The Square-barrier problem

In this section, we will analyze an elementary problem in quantum mechanics, the square barrier. The purpose is twofold. First, such an analysis can provide physical insight into the process, to gain a conceptual understanding. Second, analytically soluble models are indispensable for assessing the accuracy of approximate methods, such as the MBA. 

From a semiclassical point of view (Landau and Lifshitz, 1977), there are two qualitatively different cases, as shown in Fig. 2.5. When the top of the barrier is higher than the energy level, the barrier is classically forbidden, and the process is called tunneling. When the top of the barrier is lower than the energy barrier, the barrier is classically allowed, and the process is called channeling or ballistic transport. We will show that for square potential barriers of atomic scale, the distinction between the classically forbidden case and the classically allowed case disappears. There is only one unified phenomenon, quantum transmission. 

In the problem of quantum transmission, three traveling waves are involved the incoming wave sxp(iqx), the reflected wave A exp( - iqx), and the transmitted wave B exp(iqx), where 

This convention more conveniently connects with perturbation theory, which is developed later in this chapter. If the barrier is classically forbidden, i. e., E Ub or l l I t/fil, the transmission coefficient is 

Taking the semiclassical limit, these two cases become very different. For thick, classically forbidden barriers, Eq. (2.8) becomes 

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We will test this method with the exact solutions of the square-barrier problem in the following subsection. 

Fig. 2.9. Perturbation treatment for the square-barrier problem, (a), Original problem, (b) and (c) The potentials of the subsystems for a perturbation treatment.

Fig. 2.10. Different approximate methods for the square harrier problem. (Parameters used W= 2 A = 4 eV Utf= 16 eV.) The original Bardeen theory breaks down when the barrier top comes close to the energy level. The modified Bardeen tunneling theory is accurate with separation surfaces either centered L = lV/2) or off-centered L = VV73). By approximating the distortion of wavefunctions using Green s functions, the error in the entire region is only a few percent.

In reality, as the barrier becomes narrower, it deviates from the square shape. One often used model is the parabolic barrier (dashed line in Fig. 1). When the barrier is composed of molecules, not only is the barrier shape difficult to predict, but the effective mass of the electron can deviate significantly from the free-electron mass. In order to take these differences into account, a more sophisticated treatment of the tunneling problem, based on the WKB method, can be used [21, 29-31]. Even if the metals are the same, differences in deposition methods, surface crystallographic orientation, and interaction with the active layer generally result in slightly different work functions on either side of the barrier. 

Fig. 2.7. Apparent barrier height calculated from the exact solution. Variation of the apparent barrier height 0.95(

Using the EAM barriers, for typical parameters T = 320 K and Z = 401 the values for the return probabilities were peq = 1 — 2.4 x 10-7, p[Krp = 1.1 x 10 1, popp = 4.2 x 10 9, and the recombination probability prcc = 1.1 x 10 s. These values depend weakly on Z (e.g. the dependence of the mean square displacement, calculated later in this chapter from the return probabilities, is logarithmic (r2) a log (Z/Zo). This is a consequence of the fact that 2 is the marginal dimension for the return problem of the random walker. In higher-dimensional space the vacancy does not necessarily return and the return probability is asymptotically independent of the lattice size [43]. 

For the present problem, we expect the form of yj to be essentially exp(— 2/0 for long times. An exact initial form of y is difficult to estimate, but it must begin with rotation through ir in a time dependent mainly on the angular speed of the molecule at the crest of the potential barrier. The mean-square speed at this point will have the equipartition value kTII, so we fit this with the artificial... 

This is offset somewhat by an increase in the surface area of the edge, u. As in traditional 2-D nucleation theories, the squared-in-radius term overweighs the linear-in-radius term and the F a, b a)) curve must have a maximum zX a = a. Hence, the free energy surface F(a, b) of the normal film must have a saddle point, a maximum versus a and a minimum versus b. The problem of finding the energy barrier against coalescence is then reduced to determining the saddle point of this surface ... 

Figure 7.12 shows a plot of the behavior of the survival probability for a quadruple characterized by parameters typical of semiconductor heterostructures [61] barrier heights Vo = 200 meV, barrier widths, bo = 4.0 nm, well widths, Wo = 5.0 nm. The effective mass is m = 0.067 with the electron mass. Here the first triplet of complex poles [k ] and resonant states m (x) of the problem suffice to describe the behavior of the survival probability, namely Eq. (141) with M = 3. The initial state V (x,0) is taken by simplicity as a square infinite box state. One sees clearly the oscillating nonexponential behavior of the survival probability in this system. 

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