## Appendix A  
Consider the behavior of electrons transmitted over the potential barrier of rubber as a one-dimensional Schrodinger equation in the \(x\)-direction, which is the direction of compression of the MCF rubber. The process of the transmitting electric current, through a pair of metal regions of \(\gamma\) with a range of \(2a_{L}\) and of \(\gamma_{1}\) with a range of \(2a_{R}\), and one rubber region \(k\) with a range of \(b\) can be considered as a tunnel barrier problem on double barriers, and the diagram of potential energy \(V\) is shown in Figure 11[25; 29]. MCF rubber is organized as \(n\) pairs of two metal and one rubber regions. The wave function, \(\Psi\), is presented by Equation (11), where \(\hbar\) is the \(h/2\pi\), \(h\) the [PERSON]'s constant, \(m\) the mass of electron, \(V_{o}\) the potential energy at regions \(\gamma\), \(\gamma_{1}\) or \(k\), and \(\varepsilon\) the energy of the electron. \[-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\Psi}{\partial x^{2}}+V_{o}(x)\Psi(x) =\varepsilon\Psi(x) \tag{12}\]  
From Equation (12), \(\Psi\) is resolved as Equations (13) and (14), respectively, for the regions \(k\) and \(\gamma\) where \(A\), \(B\), \(C\), and \(D\) are arbitrary constants that can be defined from the continuity of differential coefficients on the boundary between \(k\) and \(\gamma\) (\(\gamma_{1}\)) regions as a well-known ordinary quantum problem, as follows:  
\[\Psi(x)=Ae^{ikx}+Be^{-ikx}; \tag{10}\]  
\[\Psi(x)=Ce^{\gamma x}+De^{-\gamma x}, \tag{11}\]  
where  
\[k=\sqrt{\frac{2m}{\hbar^{2}}}\varepsilon,\gamma=\sqrt{\frac{2m}{\hbar^{2}}(V_ {o}-\varepsilon)}. \tag{12}\]  
On the one pair of double barriers shown in Figure 11a, the arbitrary constants \(A\) and \(B\) exist in relation to constants \(E^{\prime}\) and \(F^{\prime}\) as shown in Equation (10). \[\left[ \begin{array}{c}A\\ B\end{array} \right]=M_{L}M_{W}M_{R}\left[ \begin{array}{c}E^{\prime}\\ F^{\prime}\end{array} \right], \tag{13}\]  
where  
\[M_{L}=\left[ \begin{array}{cc}m_{L11}e^{i\theta_{L11}}&m_{L12}e^{i\theta_{L12 }}\\ m_{L12}e^{-i\theta_{L12}}&m_{L11}e^{-i\theta_{L11}}\end{array} \right],\]  
\[M_{W}=\left[ \begin{array}{cc}e^{-ik_{1}b}&0\\ 0&e^{ik_{1}b}\end{array} \right], \tag{14}\]  
\[M_{R}=\left[ \begin{array}{cc}m_{R11}e^{i\theta_{R11}}&m_{R21}e^{-i\theta_{ R21}}\\ m_{R21}e^{i\theta_{L21}}&m_{R11}e^{-i\theta_{L11}}\end{array} \right]\]  
\[m_{L11}=\sqrt{\frac{1}{4}\Big{(}1+\frac{k_{1}}{k_{1}}\Big{)}^{2}\cosh^{2}(2 \gamma a_{L})+\frac{1}{4}\Big{(}\frac{k_{1}-\gamma^{2}}{k_{1}}\Big{)}^{2} \sinh^{2}(2\gamma a_{L})},\]  
\[m_{L12}=\sqrt{\frac{1}{4}\Big{(}\frac{k_{1}+\gamma^{2}}{k_{1}}\Big{)}^{2} \sinh^{2}(2\gamma a_{L})+\frac{1}{4}\Big{(}\frac{k_{1}}{k}-1\Big{)}^{2}\cosh^{ 2}(2\gamma a_{L})},\]  
\[m_{R11}=\sqrt{\frac{1}{4}\Big{(}1+\frac{k_{2}}{k_{1}}\Big{)}^{2}\cosh^{2}(2 \gamma_{1}a_{R})+\frac{1}{4}\Big{(}\frac{k_{1}k_{2}-\gamma_{1}^{2}}{k_{1} \gamma_{1}}\Big{)}^{2}\sinh^{2}(2\gamma_{1}a_{R})},\]  
\[m_{R21}=\sqrt{\frac{1}{4}\Big{(}\frac{k_{1}k_{2}+\gamma_{1}}{k_{1}\gamma_{1}} \Big{)}^{2}\sinh^{2}(2\gamma_{1}a_{R})+\frac{1}{4}\Big{(}\frac{k_{2}}{k_{1}}-1 \Big{)}^{2}\cosh^{2}(2\gamma_{1}a_{R})},\]  
\[\theta_{L_{11}}=-\tan^{-1}\Big{[}\frac{k_{1}-\gamma^{2}}{(k+k_{1})\gamma} \tanh(2\gamma a_{L})\Big{]}+(k+k_{1})a_{L}\]  
\[\theta_{L_{12}}=-\tan^{-1}\Big{[}\frac{k_{1}+\gamma^{2}}{(k-k_{1})\gamma} \tanh(2\gamma a_{L})\Big{]}+\pi+(k-k_{1})a_{L}\]  
\[\theta_{R_{11}}=-\tan^{-1}\Big{[}\frac{k_{1}k_{2}-\gamma_{1}^{2}}{(k_{1}+k_{2} )\gamma_{1}}\tanh(2\gamma_{1}a_{R})\Big{]}-(k_{1}+k_{2})a_{R}\]  
\[\theta_{R_{21}}=\tan^{-1}\Big{[}\frac{k_{1}k_{2}+\gamma_{1}^{2}}{(k_{2}-k_{1} )\gamma_{1}}\tanh(2\gamma_{1}a_{R})\Big{]}+\pi+(k_{1}-k_{2})a_{R}\]  
The transmitted probability can be given by Equation (12). \[T=\left|M_{T_{11}}\right|^{2} \tag{13}\]  
One pair of double barriers is expanded to \(n\) pairs of double barriers as multi-barriers.