Andreev reflection and Josephson effect in the $α$-T3 lattice

Abstract

We investigate the Andreev reflection and Josephson effect in the $α$-T3 lattice which falls between graphene ($α= 0$) and the dice lattice ($α= 1$) by adjusting the parameter $α$. In the regime of specular Andreev reflection, when the incident energy of electron is small, the probability of Andreev reflection decreases as the parameter $α$ increases. On the contrary, when the incident energy is large, the probability of Andreev reflection increases as the parameter $α$ 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 $α= 1$. In the regime of Andreev retro-reflection, when the parameter $α$ 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 $α$. We also give the differential conductance in these two regimes and find that the differential conductance increases as the parameter $α$ increases generally. In addition, the $α$-T3 lattice-based Josephson current increases as $α$ increases. When the length of junction approaches zero, the critical Josephson currents in the different values of $α$ approach the same value.