Resonant tunneling and the transmission over a non-square barrier in α-T₃ lattice

We have derived the semiclassical WKB (Wentzel-Kramers-Brillouin) electronic states, and the
corresponding wavefunction, for an arbitrary α-T₃ model based of a series expansion over the powers
of the Planck constant of the sought wave function. We have derived the set of recurrent equations
connecting the terms from the two consequent orders of this expansion. We have investigated the
tunneling of the obtained electron states over a random and not necessarily square potential barrier,
including the barriers with a nite slope of their potential. We have also found the classical action
in the case of a nite bandgap in α-T₃ which could be induced by a substrate of by applying
external circularly polarized irradiation.