Double Andreev reflections at surface states of the topological insulators with hexagonal warping

We study the Andreev reflection (AR) at the interface of the topological insulator with hexagonal warping and superconductor junction. Due to the hexagonal warping effect, the double ARs are found in a certain range of the incident angle, where for one incident electron beam, two beams of holes are reflected back. Interestingly, both the beams of holes are reflected as retro-AR on the same side of the normal line of the interface but with different reflection angles, different from the previously reported double AR with one retro-AR and one specular-AR. The double reflections owing to the warping effect show the optics-like property of the Dirac fermion and can stimulate the double reflections of light in anisotropic crystals. In addition, we find that the double ARs are dependent on the hexagonal warping parameter nonmonotonically, and in an intermediate strength the double AR phenomenon is prominent, providing a possibility to explore the warping parameter of topological insulators.