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Applied Sciences

Multiple Fano Resonances in Dynamic Resonant Tunneling Processes

Gilad Zangwill, Er’el Granot

Published Jun 2, 2023• 2 citations
Publisher

Abstract

The existence of Fano resonances in dynamic resonant tunneling (RT) systems has been investigated. Fano resonances are characterized by the appearance of a 100% reflection coefficient in proximity to a high transmission coefficient. For a Fano resonance to appear, a bound state must exist. On the other hand, a resonant tunneling process is characterized by a high transmission and the existence of a quasi-bound state (QBS) instead of a bound one. It has been shown that, by narrowing the width of the barrier, the resonance energy of the QBS gradually decreases and eventually turns into a bound state. Consequently, in a dynamic RT process, there are two scenarios: either a bound state exists, in which case, Fano resonances exist for any barrier width, or a QBS exists, and the barrier should be narrow enough for the Fano resonance to appear. In both cases, the incoming particle’s frequency must be lower than the oscillating well’s frequency. In this work, these resonances are investigated in detail, and both exactly numerically and approximated analytical expressions are derived for both the weak and strong oscillating amplitude regimes. One of the conclusions is that, when the oscillating frequency is low enough, multiple Fano resonances can appear by varying the barrier’s width. Since these resonances are very sharp and zero transmission can easily be detected, this property can be used as a very accurate method for measuring the barrier’s width, even when the particle’s de-Broglie wavelength is much larger than the barrier’s width.

DOI
10.3390/app13116767

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The existence of Fano resonances in dynamic resonant tunneling (RT) systems has been investigated. Fano resonances are characterized by the appearance of a 100% reflection coefficient in proximity to a high transmission coefficient. For a Fano resonance to appear, a bound state must exist. On the other hand, a resonant tunneling process is characterized by a high transmission and the existence of a quasi-bound state (QBS) instead of a bound one. It has been shown that, by narrowing the width of the barrier, the resonance energy of the QBS gradually decreases and eventually turns into a bound state. Consequently, in a dynamic RT process, there are two scenarios: either a bound state exists, in which case, Fano resonances exist for any barrier width, or a QBS exists, and the barrier should be narrow enough for the Fano resonance to appear. In both cases, the incoming particle’s frequency must be lower than the oscillating well’s frequency. In this work, these resonances are investigated in detail, and both exactly numerically and approximated analytical expressions are derived for both the weak and strong oscillating amplitude regimes. One of the conclusions is that, when the oscillating frequency is low enough, multiple Fano resonances can appear by varying the barrier’s width. Since these resonances are very sharp and zero transmission can easily be detected, this property can be used as a very accurate method for measuring the barrier’s width, even when the particle’s de-Broglie wavelength is much larger than the barrier’s width.
We consider the analytically solvable model of a Gaussian pulse tunneling through a transmission resonance with a general Fano characteristic. It is demonstrated that the transmitted pulse contains enough information to determine uniquely all parameters defining the Fano resonance. This is in contrast to the measurement of the static conductance. Our analytical model is in agreement with numerical data published recently for the limit of a Breit-Wigner resonance. We identify two opposite pulse propagation regimes: if the resonance is broad compared to the energetic width of the incident Gaussian pulse a weakly deformed and slightly delayed transmitted Gaussian pulse is found. In the opposite limit of a narrow resonance the dying out of the transmitted pulse is dominated by the slow exponential decay characteristic of a quasi-bound state with a long life time. In this regime we find characteristic interference oscillations.
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.
The quantum tunneling current via an opaque barrier with an oscillating well reveals a wealth of physical phenomena: eigenstate-assisted activation, the elevator effect, coherent destruction of tunneling, suppression of activation, and the Sisyphus effect are a few examples. In this paper, we investigate these effects from a different perspective---transmission via a quasibound super state (QBSS). It is shown that an oscillating well supports a QBSS, which consists of numerous quasibound substates. Each one of these substates has a finite spectral width, which corresponds to the escape probability. However, they construct a unique spectrum, which consists of activated and suppressed quasi substates all of which are simultaneously excited. Thus, when the oscillating well is integrated into an opaque barrier, one can borrow an analogy from stationary resonant tunneling (RT). In the stationary RT scenario, current flows via a quasibound state. In the oscillating RT scenario, current flows via a QBSS. This analogy can easily explain many of the system's complex behaviors: the symmetry between the current's sensitivity to the incoming energy and the outgoing one, and even why some frequencies induce activation while others suppress it. This analogy can be applied to improve the sensitivity of the system when used for a frequency-controlled transistor for it predicts that when the incoming energy is shifted from the central resonance of the QBSS, the device's current becomes exponentially sensitive to the applied frequency. While the oscillations' frequency mainly determines the spectral distance between substates, the oscillations' amplitude determines the spectral width and center of the QBSS. Furthermore, it is suggested that this analogy can be applied to investigate microbiological systems (the olfactory system) and optical devices.
We consider the analytically solvable model of a Gaussian pulse tunneling through a transmission resonance with a general Fano characteristic. It is demonstrated that the transmitted pulse contains enough information to determine uniquely all parameters defining the Fano resonance. This is in contrast to the measurement of the static conductance. Our analytical model is in agreement with numerical data published recently for the limit of a Breit-Wigner resonance. We identify two opposite pulse propagation regimes: If the resonance is broad compared to the energetic width of the incident Gaussian pulse a weakly deformed and slightly delayed transmitted Gaussian pulse is found. In the opposite limit of a narrow resonance the dying out of the transmitted pulse is dominated by the slow exponential decay characteristic of a quasibound state with a long lifetime. In this regime we find characteristic interference oscillations.
The physics of dynamic resonant tunneling is investigated.First, the resonant tunneling effect through an opaque barrier via a delta-function well is illustrated.Then, it is shown that, even in the adiabatic regime, where the dynamics can be governed by an analytic solution, the particle can be activated to higher energies.If the well varies quickly enough that the particle cannot escape from the well during the energetic elevation, the activation can be enhanced, as was anticipated by Azbel.However, and this is the main result of this work, the quasi-bound state of the well can even "reduce" the activation.In fact, because the resonant energy of the well matches twice the incoming particle's energy, and if the contribution to the wave function from both parts destructively interferes, then the particle cannot dwell in the well and activation is suppressed.This effect can be utilized in frequency-controlled transistors, and it is even speculated that it may explain the reason that humans can distinguish between tens of thousands of different odors with merely few hundreds of odor receptors.Lastly, the short time dynamics of a very fast perturbative well is also discussed.