Application of the WKB Theory to Investigate Electron Tunneling in Kek-Y Graphene

In this paper, we have constructed a WKB approximation for graphene having a Y-shaped Kekule lattice distortion and a special folding of the K and K' valleys, which leads to very specific linear energy dispersions with two non-equivalent pairs of subbands. These obtained semi-classical results, which include the action, electron momentum and wave functions, are utilized to analyze the dynamics of electron tunneling through non-square potential barriers. In particular, we explore resonant scattering of an electron by a potential barrier built on Kekule-distorted graphene. Mathematically, a group of consecutive equations for a semi-classical action have been solved by following a perturbation approach under the condition of small strain-induced coupling parameter Delta(0)<< 1 (a good fit to its actual value Delta(0) similar to 0.1). Specifically, we consider a generalized model for Kek-Y graphene with two arbitrary Fermi velocities. The dependence of the electron transmission amplitude on the potential profile V(x) and band parameters of Kekule-patterned graphene has been explored and analyzed in detail.