Barrier tunnelling

In classical physics the kinetic energy cannot be negative in other words, the total energy E cannot be less than the potential energy V. Schrodinger s equation can, however, be solved when this is the case. In some situations, these solutions are physically significant. 

The trial-and-error method for differential equations used in Section 2.3 suggests that we should look for a function which, when differentiated twice, gives a positive multiple of itself. The exponential function has this property, and so we try 

Substituting into eqn 3.1 shows that this works, provided that 

As both positive and negative values of k are possible, we can write a general 

This shows that the tunnelling probability decreases sharply with 

---

The transmission coefficient T in equation (2.58) is the relative probability that a particle impinging on the potential barrier tunnels through the barrier. The reflection coefficient R in equation (2.59) is the relative probability that the particle bounces off the barrier and moves in the negative v-direction. Since the particle must do one or the other of these two possibilities, the sum of T and R should equal unity... 

The invariance of IETS in an M-A-M junction vs an M-I-A-M device is exceptionally well demonstrated by the work of Reed [30], Figure 7 shows the Au-alkanedithiol-Au structure he used to create a single barrier tunnel diode. The IET spectra obtained from this device were stable and repeatable upon successive bias sweeps. The spectrum at 4.2 K is characterized by three pronounced peaks in the 0-200 mV region at 33,133, and 158 mV. From comparison with previously reported IR, Raman, and high-resolution electron energy-loss (HREEL) spectra of... 

Li B, Zeng C, Zhao J, Yang J, Hou JD, Zhu Q (2006) Single-electron tunneling spectroscopy of single C60 in double-barrier tunnel junction. J Chem Phys 124 064709-064720... 

In the second experimental approach, the Cgo molecules are deposited on top of an insulating self-assembled monolayer, thus creating a double barrier tunnel junction coimected in series and sharing an electrode [66, 67]. Under these conditions current steps in the I-V graph are observed, because when a potential is applied the capacitances of each junction has to be charged to a threshold potential before an electron can tunnel through the junction and when it is favorable for an electron to sit in the middle electrode the amount of current that flows through the junctions increases [68],... 

Fig. 2.6. Quantum transmission through a thin potential harrier. From the semi-classical point of view, the transmission through a high barrier, tunneling, is qualitatively different from that of a low barrier, ballistic transport. Nevertheless, for a thin barrier, here W = 3 A, the logarithm of the exact quantum mechanical transmission coefficient (solid curve) is nearly linear to the barrier height from 4 eV above the energy level to 2 eV below the energy level. As long as the barrier is thin, there is no qualitative difference between tunneling and ballistic transport. Also shown (dashed and dotted curves) is how both the semiclassical method (WKB) and Bardeen s tunneling theory become inaccurate for low barriers.

First, as the donor binding energy decreases from Fe1 to Zn1, the electronic mixing, expressed in the dependence of rate on distance k exp-(oR) should change. In tl simplest barrier tunneling theory a s (IP. - IP. . ) For this theory, then,... 

In Sect. 7, we raised the question of what were the chemical stimuli to which the reactivity indices defined in Sect. 6, the softness kernels, were presumed to be the responses, our seventh issue. Now there are various broad categories of reactions to be considered, unimolecular, bimolecular, and multimolecular. The former occur via thermal activation over a barrier, tunneling through the barrier, or some combination of both. There is no stimulus, and the softness kernels defined as responses of the electron density to changes in external or nuclear potential are irrelevant. For the study of unimolecular reactions, one needs only information about the total energy in the relevant configuration space of the molecule. 

CLASSICAL MECHANISM OF MULTIDIMENSIONAL BARRIER TUNNELING 421 expression of the time-dependent energy,... 

A. Shudo, T. Onishi, K. S. Ikeda, and K. Takahashi, Dynamical and Energetic Barrier Tunneling in the Presence of Chaos in Complex Phase Space in Fundamental Aspects of Quantum Physics, L. Accardi and S. Tasaki, eds., World Sientihc, Singapore, 2003, pp. 157-176. 

---