Electronic transmission through p-n and n-p-n junctions of graphene

In this paper, we first evaluate the electronic transmission of Dirac fermions into a p-n junction of gapped graphene and show that the final result depends on the sign of the refractive index, n. We also, by considering the appropriate wavefunctions in the region of the electrostatic potential, show that both transmission and the reflection probability turn out to be positive and less than unity instead of the negative transmission and higher than unity reflection coefficient commonly referred to as the Klein paradox. We then obtain the transmission probability corresponding to a special p-n junction for which there exists a region in which the low energy excitations of graphene acquire a finite mass and, interestingly, find that in this case the transmission is independent of the index of refraction, in contrast with the corresponding result for gapped graphene. We then discuss the validity of the solutions reported in some of the papers cited in this work which, considering the Büttiker formula, turn out to lead to the wrong results for conductivity.