Tunneling through the Potential Barrier

FIGURE 2. Electron tunneling through barrier formed by field and image potential at the metal/vacuum interface/  

This expression is the Wentzel-Kramer-Brillouin (WKB) tunneling probability for a parabolic barrier and n E) dE is the Matthews and Khan modified form of the expression for available electrons. The complete photocurrent must include contributions for tunneling, as well as over-the-barrier currents. 

The barrier height Em with respect to the bottom of the conduction band in the presence of an applied field is shown to be 

Photoemission from metals into vacuum under applied electrical field should resemble the electrochemical system better than the simple Fowler s law does. However, tunneling processes have not always been invoked to explain photoinjection from metals into electrolytes. In fact, one of the most accepted theories is the rate law, to be described below. This theory does not take into account tunneling effects. So, although Fowler s theory has been accepted universally for photoemission into vacuum in the threshold frequency range, the improvement in the wave mechanical approach has been to solve for the partial flux more precisely, using methods for threshold creation phenomena.  

3 Quantum Mechanical Photoemission Theories for the Metal/Vacuum and 

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In some of the metal-insulator transitions discussed here the use of classical percolation theory has been used to describe the results. This will be valid if the carrier cannot tunnel through the potential barriers produced by the random internal field. This may be so for very heavy particles, such as dielectric or spin polarons. A review of percolation theory is given by Kirkpatrick (1973). One expects a conductivity behaving like... 

The kinetics of electron transfer reactions at electrodes can be explained either by surmounting an activation barrier due to the chemical reorganization of the reactants or by tunnelling through the potential barrier across the electrode—solution interface. 

The quasi-bound states can dissociate by tunneling through the potential barrier and/or by vibrational quenching. In the latter case energy... 

FIGURE 1.16. Circumstances in which electron transfer does not oecur between the energy levels at the band edge on the surface and those in the electrolyte, (a) Electron transfer via surface states (b) electron tunneling through the potential barrier in the space charge layer. 

Because typical metal-ligand stretching frequencies are ca. 2 kT at room temperature (T), the possibility that the inner-shell nuclei will tunnel through the potential barrier needs to be considered. This is allowed for through the nuclear-tunneling factor, T j, which is defined by ... 

Classical physics dictates that a particle constrained by an energy barrier can become free only if it acquires an energy greater than the height of the barrier. In quantum mechanics, this restriction is eased. For example, quantum mechanics allows an electron to escape from the interior of a metal by tunneling through the potential barrier that confines it. The height of this barrier is called the work function of the metal (). The work function is a property of a metal surface which can be locally modified by the presence of an adsorbate. For a clean metal surface, 4>= 1-6 eV... 

That this transformation takes place at an energy below the potential barrier, which would be forbidden in classical mechanics, permits one to speak of a quantum mechanical tunneUng effect (pictorially, as though there were a tunnel through the potential barrier). 

The second term of Eq. (55) characterizes a decrease in modulus, caused by the atoms tunneling through the potential barrier because of the zero-point twisting and deformation vibrations and the third term is that caused by thermally activated atoms overcoming the potential barrier by means of the energy of the twisting vibrations. 

For some spectroscopic problems it is necessary to use three lasers in order to populate molecular or atomic states that cannot be reached by two-step excitation. One example is the investigation of high-lying vibrational levels in excited electronic states, which give information about the interaction potential between excited atoms at large internuclear separations. This potential V R) may exhibit a barrier or hump, and the molecules in levels above the true dissociation energy V(R = 00) may tunnel through the potential barrier. Such a triple resonance scheme is illustrated in Fig. 5.42a for the Na2 molecule. A dye laser Li excites the selected level (v J )... 

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