We investigate the Andreev reflection and the Josephson effect in the $\alpha -T_3$ lattice, which falls between graphene ($\alpha =0$) and the dice lattice ($\alpha =1$), by adjusting the parameter $\alpha$. In the regime of the specular Andreev reflection, when the incident energy of the electron is small, the probability of Andreev reflection decreases as the parameter $\alpha$ increases. On the contrary, when the incident energy is large, the probability of Andreev reflection increases as the parameter $\alpha$ increases. Interestingly, when the incident energy approaches the superconducting energy gap function, the Andreev reflection with approximate all-angle perfect transmission happens in the case of $\alpha =1$. In the regime of Andreev retroreflection, when the parameter $\alpha$ increases, the probability of Andreev reflection increases regardless of the value of incident energy. When the incident energy approaches the superconducting energy gap function, the Andreev reflection with approximate all-angle perfect transmission happens regardless of the value of $\alpha$. We also give the differential conductance in these two regimes and find that it increases as the parameter $\alpha$ increases generally. In addition, the $\alpha -T_3$ lattice-based Josephson current increases as $\alpha$ increases. When the length of the junction approaches zero, the critical Josephson currents in the different values of $\alpha$ approach the same value.

中文翻译：

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α−T3 晶格中的 Andreev 反射和约瑟夫森效应

我们研究了安德列夫反射和约瑟夫森效应 $\alpha -{吨}_3$ 晶格，介于石墨烯（$\alpha =0$) 和骰子点阵 ($\alpha =1$)，通过调整参数 $\alpha$. 在镜面 Andreev 反射的范围内，当电子的入射能量较小时，Andreev 反射的概率随着参数的增加而减小$\alpha$增加。相反，当入射能量较大时，Andreev反射的概率随着参数的增加而增加$\alpha$增加。有趣的是，当入射能量接近超导能隙函数时，近似全角完美透射的 Andreev 反射发生在$\alpha =1$. 在 Andreev 逆反射范围内，当参数$\alpha$增加，Andreev 反射的概率增加，而与入射能量的值无关。当入射能量接近超导能隙函数时，无论$\alpha$. 我们还给出了这两种情况下的微分电导，并发现它随着参数的增加而增加$\alpha$普遍增加。除此之外$\alpha -{吨}_3$ 基于晶格的约瑟夫森电流随着 $\alpha$增加。当结的长度接近零时，不同值的临界约瑟夫森电流$\alpha$ 接近相同的值。