Tunneling matrix elements

Beratan D N and Hopfield J J 1984 Calculation of electron tunneling matrix elements in rigid systems mixed valence dithiaspirocyclobutane molecules J. Am. Chem. Soc. 106 1584-94... 

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... 

The solution of the spin-boson problem with arbitrary coupling has been discussed in detail by Leggett et al. [1987]. The displacement of the equilibrium positions of the bath oscillators in the transition results in the effective renormalization of the tunneling matrix element by the bath overlap integral... 

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. 

Comparing this expression with (3.81), one obtains the formal definition of the quasiclassical tunneling matrix element... 

The operator U shifts the qj oscillator coordinate to its equilibrium through the distance QoCj/cOj, the sign depending on the state of the TLS. All the coupling now is put into the term proportional to the tunneling matrix element and the small parameter of the theory is zIq rather... 

Substitution of this for the golden-rule expression (1.14) together with the renormalized tunneling matrix element from (5.60) gives (5.64), after thermally averaging over the initial energies E-,. In the biased case the expression for the forward rate constant is... 

So far we took the tunneling matrix element /Io to be independent of the vibration coordinates. In terms of our original model with extended tunneling coordinate Q this assumption means that... 

The analytic results for the spin-boson Hamiltonian with fluctuating tunneling matrix element (5.67) are investigated in detail by Suarez and Silbey [1991a]. Here we discuss only the situation when the qi vibration is quantum, i.e., (o P P 1. When the bath is classical, cojP, j 1, the rate... 

It is to be emphasized that, despite the formal similarity, the physical problems are different. Moreover, in general, diabatic coupling is not small, unlike the tunneling matrix element, and this circumstance does not allow one to apply the noninteracting blip approximation. So even having been formulated in the standard spin-boson form, the problem still remains rather sophisticated. In particular, it is difficult to explore the intermediate region between nonadiabatic and adiabatic transition. 

The STM postulated tunneling matrix element distribution P(A) oc 1 /A implies a weakly (logarithmically) time-dependent heat capacity. This was pointed out early on by Anderson et al. [8], while the first specific estimate appeared soon afterwards [93]. The heat capacity did indeed turn out time dependent however, its experimental measures are indirect, and so a detailed comparison with theory is difficult. Reviews on the subject can be found in Nittke et al. [99] and Pohl [95]. Here we discuss the A distribution dictated by the present theory, in the semiclassical limit, and evaluate the resulting time dependence of the specific heat. While this limit is adequate at long times, quantum effects are important at short times (this concerns the heat condictivity as well). The latter are discussed in Section VA. 

There appears to be a low barrier between adjacent sites for normal muonium in Si but a substantial barrier and/or a small tunneling matrix element between adjacent bond-centered sites. In addition there is an appreciable barrier between BC and T sites. These features are consistent with experiment and with most of the theoretical calculations. 

The energy Ea is a quantum term associated with the proton reaction coordinate coupling to the Q vibration, Ea = h1 /2m. and Co is the tunneling matrix element for the transfer from the 0th vibrational level in the reactant state to the 0th vibrational level in the product state. The term AQe is the shift in the oscillator equilibrium position and F L(Eq, Ea, Laguerre polynomial. For a thorough discussion of Eq. (8), see [13],... 

In the above expression, vR and Vp are the frequencies associated with the rath level of reacting proton in the reactant state and the mth level in the product state, and q is the frequency associated with the low-frequency mode developed in the BH model. The term AQ reflects the change in equilibrium distance between the reactant and product states and Cmpm(Q) is the tunneling matrix element from the nth level in the reactant state to the mth level in the product state. An explicit evaluation of the tunneling matrix element Cmpm(Q) is obtained within the WKB semiclassical framework and is given by... 

The delta function, 5, limits the analysis to elastic processes. The tunneling matrix element, M, is determined by the overlap of the surface wave functions of the two metal subsystems at a particular separation surface, which also reflects the energy-lowering resonance associated with the interplay of the two states. The tunneling current may be found by summing over... 

Bardeen considers two separate subsystems first. The electronic states of the separated subsystems are obtained by solving the stationary Schrodinger equations. For many practical systems, those solutions are known. The rate of transferring an electron from one electrode to another is calculated using time-dependent perturbation theory. As a result, Bardeen showed that the amplitude of electron transfer, or the tunneling matrix element M, is determined by the overlap of the surface wavefunctions of the two subsystems at a separation surface (the choice of the separation surface does not affect the results appreciably). In other words, Bardeen showed that the tunneling matrix element M is determined by a surface integral on a separation surface between the two electrodes, z = zo. 

In the interpretation of the experiment of Giaever (1960), Bardeen (1960) further assumed that the magnitude of the tunneling matrix element M does not change appreciably in the interval of interest. Then, the tunneling current is determined by the convolution of the DOS of two electrodes ... 

In this subsection, we show that using Schrbdinger s equations, the tunneling matrix element can be converted to a surface integral similar to Bardeen s. Using Schrbdinger s equation for the tip states, Eq. (2.25), the matrix element is converted into... 

The problem of evaluating the tunneling matrix elements has been investigated by many authors (Tersoff and Hamann, 1983, 1985 Baratoff, 1984 Chung, Feuchtwang, and Cutler, 1987 Chen 1988, 1990, 1990a Lawunmi and... 

Fig. 3.1. Derivation of the tunneling matrix elements. A separation surface is placed between the tip and the sample. The exact position and the shape of the separation surface is not important. The coordinates for the Cartesian coordinate system and spherical coordinate system are shown, except y and 4>. (Reproduced from Chen, 1990a, with permission.)...

From Eq. (3.1), the tunneling matrix elements are determined by the wavefunctions of the tip and the sample at the separation surface, which is located roughly in the middle of the vacuum gap, as shown in Fig. 3.1. For both tip states and sample states near the Fermi level, the wavefunctions on and beyond the separation surface satisfy Schrodinger s equation in the vacuum. 

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