Study On Klein Tunneling In The ?-T₃ Model With An Effective Mass Term

The?-T₃model refers to a honeycomb lattice with an additional atom in the center of each hexagon.This additional atom is coupled with the nearest neighbor sites in the hexagonal lattice,and the coupling strength is?.The?-T₃model interpolates between a Dirac-Weyl system with pseudospin-1/2 and the T₃or dice lattice with pseudospin-1 by allowing the parameter?to vary from 0 to 1.Here we mainly study the quantum transport of the?-T₃model under the conditions of a pseudo-field and a compressed uni-axial deformation.Firstly,we investigate the quantum tunneling properties through a square potential barrier in the?-T₃model with an effective mass term.The additional mass term in the Hamiltonian produces an energy gap with a flat band in it.Such an effective mass term can be induced by the effective magnetic field or the site energy difference on different sub-lattices,in which the flat band locates at the center of the band gap or at the top(bottom)of the valence(conduction)band,respectively.The linear dispersion in the?-T₃model is modified in the presence of the mass term,leading to the destroying of the Klein tunneling.For the dice lattice that?=1,it is found that the super-Klein tunneling is unaffected when the the flat band locates at the center of the band gap.While for another case that the flat band locates at the band edge,the super-Klein tunneling and the resonant tunneling are considerably suppressed with the increase of the energy gap.For??1,the perfect transmission disappears for the normal incidence no matter where the flat band position is.Furthermore,it is demonstrated that additional mass term in general suppresses the transmission.Then we also studied the quantum tunneling effect when electrons through a potential step and a potential barrier in the?-T₃model under a compressed uni-axial deformation.When compressing the?-T₃lattice along the armchair direction,a pair of Dirac points in the energy band structure may merge into a single one along the zigzag direction.If the strain is strong enough,the energy gap will open,and the dispersion relation will be linear in the armchair direction but quadratic in the zigzag direction.The merging of Dirac points represents a topological phase transition from a metallic to an insulating phase.We found that in the armchair direction,the tunneling properties are similar to the undeformed?-T₃lattice.For example,when the incident particle is normal to the potential barrier,we find Klein tunneling effect independent of?,and for the dice lattice with the incident energy is equal to half of the barrier height,there occurs super-Klein tunneling.While in the zigzag direction,we get the opposite conclusion in the deformed?-T₃lattice.The Klein tunneling effect becomes the anti-Klein tunneling effect,and the super-Klein tunneling effect is transformed into the anti-super Klein tunneling effect,and the conductance with gap opens can be turned off using strain applied along the armchair direction.