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The tunnelling current through an oscillating resonance level is thoroughly investigated exactly numerically and with several approximations—analytically. It is shown that while the oscillations can increase the tunnelling current (and in several cases the increase is exponentially large), their main effect is to reduce it dramatically at certain energies. In fact, the current in the presence of the oscillations cannot increase the maximum current of the adiabatic solution. That is why, while the elevator effect does occur in this system, the Sisyphus effect is the more dominant and prominent one.

Resonant tunnelling (RT) is a surprising quantum effect despite being common in quantum heterostructures [

But it was then realized that dynamic resonant tunnelling is even more interesting. For example, when the perturbation’s (the changing well) time-scale is shorter than the quasi-bound-state’s life-time, the particle can be trapped inside the well, at the resonance level. As a result, when the potential well changes, the particle can be lifted energetically. This process was termed eigenstate assisted activation (EAA) and the elevator effect (EE) by Azbel [

It is the object of this paper to investigate the current (and not only the activation energy) dependence on the resonance energy and oscillation’s frequency and to show that current suppression is the main effect of the varying eigenstate. The largest current increase occurs when the incoming particle’s energy is lower than the minimum resonance energy level. But even then the activated current is always

The system is presented in Figure

Schematic illustration of the system’s dynamics. For most energies activation occurs; that is,

The system’s Schrödinger equation is then

Let

Given the initial energy of the incoming particle is

The stationary state solution for (

The maximum average current (

Below and above the resonance, the current can be approximated by the following.

Below the resonance value, that is, for

Beyond the resonance, that is, for

The adiabatic approximation (solid curve) and its approximations (the dotted and dashed curves). The lower panel is a zoom-in of the transition zone. The simulations parameters were

These approximations are very useful, especially for the lower values of the amplitude, since it is independent of the exact shape of the barrier and even independent of its width.

In Figure

The exact numerical solution of the current (solid curve) and the adiabatic approximation (dotted curve). The simulations parameters were

In the weak modulation regime, that is,

Beyond the transition threshold

This fact suggests a peculiar behaviour that an EAA occurs mainly when the incoming particle’s energy

When the oscillations amplitude increases beyond the resonance level, that is, when the minimum value of the eigenstate energy is lower than the incoming energy, then the quasi-eigenstate mostly

It was shown [

Using the same logic, the maxima of the current occur for

Since, beyond the resonance value

It is also of interest to mention that the smaller the oscillating frequency

The deviation from the adiabatic approximation occurs when the oscillation amplitude reaches the quasi-resonance

The energies for which constructive (solid horizontal lines) and destructive interference (dashed horizontal lines) occur. The curve stands for

From this reasoning it is possible to formulate an approximation for the frequency dependence of the mean current. The main contributions to the wave function (and therefore, for the current) come from the points, in which the incoming energy

Therefore,

The dependence of the mean current on the oscillating frequency. The solid line represents the exact numerical solution, while the dashed (red) curve stands for the approximation. Equation (

The process, in which the solution converges to the adiabatic one, is illustrated in Figure

The average current as a function of the oscillation’s amplitude for different oscillation’s frequencies (same numerical parameters as in Figure

Similarly, the oscillations frequencies, for which activation is suppressed, are ^{−4} in the case presented in Figure

The ratio between the mean current and the stationary current as a function of the perturbation frequency in linear (b) and logarithmic (a) scales. The parameters are

Due to the sensitivity of the current on the oscillating frequency, it is natural to identify such processes in microscopic tunnelling structure, such as odour receptors (see, e.g., [

However, current nanoscopic electronics allow fabricating such devices, where the current is controlled by the bias frequency. A possible realization of this device is a semiconductors heterostructure, where AlGaAs and GaAs are used alternately for the wells and the conductors/well. When the aluminium mole fraction is about 0.4, then the barrier height (between the two materials) is approximately 0.4 eV (see, e.g., [

Thus, an approximately 1 mV variation in the amplitude of the oscillating voltage will increase/decrease the current by at least a factor of

These performances suggest that such a device can be used as a frequency effect transistor. These devices can be much more accurate than ordinary transistors since frequency is a parameter, which can be controlled with great precision (much greater than voltage, e.g.).

The current through an opaque barrier with an oscillating well was calculated both exactly numerically and approximately analytically for different regimes. In particular the exact solution was compared to the adiabatic solution. The main conclusions are as follows.

Despite the fact that the adiabatic analysis neglects activation processes, the adiabatic approximation is a good evaluation of the

When the incoming energy crosses the resonance energy, the incoming particle can be trapped in the well and then activated to higher energies (EE); however, when it comes to the current, the increase is relatively small. The main effect of the eigenbound state is current reduction when there is a destructive interference inside the well (the Sisyphus effect).

In general, the dependence of the mean current on the oscillation amplitude

These results suggest that activation is not an optimal method to increase the mean current; however, they do show that the current can easily be controlled by changes in the frequency and therefore may be used in frequency effect devices [

The author declares that there are no conflicts of interest regarding the publication of this paper.

The author would like to thank Chene Tradonsky for helping with the derivation of the mean adiabatic value (