Theoretical study of superconducting proxin1ity- contact systems
by use of the quasi- classical C~reen's function

Yasushi Nagato
Graduate School of Biosphere Sciences,
Hiroshima University

January, 1994

Contents
1

Introduction

1

2

The quasi- classical Green's function

8

2.1

3

2.1.1

The Bogoliubov- de Gennes equation

8

2.1.2

The Andreev equation . . . . . . . .

12

2.2

The quasi- classical Green's fUIlction for a finite triple layer systelTI

16

2.3

The quasi - classical Green's function of a semi - infinite triple layer system

22

2.4

The Green's function and physical quantities

25

2.5

The self- consistent pair potential . . . . . .

29

Andreev reflection in the normal-normal-superconducting triple layer
system4

8

The Bogoliubov- de Gennes equation and the Andreev equation

35

3.1

Andreev reflection in the normal- superconducting double infinite systelTI

37

3.2

Norn1a.l- nonnal- superconducting proximity contact system . . . . . . . .

46

The local density of states of the superconducting-normal-superconducting
triple layer systenl

59

4.1

The de Gennes- Saint-James bound state . . . . . . . . . . . . . . . . . ..

59

4.2

The superconducting- normal- superconductin~~ semi- infinite triple layer systern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

63

5

The infinite double layer systelu with s- wave and d-wave superconductor 73

6

Summary

94

7

Acknowledgluent

971Introduction

In 1911, Kalneringh Onnes discovered the superconductive state in mercury at liquid heliu111
temperature. After that, nUlnerous superconducting rnaterials were discovered. However,its microscopic 111echanis111 was not realized for a long time. In 1957 Bardeen, Cooper andSchrieffer proposed their theory of superconductivity as the first successful explanation forthe microscopic origin of th is phel1Olnenon[l].
It has been known that when a nonnal Inetal is connected to a superconductor, thenormal metal bears superconducting properties.

This pheno111enon is called proxilnity

effect because the Cooper pair in the superconductor is considered to leak into the normal
metal even if the normal l1letal has no pairing interaction.

The proximity contact has

been one of the very important subjects in the study of superconductivity ( for a review,
see Ref. [2]). Recent progress in technologies fabricating artificial materials has Inade it
possible to study very clean and thin proximity-contact layers. In such a clean systen1,
many interesting ph nomena which cannot be expected in an isolated normal 111etal can beobserved.
At the normal- superconducting ( N - S ) interfacial boundary, there occurs the so called
Andreev reflection , that is, an incident electron from the N region is reflected back into ahole[5]. The N - S junction shows non- Ohmic current··voltage behavior[8][9], because thereflected hole carries away a positive charge and the current isnhanced. A theoreticalattempt to explain the non-Ohnlic behavior in the language of the Andreev reflection was
proposed by Blonder, Tinkham and Klapwijk[10]. They have taken into account the effect ofnon- ideal interface but assumed a step-like pair potential. In the proximity contact systeln,
the pair potential is in general depressed near the interface. van Son et al. [6] considered the1effects by the depression on the Andreev reflection by assuI11ing a siluple analytical fonn forthe spatial variation of the pair potcntia.\. Bruder(l2] studicd the Andreev scattering for ad- wave pairi ng su pcrcolld uctor, using a self-consistcntly sol vcd pair potential and taking
account of the interfacial rcflcction coefficient R. Nagato et al.[15] studied thc Andreev
rcflection in thc doublc infinite N - 5 systcm, and obtained the relation between thc Alldreev
reflection coefficient and the dcnsity of states at the interface. Usually, point contact device
is used in the attempt to observe the Andreev reflection through the J- V characteristics. In
this system, it is expected that the voltage drop occurs at the interface between the electrodeand the normal luetal. It 111eans that there is also finite reflection between the electrodeand the normal metal. As a result, the multiple reflection process within the normal layer
will be important and the J- V characteristics will show a geometrical resonance effect .
Taking account of the geometrical resonance effect, we discuss the Andreev reflection andthe differential conductance.
In a finite width normal metal connected to a superconductor, the density of states has
significant structure due to the finiteness of the layer, in the energy range around themagnitude of the pair potential[2S][29][30][31 ][32][33][34 ][35] . de Gennes and Saint-James
first calculated the densi ty of states in the normal metal connected to a superconductor
with a spatially constant pair potential. They found that there is a state with the energy
below the energy gap of the superconductor, which occurs because of the finiteness of thenormal layer and the Andreev reflection at the N -5 interface. This state is the so called"de Gennes- Saint-J ames bound state" . The existence of the de Gennes- Saint-J an1es boundstate is the origin of the effective energy gap in the normal rnetal.
The density of states can be detected by a tunneling experin1ent, by a scanning tunnelingspectroscopy(STM) and so on. In particular, the STM experiment is able to study a spatial2

dependence of the density of states. Recently, Inoue and Takayanagi[36] reported the STM
measurement of Nb/InAs/Nb proximity contact system . Their data indicate that the local
density of states in the InAs region has a.n effective energy gap and the gap varies spatia.lly.
Tanaka et al.[34] studied the density of states of the S-N-S system with ideal interfaces.
Hara et al.[35] studied the N - S finite system in which the N - S interface has finite reflection
but the N region has no pairing interaction . We study the density of states of the nonnal
region in the S- N - S system taking account of the effects by the finite interfacial reflection
and by a pairing interaction in the normal region. The finite reflection will give rise to thegeometrical resonance effect and the pairing interaction will lead to the spatial variation of
the density of states.
To study the effect of the geometrical resonance on the Andreev reflection and to study
the density of states in the S-N- S system, we treat the semi- infinite triple layer systeln
a is depicted in Fig.I.

~(Z)

R

L

--------------~--~-------------~>

o

Z

L

Figure 1: The semi-infinite triple layer system.

3

In the original BCS theory, an attractive interaction which leads to the Cooper instability
is originated by the electron- phonon interaction and is an s- wave pairing interaction ( spin
singlet pairing interaction ). The electron- phonon interaction lnay not be the only possible
mechanism to obtain an attractive interaction . The possibility of unconventional pairingin superconducting nletals was first discussed by Anderson and Morel[37] and Balian andWerthalner[38]. The recent discovery of" heavy fermion superconductor" is the experilnental realization of a class of superconductors with unconventional pairing. Pals et a1.[39]investigated the Josephson contact between a singlet and a triplet superconductor within
the tunneling Hamilton ian n10del. Poppe[40] observed a Josephson current between an swave superconductor Al and a heavy fermion superconductor CeCu2Si2 ' Ashauer et a1.[41]studied a thin fihn of standard superconductor in proxirnity contact with a bulk unconventional material It is an interesting problem to study the properties of proximity contact
superconductors with d iff rent kind of pairing symmetry.
Theoretical treatments to study such proximity systerDs so far reported are mostly basedon the ideal model. In the ideal model the pair potential in the superconductor regionis assumed to be constant and the interfaces are assUJmed to have electron translnission
coefficients of unity. The pair potential, however, cannot be constant, since it is depressednear the N -S interface due to the proxin1ity ffect. We have to treat the spatial variationon the scale of the coherence length. Moreover, the finite reflection that occurs in generalat the interface of different kind metals sho uld be tak1vn into account. Even if the wave
function can be connected sn100thly at the interface, the difference of the F nni velocity
yields a finite reflection coefficient(1)

4

The quasi- classi al Green's fUIlction lllethod[18][19] is able to treat the spatial vanation of the pair potelltia.l and. other physical observables. Thc conventional quasi- classicalfornlulation[25] consists of the Eilenberger equation for the quasi- classical Green's functiong, the boundary condition for 9 and the nonnalization condition g2 = -1. When one studies
the proxin1ity contact systcln by usc of the conventional rnethod, one has to solve the Eilenberger equation with the boundary condition and the normalization condition numerically.
Since the Eilenberger equation includ.cs a solution exploding at infinities, the nun1ericalcalculation needs sophisticated techniques. Calculations by use of the conventional quasiclassical Green's function have been reported[12][25][26][33][41]. Kieselmann[33] obtained
the self-consistent pair potential and the tunneling density of states in a N - 5 contact of anormal metal film with a semi- infinite superconductor. Bruder[12] studied Andreev scattering under the self- consistent pair potential in an unconventional superconductor. In the
triplet superfluid 3He, Kurkijarvi and Rainer [26] also studied Andreev scattering by thc
wall. In the finite double layer systerns, however, the conventional quasi- classical techniquecannot be applied[20] because the nonnalization condition Inentioned above is valid only
in bulk system.
Ashida et al.[20] have proposed a new quasi- classical formulation which can be applied tofinite double layer systems. They obtained an explicit forn1ula of the quasi- classical Green's
function which already satisfies the boundary condition. The formulation is written in aform including the evolution operators together with reflection coefficient R at the interface.
This new quasi- classical Green's function ( AAHN Green's function) has been applied tosome systems. Ashida et al.[21] studied the transition ten1perature of the N - 5 bilayer. Haraet al.[35] studied the local density of states in the N - 5 bilayer. Nagato et al.[15] extended
AAHN Green's function to the systeln including infinite layer and discussed the Andreev

5reflection and the local density of states.

IIigashita,ni et al.[42] discussed the Meissner

effect in the norn1al rneta,} connected to the superconductor. The AAHN Green's function
is a powerful tool to study the proximity contact systell1. In contrast to the conventional
qua.si - c1assical formula.tioll , t.he AAlIN Grccn's function is explicitly cxpresscd. The explicit
expression already satisfics thc boundary condition. To study the proxilnity contact systemof present interest, we extend the AAIIN Green's function to the semi- infinite proxilnity
contact systeIn.
This thesis is organized as follows:
In Sec.2 we ll1ainly discuss the fran1cwork of the quasi- classical Green's function for studying the proximity contact systeIn. We first begin with the Bogoliubov-de Gennes equationand define the Gor'kov Green's function.

Using the ,Andreev approximation, we derive

the quasi - classical Green's function. Following AAHN, we first obtain the quasi- classicalGreen's function for the finite triple layer system . Taking the limit of the layer size to infinity, we obtain the Green's function for the sell1i- infinite triple layer systell1. The relation
between the Green's function and the physical quantities is discussed. We also discuss on
the calculation of the self- consistent pair potential.
In Sec.31""'V5 we apply the obtained quasi - classical Green's function to S0111e proxinlity contact systems. In Sec.3, we study the Andreev reflection for the point contact systenl. To
explain the Andreev reflection, we first discuss it in the nornlal- superconducting infinite
double layer.

Secondly, we consider the normal- normal- superconducting Inodel for the

point contact experiment. We calculate the Andreev reflection and the differential conductance by use of the BTK formula.
In Sec.4, we study the local density of states in the normal- superconducting proximity contact system. We first discuss the de Gennes- Saint-James bound state. We then obtain the6local density of states in the superconducting- norn1al- superconducting systen1. In Sec.5,
we discuss the supercurrent across the interface between superconductors with different
pairing symmetries.
Throughout this thesis, we use the units h = kB = 1.

7

2

The quasi-classical Green's ft:Lnction

In this chapter, wc dcscribe an outiille of the Inethods to study the superconducting proximity contact SystClTI in thc clcan linlit. We start from the BCS model Hamiltonian andderive the Bogoliubov de Gellnes cquation which is useful to treat an inhomogcneous superconducting systenl. Within the quasi - classical approxirnation, the Bogoliubov de Gennes
equation is reduced to the Andrccv cquation, and a formal solution of the Andreev equation
is expressed by a spatial volution opcrator introduced by Ashida et a1.[20]. To obtain the
superconducting pair potential, thc dcnsity of states and so on, we use the Grcen's fUllction nlethod. In accordance with tIl quasi - classical approximation in the Bogoliubov- deGennes equation, the samc approximation to the Green's function leads to the quasi classical Green's function used in this thesis.

We construct the quasi- classical Green's

function for the finite triple layer system, following Ashida et a1.( AAHN )[20]. The quasi classical Green s function can be obtained in a form including the spatial evolution operator
and the reflection coefficients at the interfaces. Starting from the Green's function of thefinite width triple layer, we obtained the Green's function in the semi- infinite triple layer
system of present interest.

2.1

The Bogoliubov-de Gennes equation and the Andreev equation

2.1.1

The Bogoliubov-de Gennes equation

In the BCS theory, it is shown that a weak attractive interaction betwcen electrons, suchas that caused in second order of the electron-phonon interaction, causes an instability ofthe ordinary Fermi-sea ground state of the electron gas. The BCS state is characterized by8

an energy gap in the quasi-particle excitations of the systelu. The superconducting stateis characterized by the energy gap ( the pair potential)" Since we treat the electrons near
the Fermi surface, we choose the Fernli energy as the origin of energy. Using a creation
and annihilation operator of FerrnioI1 (electron ) 'ljJl( r), 'ljJex( r) , the weak coupling BCS
Hamiltonian is written as
1-{

Jdr'IjJ~(r)e(V)'ljJex(r)+ ~ :L Jdrdr' v( Ir - r'l) ,p ~ (r ),p1 (r'),piJ( r') ,p", (r),
ex,{3

=L
ex

(2)

1 2
e(V) = - 2m V - /1,
where cx, f3 are spin indices and /1 is the cheluical potential.

One applies the Gor'kov

approximation, i.e., mean field approximation, to Eq.(2) to obtain
1-{Gor'kov=

L Jdr'IjJ~(r)e(V)'ljJex(r)
ex

+~:L
ex,{3

Jdrdr'v(lr - r'I)[(,p~(r),p1(r')),piJ(r'),p",(r)+'IjJ l (r) 'IjJ1 (r') ('IjJ{3 ( r') 'IjJ ex (r)) - ('IjJ l (r) 'IjJ1 (r') ) ('IjJ {3 ( r') 'IjJ ex (r) )],(3)

The last term ( ) ( ) is a correction for the overcounting of the interaction. We define~ex{3(r,r') = v(l r -r'I)('IjJ{3( r') 'ljJex( r)) ,(4)~~(3(r,r') = v(l r - r'I)('ljJ1(r') 'IjJ~( r))(5)and obtain
1-{Gor'kov

Jdr'IjJl (r)e(V)'ljJex (r)+~:L Jdrdr'[6~iJ(r, r'),p", (r),piJ(r') +,p~(r),p1(r')6"'iJ(r, r')
ex,{3

= L
ex

-v(r - r')('ljJl(r)'ljJ1(r')) ('IjJ{3 (r')'ljJex (r))).
9(6)

The Heisenberg operators 'l/Jcx( r, t) a nd 'l/J;( r, t) obey equations of 111otion :iOt'l/JQ(r, t) = [ 'l/JQ (r, t), 1-{Gor'kolJ ]=Jdr'{~(\7)8(r- r') 'l/JQ( r',t) + 2:= ilQ,G( r,r') 'l/Jh( r', t)} ,

(7)

tl

iot'l/Jl( r, t) = [ 1jJl( r, t), 1-{ Gor'kolJ ]=Jdr'{-~(\7)8(r - r') 'l/Jl (r',t) + Lil~,G(r,r') 'l/J,G( r',t)}.
,G(8)

Thus we can write in a 111atrix form as

Jdr'£(r, r')W(]l', t),(9)il(r , r')(10)iotW(r, t) =
where

£(

r,r

') = (~(\7)8(r - r')ilt(r , r')W(r,t) =-~(\7)8(r - r')),(~t1~::ll '(11 )'l/JI( r , t)

A(
Ll

r, r

') = (ilTT(r,r')A ( ')
Ll!l r , rilTl(r,r')) At( ') _ (iltT(r, r')A ( ') , Ll r, r - illt T(r, r')
Ll 11 r, rilL(r,r'))il t11 (r, r') .

( 12)

In order to solve Eq.(9), we have only to consider the Bogoliubov d e-Gennes equation

Jdr' £(r , r')W(r') = EW(r) .

(13)

where we have regarded W(r, t) in Eq.(9) as c-nulllber W(r, t) and have substituted W(r, t) =W(r) exp (-iEt) into Eq.(9) .
When one studies layered proximity contact systems where the interface and the boundaries have translat ion al symmetry in the x, y plane, the IllomentUl11 component parallel tothe boundary is a conserved quantity. For a given parallel momentulll p, Eq.(13) is reduced

10

to

f-a;]8(z - z')~(p, z, z')) ,T,( ') E,T,()_[~"
2~ 8;]8(z - z')=
Jdz' ([~" - ~~P'z, z')
_

'J:'

Z

'J:'

Z ,

(14 )

where ~II = p2/2771, - fJ,

w(z) = W(p, z) =~(p, z, z') =Jdxe-ipoXw(r),(15 )Jd(x - x')e-ipo(x-x /)~(r, r')

(16)

and
~(Pl Z, z') =L ~(Pfr, z )eiPz(z-z') + L ~(PF' z)eipz(z-z')
Pz>O

( 17)

Pz<O

with 6.(P~, z) the position dependent pair function at the Fermi momentum p~. In Eq.(17),
we have defined two Fermi momenta Pfr and PF associated with the parallel mon1entulTIp, as depicted in Fig.2.

(x-y)

____

~~_ _ _ _ _ _~_ _ _ _~o~_ _ _ _~>

o

Z

PFz

Figure 2: Fermi momenta PF( a = ±) associated with the momentulTI P parallel to theinterface and to the walls.

112.1.2The Andreev equation

We first derive the Andreev equation from the Bogoliubov- de Gennes equation Eq.(14) using the quasi - classical (WKBJ) a.pproxirnation[5]. Also, we discuss the boundary condition
for the Andrecv anlplitude, following Zaltsev[23], Shelankov[22] and Ashida et aJ[20].
When the characteristic size of the systen1 is n1uch longer than the Fenni wave lengthl/PF, the quasi - c1assica.l approxilnation introduced by Andreev [5] is useful.

Following

Andreev, we write the Nambu alTIpliiude as(18)
where PFz = PF cos e is the z component of the Fermi mOlnentum pt. The slowly varying
amplitude <I> o(z) obeys the Andreev equaiion[5]z)) <I> = E <I>

6(PF'
. a

azvFz z

01

1 0/'

(19)

w here v Fz = v F cos () = PF cos () / m. The slowly varying ampli tudes are expected to fornl acomplete set near the given Fermi momentum in a sense that(20)
Since the Andreev equation is a first order differential equation, its formal solution canbe written as(21)
where Vo(z, z', E) is a spatial evolution operator wh ich obeys
(22)
(23)
12

and

(24)
In the above, PI, P3 a.re the Pauli matrices in the particle-hole space. The properties of theevolution operator is discussed in detail in the appendix .

6(Z)

c

L

o

R

L

Figure 3: The finite triple layer system.

To discuss the semi - infinite triple layer system, we start frorn the finite width triple layer
system as is depicted in Fig.3 . Let us discuss the boundary condition at the L - C, C - R
interfaces and the layer ends at z = -LL' z = L + L R . We use superscripts or subscripts

L, C and R to denote the quantity in the L, C and R layer, respectively. Since the parallel
momentum p is conserved,
L'()
c ·()
R'O
ppsln
-L = PF s1n C = PF s1n R·(25)

The boundary conditi ons for <I> 0' 's at the interfaces[20][22][23] can be obtained by the following arguments.

13

For the L-C interface boundary located at z = 0, the proximity interface is assulned tobe confined within a narrow range -8 < z < 8, where 8 is of order of l/p}.. It follows that
the Na.mbu alnplitude \}J should be the aSYlnptotic solution of the interface probleln. Since
the interface process is a high-energy (I"JBF ) and short range (I"Jl/PF ) process, it isgoverned by the rapidly varying part eiPFZ of the Nalnbu amplitudes and is consequently
common to both the superfluid and the normal phases. The interface process is, therefore,characterized by the two independent asyrnptotic solutions WI and \lI 2 for the electrons at
the Fenni level :
. L

. L

e1 P F z Z + 1'0 e- 1 PF z Z

for z < -8

doeip~zz

for z > 8,
for z < -8
for z > 8,

(26)
(27)

where ro and do are the reflection a.nd the transmission anlplitude at the L-C interface,respectively, and do = dov~z /v~z , 7""0 = -r;do/ d; which are obtained by use of the constantness of the Wronskian. Noting that the Nambu amplitudes wL and \lie are given byappropriate linear combinations of WI and W2,

we obtain the boundary conditions. For the C- R interface boundary located at z = L, theboundary condition can be obtained in the same way. Thus, the boundary conditions for
the Andreev amplitudes at z = 0 and z = L are written as

(28)
(29)
(30)
14

<I>C(L) =-7'L<I>R(L) + ~<I>R(L)

dL +

d*L

-

,

(31)

where 7" and d are the reflection and the translTlission anlplitude for the electronic state at
the Fermi level and they can be expressed in tern1S of the reflection coefficients R o, RL of
the interfa.ce as

(32)
do =

(33)

(34)

(35)
where Or, Od are the phase of I, d, respectively. As we shall eventually show, the final results
depend only on the reflection coefficients Ro = 1'012, RL = I'LI2 and not on the phases Or

The boundary conditions at the ends of the layers are written as<P~( -LL)C-ip~zLL<P~(L + LR)eiP%z(L+LR)+iTJR+ <P~( _LL)CipIJ,zLL+iTIL = 0,(36)

+ <P:!(L + LR)e-ipljz(L+LR) = 0,(37)where TIL, TJR are possible phase shift at the boundaries. When exp i77 = 1, one has a fixedend condition and when exp iTJ = -lone has a free end condition. As we shall see, thephysical quantities does not depend on these phases. One can study an inhomogeneous
superconducting system by use of the Bogoliubov de Gennes equation Eq.(14) or by use of
the Andreev equation Eq.(19) derived here. In this thesis, to discuss the proximity system,
we have mainly treat it by use of the quasi- classical Green's function discussed in later
sections.

15

2.2

The quasi-classical Green's function for a finite triple layer
system

In this section we closely follow the prescription proposed by Ashida et al.[20].
We consider a geOlnetry as shown in Fig. 3 . The widths of the layers LL, Land LR
a.re assulned to be still ITIuch longer than the Fermi wavelength l/PF, but not necessarily
longer tha.n the coherence length ~. The interface and boundaries are assumed to havetranslational synllnetry in the x, y plane . The mOlnentum component parallel to be theboundary is, therefore, a conserved quantity. From now on, we use superscripts or subscripts

L, C and R to denote the quantity in the L, C and R layer, respectively.
The Green's function method [3][4] is a useful method to study the superconducting
proximity contact system. The Gor'kov Green's function is written in terms of the Nambu
anlpli tude 'l! (r) as, ) = ~ 'l!, (r) 'l! 1(r')
G( r ,r,c
L.-J
E·
,

c-

(38)

-1,

Since the momentUl11 component p parallel to the interface is a conserved quantity in thepresent system, we have only to consider the spatial variation in z- direction. HenceG( r,r , ,c ) -- _1_(27r)2

JJ P
d2

G( p , z,z ',c) eip·(x-x') .

(39)

Then, the Green's function, which depends on the z direction, is written as(40)
which satisfies the Gor'kov equationf-8;]5( z - Zll).6(p, Z, Zll))] G(r(') ( )_[~I I _ 2~a;]8(z _ Zll)
Z ,Z ,c z - z . 41
JdZ" [c _ ([~II - .6~p,Z,Z")
II

,

)

_

u

Substituting Eq .(18) into Eq.(40), one finds that Gor'kov function is decomposed as

G( p, z, z',c) -- G++ ( z, z ,,c ) eipFz(Z-Z')

16

(42)
where
(43)
which satisfies the equation

6(PP'Z))](' r( Z - Z') .
. a Gcx(3 ( z, Z,,c ) = ucx(3uazvFz z(44)

It is useful for later use to introduce the notion of "directional space" which IS a twodimensional space spanned by a = ± . Fr0111 Eq.( 44), one finds that the diagonal ( in the
directional space) elements G++ and G __ have a jump at z = z', i.e.,(45)
but off-diagonal elements have no jump. Here P3 is a Pauli Inatrix in particle- hole space.
Noting the jump given by Eq .( 45), we define the quasi- classical Green's function 9cx(3 by(46)
where {3 is a Pauli matrix in the directional space. Then, the position diagonal element of
the Gor'kov Green's function is written in terms of the quasi - classical Green's function as
1

G(p, z, z,c) = - - P 3 [9++(Z)

2VFz

].
17

(47)

The quasi - classical Green's function 9+_ and 9_+ do not contribute to observed quantities
because they are accOIllpaIlied by a rapidly changing phase factor e±2ipF z Z that vanishes
under the p SUI11nlation. In the conventional quasi-classical Green's function theory,[25]therefore, only 9++ and.rJ-- arc considered. To solve the boundary problenl, however, it

is more convenient to treat aJl the components of 9a!3(z) on equal footing, because theboundary problelTI tben can be solved witbin linear algebra.
The generalized set of the quasi -classical Green's functions obeys the Eilenberger
equation[lS] ,[19](4S)
Since the Eilenberger equation is a first order differential equation, a formal solution canbe expressed in terms of the evolution operator given by Eq.(23) in the previous section as
follows:(49)
In treating the Green's function, we use the evolution operator with cOlnplex c.
The boundary conditions for the quasi-classical Green's [unction at the interfaces havebeen given by Zaltsev[23] Millis et al.[24], Shelankov[22] and Ashida et al.[20]. Using theboundary condition Eq.(31) for the amplitude of the Andreev equation and the definitionof the quasi- classical Green's function, one can obtain the boundary condition of 9 at theinterface boundaries as follows:

(50)

IS

where Mo and ML are matrices

~~ )

1
d*o

'L
d*L )
1

(51 )

'

.

( 52)

d*L
In the same way, using Eq.(37), the boundary conditions at the ends of the layers areg~+( -L£) ± i = g~_( -LL) ± i = -e=fi~Lg~=f( -L£),( 53)

g~+(L + L R ) ± i = g~_(L + L R) ± i = -e=fi~Rg~=f(L + L R ),(54)

where ~L = 2p}zLL + T/L, 1JR = 2p~z(L + L R) + T/R and T/L, T/R are possible phase shift at theboundaries.
U sing the boundary condi tions Eqs. (50 )- ( 54) and the spatial evolu tion operator of the 9given by Eq .(49), one can obtain the quasi- classical Green's function at arbitrary position.
The explicit expression of the quasi - classical Green's function can be written as follows:g~o(z) = U~(z, -LL) hL u~( -L£, z),(55)

g:o(z) = U:;(z, L + L R) hR U:;(L + L R, z),(56)

g~o(z) = U~ (z, L) h~ U~ (L, z),

(57)

where

hL = (-i) etch + AL ,etch - ALA$0 i<pc
A = (UL)-l C oe U£L+
e10<p c - $0
RoAc - ,VJ[i i<PRA
Ac = U~ H £e (U~)-l,e1<PR - VJ[iA R(59)

AR = U:(U!:)-l

(61)

19

(58)

(60)

a.nd

(62)
(63)

(64)

(65)
a.nd
h~ = (1 -RL)-l [U:;hR( U~~)-l RLU:hR( U::)-l-[ii:(e i¢RU!3(h R + i)(U:)-l + e-i¢RU!:(h R - i)(U!3)-l)]

(66)
(67)

where

u~ = U~(o, -LL),(68)

uf = Uf(O, L),(69)

U: = U:;(L, L + L R ),

(70)

(71)
(72)
(73)
The phase factors ei¢L, ei¢c and ei¢R are rapidly varying functions of layer sizes LL, LandLR and also of the polar angles of the Fermi momentum, because PFL » 1. We are notinterested, however, in the size accuracy of order l/PF nor in the accuracy of polar angle oforder 1/ PFL . Apart from the correction of order 1/ PF L, therefore, the physical quanti tiesof interest are obtained from the quasi- classical Green's function averaged over the phases20

~L, ~c and

cPR. Moreover, in actual systems the phases Oro(rL)' Odo(d L ) and 7lL(R) will berandom variables reflecting the lnicroscopic irregularities at the interfaces and at the walls.
It follows that the averages over the pha.ses cPL, cPc and cPR can be per[ornled independently

in spite of the fact. that the pola.r angles of the Fenni rnonlenta in the L, C and R layers
are connected by Eq.(25). As a result, the averaged Green's [unction do not depend on
the pha.se of the interface reflection amplitudes but are determined only by the reflection
coefficients Ro and R L .
Then, to calculate the physical quantities from the quasi- classical Green's function thefinite triple layers system, one has to average it over the phases.

AAHN analyticallyaveraged the Green's function of the finite double layers over the two phases. In this thesis,
however, we do not consider the finjte triple layers systen1, but are interested jn serni-

infinite triple layers systelTIS, such as one point contact system and so on. In the next
section to treat the selTIi- infinite triple systems, therefore, we take the li111it of LL and

LR to infinity in the Green s function obtained in this section. As a result, the Green's
functions taken the limit do not depend on the phases cPL and cPR . We have only to average
those over the phase cPc.
From now on, we confine ourselves to the singlet superconducting proxin1ity syste111
although the quasi- classical Green's function obtained in this section can also be used
to investigate the triplet superconducting system. For a singlet superconductor, the pairpotential is written in the spin space as.6 Tl )

o

'

(74)

therefore, we can decouple the spin space. For a singlet superconductor, we have only totreat a 2 x 2 matrix in the particle-hole space. Properties of the evolution operator in the21

2 x 2 Inatrix space are discussed in the appendix.
In equilibriull1 states without supercurrent, we can take the pair potential real. Then, it
can be shown that the evolu tion operator satisfies(75)( See appendix. ) Using this relation Eq.(75), we findA]'....

9++ = P39--P3·(76)

Therefore one has only to treat 9++.

2.3

The quasi- classical Green's function of a semi-infinite triple
layer system

In this section we derive the quasi-classical Green's function for the semi- infinite geometry
as shown in Fig. 1 [15]. As we have noted at the end of the previous section, we consider asinglet superconducting proximity contact systen1 in this thesis. We assume that the pairpotential is real. The case when the pair potential is not real will be discussed in Sec.5.
To obtain the quasi - classical Green's function of the semi- infinite triple layer system, wetake the limit of LL, LR to infinity in the Green's function of the finite triple layer systelTI
obtained in the previous section[15]. In this case, the pair potentia.! 6( z) will tend to thebulk value 6bulk at sufficiently large Izl. Hence, the evolution operator U can be divided
into a growing part and a damping part:' ) -- A+z,z( ') e -il'>(z-z') + A_ ( z,z ') eil'>(z-z') ,(U+Z,z,c(77)(78)
where K, = D/VFz == J c 2 -6;ulk/VFzand the square root is defined to have a positive

imaginary part in the complex c plane. The operator A± has a projection operator like22

properties as is shown ill detail in the appendix.
When LR ---t (X) , since Im K, > 0, the evolution operator V+(L, L + Ln) beC0l11CS

a divergent tenn.
Retaining the most divcrgent tern1S when LL

---t

00

and LR ---t(X) ,we can find thequasi - classical Green's function of the sen1i- infinite triple layer system.

L side ( z < 0 )
AL

g

()

z

++

- trAr(z)= z.2Ar(z)trAr(z)'(79)

Ar(z) = [J.;-(z, O)(Ac + ROP2TAC P2 + 2JRoR£trAR cos cPc)
XPli\~(O, -(0)P3Ti\~(z, -(0)P3Pl,(SO)

Ac = U~(O, L)(A~ + RLP2TA~P2)P3TU~(0, L)P3,(Sl )AR = J~(L)I¢~(L)p3'(S2)

R side ( z > L )
AR (

) _

g++ z -(S3)

.2BJ!(z) - trBR(z)trBJf(z),Z

Bc;(z) = i\~( z, 00 )P3TA~(L, 00 )P3
x (Be + R LP2 TB e P2 + 2J RoRLtrBr cos cPc )P2 TU :(Z, L )P2,

Be = P3T(;~(0,L)p3(Br + RoP21J3rp2)(;~(0,L),

(S4)
(S5)

an d C si de ( 0 < z < L )c.2C (z) - trCc(z) + 2i~trARtrBL sin cPc
g++ z = Z
trCc(z) + 2JR oRLtrARtrBr cos cPc
'
AC

()

23

(S7)

(88)
where

(89)

(90)
(91)
and ~+ and ~_ are deco111posed elernents of the evolution operator defined in the appendix .
In the course of taking the limits, the phase factors, eitPL and eitPR , disappear. Since the
C layer has finite width, the phase factor eitPc remains in the Green's function .

(92)
As we have noted in the previous section, this phase fa.ctor is a rapidly varying function oflayer size Land a.lso of the polar angles of the Fenni m0111entU111 . We are not interested,
however, in the size accuracy of order 1/ PF and in the accuracy of polar angle of orderl/P F L. Apart from the correction of order l/PFL, therefore, the physical quantities ofinterest are obtained frorn the quasi- classical Green's function averaged over the phase~c.

As a result, the averaged Green's function do not depend on the phase of the interface

reflection amplitudes but are determined only by the reflection coefficients Ro and R L . The
averaged Green's function is defined by

(93)

24

For exalnple, the averaged Green's functioll of the C layer is written as
A
(

())

g++ z

_

•

c

2C (z) - trCc (z )v(trCc (z ))2 - 4RoRL(trARtrBf)2- z---;=.============

We study some proxilnity contact systenls by use of the averaged Green's function.

2.4

The Green's function and physical quantities\Ve discuss how the physical quantities can be calculated frol11 the Green's function. We
start from the Bogoliubov- de Gennes equation Eq.(13) ,(95)
Using positive energy solution of this equation,(96)

we define a n1atrix VI , which diagonalize the Hamiltonian Eq.(6),(97)

This matrix satisfies

L' VI(r)V/(r') = 8(r - r'),I

where the symbol 2:/ indicate the sum over positive energy states . Also, n1atrix E(r, r')
can be reduced toE(r, r') = L' U/(r) (~/_~/) U/(r').

(98)

I

Using Eq.(lO), we reduce Eq.(6) to1-{Gor 'k o1J=

Jdr Jdr'~t(r)E(r, r')~(r') +25

const ..

(99)

The Hamiltonian can be rewritten by Eq.(98), i.e.,

1{Gor'k ov=

2:' Jdr(U/(r)tl-(r))t (~I _~I) Jdr'Ui(r')~(r') + const.
Io )"II,-EI( 100)(101 )" ('I)
Il 'where II and ,t are an annihilation and a creation operators of the Bogolon ( a quasi-

II =particle ), respectively. In tenDS of the operator ;0, the electron operator ~ is written
as
(102)

IDore explici tly'feAr) = L'(UI,a(r),1 + V/,a(r)*,1),

(103)

I

'fl(r) = L'(V/,a(r),1 + U/,a(r)*,t).

(104)

I

One calls these equations the Bogoliubov transforn1ation.

(i) The gap equation

The pair potential 6 aj3 (r, r') was defined by(105)

The gap equation can be rewritten in terms of the new operators It and

1,1'

26

11,

= v(l r - r'l) I:' {UI,/3 (r')VI,cA r r('"'fI,?)I+vl,cA r'r UI/3(r) ('?'l) }= v(lr - r'l) I:' {UI,{J (r')VI, cy (r)*(1- f(E I)) + VI,cy(r')*UI,/3(r)J(Ed}I= Tv(lr - r'l) 2:: 2::' {UI,~(r)VI'/3( r')* + vl,cy.(r)*UI,/3(r')} ,Wn

I

1,W n -

El

1,W n

+ El

(106)

where T is the temperature, and we have used

(107)
Wn

= 7fT(2n + 1) is the Matsubara frequency ( n = 0, ±1, ±2, ... ), and eiwn8 is a convergent

factor omitted above. Rewriting the sum of the second term of Eq.(106) to the sum overthe negative energy state, one can obtain
(108)
where

I: means the sum over the all states.
I

Comparing Eq.(108) with the definition of the Gor'kov Green's function,

WI(r)W;(r')
G( r , r,, c ) = '"
~
E'
c-

I

/

(109)

one can obtain the gap equation in terms of the Gor'kov Green's function G, i.e.,~(r, r') = Tv(lr - r'l)

I: G(r, r', iWn )h,2'

(110)

Wn

where Gh,2 is an off- diagonal (1,2) elen1ent of G in the particle-hole space.
When the system has the translational symmetry in the x - y plane, using Eqs.(16)(17),
Eq.(39) and Eq.( 4 7), one can obtain the gap equation in tern1S of the quasi- classical Green's
27

function

where 00' a.nd cPa a.re the polar angles of the Penni nlonlentum PF, v pF .P } is the pairing
interaction, and N(O) is the density of states at the Fernli surface.

(ii) The electric current

We derive an expression of the electric current in ternlS of theGreen's function. As is well known , the electric current J (r) is defined as(112)
where A is a vector potential, 'ljJa and 'IjJ~ are operators of an electron with spin ex and thesymbol ( ) denotes the thennal average . Fronl now on, we ignore the last ternl, which is
proportional to a vector potential A, and discuss on(113)
The ternl ('IjJ t (r') 'IjJ ( r)) is rewritten using the quasi- particle operators as follows:

a

a

1,1'

= T L L L' [UI, a.(r')*UI,a(r) + VI,a.(r')VI,a(r)*]
Wn

a

I

ZWn -

EI

ZWn

= TLtrs G(r,r',iwn)h,l,+ E,(114 )

Wn

where we have used Eq.(107), Gh,l is a diagonal (1,1) elenlent of G in the particle- holespace, and trs Gil,} denotes a trace of Gh,l in the spin- space. Then, one finally obtains
the electric current in terms of the Gor'kov Green's function as

(115)

28

In ternlS of the quasi- classical Green's function one can write the z-component of theelectric current as( 116)

2.5

The self-consistent pair potential

In the superconducting proxinlity cOlltact system, the pair potential near the interface
varies on the scale of the superconducting coherence length~. To study the proxin1ity
effect, one has to take into account the self- consistency for the pair potential.
To obtain the self- consistent pair potential, we have to solve the gap equation . Fron1
Eq.(111), the gap equation can be written as

L'>.(PF' z) = T N(O)4

L L l' d¢o
Wn

a=± a

t

O

sin OOdO (21 + 1)g/P/(PF . PF )9aa(iwn' z) 11 ,2, (117)awhere N (0) is the densi ty of states at the Fermi surface, and gl denotes the strength of thepairing interaction for the [- th partial wave . We have defined g[ such that is posibve when
the interaction is attractive. Here, I determines the syITIInetry type of the order parameterof the superconductor, i.e., l = 0 for s-wave, [ = 2 for d-wave superconductivity. PI are
Legendre polynomials and are expanded in terms of spherical har1110nic functions l~m i.e.,
1

P/(PF . PF) = 214: 1~/ Ylm(/J, ¢)lJ:(O°, r),

(118)

m=-/

where

e and rp are the polar angles of the Ferrni mornentU111. We confine ourselves to treat

the singlet superconductors.
For the Matsubara frequency c = iW n1 if only 6(p}) = 6(PF), the evolution operator
satisfies

U+(W n) = p 2U-(Wn)*P2

U+ (w n ) = PI U+ ( - W n )* PI,
29

and

(119)
(120)

even when the pair potential is not real FrOI11 these relations, we can easily findand(121 )( 122)

To eliminate the coupling constant g, fronl the gap equation in favor of the transition
telTIpera.ture T e , we introu lice Te by letting ~ ~ 0 in the gap equation:
Wn c =

7rTe(2n + 1)

1

1
n=O

n

+ 1/2
1

T

we 121fT

1

log-+
Te

n =O

n+l/2

L

(123)

Then , we obtain(124)

We obtain the self- consistent paIr potential by solving this equation iteratively. The
right hand side of this equation depends on the cutoff energy We but converges for large
We. So far the We used in the conventional quasi- classical theory, was at lTIOSt We rv 10Te .

In this thesis, we chose it 30Te except for section 5 in which we used 10Te . In contrast to
the conventional quasi- classical method in which one should solve the Eilenberger equation
Eq.( 48) numerically under the boundary condition and nornlalization condition, we have
only to solve the equation for the evolution operator in the present fornlulation since this30

formulation already satisfies the boundary condition. Numerical work have been considerably reduced in the present fornlula,tion. The iteration procedure is repeated until thedifference of ~( z )/ ~bulk at subsequent iteraUons becornes smaller than 10- 5 at every point.

Appendix A. Properties of the evolution operator
In this appendix, we consider the properties of the evolution operator, which is a 2 x 2matrix in the particle- hole space. Here, we aSSU111e the pair potential to be real. The caseof complex pair potential will be discussed in the appendix of Sec.5.
The evolution operator obeys an equation(a = ±1)( 125)

L'>~:) )(126)

and the boundary condition,(127)
From the structure of the equations, it can be proved thatdet UO'(z, z') = 1.

(128)

Since P1Apl = -A, therefore(129)
we have only to consider U+(z, z'). The matrix A also has an important property- _T

A = P2 AP2 = -A,
where T A is the transpose of A and A is just the cofactor matrix of A.

31

(130)

Let us consider a differential equation(131 )
It has two independent sol u tions( 132)
with the nonvanishing Wronskian

The evolution operator U+ (z , z') can be written in a linear con1bination of the above
linearly independent solutions as

U+(z, z') = Q+(z, z') + Q_(z, z'),

(134)

Q+( z, z') = ~<P+( z) '¢-( Z')P2'(135)

Q_( z, z') = ~ <p_( z)T<p+( Z')P2( 136)

It follows that( 137)(138)(139)
and also thatdet Q±(z, z') = o.

(140)

From the above results together with Eq.(127), it can be readily shown that Q + (z, z) and

Q_(z, z) are the projection operators that project out 1;+ and 1;_, respectively.
32

Moreover, we can show that(141 )
from which the inverse operator of U+( z , z') is given by

U+. 1 (z, z') = U+( z', z ) = Q+( z', z) + Q_(z', z)( 142)= Q_(z,z')+Q+( z , z')(143)(144 )

This is a nat ural res ult when we recall the fact that det U+ (z, z') = l.
Now we consider a system in which the superconductor has semi-infinite width. The pairpotential ~(z) tends to the bulk value ~bulk at z - t 00. In that case, we can choose theindependent solutions so that(145 )~--+ const. x ( c =t= n )
( U±)
A

V±

(146)

.

-UbulkIn the above, we have defined the square rootn = Jc2-~~ulk such that has positive

imaginary part in the complex c-plane. Thus, ¢+ and ¢_ are growing and damping solutionsof Eq. (131), respectively. In this case,

U+(z, z') = A+(z, z' )e-iK(z-z') + A_(z, z')eiK(z- z'),
-1 ~

T~

1 -

T-

A+(z,z' ) = W¢+(z) ¢-(Z' )P2,
A_(z, z') = W¢-(z) ¢+(Z')P2 '
33

(147)
(148)
(149)

All the properties of Q± shown above are preserved by A±.
Froin Eqs. (125) and (145), it can be shown thata (U±) = _i (c± n
z

v±

VFz

-~ ( z )~(z) ) (~±).

-c ± nv±(150)

Since in the theory of the Green's function in the sen1i-infinite geometry only the ratiov±/u± is important, we define__ . v± (z )V ±z( )z~ ()'
11± Z

(151 )

which satisfies(152)
and at Z ---+ CXJ( 153)

34

3

Andreev reflection in the normal-normal-superconducting triple layer system

The current- voltage I - V characteristics of a nornlal rnetal- superconducting contact showa non - Ohmic behavior. The non- Ohmic properties of the N - S contact are considered tobe due to the Andreev reflection in which an incident electron ( hole) from the N region
is retroreflected into a hole ( electron ). ( See Fig.4. ) Most of non-Ohrnic properties have\/
Figure 4: The Andreev reflection ( right- hand side) and the normal reflection ( left- handside). The arrows give the directions of the group velocity of incolning and outgoingparticle.

been analyzed using the theory by Blonder, Tinkham and Klapwijk ( BTK )[10]. We briefly
review the BTK formula for a differential conductance of the nonnal- superconducting
junction.
On the basis of the semiconductor 1110del, the electrical current Inn through the nonnal
metal- normal metal ( N - N ) junction applied the potential difference eV between the two
metal is written as

rm = AN(O)evF

JdE{ f(E - eV) - RN f(E - eV) - (1 - Rt,?)f(E)}(154)

in the left side,
where f(E) IS the Fenni function, Rr; is the (normal) reflection coefficient and A IS a35

constant to be detern1ined by the ge01uetry of the junction. Here, it is assulued that the
distribution functions of all the incOInillg particles are given by equilibriurn Ferrni function
detern1ined by the reservoirs. In Sq.(154), three contributions to the electric current Illean
the distribution of right going electrons, of reflected left going electrons, and of electrons
injected into the left side fron1 the right side, respectively. The current is rewritten as

Inn = AN(O)evF

JdE(1 - R7,?){f(E - eV) - f(E)} .

(155 )

An intrinsic difference between the nonual- superconducting junction and the N - N junction is the existence of the Andreev reflection in which an incident electron is reflected
into a hole at the N - S interface. Since the reflected hole has positive charge, the electric current is enhanced by the existence of the Andreev reflection.

Taking account of

the Andreev reflection, Blonder et al. [10] obtained the electrical current Ins through thenormal- su percond ucting junction

Ins = A1V(0) evF

JdE(l- RN(E) + RA(E))(f(E - eV) - J(E)) ,(J 56)

where RA(E) and RN(E) are the Andreev reflection and the normal reflection coefficient,
respectively. It is noted that the existence of the Andreev reflection increases the electric
current. This expression was also obtained by Furusaki[II].
From the above equation, the differential conductance can be written as

dI( 157)

dV

where R(O) is the nonnal state resistance. For instance, in a cornpletely transmissive contact, since RA = 1 and RN = 0 below the gap voltage (6. bu1k /e), the differentia.l conductancebecomes twice the normal conductance.

36

We first discuss the Andreev rcflection in the double infinite proxinlity contact systenl
in Sec.3.1. And, taking a.ccount of a geoDlctrica.l resonance effect in point contact devicc,
we discuss the Andreev reflection coefficient and the differential conductance in Sec.3 .2.

3.1

Andreev reflection in the normal-superconducting double
infinite system

6(Z)

N

5

o

Z

Figure 5: The normal- superconducting infinite double layer system.

First we discuss the Andreev reflection in the normal- superconducting ( N - 5 ) double
infinite layer system[14][15]. The left ( right) side region is assumed to be a normalmetal ( a superconductor) as depicted in Fig.5. We consider a scattering problelTI for an
incident electron with an incident energy E from the N region toward the N - 5 interface.
Solving the Andreev equation(19) with appropriate boundary conditions, we can discuss
the Andreev scattering. In the N region ( z < 0 ), since the pair potential will tend to zeroat sufficiently large Izl even if the pairing interaction is non-zero, the asymptotic behaviorof the N region wave function can easily be solved. ( If the pairing interaction 9 is zero,
the pair potential 6 is also zero, since 6= gF , where F is pair amplitude. If the pairpotential is zero, the Andreev equation can easily be solved. ) The asymptotic behavior of37

the wave function <P;: with the energy E are written as<P~ (z)<P~ (z)where ",Nz-+ oo
~

1'N)
(

o

e

-i",Nz,( 158)= E / v~z' 1'A a.nd 1'N are the Andrccv and the normal reflection arnplitude,

respectively. In the S- side ( z > 0 ) , the pair potential will tend to be the bulk value atsufficiently large distance. The asymptotic solutions of the wave function are written as
z-+ oo~Z---+OO( 159)~

where ",5 = n/v~z ==JE2 - 6S~ulk/V~z , PI is the Pauli matrix and we have defined thesquare root to have positive imaginary part.
Since the Andreev equation is a first order differential equation, its formal solution canbe written as( 160)
where U is the spatial evolution operator which we have defined in the previous chapter.
Since the pair potential have been assumed to be real in this section, the evolution operator
satisfies(161 )
Using the asymptotic solutions Eqs . (158)( 159), the interfacial boundary condition Eqs. (31)and the evolution operator, we can find the formal solution of the wave functions. One has
only to solve a following equation:<P~(-oo)) _(<P~(O))( <P~(-00) - Mo <P~ (0)A

U(O, -00)

38

_ ~(<l?~(oo))- MoU(O, 00) <l?~(oo) ,

(162)

where

(163)
where we have used Eq.(161). Solving this equation, it can be found that ea,ch amplitude
coefficients of the wave function are generally( 164)(165)( 166)

(167)
where

~:),

(168)

u~(O, 00) (;:) .

( 169)

u:(O,-oo) = (}(;) =Using the time dependent Andreev equation,~(Z) ). a <I>o(Z),aZVFz z(170)

the flux of the quasi - particles are given by

I: avFzlm (<I>~P3az<I>O)'(171 )o=±Using this expression for the flux and noting that lal 2 -1,812 is a conserved quantity in the

Andreev equation, we obtain the reflection and transmission coefficients39RA = 1(a,* + (38*)(a8 + (3,) - Ro(a, + (38)(a8* + (3,*) 12(a,* + (38*)2 - Ro( a8* + (3,*)2(1 72)

Andreev reflection coefficient,( 173)
Nonnal reflection coefficient,(174 )
transmission coefficient for a quasi- electron,(l a l2 - 1(312)la8* + (3,*1 2
TH = Ro(1 - Ro) l(a8* + (3,*)2 _ Ro(a8* + (3,*)212( 175)

transmission coefficient for a quasi- hole,
These coefficients satisfy the flux conservation law, i.e.,(176)
Using this expression, the Andreev reflection coefficients can be calculated numerically.
Typical results are shown in Figs.6 and 7. In Fig.6, we show the results when the N region
has no pairing interaction. The self- consistent pair potential is plotted in Fig.6(a). Theenergy dependence of the Andreev reflection shows no qualitative difference from the BTK
results, except for a small shift of the peak energy. The origin of the peak shift has been
discussed in detail by Nagato et al.[15]. In Fig.7, the results when the N region has finite
pairing interaction ( attractive and repulsive). Due to the proximity effect, the nor111al
region has a pair potential when the N region has a pairing interaction. The pair potential

40

I

(a)

O.G -

-

-.
/,'
I,'

I,'
Ii

Ro = 0.5 I,'

\J)C)

h

~

/:

0.1 -

/:

.............

Ro = 0.2,--.......

~

-

f

'---'

<1

--

0.2 -

CL 5

ito = 0.0

-

I

o

5

Z/~

Figure 6: The Ro dependence of (a) the pair potential and of (b) the Andreev reflection
for the N-S infinite double layer. Temperature T = 0.2Tg, the critical tClnperature of thenormal region t~ == Tt ITg = 0.0. The superconducting coherence length ~ == vF I7fTg .

41

(a)

0.6

I

I

>-

0.4 -

-

t ~ = 0.01

-----

-

tJ)C)

h

-

t ~ = 0.0\::::---N 0.2 ,...--..,.

---------- gN 1g5 = -0.5'-----"

<J
l-

I

-

/
-----'"

0

---

./

-

l-

I

I

- 10

-5

0

-

5

Z/~

(b)

1

t~ = 0.01/--- ----------

~~~-

./

~

~

t~ = 0.0

gN Ig 5 =-0 .5

o1

E 1t6. bu1k

Figure 7: The pairing interaction dependence of (a) the pair potential and of (b) the Andreev reflection for the N - 8 infinite double layer. Temperature T = 0.2Tg, the interfacialreflection coefficient Ro = 0 .2.

42

of the nonnal region increases as the interfacial reflection Ro decreases and as the criticaltemperature of the N increases. When the pairing interaction of the N is repulsive, theN metal has the pair potential of opposite sign to that of the 5. In the case Ito = 0,
the Andreev reflection coefficient RA at the lower energies than ~bulk, is unity. The RA
decreases as Ro increases at lower energies. When the pair interaction is repulsive, theAndreev reflection at low energy is enhanced. When the pair interaction is attractive,
it is suppressed inversely. In particular, at the zero energy limit, the Andreev reflection
coefficient RA can be estimated to be
1 - Ro - (1 + Ro) tanh 28 2
1 + Ro - (1 - Ro) tanh 28 '
oo
1
1

1

r- dz~N(z).

IT Jo

vFz

(177)
(178)

0

The Andreev reflection RA at lower energies is given in terms of Ro and of the integral of
the pair potential over the entire region of the N side. By use of Eq.(157), the differential
conductances corresponding to the cases in Figs.6 and 7 are shown in Fig.8.

We haveplotted the normalized differential conductance K such that tends to unity at the highenergy limit, i.e.,dI/dV
K(eV) == dI/dVlev-+oo

1 - RN(eV) + RA(eV)

1- Ro

for the N - 5 case.

(179)

Before concluding this subsection, we make some comlnents on the wave packet picture of
the Andreev reflection proposed in Refs. [11], [13] and [7] to explain the stair-like structure
observed in the differential conductance. According to this picture, an electron wave packet
with energy E changes the sign of its group velocity at the position that satisfies E = ~(z))
thus is converted to a hole wave packet. It follows that, if some structure in the energy
dependence of the Andreev reflection is found, the structure can be assigned to singular
43

(a)
4

~

/1
/1
/1

Ro = 0.0~
Q)'---""

~

/1
/ I
/ I
/ \
I,' -, \
,1 :\

2

I
/

Flo = 0,.,2-"

/

/
/
/

Flo = 0.5.,./--- ---/'

01

eVj 6 bulk

(b)
2

1

--r

t~ = 0.0

gNj g s =- 0.5

o1

eVj 6bulk

Figure 8: The differential conductance (a) and (b) for the cases shown in Figs . 6 and 7 ,respectively.

44

points in 6.( z), say 6. (0). The peak around 6. (0) may be interpreted in this way. Theysuggested that the conductance enhancelnent at lower energies is due to the wave packet
reflection by the induced pair potential in the N region. Such a picture, however, cannot
be applied to the N-S contact systeIll, in particular at low temperatures. In order to forill awave packet of the Bogoliubov quasi -particle which has a group velocity with definite sign,
the range of the momenta involved should be sufficiently slnall, i.e., less than TC/VF. As aresult, the width of the wave packet becoilles longer than the coherence length ~ = VF /,rrTc .
On the other hand, the depression range of the pair potential 6.( z) is of order ~ as can beseen from Fig. 7. It is difficult, therefore, to say at which point the wave packet is reflected.
In fact, the present analysis shows that the induced pair potential in the N region does notlead to a significant enhancement of the Andreev reflection at lower energies .

45

3.2

Normal-normal-superconducting proximity contact system

The experiments to observe the Andreev reflection are usually performed using a point
contact system. A Bccdlc-like elcctrodc is attached to the norll1al layer dcposited on ansemi-infinite supcrconductor. A typical gcoHlCtry of the systern is shown in Fig.9. Son1eof the experin1cntal results of the differential conductance shows a stair-likc structure In
the volta.ge depcndencc, which can not be explained by the BTK fran1ework discussed In
the last section. There are SOIne attempts to explain the stair-like structure by taking into
account the possible finite pair potential in the N -side. But we have shown in the lastsection that the effects by the fini te pair potential is not so significant as to change thestructure in the voltage dependence.
We instead are interested in the effect by the finite reflection of electrons at the point
contact. Since the voltage drop is expected to occur at the point contact, electrons andholes which traverse the normal layer will be reflected not only by the N -S interface but
also by the point contact. The multiple reflection within the normal layer is expected tolead to some geoll1etrical resonance effects in the differential conductance.
We consider a 1110del norIl1al-norI11al-superconducting triple layer systen1 as shown in
Fig.lO, to study the geometrical resonance effects. For simplicity, we aSSUlTIe the electrodeto be a normal metal. The pairing interaction 9 in the norll1al metal is not necessarily
9 = 0 but can be 9

# O.

Let us consider the Andreev scattering in the normal- normal- superconducting triple
layer ( N - N'- S ). Using the same methods as in the N - S case discussed in Sec.3.1, we canfind the formal solution of the wave fun ctions. We have only to solve a following equation,46

electrodeN

S

Figure 9: The point contact device.

~(Z)

Ro

RL

N'

N

S

( electrode )

o

L

z

>

Figure 10: The normal- normal- superconducting ( N - N'- S ) proxin1ity contact system.

47

instead of Eq.(162),

U(O , -00)

- oo)) _(<I>~/(O))( <I>~(<I>~ (- 00) - Alo <I>~' (0)A

A=

MoU(O,L)_(<I>NI(L))<I>~/(L)A

A- MoU(O, L )ML(<I>!(L))<I>~ (L)- AloU(O , L )MLU(L, 00)_

A

A(<I>!(oo))<I>~ (00),( 180)

where <I>~ (-00) and <I>~ (00) are deft ned by Eqs.(158) and (159). From this equation, we
can obtain the anlplituci es of the wave [unction.

7'A= {(l/" jJ + f-La)(f-L*jJ + I/a) - RO(T*jJ + a-a)(a-*jJ + Ta)-RL(I/*a + f-LjJ)(f-L*a + I/jJ) + RoRL(T*a + (J"jJ)((J"*a + TjJ)
+2/RoR L( r( ( - "1(*)( a 2 - jJ2) cos cPc} / D,r N = eiBro { ~(ei<fc(181 )+ R ; i4>C )+~[a2(,2 _ 82) _ jJ2(,*2 _ 8*2) + 2ajJ(,8* -,*8)]+~RL[a2(,*2 - 8*2) - jJ2(,2 - 82) + 2ajJ(,*8 -,8*)]} / D,
c+ = do dL( 1/ a + f-L * jJ + / Ro R Le - i<f c ( (J" jJ + T* a ) ) / D ,c=-rLdodL(v'a + J1(3 +J~e-;¢C(TQ+ 7'(3))/ D,(182)( 183 )

(184)

D = {(I/*jJ+f-La)2 -Ro(T*jJ+a-a?
-RL(I/*a + f-LjJ)2 + RoRL(T*a + a-jJ)2
(185)
where
1/ =

48

(8 - "1"(186)(187)(188)
The above equations cOlltain now the phase factor eirPc . This describes the multiple reflection processes in the finite layer. Apart from a correction of order 1/ ppL the Andreev
scattering coefficients are obtained by averaging over ~c. All the coefficients of the Andrcev
scattering are obtained in the fonns( 189)(190)(191 )(192)(193)
where we note that 100z 12 -1.BzI 2 is a quasi - particle flux conserved in the Andreev equation.
These coefficients satisfy the flux conservation law, i.e.,(194)
Typical results are shown in Figs.11. The differential conductance K normalized by its
high energy limit(195 )

49

I

I

I

(a)
RL = 0.5
0.4 f--

Ro = 0.2

C/)u
h

~

---N

----------_.

.....--.....
'--'"

<1

Ro = 0.5-

0.2 f------~

I..-c-«-

I

0_ 4

Ro = 0.8

-2

I

2

0

4

Z/~

(b)
0.6

I~

0.4

I,

",
I

"'<:

':

~

I
I
I
I

I
I
,
,

: ,:

0.2

". n:
------ ...
/

/

/'"

"-

" /I:

J\~
I \"

/ I:
"'- .......

--0

/

0.4

--_.-"'" /

0.8

\ \,

\

1.2

E / 6bulk

Figure 11: The typical result of (a) the pair potential, of (b) the Andreev reflection andof (c) the differential conductance for the N - N '- S systen1 . The Andreev reflection as afunction of the incident energy E has double peak structure. T = 0.2Tg, the scaled N'
layer size l == L/ ~ = 1.0 and t~ = t~' = 0.01.
50

(c)

~

~

2

Q.l
"---"'

~

1
/
/
/
./

0

0.4

0.8

1.2

eV/6.~ulk

is plotted. The Andreev reflection coefficients and the differential conductance in the energy
range below 6. bu1k has double peak structure when both the interfaces have finite reflection.
When either reflection coefficient ( Ro or RL ) vanishes, the system is equivalent to that of
the infinite fl-S system discussed in the last section .
The double peak structure can be interpreted as follows. In the N' region, a right going
electron ( hole) is partially reflected into a hole ( electron) at the N'-S interface by the
Andreev process . The reflected hole ( electron) is reflected into a hole ( electron) at the
N -N' interface by the normal reflection process. The hole ( electron) is again converted
into an electron ( hole) at the N'-S interface and so on. This sequence leads to the deGennes- Saint-J ames bound state which will be discussed in detail in the next section. The
bound state is broadened by the finiteness of the reflection at both the interfaces. The
double structure found in the Andreev reflection can be interpreted as resonance states
formed in a valley between the partially reflecting interfaces.
Kieselmann[33] has calculated the density of states of the finite width norn1a.l layer on51

the serni- infinite supercollductor by use of the conventional quasi- classical Green's fUllction
method. lIe showed that tlle density of states also has double peak structure at the lower
energies than 6 bu1k ' This double peak is also rclated to the broadened de Gennes- SaintJames bound statc. Ashida et al.[20] have shown the existence of the silnilar bound sta.tes
in the finite N - S double layer. van Son ct al.[13] have studied the case when Ro i- 0 and

RL = 0 using a n10del pair potential.
The peak structure shifts to the lowcr cnergy as the N' layer size L increases as is shownin Fig.12. When the layer size L is sufficiently large, there appear several double peaks.
We consider a role by the pairing interaction in the norn1al region for the probability ofAndreev reflection. Figure 13 is for the case that t~ = 0 and t~' i- O. Energy level of thevirtual bound state in the N' is lifted by the pair potential in the N' . The two peaks ofRA are shifted to higher energy as t~' increases. Figure 14 is for the case that t~ i- 0 andt~' = O. In a similar manner to the N - S infinite double layer system, the probability of

the Andreev reflection in low energy region is slightly enhanced as t~ increases. But theeffect by finite t~ or t~' is not so large as to change the structures.
In the one point contact experiment, the differential conductance has a stair-like structure with respect to an incident energy[6][7] . In the N - N'- S system with finite reflection
coefficients, we obtained the probability of Andreev reflection having two peak as a functionof an incident energy. However, the enhancement of the differential conductivity at zero
bias in the above experiments can not be explained . We can not reproduce the stair-likestructure found in the point contact system. Nevertheless, we expect that the double peakstructure will be observed in the coplanar geometry. We have also shown that the finite
pairing interaction in the N ' layer does not explain the stair-like structure in contrast to
the claim by Ref. [11], [13] and [7].

52

(a)

0.4

1=21 = '1=5
1
0.2

I

/1\1

,/ I ~...\ ... / \

,/

" ',I

\ ' I'...... - ' / "-,--~~-

, ../

'""

I
I

~',\

"...... "
\ ,

o

-

"

-'-'-.,-------

1

(b)

1

o

1

Figure 12: The N ' layer size L dependence of (a) the Andrcev reflection, and of (b) th edifferential conductance. T = 0.2Tg, Ro = RL = 0.5 and t~ = t~' = 0.0l.

53

We did not consider a possible supercurrent flowing across the N '-S interface. When thenormal metal has a pairing interaction, there exists a pair potential in the normal region
a.s was shown above. It is possible that Josephson like supercurrent 111ay flow across the
N-S interface in addition to the excess current carried by the Andreev reflected hole. The
excess low voltage conductance observed in some experiments[16] may be interpreted as
being due to the supercurrent. In order to discuss this possibility in a unified way, it isnecessary to develop a non-equilibriulTI quasi-classical formulation. This is an interesting
problen1 to be further studied.

54

(a)

0.4
tc

Nt

= 0.01Nt= 0.0Cl')CJhto:::---t"J~tc

0.2

gN'jgS = -0.5

"---'

<l

0

"' \

-4

-2

0

2

4

Z/~

(b)
0.4

0.2
...

...........

"o-... - _....

-.....--_/ '/'

0.4

0.8

1.2

E / D.fulk

Figure 13: The N' layer has a pairing interaction. (a) The pair potential, (b) the Andreev
reflection, and (c) the differential conductance. T = 0.2Tc, tIJ = 0.0, I = 1.0 and Ro =RL = 0.5 .

55

(c)

2

1

o

56

(a)

(b)
0.4

0.2
- ... -----..,.

oFigure 14 : The N layer has a pairing interaction. (a) The pair potential, (b) the Andreev
reflection, and (c) the differential conductance. T = 0.2Tc, t~' = 0.0, I = 1.0 and Ro =RL = 0.5.

57

(c)

2

tIJ = 0.01.....--...
~~'---'"

~

1

It~_=

0.0

~
gN /g5 = -0 .5

0

0.4

0.8

eV/6fulk

58

1.2

4

The local density of states of the superconductingnormal-supercol1.ducting triple layer system

Recently, in the supcrcond llcting- nonna.lll1ctal- supcrconducting ( 5- N - 5 ) proxilllity COlltact system the local density of states was studicd using a scanning tunneling ITlicroscopc
by Inoue and TakayaIla.gi[36]. Their data indicates that the local density of states in thenormal region has an cffccti ve cIlcrgy gap an d dcpcnds on a distance f1'On1 the S - N interface.
Tanaka et a1.[34] studied the density of states of the S- N - 5 semi- infinitc systeln with theself- consistently solved pair potential but with ideal interfaces by nurnerically solving the
Bogoliubov- de Gennes equation. They showed that the density of states spatially varies
when there exists the pairing interaction in the N region. Also, Hara et al. [35] investigatedthe density of states in the N - 5 finite double layer system in which the N - 5 interface
has finite reflection but thc N region has no pairing interaction. They showed that thedensity of states has an energy gap- like structure, although their model does not have apair potential because having no pairing interaction in the N region. We study the densityof states of the 5-N-5 system having realistic boundaries based on the self- consistent pairpotential.

4.1

The de Gennes-Saint-James bound state

We first consider the bound state, which was discussed by de Gennes and Saint-J ames [28],in a finite width normal layer connected to a bulk superconductor as depicted in Fig.15. It
was assumed that the pair potential is spatially constant and the N - 5 interface is perfectlytransmissive. They considered the states below the superconducting pair potential energy

6.

59

------------

~

5

N

o

z

L

Figure 15: A nonnal ll1etal connected to a bulk superconductor with a constant pair potential. The width of the nonnal metal is L. The nonnal- superconducting interface isassumed to be perfectly transmissive.

Within the Andreev approximation, general solutions \If of the Bogoliubov de Gennes
equation Eq.(13) with the energy E can be easily solved as

(197)
(198)
(199)
(200)
where a, b, c, d, e and f are constants and <I>~ and <I>~ have been chosen to tend to zero at
sufficiently large z . On the other hand, the boundary conditions at the nonnal layer end

60z = 0 and at the N - S iuterface z = L are written as<I> ~ (0)+ e i <I> ~ (0) = 0,(201 )

Tlo<I>~(L) = <I>~(L),(202)

where 770 is a pha.se shift at the N layer end. Fronl the general solutions and the boundary
conditions, one can get a.n equation to determine the bound states.

2L( n 71" + rp) cos 0 = -~ocos rpn : integerE
cos rp =where ~o = VF /~.

~'Consider first the case 0 = 0 (propagation along z).

(203)
For fixed L,

equation (203) has a finite number of solutions rpn(En(O)) below the energy gap ~. The
number of solutions increases with L / ~o. There is always at least one solution below ~ (even when L/~o « 1 ).
The total density of states including all the polar angles 0 of the Fermi mornenta isdefined as(204)
where A is the area of the interfacial surface. Performing integration of k x ) ky ) one canobtain2L1(
cot rp )n(w) = 2N(0)71"T L cos rp ()2 1 +,~o n=n c
n71" + rpn71" + rp

(205)

w

cos rp =~'where nc is a lower limit of n - sulnmation, i.e., nc is the s111allest integer which satisfies

2Ln71" + rp ~ ~ cos rp .
61

de Gennes and Saint-James obtained the density of states with discontinuities at E = En(O)( Fig.16 ).

y = 2L/~oy = 2.58

Y = 8.0
Y = 20.0

/1

j

4

/'
/'
,':

/1 ,':
II'
II " :
II,' :
I I,'
:/

I"

Y \

,',

., /.. :,/ ,/ } " /' 1l---::>'C--

I I'I I I 1,/,,' \.-/'V.....
I / 1/ ' ,,'I

IV., . . , . . ~. . . - '1 /-;. ......-

I, __ --:: ~/I-;",~ ... "

o

0.5

1

w/~

Figure 16: The total density of states below the gap energy 6 in the normal region of thesystem shown in Fig. 15 .

w

wL

When ~ ~ 1 and ~Z ~ 1, the density of states n( w) is
0

n(w)

Lw

I"'V

2N(0)7r ~o ~.

The density of states is linear with respect to the energy and to the nonnal layer sizearound the low energy. Similar results was obtained by Hara et al. [35] in the finite normalsuperconducting double layer solved self- consistently with a finite interfacial renection.
When the layer size L is large, there occur several discontinuities in the density of stateswL
as is shown in Fig.16. When -w ~ 1 and ~

~ ~0

n(w)

I"'V

~ I, t2N(0) .

62h e denslty. 0 f states IS.

In this ca.se, the densi ty of states becomes the normal state density of states for bothprojections.

It is found tha.t the density of states in the normal regIOn with a superconductor ha.senergy spectrum below the gap energy 6. because of the finiteness of the nonnal layer.

4.2

The superconducting- normal- superconducting semi-infinite
triple layer system

In this section, using a self- consistently solved pair potential, we discuss the density ofstates in the S- N - S system with finite interfacial reflection as depicted in Fig.17. We
assume R == flo = R L . We also take account of the pairing interaction in the N region . In
terms of the diagonal elements of the Green's function for the real frequency c = w, thelocal density of states associated with the Fermi momentuD1 PF are defined byn(w, p~, z ) ==L 8(w - EI)<I> o,[(Z)<I>~I(Z)I~

47rZVFz

P3 [9cw(W + iO, z) - 9cw(w - iO, z)] .

(206)

From the expression for the quasi- classical Green's function Eqs .(79) - (88), the Green's
function can be in general written in such a form as

where X is a matrix . For the real frequency c = w, the evolution operator satisfies(207)
From this equation, it can be found that the real frequency Green's function satisfies

(208)

63

0.6
/--=

/,'
I,'

I,'
I,'

Ii
I:

0.4

1/
I
I

tfl c')

h

I:::

S

N

S

....--...

---~'--.-/

<J

R=

0.2

0

0.5
0.2
0.0
2

0

-2

4

Z/~

Figure 17: The pair potential for the superconducting- nonnal-superconducting Sel1llinfinite triple layer. T = 0.2TJ, t~ = 0.0, I == L/~ = 1.0 and ~ = vF/7rTJ.

II

"

II

::

II
I
I
I
I

::
:~
: ~,
,
,
I

\

" ,II
,II

\

\

2

\
,

:/1
I

\ '

\:

":'-,

~I

)1

,'II

I
I

R=
0.0
0.2
0.5
__ ------::-:.-::-:..-=---

1/,,/

1/
I

I
I

o

2

Figure 18: The density of states as a function of the energy w for the case shown in Fig. 17 .
This density of states is for the Fermi 1110mentuln normal to the interface, so called thetunneling density of states.

64

Then, g(w + iO) - g(w - iO) is written in a [orn1~a) - iPl (:_ . (a + a* b - c* )gcw (w + iO) - goo (w - iO) = i (~-Zc - b*b )*PI-a-a - a*(209)

To obtain the density o[ sta.tcs, thcrcfore , wc havc only to trcat the Grcen 's fun ctioll [orthe real frequency c = w + iO and we obtainn(w, Pp, z ) = _1-P3Imgaa(w + iO).

(210)

27fVFzUsing the self- consistently solved pair potential, wc nUlTlerically calculated the density ofstates when the polar angle 0 = O. When the interfacial reflection coefficients are finite,
the density of states has double peak as a function of the energy w as is shown in Fig.18.
In SOlne systems this double peak structure of the density of states has been reported
theoretically. Kieselmann[33] obtained the density of states in a N - S contact of a finite
normal layer with a semi - infinite superconductor. Ash ida et al.[20] and Hara et al.[35]obtained the density of states in the finite N - S contact system.
The density of states at the lower energies than 6.bu lk are related to the de Gennes- SaintJames bound states[28] discussed in the previous section. The de Gennes- Saint Jan1es
bound states for the polar angle 0 = 0 becomes considerably broad when the interfacialreflection coefficients are finite as is shown in Fig .18. The width of the broadened boundstates increases with the reflection coefficients R .
When the pairing interaction of the normal region is zero, it can be easily shown thatthe density of states of the N region is spatially constant as follows : the spatial variationof the density of states is expressed by the evolution operator U, i .c.,65

The V± of the nonnal region without pair potenLial is written as(211 )
Then, in the nonna.l region the density of states does not spatially change.
On the other hand, if the pa.iring interaction is finite in the nonnal region, the densityof states spatially changes, buL its peak position does not change spatially. Figure 19
shows the density of states at the cenLer of the N region when the N Inetal has a pairing
interaction. When the pairing interaction is attractive (repulsive), the peak level of thedensity of states shifts to higher (lower) energy side.
The bound state level shifts down as increasing the normal layer size L in Fig.20. It
corresponds to the usual bound state problem of an electron in a potential well. When thelayer size is sufficiently large, several broad bound states appear, in a sin1ilar n1anner tothe de Gennes- Saint-James' result shown in the previous subsection.
It has been predicted that the density of states of the normal region have energy gap at
the lower energy than llbu lk . This state can be detected by use of a tunneling experiment,a scanning tunneling microscope ( STM ) measurement and so on.
In the above, we calculated the densi ty of states for the polar angle 0 = 0 of th

Fern1imomentum, so called' the tunneling density of states" which is detected by the tunnelingexperiment[17]. The STM experiment, however, can detect rather" the total density ofstates" which is averaged over all direction of the Fenru mornentun1, than the tunnelingdensity of states. Next, we calculate the total density of states in the normal region.
The total density of states is defined by( -rr /2
ntotal(Z) == N(O) Jo dO sin 0 Img(z,w + iO).

(212)

The total density of states was first obtained by de Gennes and Saint-J ames in the ideal66

0 .6~--~--~--~--~--~--~--~

(a)

0.4
CJJC)

h

t=::

---~

....--..,

0.2,.-----tIJ'-...-/

<]

0

9

NI

S

9 =-

0 r::

.0 - - - -

=

0.01

t eN = 0.0_____________ f - -- \

I

\

o

-2

2

4

Z/~

(b)

II

5 gN I g S = -0.5[:

~:,

o

1.5

Figure 19: (a) The pair potential and (b) the N region tunneling density of states when
the N metal has a pairing interaction. T = 0.2Tg, I = L I ~ = 1.0 and R = 0.2.

6710'---~----~,~~~----~,----~--~

-

iI
I

I

-

I
I

I

I

I

,,
I

\
\

,,

\
\

I= 2

-

I

\
\

~

I

"

f

I
I

::

~ :,

-

1=1

~

I

"
"

I-

-

,I
II

"
,1
"

-

II
, I

,
I-

, -

--'

-

-

-

: r-I

I

_

I

I

o

0.5

wi LJ.~ulk

1

l.5

Figure 20: N layer size dependence of the density of states. I = L/~, T = 0.2Tg, tlJ = 0.0
and R = 0.2.

N - 5 contact system ( see the previous section. ) Recently, the total density of states wascalculated by Hara et al.[35] in the finite N - 5 contact with finite interfacial reflection coefficient and by Tanaka et al.[34] in the semi- infinite S- N - S contact without the interfacialreflection coefficients.
Typical result is shown in Fig.21. In the norn1al region the particle with the polar angle() -=I- 0 moves effectively the more long distance than the normal layer width

L. Since

the total density of states includes contributions from all the polar angles of the Fermi
momentum, it includes the states with lower energy than the lowest energy level of thetunneling density of states. The total density of states has a cusp and a shoulder. The
cusp and the shoulder are located at the two peak positions in the tunneling density ofstates.
The finite density of states 1n the energy range below LJ.bulk is originated f1'on1 the deGennes- Saint-J ames bound state in the N - 5 system discussed in the previous subsection .

68

,
,, ,,
,
,,, ,,
,

[)

"

"
"
"
"

"

"
"

,," ,,

' ,

,,
,,
,

0
'---"

,: '
I
I

~

I

\

---2- 2.5

I.---...

"Ci
"0

~

o

0.5

1

wi 6~ulk

Figure 21: The total density of states including all the Fermi momentum components in
the N region. A solid line shows the total density of states and a dashed line shows thetunneling density of states. T = 0.2Tg,t~ = 0.0, I = L/~ = 1.0 and R = 0.2.

4t{! = 0.1.---...
0
'---"

~

3

t{! = 0.0---~.---...
<.J.../>
l.()

-----

gN I gS = -0.5 /'

0

II
~

2

/1

-2
]

.3

~

/

1

//

/,'
I,'
/

\

----,

\
\

\
\

I
I

,/
,/

,,

\ '-.,'" " _______

II.....-:

I,

," ,
,, ,,
,

I '

.....-:

I~

\

/

\

/
/

".....-:

0

0.5

1

wi 6~ulk

Figure 22: The total density of states when the N metal has a pairing interaction. Z = 0.5~,
T = 0.2T8, I = L/~ = 1.0 and R = 0.2.

69

In the N regIon, a particle is reflected into a hole at the N - S interface by the Andreev
reflection. The backward hole is reflecteJ iJlto all electron at the S- N interface. The deGennes- Saint Jarncs bound state is nothing but this closed sequence of particles a.nd holes.
When the interfacial reflection coefficient is zero, the phase of the wave function earneda.long the closed path is rather sn1a11 because of a cancellation between the particle pathand the hole path. When the interfacial reflection coefficient is not zero, however, a closed
path including only particle andj or only hole is 111ixed. The latter path earns a phasean amount of 2p F L cos O. When R is finite, therefore, the de Gennes- Saint Jarnes boundstate energy becomes very sensitive to the polar angle of the Fenni momentum. It forms
a band with width of order VF7r j L when contributions from very small polar angle range
Icose -11::; 7rjP F L are taken into account.
When the normal region has the attractive pairing interaction, the density of states in
the normal region has an energy gap as is shown in Fig.22. When the normal region has
the repulsive pairing interaction, the energy level shifts to the lower energy side.
For any pairing interaction, it can be analytically shown that the density of states at w =IS zero.

°When w = 0, the evolution operator can be written as

U+ ( z , z ') =( cosh 8z ,z '. . hr1, SInUz,z'-i sinh 8z ,z ' )hr,cos uz, z '8z ,z' = - _1_ ( Zdz" f1( z").

(213)
(214)

VFz } z'Using this expression, we can obtain the quasi- classical Green's function in the C layerg~+(z,w = 0) = i

2C(z) - trC(z) ,v(trC( z))2 - 4R oR£(215)

C(z) = ((1 + RoR£) cosh 280 ,£ + (1 - RoR£) sinh 280 ,£) po+ ((1 - RoR£) cosh 280 ,£ + (1 + RoR£) sinh 280 ,£) P2·
70

(216)

The Green's function with w = 0, therefore, does not depend on the position. It is also
shown that the density of states, which is obtained from the above Green's function, at
w = 0 is zero for any pair potential. This result is different frorn nUll1crical results obtainedby Tanaka et a.l. [3 /1]. They calculated t he total dellsi ty of states in the 5- N - 5 systcln wit hideal interfaces by nUll1crica.lly solving thc Bogoliubov- de Gennes equation. Their results
show that when the N ll1ctal has a repulsive interaction, the density of states at w = 0 is
not zero and depends 011 the position. In contrast to their nUll1erical calculation, we can
explicitly show that the density of states at w = 0 is zero and does not depend on theposition.
As we have noted, when the pairing interaction of the normal metal is zero, the densityof states of the normal metal is spatially constant. When the pairing interaction is not
zero, however, the density of states spatially varies as is shown in Fig.23. The density ofstates is larger at the point of which the pair potential is smaller. Also, when the layer size

L of the central normal ll1etal is large, the density of states has several peaks at the lower
energy than 6. bu 1k as is shown in Fig.24.
Recently, Inoue and Takayanagi studied the local density of states of Nb/InAs/Nb systen1
by use of STM . It was reported that the local density of states in the InAs rcgion has anenergy gap. Also, It was reported that the local density of states depends on a posi tion in
the InAs region. In the present model, when the normal metal has a pairing interaction,
the density of states depends on the position in the normal region. The InAs, therefore,
may have a finite pairing interaction. Since a realistic systern is not in the clean lill1it,
however, it seems necessary to take account of the ill1purity effect for 1110re quantitative
comparison with the experill1ents .

71

3----..

Z = 0.5e

<=>

"---'

~

-----

2

2

----------

Z = 0.16e-----

Z = Oe

d

3

~

1

o

1

0.5

w / t6.~ulk

Figure 23: Position dependence of the N region total density of states when the N n1eLal
has a pairing interaction t~ = 0.1. T = 0.2Tg, I = L/ = 1.0 and R = 0.2.
.

e

4

"
"
"

"
"

"

"
"

"

' \

----..
<=>

"
"

"
"
"
"

"
"
"
"
"
," I

,, ,,
,, ,,

"---'

~
----..
-----

I

,

, ,,
,

"

2

2

" -\

"
"
"
"

,, ,

I
I
I

,,
,,

"

,
,, ,

If)

II
N

I

: ,,

'-'..P

,
,

I

, :,
,
I

\

I

I

\

d

~

~

o

0.5

1

w / t6.~ulkeFigure 24: The density of states when the N layer size L is long, L/ = 10.0. A solid line
shows the total density of states and a dashed line shows the tunneling density of states.
Z = 5e, T = 0.2Tg, t{j = 0.1 and R = 0.2.

72

5The infinite double layer system with s-wave andd-wave superconductor

So far we discussed s- wave superconducting proximity contact systems. In this last section,
we study the proximity contact system which consists of superconductors with different
symmetry.

It is believed that "heavy fermion superconductor" and "high Tc superconductor", whichwere recently discovered, are classified as superconductors with unconventional pairing.
Pals et al. [39] investigated the Josephson contact between a singlet and a triplet superconductor using the tunneling IIalniltonian. Poppe[40] observed a Josephson current between
an s- wave superconductor Al and a heavy fermion superconductor CeCu2Si2. Ashauer etal.[41] studied theoretically a thin film of standard superconductor in proxin1ity contact
with a bulk unconventional material by use of the conventional quasi- classical method.

d-waves-waveoz

Figure 25 : The s- wave superconducting and the d- wave superconducting double infinite
proximity contact systern.

73

We study the double infinite proxin1ity contact systen1 between superconductors withdifferent pairing interactions, that is, with s- wave pairing and d-wave pairing interaction
as depicted in l~ ig.25.

Since both the superconductors are singlet superconductor, we have

only to treat 2 x 2 rnatrices. The d- wave pairing interaction is expanded in tenns of thespherical harmonic functions Y2m,

m=2

= -47rg2 L Y2m (0, ¢;)Y2*m(O', ¢;'),

(217)

m=-2

where 0, ¢; are the polar angles of the Fermi rnomentum. Therefore, the pair potential ofthe d-wave superconductor can be written in the fOrIn~(pF' z) =m=2

L ~m(Z)Y2m(e, ¢;).

(218)

m=-2

The d-wave superconducting pair potential in general can not be chosen real. The evolution
operator Uex , therefore does not satisfy the relation Uex = PIU-exPI . Instead, it can be foundthat it satisfies for the Matsubara frequency c = iW n

Uo:(W n ) = P2 U-ex(W n )* P2,(219)-Wnr Pl·(220)

Uex(w n ) = PI Uex (Therefore, the quasi-classical Matsubara Green's function satisfies(221 )(222)
even when the pair potential is not real.
Setting RL = 0 in the quasi-class ical Green's function of the finite triple layer systen1 andtaking the limits of LL, LR to infinity, the quasi-classical Green's function for the double

74infinite system can be obtained. Here, we rewrite Ro to R. Noting the above relations valid
for a complex pair potential, one can obtain the quasi- classical Green's functions, i.e.,"L ( ) _ .2AL( Z) - trAL(z)g++ z - 1,
trAL( z )
,

(223)

"n ()
.2BR( Z) - trBR(z)z = 1 , - - - - - -- -- -- ++trBn( z )
'

(224)

,V L
A£( z ) = U+(z ,O) [nV_ (0) + RP2 T V_R (0)P2 ] P2 T V+L (Z)P2'(225)

Bn( z ) = V_R(z) [P2TV!(0)P2 + RV;(O)] P2 T(;:(Z, 0)P2,(226)

V;( z ) = ¢~(z)¢~(O)t,(227)

V_R( z ) = ¢~(z)¢~(O)t,

(228)

9

where

(;+
L(z, 0) = U+
L(z, 0) eE !; z/vfrz ,

(229)
(230)

and ¢± (z) is a solution of the equationi6(z) ) ¢ (z)Wn =f En±,(231 )

with the boundary condition

We consider a simple case where the superconducting critical temperature in both thesides are same . Anderson and Morel[37] showed that in an isotropic systen1 the most stabled- wave superconductor pair potential is

(232)
75

0.1

o

4

Z/~

Figure 26: The s- wave superconductor vs the d- wave superconductor defined by Eq.(232).
The dashed line is 6. 5 and 6. AM in the case when the other side lnetal is a nonnal Inetal
without the pairing interaction. T = 0.5Tc and R = 0.0.
0.8~--~----~--~----~----~--~

0.4

o

o

-2

2

Z/~

Figure 27: The s- wave superconductor vs the d- wave superconductor defined by Eq.(233).
The dashed line is 6. 5 and 6. xy in the case when the other side Inetal is a norn1al metal
without the pairing interaction. T = 0.5Tc and R = 0.0.
76

We consider the d- wave superconductor with the pair potential defined by Eq.(232).
Figure.26 shows a profile of the pair potential ncar the interface. The solid line is 6 s (z)and 6 AM (z) in the left side and the right side, respectively. The dashed line is 6 5 and 6 AM
in the case when the other side metal is a normal Inetal without the pairing interaction.
The pair potentia.l is suppressed around the interface when R < 1 due to the proxi111ity
effect. When the other side is rather a superconducting rnetal than a nonnal 11letaJ, thesuppression of the pair potential is s111all. On the other hand, we considered another d- wave
superconductor defined by
2

6(PF, z) = 6 xy sin 0 cos 2¢= 6 xy

(233)

V967r

6v5 (Y2,2 + Y2,-2)'

In Fig.27, the pair potential is suppressed near the interface, but the suppression in thecase where the both side 111etals are superconductor is aln10st equal to that in the casewhere the other side metal is a norn1al m tal.

The d- wave superconductor defined by

Eq.(232) and defined by Eq.(233) are different in that the pair potential of Eq.232 conta.ins
s-wave component. The s- wave pair amplitude in these d- wave superconductor, F S , canbe obtained by'" {21r d)..cx{7r /2 . OCXd()CX D (cx ) ~ ( .
A d)J
F s = TN (0) '"~ ~ InIf' InsIn
FO PF . PF 9cxcx 'lW n , D
1 ,2,
4

Wn

cx=± 0(234)

0where Po = 1. In the d- wave superconductor defined by Eq.(233) the s- wave pair a111plitude, F S vanishes after the polar angle integral. On the other hand, the superconductor
defined by Eq.(232) has finite s- wave amplitudes. If the d-wave superconductor has thes-wave pair amplitude, the d- wave pair potential ncar the interface is l1l0re enhanced thanin the case where the neighbor metal is a normal metal.
77

Due to this enhance111ent of

the d- wave supercollductor, the neighbor s- wave superconductor ncar the illterface is alsoenhanced.
We nex t cOllsider the case when a s tl perctl rrent flows in this proxil11i ty contact system.
For simplicity, we only treat. the d- wave supercollductor defined by(235)
which is obtained by a linear c0l11bination of Eq.(232) with its c0l11plex conjugate. Whenthe supercurrent flows, the pair potential has a spatially varying phase factor.
d. d

6 (z) = 16 2 0(z)le1I;?z(3cos 0 - 1).

(236)

6 5 (z) = 16 5 (z)le i l;?;.

(237)

2

As is well known, when the supercurrent flows, a spatial derivative of the phase is a finite
constant value in the bulk region.
in the bulk region .

(238)

From Eq .(116), the electric current along the z direction is written in terms of the quasi classical Matsubara Green's function a.s(239)
Noting the conservation of the electric current and setting the spatial derivative q of thephase at sufficiently large z to an appropriate value, we nUl11erically calculated a selfconsistent pair potential containing a spatially varying pha.se factor.
When the supercurrent flows, the quasi - classical Green's [unctions can be written as

(240)
78

(241 )
(242)
BR(Z) = V_R (Z) [T
P2 V_L (0)P2 + RV+L (0) ] P2 TVV+R (Z, 0)P2,(243)

Vf(Z) = 0~J~(z)(0~J~(0))t,(244)

V_R(Z) = 0~J~(z)(0~J~(0))t,

(245)

~
c.p =

(e- 0<.{Jz/2 <.0)
{J z/2'
i

(246)

Ci

where

(247)

(248)
and

J± (z) is a solubon of the equation(249)

with the boundary condition

The gap equatjon can be written in terms of the off- diagonal tenns of g++.
we

7rT

( 1f /2

L Jo

sinO'dO'V5(d) [g++(iw n , Z)Jl,2 - g++(iw n , Z)*J2,1]

6(PF'Z) = ___w~n>_O____________________________________
T
we/ 21fT
1log Tc + ~ n + 1/2
Vs = 1

Vd =

(250)

for s- wave,~(3COS2B -1)(3cos 2 0'-1)479

for d- wave defined by Eq.(235).

(251)

N
~

"U
~
(ij

~

0

\0:::

-----<J

\
\

-1

o

-5

5

Z/~

Figure 28 : The pair potential when the supercurrent flows . The sohd line is an absolutevalue of the pair potential and the dashed line is the phase of the pair potential. T = 0.1 Tc ,R = 0.0 and qS~ = 0.1.

1
N
~

"U
~
(ij

~

0

\0:::

-----<J

-1
,

-4

o

-2

2

4

Z/~

Figure 29 : The pair potential when the supercurrent flows in the s- wave vs s- wave Josephson junction. T = 0.8Tc, R = 0.5 and qS~ = 0.3.

80

I

f-

;j

-

~

~

-----Nf-"'----"

~

f-

-

~

-

I

I

I

I

I

I

-5

0

I

5

Z/~

Figure 30: The position dependence of the supercurrent numerically obtained by use of theself- consist.ent pair potential. T = 0.2Tc, R = 0.0 and qS~ = O.l.

We obtained the self- consistent pair potential numerically by use of the Green's [unction.
Typical result is shown in Fig .28 . We have expressed the magnitude of the supercurrent J
by the spatial derivative of the phase qfulk~ at the bulk region of the s- wave superconductor.
The obtained self- consistellt solution shows an unexpected behavior. The phase gradientchanges its sign near the illterface. This result is quite different frol11 the usual s- wave vss- wave Josephson junction . In the s- wave vs s- wave superconductor, the spatial derivativeof the phase qz was larger than the bulk qbulk . This can be understood from the usual G- L
equation. The electric current is written as

Since I~( z) I near the interface are suppressed, qz near the interface is larger than the bulk
qbulk

as a result of current conservation . N ulnerical calculations using the quasi - classicalGreen's function shown in Fig.29 for the s- wave vs s- wave junction show the sanle results
as the G- L arguments .

81

To check the conservation of the electric current in the s- wave vs d- wave junction, we
calculated the current by use of the self- consistently solved pair potential. In Fig.30, theelectric current is found to be conserved everywhere. The junction between the different
pairing superconductors cannot be expressed in a form of the usual Josephson junction.
This phase anomaly near the interface is rcrnarkable at lower telnperatures.( Fig.31. ) Also,
under a fixed current, the phase difference at the interface does not n10notonically varies
as is shown in Fig.32, when the interfacial reflection coefficient R increases.
The physical origin of the above results arc as yet not clarified. We try to discuss whether
the pair potential profile obtained above can be obtained fron1 the Ginzburg- Landau ( G- L
) equation . By expanding the gap equation with respect to the pair potentia.l , we discuss
the systen1 in the G- L region. Higashitani[42] studied the Josephson junction of s- wave
vs s- wave superconducting junction in the G- L expansion of the quasi- classical Green's
function. We follow the prescription proposed by Higashitani. As is well known, in the G- L
region 1 - T/Tc « 1, the zeroth, the first and the second derivatives of the pair potential
are of the order of

(252)
1

where t == (1 - T / Tc ) "2
where a prime of fl' means the spatial derivative of fl. Since t ~ 1, we retain the tenns
up to the order of O( t 3 ) in the G- L expansion of the gap equation.

82

0.8~--~--~----~--~----~--~------- - T=O.lTc------- T=0. 2Tc........

""-

"\

0.4

\

,, '
1/

\

/---- ------

\ ,'/
\ /
\I
I - - - - T=0. 5Tc

o

o
Z/e

-2

2

0.8~--~--~----~--~----~--~

0.4o.,.,----------------- ---...-----------

T=0 .5Tc,----T=0.2Tc-0.4 _-----

-0 .8

-2

o
Z/e

2

Figure 31: The pair potential in the presence of the supercurrent at various temperatures. T = 0.1,0.2, 0.5Tc, R = 0.0 and qse = 0.1.

83

0.4

----- R = 0.6---------- R = 0.2
0.2--R=O.O

o

5

Z/~

0.5

---------o--R=0.2-0.5

IR = 0.6'~~R=O.O

o

-5

5

Z/~

Figure 32: Pair potential profile of the s- wave vs d- wave junction for various interfacialreflection coefficients under a fixed current. The phase difference of the pair potential at theinterface does not monotonically varies as a function of the interfacial reflection coefficient .
T = O.ITc and qS~ = 0.1.

84

We rewrite the Green's function in tern1S of ¢± given in the appendix. Since g~+ canbe shown to have the saIne form as g~_ except for the superscripts Rand L and for thesign of VFZl we only treat g~+ . Before expallding the Green's function g~+, we write it interms of ¢±where we have used the properties of ~ discussed in the appendix . It is useful to define ~±as follows:~_(z) = F_(z) CV~(z))'(256)~+(z) = F+(z)" CV\(z)") ,(257)

OzV- = _1 (2wnV_ -\.6.(z)\(l - V~)),
VFz

(258)

where V± and F± satisfy

1-

8zF- = --(w n - En + \.6.(z)\D_)F_,
VFz

(259)

azD~ = _ _1 (2wnD~ _ \.6.(z)\(l _ D~2)),

(260)

VFz

8zF~ = _1 (w n - En + \.6.(z) \ D~)F~(261 )

VFz

with the boundary condition

D- \bulk -D*\\.6. bu1k \+ bulk - wbu1k + En85-bulk_wnn-W+,VFzq1,-2-'

Using the equation(257), one can write An(z) in tenns of V and F ,AR(Z) = (0:)1 lAk( z ) + Ah(z)Ak(z) =

A~( z) =1( iV~(z)(~~~;D c2

2Enz

j

0:,

-iV~(z)*)V~(z)V~(z)*

(263)

R

1 + V_(z)V~(z)*V!:(z)( iV~(z)2

(262)

i)

-V~(z)
1 + V~(O)V~(O)*

L

(264)

K o,

Kt = [(V~(O)* - V~(O)*eitpLR)(1 + V~(O)*V~(O)e-itpLR)-R(V'}:(O)* - V~(O)*eitpLR)(1 + V~(O)*V~(O)e-itpLR)]/ [11 + V'}:(O)*V~(O)e-itpLRI2 + Rll + V~(O)*V~(O)e-itpLRI2], (265)where CPLR = CPL - CPR . Following Higashitani, from the differential equation(258) one can
expand V as follows :V+ can be obtained by changing the sign of VFz in the expansion of V_ .
The Green's function consists of two terms . (See Eqs .(253)rv(255). ) One of themspatially varies on the scale of the order of the coherence length ~T( t) rv O(t-l) since itconsists of ¢± only. The other term varies on the scale of the order of ~o = v F / 7rTc , since it
has an exponential damping factor. In the G- L region in which ~T is sufficiently larger than~o, the second term is finite only in a narrow region near the interface. The gap equation

can be written in such a form as(267)
where 91 and 92 are functions varying on the scale of the order ~T which are related to

Ak and Ah, respect ively. The short range behavior is described by the exponential factor
86

eO z

damping within the range of ~o. In the above , (( ... )) Ineans the average over the polar

angles and the SUl11 over Wn- In Eq.(267), the first tenn gl describes the long range profileof the pair potential and the second tcnn g2eOz is considered to give an inforrnation on theboundary condition. Following Higashitani's method, we consider the equations:6( z ) = ((gl( Z))),(268)

o = fo CXJ dz ((g2( z )eo

z

(269)

)) .

Equation(268) gives the ga.p equation and Eq .(269) gives the boundary condition.

Hi-gashitani obtained the saIne boundary condition from Eq.(269) as that which de Gennes
obtained in the 5- 5' contact systelll by different approach.
Substituting Eq.(266) into the Green's function, we obtain the G ~ L equation after SOllle
manipulation :-6 T(z ) = ~f [(8R coS 2 8)R(6 T(z)qT( Z)2 - 6 T(z )") + (8~)R(6T( Z ))3/V}] ,(270)- 6 1( z) = ~ f [(8 L co S2 8) L(6 1( Z)ql (Z)2 - 6 1( Z)") + (8(271 )iJ L(6 1( Z)? /v}] ,o = 26 T(z)'qT( Z) + 6 T(z)qT(Z)' ,(272)

o = 26 1(z ),ql(z) + 6 1(z )ql( z )',(273)

where we have defined(274)

8 R (L) = { 3 cos 2 0 ~2 _T -~:for S region
for d region7((3)( VF ) 2

16 10gT/Tc{1r /2

7fT

'

(275)
(276)

(...) == io dO sin 0 x VS(d ) ·· ·· ,(277)

Vs = 1

(278)

and
87

The boundary condition are

o = R(8 R cos O)R~r(o) - aR(~r(o)' + i~r(0)qr(O))(8R cos 2 O)R_e i CP LR [R(8 L cos O)R~I(O) - aLR(~I(O)' + i~I(O)ql(0))(8L cos 2 O)R], (279)o = R(8 R cos O)L~r(o) + aRR(~r(o)' + i~r(O)qr(O))(8R cos 2 O)L_e iCPLR [R(8 L cos B)L~I(O) + aL(~I(O)' + i~I(0)ql(0))(8L cos 2 O)L],(280)

R(L) _ 7((3) v:(L)
a

-

27f2

(281)

7fT

and
1- R
R=--.
1 +R

We have taken account of tenns up to order of O(t 3 ) for the G- L equation. On the other
hand, the boundary condition includes terms up to order of O(t2), in accordance with theboundary condition obtained by de Gennes or by Higashitani. The obtained boundary
conditions is linear. They can be rewritten as

DIN12) ( 1 ,) ,(a1DrDr,) = (Nll
N21 N22
a DI

(282)
1

Dr = O'rAr = [(8RcosB)L(8RCOS2B)R+R(8RcosB)R(8RCOS20)L]2 A r ,(283)1

DI = O',A, = [(8 L COS B)R(8 L COS 2 B)L + R(8 L COS 0)L(8 L COS 2 B)R] 2 AI, (284)(285)

Nll = _1_ [(8 L cos B)L(8 R cos 2 B)R + R(8 L cos O)R(8 R cos 2 O)L] ,(286)

N12 = _1_ [(8 L cos 2 B)L(8 R cos 2 O)R - R2(8 L cos 2 B)R(8 R cos 2 O)L] ,(287)

O'rO'I

O'rO'IN21 = -

1

O'rO'I[(8 L cos B)L(8 R cos B)R - (8 L cos 0)R(8 R cos B)L],

N22 = _1_ [(8 L cos 2 B)L(8 R cos O)R + R(8 L cos 2 B)R(8 R cos O)L] ,O'rO'I

88

(288)
(289)

where N satisfies det N = 1 and
~T=

(290)

fi/=

(291 )

fitT =
-

(292)

At _

(293)

D/=

We obtained the linear boundary condition sinlilar to de Gennes' oue. It is, however, foundthat when the superconductors have different pairing symmetries, this discussion is notconsistent. Froln Eq.(239), the current in the G- L telnperature range in the bulk region
can be written as
(294)
where

{7f /2( ... ) j=J0

dO sin B co sO .. , .

From the requirement that the current should be the saIne in both the bulk region, we
obtain(295)
On the other hand, using the linear boundary condition Eq.(282), we obtain the condition(296)
When the pairing symmetry of the both side superconductors is different, the current
conservation condition Eq.(295) is not equivalent to the condition Eq.(296) derived frol11
the linear boundary condition . When the pairing of both the sides is of the saIne type,89

as the 5- 5' system discussed by Higashitani, there occurs no such problem. When the
supercurrent flows across the junction between different symmetry superconductors, theabove discussed G- L argU111ent seenlS not to be consistent. We point out SOlne problerns
at issue.
1. We have derived the boundary condition Eq.(269) by dividing Eq.(267) into two parts .

It is not clear whether the way ernployed to obtain the boundary condition Eq.(269)is unique in the G- L region, although we never find out a 1110re reasonable way thanthat used above which was successful to obtain the boundary condition in the s-wave
vs s-wave syste111.
2. The source of the anomaly should be the exponential damping tenn 92eoz in Eq.(267).

The spatial derivative of this term is of order of ~Ol O(t) although a range of theanomaly is negligible in the G- L region ( ~o ~ ~T ) . The spatial derivative of thedamping term may not be negligible in the present case and we 111ay have to take
into account the higher order terms in the G- L expansion of the damping tenn.
These problems are to be examined in the future study.

Appendix B. The properties of the evolution operator ( In t h epresence of the supercurrent flows. )In this appendix, we consider the properties of the evolution operator when the pair potential is not real. When the supercurrent flows, t he phase of the pair potential spatially
vanes
~(z)

(297)

90

at the bulk region(298)

where qbulk = 8 z <pzJ blllk and a is a constant . We introduce a modified evolution operator
defined as

(299)
(300)
(301 )
Then, the evolution operator satisfies the differential equation

OzUex(Z, z') = aAexUex(z, z')
A __1-_ (Eex
ex - VFz -16(z)1_
VFzOz<Pz
Cex(a = ±1)(302)

I~~ll)(303)

2= C- a(304)

The matrix A has also an in1portant property(305)
It is to be noted that when the supercurrent flows, the evolution operator U does not satisfy

Except for this, the above defined evolution operator U has the same properties as U
discussed in the appendix of Sec.2 . Thus, [; can be decolTIposed as follows:(306);1ex ;;'~(z)'1>~(Z')P2'(307)

A~ (z, z' ) = ~.ex ;;,~ (z) '1>~(Z') P2'(308)

A~(z,z') =

(309)
91

and <P± are the solution of the equationai (co ± no1~(z)l) jo
azlf/± = -:;;;;:-1~(z)1 -co ± no If/±,(310)

Jc; - 1~(z)12 = no/VFz(311 )

)...0

",0

=

with the boundary conditionat sufficient large z,that is,<P± - t const. x(312)

In general, one can not obtain the relation between U+ and U_ if the pair potential is not
real. In the case of the Matsubara frequency c = iw n , however, one can obtain the relation

Evolution operator for the Matsubara frequency c = iW n

For the Matsubara frequency c = iw n , the evolution operator satisfies
(313)

and
(314)

when ~(pt) = ~(PF) is satisfied. Therefore, we have only to treat U+ for the Matsubara frequency.

We write down some useful properties of the evolution operator for the

Matsubara frequency E = iW n :
(315)

92

"t u~+ (Z,Z ')* P2ipz',"
U- (Z,Z ') = ipzP2

(31G)
(317)

AWn(
') = =t=1;'wn( Z ) Tjwn(')± z, ZW If/±
If/=f Z P2,(318)

W = 1J;~n p2J~n = _1J;~n p2J~n = const. # 0,(319)

and ¢,±n satisfies an equation

8 Jwn = _1 ( -wn =t= En
z

(320)

-ilL\(z)1.v Fz 8z ipz

±

VFz

Wn

= Wn + 1,(321 )

2

En = vw~ + l.0.bulkI2,(322)

with the boundary condition
;'w

If/±n

~ const. X

93( Wn =t= En )'1 A I '-1,

L.1bulk(323)

Summary

6

We have derived the qua.si- classical Green's function in the triple layer system includingsemi- infinite superconductor by extending the AAIIN forn1ulation [or double layer systen1
in the clean limit and have studied some superconducting proximity contact systelns.
Following AAHN, we first constructed a solution of the Green's function for the superconducting finite triple layer system in a forn1 including the spatial evolution operator within
the quasi-classical approxin1ation. Taking the limit of both the layer sizes LL and LR toinfinity ( keeping the center layer size L finite ), we have obtained the solution of the quasiclassical Green's function for the selTli- infinite triple layer systen1. The present formulatioll
has a great advantage in computing the self- consistent pair potential. In the conventional
quasi-classical Green's function method, one has to solve the Eilenberger equation under
the restriction of the nonnalization condition as well a.s of the boundary condition[23]. In
numerical calculations according to that program, one needs sophisticated techniques tofind converging solutions at infinities. In the present formulation, we have obtained an
explicit form of the Green's function which already satisfies the boundary condition andis written by quantities converging at infinities. This reduces the nun1erical efforts verymuch.
One of the applications of the present formulation is a study of the point contact experiment.

Taking account of the reflection coefficients at the point contact and at thenormal-s uperconducting interface, we have calculated the Andreev reflection. It is foundthat the Andreev reflection, which is closely related to the differential conductance, canhave double peak structure as a function of the incident energy below the bulk energy
gap f}.~ulk. This happens because of a finiteness of the normal layer and finite interfacial94reflection coefficients. It was found that the differential conductance also can has double
peak structure as a function of the bias voltage.
Next, we have obtained the density of states of the N region for the superconductingnormal- superconducting proximity contact system. It is found that the density of states
has a structure below the bulk energy gap ~~ulk' in a sin1ilar nlanncr to the Andreev
reflection in the N - N'- 8 systenl. This structure comes fronl the de Gennes- Saint-Janles
bound state which is originated by the Andreev reflection at the N - 8 interface. Also wehave calculated the total density of states. This total density of states can be detectedby scanning tunneling spectroscopy. The spatial variation of the density of states, which
was detected in a STM experinlent, has been realized by taking account of the pairing
interaction of the IV region.
We have also studied the s-wave superconductor and the d- wave superconductor junction. The self-consistent calculation of the pair potential can be achieved by use of thepresent quasi-classical forrnulation. The pair potential near the interface are suppressed
due to the proximity effect. When the supercurrent flows through the junction, it ha.s beenfound from the numerical results, that the spatial variation of the pha.se c.pz of the pairpotential in the s-wave vs d- wave junction is different from that in the s- wave vs s- wave
junction near the interface. A sign of the spatial derivative of the phase near the interface
is opposite to that of the bulk region. Also, even if the reflection coefficient is zero, thephase shows a jump at the interface.
To interpret these unordinary behavior, we have tried to analyze this junction by use of
the G- L expansion devised for the 8 - 8 junction. It was, however, found that an obtainedboundary condition is not consistent. The anomalous behavior of the pair potential isremarkable at lower temperature . It indicates that such behaviors may not be reproduced95

by the G-L expansion. This problen1 is still to be examined in future study.
We have studied sorne systen1s in the clean lin1it and in equilibrium. All actual systems,
however, are not in the clean limit and not in equilibrium. It is important to take accountof the impurity effect and extend the formulation to the non- equilibrium systen1.

96

7

Acknowledgment

I would like to thank Professor Katsuhiko Nagai for various instruction, suggestion, stimulating discussion and critical reading of the manuscript.

I would like to thank Docior Hiroshi Shirnahara for stilTIulaiing discussions and helpfuladvices.

I would like to thank Professor Jun'ichiro Hara of Yan1aguchi University for helpful discussIons.

I would like to thank Professor Mitsuo Watabe, Professor Hironobu Fujii and Professor
Kozo Hoshino for various advices .

I would like to thank Seiji Higashitani for various discussions .
I am very grateful to my colleagues in Professor Nagai's group for discussions, instructionand encouragement.

97

References[1] J.Bardeen, L.N.Cooper, and J.R.Schrieffer , Phys.Rev.108, 1175(1957).
[2] G.Deutcher and P.G. de Gennes, in Superconductivity ed. R.D. Parks ( Marcel Dekker,
Inc., New York, 1969) , p.1005.
[3] A.A.Abrikosov, L.P.Gorkov, and I.E.Dzyaloshinski Methods of Quantum Field Theory

in Statistical Ph ysics[5] A.F.Andreev , Zh.Eksp .Teor.Fiz.46,1823(1964)[Sov.Phys .-JETP 19,1228(1964)][6] P.C.van Son, H.van Kempen and P.Wyder , Phys.Rev.Lett.59, 2226(1987)[7] T.Nishino, M.Hatano, H.Hasegawa, T.Kure and F.Murai , Phys.Rev .B41,7274(1990)[8] P.A.M.Benistant, H.van Kempen, and P.Wyder , Phys.Rev.Lett.5l, 817(1983)[9] H.F.C.Hoevers, P.J.M.van Bentum, L.E.C.van de Leenlput , H.van Kempen, A .J.G.Schellingerhout and D.van der Marel , Physica C152, 105(1988)[10] G.E.Blonder, M .Tinkham, and T.M .Klapwijk , Phys.Rev.B25, 4515(1982)[11] A.Furusaki , Master thesis, University of Tokyo, 1990
[12] ChLBruder , Phys.Rev.B41, 4017(1990)[13] P.C.van Son, H.van Kempen and P.Wyder , Phys.Rev.B37, 5015(1988)[14] K.Nagai, Y.Nagato, S.Higashitani, M.Ashida and J .Hara , Physica C 185-]89,
1451(1991)
98

[15) Y.Nagato, K.Nagai and J.Hara , J.Low Temp.Phys.93, 33(1993)[16) A.Kastalsky, A.W.Kleinsasser, L.H.Greene, R.Bhat , F.P.Milliken
and J.P.Harbison, Phys.Rev.Lett.67, 3026(1991)[17) A.Kastalsky, L.H.Greene, J.B .Barner and R.Bhat , Phys.Rev.Lett.64, 958(1990)[18) G.Eilenberger , Z.Phys.214, 195(1968)[19) A .I.Larkin and Yu.N.Ovchinnikov , Zh.Eksp.Teor.Fiz .55, 2262(1968) [Soviet.Phys .JETP28, 1200(1969)[20] M .Ashida, S.Aoyama, J .Hara and K .Nagai , Phys .Rev .B40, 8673(1989)[21) M.Ashida, J.Hara, and K .Nagai , Phys .Rev.B45, 828(1992)[22) A.L.Shelankov, Fiz.Tverd.Tela26, 1615(1984) [Soviet Phys. -Solid State26, 981(1984)][23) A.V.ZaItsev , Zh .Eksp,Teor.Fiz .86, 1742(1984) [Soviet.Phys .-JETP59, 1015(1984)][24] A.Millis, D.Rainer, and J.A.Sauls , Phys .Rev.B38, 4504(1988)[25] J.W.Serene and D.Rainer , Phys.Rep .101, 221(1983)[26] J .Kurkijarvi and D.Rainer , in Modern Problems in Condensed !-vI aii er Science,

edited by W.P.Harperin and L.P.Pitaevskii (Elsevier,An1sterdall1,1989)[27] S.Higashitani and K.Nagai , to be published in the Proceedings of LT20 (Universityof Oregon,1993).
[28] P .G.deGennes and D.Saint-James , Phys.Lett.4, 151(1963)[29] W.L.McMillan , Phys.Rev.175, 537(1968)99

[30] O.Entin-Wohlman and J.Bar-Sagi , Phys.Rev.B18, 3174(1978)[31] G.B.Arnold , Phys.Rev.B18, 1076(1978)[32] W.J.Gallagher , Phys.Rev.B22, 1233(1980)[33] G.Kieselmann , Phys.Rev.B35, 6762(1987)[34] Y.Tanaka, H.Yamagami, and M.Tsukada , Solid state Commun.79, 349(1991)[35] J .Hara, M .Ashida, and K .Nagai , Phys.Rev.B47, 11263(1993)[36] K.Inoue and H.Takayanagi , Phys .Rev.B43, 6214(1991)[37] P.W.Anderson and P.Morel , Phys.Rev.123, 1911(1961)[38J R.Balian and N.R.Werthamer , Phys.Rev.131, 1553(1963)[39J J.A .Pals, W . van Haeringen, and M.H.van Maaren , Phys.Rev .B15, 2592(1977)[40] U.Poppe , J.Mag.Mag.Mater.52, 157(1985)[41] B.Ashauer, G .Kieselmann, and D .Rainer , J.Low TeI11p .Phys .63, 349(1986)[42] S.Higashitani , Master thesis, Yamaguchi Univ. (1991)

100