  We generalize Wentzel-Kramers-Brillouin (WKB) semi-classical equations for
pseudospin-1 $\alpha-\mathcal{T}_3$ materials with arbitrary hopping parameter
$0 < \alpha < 1$, which includes the dice lattice and graphene as two limiting
cases. In conjunction with a series-expansion method in powers of Planck
constant $\hbar$, we acquired and solved a system of recurrent differential
equations for semi-classical electron wave functions in $\alpha-\mathcal{T}_3$.
Making use of these obtained wave functions, we analyzed the physics-related
mechanism and quantified the transmission of pseudospin-1 Dirac electrons
across non-rectangular potential barriers in $\alpha-\mathcal{T}_3$ materials
with both zero and finite band gaps. Our studies reveal several unique
features, including the way in which the electron transmission depends on the
energy gap, the slope of the potential barrier profile and the transverse
momentum of incoming electrons. Specifically, we have found a strong dependence
of the obtained transmission amplitude on the geometry-phase $\phi = \tan^{-1}
\alpha$ of $\alpha-\mathcal{T}_3$ lattices. We believe our current findings can
be applied to Dirac cone-based tunneling transistors in ultrafast analog RF
devices, as well as to tunneling-current control by a potential barrier through
a one-dimensional array of scatters.
