Tunable valley polarization by a gate voltage when an electron tunnels through multiple line defects in graphene

By means of an appropriate wave function connection condition, we study the electronic structure of a line defect superlattice of graphene with the Dirac equation method. We obtain the analytical dispersion relation, which can simulate well the tight-binding numerical result about the band structure of the superlattice. Then, we generalize this theoretical method to study the electronic transmission through a potential barrier where multiple line defects are periodically patterned. We find that there exists a critical incident angle which restricts the electronic transmission through multiple line defects within a specific incident angle range. The critical angle depends sensitively on the potential barrier height, which can be modulated by a gate voltage. As a result, non-trivial transmissions of K and K' valley electrons are restricted, respectively, in two distinct ranges of the incident angle. Our theoretical result demonstrates that a gate voltage can act as a feasible measure to tune the valley polarization when electrons tunnel through multiple line defects.