## 1 Introduction
Mathematically, the WKB approximation is regarded as a solution technique for a differential equation with the highest-order derivative term multiplied by a small parameter. In quantum physics, this small parameter is just [PERSON]'s constant \(\hbar\), so that the solution for a WKB wave function can be sought as a power series of \(\hbar\). However, the WKB for Dirac electrons in graphene is found drastically different from that in the case of a regular Schrodinger electron, and therefore, we expect that the WKB solution for each novel Dirac material will be quite unique compared with all other known solutions. Moreover, the sought wave function for a new material will usually appear as a quickly oscillating function, which is further modulated by a much slower variation from an external potential \(V(x)\). Figure 1: Atomic structure of Kekulé-distorted graphene: honeycomb lattices of graphene with additional Kek-Y bond texture due to the presence of a substrate. Among multiple applications of the WKB theory, we would also like to calculate electron transmission, or tunneling probability, and investigate the various bound states of an electronic system [44; 45; 47]. In contrast to previous studies on tunneling and transport of electrons in Kek-Y graphene [48; 49] through complicated numerical computations, WKB methods, on the other hand, provide reliable estimates based on closed-form analytical expressions, which can be easily used to interpret experimental measurements on electron transport. Studying electron tunneling, often associated with calculating a transmission amplitude, becomes one of the most important issues for each of the newly discovered Dirac materials, mostly because of the so-called Klein paradox, i.e., a full tunneling of chiral Dirac electrons through a square potential barrier, independent of its width and height if the incoming electron moves in the direction perpendicular to the barrier boundary [12; 50; 51; 52]. For a finite-angle collision, however, a complete transmission also appears for an oblique (not a normal) incidence due to presence of Fabry-Perot resonances. Such a Klein-tunneling behavior was investigated for all Dirac-cone materials and nano-ribbons [49; 53; 54], including transport in Kekule-distorted graphene nanoribbons [49]. Specifically, a magic transmission was demonstrated for \(\alpha\)-\(7_{3}\) materials and dice lattice [55; 56; 57; 58], in addition to the unusual tunneling behavior of anisotropic Dirac electrons [59; 60; 61]. The unique properties of electron tunneling and the existence of the Klein paradox will depend on a few factors, and the most crucial one is the energy bandgap between the valence and conduction bands [62; 63; 64; 65; 66]. The two-dimensional materials with a gap do not show Klein tunneling. Such a bandgap in graphene and other Dirac lattices could be created by applying an off-resonance circularly-polarized dressing field, which also turns these material into Haldane-like Chern insulators [67; 68]. However, linearly polarized irradiation normally does not affect the gap, although it can induce or modify the existing anisotropy [69; 70]. Interestingly, the effects of these external fields with different polarizations could further mix with anisotropic and tilted energy dispersions of lattices, such as \(1\)T\({}^{\prime}\)MoS\({}_{2}\)[71; 72; 73]. One of the most crucial applications of WKB equations is investigating the electron tunneling for non-trivial potential barriers, such as trapezoidal barriers [74] or linearly increasing potentials without a sharp boundary. It was demonstrated for all previously known Dirac materials that the Klein paradox persists in such potential profiles. Verification of this property for Kekule-patterned graphene is one of the goals of the present paper. Since Dirac electrons moving under a linearly increasing potential can acquire a very large kinetic energy, applying WKB approximation to such a case seems to be the most adequate and efficient approach. The remaining part of our paper is organized as follows.