## 4 Electron Tunneling in Kekule-Patterned Graphene  
The primary application of the obtained WKB approximation is a good numerical estimate of the electron transmission amplitude for a fast-moving electron. This method is easily applicable to the cases of non-square potential barriers and a wide class of potential profiles as long as this profile \(V(x)\) is smooth and there is no abrupt change in the particle's momentum \(p(x)\). At the same time, WKB calculations were used to predict or verify the existence of the Klein paradox for a normal incident particle on a square barrier in graphene and other Dirac cone materials, i.e., they could be formally applied to a square barrier as well. The transmission amplitude is defined as the ratio of the electron current passing through the barrier to the incoming current. Since the total current in the direction perpendicular to a potential barrier is conserved, the reflect current could also be easily calculated as the difference between them. The biggest loss of a transmitted current occurs in the classically forbidden regions (CFR), in which the longitudinal momentum \(p_{x}(x)\) becomes purely imaginary and the moving particle is represented by an evanescent wave, whose amplitude is exponentially decreasing along the \(x\)-axis. The transmission amplitude \(\mathbb{T}(p_{y}\,|\,a,\Delta_{0})\) in the WKB theory is estimated by the integral of the purely imaginary momentum \(p_{x}(x)\) over each of the classically forbidden regions along the \(x\)-direction. This results in the following equation:  
\[\mathbb{T}(p_{y}\,|\,a,\Delta_{0})=\exp\left[-\frac{2}{\hbar}\int\limits_{ \text{CFR}}|p_{x}(x)|\,dx\right] \tag{43}\]  
as long as the condition \(\int\limits_{\text{CFR}}|p_{x}(x)|\,dx\gg\hbar\) is satisfied. Based on Equation (31), squared longitudinal momentum \(\Pi^{(S)}(x)=p_{x}^{2}(x)\) is determined from:  
\[\left[\Pi^{(S)}(x)+p_{y}^{2}\right]^{2}-2\Delta_{0}^{2}\left\{\left[\Pi^{(S)} (x)\right]^{2}-p_{y}^{4}\right\}-2[\varepsilon_{0}-V(x)]^{2}\Big{[}\Pi^{(S)} (x)+p_{y}^{2}\Big{]}\Big{(}1+\Delta_{0}^{2}\Big{)}+[\varepsilon_{0}-V(x)]^{4 }=0\;. \tag{44}\]  
The calculated locations of CFRs are presented in Figure 3 for \(\Delta_{0}=0\) and in Figure 4 for \(\Delta_{0}\
eq 0\). We see that the former case is pretty much equivalent to graphene and the boundaries of the CFR's are linear and symmetric with respect to \(x=0\). The width of the classically inaccessible region along the \(x\)-axis obviously depends on the slope of the potential profile, i.e., a higher slope \(a\) corresponds to reduced width of a CFR. When \(\Delta_{0}\
eq 0\), the equation with classically forbidden regions is much more complicated and the region itself is extended in a non-trivial way, as seen from Figure 4. However, the boundaries of CFR are still linear and symmetric with respect to \(x=0\). This is in stark contrast with gapped \(\alpha\)-\(\mathcal{T}_{3}\) materials in which the classically inaccessible regions are very complicated, asymmetric and even involve curved boundaries. Our results for the transmission are presented in Figure 5. First, we find that the expansion of the classical and accessible region leads to lower transmission so that it is more suppressed for a smaller slope \(a\) in the potential profile \(V(x)\), as found in Figure 5a. Similarly, the suppression of transmission \(\mathbb{T}(p_{y}\,|\,a,\Delta_{0})\) increases for a finite \(\Delta_{0}\) due to the  
Figure 4: Regions for the classically forbidden (or classically inaccessible regions) with \(\mathrm{Im}[p_{x}(x)]\
eq 0\) for an incoming electron with a linear potential \(V(x)=ax\). Here, we select a crossing point, corresponding to the electron-to-hole (and back) state transition \(\varepsilon_{Y}(s_{1},s_{2}\,|\,k,\Delta_{0})=0\) at \(x=0\). Panels (**a–d**) are calculated with \(a=1\) and various finite values of \(\Delta_{0}\).