Quantum tunneling through a rectangular barrier in multi-Weyl semimetals

Multi-Weyl semimetals are new types of topological semimetals whose topological charge is equal to the value of the winding number $J$. Here, we investigate the single-particle ballistic scattering on a rectangular barrier in multi-Weyl semimetals. Because this system has a crystallographic anisotropy, the scattering properties depend on the mutual orientation of the crystalline axis and the barrier. For different $J$, the vertical component of the wave vector ${k}_{\ensuremath{\perp}}$ and the corresponding probability current density ${j}_{\ensuremath{\perp}}$ satisfies ${j}_{\ensuremath{\perp}}\ensuremath{\propto}{k}_{\ensuremath{\perp}}^{2J\ensuremath{-}1}$. In the case of a barrier perpendicular to the $z$-axis, it is found that the reflectionless incident angles are determined by geometrical resonances between the barrier width and the de Broglie length of the scattered electrons in the barrier region. In the $z$-axis direction, the local minimum conductance ${G}_{\mathrm{min}}$ occurs when the chemical potential equals the barrier height and ${G}_{\mathrm{min}}\ensuremath{\propto}1/{L}^2/J$, where $L$ is the width of the barrier. Differently, in the case of a barrier perpendicular to the $x$-axis, the angular distribution of the transmission probability is no longer rotation invariant. For the double-Weyl semimetals ($J=2$), the transmission probability decreases rapidly to 0 as the barrier width $L$ increases for a normal incidence, which is similar to conventional nonrelativistic electrons. It is interesting that perfect transmission is again found for normally incident Weyl fermions for the triple-Weyl semimetals ($J=3$). In this case, the tunneling indicates a property similar to that in the case of $J=1$.