Tunneling constants

The significant variation of the barrier height observed for immersed junctions reflects the experimental difficulties associated with determining the tunneling constant, k. Two key issues are contamination of the junction and uncertainty as to the structural and electronic character of the tip [104], Recent data clearly reveal a dependence of the apparent barrier height on tip-substrate separation [7,92-94,104]. Specifically, the effective barrier is observed to diminish for resistance values below <10 Q as shown in Fig. 

Analysis of Electron-Transfer Distances and Tunneling Constants 112... 

At loadings lower than 1 MX per 20 DNA bps, the fraction of the electrons captured by the intercalator was found to follow Eq. 7 and increase with ln(f) as expected for a single-step tunneling process. The distances of electron transfer and the values of the tunneling constant for all the studied in-tercalators are compiled in Table 1, along with gas-phase estimates of the electron affinities (EA) of intercalators calculated by density functional theo-... 

Table 1 Summary of best estimates of electron- transfer distances, tunneling constants, and calculated electron affinities [7aj. rReprinted with permission from the J. Phys. Chem. Copyright (2000) American Chemical Society...

Messer A, Carpenter K, Forzley K, Buchanan J, Yang S, Razskazovskii Y, Cai Z, Sevilla MD (2000) Electron spin resonance study of electron transfer rates in DNA determination of the tunneling constant (1 for single-step excess electron transfer. J Phys Chem B 104 1128-1136 Meunier B (1992) Metalloporphyrins as versatile catalysts for oxidation reactions and oxidative DNA cleavage. Chem Rev 92 1411-1456... 

Figure 90 Injection-limited current j (normalized to jo) vs. applied electric field Fo for (a) l = 0.01, and (b) 0.5 nm, and different surface recombination rates tir(0)/is (as given in the figure). The tunneling constant for the surface recombination o<2 = 10 nm-1, other parameters as in Fig. 87. After Ref. 408. Copyright 1994 Wiley-VCH, with permission.

Miller W H 1979 Tunneling corrections to unimolecular rate constants, with applications to formaldehyde J. Am. Chem. See. 101 6810-14... 

Figure Bl.19.2. The two modes of operation for scanning tunnelling microscopes (a) constant current and (b) constant height. (Taken from [214], figure 1.)...

The constant height mode of operation results in a faster measurement. In this analysis, the tip height is maintained at a constant level above the surface and differences in tunneling current ate measured as the tip is scaimed across the surface. This approach is not as sensitive to surface irregularities as the constant current mode, but it does work well for relatively smooth surfaces. 

Chemical and biological sensors (qv) are important appHcations of LB films. In field-effect devices, the tunneling current is a function of the dielectric constant of the organic film (85—90). For example, NO2, an electron acceptor, has been detected by a phthalocyanine (or a porphyrin) LB film. The mechanism of the reaction is a partial oxidation that introduces charge carriers into the film, thus changing its band gap and as a result, its dc-conductivity. Field-effect devices are very sensitive, but not selective. 

The above discussion concerning eq. (2.1) implied that tunneling transitions were incoherent and characterized by a rate constant. This is predetermined by assumption (ii) mentioned at the... 

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

In this section we shall consider in some detail the mechanism of coherence breakdown due to the bath, in order to clarify the physical assumptions which underlie the concept of rate constant at low temperatures. The particular tunneling model we choose is the two-level system (TLS) with the Hamiltonian... 

This simple gas-phase model confirms that the rate constant is proportional to the square of the tunneling matrix element divided by some characteristic bath frequency. Now, in order to put more concretness into this model and make it more realistic, we specify the total (TLS and bath) Hamiltonian... 

Note in passing that the common model in the theory of diffusion of impurities in 3D Debye crystals is the so-called deformational potential approximation with C a>)ccco,p co)ccco and J o ) oc co, which, for a strictly symmetric potential, displays weakly damped oscillations and does not have a well defined rate constant. If the system permits definition of the rate constant at T = 0, the latter is proportional to the square of the tunneling matrix element times the Franck-Condon factor, whereas accurate determination of the prefactor requires specifying the particular spectrum of the bath. 

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