# Half-Spectrum Suppression in Dynamic Resonant Tunneling
## 1 Introduction
This model was used to investigate the dependence of the current's sensitivity on the incoming particle's energy. In particular, it explains the current suppression at certain energies [22]; however, it cannot predict the energies for which the current vanishes completely. When \(\Omega=E_{n}\), the current can be suppressed by orders of magnitude but cannot reach zero. This behavior can be explained below. When the well's energy varies in time, the resonance level varies as well: \(\Omega_{R}(t)\). It has been shown that when the incoming energy solves the following equation,  
\[\frac{1}{2}\int\limits_{t_{1}}^{t_{2}}\left|\Omega-\Omega_{R}\left(t^{\prime} \right)\right|dt^{\prime}=\pi\left(n-\frac{1}{4}\right) \tag{3}\]  
where \(\Omega_{R}(t)\) is the instantaneous resonant energy, \(t_{1,2}\) are the solutions of \(\Omega_{R}(t_{1,2})=\Omega\) and \(n\) is a positive integer, the current is considerably suppressed [23]. In cases where the well's energy levels varies, like a hill, i.e., initially it increases and eventually decreases (or vice versa), Equation (3) shows the high prediction of full current suppression. In this category, one can include Gaussian, Secant and even Lorentzian perturbations. In all these cases, the resonance energy \(\Omega_{R}(t)\) crosses the incoming particle's energy \(\Omega\) exactly twice and therefore the instances \(t_{1,2}\) are well defined (see the left panel of Figure 2). However, in the case where the well oscillates harmonically, then in every period there are two extrema--one minimum and one maximum (see the right panel of Figure 2). Now,  
Figure 1: The system under study. An energetic particle tunnels through an opaque barrier with an oscillating well at its center. The red area inside the thick arrow (the well) represents the spectral width of the generated spectrum. The well oscillates with frequency \(\omega\) (red arrow). \(U\) is the barrier height, \(\Omega\) is the incoming energy, and \(\Omega_{out}\) are the outgoing energies (arrows’ length indicates different transmission probabilities). if \(\Omega=E_{n}\), then the same criterion (3) can be applied for both minima and maxima. In that case, then  
\[\frac{1}{2}\int\limits_{\tau}^{\tau+2\pi/\omega}\big{[}\Omega-\Omega_{R}\big{(}t ^{\prime}\big{)}\big{]}dt^{\prime}=\frac{\pi}{\omega}(\Omega-\langle\Omega_{R} \rangle)=n\pi \tag{4}\]  
for any \(\tau\). Therefore, if  
\[\frac{1}{2}\int\limits_{t_{1}}^{t_{2}}\big{|}\Omega-\Omega_{R}\big{(}t^{\prime }\big{)}\big{|}dt^{\prime}=\pi\bigg{(}n_{1}-\frac{1}{4}\bigg{)} \tag{5}\]  
then  
\[\frac{1}{2}\int\limits_{t_{2}}^{t_{1}+2\pi/\omega}\big{|}\Omega-\Omega_{R} \big{(}t^{\prime}\big{)}\big{|}dt^{\prime}=\pi\bigg{(}n_{2}-\frac{1}{4}\bigg{)} \tag{6}\]  
where \(n_{2}=n+n_{1}\), since, without loss of generality \(\Omega>\Omega_{R}(t)\) for \(t_{2}<t<t_{1}+2\pi/\omega\), then \(\Omega<\Omega_{R}(t)\) for \(t_{1}<t<t_{2}\),  
\[\frac{1}{2}\int\limits_{t_{2}}^{t_{1}+2\pi/\omega}\big{|}\Omega-\Omega_{R} \big{(}t^{\prime}\big{)}\big{|}dt^{\prime}=\frac{1}{2}\int\limits_{t_{1}}^{t_ {1}+2\pi/\omega}\big{[}\Omega-\Omega_{R}\big{(}t^{\prime}\big{)}\big{]}dt^{ \prime}+\frac{1}{2}\int\limits_{t_{1}}^{t_{2}}\big{|}\Omega-\Omega_{R}\big{(}t ^{\prime}\big{)}\big{|}dt^{\prime}=\pi\bigg{(}n+n_{1}-\frac{1}{4}\bigg{)} \tag{7}\]  
It should be noted that in (4) \(n\) can be any integer, while \(n_{1}\) and \(n_{2}\) must be positive integers. The same reasoning, of course, applies to high currents.