# Multiple Fano Resonances in Dynamic Resonant Tunneling Processes
## 2 The Transition from a Resonant State to a Bound State and Its Dependence on the Barrier's Width  
In a stationary Resonant Tunneling process, particles tunnel through an opaque barrier via a quasi-bound state (QBS). A QBS can be formed by locating a potential well within a barrier, as described by the following stationary Schrodinger equation:  
\[\left[-\frac{\partial^{2}}{\partial x^{2}}+U(x)-\Omega+\alpha\delta(x)\right] \psi(x)=0 \tag{1}\]  
where \(U(x)=\left\{ \begin{array}{ll}U-L<x<L\\ 0&else\end{array} \right.\) is the potential barrier and the potential well is described by a delta function potential. Hereinafter, we use, for simplicity, but without a loss of generality, the units \(\hbar=1\) and \(2m=1\) for the Planck constant and the electron's mass, respectively. The system schematic is presented in Figure 1.  
Figure 1: Particles with energy \(\Omega\) propagate through a rectangular potential barrier with a delta function potential well at its center. The barrier height is \(U\) and its width is \(2L\). It should be noted that a delta function potential is an excellent approximation for a narrow well, as long as the width of the well is narrower than the De-Broglie wavelength of the QBS [32]. The solution of Equation (1) is:  
\[\psi(x)=\psi_{h}(x)+\frac{\alpha\psi_{h}(0)}{1-\alpha G_{\Omega}^{+}(0)}G_{\Omega }^{+}(x>L) \tag{2}\]  
where \(\psi_{h}(x)\) is the homogeneous solution (without the well) and \(G_{\Omega}^{+}(x)\) is the outgoing Green function of Equation (1) without the well. For a rectangular barrier, the Green function at \(x=0\) is [11; 33]:  
\[G_{\Omega}^{+}(0)=-\frac{1}{2\rho}\coth\left(\rho L-i\arctan\left(\frac{k}{ \rho}\right)\right) \tag{3}\]  
where \(k\equiv\sqrt{\Omega}\) and \(\rho\equiv\sqrt{U-\Omega}\). In the case of an extremely opaque barrier, i.e., \(L\sqrt{U}\gg 1\), the Green function at \(x=0\) is approximately \(G_{\Omega}^{+}(0)\cong-1/2\rho\) and the Resonance Energy of the Opaque Barrier (REOB) is simply:  
\[\Omega_{R}(L\rightarrow\infty)\equiv U-\frac{\alpha^{2}}{4} \tag{4}\]  
When the incoming particle's energy is equal to this resonance energy, the particle will tunnel through the barrier via the QBS with a high probability. Equation (4) is derived for a completely opaque barrier. In general, the resonance energy is a function of the barrier's parameters (potential height \(U\) and width \(L\)). In particular, the resonance energy decreases when the barrier becomes narrower. The relation between the resonance energy and the barrier's width can be derived from the denominator of Equation (2) (the real part determines the energy and the imaginary part determines its spectral width), i.e., the energy that solves \(1-\alpha G_{\Omega}^{+}(0)=0\) is the resonance energy. In solving for \(L(\Omega_{R})\) to obtain a generic expression for the barrier's width as a function of the resonance energy, we find:  
\[L(\Omega_{R})=-\frac{1}{\rho_{R}}\left(\text{arctanh}\left[\frac{\alpha}{2\rho _{R}}\right]+\text{arctanh}\left(-i\frac{k_{R}}{\rho_{R}}\right)\right) \tag{5}\]  
where \(k_{R}\equiv\sqrt{\Omega_{R}}\) and \(\rho_{R}\equiv\sqrt{U-\Omega_{R}}\). Equation (5) shows that, for \(\Omega_{R}=U-\alpha^{2}/4\), the barrier's width is infinite; however, for any \(\Omega_{R}<U-\alpha^{2}/4\), there is a barrier with a certain width, which supports this resonance energy, and the lower the resonance energy, the narrower the barrier must be. In particular, at a certain width, the barrier's resonance energy is zero:  
\[L_{0}\equiv L(\Omega_{R}=0)=U^{-1/2}\text{arctanh}\left(|\alpha|/2\sqrt{U}\right) \tag{6}\]  
Below this value, i.e., for \(L<L_{0}\), the resonance energy becomes negative and the resonance state turns into a bound state. In Figure 2, the transmission probability (a contour presentation) is presented as a function of both the width of the barrier \(L\) and the particle's energy \(\Omega\). Equation (5) is represented by a red dashed curve. As can be seen, the resonance energy presented by Equation (5) agrees with a high transmission coefficient.