# Theory of Tuning ‘Tunnelling-Probability’ through Potential Barrier by Acoustically Augmented Phonons (AAP): An Advanced Study

The goal of this chapter is to show that by superimposing an Ultra High Frequency (UHF) acoustic wave over the source of phonon transmission, it is possible to manually control the tunnelling probability of phonons via potential barriers, resulting in Acoustically Augmented Phonons (AAP). Its goal is to develop analytical relationships between the barrier height, thickness, penetrating wave kinetic energy, acoustic damping factor, and the imposed acoustic wave number in influencing tunnelling and reflection. Finally, tuneable high fidelity acoustic piezoelectric transducer emitters could be utilised to tune tunnelling.

If P(R_{p}) is the probability of phonon reflectivity (P) while tunnelling over the potential barrier, and P(R_{AAP}) is the probability of Acoustically Augmented Phonons (AAP), then it may be calculated analytically that P(RAAP) = ∝ P(RP)

Where a is dimensionless and is defined as the Augmentation Probability Factor(APF), which is determined by structure-based damping coefficient, UHF acoustic wave wave number, and source kinetic energy. The increase in tunnelling probability caused by mechanically superimposing a UHF acoustic wave over in situ phonons is independent of the initial amplitude of the UHF acoustic wave. It has been shown that when the kinetic energy of phonons grows, the range of tuneable frequencies of UHF expands until it hits a limit. Theoretically, Acoustically Augmented Phonons (AAP) have a nearly zero reflectivity of tunnelling probability through the potential barrier. To get very low phonon tunnelling probability reflectivity, the’structural damping factor’ must be smaller than the superimposed UHF of the acoustic wave.

The effect of superimposing an ultra high frequency (UHF) acoustic wave over the source of the incoming particle wave function on the tunnelling probability via a potential barrier has been studied theoretically. The natural energy unit makes V0 non-dimensional Δ1 as y [= V_0/∆_1 ] . The likelihood of tunnelling vs the kinetic energy fraction graph [(E/V0) = x] A line of inflection at a non-dimensionalized critical height is seen on the particle yc ≈ 3.12879, yc denotes the universal tunnelling constant (UTC). The reflection increases as the barrier height(y) is increased more (y > yc), and the tunnelling probability significantly decreases in general. The inherent particle kinetic energy, the superimposed wave number n, and the material parameter y all influence the lowest feasible value of y. With a decrease in the wave number n, the ‘gradient of the increase in probability’ rises and becomes larger at higher values of y. A larger ratio (E/V0) of the applied UHF acoustic wave, along with a permissible-smaller wave number (n), leads to a higher tunnelling likelihood. With increasing UHF wave numbers and decreasing x-values, the potential barrier to tunnelling becomes progressively opaque. The tunnelling opacity of the potential barrier increases as the value of y increases. When the orders of and k are comparable, the tunnelling probability is maximum (=0.98673) at y = 4, x = 0.9.

**Author(S) Details**

**P. Kumar
**Department of Mathematics, MIT World Peace University, Pune-411038, India.

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