Tunneling models

In realistic systems, the separation of the modes according to their frequencies and subsequent reduction to one dimension is often impossible with the above-described methods. In this case an accurate multidimensional analysis is needed. Another case in which a multidimensional study is required and which obviously cannot be accounted for within the dissipative tunneling model is that of complex PES with several saddle points and therefore with several MEPs and tunneling paths. 

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... 

Electron-tunneling Model. Several models based on quantum mechanics have been introduced. One describes how an electron of the conducting band tunnels to the leaving atom, or vice versa. The probability of tunneling depends on the ionization potential of the sputtered element, the velocity of the atom (time available for the tunneling process) and on the work function of the metal (adiabatic surface ionization, Schroeer model [3.46]). 

Coulomb blockade effects have been observed in a tunnel diode architectme consisting of an aluminum electrode covered by a six-layer LB film of eicosanoic acid, a layer of 3.8-nm CdSe nanoparticles capped with hexanethiol, and a gold electrode [166]. The LB film serves as a tunneling barrier between aluminum and the conduction band of the CdSe particles. The conductance versus applied voltage showed an onset of current flow near 0.7 V. The curve shows some small peaks as the current first rises that were attributed to surface states. The data could be fit using a tunneling model integrated between the bottom of the conduction band of the particles and the Fermi level of the aluminum electrode. 

In the (semi-)classical models of ETR (Marcus the Russian school), redox orbitals of reactants overlap at a close separation, followed by swift electron transfer. The activated complex, considered in equilibrium with the reactants, consists of these overlapping orbitals. In the tunneling model, the electron penetrates... 

Even Anderson et al. [39] pointed out that an important consequence of the tunnelling model was the (logarithmic) dependence of the measured specific heat on the time needed for the measurement of c. The latter phenomenon was due to the large energy spread and relaxation time of TLS. In 1978, Black [45], by a critic revision of the tunnelling theory, has been able to explain the time dependence of the low-temperature specific heat. 

On the other hand, the low-conductance values (L) give a poor linear correlation of the molecular length with an approximate decay constant fiN 0.45 0.09, distinctively different from the H and M sequences. The estimated value of fiN(L) is rather close to results reported by Cui [28] and Haiss [243]. Haiss et al. [244] found a pronounced temperature dependence of these L values, which scales logarithmically with 7 1 in the temperature range 293-353 K, indicating a transport mechanism different from a simple tunneling model. 

Vilan A (2007) Analyzing molecular current-voltage characteristics with the Simmons tunneling model scaling and linearization. J Phys Chem C 111 4431-4444... 

Nieminem J, Lahti S, Paavilainen S (2002) Contrast changes in STM images and relations between different tunneling models. Phys Rev B 66 165421... 

Chart 2. A simplified account of the Bell tunneling model ... 

Table 1 Experimental studies that led to the coupled motion and tunneling model... 

Bell quantum tunneling model, 33, 34-35, 72 (3-Cyclodextrin/amino acid complexes, 220t Bidentate ligands, 153 Biomacromolecules... 

In the second or resonant tunneling model, the intermediate state is associated with the formation of a solvation cage for the hydrated electron [96]. This scheme has largely been discounted since organizing the state... 

Brief History of Models and the Proton-Tunneling Model. 163... 

Static dielectric measurements [8] show that all crystals in the family exhibit a very large quantum effect of isotope replacement H D on the critical temperature. This effect can be exemphfied by the fact that Tc = 122 K in KDP and Tc = 229 K in KD2PO4 or DKDP. KDP exhibits a weak first-order phase transition, whereas the first-order character of phase transition in DKDP is more pronounced. The effect of isotope replacement is also observed for the saturated (near T = 0 K) spontaneous polarization, Pg, which has the value Ps = 5.0 xC cm in KDP and Ps = 6.2 xC cm in DKDP. As can be expected for a ferroelectric phase transition, a decrease in the temperature toward Tc in the PE phase causes a critical increase in longitudinal dielectric constant (along the c-axis) in KDP and DKDP. This increase follows the Curie-Weiss law. Sc = C/(T - Ti), and an isotope effect is observed not only for the Curie-Weiss temperature, Ti Tc, but also for the Curie constant C (C = 3000 K in KDP and C = 4000 K in DKDP). Isotope effects on the quantities Tc, P, and C were successfully explained within the proton-tunneling model as a consequence of different tunneling frequencies of H and D atoms. However, this model can hardly reproduce the Curie-Weiss law for Sc-... 

The soft-mode spectra in the FE phase was investigated within the same study [ 19] and a well-defined peak (S-peak) was found at 150 cm for T Tc as the lowest frequency peak in the spectra. The frequency of S-mode decreases with the increase in pressure, indicating that the S-mode is the soft mode and that the phase transition is of the displacive type, which is in accordance with the proton-tunneling model. Furthermore, Raman scattering experiments on deuterated crystals showed the disappearance of the S-peak in DKDP [20]. Since this phenomenon can also be explained by the protontunneling model, it is taken as another important piece of evidence for this model. 

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