Anisotropic Andreev reflections in the three dimensional nodal-line semimetals-superconductor system

Quantum transport is the process by which a particle changes from one state to another in a mesoscopic system. We construct a hybrid system of the anisotropic three dimensional topological nodal-line semimetals-superconductor (TNLSMs-SC), and the Andreev reflection of the hybrid system under two non-equivalent models ($k_x$-$y$ and $k_z$-$y$ models) is studied by using the non-equilibrium Green’s function. The results show that the Andreev reflection of the hybrid system is obvious, and the Andreev reflection gradually enhances with the increase of Fermi energy $E_F$. In addition, it is found that only the bulk bands of the TNLSMs are involved in the Andreev reflection of the hybrid systems, and the Andreev reflection is closely related to the anisotropy of the topological nodal-line semimetals, which shows that TNLSMs has anisotropic charge shielding behavior. The strongly anisotropic behaviors of the Andreev reflection can be used as the characterization of TNLSMs in experimental detections. For the $k_x$-$y$ model, it is shown that band structure of the TNLSMs with each given $k_x$ moves in the direction $k_y=0$ and behaves like two Weyl nodes, resulting in the contribution of bulk bands at each given $k_x$ to the Andreev reflection of the hybrid system. For the $k_z$-$y$ model, only the bulk bands for $k_z=0,\pm \frac{\pi }{20},\pm \frac{\pi }{10}$ involve Andreev reflection of the hybrid system. The presence of the mass term $m$ opens a gap in the band structure of the topological nodal-line semimetals. With the increase in the mass term $m$, the number of bulk bands involved in the Andreev reflection of the hybrid system decreases, leading to weaker Andreev reflection within the superconducting gap of the hybrid system. As the on-site energy ${\varepsilon }_z$ increases, the nodal line decreases, causing the Andreev reflection of the hybrid system to be gradually dominated by the bulk bands where $k_x$ approaches zero. These rich physical properties provide theoretical guidance for constructing superconducting information storage carriers in hybrid systems.