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

The objective of this chapter is to theoretically establish the possibility of manually controlling the tunnelling probability of phonons through potential barrier by superimposing Ultra High Frequency (UHF) acoustic wave over the source of phonons transmission which results in to Acoustically Augmented Phonons (AAP). It aims to establish analytical relations between the barrier height, its thickness, kinetic energy of penetrating wave, acoustic damping factor with the imposed acoustic wave number in influencing tunnelling and its reflection. Tuneable high fidelity acoustic piezoelectric transducer emitters may be used to finally tune the tunnelling. If the probability of reflectivity of phonon (P) during tunnelling through the potential barrier is given by P(Rp) and that of Acoustically Augmented Phonons(AAP) is P(RAAP) then it is analytically obtained that P(RAAP) = \(\propto\) P(RP) Where \(\propto\) is dimensionless and is defined as the Augmentation Probability Factor(APF) which is dependent on structure based damping coefficient, wave number of UHF acoustic wave and source kinetic energy. Enhancement of tunnelling probability by mechanically superimposing UHF acoustic wave over the in situ phonons is independent of initial UHF acoustic wave’s amplitude. It is noted that the range of tuneable frequencies of UHF increases with increase in the kinetic energy of phonons and gradually it reaches a limit. Thus, existence of almost zero reflectivity of tunnelling probability through the potential barrier by Acoustically Augmented Phonons (AAP) is theoretically obtained. A general observation is that to achieve very low reflectivity of phonon tunnelling probability ‘structural damping factor’ must be small enough than superimposed UHF of acoustic wave. The optimization of the tunnelling probability through a potential barrier by superimposing ultra high frequency (UHF) acoustic wave over the source of the incoming particle wave function has been examined theoretically. V0 is non-dimensionalized by natural energy unit \(\Delta\)1 as y [=\(V_o\over \Delta_1\)]. 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 \(\approx\) 3.12879, where yc is universal tunnelling constant (UTC). As the barrier height(y) is increased further ( y > yc), the reflection increases, and the tunnelling 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 \(\beta\). 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 tunnelling probability. For increasing values of the UHF wave numbers and decreasing x-values, the potential barrier becomes increasingly opaque to tunnelling. The higher the value of y is, the higher the tunnelling opacity of the potential barrier becomes. The tunnelling probability is highest (=0.98673) at y = 4, x = 0.9, when the orders of \(\beta\) and k are comparable.