## 4 Electron Tunneling in Kekule-Patterned Graphene
Figure 3: 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 \(x=0\). All panels (**a–d**) are calculated for \(\Delta_{0}=0\) but different values of \(a\). appearance of extended CFRs, However, its dependence on \(\Delta_{0}\) is much smaller compared to that on the potential profile slope \(a\), as can be verified by Figure 5a. It is important to mention that the WKB approximation provides a fairly satisfactory estimate for the electron transmission in the cases as standard techniques are either not applicable or would lead to tremendously complicated equations. This is exactly the case for Kek-Y graphene in which similarly to tunneling of Rashba electrons, there are two non-equivalent solutions for wave vector \(k\) at a given energy [76]. However, in the case of Kek-Y graphene, we would need to solve a quartic equation so as to obtain the \(k\) value for a given energy \(\varepsilon_{0}\), and then use these expressions to find the transmission, which seems hardly possible. As mentioned above, the main application of the WKB theory in any existing material is investigating electron tunneling over various potential barriers or regions in which the potential \(V(x)\) of the incoming particle is larger than its kinetic energy so that a classical particle could not even be present in such regions. These locations are also referred to as classically forbidden or classically inaccessible regions. Another possible application of WKB is investigating electronic bound states with quantized momentum, which are rarely considered for the Dirac lattices. A general condition for applicability of WKB theory for graphene or another Dirac cone material is given in terms of the momentum \(p(x)\) as:  
\[\hbar\Big{|}\frac{dp(x)}{dx}\Big{|}\ll p^{2}(x)\,, \tag{45}\]  
which is evidently equivalent to a slow change of the amplitude of a de Broglie wavelength \(|lambda(x)\) of a particle:  
\[\frac{p(x)^{2}}{p_{x}(x)^{2}}\,\Big{|}\frac{d\lambda(x)}{dx}\Big{|}\ll 1\,. \tag{46}\]  
The change of the electron momenta is directly related to the slope of the potential energy profile, such as:  
\[\Big{|}\frac{dp(x)}{dx}\Big{|}=\frac{m}{p(x)}\,\Big{|}\frac{dV(x)}{dx}\Big{|} \tag{47}\]  
for a regular Schrodinger electron or  
\[\Big{|}\frac{dp(x)}{dx}\Big{|}=\frac{1}{v_{F}}\,\Big{|}\frac{dV(x)}{dx}\Big{|} \tag{48}\]  
for graphene, with Fermi velocity \(v_{F}\), which implies two important limitations of any WKB approximation: a sharp, sudden change of the potential \(V(x)\) and the regions in the vicinity of the turning points, where \(p(x)\to 0\). At the same time, all the results for Klein tunneling over a square barrier with sharp edges and \(V(x)=V_{0}\,\Theta(x)\,\Theta(W_{B}-x)\), where \(W_{B}\) is constant, are faithfully reproduced by WKB theory, which appears to be valid much beyond its known range of applicability. At the time, WKB methods provide a  
\[\left[\Pi^{(S)}(x)+p_{y}^{2}\right]^{2}-2\Delta_{0}^{2}\left\{\left[\Pi^{(S)} (x)\right]^{2}-p_{y}^{4}\right\}-2[\varepsilon_{0}-V(x)]^{2}\Big{[}\Pi^{(S)} (x)+p_{y}^{2}\Big{]}\Big{(}1+\Delta_{0}^{2}\Big{)}+[\varepsilon_{0}-V(x)]^{4 }=0\;. \tag{44}\]0 Transmissions amplitudes \(\mathbb{T}(p_{y}\,|\,a,\Delta_{0})\) calculated by using Equation (43). Panel (**a**) presents \(\mathbb{T}(p_{y}\,|\,a,\Delta_{0})\) as a function of transverse momentum \(p_{y}\) for \(\Delta_{0}=0\) and different slopes \(a=0.2\) and \(0.5\). Panel (**b**) displays calculated \(T(p_{y}\,|\,a,\Delta_{0})\) as a function of \(p_{y}\) for \(a=0.2\) and different energy gaps \(\Delta_{0}=0\) and \(0.5\). straightforward and reliable estimate of the electron transmission amplitudes for the cases when traditional techniques are impossible or extremely complicated, such as for our case of Kek-Y-patterned graphene.