these algebraic equations numerically by a systematic truncation method described in Ref. 41.

In Model II, $\delta u(x)$ and $\delta a(x)$ are not independent, but proportional to each other. This condition leads to completely different equations for $Z_{n}$ and $\langle \ln T\rangle$. The invariant imbedding equation for $Z_{n}$ in this case is written as $$\begin{aligned}
&&\frac{1}{k}\frac{dZ_n}{dl}=\Big\{2in\left(\cos\theta-u_0\sec\theta-a_0\tan\theta\right)\nonumber\\&&~~
\left. +g_u n^2\left[1-f^2-3\left(\sec{\theta}+f\tan{\theta}\right)^2\right]\right\}Z_n\nonumber\\
&&~~-ne^{i\theta}\left[C_0-i(2n+1)D_2\right]Z_{n+1}\nonumber\\
&&~~+ne^{-i\theta}\left[C_0-i(2n-1)D_2\right]Z_{n-1}\nonumber\\
&&~~+\frac{1}{2}n(n+1)e^{2i\theta}D_1Z_{n+2}\nonumber\\
&&~~+\frac{1}{2}n(n-1)e^{-2i\theta}D_1Z_{n-2},
\label{eq:iie3}\end{aligned}$$ where $D_1$ and $D_2$ are defined by $$\begin{aligned}
D_1&=&g_u\left(\tan{\theta}+f\sec{\theta}\right)^2,\nonumber\\
D_2&=&g_u\left(\sec{\theta}+f\tan{\theta}\right)\left(\tan\theta+f\sec\theta\right).\end{aligned}$$ The equation for the localization length takes the form $$\begin{aligned}
&&\frac{1}{k\xi}=-\frac{1}{k}\lim_{l\rightarrow\infty}\left(\frac{{d\langle \ln T\rangle}}{dl}\right)\nonumber\\
&&~~~~=D_1+{\rm Re}\left[2e^{i\theta}\left(C_0
- iD_2\right)Z_1\left(l\rightarrow\infty\right)\right.\nonumber\\&&\left. ~~~~~~~-e^{2i\theta}D_1Z_2\left(l\rightarrow\infty\right)\right].
\label{eq:iie4}\end{aligned}$$

Interpretation of Klein tunneling and related delocalization phenomena using the concept of wave impedance {#sec:impedance}
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Effective wave impedance
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Klein tunneling originally refers to the phenomenon that Dirac fermions entering a large potential barrier are transmitted almost completely as the barrier becomes higher and wider.[@klein] In the case of massless Dirac fermions such as the electrons in monolayer graphene, the transmission is perfect regardless of the strength and the shape of the potential when electrons are incident normally on it.[@kats2] Such perfect transmission occurs even in the case where the potential is a 1D random function of the position. This unique delocalization phenomenon, which we call 