On the 2D quantum tunneling

In this paper, we have solved a quantum tunneling problem for 2-dimensional systems, including electron gas and graphene. In spite of the one-dimensional scattering problems, in two dimensions, we observe phenomenon of tunneling at energies above the barrier. This effect is an analogue to the total internal reflection in optics. The scattering amplitudes inside the barrier region exhibit decaying behavior corresponding to optical evanescent-wave coupling, not only in energies below barrier height, but also above barrier. Velocity-selecting transmission, corresponding to angle-resolved beam filtering effect is one of the achievements of the paper. The famous Hartman effect which occurs normally at sub-barrier energies and has previously been studied for graphene is also addressed. The results manifest occurrence of the Hartman effect for over-barrier energies, as well.