Phonon-assisted tunneling

Electrical conduction in such disordered materials proceeds by a process of phonon-assisted tunnelling in which the probability of transfer between two sites is given by... 

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... 

Flynn and Stoneham [1970] were the first to suggest that diffusion of the light impurity atom is enhanced by symmetrically coupled vibrations. To incorporate this effect, which is called fluctuational barrier preparation, the authors have proposed to take into account the dependence of the tunneling matrix element on displacements of the heavy nuclei. This approach goes beyond the familiar Condon approximation. In this version of phonon-assisted tunneling, the phonon-dressed incoherent transitions are also induced by a suitable reduction in the barrier height via emission or absorption of phonons. 

Dick [1977] explained this behavior within the framework of a phonon-assisted tunneling mechanisms using the TLS approximation and golden rule formalism (see Sections 2.3 and 6.4). One-phonon transitions dominate the mechanism at low temperatures, resulting in a linear dependence of k with 7 this follows directly from relation (6.27) when j3/Wl. At higher temperatures, the main contribution comes from Raman processes, leading to a T4 dependence of the rate constant. This predicted T4 temperature dependence for RbBr OH- is analogous to results obtained by Silbey and Trommsdorf [1990] for two-proton transfer in benzoic acid crystals (see Section 6.4). 

K. The transition time is 102 s at 1.3 K. Kapphan and Luty proposed a phonon-assisted tunneling mechanism for this process. Although the notion that a mass as heavy as Ag+ can tunnel 0.9 A through a 0.17-kcal/mol barrier at 1.3 K is quite surprising, the experimental data and analysis clearly support this conclusion. 

The dispersionless optical phonon mode />, splits the degenerated unperturbed electron level (j = 1,2) while the mode b2 mediates the electron transitions between the levels. This latter term represents phonon-assisted tunneling, a mechanism of the... 

Let us note that the quantum phonon assistance of the electron tunneling (/3-term in equations (1) and (2)) constitutes the difference of the model from the related dimer and exciton quantum models where instead of /3 Y.n (h2n + 2n)° n °f equation (1) there stands Acrxn, where A is the distance between the levels [4,5],... 

In two-level models with phonon assisted tunneling, there appears coupling of both phonon modes mediated by Rabi oscillations. Therefore, also the interlevel... 

SLR time results provide novel information on the temperature dependence and distribution of local RFs and their dynamics. A direct evidence for incoherent phonon-assisted deuteron tunnelling in H-bonded systems has been observed from NMR experiments. The use of quadrupole-perturbed ID 2H NMR spectroscopy and in particular 2D 2H NMR exchange spectroscopy has found to be very promising as regards elucidating the details of the hydrogen dynamics. 

Horsewill et al. (1994) examined the hydrostatic pressure effect on the proton transfer in crystals of a carboxylic acid dimer. Under a hydrostatic pressure, the distance of hydrogen bonds becomes shorter, and this is accompanied by a decrease in the potential barrier to proton transfer. The temperature dependence of the rate of the proton transfer turns out to be of a non-Arrhenius type. The influence of phonon-assisted tunnelling becomes evident as the external pressure increases, especially at lower temperatures. 

PHONON-ASSISTED TUNNELING THEORIES APPLIED TO ELECTRONIC CONDUCTION IN NANOWIRES OF INORGANIC COMPOUNDS... 

Phonon-assisted tunneling model is applied for explication of temperature-dependent I-V characteristics measured by other authors for (Bii-xSbs)2Te3 and ZnSn03 nanowires. The fit of exper imental data of the current dependence on applied volta measured at different temperatures with two theories of phonon-assisted tunnelling explains well the variation of I-V curves with temperature as well as the temperature dependence of conductivity of nanowire. 

Phonon-assisted tunneling (PhAT) model has been shown to explain properly nonlinear current-voltage characteristics and temperature dependence of conductivity in carbon nanotubes and other nanostructures of organic materials [1,2], In this paper, we want to show that this model is workable in explanation of I-V characteristics of inorganic nanodevices (Bi xSbx)2Te3 nanowires measured in a wide temperature range, 1.75-350 K, by Xiao et al. [3] and ZnSnOs nanowires presented in [4]. 

We assume that a source of charge carriers is the local electronic states (traps) in the electrode-nanowire interfaee layer, whose electrons emerge to the conduction band of the erystal due to an electrical field, induced phonon-assisted tunneling. 

In conclusion, the phonon-assisted tunnelling model describes well both the temperature-dependent I-V data and dependence of resistance on temperature in a wide range of temperatures. The electric field strength at which tunnelling occurs was estimated to be in the range of 1-5 MV/m. The densities of localized states in the interface layer were found to be about lO cm in the Bi-rich and 3x10 cm in the Sb-rich BiSbTe nanowires. 

Phonon-assisted tunneling theories applied to electronic... 

The phonon assisted tunneling on the other hand has a correlation time i2 given by ... 

For g(x) to be proportional to 1/x, it requires the relaxation time to be an exponential function of some random variable such that x = xq exp ( ), where itself has a flat distribution. It means that ( ) = constant, and rt(x) = n ). (d /dx) oc x. If a, the polarisability, is also a function of then it can lead to a sub-linear frequency dependence of a (co). The functional form given for the variation of x can arise from two different relaxation mechanisms. The first is a classical barrier hopping, in which two energetically favourable sites like in a double well potential are separated by a barrier fV and = W/kT. The second mechanism is a phonon assisted quantum tunneling through a barrier, which separates two equilibrium positions, in which case = 2aR, where a is the localization length and R is the separation between the sites. In the first case, by treating JV as independent of R, it has been shown (Poliak and Pike, 1972) that... 

Tunnelling of protons through their H-bonds is at the origin of disorder. This tunnelling is phonon-assisted, that is, they appear more easily when vibrational modes of the crystal, called phonons, are excited. When temperature increases from 0 K, such low-frequency phonons... 

What differentiates the models is how the double well potential is linked to other vibrations of the molecule or lattice. The two approaches are, first the phonon assisted tunnelling model and, second the independent proton model. 

Fig. 9.12 Diagrammatic representations of the potential wells for the hydrogen atom in a H-bond as understood within the phonon assisted tunnelling model, shown above, and the independent proton model, below.

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