Rectangular barrier

Hie possibility that a particle with energy Jess than the barrier height can penetrate is a quantum-mechanical phenomenon known as the tunnel effect. A number of examples are known in physics and chemistry. The problem illustrated here with a rectangular barrier was used by Eyring to estimate the rates of chemical reactions. ft forms the basis of what is known as the absolute reaction-rate theory. Another, more recent example is the inversion of the ammonia molecule, which was exploited in the ammonia maser - the fbiemnner of the laser (see Section 9.4,1). 

Figure 2.22 Schematic representation of an electron, of total energy E, tunnelling through a rectangular barrier of height V0. From Christensen (1992).

Equation (1) suggests that tunnel junctions should be ohmic. This is true only for very small bias. A much better description of the tunneling current results when the effects of barrier shape, changes in barrier with applied potential, and effective mass of the electron are all included. An example of such an improved relationship is given by (2), where / is the current density, a is a unitless parameter used to account empirically for non-rectangular barrier shape and deviations in the effective electron mass, and barrier height given by B = (L + work function of the left-hand metal ... 

Width of the rectangular barrier Donor-acceptor tunneling... 

Landauer-Buttiker tunneling time for the rectangular barrier Landauer-Buttiker tunneling time in a molecular orbital representation Timneling time... 

Height of the rectangular barrier through which the particle is turuieling... 

The Landauer-Buttiker time was derived for tunnehng through a rectangular barrier. In the case of a series of electronic levels, such as occur for DNA in the models of Fig. 1, one can generahze the Landauer-Buttiker argument to a molecular orbital representation. In the hmiting case that is appropriate to most DNA-type problems, the time in Eq. 10 holds [88]... 

Fig. 9. Schematic representation of barriers for electron tunneling, (a) A Coulomb barrier (b) a rectangular barrier.

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... 

Simmons21 has shown that the current I, as a function of applied voltage V, that traverses a molecule considered simply as a rectangular barrier of energy B and width d, in the direct tunneling regime V < B e X [12,13] is given by... 

Here N(x) is the potential energy barrier between reactant and product state of the hydrogen and E is the particle energy. For a static square barrier the theory predicts a huge non-realistic isotope effect and its non-sensitivity to temperature. The thermal fluctuations produce a thermal distribution of the transfer distance, /. For a rectangular barrier and low frequency vibration of substrate and medium and harmonic behavior of l ... 

Coy, A., and Callaghan, P. T. (1994b). Pulsed gradient spin echo nuclear magnetic resonance for molecules diffusing between partially reflecting rectangular barriers. J. Chem. Phys. 101, 4599-4609. 

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 ... 

Figure 20. Elastic tunneling of an electron between two metal phases separated by vacuum (rectangular barrier). Shown are the wave functions of a free electron propagating in a direction perpendicular to the interface. The wave function decays exponentially in the vacuum. The tunneling probability is related to the amplitude of the free electron wave functions (see section 6).

Very few potential barrier models, including the rectangular barrier model discussed above, yield exact results for the tunneling problem. In general one needs to resort to numerical calculations or approximations. A very useful approximation is the WKB formula, which generalizes the solution exp( zhr) of the free particle Schrodinger equation to the form... 

Schrodinger equation with the appropriate Bloch s boundary conditions. For the general case an analytical solution is not possible and one needs to solve the equation numerically. But we found that, for our purposes, one can get qualitatively the same results if the grained films considered, which have the hexagonal symmetry, are represented by an effective 2D set of interacting square wells with rectangular barriers. Thus, to estimate the values of and I y, for the square lattice inside the unit cell one has two independent Kronig-Penney equations for both electron and hole subsystems. 

In order to exemplify the above results we calculate the transmission coefficient vs energy in multibarrier resonant tunneling structures which may be modeled by a one-dimensional system formed by alternating N-1-1 rectangular barriers with N rectangular wells. We assume that the electrons possess the same effective mass through the system and that the tunneling process is coherent i.e., elastic [61]. 

The essential nature of the problem is illustrated in a simple version (due to Kronig and Penney) where V is assumed to be of the form shown below, and to consist of rectangular barriers of height Vq and width b separated by intervals of length a. 

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