27 is the conclusion that the parameter y depends not only on the properties of the donor, but also on those of the acceptor. Analysis of the experimental data shows that, for many electron tunneling reactions, the parameter y depends rather strongly upon the nature of the acceptor (see Chaps. 6 and 7). However, strictly speaking, it is not possible to conclude that this is the consequence only of non-adiabatic effects since the parameter y can also depend on the properties of the acceptor within the scope of the traditional description of the electron tunnel transfer (see Sect. 4). 

As for the samples which were not only stored but also irradiated at different temperatures (in particular, at 77 and 120 K), further, more detailed investigations [39] of the effect of temperature on the kinetics of reaction (4) atT > 77 K, as well as a comparison of these data with those on the reaction etr with other acceptors in water alkaline glasses [39,40], showed the non-activated electron tunneling to be the predominant channel of reaction (4) only at sufficiently low temperatures (T < 93 K). At 93 K and above, the dominant channel appears to be the activated tunnel transfer with the activation energy Ea = 3.1 0.4kcalmol (Chap. 5, Sect.2.3). 

Chapter 3 describes radiationless transitions in the tunneling electron transfers in multi-electron systems. The following are examined within the framework of electron Green s function approach the dependence on distance, the influence of crystalline media, and the effect of intermediate particles on the tunneling transfer. It is demonstrated that the Born-Oppenheimer approximation for the wave function is invalid for longdistance tunneling. 

The theory of electron transfer in polar media is considered in detail in Kuznetsov s book [13] where numerous references on the different problems are cited. It should be noted that the tunneling transfer of protons accompanied by the reorganization of the vibration degrees of freedom may be examined in the same way as the electron transfer. The theory of this phenomenon is investigated in details in the monograph of Gol danskii, Trakhtenberg and Flerov [28], and also in Ref. [13]. 

Some important problems of the theory of multi-phonon electron transition were not touched upon in this chapter. These are, first, the calculation of the expression for the electron matrix element at the tunneling transfer, second, the influence of medium on the electron matrix element, and, finally, the investigation of the applicability of Born-Oppenheimer approach in the electron tunneling transfer. These issues will be considered in the next chapter. 

Violation of Born-Oppenheimer s Approach in Electron Tunneling Transfer. 54... 

So, the expression for the matrix element for the electron tunneling transfer in a crystal medium has the former form (see Eq. (18)), and the crystal influence is reduced to the strong change in the tunneling exponent (compare expressions (26) and (16)) and, besides, to some changes in the preexponential factor (compare expressions (19) and (27)). 

The consideration of the reactions of the electron tunneling transfer was until now based on Born-Oppenheimer s adiabatic approach (see Section 2 of Chapter 2) that was used for the description of the wave functions of the initial and final states. The electron tunneling interaction V results in the non-adiabatic transition between these states, if the matrix element Vtf... 

The amplitude of electron tunneling transfer with simultaneous change of the vibrations in the case of the non-adiabatic asymptotics (56) may be found, if to substitute the expression (56) instead of F ) to the definition (9) and the acceptor wave function in the matrix element (9) should be its total expression... 

The rate constant of the electron tunneling transfer is then calculated on the standard formula (see expression (9) of Chapter 2) ... 

So, the description of the theory of electron tunneling transfer is logically accomplished in this chapter - it describes the methods of calculation of the electron matrix element, whereas the methods of calculation and the form of the vibration part of the transition probability was represented in Chapter 2. Besides, in Chapter 3 the procedure of the calculation of the rate constant of tunneling transfer in the conditions of the violation of Born-Oppenheimer s approach is examined. The basic results of this chapter may be formulated as follows. 

The formula for the electron matrix element Fy-at the tunneling transfer is strongly realized where it is given the exact expression for the pre-exponential factor. The exponent is defined by the electron energy. 

Probably, in these reactions as well as in reaction under action of Cu-PPX film [116], catalytic activity increases due to appearance of negatively charged Cu nanoparticles formed by the tunnel transfer of electrons between... 

Experimental data relating to the conductivity of composite films with M/SC nanoparticles are described by the classical percolation model in terms of tunnel processes. Chemisorption of chemical compounds on the surface of M/SC nanoparticles in films and the subsequent reactions with participation of chemisorbed molecules change the concentration of conducting electrons and/or barriers for their tunnel transfer between the nanoparticles with the result of strong influence on the film conductivity. Such films are used as conductometric sensors for detecting various substances in an atmosphere. 

The concept of tunneling in chemical kinetics was initiated by Hund [2] in discussing the problem of delocalization of electron between two potential wells. It was stated for the first time in this work that for a double-well potential with typical vibration frequency at in the wells and barrier height V , the tunneling transfer probability decreases exponentially with growing 2V jh(o, and therefore, for typical values of of 0.2-2 eV and fico =0.1 eV, it... 

The consistent theory of the tunneling transfer of electrons was developed by Ivanov and Kozhushner [47]. The following relation was obtained for the transition probability ... 

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