Developing semiclassical Wentzel-Kramers-Brillouin theory for α−T3 model

We developed a complete semiclassical Wentzel-Kramers-Brillouin (WKB) theory for $\alpha \text{−}{\mathcal{T}}_3$ model which describes a wide class of existing pseudospin-1 Dirac cone materials. By expanding the sought wave functions in a series of powers of Planck's constant $h$, we obtained the leading-order expansion term which is the key quantity required for calculating the electronic and tunneling properties of a semiclassical electron in $\alpha \text{−}{\mathcal{T}}_3$ materials. Based on this, we further derived the WKB transport equations relating two consecutive orders of the wave-function expansion and solved them analytically to obtain the first-order WKB wave function. Meanwhile, we discussed the applicability of the employed approximation as well as how these results could be applied to study various tunneling and transport properties of $\alpha \text{−}{\mathcal{T}}_3$ materials with nontrivial potential profiles. Finally, our results could also be useful for designing next-generation electronic transistors and devices with help from innovative flat-band Dirac materials.