Tunneling frequency, electronic

The K parameter, the electronic transmission coefficient, is related to the extent of overlap between the donor and acceptor orbitals. When this overlap is very small, electron tunneling frequency determines the pre-exponential factor, the reaction is nonadiabatic and k <1. Such overlap may be diminished if the electrode-reactant distance, in the course of the charge transfer, is increased due, for instance, to the presence of a blocking film on the electrode. On the other hand, when the overlap is relatively large, k is close to 1. Only when the reactant is near the electrode surface does significant overlap of donor and acceptor orbitals occur. 

Fig. 6 Calculated inelastic electron tunneling spectra with the different stretching distances offset by 2.5 V-1. The geometries with gauche defects are marked in the right margin for stretching 4.0-7.0 A. Highlighted frequency bands are (1) top, 82 (degenerate C-S) and 130 (C-C) meV, and (2) bottom, 75 (nondegenerate C-S), 95 (rock), 165 (wag), and 365 (C-H) meV. (Reprinted with permission from [60])...

By the same token, electron transfer involves transfer of a particle between electronically coupled chemical sites and can be described as a tunneling process. In that sense, every electron transfer process involves electron tunneling with a tunneling frequency given, in the classical limit, by equation (31). 

In 1958, Franz [45] and Keldysh [46] independently theoretically predicted the absorption by a semiconductor, placed in an electric field, of light quanta which have an energy less than the width of the forbidden gap. The effect is connected with interband tunneling (Fig. 20). The valence band electron tunnels from point xl to point 3c, then it absorbs a quantum with a frequency lo < Eg (Eg is the width of the forbidden gap) and further tunnels to point x2. Using the law of conservation of energy and the law of conservation of imaginary momentum (see the previous section), it is easy to show that light absorption at point 3c, which lies exactly between points acj and x2, is optimal. Consequently... 

There exists an opinion that, in the case of electron tunneling, the frequency factor does not depend on temperature and has the order of the frequency of electron motion in atoms, i.e. v 1016s 1, while y is connected with the ionization energy, I, of a donor by the ratio... 

The possibility of the anomalous isotope effect for electron tunneling reactions was first noted by Ulstrup and Jortner [7]. This effect becomes possible when the reorganization energy is approximately equal to the reaction exothermicity. If, in this case, for example, the relationship Er - J + a) — 0 is satisfied, where co is the vibrational frequency for a heavy isotope, then from the viewpoint of the activation energy [see eqn. (42)1, the transition (0 - 1) is optimal for the heavy isotope. Compared with this transition for the heavy isotope, both the transitions (0 - 0)and(0 - 1) for the light isotope contain the additional activation multiplier. In this situation the anomalous isotope effect will be observed, provided that the Franck Condon factor for the transition (0 -> 1) of the heavy isotope is not too small compared with that of the light isotope. An example of the electron tunneling reaction for which the anomalous isotope effect is observed experimentally will be considered in Chap. 7, Sect. 4. 

Note that different absorption frequencies correspond to different electron transfer mechanisms with the participation of radiation. In principle, this difference can be used to distinguish between various possible mechanisms. A comparison of the efficiency of various electron tunneling processes with the participation of radiation has been carried out in ref. 31. 

The gate voltage dependence of the frequency is a stepwise function, as shown in the inset of Fig. 3. Steps occur whenever an additional electron tunnels onto the tube. For the E-nanotube, their height is MHz, which is measurable. Note, that the present submicron silicon devices are always in the weak-bending regime so that corrections due to the second term in Eq. (16) are too small to be measured. Furthermore, one should realize that frequency quantization is only observable if the frequency itself is greater than the inverse tunneling time for electrons. 

Fig. 3. Gate voltage dependence of the frequency ujq of the fundamental mode for three different values of the residual stress. Numbers are taken for the E-nanotube (see Fig. 2). The fundamental mode of an unstressed tube is 140 MHz (thin horizontal line). The inset is an enlargement of the To = 0 curve of the main figure showing step-wise increases of vn whenever an additional electron tunnels onto the tube.

Covalon conduction is a spin-paired two-electron charge transfer that can take place when, the anti-symmetric normal mode of vibration, o, matches , the electron pair tunneling frequency. 

Temperature independent electron tunneling was observed also in Ref. [319], where the rate of electron transfer over large distance in mixed-metal hemoglobin hybrids [MP, FeIU(CN )P], where M = Zn or Mg, was measured in the temperature range from ambient to 100 K. The electron transfer from the triplet state of MP to Fera was not effected by the freezing of the cryosolvent, which may indicate that coupling of electron transfer to low-frequency solvent modes may be minimal. For both M, but especially for M = Mg, the rate constant of the back reaction is nearly temperature independent. 

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