Bardeen matrix element

The first order Green s function, from the Dyson series and taking into account the exponential decay of electron states, contains two discrete terms, which can be written in terms of the Bardeen matrix elements... 

It is evident that each subsequent iteration in the interface Green s function can also be formulated in terms of Bardeen matrix elements in principle, the Green s function and thus the current can therefore be evaluated to any order. 

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... 

By writing down the kinetic energy term T = — (ftV2m)V explicitly, and using Green s theorem, the transition matrix element is finally converted into a surface integral similar to Bardeen s, in terms of modified wavefunctions ... 

The matrix element has the dimension of energy. In Chapter 7, we will show that the physical meaning of Bardeen s matrix element is the energy lowering... 

In Chapter 2, we showed that the tunneling current can be determined with a perturbation approach. The central problem is to calculate the matrix elements. Those are determined by the modified Bardeen surface integral, evaluated from the wavefunctions of the tip and the sample (with proper corrections) on a separation surface between them, as shown in Fig. 3.1 ... 

As shown in Eq. (3.1), the transmission matrix elements for different tip states are determined by the Bardeen integral on a surface separating the sample and the tip with one of the tip states. For an s -wave tip state, using Eq. (3.11),... 

Bardeen showed that to derive the tunneling matrix element, which represents the amplitude of electron transfer between the sample and tip, explicit expressions for the wavefunctions of the tip and sample were... 

Hence to produce a gap, one has to devise a physically meaningful wave function that has phase coherence and equal amplitudes for the and a potential U which is attractive (i.e., leads to matrix elements U = -V). In the next section, we review how Bardeen, Cooper and Schrieffer (7) (BCS) met this challenge. 

Electron tunneling was first analyzed by Bardeen [12] and Cohen et al. [13] using the perturbative transfer Hamiltonian (TH) approach and more recently by many other authors [14-16]. Although the TH gives, in many cases, a good description of the observed effects, it lacks a firm first principles theoretical basis and does not account properly for many-body effects [17]. An improved form of TH [18] that involved energy dependent transfer matrix elements was used to incorporate many-body effects. However, this model does not describe the electron-phonon interaction properly [19]. 

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