Oblique and asymmetric Klein tunneling across smooth NP junctions or NPN junctions in 8-Pmmn borophene

The tunneling of electrons and holes in quantum structures plays a crucial role in studying the transport properties of materials and the related devices. 8-Pmmn borophene is a new two-dimensional Dirac material, which hosts tilted Dirac cone and chiral, anisotropic massless Dirac fermions. We develop the transfer matrix method to investigate the Klein tunneling of massless fermions across the smooth NP junctions and NPN junctions of 8-Pmmn borophene. Like the sharp NP junctions of 8-Pmmn borophene, the tilted Dirac cones induce the oblique Klein tunneling. The angle of perfect transmission to the normal incidence is 20.4 degrees, a constant determined by the Hamiltonian of 8-Pmmn borophene. For the NPN junction, there are branches of the Klein tunneling in the phase diagram. We find that the asymmetric Klein tunneling is induced by the chirality and anisotropy of the carriers. Furthermore, we show the oscillation of electrical resistance related to the Klein tunneling in the NPN junctions. One may analyze the pattern of electrical resistance and verify the existence of asymmetric Klein tunneling experimentally.