Exploring Electron Tunneling in Kek-Y Patterned Graphene Using the WKB Theory

We have developed a comprehensive semi-classical WKB theory for graphene with a recently-discovered Y-shaped Kekulé (Kek-Y) distortion pattern and a unique folding of the K and K' valleys. This lattice distortion results in a highly specific linear energy dispersion characterized by two non-equivalent Dirac cones, each described by different Fermi velocities, v_F(1 ± Δ₀). We derive the semi-classical action, electron momentum, and wave functions to analyze electron tunneling dynamics and resonant scattering through non-square potential barriers. Additionally, we have formulated and solved a set of transport equations that connect successive pairs of our model wavefunctions using a perturbative approach, assuming a small, strain-induced coupling parameter, Δ₀. These equations enable us to determine electron transmission amplitudes into regions that are classically inaccessible to electrons. We have conducted a detailed exploration of the dependence of electron transmission amplitudes on the potential and the band parameters of Kek-Y-patterned graphene. Our findings have practical applications in the development and operation of the next generation of opto-electronic and valleytronics devices, as the derived electron transmission amplitudes directly impact quantum transport in this model material.