Theory of proximity effect in normal metal/$d_{x^2-y^2}$-wave superconductor interface in the presence of subdominant components of the pair potentials

Superconducting proximity effect in normal metal (N) / $d_{x^2-y^2}$-wave superconductor (D) junctions in the presence of attractive interelectron potentials which can induce subdominant s-wave pair potentials both in N and D sides, is studied based on the quasiclassical Green's function theory, where spatial dependencies of the pair potentials are determined self-consistently. In the N/D junctions with orientational angle with $\theta = 0$, the s-wave component is induced in the N side by the proximity effect only for high transparent case, where the induced s-wave components in both the N and D sides do not break the time reversal symmetry (TRS). For fully transparent case, the resulting local density of states has a very sharp zero-energy peak (ZEP), the origin of which is the sign change of the pair potentials felt by the quasiparticles between the s-wave component in the N side and $d_{x^2-y^2}$-wave dominant component in the D side through Andreev reflections. On the other hands, for $\theta = \pi/4$, the subdominant s-wave component which breaks the TRS appears near the interface. Besides, for lower transparent cases, the subdominant imaginary s-wave component is also induced near the interface in the N side. The proximity induced s-wave component in the N side does not enhance the magnitude of the s-wave component of the pair potential which break the TRS in the D side. The resulting LDOS at the interface has the ZEP or its splitting depending on the transparency of the junction.