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
Y. Tanuma
$^1$ Y. Tanaka
$^{2,3}$ and S. Kashiwaya$^4$
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. %
paperv1.tex
%==========================================================
%   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
%----------------------------------------------------------
%   submitted in cond-mat. 04/12/14
%    Y. Tanuma, Y. Tanaka, S. Kashiwaya
%==========================================================
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%-----------------------------------------------------------
\begin{document}
\preprint{}

%--- title ---
\title{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
}

%--- author ---
\author{
Y. Tanuma,$^1$
Y. Tanaka,$^{2,3}$ and S. Kashiwaya$^4$
}%
%
%--- address ---
\affiliation{
$^1$Institute of Physics, Kanagawa University,
Rokkakubashi, Yokohama, 221-8686, Japan \\
%
$^2$Department of Applied Physics, 
Nagoya University, Nagoya, 464-8603, Japan \\
%
$^3$CREST, Japan Science and Technology Cooperation (JST),
Nagoya, 464-8603, Japan  \\
%
$^4$National Institute of Advanced Industrial Science
and Technology, Tsukuba, 305-8568, Japan
}
%
%--- date ---
\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified

%-----------------------------------------------------------
%   Abstract
%-----------------------------------------------------------
\begin{abstract}
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.
%
\end{abstract}

%-----------------------------------------------------------
\pacs{74.50.+r, 74.20.Rp, 74.72.-h}
                             % PACS, the Physics and Astronomy
                             % Classification Scheme.
%\keywords{Suggested keywords}%Use showkeys class option if keyword
                             %display desired
\maketitle

%-----------------------------------------------------------
\section{Introduction}
%-----------------------------------------------------------
To determine the pairing symmetry in unconventional
superconductors is an interesting problem to
understand the pairing mechanism of superconductivity.
%
Nowadays, it is widely accepted that
the superconducting pair potential of high-$T_c$ cuprates
has a $d_{x^{2}-y^{2}}$-wave symmetry
in the bulk state.
\cite{SR92,SR95,Scalapino,Harlin,Tsuei}
%
In order to determine the pairing symmetry,
several phase-sensitive probes have been used
\cite{SR92,SR95,Harlin,Tsuei}.
%
Among them tunneling spectroscopy via
Andreev bound states (ABS's),
\cite{Buch,Hu,TK95,KT95,Matsu1,FRS97,KT00,KT96,Barash,Lofwander,ATS04}
help us to detect the internal phase
in the pair potential.
%
The formation of ABS's at the Fermi energy (zero-energy),
which is originated from the interference effect
in the effective pair potential of
$d_{x^{2}-y^{2}}$-wave symmetry
through the reflection at the surface/interface,
plays an important role 
when the angle $\theta$ between the lobe direction of the
$d_{x^{2}-y^{2}}$-wave pair potential
and the normal to the surface/interface is nonzero.
\cite{Hu}
%
In fact, the existence of ABS's,
which manifests itself as a distinct conductance peak
at zero-bias in the tunneling spectrum (ZBCP),
has been actually observed.
%
Up to now, the consistency between theories
\cite{Tanu1,Tanu2,Tanu2',Zhu,W1,W2,asa1,asa2,r1,r2,r3,r4,r5,r6,r7,r8,r9,Lubi03}
and experiments
\cite{e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,Koren,Boston,e11,e12,e13}
has been checked in detail.
\par
%
On the other hand, 
at the surface/interface of $d_{x^{2}-y^{2}}$-wave 
superconductor, it is known 
that the time-reversal symmetry (TRS) may be broken.
\cite{SBL95,Kuboki,Sig98}
%
The reduction of the amplitude 
of $d_{x^{2}-y^{2}}$-wave pair potential near the surface/interface
at low temperatures 
allows to induce the subdominant
$s$-wave [$d_{xy}$-wave] component
in the imaginary part of the pair potential,
i.e.,
$d$+i$s$-wave \cite{FRS97,Matsu2,Matsu3,Tanu2,Tanu2',Zhu}
[$d$+i$d'$-wave \cite{Krishana,Balatsky,Laughlin} ] state.
%
The amplitude of the induced $s$-wave component 
has a maximum at $\theta =\pi/4$, which 
blocks the motion of quasiparticles. 
%
Then, the energy levels of ABS's shift from zero
and the resulting tunneling spectra has a ZBCP splitting
even in zero magnetic field.
%
The observation of the ZBCP splitting without magnetic fields
was believed to be one of the evidence
for the broken time reversal symmetry states (BTRSS).
%
However, actual tunneling experiments still
remain to be controversial.
%
In fact, some groups \cite{e5,e6,e9,e10,Koren,Boston}
have reported the ZBCP splitting
and they ascribed
the origin of the ZBCP splitting
to the above BTRSS. 
%
At the same time,
there are other experiments which do not show
the ZBCP splitting, and in these experiments,
the ZBCP survives even at low temperature.
\cite{e3,e4,e7,e8,e11,e12,e13}
%
Moreover, 
a critical current measurement of
grain boundary junctions in high-$T_c$ cuprates
concluded the absence of BTRSS at the interface.
\cite{Neils}
%
It has been studied in detail that 
the ZBCP splitting  in the tunneling spectra
is sensitive to several factors:
(i) transmission probability of the junctions,
\cite{Tanu5}
(ii) roughness at the interface,
\cite{Tanu1,Tanu2,Tanu2'}
and
(iii) effect of impurity scattering
in normal metals or superconductors.
\cite{r1,asa1}
%
Recent studies by Asano \textit{et al.},
\cite{asa1} revealed that the existence of the 
impurity scattering can induce the 
ZBCP splitting even without BTRSS.
%
Taking account of these situations,
one can not conclude that 
the observation of the ZBCP splitting
by tunneling experiments 
directly indicate the presence of the BTRSS
near the surface/interface.
\par
%
More recently, 
it is proposed that
the induced $s$-wave component of the pair potential 
by proximity effect in normal metals (N) may
enhance the magnitude of 
the subdominant $s$-wave component in
$d_{x^2-y^2}$-wave superconductors (D),
which forms the BTRSS based on the analysis of
the tunneling experiments.
\cite{Kohen}
%
%Furthermore, they interpret that
%the penetrated $s$-wave component into normal
%metals leads to the enhancement of i$s$-wave one
%induced in the vicinity of the interface
%of $d_{x^2-y^2}$-wave superconductors.
%
It is of great interest to study
whether the induced $s$-wave component
in the N side by the proximity effect has the influence on
the subdominant component in the D side
which forms the BTRSS.
%
The proximity effect in the N/D junction without the BTRSS
was theoretically studied by Ohashi.
\cite{Ohashi}
%
It is shown that the amplitude of the subdominant i$s$-wave is
the largest at $\theta = \pi/4$.
%
For $\theta=0$ with fully transparent junctions,
local density of states (LDOS) at the interface
has a zero energy peak (ZEP), the origin of which is not 
ABS but the sign change of the pair potentials 
between N and D sides felt by
Andreev reflection of quasiparticles.
%
%However as far as we know,
%the relevance between the BTRSS
%and the induced $s$-wave component in N
%have not been clarified yet. 
%
It is necessary to study the interplay between the 
proximity effect and the BTRSS by changing the orientational angle $\theta$.
%
At the same time,
it is also interesting to clarify the relationship
between the different origin of two kinds of
ZEP's.
%
One is the ZEP which originates from the formation of ABS
for $\theta \neq 0$.
%
The other is due to the proximity effect expected for
fully transparent junctions for $\theta=0$.
\cite{Ohashi}
\par
%
In the present paper, 
we study the proximity effect
in N/D junctions on the basis of
quasiclassical Green's function methods.
%
We assume attractive interelectron potentials 
which induce subdominant $s$-wave components
in the N side as well as the D.
%
The spatial dependencies of the pair potentials 
both in the N and D sides are determined self-consistently. 
%
The LDOS at the interface of the N/D junctions 
are studied in detail by changing $\theta$ and 
the transparency of the junction. 
%
For $\theta = \pi/4$, the magnitude of the 
$s$-wave component is induced even in the N side,
while the subdominant $s$-wave component
which breaks TRS exists near the interface
on the D side.
\cite{Lof}
%
On the other hand, for $\theta = 0$, 
$s$-wave component is induced in the N side
by the proximity effect with the increase of
the transparency of the junction. 
%
The induced $s$-wave component in the N side and that
in the D side do not break the TRS. 
%
It is revealed that the 
proximity induced $s$-wave component in the N side
does not enhance the BTRSS in the D side. 
%
\par
%
The organization of the present paper is as follows.
In Sec.~\ref{sec:02},
a theoretical formulation to calculate
the spatial dependence of the pair potential,
and local density of states is presented.
%
In Sec.~\ref{sec:03},
results of the numerical calculations 
are discussed in detail.
%
Finally,
we summarize the paper in Sec.~\ref{sec:04}.
\par
%
%----- figure 1 ------
\begin{figure}[htb]
\begin{center}
\scalebox{0.60}{
\includegraphics{profig01.eps}}
\caption{Schematic illustration of a N/D junction
with a spatially varying pair potential. 
The misorientation angle $\theta$ denotes the 
angle between the crystal axis of $d_{x^2-y^2}$-wave superconductor 
and the normal to the interface. 
\label{fig:01}}
\end{center}
\end{figure}
%---------------------
%
%-----------------------------------------------------------
\section{Theoretical model}
\label{sec:02}
%-----------------------------------------------------------
%
In the present paper, 
we consider the N/D junction
separated by an insulating interface at $x=0$,
where the normal metal is located at $x<0$
and the $d_{x^{2}-y^{2}}$-wave superconductor
extends elsewhere (Fig.~\ref{fig:01}). 
%
For the simplicity,
two dimensional system is assumed and
the $x$ axis is taken perpendicular to the interface.
%
When quasiparticles are in the $xy$ plane,
a transmitted electron like quasiparticle
and holelike quasiparticle in the D side feel 
different effective pair potentials
$\Delta_{d}(\phi_{+})$ and $\Delta_{d}(\phi_{-})$,
with $\phi_{+}=\phi$ and $\phi_{-}=\pi-\phi$.
%
Here $\phi$ is the azimuthal angle
in the $xy$ plane 
given by $(k_{x}+{\rm i}k_{y})/|\bm{k}|={\rm e^{i\phi}}$.
%
Besides, a cylindrical Fermi surface is assumed 
and the magnitude of the Fermi momentum 
and the effective mass are chosen to be equal 
both in the N and D sides.
\par
%
In order to study the proximity effect
in the N/D junction,
we determine the spatial variation
of the pair potentials self-consistently.
%
For this purpose,
we make use of the quasi-classical Green's function procedure
\cite{Eilen} developed by Nagai and co-workers.
\cite{Nagai,Ashida}
%
In the following,
we briefly summarize this scheme we employed.
%
We introduce the generalized Eilenberger equation,
\cite{Eilen}
%
\begin{align}
{\rm i}|v_{{\rm F}x}|\frac{\partial}{\partial x}
g_{\alpha \beta}(\phi,x) = &-\alpha
\left [ {\rm i}\omega_{m}\hat{\tau}_{3}
+\hat{\Delta}(\phi_{\alpha},x) \right ] 
g_{\alpha \beta}(\phi,x)
\nonumber \\
&+ \beta  g_{\alpha \beta}(\phi,x)
\left [ {\rm i}\omega_{m}\hat{\tau}_{3}
+\hat{\Delta}(\phi_{\beta},x) \right ] , \\
%
\hat{\Delta}(\phi_{\alpha},x) =&
\left (
    \begin{array}{cc}
        0 & \Delta(\phi_{\alpha},x) \\
  -\Delta(\phi_{\alpha},x)^{*} & 0
    \end{array}
\right ),
\end{align}
%
where $v_{{\rm F}x}=v_{\rm F}\cos \theta$
and $\hat{\tau}_i (i=1,2,3)$ stand for
the $x$ component of the Fermi velocity
and the Pauli matrices, respectively.
Here $\omega_m=\pi T(2m+1)$ ($m$: integer)
is the Matsubara frequency.
\par
%--------------------------------------------------------
%
Now,
considering a semi-infinite N/D junction geometry,
the pair potential in D [N] side
tend to the bulk value [zero]
$\Delta^{\rm D}(\phi_{\alpha},\infty)$
[$\Delta^{\rm N}(\phi_{\alpha},-\infty)$]
at sufficiently large $x$.
%
In semi-infinite limit,
we can find the quasi-classical Green's function
$\hat{g}_{\alpha \alpha}^{l}(\phi_{\alpha},x)$
for $l(=\textrm{N,D})$ regions given by
\cite{Ohashi,Nagai,Nagato}
%
\begin{align}
\hat{g}_{\alpha \alpha}^{l}(\phi_{\alpha},x)
={\rm i}
\left (
\frac{2\hat{A}^{l}_{\alpha}(x)}
{{\rm Tr}[\hat{A}^{l}_{\alpha}(x)]}
-\bm{1}
\right ),
\end{align}
%
Following equations are satisfied 
for the N side,
%
\begin{align}
\label{an+}
\hat{A}^{\rm N}_{+}(x)
&=
\tilde{U}^{\rm N}_{+}(\phi_{+},x,0)
\hat{R}_{\rm D}
\hat{\lambda}^{\rm N}(0,x),
\\
\hat{A}^{\rm N}_{-}(x)
&=
\hat{\lambda}^{\rm N}(x,0)
\hat{R}_{\rm D}
\tilde{U}^{\rm N}_{-}(\phi_{-},0,x),
\\ \nonumber
%
\hat{\lambda}^{\rm N}(x,x^{\prime}) &=
\left (
\begin{array}{cc}
                1 &
\mathrm{i}/{\cal G}^{\rm N}_{+}(x^{\prime}) \\
      \mathrm{i}{\cal G}^{\rm N}_{-}(x) &
      -{\cal G}^{\rm N}_{-}(x)/
      {\cal G}^{\rm N}_{+}(x^{\prime}) \\
\end{array}
\right ),
\end{align}
%
and for the D side,
%
\begin{align}
\label{ad+}
\hat{A}^{\rm D}_{+}(x)
&=
\hat{\lambda}^{\rm D}(x,0)
\hat{R}_{\rm N}
\tilde{U}^{\rm D}_{+}(\phi_{+},0,x),
\\
\label{ad-}
\hat{A}^{\rm D}_{-}(x)
&=
\tilde{U}^{\rm D}_{-}(\phi_{-},x,0)
\hat{R}_{\rm N}
\hat{\lambda}^{\rm D}(0,x),
\\ \nonumber
%
\hat{\lambda}^{\rm D}(x,x^{\prime}) &=
\left (
\begin{array}{cc}
                1 &
\mathrm{i}/{\cal G}^{\rm D}_{-}(x^{\prime}) \\
      \mathrm{i}{\cal G}^{\rm D}_{+}(x) &
      -{\cal G}^{\rm D}_{+}(x)/
      {\cal G}^{\rm D}_{-}(x^{\prime}) \\
\end{array}
\right ), 
\end{align}
respectively. 
%
In the above,
$\hat{R}_{\rm N}$ and $\hat{R}_{\rm D}$ are matrices
with reflection probability $R$ at the interface
given by
\cite{Ohashi}
%
\begin{align}
\hat{R}_{\rm N} & \propto
\left (
\begin{array}{cc}
  {\cal G}^{\rm N}_{+}(0) - R{\cal G}^{\rm N}_{-}(0) &
            \mathrm{i}(1-R) \\
   \mathrm{i}(1-R){\cal G}^{\rm N}_{-}(0)
                  {\cal G}^{\rm N}_{+}(0) &
  {\cal G}^{\rm N}_{+}(0)R - {\cal G}^{\rm N}_{-}(0) \\
\end{array}
\right ),
\\
%
\hat{R}_{\rm D} & \propto
\left (
\begin{array}{cc}
  {\cal G}^{\rm D}_{-}(0) - R{\cal G}^{\rm D}_{+}(0) &
            \mathrm{i}(1-R) \\
   \mathrm{i}(1-R){\cal G}^{\rm D}_{+}(0)
                  {\cal G}^{\rm D}_{-}(0) &
  {\cal G}^{\rm D}_{-}(0)R - {\cal G}^{\rm D}_{+}(0) \\
\end{array}
\right ).
\end{align}
%
Here, we define ${\cal G}^{l}_{\alpha}(x)$ given by
%
\begin{align}
\label{gg}
{\cal G}^{\rm N}_{\alpha}(x)=
-{\rm i}\frac{v_{n}^{{\rm N}(+)}(\phi_{\alpha},x)}
{u_{n}^{{\rm N}(+)}(\phi_{\alpha},x)}, \quad
{\cal G}^{\rm D}_{\alpha}(x)=
-{\rm i}\frac{v_{n}^{{\rm D}(-)}(\phi_{\alpha},x)}
{u_{n}^{{\rm D}(-)}(\phi_{\alpha},x)}.
\end{align}
%
\par
%
Next, in order to obtain the quantities
$\tilde{U}^{l}_{\alpha}(\phi_{\alpha},x,x^{\prime})$
in Eqs.~(\ref{an+})-(\ref{ad-}),
we rewrite $\hat{A}^{l}_{\alpha}(x)$ as
\cite{Nagai,Nagato}
%
\begin{align}
\hat{A}^{\rm N}_{+}(x) &=
\left (
\begin{array}{c}
      X^{\rm N}_{+}(x) \\
      Y^{\rm N}_{+}(x) \\
\end{array}
\right )
%
\left (
\begin{array}{cc}
      u_{n}^{{\rm N}(+)}(\phi_{+},x) &
      v_{n}^{{\rm N}(+)}(\phi_{+},x) \\
\end{array}
\right )
\hat{\tau}_{2},
\\
\hat{A}^{\rm N}_{-}(x) &=
\left (
\begin{array}{c}
      u_{n}^{{\rm N}(+)}(\phi_{-},x) \\
      v_{n}^{{\rm N}(+)}(\phi_{-},x) \\
\end{array}
\right )
%
\left (
\begin{array}{cc}
      X^{\rm N}_{-}(x) &
      Y^{\rm N}_{-}(x) \\
\end{array}
\right )
\hat{\tau}_{2},
\\
%
\hat{A}^{\rm D}_{+}(x) &=
\left (
\begin{array}{c}
      u_{n}^{{\rm D}(-)}(\phi_{+},x) \\
      v_{n}^{{\rm D}(-)}(\phi_{+},x) \\
\end{array}
\right )
%
\left (
\begin{array}{cc}
      X^{\rm D}_{+}(x) &
      Y^{\rm D}_{+}(x) \\
\end{array}
\right )
\hat{\tau_{2}},
\\
\hat{A}^{\rm D}_{-}(x) &=
\left (
\begin{array}{c}
      X^{\rm D}_{-}(x) \\
      Y^{\rm D}_{-}(x) \\
\end{array}
\right )
%
\left (
\begin{array}{cc}
      u_{n}^{{\rm D}(-)}(\phi_{-},x) &
      v_{n}^{{\rm D}(-)}(\phi_{-},x) \\
\end{array}
\right )
\hat{\tau}_{2},
%----------------------------------
\end{align}
with
\begin{align}
%----------------------------------
\left (
\begin{array}{c}
      X^{\rm N}_{-}(x) \\
      Y^{\rm N}_{-}(x) \\
\end{array}
\right )
= &
\tilde{U}^{\rm N}_{+}(\phi_{+},x,0)
\hat{R}_{\rm D}
\left (
\begin{array}{c}
      u_{n}^{{\rm N}(+)}(\phi_{-},0) \\
      v_{n}^{{\rm N}(+)}(\phi_{-},0) \\
\end{array}
\right ),
\\
%
\left (
\begin{array}{cc}
      X^{\rm N}_{-}(x) &
      Y^{\rm N}_{-}(x) \\
\end{array}
\right )
\hat{\tau}_{2} =&
\left (
\begin{array}{cc}
      u_{n}^{{\rm N}(+)}(\phi_{+},0) &
      v_{n}^{{\rm N}(+)}(\phi_{+},0) \\
\end{array}
\right ) \hat{\tau}_{2} \nonumber \\ &\times
\hat{R}_{\rm D}
\tilde{U}^{\rm N}_{-}(\phi_{-},0,x),
\\
%
\left (
\begin{array}{cc}
      X^{\rm D}_{+}(x) &
      Y^{\rm D}_{+}(x) \\
\end{array}
\right )
\hat{\tau}_{2} =&
\left (
\begin{array}{cc}
      u_{n}^{{\rm D}(-)}(\phi_{-},0) &
      v_{n}^{{\rm D}(-)}(\phi_{-},0) \\
\end{array}
\right ) \hat{\tau}_{2} \nonumber \\ &\times
\hat{R}_{\rm N}
\tilde{U}^{\rm D}_{+}(\phi_{+},0,x),
\\
\left (
\begin{array}{c}
      X^{\rm D}_{-}(x) \\
      Y^{\rm D}_{-}(x) \\
\end{array}
\right )
= &
\tilde{U}^{\rm D}_{-}(\phi_{-},x,0)
\hat{R}_{\rm N}
\left (
\begin{array}{c}
      u_{n}^{{\rm D}(-)}(\phi_{+},0) \\
      v_{n}^{{\rm D}(-)}(\phi_{+},0) \\
\end{array}
\right ).
\end{align}
%
Then we define ${\cal F}^{l}_{\alpha}(x)$ given by
%
\begin{align}
\label{ff}
{\cal F}^{\rm N}_{\alpha}(x)=
{\rm i}\frac{X^{\rm N}_{\alpha}(x)}
{Y^{\rm N}_{\alpha}(x)}, \quad
{\cal F}^{\rm D}_{\alpha}(x)=
{\rm i}\frac{X^{\rm D}_{\alpha}(x)}
{Y^{\rm D}_{\alpha}(x)}.
\end{align}
%
\par

%----- figure 2 ------
\begin{figure}[htb]
\begin{center}
\scalebox{0.55}{
\includegraphics{profig02.eps}}
\caption{(a) Spatial dependence of the pair potentials 
in a normal metal /$s$-wave superconductor (N/S) junction
with $T=0.02T_{s}$ and $T_{\rm N}/T_{s}=10^{-3}$.
(b) The corresponding local density of states
(LDOS) at the S side of the interface.  
\label{fig:02}}
\end{center}
\end{figure}
%---------------------

%
Here, ${\cal G}^{l}_{\alpha}(x)$ and
${\cal F}^{l}_{\alpha}(x)$
in Eqs.~(\ref{gg}) and~(\ref{ff}), 
obey the following Riccati type equations:
%
\begin{align}
\label{req}
\hbar |v_{{\rm F}x}|
\frac{\partial}{\partial x}{\cal G}^{l}_{\alpha}(x)
= & \alpha \left [
2\omega_{m} {\cal G}^{l}_{\alpha}(x) +
\Delta^{l}(\phi_{\alpha},x)
{\cal G}^{l}_{\alpha}(x)^{2}
\right .
\nonumber \\
&- \left . \Delta^{l}(\phi_{\alpha},x)^{*}
\right ],
\\
\hbar |v_{{\rm F}x}|
\frac{\partial}{\partial x}{\cal F}^{l}_{\alpha}(x)
= &-\alpha \left [
2\omega_{m} {\cal F}^{l}_{\alpha}(x)
- \Delta^{l}(\phi_{\alpha},x)^{*}
{\cal F}^{l}_{\alpha}(x)^{2}
\right . \nonumber \\
&+ \left . \Delta^{l}(\phi_{\alpha},x)
\right ].
\end{align}
%
We can write the quasi-classical Green's function
in a compact form \cite{Nagai}
%
\begin{widetext}
\begin{align}
\label{gn--}
\hat{g}^{\rm N}_{--}(\phi_{-},x) &=
\frac{{\rm i}}{1-{\cal G}^{\rm N}_{-}(x)
{\cal F}^{\rm N}_{-}(x)}
\left (
\begin{array}{cc}
1+{\cal G}^{\rm N}_{-}(x){\cal F}^{\rm N}_{-}(x) &
2{\rm i}{\cal F}^{\rm N}_{-}(x) \\
2{\rm i}{\cal G}^{\rm N}_{-}(x) &
-1-{\cal G}^{\rm N}_{-}(x){\cal F}^{\rm N}_{-}(x)
\\
\end{array}
\right ),
\\
%
\label{gd++}
\hat{g}^{\rm D}_{++}(\phi_{+},x) &=
\frac{{\rm i}}{{\cal G}^{\rm D}_{+}(x)
{\cal F}^{\rm D}_{+}(x)-1}
\left (
\begin{array}{cc}
{\cal G}^{\rm D}_{+}(x){\cal F}^{\rm D}_{+}(x)+1 &
2{\rm i}{\cal F}^{\rm D}_{+}(x) \\
2{\rm i}{\cal G}^{\rm D}_{+}(x) &
-{\cal G}^{\rm D}_{+}(x){\cal F}^{\rm D}_{+}(x)-1
\\
\end{array}
\right ),
\end{align}
\end{widetext}
%
with $\hat{g}^{l}_{--}(\phi_{-},x)
=-\hat{g}^{l}_{++}(-\phi_{+},x)^{\dagger}$.
%
Initial conditions of these equations are as follows, 
%
\begin{align}
{\cal G}_{-}^{\rm N}(-\infty)=0, \quad
{\cal G}_{\alpha}^{\rm D}(\infty)=
\frac{\Delta^{\rm D}(\phi_{\alpha},\infty)^{*}}
{\omega_{m} + \alpha \Omega_{\alpha}^{\rm D}},
\end{align}
%
with $\Omega_{\alpha}^{l}=\sqrt{\omega_{m}^{2}
+|\Delta^{l}(\phi_{\alpha},\infty)|^{2}}$.
%
Moreover, the boundary condition of 
the ${\cal G}_{\alpha}^{l}(0)$ and
${\cal F}_{\alpha}^{l}(0)$
at the interface $x=0$
are 
%
\begin{align}
\label{fn+1}
{\cal F}^{\rm N}_{+} &=
\frac{{\cal G}^{\rm D}_{-} - R{\cal G}^{\rm D}_{+}
- (1-R){\cal G}^{\rm N}_{-}}
{[R{\cal G}^{\rm D}_{-} - {\cal G}^{\rm D}_{+}]
{\cal G}^{\rm N}_{-}
+(1-R){\cal G}^{\rm D}_{+}{\cal G}^{\rm D}_{-}},
\\
%
\label{fn-1}
{\cal F}^{\rm N}_{-} &=
\frac{R{\cal G}^{\rm D}_{-} - {\cal G}^{\rm D}_{+}
+ (1-R){\cal G}^{\rm N}_{+}}
{{\cal G}^{\rm N}_{+}
[{\cal G}^{\rm D}_{-} - R{\cal G}^{\rm D}_{+}]
-(1-R){\cal G}^{\rm D}_{-}{\cal G}^{\rm D}_{+}},
\\
%
\label{fd+2}
{\cal F}^{\rm D}_{+} &=
\frac{R{\cal G}^{\rm N}_{+} - {\cal G}^{\rm N}_{-}
+ (1-R){\cal G}^{\rm D}_{-}}
{{\cal G}^{\rm D}_{-}
[{\cal G}^{\rm N}_{+} - R{\cal G}^{\rm N}_{-}]
-(1-R){\cal G}^{\rm N}_{+}{\cal G}^{\rm N}_{-}},
\\
%
\label{fn-2}
{\cal F}^{\rm D}_{-} &=
\frac{{\cal G}^{\rm N}_{+} - R{\cal G}^{\rm N}_{-}
- (1-R){\cal G}^{\rm D}_{+}}
{[R{\cal G}^{\rm N}_{+} - {\cal G}^{\rm N}_{-}]
{\cal G}^{\rm D}_{+}
+(1-R){\cal G}^{\rm N}_{-}{\cal G}^{\rm N}_{+}}.
\end{align}
\par
%
The pair potentials for both N and D sides are given by
\cite{Matsu2,Ohashi,Ashida,Nagato,Bruder}
%
\begin{align}
\label{peq}
\Delta^{l}(\phi,x) =&
\sum_{0 \leq m < \omega_{c}/2\pi T}
\frac{1}{2\pi}
{\displaystyle
\int ^{\pi/2}_{-\pi/2}
d\phi^{\prime}\sum_{\alpha}
V^{l}(\phi,\phi^{\prime}_{\alpha})}
\nonumber \\
& \times
[\hat{g}^{l}_{\alpha \alpha}(\phi^{\prime}_{\alpha},x)]_{12},
\end{align}
%
where $\omega_{c}$ is the cutoff energy and
$[\hat{g}^{l}_{\alpha \alpha}(\phi_{\alpha},x)]_{12}$
means the 12 element of
$\hat{g}^{l}_{\alpha \alpha}(\phi_{\alpha},x)$.
%
Here $V^{l}(\phi,\phi_{\alpha})$
is the effective inter-electron
potential of the Cooper pair in the $l$ side.
%
In our numerical calculations,
new $\Delta^{l}(\phi_{\alpha},x)$ and 
$\hat{g}^{l}_{\alpha \alpha}(\phi_{\alpha},x)$ are obtained
using Eqs.(\ref{req})-(\ref{gd++}) and (\ref{peq}).
%
We reiterate this process until the convergence is sufficiently
obtained.
\par
%
Based on the self-consistently determined
pair potentials, the LDOS can be calculated
as,
%
\begin{align}
%
N_{l}(E,x) &= \frac{1}{\pi}
{\displaystyle
\int ^{\pi/2}_{-\pi/2}d\phi
n_{l}(E,\phi,x)}, \\
\label{eq:LDOS}
%
n_{l}(E,\phi,x) &={\rm Im} \left \{
\frac{N_{0}}{2}
{\rm Tr}\left [
\hat{g}_{\alpha \alpha}^{l}
(\phi_{\alpha},x)\hat{\tau}_{3}
\right ] \right \}
_{{\rm i}\omega_m
\rightarrow
E+{\rm i}\delta},
\end{align}
%
where $N_{0}$ means the density of states (DOS)
in normal states, and $\delta$ is infinitesimal.
%
In this paper, 
we choose  the temperature $T$ as  $T/T_{d}=0.02$,
where $T_d$ is the critical temperature
of the bulk $d_{x^2-y^2}$-wave superconductor.
\par

%----- figure 3 ------
\begin{figure}[htb]
\begin{center}
\scalebox{0.53}{
\includegraphics{profig03.eps}}
\caption{Spatial dependence of the pair potentials in 
N/D junctions with $\theta = \pi/4$ and
$T_{\rm N}/T_{d}=0.01$.
(a) $R=0$ and $1$,
(b) $R=0.25$, $0.5$, and $0.75$.
\label{fig:03}}
\end{center}
\end{figure}
%---------------------

%-----------------------------------------------------------
\section{Results of numerical calculations}
\label{sec:03}
%-----------------------------------------------------------
In this section, 
we show our results of numerical calculations on
the spatial dependence of the self-consistently determined
pair potentials and the corresponding LDOS.
%
The spatial variation of the 
pair potentials in the N/D junction
is expressed as
%
\begin{align}
\label{del:N}
\Delta^{\rm N}(\phi,x) &=\Delta_{\rm N}(x),
\\
\label{del:D}
\Delta^{\rm D}(\phi,x) &=\Delta_{d}(x)\cos2(\phi - \theta)
+\Delta_{s}(x),
\end{align}
%
where $\Delta_{\rm N}(x)$, $\Delta_{d}(x)$, and $\Delta_{s}(x)$
denote the amplitude of $s$-wave component in the N side,
$d_{x^2-y^2}$-wave component in the D side,
and subdominant $s$-wave component 
in the D side, respectively.
%
The attractive potentials $V^{l}(\phi,\phi^{\prime})$
with $l= {\rm N,D}$, 
are given by
%
\begin{align}
V^{\rm N}(\phi,\phi^{\prime}) &= V_{\rm N},
%
\\
V^{\rm D}(\phi,\phi^{\prime}) &= 2V_{d}
\cos (2\phi-2\theta)\cos (2\phi^{\prime}-2\theta) + V_{s},
\end{align}
%
and
%
\begin{align}
V_{\rm N} &= \frac{2\pi k_{\rm B}T}
{\displaystyle \ln \frac{T}{T_{\rm N}}
+ \sum_{0 \leq m < \omega_{c}/2\pi T}\frac{1}{m+1/2} },
%
\\
%
V_{d} &= \frac{2\pi k_{\rm B}T}
{\displaystyle \ln \frac{T}{T_{d}}
+ \sum_{0 \leq m < \omega_{c}/2\pi T}\frac{1}{m+1/2} },
%
\\
%
V_{s} &= \frac{2\pi k_{\rm B}T}
{\displaystyle \ln \frac{T}{T_{s}}
+ \sum_{0 \leq m < \omega_{c}/2\pi T}\frac{1}{m+1/2} }.
\end{align}
%
Here, $T_{\rm N}$ denotes the
transition temperature of $s$-wave pair potential 
in the N side.
%
$T_{s}$ denotes the transition temperature
of $s$-wave pair potential 
in the D side without $d_{x^2-y^2}$-wave
attractive potential.
\par
%
First,
in order to understand the role of
the proximity effect clearly, let us check the case of
normal metal/$s$-wave superconductor (N/S) junctions.
%
Figure~\ref{fig:02}(a) shows
the obtained spatial dependence of the pair potentials
in the junctions for extremely
low ($R=1$) and high ($R=0$) transparency cases.
%
The $x$-axis of Fig.~\ref{fig:02}
is normalized by $\xi_{s}=\hbar v_{\rm F}/\pi \Delta_{s0}$,
which is the coherence length of
$s$-wave superconductor.
%
For $R=1$,
the $s$-wave pair potential
remains to be constant in S side,
whereas the pair potential in the N side is zero.
The spatial variation of the pair potentials
is represented  as a step function.
%
The resulting LDOS at the interface
reproduces the bulk U-shaped DOS
[see Fig.~\ref{fig:02}(b)].
%
On the other hands, for $R=0$,
the pair potential $\Delta_{\rm N}$ in the N side
survives toward the inside.
%
We can see that the LDOS at the interface
in the presence of the proximity effect
is different from the bulk DOS.
%
\par
%

%----- figure 4 ------
\begin{figure}[htb]
\begin{center}
\scalebox{0.65}{
\includegraphics{profig04.eps}}
\caption{The LDOS for the N/D junction with $\theta = \pi/4$
and $T_{\rm N}/T_{d}=0.01$.
(a) $R=0$, $0.25$, and $0.5$,
(b) $R=0.75$ and $1$.
\label{fig:04}}
\end{center}
\end{figure}
%---------------------

%----- figure 5 ------
\begin{figure}[htb]
\begin{center}
\scalebox{0.53}{
\includegraphics{profig05.eps}}
\caption{Spatial dependencies of the pair potentials 
in N/D junctions with $\theta = 0$ and
$T_{\rm N}/T_{d}=0.01$.
(a) $R=0$ and $1$,
(b) $R=0.25$, $0.5$, and $0.75$.
\label{fig:05}}
\end{center}
\end{figure}
%---------------------


%----- figure 6 ------
\begin{figure}[htb]
\begin{center}
\scalebox{0.45}{
\includegraphics{profig06.eps}}
\caption{(a) The  LDOS at the interface 
by its value in the normal state for various $R$ 
with $\theta = 0$ and $T_{\rm N}/T_{d}=0.01$.
(a) The N side, and (b) the D side.
%
\label{fig:06}}
\end{center}
\end{figure}
%---------------------
%

%-----------------------------------------------------------
\subsection{N/D junction with $\theta=\pi/4$}
\label{sec:03-1}
%-----------------------------------------------------------
%
In this subsection, 
we focus on the N/D junction 
with (110)-oriented interface ($\theta = \pi/4$). 
%
As the parameter of the subdominant $s$-wave 
component, we take $T_{s}/T_{d}=0.3$ in the following. 
%
In Figs.~\ref{fig:03}(a) and \ref{fig:03}(b),
the spatial variations of the pair potentials
is shown for $T_{\rm N}/T_{d}=0.01$
for $R=0$, $1$, and $0.25$, $0.5$, $0.75$,
respectively.
%
Here,
Re[$\Delta_{d}(x)$] and Im[$\Delta_{{\rm N},s}(x)$]
denote real part of $\Delta_{d}(x)$ and imaginary
part of $\Delta_{{\rm N},s}(x)$, respectively.
%
The $x$-axis of Fig.~\ref{fig:03}
is normalized by the $d_{x^2-y^2}$-wave
coherence length $\xi_{d}=\hbar v_{\rm F}/\pi \Delta_{d0}$,
where $\Delta_{d0}$ is $d_{x^2-y^2}$-wave pair potential
in bulk states.
\par
%
First,
we present our results in the light of
previous theories. \cite{Hu,Tanaka9371,Matsu2}
%
The reduction of Re[$\Delta_{d}(x)$]
originates from a depairing effect that the
effective pair potentials $\Delta^{\rm D}(\phi_{+},0)$
and $\Delta^{\rm D}(\phi_{-},0)$ have reversed
contribution to the pairing interaction
for certain range of $\phi$.
%
Re[$\Delta_{d}(x)$] is suppressed at the interface
in the D side. 
%
At the same time, 
the quasiparticle forms the ABS with zero-energy
at the interface. \cite{Hu,Tanaka9371}
%
When the magnitude of the reflection probability
$R$ approaches unity, 
the ABS becomes unstable with the introduction of
the subdominant $s$-wave attractive potential in the D side.
%
And then Im[$\Delta_{s}(x)$] is induced
in the vicinity of the interface in the D side
in low transparent cases with large magnitude of $R$.
\cite{Matsu2}
%
As regards the proximity effect in the N side,
the pair potential $\Delta_{{\rm N}}(x)$
is not induced for fully high and low
transparent cases.
%
However,
the imaginary component of $\Delta_{{\rm N}}(x)$
is enhanced nearly in $R=0.75$
[see Fig.~\ref{fig:03}].
%
We can recognize that
the existence of Im[$\Delta_{s}(x)$]
near the interface of D side
can allow the enhancement of the Im[$\Delta_{{\rm N}}(x)$]. 
%
This fact is more recently found
by L\"{o}fwander, \cite{Lof}
and our results are consistent with his work.
\par
%-----------------------------------------------------------
In Figs.~\ref{fig:04}(a) and \ref{fig:04}(b),
we show the corresponding LDOS
at the D side of the interface, 
where the same parameters of Figs.~\ref{fig:03}(a)
and \ref{fig:03}(b) are used, 
respectively.
%
For $R=0$, $N_{\rm D}(E,0)$ is equal to $N_{0}$
and has no $E$ dependence.
%
If we neglect the induced $s$-wave component in the N side,
the resulting LDOS has a zero-energy enhanced line shapes
for small $R$.
\cite{Tanu5}
%
This is because that the magnitude of the subdominant 
$s$-wave component, \textit{i.e.}, Im[$\Delta_{s}(x)$]
is very small and the incident and
reflected quasiparticles normal to the interface
can feel opposite signs of the $d_{x^2-y^2}$-wave
pair potentials.
%
However, as shown in Fig.~\ref{fig:03}(b),
not only Im[$\Delta_{{\rm N}}(x)$]
but also Im[$\Delta_{s}(x)$]
remain to be non-zero due to the proximity effect.
%
In this case, the resulting LDOS has complex 
line shapes.
%
For $R=1$, due to the enhancement of the magnitude of Im[$\Delta_{s}(x)$], 
the LDOS has the ZEP splitting, which is consistent with the 
previous works \cite{Tanu5,FRS97,Matsu2}. 
%
\par

%----- figure 7 ------
\begin{figure}[htb]
\begin{center}
\scalebox{0.50}{
\includegraphics{profig07.eps}}
\caption{The LDOS at the interface near zero energy with $R=0$
for various $\delta$.
\label{fig:07}}
\end{center}
\end{figure}
%---------------------


%----- figure 8 ------
\begin{figure}[htb]
\begin{center}
\scalebox{0.44}{
\includegraphics{profig08.eps}}
\caption{The LDOS at $x<0$ with $\theta = 0$.
(a) $R=0$ and (b) $R=0.1$ for various $x$.
Here
$\xi_{\rm N}=v_{\rm F}/2\pi T$ is the 
coherence length in the N side with $T=0.02T_{d}$. 
%
\label{fig:08}}
\end{center}
\end{figure}
%---------------------

%----- figure 09 ------
\begin{figure}[htb]
\begin{center}
\scalebox{0.50}{
\includegraphics{profig09.eps}}
\caption{Angle resolved LDOS at the interface
for $\theta = 0$ and $R=0$.
\label{fig:09}}
\end{center}
\end{figure}
%---------------------

%-----------------------------------------------------------
\subsection{N/D junction with $\theta=0$}
\label{sec:03-2}
%-----------------------------------------------------------
In this subsection, 
we focus on the N/D junction with $\theta = 0$
and $T_{\rm N}/T_{d}=0.01$.
%
The spatial dependencies of the pair potentials
Re[$\Delta_{\rm N}(x)$], Re[$\Delta_{d}(x)$],
and Re[$\Delta_{s}(x)$],
are plotted in Figs.~\ref{fig:05}(a) and \ref{fig:05}(b),
for $R=0$, $1$, and $0.25$, $0.5$, $0.75$,
respectively.
%
For the low transparent limit, $R=1$, 
[see Fig.~\ref{fig:05}(a)],
the amplitude of the pair potential is constant.
%
This situation is similar to the case
of the N/S junctions with $R=1$
[see Fig.~\ref{fig:02}(a)].
%
With the decrease of $R$, 
Re[$\Delta_{d}(x)$] is suppressed near the interface,
while Re[$\Delta_{s}(x)$] is slightly mixed
in the vicinity of the interface.
%
Since Im[$\Delta_{s}(x)] =0$ is satisfied,
the TRS is not broken
near the (100) interface.
%
On the other hand for the N side,
we can readily see that the pair potential
Re[$\Delta_{\rm N}(x)$] is induced. 
%
This indicates that the superconducting
pair potential penetrates into the N side
due to the proximity effect.
\cite{Ohashi}
%
With increasing the magnitude of $T_{\rm N}$,
the amplitude of Re[$\Delta_{\rm N}(x)$] becomes
larger.
%
However, both Im[$\Delta_{s}(x)] =0$ and
Im[$\Delta_{\rm N}(x)] =0$ are satisfied. 
%
This situation is 
significantly different from the corresponding 
case of $\theta = \pi/4$ where the TRS is broken. 
\par
%
Next, let us look at the corresponding LDOS
at the interface. 
%
We show that the LDOS at the interface in 
the N and D sides 
as shown in Fig.~\ref{fig:06}.
%
For $R=1$, the LDOS on the D side has a V-shaped structure
similar to the bulk $d$-wave density of states.
\cite{Tanu5,FRS97,Matsu2}
%
The corresponding LDOS in the N side is constant. 
%
With the decrease of the magnitude of $R$, 
the magnitude of LDOS in the D side 
around zero energy is enhanced
while that around $\pm \Delta_{d0}$ is suppressed.
%
For $R=0.25$, the LDOS has a small dip like structure 
both in the N and D sides.
%
The fine structure around zero energy
is due to the induced 
pair potential $\Delta_{\rm N}(x)$ in the N side. 
%
The extreme case is $R=0$, where normal reflection 
is absent.  
%
The pair potential Re[$\Delta_{\rm N}(x)$]
induces the ZEP in the LDOS [see Fig.~\ref{fig:06}]
through Andreev reflection. \cite{Ohashi}
%
%By comparing Figs.~\ref{fig:06}(a) and ~\ref{fig:06}(b),
%we can see that the dip and peak structure in the LDOS
%becomes much clear 
%with the increase of the magnitude of $T_{N}$ 
%due to the enhancement of the amplitude of $\Delta_{\rm N}(x)$.
\par
%
Now we concentrate on the width of this ZEP.
%
We see that the relevance of the peak width 
and the infinitesimal number $\delta$ 
introduced in Eq.(\ref{eq:LDOS}) in order to avoid 
divergence in the actual calculation where 
the inverse of $\delta$ can be regarded
as a life time of quasiparticles. 
%
As shown in Fig.~\ref{fig:07}, the width of the ZEP becomes narrow
with the decrease of the magnitude of $\delta$, 
while the height of it increases monotonically. 
%
For $\delta \rightarrow 0$ limit, 
the ZEP is reduced to be expressed by the $\delta$-function.
\par
%
In Fig.~\ref{fig:08}, 
we show the spatial dependence of LDOS in the N side ($x<0$). 
%
For $R=0$, the zero energy states formed at the 
interface penetrate into the N side.
%
The width and the height of the ZEP is reduced 
with the increase of the magnitude of $x$. 
%
For $R=0.1$, the LDOS has a mini gap. 
%
The width of the gap has a spatial dependence
since the induced pair potential
Re[$\Delta_{\rm N}(x)$] depends on $x$.
%
The width of the mini gap is reduced  with the 
increase of the magnitude of $x$. 
%
\par
%
At the end of this subsection, 
let us remark on the difference
on the origin of ABS's between
$\theta =\pi/4$ and $\theta =0$.
%
As mentioned in Sec.~\ref{sec:03-1},
the formation of ABS's is originated from
the opposite sign of the pair potentials 
between $\Delta_{d}(\phi_{+},0)$ and
$\Delta_{d}(\phi_{-},0)$ felt by 
quasiparticles in the D side.
%
However, in the case of $\theta =0$,
there is no sign change between 
$\Delta_{d}(\phi_{+},0)$ and
$\Delta_{d}(\phi_{-},0)$ for any $\phi $.
%
This manifestation of the ZEP in the LDOS
is due to the another origin. 
%
It is due to the sign change of the pair potentials 
between Re$[\Delta_{\rm N}(x)]$
and Re$[\Delta_{s}(x)]$
or that between Re$[\Delta_{\rm N}(x)]$ 
and Re$[\Delta_{d}(x)]$ for $|\phi |>\pi/4$
through the Andreev reflection.
%
Actually,
by calculating
the angle resolved LDOS $n_{\rm D}(E,\phi,x)$
in Eq.(\ref{eq:LDOS}), 
$n_{\rm D}(E,\phi,x)$ has the ZEP for $\phi =\pi/3$
while it does not for $\phi =\pi/6$
as shown in Fig.~\ref{fig:09}.
%
\par
%

%----- figure 10 ------
\begin{figure*}[htb]
\begin{center}
\scalebox{0.90}{
\includegraphics{profig10.eps}}
\caption{Spatial dependencies of the pair potentials  for 
the N/D junctions with $\theta = \pi/6$
and $T_{\rm N}/T_{d}=0.01$.
%
In (a), Re$[\Delta_{\rm N}(x)]$,
Re$[\Delta_{d}(x)]$, and
in (b), Im$[\Delta_{\rm N}(x)]$, Im$[\Delta_{d}(x)]$,
Re$[\Delta_{s}(x)]$, Im$[\Delta_{s}(x)]$,
are plotted, respectively.
\label{fig:10}}
\end{center}
\end{figure*}
%---------------------
%

%----- figure 11 ------
\begin{figure}[htb]
\begin{center}
\scalebox{0.65}{
\includegraphics{profig11.eps}}
\caption{The LDOS for the N/D junctions
with $\theta = \pi/6$ and $T_{\rm N}/T_{d}=0.01$.
(a) $R=0$, $0.25$, and $0.5$,
(b) $R=0.75$ and $1$.
\label{fig:11}}
\end{center}
\end{figure}
%---------------------
%-----------------------------------------------------------
\subsection{N/D junction with $\theta=\pi/6$}
\label{sec:03-3}
%-----------------------------------------------------------
In this subsection, we concentrate on the case
with intermediate angle $\theta$,
\textit{e.g.}, $\theta = \pi/6$.
%
The spatial dependencies of the pair potentials 
become very complicated as shown in  Fig.~\ref{fig:10},
where we choose $T_{\rm N}/T_{d}=0.01$ for various $R$. 
%
For $R=0$, 
the amplitude of Re[$\Delta_{\rm N}(x)$] is induced 
toward the inside of N.
This situation is similar to the case of $\theta =0$
by the proximity effect.
%
With the increase of $R$,
the amplitude of Re[$\Delta_{d}(x)$] is significantly
suppressed at the interface, 
due to the destructive interference originating
from the sign changing 
nature of $d_{x^{2}-y^{2}}$-wave pair potential. 
%
The amplitude of Re[$\Delta_{\rm N}(x)$] is suppressed 
due to the reduction of the proximity effect. 
%
On the other hand, amplitudes of Im[$\Delta_{d}(x)$], 
Re[$\Delta_{s}(x)$], and Im[$\Delta_{s}(x)$] are enhanced. 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As regards the subdominant components,
the amplitude of Im[$\Delta_{s}(x)$] for lower transparent
cases $R=0.75$ and $R=1$,
is one order larger than those of
Im[$\Delta_{d}(x)$], and Re[$\Delta_{s}(x)$]
[see Fig.~\ref{fig:10}(b)].
%
In particular, for $R=0.75$, the situation that
the amplitude of Im[$\Delta_{\rm N}(x)$] is induced near
the interface of N side
is similar to the case of $\theta =\pi/4$.
%
Subdominant components which break TRS
is induced also by the proximity effect
and depends on the transparencies of the junction.
\par
%----------------
The corresponding LDOS for $\theta = \pi/6$
are plotted in Figs.~\ref{fig:11}(a)
and \ref{fig:11}(b).
%
Here, we choose $T_{\rm N}/T_{d}=0.01$,
which is the same parameters
used in Fig.~\ref{fig:10}.
%
The resulting line shapes of LDOS
are complex reflecting on the complicated
spatial dependence of the pair potentials.
%
For $R=0$,
we can see that the zero energy states
are formed at the interface
and the resulting LDOS has a ZEP 
due to the \textit{proximity effect}. 
%
This ZEP, the origin of which is similar
to the case of $\theta =0$ [see Fig.~\ref{fig:06}(b)],
arises from the sign change of the pair potentials 
between Re[$\Delta_{\rm N}(x)$] and Re[$\Delta_{d,s}(x)$]
through Andreev reflections due to proximity effect.
%
With the increase of $R$, the LDOS has a rather broad ZEP.
due to the formation of \textit{ABS's}.
%
This ZEP for intermediate value of $R$ originates from
the formation of ABS's peculiar to unconventional
superconductors \cite{TK95,KT00},
\textit{i.e.} with sign change of the pair potentials
between $\Delta_{d}(\phi_{+},0)$ and $\Delta_{d}(\phi_{-},0)$.
%
With the further increase of $R$, (see the curve with  $R=0.75$),
since the imaginary components of both $\Delta_{\rm N}(x)$
and $\Delta_{s}(x)$ are induced in the vicinity of the interface,
the line shape of the LDOS has a complicated structure
reflecting on the complex spatial dependence of pair potentials.
%
%
For $R=1$, the LDOS has the ZEP splitting
due to the existence of subdominant
$s$-wave component which breaks TRS in the D side,
\textit{i.e.}, Im[$\Delta_{s}(x)$].
%
As a results,
in the N/D junctions with $\theta = \pi/6$,
we can conclude that the line shapes of the LDOS
around zero energy
change from  
(I) \textit{the ZEP due to proximity effect},  
(II) \textit{the ZEP due to the ABS},
and (III) \textit{the ZEP splitting due to the formation of the BTRSS}, 
with the increase of the magnitude of $R$.
%
\par
%
Finally,
we look at the angle resolved LDOS
$n_{\rm D}(E,\phi,x)$ in order to understand the basic features
of the line shapes of $N_{\rm D}(E,0)$.
%
In Figs.~\ref{fig:12} and \ref{fig:13},
the angle resolved LDOS $n_{\rm D}(E,\phi,x)$
at the interface is plotted for $\theta = \pi/6$ and $R=0.75$.
%
In order to understand the role of the induced pair potential 
in N region,  we intentionally neglect the pair potential
$\Delta_{\rm N}(x)$ in the N side in the actual calculation of LDOS 
as shown in the upper panel of Fig.~\ref{fig:12} and Fig.~\ref{fig:13}. 
%
We also consider the case, where induced $s$-wave component 
$\Delta_{s}(x)$ is neglected in the calculation of LDOS.
%
First we concentrate on the upper panels
where $\Delta_{N}(x)$ is absent.
%
If only the pure $d_{x^2-y^2}$-wave component exists
in the D side, 
the quasiparticles form the ABS's at zero energy.
%
For $\phi = \pm \pi/3$,
since $\Delta_{d}(\phi_{+},0) \Delta_{d}(\phi_{-},0)<0$
is satisfied, $n_{\rm D}(E,\phi,0)$ has the ZEP
[see Figs.~\ref{fig:12}(a) and \ref{fig:12}(b)].
%
In the presence of the subdominant $s$-wave component 
which breaks TRS in the D side,
the resulting $n_{\rm D}(E,\phi,0)$ is modified.
%
In this case, the line shape of $n_{\rm D}(E,\phi,0)$
is not asymmetric around $E=0$, \textit{i.e.},
where $n_{\rm D}(E,\phi,0) \neq n_{\rm D}(-E,\phi,0)$, 
due to the formation of the BTRSS.
%
Therefore, the peak position of $n_{\rm D}(E,\phi,0)$
is shifted from zero energy.
%
On the other hands, for $\phi = \pm \pi/20$,
as $\Delta_{d}(\phi_{+},0) \Delta_{d}(\phi_{-},0)>0$
is satisfied, $n_{\rm D}(E,\phi,0)$ has gap structures
[see Figs.~\ref{fig:13}(a) and \ref{fig:13}(b)].
%
These results are consistent with those
by Matsumoto and Shiba.
\cite{Matsu3}
%
Next, we look at lower panels of  Figs.~\ref{fig:12} and \ref{fig:13}. 
%
Since  induced Im[$\Delta_{\rm N}(x)$] breaks time reversal symmetry, 
the resulting line shape of LDOS is no more symmetric around $E=0$, 
even for $\Delta_{s}(x)=0$. 
%
As shown in the lower panels, the presence of $\Delta_{\rm N}(x)$
forms a minigap around zero energy $E=0$.
\par
%------
In Fig.~\ref{fig:14},
the LDOS $n_{\rm D}(E,\phi,0)$ is plotted
for $R=0$, where the Andreev reflection only exists. 
%
In this case, the line shape of the LDOS can be understood only 
by taking into account the pair potentials
$\Delta_{\rm N}(x)$ and $\Delta_{d}(x)$. 
%
For $\phi = \pi/3$ and $\phi = \pm \pi/20$,
the LDOS has no ZEP since
$\Delta_{\rm N}(\phi_{+},0) \Delta_{d}(\phi_{+},0)>0$
is satisfied. 
The resulting  $n_{\rm D}(E,\phi,0)$ has a gap structure
[see Figs.~\ref{fig:14}(a) and \ref{fig:14}(b)].
%
If the quasiparticle's scattering
feel different sign of the pair potentials
between the N and D sides,
we can expect that the $n_{\rm D}(E,\phi,0)$ has the ZEP even for $R=0$.
%
For $\phi = -\pi/3$, since 
$\Delta_{\rm N}(\phi_{+},0) \Delta_{d}(\phi_{+},0)<0$
is satisfied, then the  $n_{\rm D}(E,\phi,0)$
has the ZEP [see Fig.~\ref{fig:14}(b)].
%
These situations are similar to those studied in Sec.~\ref{sec:03-2}.
%
%
%The mechanism of the formation of ABS
%in the $N/D$ junction
%is illustrated in Fig.~\ref{fig:14}.
%
\par

%----- figure 12 ------
\begin{figure*}[htb]
\begin{center}
\scalebox{0.88}{
\includegraphics{profig12.eps}}
\caption{The angle resolved LDOS $n_{\rm D}(E,\phi,x)$
at the interface with $\theta = \pi/6$ and $R=0.75$.
(a): LDOS without subdominant $s$-wave pair potential in the N side.
%
(b): LDOS in the presence of subdominant $s$-wave pair potential
in the N side.
$\phi = \pi/3$ for the left panel and 
$\phi = -\pi/3$ for the right panel.
%
\label{fig:12}}
\end{center}
\end{figure*}
%---------------------


%----- figure 13 ------
\begin{figure*}[htb]
\begin{center}
\scalebox{0.88}{
\includegraphics{profig13.eps}}
\caption{The angle resolved LDOS $n_{\rm D}(E,\phi,x)$
at the interface with $\theta = \pi/6$ and $R=0.75$.
(a): LDOS without subdominant $s$-wave pair potential in the N side.
%
(b): LDOS in the presence of subdominant $s$-wave pair potential
in the N side. $\phi = \pi/20$ for the left panel and 
$\phi = -\pi/20$ for the right panel.
%
\label{fig:13}}
\end{center}
\end{figure*}
%---------------------

%----- figure 14 ------
\begin{figure}[htb]
\begin{center}
\scalebox{0.65}{
\includegraphics{profig14.eps}}
\caption{The angle resolved LDOS $n_{\rm D}(E,\phi,x)$
at the interface with $\theta = \pi/6$ and $R=0$.
(a) $\phi = \pi/3$, $\pi/20$, and
%
(b) $\phi =-\pi/3$, $-\pi/20$. 
%
\label{fig:14}}
\end{center}
\end{figure}
%---------------------

%-----------------------------------------------------------
\section{Conclusions}
\label{sec:04}
%-----------------------------------------------------------
%
In this paper, 
we have studied quasiparticle properties in the N/D junctions,
in the presence of the proximity effect
as well as the BTRSS in the D side. 
%
We assume the attractive interelectron potentials 
which induces subdominant $s$-wave components
both in the N and D sides.  
%
The spatial dependencies of the pair potentials 
in the N/D junctions are determined self-consistently. 
%
The LDOS at the interface of the N/D junctions 
are studied in detail by changing orientational angle $\theta$
and the transparency of the junction. 
%
Our main results are summarized as follows.
\par
%---------
i)
For (110) oriented junction with $\theta=\pi/4$, 
the predominant $d_{x^2-y^2}$-wave component in the D side
is reduced at the interface. 
%
The subdominant $s$-wave component which breaks TRS
appears near the (110) interface in low transparent cases.
%
In the N side, subdominant imaginary $s$-wave component 
is induced near the interface \cite{Lof}. 
%
The resulting LDOS at the interface in the N side 
has the ZEP or its splitting
depending on the transparency of the junction.
%
These results are consistent with previous theoretical
works. \cite{Tanu5,FRS97,Lof,Matsu2,Matsu3}
\par
%
%--------
ii) For (100) oriented junction, 
the subdominant $s$-wave component becomes 
real number and does not break TRS \cite{Lof}. 
%
With the increase of the transparency of the junction, 
the magnitude of the $s$-wave component in the N side
is enhanced by the proximity effect. 
%
For fully transparent case, 
the LDOS has very sharp ZEP due to the formation of 
zero energy states.
%
This ZEP originates from the fact that
quasiparticles feel different sign
of the pair potentials between
the N and D sides through Andreev reflection.
%
\par
%-------
iii) For $\theta = \pi/6$,
the spatial dependencies of the pair potentials
become very complicated. 
%
The resulting LDOS has a wide variety.
For high barrier limit, the LDOS has the ZEP splitting.
%
\par
%
In the light of our results,
although the penetration of the pair potential into the N side
by the proximity effect 
is expected for (100) oriented junctions, 
the subdominant $s$-wave components both in 
the N and D sides are real numbers. 
%
Thus, the BTRSS is not formed in the (100) junctions.
%
The present results are inconsistent with 
previous prediction based on 
tunneling experiment on high-$T_{c}$ cuprates
by Kohen \textit{et al.}
\cite{Kohen}
%
%
In order to understand their experimental results,
we must seek for other possibilities. 
\par
In the present paper, we study proximity effect in 
N/D junctions in the ballistic limit.
In the actual junctions, we can not neglect 
impurity scattering effect.
%
To reply this issue, two of the present author 
present theoretical works of charge transport in 
N/D junctions in the presence of the ABS 
where the N region is a diffusive metal. 
\cite{r2,Tanaka2004}
%
It is an interesting problem to extend the present theory 
in the diffusive regime based on the Keldysh Green's function
formalism.
\cite{Tanaka2004}
%
Although there are many works about ABS's in unconventional superconductor
junctions up to now, \cite
{a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,o1,o2,o3,o4,o5,K95,m1,m2,m3}
proximity effects both in the presence of induced pair potential
in the N region and the ABS's are not fully studied. 
%
Recent study by L\"{o}fwander is remarkable 
where the proximity effect is studied in the 
presence of both impurity scattering
and induced subdominant component of the 
pair potentials. \cite{Lof}
%
Based on his detailed calculation,  
for (100) junction, a real combination of the proximity effect 
is always found. 
%
For (110) orientation, the $s$-wave component induced
by the proximity effect in the N side breaks TRS. 
\cite{r4,Lof}
These results are consistent with the present results. 
\par
%
%
As a future problem, to clarify the charge transport property
in Josephson junctions both in the presence of ABS
\cite{J1,J2,J3,J4,J5,J6,J7,J8,J9,J10,J11,J12,J13,J14,J15,J16,J17,J18,J19}
and proximity effect is interesting.
%
We want to extend the present theory towards these directions. 
\par
%-----------------------------------------------------------
\section*{Acknowledgments}
%
We would like to sincerely thank Dr. T. L\"{o}fwander
for critical reading of the manuscript 
and valuable discussions.
The computations have been performed
at the Supercomputer Center of
Yukawa Institute for Theoretical Physics,
Kyoto University.

\appendix

%-----------------------------------------------------------
\section{Bogoliubov-de Gennes equation
and quasi-classical approximation}
\label{app:01}
%-----------------------------------------------------------
Our theoretical formalism is started from
the Bogoliubov-de Gennes (BdG) equation
for unconventional spin-singlet superconductors,
%
\begin{align}
E_{n}\tilde{u}_{n}(\bm{r}) &= H_{0}\tilde{u}_{n}(\bm{r})
+ \int d \bm{r^{\prime}}
\Delta(\bm{r},\bm{r^{\prime}})
\tilde{v}_{n}(\bm{r^{\prime}}),
\\
E_{n}\tilde{v}_{n}(\bm{r}) &= -H_{0}\tilde{v}_{n}(\bm{r})
+ \int d \bm{r^{\prime}}
\Delta(\bm{r},\bm{r^{\prime}})^{*}
\tilde{u}_{n}(\bm{r^{\prime}}), \\
\nonumber
%
H_{0} &= -\frac{\hbar^{2}}{2m}\nabla^{2} - \mu,
\end{align}
%
where $\mu$ is the chemical potential,
while $\tilde{u}_{n}(\bm{r})$ and $\tilde{v}_{n}(\bm{r})$
denote the electron like and holelike components
of the wave function
%
\begin{align}
\tilde{\Psi}_{n}(\bm{r}) &=
\left (
    \begin{array}{c}
        \tilde{u}_{n}(\bm{r}) \\
        \tilde{v}_{n}(\bm{r})
    \end{array}
\right ), \nonumber \\
&\equiv
\left (
    \begin{array}{c}
        u_{n}(\hat{\bm{k}},\bm{r}) \\
        v_{n}(\hat{\bm{k}},\bm{r})
    \end{array}
\right ) e^{{\rm i}k_{\rm F} \hat{\bm{k}} \cdot \bm{r}}
%
=\Psi_{n}(\hat{\bm{k}},\bm{r})
e^{{\rm i}k_{\rm F} \hat{\bm{k}} \cdot \bm{r}}.
\end{align}
%
Here the quantities
$\hat{\bm{k}}$ and $\bm{r}$
stand for the unit vector of
the wave number of the Cooper pair which
is fixed on the Fermi surface
$(\hat{\bm{k}}=\bm{k}_{\rm F}/|\bm{k}_{\rm F}|)$,
and the  position of the center of mass of Cooper pair,
respectively. 
%
After applying the quasi-classical approximation,
the BdG equation is reduced to the Andreev equation
\cite{Hu,Bruder,Kurki},
%
\begin{align}
%
E_{n}\Psi_{n}(\hat{\bm{k}},\bm{r})
   &= -\left [ {\rm i}\hbar v_{\rm F}\hat{\bm{k}} \cdot \nabla
      + \hat{\Delta}(\hat{\bm{k}},\bm{r}) \right ]
       \hat{\tau}_{3}\Psi_{n}(\hat{\bm{k}},\bm{r}), \\
%
\hat{\Delta}(\hat{\bm{k}},\bm{r}) &=
\left (
    \begin{array}{cc}
        0 & \Delta(\hat{\bm{k}},\bm{r}) \\
  -\Delta(\hat{\bm{k}},\bm{r})^{*} & 0
    \end{array}
\right ).
\end{align}
%
The wave function $\Psi_{n}(\hat{\bm{k}},\bm{r})$
is obtained by neglecting the rapidly oscillating
plane-wave part following the quasi-classical approximation
\cite{Bruder,Kurki}.
%
The $\hat{\bm{k}}$ dependence of
$\Delta(\hat{\bm{k}},\bm{r})$
represents the symmetry of the pair potential.
\par
%
In the present study, we consider the case where a secularly
reflecting surface or interface runs along the $y$ direction.
Then, 
the pair potential depends only on $x$
since the system is homogeneous along the $y$ direction.
%
The wave function $\Psi_{n}(\hat{\bm{k}},\bm{r})$ 
can be written in the following directional
notation:\cite{Matsu2,Ashida}
%
\begin{align}
\Psi_{n}(\hat{\bm{k}},x,y) =& \left [
 \Phi_{n}^{(+)}(\phi_{+},x)e^{{\rm i}|k_{{\rm F}x}|x}
\right .
\nonumber \\
&+ \left .
\Phi_{n}^{(-)}(\phi_{-},x)e^{-{\rm i}|k_{{\rm F}x}|x}
\right ] e^{{\rm i}|k_{{\rm F}y}|y},
\nonumber \\
\Phi_{n}^{(\alpha)}(\phi_{\alpha},x) =&
\left (
    \begin{array}{c}
        u_{n}^{(\alpha)}(\phi_{\alpha},x) \\
        v_{n}^{(\alpha)}(\phi_{\alpha},x)
    \end{array}
\right ).
\end{align}
%
Here $\pm$ represents
the sign of the $x$ component of the Fermi wave number
$k_{{\rm F}x}$ and $\alpha (\beta) = \pm$.
We define a Green's function
$G_{\alpha \beta}(\phi,x,x^{\prime})$
and a quasi-classical Green's function
$g_{\alpha \beta}(\phi,x)$
%
\begin{align}
G_{\alpha \beta}(\phi,x,x^{\prime})
&= \sum_{n}\frac{\Phi_{n}^{(\alpha)}(\phi_{\alpha},x)
\Phi_{n}^{(\beta)}(\phi_{\beta},x^{\prime})^{\dagger}}
{{\rm i}\omega_{m} -E_{n}}, \\
%
g_{\alpha \beta}(\phi,x) \pm {\rm i}
(\hat{\gamma}_{3})_{\alpha \beta}
&= -2\hbar|v_{{\rm F}x}|\hat{\tau}_{3}
G_{\alpha \beta}(\phi,x\pm 0,x).
\end{align}
%
where $\hat{\gamma}_{3}$ is the Pauli matrix
in the directional space\cite{Ashida}.
\par

%-----------------------------------------------------------
\section{Evolution operator
$\tilde{U}^{l}_{\alpha}(\phi_{\alpha},x,x^{\prime})$}
\label{app:02}
%-----------------------------------------------------------
The quasi-classical Green's function
can be written by the following evolution operator
$U_{\alpha}(\phi_{\alpha},x,x^{\prime})$ as
%
\begin{eqnarray}
g_{\alpha \beta}(\phi,x)
   =U_{\alpha}(\phi_{\alpha},x,x^{\prime})
     g_{\alpha \beta}(\phi,x^{\prime})
      U_{\beta}^{-1}(\phi_{\beta},x,x^{\prime}),
\end{eqnarray}
%
where $U_{\alpha}(\phi_{\alpha},x,x^{\prime})$
satisfies the Andreev equation
%
\begin{align}
{\rm i}\hbar |v_{{\rm F}x}| &
\frac{\partial}{\partial x}
U_{\alpha}(\phi_{\alpha},x,x^{\prime})
\nonumber \\
&=-\alpha
\left [ {\rm i}\omega_{m}\hat{\tau}_{3}
+\hat{\Delta}(\phi_{\alpha},x)\right ]
U_{\alpha}(\phi_{\alpha},x,x^{\prime}),
\end{align}
%
with $U_{\alpha}(\phi_{\alpha},x,x)=1$.
\par
Hence,
the evolution operators in the $l$ side
can be divided
into a growing part and a decaying part
%
\begin{align}
U_{\alpha}^{l}(\phi_{\alpha},x,x^{\prime}) =&
\Lambda_{\alpha}^{l(+)}(\phi_{\alpha},x,x^{\prime})
e^{{\kappa^{l}_{\alpha}}(x-x^{\prime})}
\nonumber \\
&+ \Lambda_{\alpha}^{l(-)}(\phi_{\alpha},x,x^{\prime})
e^{-{\kappa^{l}_{\alpha}}(x-x^{\prime})},
\\
%
\Lambda_{\alpha}^{l(+)}(\phi_{\alpha},x,x^{\prime})
=& -\frac{1}{W^{l}_{\alpha}}
\Phi_{n}^{l(+)}(\phi_{\alpha},x)
{}^{\rm T}\Phi_{n}^{l(-)}(\phi_{\alpha},x^{\prime})
\hat{\tau}_{2},
\nonumber \\
\Lambda_{\alpha}^{l(-)}(\phi_{\alpha},x,x^{\prime})
=& \frac{1}{W^{l}_{\alpha}}
\Phi_{n}^{l(-)}(\phi_{\alpha},x)
{}^{\rm T}\Phi_{n}^{l(+)}(\phi_{\alpha},x^{\prime})
\hat{\tau}_{2},
%
\end{align}
where
%
\begin{align}
\kappa^{l}_{\alpha} &=
\frac{\Omega^{l}_{\alpha}}{|v_{{\rm F}x}|},
\quad
\Omega_{\alpha}^{l}=\sqrt{\omega_{m}^{2}
+|\Delta^{l}(\phi_{\alpha},\infty)|^{2}},
\\
W^{l}_{\alpha} &=
{}^{\rm T}\Phi_{n}^{l(+)}(\phi_{\alpha},x)\hat{\tau}_{2}
\Phi_{n}^{l(-)}(\phi_{\alpha},x)
\nonumber \\
&=-{}^{\rm T}\Phi_{n}^{l(-)}(\phi_{\alpha},x)\hat{\tau}_{2}
\Phi_{n}^{l(+)}(\phi_{\alpha},x)
\nonumber \\
&= \mathrm{const.}
\end{align}
%
In the above, $^{\rm T}\Phi_{n}^{l(\alpha)}(\phi_{\alpha},x)$
denotes the transposition of
$\Phi_{n}^{l(\alpha)}(\phi_{\alpha},x)$. 
In the actual numerical calculations,  
we use $\tilde{U}^{l}_{\alpha}(\phi_{+},x,0)$ and 
$\tilde{U}^{l}_{\alpha}(\phi_{+},0,x)$
given by 
%
\begin{align}
\tilde{U}^{\rm N}_{+}(\phi_{+},x,0)
& = U^{\rm N}_{+}(\phi_{+},x,0)e^{\kappa^{\rm N}_{+}x},
\\
\tilde{U}^{\rm D}_{+}(\phi_{+},0,x)
& = U^{\rm D}_{+}(\phi_{+},0,x)e^{-\kappa^{\rm D}_{+}x}.
\end{align}
\par
%-----------------------------------------------------------
\section{For avoiding divergence
in our calculation}
\label{app:03}
%-----------------------------------------------------------
In particular,
if $\Delta^{\rm N}(\phi_{+},-\infty)$
$[\Delta^{\rm D}(\phi_{-}, \infty)]=0$,
we can find ${\cal G}^{\rm N}_{+}$
$[{\cal G}^{\rm D}_{-}] \rightarrow \infty$.
%
In this case, instead of Eq.~(\ref{req}),
we solve the following equation as
%
\begin{align}
\hbar |v_{{\rm F}x}|
\frac{\partial}{\partial x}
\left ( \frac{1}
{{\cal G}^{l}_{\alpha}(x)}
\right )
=& -\alpha \left [
2\omega_{m} \left ( \frac{1}
{{\cal G}^{l}_{\alpha}(x)}
\right ) \right . \nonumber \\
+ \Delta^{l}(\phi_{\alpha},x)^{*}
& \left . \left ( \frac{1}
{{\cal G}^{l}_{\alpha}(x)}
\right )^{2}
- \Delta^{l}(\phi_{\alpha},x)
\right ],
\end{align}
%
under initial condition,
%
\begin{align}
\frac{1}{{\cal G}^{l}_{\alpha}(x)}
= - \frac{\Delta^{l}(\phi_{\alpha},-\alpha \infty)}
{\omega_{m}+ \Omega^{l}_{\alpha}}.
\end{align}




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\end{thebibliography}
%===============================================================
\end{document}

%
cond-mat0412357.extracted_bib
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\end{thebibliography}