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

We show that under compressive uniaxial deformation of the three-band $α-T_3$ 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 $α-T_3$ 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 ($α=1$). For all values of $α$, this transition manifests as a change from conventional Klein tunneling to anti-Klein tunneling.

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