Klein tunneling in deformed honeycomb-dice lattice: from massless to massive particles

We show that under compressive uniaxial deformation of the three-band $\alpha-T_3$ lattice, the Dirac cones move toward each other, merge, and a gap opens, while the flat band remains unchanged. Consequently, the low-energy spectrum transitions from linear to quadratic dispersion, indicating the shift from massless to massive Dirac particles. Here, we theoretically investigate the tunneling properties of particles through a sharp $np$ junction in a deformed $\alpha-T_3$ lattice, focusing on the case where the particle energy is half the junction height. We show that this transition from massless to massive particles leads to a change from omnidirectional total transmission, known as super-Klein tunneling, to omnidirectional total reflection, referred to as anti-super-Klein tunneling, in the case of the dice lattice ($\alpha=1$). For all values of $\alpha$, this transition manifests as a change from conventional Klein tunneling to anti-Klein tunneling.