Josephson and spontaneous currents at the interface between two d-wave superconductors with transport current in the banks
A stationary Josephson effect in the ballistic contact of two d-wave superconductors with different orientation of the axes and with transport current in the banks is considered theoretically. We study the influence of the transport current on the current–phase relation of the Josephson and tange...
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nasplib_isofts_kiev_ua-123456789-1194772025-02-09T18:02:00Z Josephson and spontaneous currents at the interface between two d-wave superconductors with transport current in the banks Kolesnichenko, Yu.A. Omelyanchouk, A.N. Shevchenko, S.N. Свеpхпpоводимость, в том числе высокотемпеpатуpная A stationary Josephson effect in the ballistic contact of two d-wave superconductors with different orientation of the axes and with transport current in the banks is considered theoretically. We study the influence of the transport current on the current–phase relation of the Josephson and tangential currents at the interface. It is demonstrated that the spontaneous surface current at the interface depends on the transport current in the banks due to the interference of the angle-dependent condensate wave functions of the two superconductors 2004 Article osephson and spontaneous currents at the interface between two d-wave superconductors with transport current in the banks / Yu.A. Kolesnichenko, A.N. Omelyanchouk, S.N. Shevchenko // Физика низких температур. — 2004. — Т. 30, № 3. — С. 288-294. — Бібліогр.: 21 назв. — англ. 0132-6414 PACS: 74.50.+r, 74.76.Bz https://nasplib.isofts.kiev.ua/handle/123456789/119477 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная Kolesnichenko, Yu.A. Omelyanchouk, A.N. Shevchenko, S.N. Josephson and spontaneous currents at the interface between two d-wave superconductors with transport current in the banks Физика низких температур |
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A stationary Josephson effect in the ballistic contact of two d-wave superconductors with different
orientation of the axes and with transport current in the banks is considered theoretically.
We study the influence of the transport current on the current–phase relation of the Josephson and
tangential currents at the interface. It is demonstrated that the spontaneous surface current at the
interface depends on the transport current in the banks due to the interference of the angle-dependent
condensate wave functions of the two superconductors |
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Kolesnichenko, Yu.A. Omelyanchouk, A.N. Shevchenko, S.N. |
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Kolesnichenko, Yu.A. Omelyanchouk, A.N. Shevchenko, S.N. |
| author_sort |
Kolesnichenko, Yu.A. |
| title |
Josephson and spontaneous currents at the interface between two d-wave superconductors with transport current in the banks |
| title_short |
Josephson and spontaneous currents at the interface between two d-wave superconductors with transport current in the banks |
| title_full |
Josephson and spontaneous currents at the interface between two d-wave superconductors with transport current in the banks |
| title_fullStr |
Josephson and spontaneous currents at the interface between two d-wave superconductors with transport current in the banks |
| title_full_unstemmed |
Josephson and spontaneous currents at the interface between two d-wave superconductors with transport current in the banks |
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josephson and spontaneous currents at the interface between two d-wave superconductors with transport current in the banks |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2004 |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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https://nasplib.isofts.kiev.ua/handle/123456789/119477 |
| citation_txt |
osephson and spontaneous currents at the interface between two d-wave superconductors with transport current in the banks / Yu.A. Kolesnichenko, A.N. Omelyanchouk, S.N. Shevchenko // Физика низких температур. — 2004. — Т. 30, № 3. — С. 288-294. — Бібліогр.: 21 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT kolesnichenkoyua josephsonandspontaneouscurrentsattheinterfacebetweentwodwavesuperconductorswithtransportcurrentinthebanks AT omelyanchoukan josephsonandspontaneouscurrentsattheinterfacebetweentwodwavesuperconductorswithtransportcurrentinthebanks AT shevchenkosn josephsonandspontaneouscurrentsattheinterfacebetweentwodwavesuperconductorswithtransportcurrentinthebanks |
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Fizika Nizkikh Temperatur, 2004, v. 30, No. 3, p. 288–294
Josephson and spontaneous currents at the interface
between two d-wave superconductors with transport
current in the banks
Yu.A. Kolesnichenko, A.N. Omelyanchouk, and S.N. Shevchenko
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of
Sciences of Ukraine, 47 Lenin Ave., 61103, Kharkov, Ukraine
E-mail: omelyanchouk@ilt.kharkov.ua
Received November 6, 2003
A stationary Josephson effect in the ballistic contact of two d-wave superconductors with dif-
ferent orientation of the axes and with transport current in the banks is considered theoretically.
We study the influence of the transport current on the current–phase relation of the Josephson and
tangential currents at the interface. It is demonstrated that the spontaneous surface current at the
interface depends on the transport current in the banks due to the interference of the angle-de-
pendent condensate wave functions of the two superconductors.
PACS: 74.50.+r, 74.76.Bz
1. Introduction
It was shown that in the ground state of the contact
of two d-wave superconductors with different orienta-
tion of the axes there is a current tangential to the
boundary [1–8]. For the particularly interesting case
of �/4 misorientation the ground state is twofold de-
generate: there are the tangential currents in opposite
directions at � �� � 2 in the absence of Josephson cur-
rent. The probabilities of finding the contact in one of
the two states are equal and the corresponding tangen-
tial current is referred to as the spontaneous one. It
was proposed to use such two-state quantum systems
for quantum computation [9–11]. It is of interest to
study the possibility of controlling this system by the
external transport current, which is the motivation of
the present work.
In the described problem of the Josephson contact
of two d-wave superconductors with transport current
in the banks the resulting tangential current is not a
sum of the spontaneous and transport current. In the
paper [12] we have studied the simpler case of the con-
tact of two s-wave superconductors with a transport
current flowing in the banks. It was shown that the
presence of magnetic field [13–16], of transport super-
conducting current [12], or of the current in normal
layer [17,18] in a mesoscopic Josephson junction can
significantly influence the current–phase characteris-
tics, current distribution etc.
In the present problem the Josephson current is de-
fined by the interference of the angle-dependent con-
densate wave functions of the two superconductors.
There are two factors of anisotropy which define the
angle dependence of the order parameter: the pairing
anisotropy and the transport current. Thus, it is natu-
ral to expect that the resulting interference current
(which has both normal and tangential components)
is parametrized by the external phase difference � and
by the value of the transport current (or by the
superfluid velocity vs). The presence of these two con-
trolling parameters can be useful in the applications of
Josephson junctions of high-Tc superconductors.
In Sec. 2 we derive basic equations to describe the
ballistic planar Josephson junction of two differently
orientated d-wave superconductors with homogeneous
current in the banks. These equations are solved ana-
lytically in Sec. 3. Then we study in Sec. 4 the influ-
ence of the transport current on the Josephson current
and vice versa at the interface. In the Appendix the or-
der parameter and the current density in the homoge-
neous situation are considered.
© Yu.A. Kolesnichenko, A.N. Omelyanchouk, and S. N. Shevchenko, 2004
2. Model and basic equations
We consider a model of the Josephson junction as
an ideal plane between two singlet (particularly,
d-wave) superconductors with different orientation of
the axes (see Fig. 1). The pair breaking and the scat-
tering at the junction as well as the electron scattering
in the bulk of metals are ignored. We did not take into
account the possibility of the generation of a sub-
dominant order parameter, which results in decreasing
of the amplitude of the current [7]. The c axes of both
superconductors are parallel to the interface. The c
axis direction is chosen as the z axis. The a and b axes
are situated in the xy plane. In the banks of the con-
tact a homogeneous current flows with a supercon-
ducting velocity vs. We consider the superfluid veloc-
ity vs in the left (L) and right (R) superconducting
half-spaces to be parallel to each other vsL||vsR and to
the boundary; we choose the y axis along vs and the x
axis perpendicular to the boundary; x = 0 is the
boundary plane.
We describe the coherent current state in the super-
conducting ballistic structure in the quasiclassical ap-
proximation by the Eilenberger equation [19,20]
v
r
F G G
�
�
� � �� [~ � � , �]��
0, (1)
where ~� �� �n i F sp v , � �n T n� �( )2 1 are the
Matsubara frequencies, � � ( )G G ,
g f
f g
F� �
�
�
�
�
�
��� v r is
the energy-integrated Green function, and �
�
�
�
�
�
�
��
0
0
. Equation (1) should be supplemented by
the equation for the order parameter (the self-con-
sistency equation):
( , ) ( , ) ( , )v r v v v r vF F F F F
N T V f� � � � �� ��
�
0 , (2)
N0 is the density of states at the Fermi level and � �... vF
denotes averaging over directions of v F ;V F F( , )v v � is a
pairing attractive potential. For the bulk d-wave su-
perconductor it is usually assumed that
( )� �
�
0 2( , ) cosT sv �, V , VF F d( ) cos cosv v � � �2 2� � ,
where the angle � defines a direction of the velocity
v F . Solutions of Eqs. (1), (2) must satisfy the condi-
tions for the Green functions and gap function in the
banks far from the interface:
g
L R
L R
( )
,
,
�� �
�
�
, (3)
f
L R
( )
( )
,
�
�
� �
�
�
, (4)
( ) exp( ),�� � �L R i� 2 . (5)
Here � �L R n F s L Ri, ; , ,� � p v �
L R L R L R, , ,� ��2 2 ;
� is the phase difference between the left and right
superconductors, which parametrizes the Josephson
current state. The angles � L R, define the orientation
of the crystal axes a and b in the left and right
half-spaces (see Fig. 1). The angle between the axes
of the right and left superconductors (the
misorientation angle) is �� � �� �R L .
Provided we know the Green function �G, we can
calculate the current density:
j r v v r v( ) ( , )� � � ��2 0�
�
ieN T gF F F
. (6)
For singlet superconductors it is usually assumed
that
( ) ( )� �v vF F , and we therefore have:
f f fF F F
� � �� � � �( , ) ( , ) ( )� � ��v v v , (7)
g g gF F F( , ) ( , ) ( , )� � �� � � � � �v v v . (8)
Making use of Eq. (8), we can rewrite Eq. ( 6) as
j r v r v( ) � Im ( )� � � �
�
�j
T
T
g
c
FF0
0�
,
j e N v TF c0 4 0� �| | ( ) .
(9)
Josephson and spontaneous currents at the interface between two d-wave superconductors
Fizika Nizkikh Temperatur, 2004, v. 30, No. 3 289
�
b b
a
a
R
�
y
x
s;Lv s;Rv
L
Fig. 1. Geometry of the contact of two superconductors
with different orientation of the axes and different trans-
port currents (superfluid velocities vs;L,R) in the banks.
3. Analytical solution of the Eilenberger
equation
In this paper we consider the problem non-self-con-
sistently: we assume the superconducting velocity v s
is homogeneous and that the order parameter
is con-
stant in the two half-spaces:
v r
v
v
rs
s L
s R
L
R
x
x
i x
( )
,
,
, ( )
exp( ),
exp
;
;
�
�
�
�
!
�
�0
0
2 0
�
( ),
.
� �
�
! i x� 2 0
(10)
As was shown in the papers [7], the self-consistent
consideration of a Josephson junction of d-wave su-
perconductors does not differ qualitatively from the
non-self-consistent one. In the paper [7] the authors
compare numerically the self-consistent solution with
the non-self-consistent one. The self-consistency of the
solution allows one to take into account the suppres-
sion of the order parameter at the interface; the major
effect of this is the reduction of the current [7].
Equation (1) together with Eqs. (3)–(5) and (10)
yields for the left and right superconductors:
g x C
x
v
L R
L R
L R
L R
x
L R,
,
,
, ,( ) exp
| |
| |
� � �
�
�
�
�
�
�
�
�
�
�
2
, (11)
f x C
x
L R
L R
L R
L R L R
L
x i
L R,
,
,
, ,
( )
sgn( )
sgn( )
,� �
�
�
�
�
e �
" �
2
,R
#
# �
�
�
�
�
�
�
�
�exp
| |
| |
,
sgn( )2 2x
v x
L R
x i� e � , (12)
where " � sgn( )v x . Making use of the continuity con-
dition, we obtain the expression for the g function at
the interface:
g
iL R R L L R
L R L R L R
( )
sin
cos
0 �
� �
� �
� �
� �
� � " �
� � �
. (13)
Equations (9) and (13) allow us to calculate the
Josephson current j j xJ x� �( )0 and the tangential
current j xy ( )� 0 at the interface. We emphasize that
these equations are valid for describing the current at
the interface of two singlet superconductors with dif-
ferent orientation of the axes and with different trans-
port currents in the banks. The contact of conven-
tional superconductors was considered in [12] and in
the present paper we study the contact of d-wave
superconductors, for which the order parameter is
L R s L R L RT, ; , ,( ) ( , ) cos ( )� � �� �0 2v . The consi-
deration presented here can be also used to consider
the contact of g-wave superconductors or an
s-wave/d-wave contact, etc.
As we restrict ourselves to the non-self-consistent
model we should calculate the order parameter
0 0� ( , )T sv in the bulk d-wave superconductor.
That is the subject of the Appendix.
In the particular case considered below in detail we
have v v vs L s R s; ;� � and denote ~� �� �n F sip v ,
�
L R L R, ,
~� ��2 2 ; in this case we obtain
g
iL R L R
L R L R
( )
~( ) sin
~ cos
0
2
�
� �
� �
� " �
� �
� �
� �
. (14)
In the absence of the transport current (v s � 0) in
this expression: ~� �� n [7].
We should also clarify the sign of the square root in
�L R, . To make the solution (11) convergent, we must
require Re ,�L R � 0, which fixes the sign of the square
root in� L R, to be sgn ( ); ,�p vF s L R . Moreover, this re-
quirement, as can be shown, provides the supplemen-
tary condition on Re g: sgn(Re ) sgn( )g � � .
4. Influence of the transport current on the
Josephson and spontaneous
currents at the interface
Further we study the Josephson contact for the de-
finite case: v v vsL sR s� � and � L � 0 and � �R � 4.
For small values of v s (in the approximation linear
in p v TF s c ) we can state the following approximate
relations (which are valid for values of � in the vicin-
ity of � � 2):
j J s( , )�v � � j J s( , )v � ,
j y s( , )�v � � � �j y s( , )v � ,
and for the difference �j j js s$ � �( ) ( )v v 0 :
� �j J ( )� � �� �j J ( ), � �j y ( )� � � �j y ( ),
while at v s � 0
j jJ J( ) ( )� � �� � , j jy y( ) ( )� � �� � .
In the linear approximation the shift current �j y is an
even function of �, in contrast to j y s( )v � 0 . For the
spontaneous current (at � � �� 2) the shift currents
�j y are equal:
j j jy S y( )� � �� � � �2 . (15)
j j j jS S y y( ) ( ), ( ) ( )� � � � �� � � � � �2 2 2 2 .
In a nonlinear treatment these shift currents are dif-
ferent for the two cases and are discussed below.
In Figs. 2, 3 we plot the normal (Josephson) and
tangential components of the current densities at the
interface plane as functions of the phase difference
290 Fizika Nizkikh Temperatur, 2004, v. 30, No. 3
Yu.A. Kolesnichenko, A.N. Omelyanchouk, and S.N. Shevchenko
�
� at low temperature. In the absence of the transport
current: (i) j is an odd function of �; (ii) the normal
component of the current (Josephson current) is �-pe-
riodic; (iii) in the equilibrium state at � �� � 2:
j J � 0, j j jy SS( ) | |� � �� 2 � . In the latter case the
tangential current exists in the absence of the
Josephson current; for that reason it is referred to as
the spontaneous current. The presence of the transport
current breaks the symmetry relations (i)–(iii). There
is a nonzero Josephson current at � � 0, �. How the
transport current influences the spontaneous current
(i.e., the tangential current at � �� 2 and � �� � 2) is
shown in Fig. 4. The shift of the two values of the cur-
rent for small values of v s (in the linear in p v TF s c
approximation) is equal (see Eq. (15)); however, at
values v ps F~ .0 2 00
the shift current (i.e., the dif-
ference j jy s S s( ) ( )v v� � 0 ) is of different sign for
the two currents and in the directions opposite to jS .
We also note the following relations for v s % 0:
1 0 0) ( ) ( )j jJ J� � �� � � � % (the presence of the
transport current induces a nonzero Josephson current
in the absence of an external phase difference);
2
2
0) j J �
�
� ��
�
�
�
� � ,
dj
d
J
�
�
�
� ��
�
�
�
� �
2
0 (the transport
current does not change the values of equilibrium
phase difference, at � �� � 2); 3) j y ( )� �� �
� � %j y ( ) .� 0 0 This last relation concerns the intere-
sting phenomena studied in [12]: for some values of
phase difference (here in the vicinity of � � 0, �) the
interference of the angle-dependent condensate wave
functions results in the appearance of an additional
tangential current with the direction opposite to the
transport current in the banks. We emphasize that the
resulting tangential current is not the sum of the spon-
taneous current and the transport current [12]. Thus,
the transport current drastically influences both the
tangential (spontaneous) and Josephson currents.
We can write down explicitly an expression for the
current for temperatures close to the critical (so close
that
0 ,p v TF s c�� ). From Eq. (14) we have:
Im ( )g 0 �
L R
F s
n n
� � �
&
'
(
(
"
�
�
�
�
1
2 2 3
sin cos
p v
� �
)
*
+
+
"
�
� "
�
�
,
3
2 8
2
2
4
( )p vF s L R
n n
sin sin
. (16)
At � L � 0 and � �R � 4 this results in the following:
j j j j� � �J S
~, (17)
Josephson and spontaneous currents at the interface between two d-wave superconductors
Fizika Nizkikh Temperatur, 2004, v. 30, No. 3 291
0 0.2 0.4 0.6 0.8 1.0
–0.04
–0.02
0
0.02
0.04
j J
/
j 0
�/2�
vs=0
p
F
vs
/
00=0.2
p
F
vs
/ =0.4
00
Fig. 2. Josephson current density through the interface jJ
versus phase � (� L � 0, � �R � 4, T Tc� 01. );
00 �
� � � �
0 0 0 214( , ) .T v Ts c .
–0.06
j y
/
j 0
vs=0
p
F
vs
/
00=0.2
=0.4p
F
vs
/
00
0 0.2 0.4 0.6 0.8 1.0
�/2�
–0.04
–0.02
0
0.02
0.04
Fig. 3. Tangential current density at the interface jy ver-
sus phase �(� L � 0,� �R � 4, T Tc� 01. ).
j y
/j
0
pFvs /
00
j
y
(�= � /2)
j
y
(�= –�/2)
0 0.2 0.4 0.6 0.8–0.10
–0.05
0
0.05
0.10
Fig. 4. Tangential current density at the interface jy for
two values of the phase difference (spontaneous current)
versus superfluid velocity vs (� L � 0,� �R � 4, T Tc� 01. ).
j eJ
c
xj
T
� � -
1
3024
20
0
4
4�
�
sin , (18)
j eS
c
yj
T
� � -
1
60 0
0
2
2�
�
sin , (19)
~ ( )
sinj e� -
3
560 0
0
2
2
2
2�
�j
T
p v
Tc
F s
c
y
. (20)
Here
0 0� ( , )T v s and is defined by Eq. (25). In
particular, at v s � 0 this gives:
j j
T
T
J
c
� � - �
�
�
�
�
�
�
�
�17 10 1 22
2
0. sin �, (21)
j j
T
T
S
c
� � - �
�
�
�
�
�
�
�
�6 6 10 12
0. sin�. (22)
We note that ~ ( )
j j� �
9
28
2
2
p v
T
F s
c
S . It follows that
the effect of transport current on the spontaneous
tangential current at T Tc~ is to reduce its value by
a small shift. It is remarkable that the current tan-
gential to the boundary contains only corrections of
the second order in the parameter p v TF s c .* If
� L � 0 and � �� � ,R � % , the integration of the
second term in Eq. (16) would give us the factor
� �� �cos2 2� , which is zero for �� �� 4; this term
at �� � 0 and � � 0 gives the homogeneous current
density (Eq. (26)).
The integration of the first term in Eq. (16) gives
us the factor cos2�� for the x component of the cur-
rent and sin2�� for the y component. In the case of
�� �� 4 this term gives only the tangential component.
As a consequence j jS J�� (see Eqs. (21), (22)).
It was discussed above that the terms linear in
p v TF s c result in a uniform shift of jS . We can see
that nonlinear terms result in a shift of different sign,
and in both cases in the direction opposite to jS (see
Eq. (20)). This in part explains the nonmonotonic be-
havior of j y (see Fig. 4). The fact that the presence of
the transport current significantly changes the tangen-
tial (spontaneous) currents might be used for its con-
trol, which is important in view of their possible appli-
cation for quantum computation [9–11].
5. Conclusion
We have studied influence of the transport current,
which flows in the banks, on the stationary Josephson
effect in the contact of two d-wave superconductors.
We have derived equations which allow general con-
sideration of the contact of two singlet superconduc-
tors with different orientation of the axes and with
different transport currents in the banks. In particu-
lar, we have studied the planar contact of two d-wave
superconductors in the case of � 4 misorientation with
equal transport currents in the banks. It was demon-
strated that the current–phase relation drastically de-
pends upon the value of the transport current. The
ground state degeneracy in the absence of transport
current (at � �� � 2) is lifted at v s % 0 . The depend-
ence of the shift current (which is the difference of the
tangential current and the spontaneous one) on vs is
shown to be nonlinear. It is proposed to use the trans-
port current for the control of qubits based on the con-
tact of two d-wave superconductors.
We acknowledge support from D-Wave Systems
Inc. (Vancouver).
Results of the present study were reported at the In-
ternational Conferences: «Applied Electrodynamics of
high-Tc Superconductors», IRE, Kharkov, Ukraine
(May 2003) and «Basic Studies and Novel Applica-
tions», Jena, Germany (June 2003).
6. Appendix. Order parameter in the
homogeneous current state
In this Section we study the homogeneous current
state in the bulk d-wave superconductor (see also in
[21]). We note that the order parameter
0 is a func-
tion of temperature T, superfluid velocity vs, and the
angle � between the crystallographic axis a and the
direction of the superfluid velocity v s . For that we
should solve Eqs. (2) and (9) with g and f given by
Eqs. (3) and (4):
1
2
0 2
2 2
0
2
.
�
�
� �
�
�
� �
� /T d Re
( )
�
,
j
v
j
T
T
d
c
F
0 0 2
2
� �
� �
� /�
�
�
� �
�
� Im
~
�
.
Here . � N Vd0 , ~� �� �n F sip v , �
� �~�2 2 ,
( ) ( , ) cos ( )� � �� �0 2T sv .
292 Fizika Nizkikh Temperatur, 2004, v. 30, No. 3
Yu.A. Kolesnichenko, A.N. Omelyanchouk, and S.N. Shevchenko
* There is also a term with the factor
p v
T T
F s
c c
,0
4
, which is neglected here. This term results in equal shifts of jS for
� �� � 2.
For T = 0 (replacing � �T � by the integral / d�) we
obtain the equations for the order parameter
0 and
the current density j:
ln
( )
ln
( )
00
0 0
2
2
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�/� �
�
�
d
s F sv p v pF
( )
,
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
2
1
(23)
here
00 0 0 0� � � �( , )T v s c0� .e–2 , 0 � 4e–1 2,
j
j
v p
T
s F
c0
1
4
� � �
�
�
�
�
�
�
�
�
�
�
�
��
�
�
��/
1
2 2
2 2
�
� �
�
d
T T
s F
cc
| cos |
( )v p
. (24)
In Eqs. (23) and (24) the integration is performed in
the region where
( ) ( )� 2 2� v ps F for � 1 ��
�
�
�
�
� �
2 2
, .
In Figs. 5 we plot the order parameter
0 ( , )T sv
and the current density versus the superfluid velocity
vs for different angles � at low temperature. For com-
parison we also plot the curves for the s-wave super-
conductor. A numerical analysis at low temperature
shows that in spite of the strong anisotropy of the
pairing potential, the order parameter
0 , the critical
velocity vs
cr , and the critical current j c depend
weakly on the angle � between v s and the crystallo-
graphic a axis (see Figs. 5 and in Ref. 21). Namely,
the respective difference is maximal for � � 0 and
� �� 4 and does not exceed 0.1. For small values of
the superfluid velocity, i.e., in the approximation lin-
ear in the parameter v p Ts F c , both
0 and j are
independent of �.
For a temperature close to Tc c� �2� .e–2
� 0 47 00.
, where 2 �� (2 )eC (C is the Euler con-
stant), both the gap function
0 and current density j
are independent of the angle �:
0
2
3
2 232
21 3
1
4
3
� �
�
��
�
�
�� �
�
3( )
( ) ,T
T
T
p vc
c
F s (25)
j
j T
p v
T
c
F s
c0
7 3
32 3
0
2
2
� �
3
�
( )
. (26)
The temperature dependence of critical velocity vs
cr
follows from Eq. (25):
p v
T
T
T
F s
c c
cr
� �
8
7 3
1
2�
3( )
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a
0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.2
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