Quantum transmission probabilities

In order to obtain estimates of quantum transport at the molecular scale [105], electronic structure calculations must be plugged into a formalism which would eventually lead to observables such as the linear conductance (equilibrium transport) or the current-voltage characteristics (nonequilibrium transport). The directly measurable transport quantities in mesoscopic (and a fortiori molecular) systems, such as the linear conductance, are characterized by a predominance of quantum effects—e.g., phase coherence and confinement in the measured sample. This was first realized by Landauer [81] for a so-called two-terminal configuration, where the sample is sandwiched between two metalhc electrodes energetically biased to have a measurable current. Landauer s great intuition was to relate the conductance to an elastic scattering problem and thus to quantum transmission probabilities. 

A and A = 0.1 eV. The adiabatic ground potential energy surface is shown in Fig. 11. The present results (solid line) are in good agreement with the quantum mechanical ones (solid circles). The minimum energy crossing point (MECP) is conventionally used as the transition state and the transition probability is represented by the value at this point. This is called the MECP approximation and does not work well, as seen in Fig. 10. This means that the coordinate dependence of the nonadiabatic transmission probability on the seam surface is important and should be taken into account as is done explicitly in Eq. (18). 

Fig. 6.2 (a) Bell (parabolic) and Eckart barriers, both widely used in approximate TST calculations of quantum mechanical tunneling, (b) Transmission probability (Bell tunneling) as a function of energy for two values of the reduced barrier width, a... 

The expressions presented above are restricted since we used the parabolic barrier transmission probabilities. Extension of the theory to temperatures below the crossover temperature may be foimd in Ref 136. More sophisticated quantum rate theories will be discussed in Section V. 

There are two main ingredients that go into the semiclassical turnover theory, which differ from the classical limit.51 In the latter case, a particle which has energy E > 0 crosses the barrier while if the energy is lower it is reflected. In a semiclassical theory, at any energy E there is a transmission probability T(E) for the particle to be transmitted through the barrier. The second difference is that the bath, which is harmonic, may be treated as a quantum mechanical bath. Within first order perturbation theory, the equations of motion for the bath are those of a forced oscillator, and so their formally exact quantum solution is known. 

Figure 6 A scheme of the three possible resonances in OOTF. i) Global resonance (A). Very weak electron-vibration interaction is expected ii) Localized resonances or traps (B).Usually the LEPS experiments are not detecting electrons trapped in these resonances and they appear as a reduction in the transmission probability, iii) Quantum well structure (C). Here the electron is localized in one dimension, while it is delocalized in the other two dimensions. There is a significant electron-vibration coupling.

The objective is now to modify this equation such that quantum dynamical corrections to the classical transmission probability, Eq. (6.15), are introduced. 

In the present context, it is relevant to consider the barrier penetration that is associated with the traditional (one-dimensional) picture of tunneling. When we consider the time-dependent description of tunneling where a (broad) wave packet hits, e.g., a rectangular barrier, one finds that the center of the wave packet moves as a classical particle. The part of the packet that penetrates the barrier and tunnels through is not slowed down, i.e., it has exactly the same position and velocity as a wave packet that did not experience a barrier (see also [7] for a general discussion of the time-dependent picture of tunneling). The classical transmission probability of Eq. (6.15) is now replaced by the quantum mechanical transmission probability Pqm(E) (see Fig. 6.4.2). Thus, as a natural extension of the conventional formulation based on classical mechanics, in the derivation above we replace Pc by Pqm. That is, we can replace Eq. (6.21) by... 

Note that this result is identical to Eq. (6.24). Although the quantum mechanical transmission probability at kinetic energies above the barrier energy is less than one, that is, particles are reflected above the barrier, the transmission in the tunneling region dominates in the integral due to the Boltzmann factor exp(—E/kBT). 

We consider two metallic free-electron systems, with atomically flat surfaces separated by vacuum over a distance Ax (Figure 20). In fact, the model system is an extension of the metal surface considered in Section 4.5. The complex potential energy barrier at a metal surface, discussed in Section 4.5 is simplified here to a rectangular barrier. We look for the quantum-mechanical probability that an electron in phase A is also present in phase B. This probability is given by the ratio of squared amplitudes, and A, of the free-electron wave function in phase B and A, respectively. It is quantified by the transmission coefficient ... 

Quantum mechanical tunneling is a result of the wavelike nature of particles which allows transmission through a reaction barrier. The quantum mechanical transmission probability for energies below is governed by tunneling and reflection at the barrier. The transmission is larger than zero even well below the barrier and will depend crucially on the barrier width. In... 

Quantum mechanical tunneling is a result of the finite potential barrier at the metal-vacuum interface. The electronic wave function ip and its first derivative dip/dz are continuous across this interface (or finite potential discontinuity), the electron wave function decaying exponentially, e in the forbidden region where the barrier exceeds the total electron energy. In this context, k is approximately related to the apparent work function or mean local tunneling barrier, = 2m((p — E)/ h/2nf-. Thus, the tunneling current. It, or transmission probability also decays with barrier width, z [6.10-6.13]... 

A numerical analysis, based on a Green s function approach, has been carried out to explain the interference patterns of an electron beam injected and detected via quantum point contacts. The calculations show the profound influence of back-scattering from potential fluctuations located close to the injector or detector on the transmission probability of the propagated electron beam. The interference patterns are sensitive to even small changes of the scatter location. 

If the specific state considered is an outgoing translational state on the adiabatic channel a) (Figure 4), one obtains an expression (51) for the specific rate constant on this adiabatic channel, if the quantum mechanical transmission probability for leaving the complex boundary is 7a. the rate constant being the frequency of generating the outgoing wave multiplied by the transmission probability ... 

At low temperatures, in a sample of very small dimensions, it may happen that the phase-coherence length in Eq.(3) becomes larger than the dimensions of the sample. In a perfect crystal, the electrons will propagate ballistically from one end of the sample and we are in a ballistic regime where the laws of conductivity discussed above no more apply. The propagation of an electron is then directly related to the quantum probability of transmission across the global potential of the sample. 

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