Lieb lattices and pseudospin-1 dynamics under barrier- and well-like electrostatic interactions

This work considers the confining and scattering phenomena of electrons in a Lieb lattice subjected to the influence of a rectangular electrostatic barrier. In this setup, hopping amplitudes between nearest neighbors in orthogonal directions are considered different, and the next-nearest neighbor interaction describes spin-orbit coupling. This makes it possible to confine electrons and generate bound states, the exact number of which is exactly determined for null parallel momentum to the barrier. In such a case, it is proved that one even and one odd bound state is always generated, and the number of bound states increases for non-null and increasing values of the parallel momentum. That is, bound states carry current. In the scattering regime, the exact values of energy are determined where the resonant tunneling occurs. The existence of perfect tunneling energy in the form of super-Klein tunneling is proved to exist regardless of the bang gap opening. Finally, it is shown that perfect reflection appears when solutions are coupled to the intermediate flat-band solution.