Electron tunneling amplitude

If the localized electron tunnels out through the barrier (state 1 in Fig. 12 b) a certain amount of f-f overlapping is present. States like 1 in Fig. 12 b are called sometimes resonant states or "virtually bound" states. In contrast with case 2 in Fig. 12b, which we may call of full localization , the wave function of a resonant state does not die out rapidly, but keeps a finite amplitude in the crystal, even far away from the core. For this reason, overlapping may take place with adjacent atoms and a band may be built as in ii. (If the band formed is a very narrow band, sometimes the names of localized state or of resonance band are employed, too. Attention is drawn, however, that in this case one refers to a many-electron, many-atoms wave function of itinerant character in the sense of band theory whereas in the case of resonant states one refers to a one-electron state, bound to the central potential of the core (see Chap. F)). 

Since the probability amplitude for tunneling is exponentially sensitive to the position of the dot, the maximum of the tunnel exchange interaction between an electron on the dot and an electron in one lead occurs when the tunneling coupling to the other lead is negligible. This is why we will assume the following property of tunneling amplitude ALjR t) to be fulfilled ... 

The amplitude of electron tunneling transfer with simultaneous change of the vibrations in the case of the non-adiabatic asymptotics (56) may be found, if to substitute the expression (56) instead of F ) to the definition (9) and the acceptor wave function in the matrix element (9) should be its total expression... 

When a tunneling calculation is undertaken, many simplifications render the task easier than a complete transport calculation such as the one of [32]. Let us take the formulation by Caroli et al. [16] using the change induced by the vibration in the spectral function of the lead. In this description, the current and thus the conductance are proportional to the density of states (spectral function) of the leads (here tip and substrate). This is tantamount to using some perturbational scheme on the electron transmission amplitude between tip and substrate. This is what Bardeen s transfer Hamiltonian achieves. The main advantage of this approximation is that one can use the electronic structure calculated by some standard way, for example plane-wave codes, and use perturbation theory to account for the inelastic effect. In [33], a careful description of the Bardeen approximation in the context of inelastic tunneling is given, and how the equivalent of Tersoff and Hamann theory [34,35] of the STM is obtained in the inelastic case. 

In fact, this mechanism has been confirmed by a calculation of the cross sections for impact ionization and impact excitation in electron-ion collisions [36]. The latter was found to be exceptionally low for neon. The basic features of recollision excitation follow from an amplitude such as (4.1) with, however, integration over one additional time the first electron tunnels out at the time t", recollides, promotes the bound electron into an excited state with energy Eq2 > E02, and leaves the interaction region at the time t > t", while the bound electron only becomes free at the later time t > t. Figure 4.5 exhibits the results of such a model. Indeed, for a sufficiently high-lying excited state, the region around zero ion momentum is filled in. 

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. 

A typical example of a special state is as follows. Electron transfer reactions at an atom are aided by vibrations that equilibrate the interatomic distances that differ for the two oxidation states. Thus a low-energy, high-amplitude vibration is desirable. The vibration could have the further function that it provided a time-dependent fluctuation of the redox potential. As I and Goldanskii in this volume have pointed out, this allows a precise matching of the redox potential of one redox couple with another leading to tunneling of electrons. 

Fig. 1. Schematic view of the nanomechanical GMR device a movable dot with a single electron level couples to the leads due to tunneling of electrons, described by the tunneling probability amplitudes TL,n(t)), and due to the exchange interaction whose strength is denoted by JL,n(t). An external magnetic field H is oriented perpendicular to the direction of the magnetization in the leads (arrows).

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