  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.
