# Multiple Fano Resonances in Dynamic Resonant Tunneling Processes
## 3 Total Reflection via Narrow Barriers
In Figure 5, the current is presented as a function of the particle's incoming energy \(\Omega\) and the barrier's width \(L\) for the same parameters as in Figure 4 (except \(\Omega\)). The red color represents a high current, while the blue color represents a low current. As can be seen in Figures 4 and 5, when the incident particle's energy is lower than the oscillating energy \(\Omega<\omega\), multiple solutions of zero current exist, i.e., _several_ Fano resonances appear for different values of \(L\). When the current is presented as a function of both \(L\) and \(\Omega\), the full structure of these Fano resonances appears. The narrower the barrier, the thinner the Fano resonances, and the lower the oscillation amplitude, the narrower the Fano resonances. Therefore, high-order resonances are very narrow and hard to detect. The zero currents are presented by the multiple curved lines. The zero-current curve (ZCC) on the right, i.e., for \(LU^{1/2}>>1\), approximately agrees with Equation (5) for \(L(\Omega-3\omega)\). It should be noted that, since during the dynamic process, the instantaneous resonance energy varies in time, the resonance energy (Equation (4)) should be replaced with the minimum resonance energy, i.e.,  
\[\Omega_{R}\cong U-(|\alpha|+\beta)^{2}/4 \tag{16}\]  
\[\varphi_{n}^{+}\rightarrow\tau_{\Omega_{n}}\exp[ik_{n}x-i\Omega_{n}t]\ (for\,x \rightarrow\infty) \tag{9}\]0 The current (Equation (15)) as a function of the barrier’s width \(L\) is presented for \(\Omega/U=0.04\). The vertical red lines represent the analytical solution of Equation (27) for \(m=2\), \(3\dots\) 8. The other parameters are: \(\alpha U^{-1/2}=-1.8\), \(\beta U^{-1/2}=0.4\), and \(\omega/U=0.1\). This simulation is within the regime \(\Omega_{R}<0\) (Equation (16)). Moreover, the other curves approximately agree with \(L(\Omega-m\omega)\) for \(m=3,4\ldots\). That is, these ZCCs can be regarded approximately as a folded curve. The curves cannot exceed the oscillating frequency \(\Omega<\omega\). The number of ZCCs (\(N\)) is finite, which can be evaluated by dividing the bare well's (i.e., without the barrier) depth by the oscillating frequency \(\omega\), i.e., \(N\cong\alpha^{2}/4\omega\). In the case presented in Figures 4 and 5, indeed, \(N\cong 8\). Equation (5) can be used to distinguish between two scenarios. There are two regimes in which \(L\) has a real value. In the first scenario, the REOB is negative, \(\Omega_{R}<0\) (Equation (16)), in which case, a real solution exists, provided \(\Omega_{R}-U<\Omega-m\omega<\Omega_{R}\). In this scenario, the Fano resonance exists even for very opaque barriers, i.e., \(L\) can be arbitrarily large. In the second scenario, the REOB is positive, \(0<\Omega_{R}<U\), in which case, a real solution exists, provided \(\Omega_{R}-U<\Omega-m\omega<0\). In this scenario, Fano resonance exists only for finite-width barriers. Specifically, \(L\) must be narrower than \(L_{0}\) (Equation (6)). The second scenario is presented in Figures 6 and 7. It can clearly be shown that, unless the barrier's width is narrower than a certain cut-off value, no Fano resonances appear. In particular, this scenario does not support a Fano resonance (and hence no zero transmission) in the opaque barrier regime. The discrepancy between the cut-off value of the barrier's width and \(L_{0}\) (Equation (6)) is due to the fact that Equation (6) was calculated assuming \(\beta=0\). It will be shown that, when \(\beta\) is weak, there is an excellent agreement with Equation (6). \[\varphi_{n}^{+}\rightarrow\tau_{\Omega_{n}}\exp[ik_{n}x-i\Omega_{n}t]\ (for\,x \rightarrow\infty) \tag{9}\]1 The current (in logarithmic scale) as a function of the particle’s incoming energy \(\Omega\) and the barrier’s width \(L\). Red color represents high current while blue represents low current. The analytical results are presented on the lower panel.