Applications of the WKB Equations to Dirac Materials

We have derived the key WKB relations and the wave function for graphene, \(\alpha -\mathcal {T}_3\) materials, dice lattice, and some others. Now it is crucial to understand how all these results could be applied to studying the electronic properties and phenomena in those materials. There might be two major applications of the WKB theory: finding the quantized electron momentum for the bound or localized states of a semiclassical particle Semiclassical particle and calculating the tunneling probabilities for various types of potential barriers. While considering bound states in graphene is rather uncommon (Gupta and Sen, Mod Phys Lett A 24:99–107, 2009), electron tunneling remains one of the central research subjects in all Dirac materials mainly because of the so-called Klein paradox: Klein paradox a complete electron transmission through a square barrier or a step for the direct incidence. We will also apply the WKB method to investigating the electron tunneling for nonuniform potentials, Landau–Zener tunneling Landau–Zener tunneling, and the tunneling in gapped \(\alpha -\mathcal {T}_3\) materials.