Tunneling rate

Fig. 8. Arrhenius plot of dissipative tunneling rate in a cubic potential with Vq = Sficoo and r jlto = 0, 0.25 and 0.5 for curves 1-3, respectively. The cross-over temperatures are indicated by asterisks. The dashed line shows k(T) for the parabolic barrier with the same CO and Va-...

The problem of nonadiabatic tunneling in the Landau-Zener approximation has been solved by Ovchinnikova [1965]. For further refinements of the theory beyond this approximation see Laing et al. [1977], Holstein [1978], Coveney et al. [1985], Nakamura [1987]. The nonadiabatic transition probability for a more general case of dissipative tunneling is derived in appendix B. We quote here only the result for the dissipationless case obtained in the Landau-Zener limit. When < F (Xe), the total transition probability is the product of the adiabatic tunneling rate, calculated in the previous sections, and the Landau-Zener-Stueckelberg-like factor... 

From the very simple WKB considerations it is clear that the tunneling rate is proportional to the Gamov factor exp —2j[2(F(s(0) — )] ds, where s Q) is a path in two dimensions Q= 61)62 ) connecting the initial and flnal states. The most probable tunneling path , or instanton, which renders the Gamov factor maximum, represents a compromise of two competing factors, the barrier height and its width. That is, one has to optimise the instanton path not only in time, as has been done in the previous section, but also in space. This complicates the problem so that numerical calculations are usually needed. 

Again we use the ImF method in which the tunneling rate is determined by the nontrivial instanton paths which extremize the Eucledian action in the barrier. Let for deflniteness the potential V Q) have a single minimum at = 0, F(0) = 0, separated from the continuous spectrum... 

P Q-) =p Q-,Q-,p), which in the harmonic approximation is described by (3.16), PhiQ-iQ-,P) exp(— CO Q1 tanh co ). Having reached the point Q, the particle is assumed to suddenly tunnel along the fast coordinate Q+ with probability A id(Q-), which is described in terms of the usual one-dimensional instanton. The rate constant comes from averaging the onedimensional tunneling rate over positions of the slow vibration mode,... 

In the light of the path-integral representation, the density matrix p Q-,Q-,p) may be semi-classically represented as oc exp[ —Si(Q )], where Si(Q ) is the Eucledian action on the -periodic trajectory that starts and ends at the point Q and visits the potential minimum Q = 0 for r = 0. The one-dimensional tunneling rate, in turn, is proportional to exp[ —S2(Q-)], where S2 is the action in the barrier for the closed straight trajectory which goes along the line with constant Q. The integral in (4.32) may be evaluated by the method of steepest descents, which leads to an optimum value of Q- = Q. This amounts to minimization of the total action Si -i- S2 over the positions of the bend point Q. ... 

If the potential is parabolic, it seems credible that the inverted barrier frequency A should be substituted for the parabolic barrier transparency to give the dissipative tunneling rate as... 

Thus, it is concluded that the FCC structure induces an increase in the tunneling rate i.e., the resistance decreases between particles. The tunneling between adjacent particles is a major contribution to the conduction. This inhibits the Coulomb blockade in the tunneling I V) measurements, and thus the 3D superlattices yield an increased tunneling current. 

Finally, two other experimental observations were addressed. First, the small increases in rate resulting from warming were suggested to be due to matrix softening at higher temperatures, and Ar appeared to be a less-rigid matrix than N2. Second, the tunneling rate constant for rotation of 11-endo to the reactive 11-exo conformer... 

In impure metals, dislocation motion ocures in a stick-slip mode. Between impurities (or other point defects) slip occurs, that is, fast motion limited only by viscous drag. At impurities, which are usually bound internally and to the surrounding matrix by covalent bonds, dislocations get stuck. At low temperatures, they can only become freed by a quantum mechanical tunneling process driven by stress. Thus this part of the process is mechanically, not thermally, driven. The description of the tunneling rate has the form of Equation (4.3). Overall, the motion has two parts the viscous part and the tunneling part. 

Wu SW, Nazin GV, Chen X, Qiu XH, Ho W (2004) Control of relative tunneling rates in single molecule bipolar electron transport. Phys Rev Lett 93 236802/1-236802/4... 

We caution that Denteneer et al. (1989c) did not estimate a tunneling rate for the pathway they have proposed. To explain the experiment by Muro and Sievers (1986), a tunneling rate of 1013 s-1 is required. This is still 10 decades larger than the rough estimate of a typical tunneling rate made by Watkins (1989). 

As discussed in section 2.2, a mixture of AMP and AA showed two solid condensed phases above and below about 30 mN m- [5,10]. A loosely stacked structure of two porphyrins was proposed for LB films prepared at higher surface pressures than 30 mNmr1, which was caused by squeezing-out of a monomolecular structure formed at lower surface pressure [5,10]. In this section, photoelectric characteristics of LB films containing AMP and AA deposited at two solid condensed phases will be discussed in relation to multilayer structure and the anisotropic intermolecular tunneling rates [87]. Seven monolayers of 1 5 or 1 10 mixture of AMP and AA were deposited at 20 and 50 mN m-1 on an ITO plate at 18 °C to form stable Y-type LB films. Aluminum was vacuum evaporated onto LB films as sandwich-type electrodes at 10-6 Torr. Steady photocurrents were measured in a similar manner as mentioned above. 

On the basis of a Landau-Zener curve crossing formalism, Borgis and Hynes derived the nonadiabatic rate constant k, which is similar to that expressed by the DKL model but where the tunneling term Cnm found in Eq. (4) is significantly modified due to the influence of the low-frequency promoting mode Q, with a frequency 0)q, on the tunneling rate. The dependence of Cnm on Q is given by [13]... 

The structural stability of mixed-metal hemoglobin hybrids also has allowed us to study low-temperature electron transfer in this system. We first reported the temperature dependence of triplet-state quenching within the [ (ZnP), Fe (H20)P] hybrids, which we attributed to the ZnP Fe P ET reaction [7d]. The rate constant dropped smoothly as the temperature was lowered from room temperature to 200 K. Below this temperature the rate constant remained roughly constant with a tunnelling rate constant of kt 9 s (Fig. 7). 

Equation 8-19 indicates that the rate constant of electron transfer by tunneling is the same in the forward and backward directions at constant electron energy. The same tunneling rate constant of electrons in the forward and backward directions is valid not only in the equilibrium state but also in the nonequlibrium state of electron transfer [Gerischer, I960]. 

Fig. 8-16. Electron state density in a semiconductor electrode and in hjrdrated redox partides, rate constant of electron tunneling, and exchange redox current in equilibrium with a redox electron transfer reaction for which the Fermi level is close to the conduction band edge eF(sc) = Fermi level of intrinsic semiconductor at the flat band potential 1. 0 (tp.o) = exchange reaction current of electrons (holes) (hvp)) - tunneling rate constant of electrons (holes).

In the theoretical model used the overall rate constant k is represented as the product of the no-tunnelling rate constant k° by the tunnelling correction P, ... 

Finally, we should still mention work by Makram-Ebeid and Lannoo (1982a,b) that calculates (in the adiabatic approach) the effect of an electric field in enhancing, with phonon assistance, the tunneling rate. Very good... 

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