# Multiple Fano Resonances in Dynamic Resonant Tunneling Processes
## 5 Weak Modulation Regime  
In the weak modulation amplitude (\(\beta\)) regime at the vicinity of \(\alpha\cong\chi_{-m}\), the determinant of (20) can be written as approximately:  
\[\left|M_{(-1)}^{(-N)}\right|=(\alpha-\chi_{-m})\left|M_{(-m-1)}^{(-N)}\right| \left|M_{(-1)}^{(-m+1)}\right|-\frac{\beta^{2}}{4}\Big{(}\left|M_{(-m-1)}^{(- N)}\right|\left|M_{(-1)}^{(-m+2)}\right|+\left|M_{(-m-2)}^{(-N)}\right| \left|M_{(-1)}^{(-m+1)}\right|\Big{)}+O\Big{(}\beta^{4}\Big{)} \tag{24}\]  
which can also be rewritten as:  
\[\left|M_{(-1)}^{(-N)}\right|\cong\prod_{m^{\prime}=1}^{N}(\alpha-\chi_{-m^{ \prime}})\left[1-\frac{\beta^{2}}{4}\frac{1}{(\alpha-\chi_{-m})}\left(\frac{ 1}{(\alpha-\chi_{-m-1})}+\frac{1}{(\alpha-\chi_{-m+1})}\right)+O\Big{(}\beta^ {4}\Big{)}\right] \tag{25}\]Therefore, the transmission vanishes (Equation (23) is valid) up to the second order in \(\beta\), provided that:  
\[1\cong\frac{\beta^{2}}{4}\frac{1}{(\alpha-\chi_{-m})}\bigg{(}\frac{1}{(\alpha- \chi_{-m-1})}+\frac{1}{(\alpha-\chi_{-m+1})}\bigg{)} \tag{26}\]  
Since \(\chi_{n}\) is a function of \(L\), one can find an expression for the relation between the zero-transmission energy \(\Omega\) and the width of the barrier \(L\):  
\[L_{m}(\Omega)\cong\frac{\text{arctanh}\left\{\frac{1}{2\rho_{-m}}\left[\frac{ \beta^{2}}{8l}\left(\frac{1}{(k_{-m}-k_{-m-1})}+\frac{1}{(k_{-m}-k_{-m+1})} \right)-\alpha\right]\right\}+\text{arctan}\left(\frac{k_{-m}}{\rho_{-m}} \right)}{\rho_{-m}} \tag{27}\]  
In the regime where the barrier is opaque, i.e., \(LU^{1/2}>>1\), Equation (27) can be approximated by:  
\[(\alpha+2\rho_{-1})\cong\frac{1}{(\alpha+2\rho_{-2})}\left(\frac{\beta}{2} \right)^{2}\text{, and when }\beta<<\alpha\text{, it can be solved for the energy:}\]  
\[\Omega_{c}\cong U+\omega-\frac{1}{4}\bigg{[}\alpha-\frac{(\beta/2)^{2}}{\alpha +2\sqrt{\omega+\alpha^{2}/4}}\bigg{]}^{2} \tag{28}\]  
Expression (27) is presented in Figures 4-7. The discrepancy between this expression and the numerical results, which is larger for low orders, is due to the fact that Equation (27) is derived for a weak \(\beta\). In the following Figures 9-12, since \(\beta\) is smaller, there is a better agreement with Equation (27).  
Figure 9: The current as a function of the barrier’s width \(L\) for \(\Omega/U=0.04\) in the second scenario regime (\(0<\Omega_{R}<U\)). The dashed red vertical lines are the solutions for zero transmission (Equation (27) for \(m=1\), \(2\)). The other parameters are: \(\alpha U^{-1/2}=-1.8\), \(\beta U^{-1/2}=0.03\), and \(\omega/U=0.1\).  
Figure 11: The current as a function of the barrier’s width \(L\) is presented for \(\Omega/U=0.04\). The vertical dashed red lines represent the analytical solution of Equation (27) for \(m=2\), \(3\). The other parameters are: \(\alpha U^{-1/2}=-2.1\), \(\beta U^{-1/2}=0.08\), and \(\omega/U=0.1\), i.e., \(\Omega_{R}<0\).  
Figure 10: The current as a function of the particle’s incoming energy \(\Omega\) and the barrier’s width \(L\) in the second scenario regime (\(0<\Omega_{R}<U\)). The lower panel is an enlargement of the upper one. The white dashed curve is the solution of Equation (5), where \(L_{0}\) is Equation (6). The dashed black curves are the solution for zero transmission (Equation (27)). The parameters are the same as in Figure 9.  
In Figures 9 and 10, the second scenario (i.e., \(0<\Omega_{R}<U\)) is presented for a weak \(\beta\). In Figures 11 and 12, the first scenario (i.e., \(\Omega_{R}<0\)) is presented for a weak \(\beta\). As can be seen in both cases, the agreement with the theoretical prediction (Equation (27)) is excellent.