Anomalous proximity effect of a spin-singlet superconductor with a spin-orbit interaction

The anomalous proximity effect of a spin-triplet $p$-wave superconductor has been known as a part of the Majorana physics and is explained by the penetration of zero-energy states from a surface of a superconductor to a dirty normal metal. We demonstrate that a spin-singlet $d$-wave superconductor without any surface zero-energy states exhibits the anomalous proximity effect in the presence of a specific spin-orbit interaction. The results show the quantization of the zero-bias conductance in a dirty normal-metal/superconductor junction. We also discuss a relation between our findings and results in an experiment on a CoSi${}_2$/TiSi${}_2$ junction.