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Journal of the Korean Physical Society

Theory of Optimizing the Tunneling Probability Through a Potential Barrier by Acoustically Augmented Phonons

P. Pradeep Kumar

Published Nov 19, 2020• 0 citations
Publisher

Abstract

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/V 0 )= x ] of the particle shows a line of inflection at a non-dimensionalized critical height y c ≈ 3.12879, where y c is universal tunneling constant (UTC). As the barrier height ( y ) is increased further ( y > y c ), 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/V 0 ) 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.

DOI
10.3938/jkps.77.1188

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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/V 0 )= x ] of the particle shows a line of inflection at a non-dimensionalized critical height y c ≈ 3.12879, where y c is universal tunneling constant (UTC). As the barrier height ( y ) is increased further ( y > y c ), 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/V 0 ) 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.
Abstract This paper examines tunnelling through potential well protected by double barriers as well as the motion of a wave packet encountering this type of potential. Various options for the left and right barriers’ penetration factors are taken into consideration. When one of the penetration factors has limiting value of 0.5, the transmission coefficients turns out to be independent of incoming energy of the particle and the width of potential well. When both penetration factors have a limiting value of 0.5, a scenario similar to resonance tunneling is achieved. Furthermore, by doing the wave packet analysis, we found that the width and height of potential barriers have a negligible impact on the probability of finding the particle inside the potential well in the limit of large penetration factors.
The working principle of a large number of modern scientific equipment has been applied to the quantum tunneling effect. Although the explanations for the relevant principles of this effect are clarified, the calculation process is very complex and difficult to understand. Previous studies combine specific applications without theoretical model references. This study adopts the method of using the time-independent Schrodinger equation to solve the wave function, which is more convenient to obtain the expression of transmission probability. Then, one discusses the relationship between transmission probability and potential barrier width, potential barrier spacing, and incident particle energy. The experimental results show that the transmission probability decreases exponentially from 1 to 0 as the potential barrier width gradually increases. As the potential barrier width increases, the transmission probability exhibits a clear periodic oscillation relationship, the oscillation period remains unchanged, and this paper derives the relationship between the oscillation period and the potential barrier width. As the incident energy of particles gradually increases, the transmission probability exhibits periodic changes, and the minimum value of transmission probability increases with the periodic changes. Based on the analysis, the smaller the potential barrier width, the greater the incident energy, and the more likely quantum tunneling occurs.
This work is a contribution to the theory of the quantum tunneling effect. In the literature at least two barriers are studied, for which the third-order contribution to the phase - integral asymptotic approximation of the transmission coefficient do not contribute. These are the parabolic barrier and the inverse Morse barrier. In the present work we will show that with a proper choice of the so called base function there is at least one more barrier in this category namely the Eckart-Epstein potential. The fact that the third - order contribution vanishes is a good indication that we have found an optimal choice of the base function, and the treatment to find an optimal base function may be possible to generalize to other classes of potential barriers. For particles of a low energy compared to the energy near the top of the barrier we obtain a vary low transmission coefficient, which means that the probability for tunneling to occur is very low. There exist some cases, for example that with a double barrier which is transparent, even for certain relatively low energies but no evidence for this kind of transparency for a single barrier has been found. The present work does not give any such evidence. At the same time there are still speculations on cold fusion like effects, which would demand a higher probability for tunneling through for a single barrier.
The dynamics of resonant tunneling via a spatially oscillating narrow well is investigated. The well generates a quasiresonance state, which can trap the incoming particles. Four spectral regimes are found: (1) the adiabatic regime, when the vibrations' frequency is lower than the spectral width of the resonance. In this regime, the mean current is independent of the vibration's frequency, and the current decreases as a function of the vibration's amplitude. (2) When the frequency of the vibration is higher than the spectral width of the resonance, the particle is partially trapped to the moving well and the dependence of the current on the vibrations' amplitude is more moderate. (3) However, and this is the main result of this paper, beyond a certain frequency the kinetic energy of the trapped particle exceeds the spectral width of the resonance, in which case particles cannot be trapped in the moving well and the current is abruptly suppressed. (4) When the energy quanta of the vibrations are higher than the energy gap between the resonance energy and the barrier's potential height, a single phonon can be absorbed by the particle only when the final energy agrees with the resonances of the barrier, in which case current suppression is selective and occurs only for specifics frequencies. The model is solved exactly numerically, and analytical approximations are presented for the different regimes. The analytical solutions show high agreement with the numerical ones. This effect can be implemented in extremely sensitive accelerometers. Moreover, it may explain the odor receptor's sensitivity to molecular vibrations.
At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.