We investigate the quantum tunneling properties through a square potential barrier in the $$\alpha -T_3$$ model with an effective mass term. The additional mass term can be induced by the effective magnetic field or the site energy difference on different sublattices, in which the flat band is located at the center or edge of the band gap band, respectively. The linear dispersion in the $$\alpha -T_3$$ model is modified in the presence of the mass term, leading to the destroying of the Klein tunneling for the normal incidence. It is demonstrated that the additional mass term in general suppresses the transmission. When the flat band is located at the band edge, the super-Klein tunneling and the resonant tunneling are considerably suppressed with the increase in the energy gap. For the dice lattice that $$\alpha =1$$, the super-Klein tunneling is preserved when the flat band is located at the center of the band gap.

中文翻译：

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具有有效质量项的 $$\alpha -T_3$$α-T3 模型中的量子隧道

我们通过具有有效质量项的 $$\alpha -T_3$$ 模型中的方形势垒来研究量子隧道特性。附加质量项可由有效磁场或不同亚晶格上的位能差引起，其中平带分别位于带隙带的中心或边缘。$$\alpha -T_3$$ 模型中的线性色散在质量项的存在下被修改，导致法向入射的克莱因隧道破坏。结果表明，附加质量项通常会抑制传输。当平坦带位于带边缘时，随着能隙的增加，超克莱因隧穿和共振隧穿受到显着抑制。对于 $$\alpha =1$$ 的骰子点阵，