  Based on the transmission coefficient of tunneling electrons, we have
presented tunneling current and conductivity across a square-potential barrier
for both graphene and $\alpha$-$\mathcal{T}_3$ lattices under a
linearly-polarized off-resonant dressing field. The presence of such a dressing
field introduces an anisotropy factor in the energy dispersion of tunneling
electrons so that the cross section of a Dirac-cone appears as elliptical.
Consequently, the field-polarization controlled major axis of the ellipse will
be misaligned with the normal direction of a barrier layer in the tunneling
system, which exhibits an asymmetric Klein-paradox for an off-normal-direction
tunneling. The resulting tunneling current in this system is calculated by
using a transmission coefficient and a longitudinal group velocity (different
from a longitudinal momentum) of electrons. By presenting numerically
calculated tunneling conductivity modified by a laser dressing field, we
demonstrate a significant enhancement of electrical conductivity by external
laser-field intensity, which is expected to be crucial in application of
ultrafast optical modulation of opto-electronic devices for photo-detection and
fiber-optic communication.
