Semiclassical Approximation for Graphene

In this chapter, which is one of the central parts of the book, we develop the WKB theory for graphene and a few more similar materials, such as gapped graphene, silicene, and germanene, and, finally, the so-called Kekule-patterned (Kek-Y) graphene. We will go through a rigorous derivation of all the relevant WKB equations for Dirac materials, obtained by looking for the unknown spinor wave function as a power series over the Planck’s constant \(\hbar \) . Our derivations and the actual equations for each Dirac cone material are quite distinct from the case of the Schrödinger equation, discussed in Chaps. 3 and 4 . Our WKB results for graphene include a semiclassical action, which determines the type of the quantum state of our particle, the x-dependent longitudinal momentum \(P_x(x)\) , leading-order (zeroth-order) wave function, and some other crucial findings.