Wentzel-Kramers-Brillouin (WKB) semiclassical equations for alpha-T_3 materials

We have derived Wentzel-Kramers-Brillouin (WKB) semiclassical equations for the pseudospin-1 Dirac Hamiltonian for arbitrary α − T₃ materials with hopping parameters 0 < α < 1. We have both derived the analytical expression for the semiclassical states and the transport equations connecting different orders of the hbar-expansion of the sought wave function. Our obtained results have several crucial applications. These include  approximate evaluation of electron transmission and reflection over a non-square potential barrier with a non-linear potential profile.