Developing a semiclassical Wentzel-Kramers-Brillouin theory for $\alpha-\mathcal{T}_3$ model

We have developed a complete semiclassical Wentzel-Kramers-Brillouin (WKB) theory for $\alpha-\mathcal{T}_3$ model which describes a wide class of existing pseudospin-1 Dirac cone materials. By expanding the sought wave functions in a series over the powers of Planck constant $\hbar$, we have obtained the leading order expansion term which is the key quantity required for calculating the electronic and transport properties of a semiclassical electron in $\alpha-\mathcal{T}_3$. We have derived the transport equations connecting each two consecutive orders of the wave function expansion and solved them to obtained the first order WKB wavefunction. We have also discussed the applicability of the obtained approximation and how these results could be used to investigate various tunneling and transport properties of $\alpha-\mathcal{T}_3$ materials with non-trivial potential profiles. Our results could be also helpful for constructing electronics devices and transistors based on innovative flat-band Dirac materials.