Zero-Temperature Superonduting Transition in
Frustrated Josephson-Juntion Arrays
Enzo Granato
LaboratorioAssoiadodeSensoreseMateriais,
InstitutoNaionalde PesquisasEspaiais
12201-970S~aoJosedosCampos,S~aoPaulo,Brazil
Reeivedon28February,2002
The ritial behavior of zero-temperature superonduting transitions whih an our in
disor-dered two-dimensionalJosephson-juntion arrays are investigated by Monte Carlo alulation of
ground-stateexitationenergiesanddynamialsimulationoftheurrent-voltageharateristisat
nonzerotemperatures. Twomodelsofarraysinanappliedmagnetieldareonsidered: random
dilutionofjuntionsandrandomouplingswithhalf-uxquantumperplaquettef =1=2. Above
aritialvalueofdisorder,nite-sizesalingoftheexitationenergiesindiatesazero-temperature
transitionandallows anestimateoftheritial disorderandthethermalorrelation length
expo-nentharaterizingthetransition. Current-voltagesaling isonsistentwiththezero-temperature
transition. Thelinearresistaneisnonzeroatnitetemperaturesbutnonlinearbehaviorsetsinat
aharateristiurrentdensitydeterminedbythethermalritialexponent. Thezero-temperature
transitionprovidesanexplanationofthewashingoutofstrutureforinreasingdisorderatf=1=2
whileitremainsfor f=0,observedexperimentallyinsuperondoutingwirenetworks.
I. Introdution
Inhomogeneous superondutorsin the form of an
array of superonduting grains embedded in a
non-superonduting host, may exhibit many properties
that ariseessentially from thephase-ohereneamong
thegrains,duetosuperondutingoupling[1-8℄. The
oupling between the grains an our by Josephson
tunnelling through an insulating host or by
proxim-ity eet in a normal-metal host. These systems an
bephysiallyrealizedasgranularsuperondutorsand
are partiularly importantin high-T
superondutors
[2℄ where phase utuations eets from temperature
and disorder play an important role. They an be
artiially fabriated in two dimensions as arrays of
Josephsonjuntions [4,5℄,withwellontrolled
param-eters, and are also losely related to superonduting
wire networks [9, 10℄, where the nodes of the
net-work at like oupled eetive "grains". In the
sim-plest model only the phase oupling between the
su-peronduting orderparameter
j =j
j je
ij
ofpoint
grains, dened on a lattie labelled by j, are taken
into aount. Eah juntion between nearest-neighbor
grains ontributesto theenergy of the system as[11℄
E
ij = J
ij os(
i
j A
ij
). Intheaseof
Josephson-juntion arrays,theouplingJ anberelatedto the
juntion ritial urrent I
ij as J
ij = ~I
ij
=2e and for
wirenetworksitis proportionaltothe superondutor
ondensation energy of the wire material. The bond
variable A
ij =
2
o R
j
i ~
Adl is the line integral of the
vetorpotential ~
A betweengrains in units of theux
quantum
o
=h=2edue to anapplied magnetield
~
B=r ~
A. TheresultingHamiltonianoftheoupled
Josephsonjuntion arrayan thenbewrittenas
H= X
<ij> J
ij os(
i
j A
ij
) (1)
wherethesumistakenovernearestneighbors<ij >.
This model is equivalent to an XY (two-omponent)
spinsysteminwhihthegaugeeldA
ij
leadsto
frustra-tioneetsin thespinalignment[12℄. Thefrustration
introdued by themagneti eld on agiven plaquette
pof the lattie an be dened asf
p =
p =
o , where
p
is the ux threading the plaquette, giving in
gen-eralafrationalnumberofuxquanta. Inthismodel,
onealsoassumesthatquantum-phaseutuationsdue
to hargingeets [3℄an benegleted. Inagranular
material,disorderarisesnaturallyfromtherandom
lo-ationof grains leadingto randomnessin J
ij and A
ij
(orequivalentlyonf
p
). Ingeneral,dierentmodelsan
thenbeonsidered depending ontheassumedrandom
Evenwhenartiiallyfabriatedastwo-dimensional
periodilatties,inwhihasef
p
=f isaonstantand
themodel inEq. 1is periodiin f withperiodf =1,
Josephson-juntionarraysorwirenetworkswillalways
ontainsomedegreeof disorder[13℄. Forexample,
in-homogeneities in the links of a regular
superondut-ingwirenetwork ofYBa
2 CuO
7
hasbeensuggestedas
animportane soureof disorder[14℄. Asdisussed in
the present work, if suh disorder is modelled by Eq.
1with random J
ij
, it mayexplain thewashingoutof
resistaneminimaobservedathalf-integerf inthis
sys-tem,inontrasttotheasewithintegerf wheresharp
minimaisobserved.
Ontheother hand,disorderanalso be
intention-allyintroduedtostudyitseets[9,15,16℄and
om-pare with theoretial preditions [7, 13℄. In this
re-gard, random dilution of grains, in an otherwise
two-dimensional periodi array, is probably the simplest
way to introdue disorder in a ontrollable manner
[9, 16℄. Suh system an be modelled by Eq. 1
as-sumingthataonentrationxofjuntionsareremoved
randomly. Inthisase,reenttheoretialwork[17,18℄
indiates that forf =1=2, the superonduting
tran-sition should vanish above aritial valueof disorder
x
s
but belowtheperolationdilutionthresholdx
p . In
priniplethisanbeveriedexperimentallythrough
re-sistivitymeasurementsinarrays. Thedisappearaneof
strutureatf =1=2forinreasingdisorderalready
ob-servedexperimentallyindilutedwirenetworks[9℄
sup-portsthisresult.
Inexperimentsaswellinnumerialsimulations,the
superonduting transition is usually identied from
the behavior of the nonlinear urrent-voltage
hara-teristis. However,thedening propertyof the
super-onduting phaseisthevanishing linearresistivity. In
ordertoextratinformationofthelinearresponseand
underlying transition, asalinganalysis [2, 19℄ isthen
required from whih the ritial temperature and
re-latedexponentsanbedetermined. Theinterpretation
oftheexperimentaldataalsoreliesontheexpeted
be-havior of the model used to desribe the system. In
partiular,knowledgeofthesoalledlowerritial
di-mensionof themodel[20℄ isessentialforameaningful
appliation ofthe urrent-voltagesaling theory. This
is beause, below the lower ritial dimension, there
is a phasetransition only at zero temperature but its
eets an still show up at nite temperatures as a
rossoverbehavior,ratherthanathermodynami
tran-sition. Thus,itisimportanttostudyindetailthelower
ritialdimensionandtransportpropertiesofthebasi
models in order to providesupport for these
interpre-tations aswell asto makepreditionsfor therealisti
physialsystem.
Inthis work,models of two-dimensional
superon-duting arrays with possible zero-temperature
transi-tions relevant for Josephson-juntion arrays and wire
networks are onsidered. Monte-Carlo simulated
an-nealing is used to obtain the ground-state exitation
energiesin order to study thelower-ritialdimension
while Langevin-type simulation is used to study the
salingoftheurrent-voltageharateristisatnonzero
temperatures. Two dierent models of disordered
ar-rays in an applied magneti eld are onsidered:
ran-dom dilutionof juntions and random ouplingswith
half-uxquantumperplaquette. Abovearitialvalue
of disorder, nite-size saling of the exitation
ener-gies indiates that suh models are below the
lower-ritialdimensionandazero-temperaturetransitionis
expeted. At nonzero temperatures, the linear
resis-tane is nonzero but nonlinear behavior sets in at a
harateristi urrent densitydetermined by the
ther-mal ritialexponent. These resultsareapossible
ex-planationfor thewashingoutof struture at f =1=2
observedinsomewirenetworks[9,14℄.
II. Modelsand simulation
ThedilutedJosephson-juntionarrayan be
mod-elledbyEq. 1assumingajuntion dilution
onentra-tionxorrespondingtoJ
ij
beingzeroorJ
o
with
prob-abilitiesxand1 x,respetively,andisrelatedtothe
frationof juntionspresentpbyx=1 p. Sineany
losedloopsofnonzerobondsJ
ij
haveanareawhihis
anintegermultipleoftheelementaryareaS,the
prop-ertiesofthismodelremainperiodiinf =BS=
o with
period 1. Similarly, therandomoupling model is
de-nedbyEq. 1assumingJ
ij =J
o
D withequal
prob-ability. Asdened,thesetwomodelsarejustpartiular
asesofamoregeneralmodelwithrandomJosephson
ouplingswhereJ
ij
isgivenbyanarbitraryprobability
distribution. In both ases, we onsider in this work
a square lattie and an external magneti eld
orre-spondingtohalf-uxquantumperplaquette,f =1=2.
To studythe stability of the groundstate, the
en-ergy hange E(L) of a defet in the ground state,
orrespondingto alow-energyexitation,is alulated
forsmallsystemsizesLbyMonteCarlosimulated
an-nealing[17℄. Thedefet-energyrenormalization
analy-sis [20℄ is then used to inferthe behavior in the large
systemlimit. Stabilityofthegroundstateagainst
ther-malutuationsrequiresthat[E℄
d
,where[ ℄
d
denotes
a disorderaverage, is nite or inreaseswith L. In a
stronglydisorderedphase,asinthevortex-glassstate,
W L =f[(E) 2 ℄ d ([E℄ d ) 2 g 1=2
isexpetedtosaleas
W
L /L
(2)
for largeenough L. If thestiness exponent is
pos-itive there is long-range order in the system and a
nonzero ritialtemperature. However,if <0,there
exists a length sale where L
W
L
kT, beyond
whih thermal utuations destroy the order at any
nonzero temperature and so the ritial temperature
vanishes. Thislengthsaleanbeidentiedasa
orre-lation length /1=T
, with aritial exponentgiven
by
T
=1=jj. As disussed below,this ritial
expo-nentdeterminestheurrent-voltagesalingofthezero
temperaturetransition.
Theurrent-voltageharateristisanbeobtained
by numerial simulation of the resistivelly shunted
Josephson-juntion (RSJ) model for the urrent ow
betweengrains [21, 22, 23℄. The Langevin-type
equa-tionsforthismodelan bewrittenas
C o d 2 i dt 2 + 1 R o X j d( i j ) dt = I X j sin( i j A ij )+I
ext i + X j ij ;(3) whereI ext
istheexternalurrent,
ij
represents
Gaus-sianthermalutuationssatisfying
<
ij
(t)> = 0
< ij (t) k l (t 0
)> = 2k B T R o Æ ij;k l Æ(t t 0 ) (4)
and aapaitane to theground C
o
is allowed, in
ad-dition to the shunt resistane R
o
, in order to
faili-tate the numerial integration [22℄. The parameter
I R 2 o C o
= 0:5 used in the simulations orresponds to
theoverdampedregime.Hereafter,dimensionless
quan-tities are used in units where ~=2e = 1, R
o
= 1and
I
=1. Theaboveequationsanbeintegrated
numeri-allyandtheresultsaveragedoverdierentrealizations
of thedisorder.
Todeterminethenonlinearresistivity(orresistane
intwodimensions),
nl
=E=J,auniformexternal
ur-rent I is imposed with density J = I=L along oneof
theprinipal diretions ofthelattie usingutuating
boundaryonditions[24℄. TheaveragevoltagedropV
arossthesystemisomputedas
V = 1 L L X j=1 ( d 1;j dt d L;j dt ) (5)
and the average eletri eld by E = V=L. The
lin-ear resistane, R
L = lim
J!0
E=J, an also be
om-puted without nite urrent eets, diretly from the
long-timeequilibrium utuationsof the phase
dier-enearossthesystem[23℄(t)= P L j=1 ( 1;j L;j )=L as R L = 1 2T ((t) (0)) 2 =t (6)
whih an beobtained from Kuboformula of
equilib-riumvoltage-voltageutuations,
R L = 1 2T Z
dt<V(t)V(0)>; (7)
usingtheJosephsonrelationV =d=dt.
III. Resultsand disussion
IV. Defet-energy saling
To determine the stability of the ground state
against low-energy exitations, a defet is reated in
asystem of size L L by imposing ahange in the
boundary onditions in one diretion. The hange
E(L) in the ground state energy for small systems
isalulatedforalargenumberofsamplesbydiretly
searhing for the minimum energyusing Monte Carlo
simulated annealing. The defet energy is obtained
fromtheenergydiereneE =E
a E p between peri-odiE p
andantiperiodiE
a
boundaryonditionsinthe
phases
i
. Thisenergydiereneprovidesameasureof
phase oherene and an related to the renormalized
stinessonstantbyJ
(L)=2E= 2
. Inthe
thermo-dynamilimit, J
is nite in thephaseoherentstate
and vanishes in the inoherent state. In presene of
disorder,Eutuatesbetweensamples,witha
distri-butionthatanbeharaterizedbyitsmoments.
Sta-bilityof theground stateagainstthermalutuations
requiresthattheaverage[E℄,where[℄denotesa
dis-orderaverage,isnite orinreaseswithL.
Fig. 1(a)showsthebehaviorof[E℄asfuntion of
Lforthedilutedarrayonasquarelattieforinreasing
dilutionx. Thebehaviorissimilartothatobservedfor
thesamemodelontriangularlattie[17℄. Forsmallx,
it inreases with L and eventually saturates,
indiat-ing the existene of long-rangephase oherene. The
inreasing trend is a small size eet. Forlarge L, it
shouldsaleasthephasestinesswhihisproportional
toL d 1
wheredisthedimensionofthesystem. Onthe
otherhand, forsuÆiently largex itlearly dereases
for inreasing L, indiating a disordered phase. The
hangeinthebehavioryieldsanestimateofthe
valueisonsistentwiththeoneinferredfromthe
behav-iorofthezero-temperatureritialurrentin
dynami-alsimulations[18℄. Surprisingly, thesameestimateis
also obtained for the triangular lattie, x
s
= 0:14(1),
even though the perolation dilution thresholds [25℄,
x
p
=0:5 and 0:652,forthe squareand triangular
lat-tiesrespetively,arequitedierent. Sineitisknown
[25℄thatthetransitiontemperatureinabseneof
mag-neti eld, f = 0, vanishes at x
p
, there is a range of
disorderx
s
<x<x
p
wherethetransition forf =1=2
only ours at T =0 while it ours at nonzero
tem-peraturesforf =0. Measurementsofthe
temperature-magneti eld phaseboundary for inreasing disorder
indilutedwirenetworks[9℄areonsistentwithaphase
oherenethresholdx
s
forf =1=2,whihshowsupas
awashing outofthestruture at half-integerf, while
itremainsessentiallyunaetedforf integer.
0.0
0.1
0.2
0.3
0.4
0.5
0
1
2
3
( b )
( a )
x
p
x
s
f = 1/2
L=4
L=6
L=8
L=10
∆
E
x
10
1
f = 1/2
x = 0.35
θ = -1.00(2)
W
L
Figure 1. (a) Finite size behavior of defet energy [E℄
for inreasing dilution x and various system sizes L, on
asquare lattie. Arrows indiatethe phase-ohereneand
perolationdilutionthresholds. (b)Finitesize behaviorof
thewidthofdefetenergydistributionW atadilution
on-entrationx=0:35intherangex
s
<x<x
p .
The disorderedphase for x
s
< x < x
p
an be
re-garded asavortexglass, sineitalso lakslong-range
order in the vortex lattie [17℄. The stability of this
glassphaseagainst thermalutuationsisdetermined
by the size dependene of the seond moment of the
apower-lawbehaviorgivenbyEq. 2. AsshowninFig.
1(b), thepower-lawexponentforavalueofx=0:35
inthisregionisnegative. Asaonsequene,thisvortex
glassphaseisbelowitslower-ritialdimensionandthe
phasetransitiononlyoursatT =0. Fromthe
power-lawbehaviorofW(L)theexponent
T
=1=jjofthe
thermalorrelationlengthisestimated tobe
T 1.
The results of defet-energy saling for the array
withrandomJosephsonouplingsareshownin Fig. 2.
Thebehaviorissimilartothatobservedforthediluted
arraydisussedabove. Forsmalldisorder,Einreases
withL,indiatinglongrangephaseoherene,whilefor
suÆiently largeD itdereases for inreasingL,
indi-ating a disordered phase. The hange in the
behav-ioryieldsanestimateofthephase-oherenethreshold
D
s
= 0:6(4) for f = 1=2 and D
o
0:9 for f = 0.
Again,thepower-lawexponentforavalueofdisorder
D = 0:8 larger than the ritial value D
s
is negative
as shown in Fig. 2(b) and the vortex glass phasefor
D > D
s
is belowits lower-ritialdimension and
or-respondingly the phasetransition for f = 1=2 should
ouratT =0. Fromthepower-lawbehaviorofW(L)
the exponent ofthe thermalorrelation lengthis
esti-matedtobe
T
1. Asforthedilutedmodel,thereisa
rangeofdisorderD
s
<x<D
o
,wherethetransitionfor
f =1=2onlyoursatT =0whileitoursatnonzero
temperaturesforf=0. Thus,magnetoresistane
mea-surementsjustabovethef =0ritialtemperature,in
arrayswithdisorderinthisrange,should display
resis-taneminimaonlyatintegervaluesoff. Thisbehavior
hasbeenobservedinregularsuperondutingwire
net-work[14℄ofYBa
2 CuO
7
withhighinhomogeneityinthe
linkswhihinprinipleouldleadtoouplingdisorder
in thisrange.
V. Current-voltagesaling
ThenonlinearresistaneE=J asafuntion of
ur-rent density J and temperature T for the array with
random Josephsonouplingsis shownin Fig. 3(a)for
adegree ofdisorderD =0:75 abovetheritial value
D
s
estimated above. Similarbehaviorisfoundfor the
dilutedarray[18℄abovethephaseoherene threshold
x
s
. ThedatashowstheexpetedbehaviorforaT =0
superondutingtransition[2,19℄. InFig. 3(a),the
ra-tioE=J tendstoanitevalueforsmallJ,
orrespond-ing to the linear resistane R
L = lim
J!0
E=J, whih
depends strongly on the temperature. For inreasing
J, there is a smooth rossoverto nonlinear behavior
thatappearsatsmallerurrentsfordereasing
temper-atures. If the transition only ours at T =0, as
in-diated bythedefet-energysaling analysisdisussed
above, then the orrelation length should diverge for
dereasing temperature and a temperature-dependent
rossoverisexpeted. ThelinearresistaneR
L
0.0
0.2
0.4
0.6
0.8
1.0
0
1
2
3
4
D
0
D
s
f=0
f=1/2
L=4
L=6
L=8
L=10
∆
E
D
10
1
( b )
( a )
θ = -1.05(6)
f = 1/2
D = 0.8
W
L
Figure2. (a)Finitesizebehaviorofdefetenergy[E℄for
f=1=2andf=0forinreasingdisorderDandvarious
sys-temsizesLforthearraywithrandomouplingsonasquare
lattie. Arrowsindiatetheorrespondingphase-oherene
thresholds. (b)Finite size behavior of thewidthof defet
energydistributionWatavalueofdisorderD=0:75above
Ds.
R
L
/exp( E
b
=kT),where E
b
isanenergy barrier. If
one assumes that the orrelation length diverges asa
power-law
T /T
T
thenthebehaviorofthe
nonlin-ear resistivity normalized to R
L
an be ast into the
saling form[2, 19℄
E
JR
L =g(
J
T 1+
T
) (8)
in d = 2 dimensions, where g is a saling funtion
with g(0)=1. Arossoverfromlinearbehavior,when
g(x) 1, to nonlinear behavior, when g(x) >> 1, is
expetedto ourwhenx1whihleadstoa
hara-teristiurrentdensityJ
nl
atwhihnonlinearbehavior
setsinthat dereaseswithtemperatureasapowerlaw
J
nl /T
1+T
.
Oneanthenproeedto verify thesaling
hypoth-esisandobtainanumerialestimateoftheritial
ex-ponent
T
. Fig. 3(b) shows asaling plot aording
to Eq. (8)obtainedby adjusting theparameter
T so
thatabestdataollapseisobtained. Thedataollapse
supportsthe salingbehaviorof Eq. (8) and provides
an estimate of
T
=2:2. This estimate diers
signif-iantly from the value obtained by the above
defet-thepresent, theoriginof this disrepanyis notquite
lear. It is possible that the trueasymptoti value of
T
anonlybeobtainedfromthedefet-energysaling
formuhlargersystemsizes,whihwouldrequiremore
auratedetermination oftheground-stateenergy.
0.1
1
1E-3
0.01
0.1
1
f = 1/2 D = 0.75
T=0.5
T=0.4
T=0.3
T=0.25
T=0.2
E/J
J
0.1
1
10
100
0.1
1
10
100
( b )
( a )
f = 1/2 D=0.75
ν
T
= 2.2 E
b
= 1.1
1 + ν
T
T=0.5
T=0.4
T=0.3
T=0.25
T=0.2
E /
J A ex
p(
-E
b
/T
)
J / T
Figure 3. (a) Nonlinear resistane E=J as a funtion of
temperature T for the array with random ouplings, for
D = 0:75 and system size L = 64, above the
phase-oherenethresholdD
s
. (b)Salingplotofthedatain(a)
forthelowesttemperaturesandurrentdensitiesaording
toaT =0transition.
VI. Conlusions
Theresultsofthegroundstatedefet-energy
alu-lationsanddynamialsimulationsdisussedhereshow
that disorder in two-dimensional Josephson-juntion
arrays has important eets whih show up in the
urrent-voltage harateristisas amanifestation of a
zero-temperaturesuperonduting transition. It is
in-terestingtonote that evenin theabseneof magneti
eld, arrays with random ouplings an display
sim-ilarurrent-voltage saling near perolation threshold
[26℄ and,without disorder,ifthe magnetield
orre-spondstoanirrationalfrustration[27℄. Forthediluted
and random-oupling arrays studied in this work we
ndthat,inpreseneofamagnetieldorresponding
to f = 1=2 ux quantum per plaquette and above a
exi-andallowsanestimate oftheritialdisorderandthe
thermalorrelationlengthexponentharaterizingthe
transition. Correspondingly,resistivity alulationsat
nonzerotemperatureshowsthatthelinearresistaneis
nonzerobut nonlinearbehaviorsets in at a
harater-istiurrentdensitydeterminedbythethermalritial
exponent. Sinetheritialdisorderissmallerthanthe
orresponding valueforf =0,there is arangeof
dis-order where the superonduting transition ours at
nonzero temperatures for f = 0 while it only ours
at zero temperaturefor f = 1=2. Some experimental
resultsonsuperondutingwirenetworkswith
inhomo-geneous[14℄anddilutedlinks[9℄supportthese
onlu-sions in thesense that awashingout ofthe struture
at f =n=2 (n integer) is found due to disorderwhile
thestrutureatf =nisessentiallyunaeted.
Never-theless,adetailed omparison betweentheoryand
ex-periment still awaits more aurate data and further
preditionsofmeasurablequantities.
ThisworkwassupportedbyFAPESPandCNPq.
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