Theory of Optimizing the Tunneling Probability Through a Potential Barrier by Acoustically Augmented Phonons
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Theory of Optimizing the Tunneling Probability Through a Potential Barrier by Acoustically Augmented Phonons P. Kumar∗ Department of Mathematics, Maharashtra Institute of Technology World Peace University, Pune 411038, India (Received 23 June 2020; revised 11 August 2020; accepted 14 August 2020) In this research, the optimization of the tunneling probability through a potential barrier by superimposing ultra high frequency (UHF) acoustic wave over the source of the incoming particle wave function was examined theoretically and was shown to result in acoustically augmented phonons (AAP). The graph of the tunneling probability against the kinetic energy fraction [(E/V0 ) = x] of the particle shows a line of inflection at a non-dimensionalized critical height yc ≈ 3.12879, where yc is universal tunneling constant (UTC). As the barrier height (y) is increased further (y > yc ), the reflection increases, and the tunneling probability sharply declines, in general. The lowest possible value of y is guided by the inherent particle kinetic energy, the superimposed wave number n and the material parameter β. The ‘gradient of the increase in probability’ rises with a drop in the wave number n and is larger at higher values of y. A higher ratio (E/V0 ) coupled with a permissible-smaller wave number (n) of the applied UHF acoustic wave, leads to a higher tunneling probability. For increasing values of the UHF wave numbers and decreasing x-values, the potential barrier becomes increasingly opaque to tunneling. The higher the value of y is, the higher the tunneling opacity of the potential barrier becomes. The tunneling probability is highest (=0.98673) at y = 4, x = 0.9, when the orders of β and k are comparable. Keywords: Acoustically augmented phonons (AAP), Universal tunnelling constant (UTC), Tunnelling optimization, UHF acoustic waves DOI: 10.3938/jkps.77.1188
I. INTRODUCTION Tunneling is the process by which quantum particles penetrate classically forbidden regions. The probability of penetrating a barrier is given in the WKB approximation in reference [1]. Optimization of the tunnelling effect has been experimentally examined with different particle masses, temperatures and vibrationally enhanced energies in different bio-molecular, chemical, electronic and thermoelectric problems. In electron transfer reactions, including the primary events of photosynthesis, the quantum dynamics of the transferred electron is well established to be crucial for understanding reaction rates; i.e., the electron tunnelling rate [2,3]. In some electron transfer processes, the quantum character of the atomic motions at the donor and the acceptor sites are also likely to play roles in determining the temperature and the driving force dependencies of the reaction rates, even at room temperature [4–7]. In 1989, Cha et al. [8] showed that hydrogen could be transferred by means of quantum tunnelling in the yeast alcohol dehydrogenase (EC l.l.1.1 ) reaction. Thus, tunneling has multiple applications in different fields of science and technology. ∗ E-mail:
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