Download Disorder in two-dimensional Josephson junctions

PHYSICAL REVIEW B
VOLUME 55, NUMBER 21
1 JUNE 1997-I
Disorder in two-dimensional Josephson junctions
Baruch Horovitz and Anatoly Golub
Department of Physics, Ben-Gurion University, Beer-Sheva, 84105, Israel
~Received 18 July 1996!
An effective free energy of a two-dimensional ~i.e., large area! Josephson junction is derived, allowing for
thermal fluctuations, random magnetic fields, and external currents. We show by using replica-symmetrybreaking methods that the junction has four distinct phases: disordered, Josephson ordered, a glass phase, and
1
a coexisting Josephson order with the glass phase. Near the coexistence to glass transition at s5 2 the critical
current is ;( area) 2s11/2 where s is a measure of disorder. Our results may account for junction ordering at
temperatures well below the critical temperature of the bulk in high-T c trilayer junctions.
@S0163-1829~97!13617-7#
I. INTRODUCTION
Recent advances in the fabrication of Josephson junctions
have led to junctions with large area, i.e., the junction length
L ~in either direction in the junction plane! is much larger
than l, the magnetic penetration length in the bulk superconductors. Experimental studies of trilayer junctions like1
YBa2 Cu3 Ox /PrBa2 Cu3 Ox /YBa2 Cu3 Ox ~YBCO junction! or
like2
Bi2 Sr2 CaCu2 O8 /Bi2 Sr2 Ca7 Cu8 O20 /Bi2 Sr2 CaCu2 O8
~BSSCO junction! have shown anomalies in the temperature
dependence of the critical current I c . In particular in the
YBCO junction1 with an area of 50 m m2 , a zero resistance
state was achieved only below 50 K, although the
YBa2 Cu3 Ox layers were superconducting already at T c '85
K. More recent data on similar YBCO junctions3–5 with
junction areas of 102 2104 m m2 show a measurable I c only
at 20260 K below T c of the superconducting layers. An
even larger junction6 of area '105 m m2 shows a welldefined gap structure in the I-V curve, while a critical current
is not observed. In the BSCCO junction2 a supercurrent
through the junction could not be observed above 30 K, although the Bi2 Sr2 CaCu2 O8 layer remained superconducting
up to T c '80 K.
These remarkable observations are significant both as basic phenomena and for junction applications. In particular,
these data raise the question of whether thermal fluctuations
or disorder can significantly lower the ordering temperature
of two-dimensional ~2D! junctions.
We note that for both YBCO and BSCCO junctions typically l'0.2 m m at low temperatures where the junctions
order, so that the junctions above are 2D in the sense that
disorder and spatial fluctuations on the scale of l can be
important. The qualitative effect of these fluctuations depends on the Josephson length l J (l J .l) which is the
width of a Josephson vortex ~see Sec. II!. For l,L,l J
junction parameters are renormalized and become L dependent, while more significant renormalizations which correspond to 2D phase transitions occur in the regime l J ,L.
From magnetic-field dependence4 and L dependence7 of I c ,
junctions with l J ,L can be realized. The studied junctions
are 2D also in the sense the thermal fluctuations at temperature T do not lead to uniform large phase fluctuations, i.e.,
0163-1829/97/55~21!/14499~14!/$10.00
55
f 0 I c /2c,T, a condition valid for the relevant data ~see Sec.
V!; f 0 5hc/2e is the flux quantum.
The energy of a 2D junction, in terms of the Josephson
phase w J (x,y) where (x,y) are coordinates in the junction
plane, was derived by Josephson.8 It has the form
F0 5
E
dx dy
S
D
t
EJ
~ “ f J ! 2 1 2 ~ 12cosw J ! ,
16p
l
~1!
where E J is the Josephson coupling energy in area l 2 .
Equation ~1! was derived8 on a mean-field level, i.e., only
its value at minimum is relevant. It was shown, however,
~see Ref. 9 and Appendix A! that Eq. ~1! is valid in a much
more general sense, i.e., it describes thermal fluctuations of
w J (x,y) so that a partition function at temperature T(,T c )
Z5
E
Df J exp$ 2F0 @ w J ~ x,y !# /T %
~2!
is valid.
Equation ~2! implies a Berezinskii-Kosterlitz-Thoulesstype phase transition10 at a temperature T J ' t so that at
T.T J the phase f J is disordered, i.e., the cosfJ correlations
decay as a power law, while at T,T J cosfJ achieves longrange order. For the clean system, however, T J ' t is too
close to T c for either separating bulk from junction fluctuations or for accounting for the experimental data.9 A consistent description of this transition, as shown in the present
work, can be achieved by allowing for disorder at the junction, a disorder which reduces T J considerably.
Equation ~1! with disorder is related to Coulomb gas and
surface roughening models which were studied by replica
and renormalization-group ~RG! methods.11,12 We find, however, that the RG generates a nonlinear coupling between
replicas and therefore standard replica symmetric RG methods are not sufficient. In fact, related systems13,14 were
shown to be unstable towards replica symmetry breaking
~RSB!.
In our system we find a competition between long-range
Josephson-type ordering and formation of a glass-type RSB
phase. The phase diagram has four phases a disordered
phase, a Josephson phase ~i.e., ordered with finite renormalized Josephson coupling!, a glass phase, and a coexistence
phase. The coexistence phase is unusual in that it has
Josephson-type long-range order coexisting with a glass
14 499
© 1997 The American Physical Society
14 500
BARUCH HOROVITZ AND ANATOLY GOLUB
order parameter. This phase is distinguished from the usual
ordered phase, presumably, by long relaxation phenomena
typical to glasses.15
In the disordered and glass phases fluctuations reduce the
critical current by a power of the junction area, while in the
Josephson and coexistence phases the fluctuation effect saturates when the (area) 1/2 is larger than either the Josephson
length ~in the Josephson phase! or larger than both the Josephson length and a glass correlation length ~in the coexistence phase!. These predictions can serve to identify these
phases. We show that a transition between the glass phase
and the coexistence phase can occur well below the critical
temperature T c of the bulk, a result which may account for
the experimental data on trilayer junctions.1–5
In Sec. II we define the model and study the pure case. In
Sec. III we study the system with random magnetic fields
due to, e.g., quenched flux loops in the bulk and show that
the RG generates a coupling between different replicas. The
system with disorder is solved by the method of one-step
RSB ~Refs. 13,16! in Sec. IV. Appendix A derives the free
energy of a 2D junction. In particular, Appendix A 2 allows
for space-dependent external currents, a situation which, as
far as we know, was not studied previously. Appendix B
extends the one-step solution of Sec. IV to the general hierarchical case, showing that they are equivalent.
II. THERMAL EFFECTS
Appendixes A 1–A 4 derive the effective free energy of a
2D junction, in presence of an external current j ex(x,y), for
the geometry shown in Fig. 1. The presence of j ex(x,y) dictates that the relevant thermodynamic function is a Gibbs
free energy, Eq. ~A10! which for the junction becomes @Eqs.
~A26!,~A39!#
GJ $ f J % 5F0 $ f J % 2 ~ f 0 /2p c !
E
dx dy j ex~ x,y ! w J ~ x,y ! ,
~3!
where F0 is given by Eq. ~1!. The cosine term is the Josephson tunneling8 valid for weak tunneling E J ,, t and t is
found in two cases @Eqs. ~A24!,~A36!#: Case I of long superconducting banks W 1 ,W 2 @l and case II of short banks,
W 1 ,W 2 !l,
t5
5
f 20
4 p 2l
case I:W 1 ,W 2 @l
f 20
W 1W 2
2 p 2 l 21 W 2 1l 22 W 1
case II:W 1 ,W 2 !l.
~4!
Note that in case II the derivation allows for an asymmetric
junction with different penetration lengths l 1 ,l 2 and different lengths W 1 ,W 2 .
It is of interest to note that j ex breaks the symmetry
w J → w J 12 p , i.e., the external current distinguishes between
different minima of the cosine term in Eq. ~1!. For a uniform
j ex the Gibbs term reduces to the previously known form.17
Appendixes ~A1!–~A4! present detailed derivation of Eq.
~3!. This derivation is essential for the following reasons: ~i!
It shows that the fluctuations of w J decouple from phase
fluctuations in the bulk ~excluding flux loops in the bulk
55
FIG. 1. Geometry of the 2D Josephson junction. The various
components are superconductors ~S!, insulating barrier ~I!, normal
metal ~N! for the external leads, and vacuum ~V!. The dashed rectangle serves to derive boundary conditions in Appendix A 1.
which are introduced in Sec. III!. Thus Eq. ~3! is valid below
the fluctuation ~or Ginzburg! region around T c . ~ii! It shows
that Eq. ~3! is valid for all configurations of w J and not just
those which solve the mean-field equation.
It is instructive to consider the mean-field equation
d G J / d w J 50, i.e.,
EJ
t 2
f 0 ex
j .
“ w J1
2 sinw J 5
l
8p
2pc
~5!
This equation can also be derived by equating the current
j z 5(2c/4p l 2 )A z8 at z5d/2 @given, e.g., in case I by
Eqs. ~A18!, ~A23!# with the Josephson tunneling current
j J 5(2 p c/ f 0 )(E J /l 2 )sinwJ . Equation ~5!, however, is not
on a level of conservation law or a boundary condition since
configurations which do not satisfy Eq. ~5! are allowed in the
partition sum. More generally, Eq. ~5! is satisfied only after
thermal average ^ d G J / d w J & 50. An equivalent way of
studying thermal averages is to add to Eq. ~5! timedependent dissipative and random force terms. The time average, which includes configurations which do not satisfy
Eq. ~5!, is by the ergodic hypothesis equivalent to the partition sum, i.e., a functional integral over w J with the weight
exp@2GJ /T#.
Equation ~1! is the well-known 2D sine-Gordon system10
which for j ex50 exhibits a phase transition. Since the renormalization group ~RG! proceeds by integrating out rapid
variations in w J , j exÞ0 is not effective if it is slowly varying
~e.g., as in case II!.
RG integrates fluctuations of w J with wavelengths between j and j 1d j , the initial scale being l. The parameters
t5T/ t and u5E J /T are renormalized, to second order in
u, via10
du/u52 ~ 12t ! d j / j ,
dt52 g 2 u 2 t 3 d j / j ,
~6!
where g is of order 1 ~depending on the cutoff smoothing
procedure!. Equation ~6! defines a phase transition at
1/t512 g u. Note, however, that t itself is temperature dependent since l(T)5l 8 (12T/T 0c ) 21/2, where T 0c is the
mean-field temperature of the bulk. Thus the solution of
t (T)/T512 g E J /T defines a transition temperature T J
55
DISORDER IN TWO-DIMENSIONAL JOSEPHSON JUNCTIONS
which is below T 0c . However, T J is too close to T 0c and is in
fact within the Ginzburg fluctuation region around T 0c . To
see this, consider a complex order parameter c 5 u c u exp(iw)
with a free energy of the form
F5
E
14 501
~including the area integration! is larger then temperature,
i.e., in terms of I c , f 0 I c /2c.T. This condition is consistent
with experimental data ~see Sec. V!.
III. DISORDER AND RG
d r @ a u c u 1b u c u 1a j u “ c u # .
3
2
4
2
2
The Ginzburg criterion equates fluctuations with b50,
i.e., ^ d c 2 & 'T/a j 3 with u c u 2 (5 u a u /2b) in the ordered phase.
Since u “ c u 2 ' u c u 2 (“ w ) 2 Eq. ~A14! identifies a j 2 u c u 2
5( f 0 /2p l) 2 /8p , so that the Ginzburg temperature is
T Ginz5a j 3 u c u 2 5 j ~ f 0 /2p l ! 2 /8p .
~7!
Since j ,l,W in both cases I and II, T Ginz,T J . The neglect of flux-loop fluctuations, as assumed in Appendixes
A 3, A 4 is therefore not justified at T J . Thus the relevant
range of temperatures for the free energy Eqs. ~1!,~3! is
T!T Ginz, t , i.e., t!1.
The RG Eqs. ~6! can, however, be used in the range
T,T J to study fluctuation effects in the ordered region. Excluding a narrow interval near T J where u t /T21 u
, g E J /T!1 renormalization of t can be neglected and integration of Eq. ~6! yields a renormalized Josephson coupling
E RJ 5E J ( j /l) 2(12t) . Scaling stops at the Josephson length
l J at which the coupling becomes strong, E RJ ' t /8p ~the
8 p is chosen so that l J 5l 0J at T50, where l 0J is the conventional Josephson length!. Thus l J 5l( t /8p E J ) 1/[2(12t)] ;
the T50 value is l 0J 5l( t /8p E J ) 1/2. The scaling process is
equivalent to replacing (E J /l 2 ) ^ coswJ& by t /8p l 2J so that
^ coswJ&5(l0J /lJ)2 is the reduction factor due to fluctuations.
The free energy Eqs. ~1!,~3! with renormalized parameters
yields a critical current by a mean-field equation @see comment below Eq. ~10!#. The renormalized junction is either an
effective point junction (L,l J ) with the current flowing
through the whole junction area, or a strongly coupled
(E J ' t /8p ) 2D junction where the current flows near the
edges of the junction with an effective area Ll J . The meanfield critical currents18 are
I 0c1 5 ~ 2 p c/ f 0 ! E J ~ L/l ! 2 ,
L,l J
I 0c2 5c t L/2f 0 l 0J ,
L.l J .
~8!
The effect of fluctuations is to reduce E J so that the critical current is
~9!
I c 5I 0c1 ~ L/l ! 22t , L,l J .
In the second case, L.l J , the fluctuations reduce the
current density by ^ coswJ& but enhance the effective area by
l J /l 0J 5 ^ coswJ&21/2. The critical current is then
I c 5I 0c2 ~ 4 p E J / t ! t/[2 ~ 12t ! ] ,
L.l J .
~10!
Thus even if t!1 in Eqs. ~9!,~10! a sufficiently small E J can
lead to an observable reduction of I c .
Note that thermal fluctuations act to renormalize E J which
then determines a critical current by the mean-field equation.
This neglects thermal fluctuations in which w J fluctuates uniformly over the whole junction. These fluctuations can be
neglected when the coefficient of the cosine term in Eq. ~1!
There are various types of disorder in a large area junction. An obvious type are spatial variations in the Josephson
coupling E J . A random distribution of E J with zero mean is
equivalent to known systems13,14 and produces only a glass
phase. The more general situation is to allow a finite mean of
E J , and allow for another type of disorder, i.e., random coupling to gradient terms. Since the magnetization of the junction is proportional to8 ¹w J we propose that the most interesting type of disorder are random magnetic fields. Such
fields can arise from magnetic impurities, or more prominently from random flux loops in the bulk.
A flux loop in the bulk with radius r 0 has a magnetic field
of order f 0 /2p l 2 in the vicinity of the loop. A straightforward solution of London’s equation shows that the field far
from the loop depends on the ratio r 0 /l. For large loops,
r 0 .l, the field at distance r@r 0 decays exponentially while
for small loops r 0 !l, it decays slowly as 1/r 2
(l.r@z,r 0 , where r is in the loop plane and z is perpendicular to it! or as 1/z 3 (l.z@r,r 0 ). Thus, the local magnetic field has contributions from all flux loops of sizes
r 0 ,l. If P(r 0 ) is the probability of having a flux loop of
size r 0 then the local average magnetic field is of order
F
H 2s ' ~ f 0 /2p l 2 !
E
l
G
2
P ~ r 0 ! dr 0 [4s f 20 / p l 4 .
~11!
The last equality defines a measure of disorder s which
increases with the r 0 integration, say as s;l a with a .0.
The distribution of H s is therefore of the form
exp@2pH2s l4/4s f 20 # .
Consider a dimensionless random field q(x,y)
5l A8 p Hs (x,y)/4f 0 so that its distribution is
F
exp 2l 2
q2 ~ x,y ! /2s
(
x,y
G FE
5exp 2
G
q2 ~ x,y ! dx dy/2s .
~12!
The coupling of magnetic fields to the Josephson phase is
from Eqs. ~A23!,~A43! and for t of case I @Eq. ~4!#
Fs 52 ~ t / A8 p !
E
dx dy @ ẑ3“ w J ~ x,y !# •q~ x,y ! . ~13!
The fields in Eq. ~11! are in fact relevant only to case I. In
case II image flux loops across the superconducting-normal
~SN! surface reduce the contribution of loops with r 0 ,W.
Thus Eq. ~11! is valid with the r 0 integration limited by W.
Since now t 5 f 20 W/4p 2 l 2 @Eq. ~4! for symmetric junction#,
we define q(x,y)5 A8 p l 2 Hs (x,y)/4f 0 W so that the coupling Eq. ~13! has the same form. The distribution of
q(x,y) has the same form as in Eq. ~12! except that now
s;l 2 . Since l is T dependent, s is also T dependent, a
feature which is relevant to the experimental data ~see Sec.
V!.
14 502
BARUCH HOROVITZ AND ANATOLY GOLUB
We proceed to solve the random magnetic-field problem
by the replica method.15 We raise the partition sum to a
power n, leading to replicated Josephson phases w a ,
a 51, . . . ,n. The factor q(x,y) in Eq. ~13! is then integrated
with the weight Eq. ~12!, leading to
F
Z ~ n ! ;exp ~ s t 2 /16p T 2 !
S( D G
a
~14!
.
In this section we attempt to solve the system by RG
methods.11,12 We find, however, that RG generates nonlinear
couplings between replicas which eventually lead to replica
symmetry breaking ~Sec. IV!. Thus the direct application of
RG is not sufficient.
Consider first the Gaussian part
F~0n ! 5
1
2
E
dx dy
M a,b“ w a“ w b
(
a,b
G a , b ~ r! 5 ^ z a ~ r! z b ~ 0 ! &
5 ~ M 21 ! a , b
(r
K S( DL
cos
i51
h iw ai
5
S D F KS( D LG
S ( DF
(
(r cos i51
( h ix a
8
(r
S
G 2 ~ 0 ! 5G a Þ b ~ 0 ! 5
i51
h ix ai
D
8ps
dj
,
12ns/t 2 pj
dj
8ps
.
12ns/t 2 pj
~19!
The most relevant operators in Eq. ~18! are when ( iÞ j h i h j
is minimal, i.e., ( i h i 50 for even l or ( i h i 561 for odd
l . Thus,
dv
~l !
52 v
~l !
~ 12 l t ! d lnj ,
d v ~ l ! 52 v ~ l ! ~ 12 l t2s ! d lnj ,
H
E
dx dy
l odd.
2
v
cos~ w a 2 w b !
l 2 a ,Þ b
(
1
u
M “ w a“ w b2 2
2 a,b a,b
l
(
J
~17!
112
2
l
i51
h iz ai
G
dj m
1
2 G 1~ 0 ! 2
h h G ~0! ,
j
2
2 iÞ j i j 2
~18!
Note that the v term is also generated by disorder in the
Josephson coupling, corresponding to a distribution with a
mean value ;u. If u50 Eq. ~21! reduces to the well studied
case13,14 with a glass phase at low temperatures. We consider
here the more general case of uÞ0, which indeed leads to a
much more interesting phase diagram.
The initial values for RG flow are u5E J /T, v 50. Standard RG methods10 to second order in u, v lead to the following set of differential equations:
du5 @ 2u ~ 12t2s ! 22 g 8 y v t # d lnj ,
d v 5 @ 2 v~ 122t ! 1 ~ 1/2! g 8 su 2 22 g 8 t v 2 # d lnj ,
l even
~20!
Thus, as temperature is lowered, successive v ( l ) terms become relevant at t,1/l ( l even! and at t,(12s)/ l ( l
odd!.
We consider in more detail the v 5 v (2) term, the lowestorder term which mixes different replica indices. The free
energy of this model has the form
F~ n ! 5
i
1
exp 2
2
l
cos
where ( 8 denotes summation on a unit cell larger by
112d j / j and
G 1 ~ 0 ! 5G a , a ~ 0 ! 5 8 p t1
d 2 qexp~ 2iq•r! / ~ 2 p q ! 2
Defining w a 5 x a 1 z a , RG to first order is obtained by integrating z a ,
l
5
E
5 ~ M 21 ! a , b J 0 ~ r/ j ! d j /2pj .
~15!
with
l
~16!
~From here on T is absorbed in the definition of free energies, i.e., F→F/T).
We use Eq. ~15! to test for relevance of terms of the form
l
h i w a i ). These terms are generated from powers
v ( l ) cos((i51
of the ( a coswa interaction in the presence of disorder s. A
first-order RG is obtained by integrating a high momentum
field z a with momentum in the range j 21 1d( j 21 ),q
, j 21 . The Green’s function, averaged over these high momentum terms in Eq. ~15!, is
2
“wa
55
1
s
M a,b5
d a,b2
.
8pt
8pt2
(a cosw a
~21!
dt522 g 2 ~ t1s ! t 2 u 2 d lnj ,
d ~ s/t 2 ! 516g 2 t v 2 d lnj ,
~22!
where g , g 8 are numbers of order 1 ~depending on cutoff
smoothing procedure!.
Note that any uÞ0 generates an increase in v , so that
v 50 cannot be a fixed point. In contrast, v Þ0 allows for a
u*50 fixed point ~ignoring for a moment the flow of s),
with u*50, v *5(122t)/ g 8 t. This fixed point is stable in
the (u, v ) plane if t,1/2,s; however, s increases without
bound. This indicates that the v term is essential for the
behavior of the system.
We do not explore Eq. ~22! in detail since it assumes
replica symmetry, i.e., the coefficient v is common to all
55
DISORDER IN TWO-DIMENSIONAL JOSEPHSON JUNCTIONS
14 503
a , b . In the next section we show that the system favors to
break this symmetry, leading to a different type of ordering.
IV. REPLICA SYMMETRY BREAKING
The possibility of replica symmetry breaking ~RSB! has
been studied extensively in the context of spin glasses15 and
applied also to other systems. In particular, the free energy
Eq. ~21! with u50 was studied in the context of flux-line
lattices and of an XY model in a random field.13,14 In this
section we use the method of one-step replica symmetry
breaking13,16 for the Hamiltonian Eq. ~21!; in Appendix B we
present the full hierarchical solution, which for our system
turns out to be equivalent to the one-step solution.
Consider the self-consistent harmonic approximation13 in
which one finds a harmonic trial Hamiltonian
H0 5
1
2
*~ q ! ,
G a21
(q (
,b~ q ! w aw b
a,b
~23!
such that the free energy
Fvar5F0 1 ^ H2H0 & 0
~24!
is minimized. H5F /T is the interacting Hamiltonian, Eq.
~21!, F0 is the free energy corresponding to H0 , and
^ . . . & 0 is a thermal average with the weight exp(2H0 ). The
interacting terms lead to
(n)
E
d 2 r ^ cosw a ~ r! & 0 5exp~ 2A a /2!
A a5
E
5
where Î is the unit matrix, L̂ is a matrix with all entries
51, i.e., L a , b 51, and ŝ is given by
s a , b 5exp~ 2 21 B a , b ! 2 d a , b
2
1
2
(q ^ u w a~ q ! 2 w b~ q ! u &
(q @ G a , a 1G b , b 2G a , b 2G b , a # .
~25!
( Tr@ lnĜ ~ q ! 1 ~ Ĝ 21~ q ! 1M̂ q 2 ! Ĝ ~ q !#
u
l2
(a
S
D
S
D
~26!
where the TrlnĜ(q) term corresponds to F0 ~up to an additive constant! and the ˆ sign denotes a matrix in replica
space.
We define now u 0 58 p tu/l 2 , v 0 516p t v /l 2 and using
Eq. ~16! the minimum condition d Fvar / d G a , b 50 becomes
H
B a,g !.
s
Ĝ ~ q ! 58 p t @ q 2 1u 0 exp~ 2 21 A a !# Î2 q 2 L̂2 v 0 ŝ
t
~28!
J
21
,
~27!
~29!
Using L̂ 2 5nL̂ it is straightforward to find the inverse in Eq.
~27!. In terms of an order parameter z5u 0 exp(2Aa/2), Eq.
~28! with n→0 yields s 0 5(z/D c ) 2t where D c ('1/l 2 ) is a
cutoff in the q 2 integration so that z!D c is assumed. The
definition of z yields
z5u 0
1
1
v
exp 2 A a 2 2
exp 2 B a , b ,
2
l aÞb
2
(
1
2
ŝ 5 s 0 L̂2n s 0 Î.
2
Therefore
Fvar52
(g exp~ 2
Note that the sum on each row vanishes, ( b s a , b 50.
Consider first briefly the replica symmetric solution. A
single parameter s 0 defines ŝ so that the constraint
( b s a , b 50 yields
(q ^ u w a~ q ! u 2 & 5 (q G a , a ,
d 2 r ^ cos~ w a 2 w b ! & 0 5exp~ 2B a , b /2! ,
B a,b5
FIG. 2. Phase diagram of a 2D junction in terms of s, the spread
in random magnetic fields and t, which is proportional to temperature. The various phases, in terms of the Josephson order z and the
glass order D are: ~i! Disordered phase with z5D50, ~ii! Josephson phase with zÞ0, D50, ~iii! coexistence with both zÞ0, D
Þ0, and ~iv! glass phase with z50, DÞ0. The dashed line within
the coexistence phase is where D changes sign.
S D
z
Dc
t1s
exp~ s2t v 0 s 0 /z ! .
For t v 0 s 0 /z!1 a consistent z!D c solution is possible at
t,12s. ~Indeed t v 0 s 0 /z!1 since s 0 !1, except at z→0,
i.e., at t→12s.! Hence, @neglecting an exp(s) factor#
z/D c ' ~ u 0 /D c ! 1/~ 12t2s ! .
~30!
The replica symmetric solution thus reproduces the firstorder RG solution @Eq. ~20! with l 51#. The order parameter
z corresponds to 1/l 2J of Eq. ~20! where the Josephson length
l J is the scale at which strong coupling is achieved,
v (1) (l J )'1, and RG stops.
Consider now a one-step RSB solution of the form13,16
ŝ 5 s 0 L̂1 ~ s 1 2 s 0 ! Ĉ2 @ s 0 n1m ~ s 1 2 s 0 !# Î,
~31!
14 504
BARUCH HOROVITZ AND ANATOLY GOLUB
where Ĉ is a matrix with entries of 1 in m3m matrices
which touch along the diagonal and 0 otherwise; m is treated
as a variational parameter. The coefficient of Î is fixed by the
constraint ( b s a , b 50.
Equation ~31! corresponds to two order parameters,
z5u 0 exp~ 2A a /2! ,
D5 v 0 @ s 0 n1m ~ s 1 2 s 0 !# .
~32!
The inverse matrix in Eq. ~27! is obtained by using
L̂ 2 5nL̂, ĈL̂5mL̂, and Ĉ 2 5mĈ. It has the form
Ĝ5 @ a ~ q ! Î1b ~ q ! L̂1c ~ q ! Ĉ # 21 5 a ~ q ! Î1 b ~ q ! L̂1 g ~ q ! Ĉ,
~33!
and is solved by
b ~ q ! 52b ~ q !@ a ~ q ! 1mc ~ q !# 21
~ q ! 1 @ a ~ q ! 1mc ~ q !#
S D
d f 3 5 12
21
% /m.
~38!
8 p @ f 3 ~ z,D ! 2 f 3 ~ 0,0!#
5 ~ 121/m ! D2 v 0 exp@ 2 a ~ z,D ! 2 g ~ z,D !#
1 ~ 11s/t ! z.
~39!
2 v 0 ~ 12m/2t ! e 2 a 2 g 1
~34!
q
b [ ( b ~ q ! 52s ln~ D c /z ! 1 ~ 2/z ! t v 0 s 0 22s,
q
g [ ( g ~ q ! 52 ~ 2t/m ! ln@ z/ ~ z1D !# .
q
~35!
The definitions of ŝ and z identifies the parameters
s 0 5exp~ 2 a 2 g ! ,
m5
z5u 0 e s
~36!
These equations determine the order parameters z, D in
terms of m and the parameters of the Hamiltonian. The value
of m must be determined by minimizing the free energy
Fvar . @However, in the hierarchical scheme with D(m) as
function of m, the variation with respect to G a , b is sufficient
to determine the position of a step in D(m), see Appendix
B#.
Consider first the Gaussian terms F3 , i.e., the trace term in
Eq. ~26!. Since this term contains the uninteresting vacuum
energy (z5D50) it is useful to find the differential dF3 and
then integrate. Using Eq. ~33! for dĜ(q) we have
(q Tr$ @ Ĝ 21~ q ! 2M̂ q 2 #
3 @ Îd a ~ q ! 1L̂d b ~ q ! 1Ĉd g ~ q !# % .
u 0 2[ a 1 b 1 g ]/2
e
t
~40!
2tD12tz ln@ z/ ~ z1D !#
.
D12t v 0 s 0 ln@ z/ ~ z1D !#
~41!
Rewriting Eq. ~36! with Eq. ~35!, we have the following
relations:
s 1 5exp~ 2 a ! ,
z5u 0 exp@ 2 ~ a 1 b 1 g ! /2# .
2
v0
~ 12m ! e 2 a
2t
where a , b , g are functions of z and D from Eq. ~35!. Since
Eqs. ~36! are already minimum conditions, it must be
checked that ] f / ] z5 ] f / ] D50 reproduces these equations
so that m in Eq. ~40! can be taken as an independent variational parameter. The latter statement is indeed correct and
] f / ] m50 leads to the relation
a [ ( a ~ q ! 52t ln@ D c / ~ z1D !# ,
1
2
D
Integrating ] f 3 (z,D 8 )/ ] D 8 from 0 to D, and then
] f 3 (z 8 ,0)/ ] z 8 from 0 to z adds up to
Identifying a(q),b(q),c(q) from Eqs. ~27!,~31! we obtain
~after n→0)
dF3 52
S
z
z
dz
1
2 v 0s 0
2 db.
d ~ z1D ! 1
m
m
z
2t
8 p f ~ z,D ! 58 p f ~ 0,0! 1 ~ 121/m ! D1 ~ 11s/t ! z
3 @ a ~ q ! 1nb ~ q ! 1mc ~ q !# 21 ,
g ~ q ! 5 $ 2a
Performing the trace and expressing d a ,d g in terms of
dz,dD @from Eq. ~35!# we obtain for the free energy per
replica, f 5F(n) /n,
The u and v terms in Eq. ~26! lead, by using Eq. ~25!, to
;exp@2(a1b1g)# and to ; ( a s a , a 5 @ s 1 2( s 1 2 s 0 )m # ,
respectively. Finally, we have
a ~ q ! 51/a ~ q ! ,
21
55
~37!
S D S D
S DF S D G
S DS D
z
Dc
D5 v 0 m
s 05
s1t/m
z1D
Dc
z1D
Dc
z1D
Dc
2t
12
2t
t ~ 121/m !
e 2t v 0 s 0 /z ,
z
z1D
z
z1D
~42!
2t/m
,
~43!
2t/m
.
~44!
The solutions for z and D of Eqs. ~41!–~44! determine the
phase diagram. Consider first the Josephson ordered phase
zÞ0,D50. Expecting s 0 !1 an expansion of Eq. ~41! in
powers of D/z yields m'tD/z so that s 0 'e 2 (z/D c ) 2t is
indeed small. The solution for z when D→0 is equivalent to
the replica symmetric case, Eq. ~30! and is possible for
t,12 s .
Consider next an RSB solution z50,DÞ0. Equation ~41!
yields m52t and Eq. ~43! leads to
D/D c 5 ~ 2t v 0 /D c ! 1/~ 122t ! .
~45!
55
DISORDER IN TWO-DIMENSIONAL JOSEPHSON JUNCTIONS
14 505
TABLE I. Correlations in junctions of size L; c(L) determines I c via Eqs. ~50!, ~51!.
Phase
G a , a (q)
Disorder
8 p (t1s)
q2
Josephson
SD
SD
SD
L
l
8 p sz
8 p (t1s)
2 2
2
q 1z
(q 1z) 2
L
l
4 p (112s) 4 p (2t21)
1
q2
q 2 1D
Glass
Coexistence
L
l
SD
4 p (2t21) 4 p (112s)
1
q 2 1z1D
q 2 1z
4pz
2 2
(q 1z) 2
L
l
Thus a glass-type phase is possible for t,1/2. @Curiously, a
similar result is obtained for the v term in first-order RG,
l 52 in Eq. ~20!, however, G a , a ;1/q 4 at q→0, while here
G a , a ;1/q 2 #.
Finally consider a coexistence phase, where both z, D
Þ0. It is remarkable that m52t is an exact solution even in
this case, as can be checked by substitution in Eqs.
~41!,~43!,~44!. The resulting solutions are
S D
S D
z1D
v0
5 2t
Dc
Dc
1/~ 122t !
u 20
z
21
5e
Dc
2t v 0 D c
,
1/~ 122s !
.
~46!
This coexistence phase is therefore possible at t,1/2 and
s,1/2, as shown in the phase diagram, Fig. 2. It is interesting to note that D50 on some line within the coexistence
phase, i.e., D changes sign continuously across this line.
When u 0 ' v 0 this line is s5t, as shown by the dashed line in
Fig. 2. This line is not a phase transition as far as the correlation c(r) @Eq. ~47! below# or the critical currents are concerned. We expect, however, that the slow relaxation phenomena, associated with the glass order, will disappear on
this line.
The boundary s51/2 of the coexistence phase is a continuous transition with z→0 at the boundary. On the other
hand, the boundary at t51/2 is a discontinuous transition,
z1D→0 from the left, while D50,zÞ0 on the right, i.e.,
both D and z are discontinuous.
To identify the various phases we consider the correlation
function
c ~ r ! 5 ^ cosw a ~ r! cosw a ~ 0 ! & 5 @ exp~ 2 f 1 ! 1exp~ 2 f 2 !# /2,
~47!
where
f 65
E
AD c
1/L
q dq @ 16J 0 ~ qr !# G a , a ~ q ! /2p ,
c~L!; L.min~lJ ,lG!
c~L!; L,lJ ,lG
~48!
24~t1s!
SD
24~t1s!
lJ
l
SD
24~t1s!
24~t1s!
L
l
S
8!
min~L,lG
l
D
24~t1s!
22~112s!
22~2t21!
SD
lG
l
22~2t21!
min~L,lJ! 22~112s!
)
l
(
and the system size L appears as a low momentum cutoff.
Using G a , a (q)5 a (q)1 b (q)1 g (q), the various correlations are summarized in Table I. The ordered phases have
finite correlation lengths defined as l J 5z 21/2 for the Josephson length, l G 5D 21/2 for the glass correlation length and
8 5(z1D) 21/2 in the coexistence phase. It is curious to
lG
note that in the coexistence phase G a , a has a
(2t21)/(q 2 1z1D) term. Since z1D→0 much faster than
2t21→0 at the boundary t51/2, this leads to an apparent
8 ; however, f 6 is finite at t→1/2 and the
divergence of l G
transition is of first order.
The phases with z50 have power-law correlations; for
L→`, c(r);r 24t24s in the disordered phase, while
c(r);r 2224s in the glass phase. The glass phase leads to
stronger decrease of c(r) then what would have been c(r) in
a disordered phase at t,1/2; a prefactor (l J /l) 2(122t)
somewhat compensates for this reduction.
The phases with zÞ0 have long-range order. Note in particular the z/(q 2 1z) 2 terms in G a , a ; these terms do not
arise in RG since they are of higher order in z and are of
interest away from the transition line. Note that in the Josephson phase v 0 'u 0 is assumed, so that s 0 v 0 !z; otherwise the coefficient of (q 2 1z) 22 is modified.
The correlation c(L) measures the fluctuation effect on
^ coswJ& in a finite junction, i.e., ^ coswJ&'Ac(L), which is
therefore related to the Josephson critical current I c . The
results for c(L) are summarized in Table I. Consider first a
junction with L,l J ~which is always the case in the z50
phases!. The current flows through the whole junction and
the system is equivalent to a point junction with an effective
Gibbs free energy,
2
ex
G eff
J 5E J ~ L/l ! Ac ~ L ! cosw J 2 ~ f 0 /2p c ! I w J .
~49!
Here we assume ~as at the end of Sec. II! that point junction
fluctuations can be ignored, i.e., f 0 I c /2c.T and the critical
current of Eq. ~49! can be deduced by its mean-field equation
~see Sec. V for actual data!. Thus, the mean-field value I 0c1
@Eq. ~8!# is reduced by the fluctuation factor, leading to a
critical current
14 506
BARUCH HOROVITZ AND ANATOLY GOLUB
I c 5I 0c1 Ac ~ L ! , L,l J .
~50!
For L,l J ,l G the parameters D and z are no longer related
to l J or to l G ; instead they are L dependent @Eq. ~35!
should be reevaluated leading to power laws of L#. In particular z affects c(L) via the (q 2 1z) 22 terms by either a
factor exp@2sz(L)L2# ~in the Josephson phase! or
exp@z(L)L2# ~in the coexistence phase!. Although of unusual
form, these factors are neglected in Table I since zL 2 ,1.
The dominant dependence in a small area junction,
L,l J ,l G ~for all phases! is a power-law decrease of
c(L), leading to I c ;L 222t22s .
For systems with L.l J , the current flows in an area
Ll J near the edges of the junction. The mean-field value
I 0c2 @Eq. ~8!# is reduced now by a factor l 0J /l J . Using
^ coswJ&5Ac(L) and z5u 0 ^ coswJ&51/l 2J we obtain
l J 5l 0J c 21/4(L) with l 0J 5l( t /8p E J ) 1/2, as in Sec. II. The
critical current is then
I c 5I 0c2
Ac ~ L ! ,
4
L.l J .
~51!
The relevant range of temperatures T! t ~see Sec. II!, for
typical junction parameters, is most of the range T,T c , excluding only T very close to T c . Thus t!1 and our main
interest is the coexistence to glass transition at s5 21 . This
transition can be induced by a temperature change since
s5s(T) ~see Sec. III!. Thus we consider t!s for which
8 . When the transition at s5 21 is apz!D and l J @l G 'l G
proached l J diverges and for a given L the system crosses
into the regime l G ,L,l J ~which includes the glass phase!
where c(L);(L/l) 24s (l G /L) 2 and I c ;(L/l) 122s . Since
L@l we predict a sharp decrease of I c at some temperature
T J for which s(T J )5 21 ; this is the finite-size equivalent of the
L→` phase transition.
V. DISCUSSION
We have derived the effective free energy for a 2D Josephson junction ~Appendix A! and studied it in the presence
of random magnetic fields. We show that a coupling between
replicas of the form cos(wa2wb) is essential for describing
the system. This coupling is generated by RG from the Josephson term in presence of the random fields, or also from
disorder in the Josephson coupling, a disorder whose finite
mean is E J .
We find the phase diagram, Fig. 2, with four distinct
phases defined in terms of a Josephson ordering z; ^ coswJ&
and a glass order parameter D. At high temperatures thermal
fluctuations dominate and the system is disordered,
z5D50. Lowering temperature at weak disorder (s, 21 ! allows formation of a Josephson phase, zÞ0,D50. Further
decrease of temperature leads by a first-order transition to a
coexistence phase where both z,DÞ0. The Josephson and
coexistence phases have similar diagonal correlations ~see
Table I!. The main distinction between these phases is then
the slow relaxation times typical of glasses. Finally, at strong
disorder and low temperatures the glass phase with z50,
DÞ0 corresponds to destruction of the Josephson long-range
order by the quenched disorder.
Our main result, relevant to experimental data with
t!1, is the coexistence to glass transition at s5 21 . The criti-
55
cal behavior of I c (s) near this transition depends on the ratio
L/l J ; not too close to s5 21 where L.l J we have from Eq.
~46!,~51! lnIc;1/(122s) while closer to s5 21 the divergence
of l J implies L,l J with I c ;(L/l) 122s . The junction ordering temperature T J corresponds to s(T J )5 21 so that either
lnIc;2(TJ2T)21 ~not too close to T J ) or lnIc;(TJ2T)lnL/l
close to T J .
We reconsider now the experimental data1–5 where the
junctions order at temperatures well below the T c of the
bulk. In our scheme, this can correspond to a transition between the glass phase and the coexistence phase, a transition
which may occur even at low temperatures t!1 provided s
decreases with temperature. As discussed in Sec. III, s depends on a power of l, in particular s;l 2 for short junctions, the experimentally relevant case. Thus s decreases
with temperature since l is temperature dependent. We propose then that junctions with random magnetic fields ~arising, e.g., from quenched flux loops in the bulk! may order at
temperatures well below T c of the bulk.
From critical currents1,2 at 4.2 K I c '1502400 m A we
infer E J '124 K and l 0J '224 m m, the latter is somewhat
below the junction sizes L'5250 m m. For the more recent
data on YBCO junctions3–5 with I c '0.426 mA we obtain
l 0J !L and Eq. ~51! applies. In fact, magnetic-field
dependence4 and I c ;L dependence7 show directly that
l J ,L is feasible.
We note also that mean-field treatment of the effective
free energy Eq. ~49! is valid since thermal fluctuations of the
effective point junction are weak ~as assumed in Secs. II and
IV!, i.e., f 0 I c /2c.T. E.g., at 80 K f 0 I c /2c5T corresponds
to I c '1 m A, while the mean field I c at the temperatures
where I c disappears, i.e., at 0.420.8T c , should be comparable to its low-temperature values1–5 of I c 50.126 mA.
Thus f 0 I c /2c@T and point-junction-type fluctuations can be
neglected.
Other interpretations of the data assume that the composition of the barrier material is affected by the superconducting material and becomes a metal3 N or even a
superconductor5 S’. In an SNS junction the coherence length
in the metal is temperature dependent and affects I c , while
the onset of an SS’S junction obviously affects I c . Note,
however, that the SNS interpretation with lnIc;2T1/2 is consistent with the T dependence but leads to an inconsistent
value of the coherence length.3 In our scheme,
lnIc;(TJ2T)lnL/l is consistent with the data3 of the
1003100 m m2 junction showing a cusp in I c (T) near
T J '25 K. Further experimental data, and in particular the
L dependence of I c , can determine the appropriate interpretation of the data.
The increasing research on large area junctions is motivated by device applications. The design of these junctions
should consider the various types of disorder studied in the
present work. Furthermore, we believe that disordered large
area junctions deserve to be studied since they exhibit glass
phenomena. In particular the coexistence phase with both
long-range order and glass order is an unusual type of glass.
ACKNOWLEDGMENTS
We thank S. E. Korshunov for valuable and stimulating
discussions. This research was supported by a grant from the
Israel Science Foundation.
55
DISORDER IN TWO-DIMENSIONAL JOSEPHSON JUNCTIONS
APPENDIX A: FREE ENERGY OF A 2D
JOSEPHSON JUNCTION
2. Gibbs free energy
In this Appendix we derive the effective free energy of a
large area Josephson junction. In Appendix A 1 boundary
conditions and the Josephson phase are defined. In Appendix
A 2 the Gibbs free energy in presence of an external current
is derived. In Appendixes A 3, A 4 the Gibbs free energy is
derived explicitly for superconductors in the Meissner state,
i.e., no flux lines in the bulk; Appendix A 3 considers long
junctions, i.e., W@l ~see Fig. 1!, while Appendix A 4 considers short ones, W!l. Finally, in Appendix A 5 the free
energy in presence of ~quenched! flux loops in the bulk is
derived.
1. Boundary conditions
The barrier between the superconductors ~region I in Fig.
1! is defined by allowing currents j z (x,y) in the z direction
so that Maxwell’s relation for the vector potential A(x,y,z)
is
“3“3A5 ~ 4 p /c ! j z ẑ,
~A1!
where ẑ is a unit vector in the z direction. There is no additional relation between j z and A ~e.g., as in superconductors!. This allows j z to be a fluctuating variable in thermodynamic averages.
Equation ~A1! implies that the magnetic field in the barrier H(x,y)5“3A is z independent and H z 50; thus the
currents j x , j y 50 as required.
Considering the superconductors in Fig. 1 we denote all
2D fields ~i.e., x, y components! at the right and left junction
surfaces ~i.e., z56d/2) with indices 1,2, respectively.
Boundary conditions are derived18 by integrating “3A
around the dashed rectangle in Fig. 1, which since j y 50,
yields continuity of the parallel magnetic fields
H1 ~ x,y ! 5H2 ~ x,y ! .
~A2!
Integrating A along the same rectangle yields for the vector
potentials on the junction surfaces
A 1x 2A 2x 1
E
d/2
2d/2
~ ] A z / ] x ! dz5dH y ,
~A3!
and a similar relation interchanging x and y. Introducing the
phases w i (r), i51,2 for the two superconductors and a
gauge-invariant vector potential
~A4!
yields for Ai8 (x,y) on the junction surfaces
A81 ~ x,y ! 2A82 ~ x,y ! 5dH~ x,y ! 3ẑ2 ~ f 0 /2p ! “ w J ~ x,y ! ,
~A5!
where w J (x,y) is the Josephson phase,
w J ~ x,y ! [ w 1 ~ x,y ! 2 w 2 ~ x,y ! 2 ~ 2 p / f 0 !
In the presence of a given external current jex passing
through the junction we separate the system into the sample
with relevant fluctuations ~e.g., superconductors with barrier!
and an external environment in which jex is given. Thermodynamic quantities are then given by a Gibbs free energy
G(H) where H is the field outside the sample which determines jex. The situation which is usually studied is such that
jex does not flow through the sample19 so that it is uniquely
defined everywhere. We need to generalize this situation to
the case in which jex flows through the sample, a generalization which to our knowledge has not been developed previously.
In standard electrodynamics,20 in addition to the spaceand time-dependent electric and magnetic fields E and H,
respectively, one defines a free current j f , a displacement
field D and an induction field B such that
“3H5 ~ 4 p /c ! j f 1 ~ 1/c ! ] D/ ] t,
“3E52 ~ 1/c ! ] B/ ] t,
E
d/2
2d/2
A z dz.
~A6!
~A7!
and only outside the sample D5E, B5H, and j f 5j . When
the various electrodynamic fields change by a small amount,
the change in the sample’s energy is the Poynting vector
integrated over the sample surface S ~with normal ds) in
time dt
ex
2dt
c
4p
E
E3H ds5
S
EF p
V
1
1
H•dB1
E•dD
4
4p
G
1E•j f dt dV,
~A8!
where integration is changed from the surface S to the enclosed volume V by Eq. ~A7!. When jex does not flow
through the sample, j f 50 and neglect of D ~for lowfrequency phenomena! leads to the usual energy change19
dE5 * H•dB/4p .
The general situation is described by keeping the surface
integral in Eq. ~A8! and in terms of the vector potential A,
where E52(1/c) ] A] t,
dE5
E
S
A8i ~ r! 5Ai ~ r! 2 ~ f 0 /2p ! “ w i ~ r!
14 507
dA3H ds/4p .
~A9!
Thus the surface values of A and H ~parallel to the surface!
determine the energy exchange dE and there is no need to
specify an H or a j f inside the sample, where in fact they are
not uniquely determined.
Since H ~on the surface! is determined by jex @via Eq.
~A7! outside the sample# we define a Gibbs free energy
G(H) by a Legendre transform
G~ H! 5F2 ~ 1/4p !
E
S
A3H ds.
~A10!
A is determined now by a minimum condition d G/ d A50
which indeed reproduces Eq. ~A9!.
We apply now Eq. ~A10! to the Josephson-junction system. We assume a time-independent current jex, i.e.,
“3H5(4 p /c)jex outside the sample and that the same cur-
14 508
BARUCH HOROVITZ AND ANATOLY GOLUB
rent jex flows through both superconductor-normal ~SN! surfaces ~e.g., the superconductors close into a loop or that the
current source is symmetric!. Consider now the surface S 1 of
superconductor 1, which includes the superconductor-normal
~SN! surface and the superconductor-vacuum ~SV! surface.
The boundary of S 1 is a loop J which encircles the junction
surface, oriented with normal 1ẑ. In terms of the gaugeinvariant vector A8 5A2( f 0 /2p )“ w 1 , assuming jex is time
independent, ] E/ ] t50 and using
“ w 1 3H5“3 ~ w 1 H! 2 ~ 4 p /c ! w 1 jex,
Z5
E
DAs8 ~ rs !
A3H ds5
S1
FR
f0
2p
1
E
S1
J w 1 H•dl2 ~ 4 p /c !
Ew
1j
G
ex
•ds
A8 3H•ds.
~A11!
The jex•ds term for both superconductors involves the
difference w 1 2 w 2 of the phases on the two SN surfaces.
This difference17 is related to the chemical potential difference in the external circuit so that the corresponding term is
w J independent.
Consider next the insulator-vacuum ~IV! surface. Since
H z 50 in the insulator only the A z H y or A z H x terms contribute with
E
A3H ds52
IV
R
J H•dl
E
d/2
2d/2
~A12!
A z dz.
Combining Eq. ~A11!, the similar term for superconductor 2
and Eq. ~A12!, ~ignoring w J independent terms! we obtain,
G~ H! 5F2
1
4p
E
SV1SN
A8 3H•ds2
f0
8p2
Rw
J H•dl.
~A13!
3. Long superconductors
We derive here an explicit free energy, in terms of the
Josephson phase, for the case W@l 1 ,l 2 ~see Fig. 1!, where
l i ~i51,2! are the London penetration lengths of the two
superconductors, respectively. The incoming current
jex(x,y) is parallel to the ẑ axis.
Consider the free energy19 of superconductor 1 ~suppressing the subscript 1 for now!
F5
1
8p
E
z>d/2
d 3r
F S
D
G
2
1 f0
“
w
2A
1 ~ “3A! 2 .
l2 2p
~A14!
The superconductor is assumed to have no flux lines, i.e.,
w (r) is nonsingular. The vector A9 5A2( f 0 /2p )“ w has
then three independent components ~no gauge condition on
A9 ) and “3A9 5“3A. The partition function involves integration on all vectors A9 and on its boundary values
As8 (rs ) on the boundary rs of the superconductor,
DA9 ~ r! exp@ 2F$ A9 ~ r! ,As8 ~ rs ! % /T # .
~A15!
We shift now the integration field from A9 to d A8 where
A9 5A8 1 d A8 and A8 is the solution of d F/ d A8 50, i.e.,
“3“3A8 52A8 /l 2
~A16!
with A8 5As8 at the boundaries; thus d A8 (rs )50. Since F is
Gaussian, F(A8 1 d A8 )5F(A8 )1F( d A8 ) and the integration on d A8 is a constant independent of As8 (rs ). Thus
we obtain
E
E
55
Z;
E
DAs8 ~ rs ! exp@ 2F$ A8 ~ r! % /T # ,
where
F$ A8 % 5
1
8p
E F
d 3r
G
1
~ A8 2 1“3A8 ! 2 .
l2
~A17!
Note that Eq. ~A16! implies “•A8 50 and therefore
“ 2 A8 5A8 /l 2 . Note also that the currents obey
j52(c/4p l 2 )A8 .
We are interested in boundary fields at the barrier which
are 2D vectors, e.g.,
A81 ~ x,y ! [ @ A 81x ~ x,y ! ,A 81y ~ x,y !# .
The effect of these fields decays on a scale l so that for
z@l, A8 ;ẑ j ex(x,y) also obeys London’s equation
l 2 ¹ 2 j ex5 j ex. Therefore j ex is confined to a layer of thickness l near the SV surface. The solution for z>d/2 has the
form
@ A 8x ~ r! ,A 8y ~ r!# 5A81 ~ x,y ! exp@ 2 ~ z2d/2! /l # ,
A z8 ~ r! 5l“A81 ~ x,y ! exp@ 2 ~ z2d/2! /l # 2 ~ 4 p l 2 /c ! j ex~ x,y ! .
~A18!
This ansatz is a solution of London’s equation ~A16! provided that A81 (x,y) is slowly varying on the scale of l. The
corresponding magnetic fields are
~ “3A! x8 5 ~ 1/l ! A 8y 2 ~ 4 p l 2 /c ! ] y j ex1O ~ ¹ 2 A18 ! ,
~ “3A! 8y 52 ~ 1/l ! A x8 2 ~ 4 p l 2 /c ! ] x j ex1O ~ ¹ 2 A18 ! .
~A19!
Since eventually A81 ;“ w J @Eq. ~A23! below# we evaluate
F by neglecting terms with derivatives of A81 . Some care is,
however, needed in evaluating cross terms with j ex, which is
not slowly varying. Thus, * A z8 2 (r) from Eq. ~A18! involves
E
j ex“•A18 dx dy52
E
A18 •“ j exdx dy,
which cannot be neglected. Note that the line integral on the
SV surface vanishes since on this surface the perpendicular
component of A81 is zero, i.e., no currents flowing into
vacuum. The O(¹ 2 A81 ) terms in Eq. ~A19! can be neglected
since their product with j ex cannot be partially integrated
without SV line integrals.
The cross terms from squaring Eqs. ~A18!,~A19! involve
55
DISORDER IN TWO-DIMENSIONAL JOSEPHSON JUNCTIONS
E
@ j ex“•A81 1A81 •“ j ex# dx dy5
E
“• ~ j exA81 ! dx dy50.
For superconductor 2 with z,2d/2 the solution has the
form of Eq. ~A18! with A82 (x,y) replacing A81 (x,y), the z
dependence has exp@(z1d/2)/l 2 # and 2“•A82 replaces
“•A81 in the equation for A z8 . For both superconductors
(i51,2!, after z integration, we obtain
Fi 5
E
dx dyAi8 ~ x,y ! /8p l i 1O ~ ] Ai8 ! .
2
2
~A20!
Next we use the boundary conditions Eqs. ~A2!, ~A5! to
relate Ai to w J . Equations ~A2!, ~A19! yield
8
8 /l 2 2 ~ 4 p l 22 /c ! “ j ex
A18 /l 1 2 ~ 4 p l 21 /c ! “ j ex
1 52A2
2
1O ~ ] Ai8 ! 2 ,
~A21!
E
SV
A8 3H•ds5
A81 2A82 5d @ 2A81 /l 1 1 ~ 4 p l 21 /c ! “ j ex
1 # 2 ~ f 0 /2p ! “ w J .
~A22!
Since “ j is not slowly varying, the ansatz Eq. ~A18! is
consistent ~i.e., Ai8 are slowly varying! only if the junction is
ex
symmetric, j ex
1 (x,y)5 j 2 (x,y), l[l 1 5l 2 and that the limit
d/l→0 is taken. Thus,
ex
A81 52A82 52 ~ f 0 /4p ! 2 “ w J .
~A23!
J
J H•dl5 ~ 4 p l
2
/c !
dx dy ~ “ w J ! .
Rw
J
2
G5
E
SV1
A8 3H•ds52 ~ 4 p l 4 /c !
5 ~ 4 p l /c !
2
E
E
dx dy2
f0
2p
Rw
J
J H•dl.
dx dy
F S D
1
f0
4pl 4p
2
~ “ w J !22
G
f0
w j ex .
2pc J
~A26!
Adding the Josephson tunneling term ;coswJ leads to Eqs.
~1!,~3!.
4. Short superconductors
Consider superconductors with length W 1 ,W 2 !l 1 ,l 2
~see Fig. 1!. The exp(2z/l1) in Eq. ~A18! can be expanded to
terms linear in z. Since now both exp(6z/l1) are allowed at
z.0, there are two slowly varying surface fields A1 , H1 ,
8 2z“•A81 1O ~ z 2 ! ,
A z8 5A 1z
~A27!
and the magnetic field is
H5H1 ~ x,y ! 2 ~ z/l 21 ! A81 ~ x,y ! 3ẑ1O ~ z 2 , ] A z ! .
8
~A28!
The x,y components of H5Hex
1 at z5W 1 define the
boundary conditions,
H 1x 2 ~ W 1 /l 21 ! A 81y 5H ex
1x ,
8 5H ex
H 1y 1 ~ W 1 /l 21 ! A 1x
1y ,
~A24!
J“ j
ex
•dl3ẑ.
~A29!
SV1
H 2x 1 ~ W 2 /l 22 ! A 82y 5H ex
2x ,
H 2y 2 ~ W 2 /l 22 ! A 82x 5H ex
2y .
A z8 “ j ex•ds
A81 •“ j dxdy
ex
1O ~ ¹ 2 A18 , w J independent terms! ,
where “ j ex•ds is replaced by ¹ 2 j exdxdydz as j ex has dominant x,y dependence. Using Eq. ~A23! and adding terms for
both superconductors leads to
~A30!
Equations ~A29!,~A30! and the boundary conditions
~A2!,~A5! at the junction determine all the fields Ai8 ,Hi in
terms of Hex
i and w J , e.g.,
85
A 1x
~A25!
For the SV surface integration we use again HSV so that for
superconductor 1,
E
ex
and similarly Hex
2 at z52W 2 .
2
If j ex50, Eqs. ~A20!,~A23! are valid also for nonsymmetric
junctions and F has the form ~A24! with 2l replaced by
l 1 1l 2 1d.
We proceed to find the Gibbs terms in Eq. ~A13!. Since
Eq. ~A19! and the constraint of no current flowing into the
vacuum, A8 3ẑ•dl50 yield HSV52(4 p l 2 /c)“ j ex3ẑ on
the SV surface, the loop integral becomes
Rw
Jj
Finally we obtain,
The magnetic energy in the barrier is neglected since it involves d/l. The total free energy, from Eqs. ~A20!,~A23! is
then
S DE
Ew
@ A 8x ,A 8y # 5A81 ~ x,y ! 1zH1 ~ x,y ! 3ẑ1O ~ z 2 ! ,
while Eq. ~A5! yields
f0
1
F5F1 1F2 5
4pl 4p
2f0
c
14 509
l 21
l 21 W 2 1l 22 W 1 1dW 1 W 2
2 ex
3 @~ l 22 1W 2 d ! H ex
1y 2l 2 H 2y 2W 2 ~ f 0 /2p ! ] x w J # ,
A 82x 5
2l 22
l 21 W 2 1l 22 W 1 1dW 1 W 2
2 ex
3 @~ l 21 1W 1 d ! H ex
2y 2l 1 H 1y 2W 1 ~ f 0 /2p ! ] x w J # ,
H 1y 5
2
ex
l 22 W 1 H ex
2y 1l 1 W 2 H 1y 1W 1 W 2 ~ f 0 /2p ! ] x w J
l 21 W 2 1l 22 W 1 1dW 1 W 2
.
~A31!
need to be slowly varying ~of order
The boundary fields
“ w J ) so that Eq. ~A31! is slowly varying; thus H z ,
A iz ;¹ 2 w J can be neglected.
The free energy ~A17!, to leading order in W i /l i is
Hex
i
14 510
BARUCH HOROVITZ AND ANATOLY GOLUB
F1 5 ~ W 1 /8p l 21 !
E
5. Junctions with bulk flux loops
A81 2 ~ x,y ! dx dy
1O @~ W 1 /l 1 ! ~ “ w J ! , ~ W 1 /l 1 !
3
2
2
“ w J •Hex
1 #.
~A32!
Ignoring w J independent terms,
f0
F1 1F2 5
16p 2
3
H
E
dx dy
2
ex
l 21 W 2 H ex
1y 1l 2 W 1 H 2y
~ l 21 W 2 1l 22 W 1 1dW 1 W 2 ! 2
J
FI 5 ~ d/8p !
E
S D
2
dx dyH21 ~ x,y !
W 1W 2
l 21 W 2 1l 22 W 1
E
1
4p
E
SN
A8 3H•ds5
f0
8p2
E
dx dy
1
E
f
Rw
p
f0
2pc
8
J
dx dy ~ “ w J ! 2 .
~A36!
j 5
J H•dl1•••,
l 21 W 2 1l 22 W 1
~A37!
.
~A38!
The Gibbs free energy is finally,
G5F2 ~ f 0 /2p c !
E
with F given by Eq. ~A36!.
F5
dx dy j ex~ x,y ! w J ~ x,y !
~A39!
1
8p
E F
d 3r
G
1 ~ “3A9 ! 2 .
~A40!
~A41!
1
~ l 2 “3“3As 2A8 ! 2
l2
G
1 ~ “3A8 1“3As ! 2 .
l 21 W 2 1l 22 W 1
2
ex
l 21 W 2 j ex
1 1l 2 W 1 j 2
D
2
Since F is Gaussian in A9 , the integration on d A9 decouples
from that of w s and the boundary values. Define now
A9 5A8 1As where As is a specific solution of Eq. ~A41! and
A8 is the general solution of the homogeneous part of Eq.
~A41!, “3“3A8 52A8 /l 2 , which depends on boundary
conditions, i.e., on w J .
Substituting Eq. ~A41! for As in Eq. ~A40! yields
2
ex
l 21 W 2 H ex
1y 1l 2 W 1 H 2y
where higher-order terms in W i /l i and w J independent
terms are ignored, and the fact that H•l is z independent on
the SV surface is used ~this arises from zero current into the
vacuum and neglecting H z ). The current j ex is defined here
ex
as an average of the currents on both sides, j ex
i 5(“3Hi ) z
~which locally may differ!, i.e.,
ex
1 f0
“ w s 2A9
l2 2p
“3“3A9 5 @~ f 0 /2p ! “ w s 2A9 # /l 2 .
dx dy j exw J
0
2
E F S
d 3r
~A35!
3 ] x w J 2 ~ x↔y ! 1•••
52
1
8p
We shift the integration field A9 by A9 →A9 1 d A9 ~as in
Appendix A 3! where d A9 50 at the boundaries and A9 satisfies d F/ d A9 50, i.e.,
Considering next the Gibbs term, the SV surface involves
A z 8 or H z which are neglected. The SN surface involves
A8 (z5W 1 )5A1 1O(W 21 ]w J ), hence
2
F5
~A34!
precisely cancels the terms linear in “ w J in Eq. ~A34! so that
1 f0
8p 2p
We define a three-component vector A9 5A2( f 0 /2p )“ w ns
so that the free energy Eq. ~A14! is
] x w J 1 ~ x↔y ! .
The free energy in the barrier
F5
Consider a junction with flux loops in the bulk of the
superconductors. These loops induce magnetic fields which
couple to w J . To derive this coupling we decompose the
superconducting phase into singular w s and nonsingular w ns
parts, i.e.,
“3“ ~ w s 1 w ns! 5“3“ w s Þ0.
f 0 W 1 ~ W 2 l 1 ! 2 1W 2 ~ W 1 l 2 ! 2
~ “ w J ! 2 ~A33!
2 p ~ l 21 W 2 1l 22 W 1 1dW 1 W 2 ! 2
22d
55
~A42!
In the absence of flux loops “3As 50 and Eq. ~A42! reduces to the previous F(A8 ) as in Eq. ~A17!. The terms
which depend only on As represent energies of flux loops in
the bulk and affect the thermodynamics of the bulk superconductors. Here we are interested in temperatures well below T c of the bulk so that fluctuations of these flux loops are
very slow and are then sources of frozen magnetic fields. The
thermodynamic average is done only on the boundary fields
which determine A8 , and are coupled to As by the cross
terms in Eq. ~A42!,
E
p E
Fs 5 ~ 1/8p !
5 ~ 1/4 !
S
d 3 r @ 22A8 •“3“3A8 12“3A8 •“3As #
~ A8 3“3As ! •ds.
~A43!
The surface values of A8 are determined by w J . The SV
surface involves z integration of “3As with either
exp(6z/l), Eq. ~A18!, or a linear function, Eq. ~A27!. In
either case the randomness in “3As causes this integral to
vanish. The relevant surface in Eq. ~A43! is therefore the
junction surface.
55
DISORDER IN TWO-DIMENSIONAL JOSEPHSON JUNCTIONS
14 511
To find the inverse B̂5Â 21 we solve for c̃51, c(u)50
and find
APPENDIX B: HIERARCHICAL REPLICA
SYMMETRY BREAKING
In this appendix we examine the full replica-symmetrybreaking formalism ~RSB! and show that it reduces to the
one-step symmetry-breaking solution, as studied in Sec. IV.
The method of RSB is based15,16 on a representation of hierarchical matrices A ab in replica space in terms of their diagonal ã and a one-parameter function a(u), i.e., A ab
→ @ ã,a(u) # . In our case A ab is related to the inverse Green’s
function G 21
ab which was obtained by Gaussian variational
method ~GVM!.
To derive this representation, consider the hierarchical
form of a matrix Â,
b̃2b ~ u ! 5
b̃5
1
u @ ã2 ^ a & 2 @ a #~ u !#
1
ã2 ^ a &
F
12
E
( a i~ Ĉ i 2Ĉ i11 ! 1ãÎ.
~B1!
i50
Here Ĉ i is n3n matrix whose nonzero elements are blocks
of size m i 3m i along the diagonal; each matrix element
within the blocks is equal to one; the last matrix equals the
unit matrix Ĉ k11 5Î. The matrices Ĉ i satisfy relations which
are useful for finding the representation of the product of
matrices ÂB̂ . Since the hierarchy is for m i /m i11 integers,
we have
dv
1
u
,
v @ ã2 ^ a & 2 @ a #~ v !#
~B5!
2
v @ ã2 ^ a & 2 @ a #~ v !#
@ a #~ u ! [ua ~ u ! 2
E
u
0
a~ 0 !
2
2
k
Â5
E
d v@ a #~ v !
1
0
2
ã2 ^ a &
G
,
~B6!
~B7!
dva~ v !.
The inverse Green’s function is from Eq. ~27!
4 p G 21
ab ~ q ! 5
1
@ d ~ z1q 2 ! 2 v 0 s ab 2q 2 s/t # ,
2t ab
~B8!
which for ŝ → @ s̃ , s (u) # parametrizes as @ ã q ,a q (u) # with
ã q 5
1 2
@ q ~ 12s/t ! 1z2 v 0 s̃ # ,
2t
k
Ĉ i 5
~ Ĉ j 2Ĉ j11 ! 1Î,
(
j5i
Ĉ i Ĉ j 5
H
m i Ĉ j ,
j<i
m j Ĉ i ,
j.i.
a q ~ u ! 52
k
( b i~ Ĉ i 2Ĉ i11 ! 1b̃Î
i50
~B2!
F(
G F(
k
k
(
~ Ĉ j 2Ĉ j11 !
j50
j
1
(
i50
i5 j11
~ a i b j 1a j b i ! dm i 2a j b j m j11
k
a i b i dm i 1Î
i50
G
a i b i dm i 1ãb̃ ,
b̃ q 2b q ~ u ! 52t
~B3!
where dm i 5m i 2m i11 .
In the limit n→0 m i becomes a continuum variable u in
the range 0,u,1 and a i becomes a function a(u); thus the
matrix  is represented by @ ã,a(u) # . The product of two
matrices, using Eq. ~B3!, becomes ÂB̂→ @ c̃,c(u) # where
c̃5ãb̃2 ^ ab & ,
2
E
0
@ a ~ u ! 2a ~ v !#@ b ~ u ! 2b ~ v !# d v ,
and ^ a & 5 * 10 a(u)du.
1 2
@ q 1z1D ~ u !# ,
2t
~B10!
F
1
2
u @ q 1z1D ~ u !#
2
E
1
u
G
dv
.
v @ q 1z1D ~ v !#
~B11!
2
2
The GVM equation for s (u) is from
s (u)5exp@2B(u)#, where from Eq. ~25!
B ~ u ! 54 p
E
g~ u !
d 2q
2
2 @ b̃ q 2b q ~ u !# 5
u
~2p!
E
~28!
Eq.
1dvg~ v !
u
v2
.
~B12!
Equation ~B11!, after summation on q, identifies
c ~ u ! 5 ~ ã2 ^ a & ! b ~ u ! 1 ~ b̃2 ^ b & ! a ~ u !
u
ã q 2 ^ a q & 2 @ a q #~ u ! 5
where the order parameter D(u) is defined by D(u)
5 v 0 @ s # (u). From Eq. ~B5! the representation of the Green’s
function takes the form (4 p )G ab → @ b̃ q ,b q (u) # with
is found to be
ÂB̂5
~B9!
Since the sum on each row of ŝ vanishes @Eq. ~28!# we
obtain s̃ 5 ^ s & . Therefore the denominator under the integration in Eqs. ~B5!,~B6! assumes the form
The matrix product with a matrix B̂,
B̂5
1 2
@ q s/t1 v 0 s ~ u !# .
2t
~B4!
Dc
.
g ~ u ! 52t ln
D ~ u ! 1z
~B13!
Using s 8 (u)5d @ exp(2B(u)/2) # /du52 s (u)g 8 (u)/u and
the definition of D(u), D 8 (u)5 v 0 u s 8 (u) we obtain
BARUCH HOROVITZ AND ANATOLY GOLUB
14 512
F G
d D 8~ u !
D 8~ u !
52
,
u
du g 8 ~ u !
~B14!
which from Eq. ~B13! can be written as
S
1
D
1 1 dD
2
50 .
u 2t du
~B15!
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55
The solution of this equation is a step function,
i.e., D(u) jumps from zero to a constant value at u52t,
which is precisely the one-step solution.
We note that keeping finite cutoff corrections13 spoils this
correspondence. The variational method is, however, designed for weak-coupling systems and an infinite cutoff procedure is appropriate.
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15
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G. Schön and A. D. Zaikin, Phys. Rep. 198, 257 ~1990!.
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I. O. Kulik and I. K. Janson, The Josephson Effect in Superconducting Tunnel Structures ~Keter, Jerusalem, 1972!.
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P. G. deGennes, Superconductivity in Metals and Alloys ~Benjamin, New York, 1966!.
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L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous
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11
12