Table Of ContentMesoscopic conductance fluctuations in YBa Cu O grain boundary Junction at low
2 3 7−δ
temperature
A. Tagliacozzo1, F. Tafuri2,3, E. Gambale1,2, B.Jouault4, D.Born2,3,
P.Lucignano1,5, D. Stornaiuolo1,F.Lombardi6, A. Barone1, B.L. Altshuler7
1 Coherentia INFM-CNR and Dipartimento di Scienze Fisiche, Universit`a di Napoli Federico II, Italy
2 Dip. Ingegneria dell’Informazione, Seconda Universit`a di Napoli, Aversa (CE), Italy
3 NEST CNR-INFM and Scuola Normale Superiore, I-56126 Pisa,Italy
4 Universit´e Montpellier 2, Groupe d’E´tude des Semiconducteurs, and CNRS,
9
UMR 5650, cc074, pl. Eug`ene Bataillon, 34095 Montpellier cedex 5, France
0
5 SISSA Via Beirut 2-4, 34014 Trieste, Italy
0
6 Dept. Microelectronics and Nanoscience, MINA,
2
Chalmers University of Technology, 41296 Gteborg, Sweden and
n 7 Physics Dept. Columbia University, New York NY 10027 and NEC
a Laboratories America INC, 4 Independence Day, Princeton, NJ 08554,USA
J
(Dated: January 12, 2009)
2
1 The magneto-conductance in YBCO grain boundary Josephson junctions, displays fluctuations
at low temperatures of mesoscopic origin. The morphology of the junction suggests that transport
] occursinnarrowchannelsacrossthegrainboundaryline,withalargeThoulessenergy. Nevertheless
n themeasuredfluctuationamplitudedecreasesquiteslowly whenincreasingthevoltageuptovalues
o about twenty times the Thouless energy, of the order of the nominal superconducting gap. Our
c
findings show the coexistence of supercurrent and quasiparticle current in the junction conduction
-
r evenathighnonequilibriumconditions. Modelcalculationsconfirmthereducedroleofquasiparticle
p relaxation at temperatures up to 3 Kelvin.
u
s
t. I. INTRODUCTION is another important1, to which a large part of the phe-
a
nomenologyhasbeenclearlyassociated1,2,3,10,11. D-wave
m
OPS implies the presence of antinodal (high energy),
- In high critical Temperature Superconductor (HTS)
d and nodal(low energy)quasi-particlesin the conduction
junctions, including grain boundary (GB) structures,
n acrossjunctions andtheabsenceofsharpgapfeaturesin
it is well established that various interplaying mecha-
o thedensityofstatesoftheweaklink. Recently,lowtem-
c nisms contribute to transport with different weights still peratures measurements have proved macroscopic quan-
[ to be completely defined in a general and consistent
tumtunneling(MQT)inYBCOGBjunctions,stimulat-
framework1,2,3. As in all non-homogeneous systems, the
1 ing novel research on coherence and dissipation in such
v barrier region will significantly contribute to determine complex systems12,13,14.
0 thetransportpropertiesacrossthestructure. Whatispe-
5 culiar ofHTSis the complicate materialscience entering In this work we report on an investigation of magne-
5 in the formation of the physical barrier microstructure. toconductance at low temperatures for the same type of
1 This will depend on the type of device and the fabrica- biepitaxial GB junctions15, used for the MQT experi-
.
1 tion procedure. The materialscience complexity of HTS ments. These structures are very flexible and versatile,
0 may also result in different precipitates and inclusions guaranteeing on the one hand low dissipation13,14 and
9 present at interfaces and grain boundaries, and in some on the other a reliable way to pass from tunnel-like to
0 type of inherent lack of uniformity of the barriers2. All diffusive transport on the same chip by changing the in-
:
v this has turned into some uncertainty about the nature terface orientation12,15. We give direct evidence of the
i of the barrier and has originated various hypotheses on roleplayedbynarrowconductionchannelsacrosstheGB.
X
the transport properties. The most wide-spread models Thesechannelsmayhavedifferentsizesanddistributions
ar are basically all in-between two extreme ideas2,4,5,6,7,8,9: and obviously a different impact on the transport prop-
on the one hand resonant tunnelling through some kind erties. Whenincreasingappliedvoltage,mesoscopiccon-
of dielectric barrier4,5,6,7, on the other, especially in GB ductance fluctuations16,17,18,19,20 appear in our samples,
junctions, a barrier composed of thick insulating regions at low temperatures, not dissimilar from what is usually
separated by conducting channels, which act as shorts observed in normal narrow metal samples21. We expect
or microbridges8,9. In most cases the interface can be that, in our sample, typical sizes of the current-carrying
modeled as an intermediate situation between the two constrictions L and L range from 50 nm to 100 nm
x y
limits mentioned above. A transition from one extreme and, as a consequence, Thouless energy E ( see Table
c
to the other can therefore take place. What is un- I ) turns out to be quite large, when compared to the
fortunately missing is a way to describe this tuning- values usually measured in traditional normal metal ar-
transition through reliable and well defined barrier pa- tificial systems22. The mesoscopic effects persist at volt-
rameters (for instance the barrier transparency). The agesabouttwentytimeslargerthantheThoulessenergy.
predominant d-wave order parameter symmetry (OPS) ”Novel”mesoscopicissuesthatemergefromtheanalyses
2
carried out in the present work are tightly connected to gerprints of our sample in Section IIIA, we collect ev-
the natureofthe GB systems: a)asmoothcrossoverap- idences of the mesoscopic character of the conductance
pearstoexistfromthecoherentconductionmostlydriven fluctuations that we have measured. In section IIIB we
by the supercurrent, to the magnetoconductance driven derive from the ensamble average of the fluctuations the
by quantum coherent diffusion of quasiparticles across varianceoftheconductance,whichispresentedinsection
themesoscopicarea,whenthevoltageatthejunctionin- IIIC.
creases;b)inanalogytopairs,quasiparticlesalsoappear
to have a large phase coherence as proved by the shape InSectionIVweshowtheconductanceautocorrelation
of the power spectrum of the conductance fluctuations, for different magnetic fields in an intermediate voltage
up to temperatures of 3K; c) the voltage drop appears range. By analyzing the power spectral density we esti-
to be concentrated at the GB, and non equilibrium does matethephasecoherencelengthL whichisfoundtobe
ϕ
not affect substantially the mesoscopic interference over <∼ L, at intermediate voltages ( V = 14÷18 mV). The
a wide area about 1µm2. conductance autocorrelation at different voltages allows
This work builds upon a previous report, where the usto interpretthe roleofnonequilibriumbydefining the
main ideas have been illustrated23. Herein a more com- voltagedependenceofthephasecoherencelength,asdis-
plete analysis of the experimental data is carried out. cussedin Section V. The discussionof the results can be
Wehaveappliedthe”protocol”establishedinthelast20 found in Section VI. Our simple model theory well ac-
years on semiconducting and normal metal nanostruc- counts for the experimental results and clarifies the sur-
turestooursystemandwehaveextractedthecharacter- vivalofnonlocalityinthequantumdiffusioninpresence
istic lengths and scaling energies. of a large voltage bias. Table I gives additional informa-
In Section II we give some details about the sample tion on the planning of this work. Our conclusions can
fabrication. By presenting the magnetic mesoscopic fin- be found in Section VII.
TABLEI: Summary of the results:
Topic Relative measurement Extracted parameters Estimates
Junction geometry Magnetic pattern: I vs.H in Fig.(3) L ∼L ∼50nm E ∼1mV
C x y c
Transport properties RN in Fig.(4) and IcRN in Fig.(2) RN =400Ω, T >∼257mK IcRN ∼Ec/e
Mesoscopic Fingerprints Resistance fluctuationsin Fig.s(5,6) var[g]<∼1
Phase coherence length Autocorrelation vs∆H in Fig.(7), PSD in Fig.s(8,9) Lϕ <∼1µm Lϕ ∼V−1/4
Coherent phase breaking time Autocorrelation vs∆V in Fig.(10) τ ∼400ps Dτ ∼L2
ϕ ϕ ϕ
II. FABRICATION AND AVERAGE and compared with data available in literature3.
TRANSPORT PROPERTIES OF THE SAMPLE In the tilt cases J ≈ 103 A/cm2 and σ =1/(R A)
C N N
≈ 0.2(mΩcm2)−1, bothmeasuredatT = 4.2K (A is the
junctioncrosssection). TwistGBsjunctionsaretypically
TheGB Josephsonjunctionsareobtainedattheinter- characterizedby higher values of J in the range 0.1-4.0
C
facebetweena(103)YBa2Cu3 O7−δ (YBCO)filmgrown x 105 A/cm2 and σN ≈10 (mΩcm2)−1 (at T = 4.2 K).
on a (110) SrTiO substrate and a c-axis film deposited WehaveselectedYBCOgrainboundaryjunctionsand
3
on a (110) CeO seed layer (see Fig.(1)). The presence measured their I-V curves at low temperatures T, as a
2
◦
oftheCeO producesanadditional45 in-planerotation function of the magnetic field H applied in the direction
2
of the YBCO axes with respect to the in-plane direc- orthogonal to the plane of the junction. In HTS junc-
tions ofthe substrate12. The angleθ ofthe grainbound- tions, the correlation between the magnetic pattern and
ary relative to the substrate a,b axes is defined by suit- thecurrentdistributionprofilealongthejunctionismade
ablypatterninglithographicallytheCeO seedlayer(see more complicate by the d-waveorder parameter symme-
2
Fig. (1 a)). Details about the fabrication process and a try(OPS),whichgeneratesanomalousmagneticresponse
wide characterization of superconducting properties can especially for faceted interfaces1,2,3,28. Additional devia-
befoundelsewhere12,15. Theinterfaceorientationcanbe tions are expected because of the presence of the second
tuned to some appropriate transport regime, evaluated harmonic in the current-phase relation10,11. In Fig.(3)
throughthe normalstate resistanceR and criticalcur- we report the magnetic field dependence of the IV char-
N
rent density (J ). Typical values are reported in Fig.(2) acteristics of the maximum Josephson current of the of
C
3
(103) YBCO
! wave carrier
(001) YBCO
Grain boundary line
Ly
I
Lx H
Junction cross section
FIG.1: a)SketchoftheYBCObiepitaxialGBjunctionused
FIG.3: (coloronline)TheI-Vcharacteristicsasafunctionof
in this experiment. b)Geometry of the model system of the
magnetic field.
current-carryingconstriction.
approximately on an average scale of 20-30 nm, which
is remarkable if compared with most results available in
literature. Even if we cannot draw any conclusion on
the current distribution on lower length scales, we can
ruleoutthe presenceofimpurities oflargesize alongthe
widthoftheactivemicrobridge. Infact,weretheremore
than one active microbridge,the currentof eachof them
wouldaddinparallelandthepatternwouldpresentother
periodicities referring to the area enclosed between the
conduction channels (Ref.30). The I-V characteristics of
theHTSJosephsonjunctionsstillpresentfeatures,which
cannot be completely understood in terms of the classi-
cal approaches used to describe the low critical temper-
FIG. 2: (color online) The product I R is reported as a
c N
ature superconductors Josephson junctions. These are
function of thecritical currentdensity J . Data arecollected
c
from Ref.s2,3,24,25,26,27 frequentlyobservedandoftenreferredinliteratureasun-
conventional features1,2,3. Examples are3: a) I R val-
C N
uesaremuchlowerthanthegapvalue∆;b)theshapeof
thejunctionthatwehaveextensivelyinvestigatedinthis the I-V strongly depends on the critical current density;
work (with barrier orientation θ=60◦). This angle gives c) I-V curves show significant deviations from the Resis-
the maximum J 12. The magnetic response presents a tively Shunted Junction model (RSJ); d) there is a poor
C
maximum of the critical currentat zero field and two al- consistency between the amplitude of the hysteresis and
mostsymmetriclobesfornegativeandpositivemagnetic the extracted values of the capacitance, when compared
fields respectively. At higher magnetic fields (above 100 to low-TC superconductor junctions.
Gorbelow-100G),thecriticalcurrentisnegligible. The Atlowtemperatures,the resistancevs appliedcurrent
flux periodicityisroughlyconsistentwiththe sizeweex- R(I), as derived from the I-V characteristics is rather
pect for our microbridge (50-100 nm), since the London temperature insensitive, while the critical current I
C
penetration depth in the off-axis electrode is larger than maintainsasizabletemperaturedependence. Intheinset
the one in c-axis YBCO films, of the order of microns of Fig.(4) we show the resistance R(I) at zero magnetic
(see Ref.14 and29 for instance). Experimental data can field for three temperatures: 257mK, 1K and 3K. R(I)
be compared with the ideal Fraunhofer case in the crud- obviously vanishes in the Josephsonbranch and displays
est approximationwithout taking into account the pres- a sharp peak when switching to/from the finite voltage
enceofasecondharmonicoranyspecificfeatureofHTS. conductance. The data are displayed in order to show
Even if deviations from the ideal Fraunhofer pattern are the critical current at I > 0 and the retrapping current
present, they can be considered to some extent minor if at I < 0. In Fig.(4,main-panel) we show a blow up of
comparedwithmostofthe dataonHTS grainboundary the resistance R(V) in a range of voltage values V be-
Josephson junctions, which present radical differences. tween0.25mV and 30mV atzeromagnetic field andfor
We can infer an uniformity of the junction properties different temperatures T =257mK,1K,3K.
4
Measurements have been taken after different cool- v ≈7×107cm/secforoptimallydopedYBCO,wecon-
F
downsinthe time lapseoftwoyearstostudy the sample clude that the mean free path ℓ is smaller than the size
dependent properties. of the nanobridge. Therefore we argue that the trans-
The average resistance in the range of voltages V ≈ port in the junction is diffusive, which is confirmed by
10mV÷15mV hasbeenstableforabouteighteenmonths the observation of the resistance fluctuations.
at∼180Ωandhasincreasedinthelastyearuptoabout TheThoulessenergy,asderivedfromtheexpectedsize
430Ω. Thesechangesshouldbeattributedtoagingofthe of the nanoconstriction, is Ec = h¯D/L2 >∼ 1meV. This
diffusion properties at the grain boundary. However, in value is confirmed by our measurements as discussed in
the meantime, no significant change of the IC has been SectionIII.Thenumberoftransversescatteringchannels
detected. Only one sample was available with such a in the constriction for a fixed cross section A is approx-
reduced width. The steady progress in nanotechnology imately N ≈ k2A ∼ 5·104. A ∼ 100×100nm2 is
ch F
will probably lead to the realization of reliable micro- given by the product of the thickness of the film and
bridges of nominal width of a few hundred nanometers, the width of the channel. Hence, quantization of trans-
fromwhichitwillbe easierto havejunctions withtrans- verse levels in the bridge does not seem to play any role
portcarriedbyveryfew mesoscopicchannels. We finally even at the lowest temperatures investigated. Indeed,
signal a strong similarity of the I-V and dI/dV-V curves k T >> δ ∼ E /N ≈ 0.1µeV, where δ is the mean
B c ch
of our junctions with those from sub-micron YBaCuO energy level spacing. As a consequence, our system can
junctions, reported in Ref.31. be thought as a disordered bridge in the diffusive limit
ℓ<L .
x
We concentrate on the marked non periodic fluctua-
800 tions of the resistance at finite voltages, with magnetic
750 800 field in the range B =−100÷100G. An example of the
600 magnetoresistancefluctuations is reportedin Fig.(5) top
700
400 panel for T =257mK,1K,3K.
650
Thefluctuationsarenotrelatedtothemagneticdepen-
200
ΩR () 565000 0 −100 curr e n0t (µA) 100 dVen>ce2mofVth.e JosephsoncriticalcurrentIc(H) atvoltages
500 In order to avoid trapping of flux which may oc-
cur when increasing the temperature, especially at the
450
higher fields, the data shown here refer to a single cool-
400
01. 2K57 K down. Occasionally the pattern has still a slight de-
350 3 K viation from reproducibility within one single cooling
300 bath, which could be due to finite relaxation in the
70 80 90 100 110 120 130
current (µA) spin orientation of paramagnetic impurities. CONFER-
MAREThe resistancepatternderivedfromourfourter-
minal measurementdoes not show any mirror symmetry
FIG. 4: (color online) Main: zooming on the oscillations R (H)6=R (−H). Below1Kthereislittletemperature
N N
of Rvs.I above I ≈ 80µA. Inset: Resistance as a func- dependence. Theamplitudeofthefluctuationsdecreases
tion of applied bias current for three different temperatures: between T = 1K and T = 3K. They are sample depen-
257mK,1K,3K.
dent, as different cool-downs provide different patterns.
All these features, as well as the ones described below,
strengthen the conclusion that they are mesoscopic fluc-
tuations.
III. MAGNETORESISTANCE AND ThecolorplotoftheresistancefluctuationsintheI,H
MESOSCOPIC FINGERPRINTS
plane provides the fingerprints of our sample. The devi-
ation from the average δR = R−R is shown in Fig.(5)
A. Resistance fluctuations (three color-plot panels) as a color plot (color online),
for three different temperatures, 257mK,1K,3K. R is
According to what reported in the previous Section, the average resistance performed over the full range of
we figure out that most of the current in the junction magnetic fields. The pattern keeps its shape within one
substantially flows across a single nanobridge of charac- single cool down and the contrast of the colors increases
teristic size L < 100nm. Possible lack of spatial uni- inloweringthe temperature. The colorscaleis suchthat
x
formityofthe currentdistributionona scaleoflessthan thedarkredcolorreferstoresistancessignificantlylarger
20 nm does not affect the arguments developed below. thantheaverage,whilethedarkbluecolorreferstoresis-
We call x the flow direction and y the direction per- tances significantly smaller then the average. The data
pendicular to the nanobridge, as shown in Fig.(1b). have been filtered by gaussian convolution to get rid of
The diffusion coefficient D in YBCO is expected to be the underlying white noise.
∼ 20÷24cm2/sec32. By estimating the Fermi velocity Despite the small equilibrium thermal length L =
T
5
voltagedo not allowthe thermalizationofthe carriersin
the sample. Indeed, our results do not change qualita-
tivelyupto1.5K ,andtransportcanbeclassifiedasnon
equilibrium quantum diffusive because Lϕ <∼ Lx. Here
L is the phase coherence length for carriers diffusing in
ϕ
the junction area and is <∼1µm (see Section IV).
B. Ensemble average
Thevariancevar[g]istheensembleaverageofthe am-
plitude squaredof the conductance fluctuations, h(δg)2i,
where g = G/(2e2/h) is the dimensionless conductance.
FIG. 5: (color online) Color plot of the resistance fluctua-
We analyzethe fluctuationsofthe conductanceobtained
tionsδR(H,I)asafunctionoftheappliedcurrentandofthe
by averaging over runs at different magnetic fields up to
magneticfieldforthreetemperatures: 257mK,1K,3K. The
100G, at different voltages. Here we argue that this av-
right panel shows δR(H = 2.7G,I)vs.I for the three differ-
erage can be taken as an acceptable ensemble average
enttemperatures. Thethreetoppanelsshowsinglemagneto-
conductance trace for each temperature at fixed bias current and provides a bona fide information about the variance
(indicated in the labels). var[g]oftheconductanceanditsautocorrelation. Tojus-
tify our statement, we have to show that the Cooperon
contributiontothevarianceisnotsignificantlyinfluenced
¯hD/k T ∼0.14µmat1K,thereisaclearpersistence by H up to at least 100G. The variance of the conduc-
B
of the fingerprintsup to T =3K. This suggeststhat the tance atequilibrium at temperature T canbe calculated
pstrongnon-equilibriumconditionsinducedbytheapplied as33
4s2 d∆E ∆E
var[g(H,T)]= f [F (∆E,H ,H )+F (∆E,H ,H )] (1)
π4 2k T 2k T D 1 2 C 1 2
Z B (cid:20) B (cid:21) (cid:12)H1=H2=H
(cid:12)
wheres isthe spindegeneracy,f(x)=(xcothx−1)/sinh2xandFD,C arethe DiffusonandC(cid:12)(cid:12)ooperonautocorrelation
functions, respectively. They can be rewritten in terms of the eigenvalues λD,C of the diffusion equation34:
α
e 2 1 ∆E λD,C
−D −i∇~ + A~ ±A~ + −i ψ = α ψ . (2)
1 2 α α
¯hc τ ¯h τ
(cid:20) (cid:16) (cid:16) (cid:17)(cid:17) in (cid:21)
Here Dτ = l2/d ∼ 4·10−12cm2 (where d is the effective dimensionality ) and τ is the inelastic relaxation time
in
(τ >> τ ∼ 0.2psec). A ,A are the vector potentials (H and H the magnetic fields) influencing the outer and
in 1 2 1 2
inner conductance loops respectively and the +/− sign refers to the C/D propagator respectively. We have:
1 1 1
F =L−4(Dτ)2 + ℜe . (3)
D,C Xα λDα,C 2 2 λDα,C2
(cid:12) (cid:12)
At zero temperature only ∆E = 0 contributes to the int(cid:12)egral(cid:12)in Eq.(1), so that the eigenvalues become real. In the
(cid:12) (cid:12)
evaluation of the variance, H = H = H implies that the Diffuson eigenvalues become insensitive of the magnetic
1 2
field. Instead,theCooperoneigenvaluesdependon2H andcanbewritteninanalogywiththeLandaulevelsenergies.
It follows that :
3 nmax 1 3π2 H
F (∆E =0,H)∼ ∼ 1− , (4)
C 2 2 4 H
n=0 n+ 1 + 1 (cid:18) o(cid:19)
X 2 ωHτin
(cid:16) (cid:17)
where ω =4eHD/¯hc. Eq.(4) defines a decay threshold from the truncation of the sum over the orbital quan-
H
field of the Cooperon H ∼ πhc/(12eDτ), which derives
o
6
tum number n at n ≈ ¯h/mω l2 (ω = eH/mc is than those in normal constrictions. Fluctuations
max c c
the cyclotron frequency). This limitation is required by survive up to voltages which are many times the
quantum diffusion (hr2i > l2). The large value of Thouless energy.
nmax
n , In our case (∼ 102 for H < 600G), determines
max
Ho >∼ 2.5Tesla which is far beyond the field strengths Here we focus on the variance var[g] and on the auto-
correlation of the magnetoconductance as a function of
that can be applied to our sample without trapping flux
voltage at low temperatures up to V >> E /e ∼ 1mV.
due to vortices. This confirms that averaging over the c
Our data can be interpreted on the basis of models for
interval of H values H ∈ (−100÷100G) is equivalent
the quantum interference of carriers transported in the
toasampleaverage,withoutintroducingsignificantfield
narrow diffusive channel across the GB line. The ex-
dependencies. In the followingthe ensambleaveragewill
perimental findings are consistent with a large Thouless
be denoted by the symbol h...i . As it is shown in the
H
energy E and quite long dephasing times τ . A com-
next Section, the typical magnetic field scale that arises c ϕ
parison of the data with the results of our models seem
from the autocorrelation of the conductance is ∼ 10G,
to confirm that non equilibrium effects induced by the
much smaller than the interval overwhich the averageis
voltage bias V are not the source of heavy energy relax-
performed.
ation of the carriers, even at voltages V >> E /e. A
In the rest of the paper, we will generically denote c
discussion about the voltage dependence of the variance
the conductance autocorrelation, which is an extension
ofthe conductance forlargevoltagescanbe foundinthe
of Eq.(1), by K . This quantity depends on many vari-
g
Section VI. In the next Section we analyze the conduc-
ables: T,H ≡ (H + H )/2,∆H ≡ H − H ,V ≡
1 2 1 2
tance autocorrelation at finite voltage in some detail, to
(V +V )/2,∆V ≡ V −V . When no ambiguity arises,
1 2 1 2
extractinformationaboutthephasecoherencelengthL
we have taken the liberty to list just the parameters rel- ϕ
and the phase coherence breaking time τ .
evant to the ongoing discussion, in order to simplify the ϕ
notation.
IV. SAMPLING NONLOCALITY:
AUTOCORRELATION vs ∆H
C. Variance of the conductance and different
voltage regimes
The variance var[g], of the conductance fluctuations
h(δg)2i discussed in Section III.C can be derived from
The conductance is derived from the I/V character- H
the maximumat∆H ≈0 ofthe moregeneralautocorre-
istic. We have checked the behavior of the differential
lation function:
resistance,measuredthroughastandardlock-inmethod,
and we have found qualitatively similar results.
K (V,∆H)≡hδg(V,H +∆H)δg(V,H)i (5)
In Fig.(6) (upper panel ) we have reported the mea- g H
suredconductancefluctuations vs voltagebiasandmag-
The data havebeenaveragedoverH, asusual,as wellas
neticfieldatthetemperatureT =257mK,inagreyscale
overasmallintervalofvoltagevaluesaboutV. K isthe
plot. The plot shows two different regimes: g
sumofCooperonF andDiffusonF contributions. Due
C D
tothe independence ofourresultsofthetemperature we
a) Low voltages ( V < 3mV), where fluctuations ap-
considerthezerotemperaturelimitofEq. (1). InFig.(7)
pear to be very high. Fluctuations in this range
we plot the measured autocorrelation of the dimension-
mostly arise from precursive switching of the cur-
lessconductancevs. ∆H atT =257mK forthreevalues
rent out of the zero voltage Josephson state. The
ofV: 10mV(blue curve),14mV (redcurve)and18mV
analysis of this range of voltages is better dis-
(redcurve)(coloronline). The datahavebeenaveraged
cussedwithintheMacroscopicQuantumTunneling
dynamics13; it requires full account of the super- overavoltageintervalofwidthδV =0.5mV (the results
do not depend on this choice).
conductivecorrelationsandisnotaddressedinthis
Similar curves have been measured for normal metal
paper.
wires at zero voltage bias22.
b) Large voltages (V >5mV) . In this regime we ob- The autocorrelation of Fig.(7) is practically insensi-
serve some reproducible, non periodic and sample tive to increasing temperature up to about 1.5K. This
dependent fluctuations. The variance of the con- fact can be viewed as evidence that a significant contri-
ductance < (δg)2 > , is plotted vs voltage bias ( bution to transport and to the conductance fluctuations
H
Fig. (6) bottom panel) for two temperatures. The is still provided by the pair current. Its time average
scale for its magnitude is estimated according to and absolute value can be seen as rather temperature
δR/R=δg/g ≈0.01withR=410Ω(T =300mK). and voltage independent, at fractions of Kelvin, while
The variancevar[g] stabilizes aroundunity at V >∼ quasiparticlesremainratherfrozen,providedthe voltage
7mV. As the voltage increases V >∼ 18 mV, the does not increase too much35. For weak links, charac-
variance is increasingly reduced. However, small terized by and higher barrier transparency the contribu-
amplitude fluctuations seem to persist over a wide tion of the contribution of the supercurrent in the I-V
voltage range up to values which are by far larger curve can be relevant at finite voltages36. The physical
7
1.5 1
V=10mV
V=14mV 0.5
V=18mV
H)> 1 0
g(
δH) 0.5 -0.5
∆
+ -100 -50 0 50 100
H
g(
δ< 0
-0.5
-100 -50 0 50 100
∆H(Gauss)
FIG. 7: (color online) Autocorrelation < (δg)2 > vs ∆H at
T = 257mK for three different values of the voltage drop:
V = 10mV (black curve), V = 14mV (red curve), V =
18mV (green curve). Inset. Resultsofthemodelcalculation
sketched in the text for ξ = 1.05 (black curve), 1.15 (red
curve),1.4 (green curve).
particle contributions to the current which still depends
on ϕ.
TheinsetofFig.(7)showstheresultofasimplemodel
calculation of the autocorrelation function based on the
following assumptions: 1) negligible proximity effect in
the sub-micron bridge induced by the superconducting
contacts; 2) an equilibrium approach to transport, in
whichthecurrentismostlyphasedependent;3)handling
of the magnetic field H is treated as a small correction
andthereforeH onlyappearsinthegaugeinvariantform
of the phase difference.
The model (see Ref.41 for details) uses as unique fit-
ting parameter ξ = L /L , where L is the transverse
y ϕ y
sizeoftheconductionchannel. Thecurvesplottedinthe
FIG.6: (coloronline)Toppanel: grey-colorplotofthefluctu-
inset are with ξ =L /L =1.05 (black curve), 1.15 (red
ations of the dimensionless conductance as a function of the y ϕ
curve), 1.4 (green curve). The three different measured
voltageVandoftheappliedmagneticfieldHatT =257mK.
curves in the main plot of Fig.(7) refer to different bias
Bottom panel: Variance of the dimensionless conductance as
voltages and cannot be directly compared to the theo-
a function of the voltage V around zero magnetic field for
T=257mK and T=3K. retical curves in the inset of Fig.(7). The qualitative
agreement between experimental and theoretical curves
is evident, providedwe assume that ξ increases,with in-
reason is that the phase changes in a sharply nonlinear creasing voltage. This assumption is feasible since, on
mannerwiththegreaterpartoftheperiodbeingcloseto the one hand Lϕ is likely to be reduced when increasing
π/2+2nπ. Inaddition,non-equilibriumeffects37 andun- applied voltage, and the number of conduction channels
conventionalorderparametersymmetry(withanotneg- increases by changing the voltage and,as a consequence,
ligible second harmonic component in the current-phase the effective width of the bridge. If we assume that Lϕ
Josephson relation38) are possible additional sources of scales with voltage as V−1/4 ( see discussion in Section
supercurrent flowing at finite voltage. Our results seem V ), we find that the ξ’s, that have been chosen to draw
to confirm the presence of non negligible contribution of the inset of Fig.(7), are consistent with the voltages of
supercurrent at large voltages, from a different perspec- the experimental curves within 15% of error.
tive. This is consistent with Ref.s 36,37,38 and possibly The qualitative fit, based on the simple theoretical
withtheobservationoffractionalShapirostepsonYBCO model used here, gives evidence of the fact that non-
grain boundary Josephson junctions39,40. equilibriumdoesnotseemtospoiltheautocorrelationas
According to the remark made above, we can assume a function of the magnetic field, even if the voltage bias
that the currentat T ∼0 is only a function of the phase exceeds the Thouless energy: eV > E . While the de-
c
difference ϕ between the two superconducting contacts. phasingtime τ is discussedin the nextSection, here we
ϕ
This assumes little dephasing induced by inelastic scat- are in position to extract the value of the phase coher-
teringprocesses,butnotnecessarilytheabsenceofquasi- ence length L from the Power Spectral Density (PSD)
ϕ
8
of the conductance autocorrelationfunction.
2
10 257 mK
1 K
uss) 5 3 K m)
ga µ
dB 0 L (φ
g) (
δ
D( −5
PS 1
−10 0.9
4 10 20
0 0.2 0.4 0.6 0.8 bias(mV)
f(gauss−1)
FIG. 9: Logarithmic plot of L vs V derived by using the
ϕ
FIG. 8: (color online) PSD at T= 257 mK, 1K, 3K and V =
eq.(6). Full lines are a guide to the eye. The slope of the
7mV, averaged over a voltage interval of 0.5mV. The curves straight line is −1/4.
havebeen shifted for clarity.
0.8
The Powerspectral density (PSD) of the conductance ) 0.6 c)
autocorrelationisplottedinFig.(8)vsf ,theconjugate Ω
H k0.4
variable to the magnetic field ∆H , for data at T = 273 (
R0.2
mK, 1K, 3K and V= 7 mV. f has the dimension of an
H 0
inverse magnetic field. Inspection of Fig.(8) shows that 10 20 30
bias(mV)
there is a linear slope at small frequencies and a roughly 4
flat trend at larger frequencies. The latter is due to the V
∆ a) b)
whitenoiseaffectingthemeasurement. Curveshavebeen H,
>
shifted,inthefigure,forclarity. Inreality,boththelinear )
V 2
slope and the constant value are almost independent of (
g
the temperature. The linear slope allows to extract the δ
)
value of the phase coherence length L according to the V
ϕ ∆
fit42: + 0
V
(
log(PSD/PSD(f =0))=−2πf H +cnst , (6) g
H H c δ
<
-2
as correlation field Hc can be related to Lϕ as follows: -4 -2 0 2 4 -4 -2 0 2 4
L ∼ hc/2eH . From this plot we derive H ∼ 10G, ∆V(mV) ∆V(mV)
ϕ c c
that gives Lϕ <∼ 1µm. A logarithmic plot of the V de-
p
pendenceofL (V)isreportedinfig(9). Thestraightline FIG. 10: ( color online) The measured autocorrelation of
ϕ
drawn among the experimental points shows the func- theconductanceKg(V,H =0,∆V)vs. thevoltagedifference
tional dependence V−1/4. ∆V, for average voltages V ≈7.5,12.5,17.5,25.0mV ( panel
a)); theoretical fit based on Eq.(10) (panel b)); Resistance
Rvs.V (panel c)). There is correspondence between the
τ colored rangesofpanel c) andthecolors ofthecurvesofthe
V. NON EQUILIBRIUM EFFECTS ON ϕ panels a) and b).
In this Section we study the dependence of the auto-
correlation function on V and ∆V:
applied voltage V. We stress that the applied voltage is
Kg(V,∆V)≡hδg(V +∆V,H)δg(V,H)iH . (7) larger than the two natural energy scales: Ec/e∼1mV
and the nominal superconducting gap ∆/e ∼ 20 mV.
A voltage average has been performed over inter- The tail of the curves shows a large anti-correlation dip
vals centered at V, of typical size lower than the when ∆V increases and damped oscillations. The auto-
min{E ,∆V}. correlation maximum flattens when the applied voltage
c
In Fig.(10a) we report the experimental autocorrela- V increases. Fig.(10 c) emphasizes the voltage range to
tion function K (V,∆V) vs ∆V for variousvalues of the whicheachcurveofthe panelarefers,byusingthe same
g
9
color. and of the parameters τ /τ and (L/L )3. Here
ϕ C/D ϕ
We have reproduced the same trend of the data in τ−1 =e2L2D(H ±H )2/(12h¯2c2) (with the + (-) sign
C/D y 1 2
Fig(10 a),by assuming that proximity effect induced by
for the C (D) case ) is the relaxation time induced by
thesuperconductingcontactsdoesnotplayanimportant the magnetic field45. In the limit of vanishing ∆V, the
role in the transport and by adapting the quasi- one-
result for the variance is recovered, which, up to numer-
dimensionalnon-equilibriumtheoryofRef.(43,44)toour
ical factors, is given in terms of the ratio between the
case (see Fig(10 b)). We will report on the derivation of
Airy function Ai(u) and its derivative with respect to
ourtheoreticalresultselsewhere41. Wejustmentionthat, ′
the argument Ai(u):
our nonequilibrium approach gives rise to an autocorre-
lationK (V,∆V) whichis a functionofx=e∆VL2/¯hD
g
Ai 2τϕ
K (V,∆V =0)=− DτϕLϕ ℜe τν . (8)
g (cid:18) L3 (cid:19) ν=XC,D Ai′(cid:16) 2ττνϕ(cid:17)
(cid:16) (cid:17)
τ ,L and τ /τ all depend on the voltage. This for- Eq.(10)wouldbetheresultwithinlinearresponsetheory
ϕ ϕ ϕ ν
mula is similar to the thermal equilibrium result by Alt- only, except for the V dependence of L (V).
ϕ
shuler, Aronov, Khmielnitski (AAK)46, which includes The globalinterpretationof the data givenhereshows
the dephasing induced by e-e scattering with small en- thatamonotonousdecreaseofτ withincreasingvoltage
ϕ
ergy transfer. In particular AAK find is not achieved. A non monotonousdecrease ofvar[g] vs
V isindeedfound,ascanbeseenfromFig.(6). According
1 e2k T
τϕLϕ = ¯h2Bσ (9) tdoectrheeacsoinrgrestpreonnddeonfceKDg(τVϕ,=∆VL)2ϕ,waintdhtioncFriega.(s9in)gavgoenlteargael
couldtakeoveronlyabove20mV.InEq.(8)derivedfrom
whereσistheconductivity. Atintermediateappliedvolt-
ages,ourzerotemperatureprefactorDτ L /L3inEq.(8) our model calculation, we have found h¯/τϕ in place of
ϕ ϕ
looks similar to the AAK prefactor h¯DL /(3πL3k T) if kBT appearing in the AAK result of Eq.(9). At larger
ϕ B voltages the expected substitution in Eq.(9) is44 k T →
¯h/τ replaces k T. B
ϕ B
eVL /L and, by requiring the consistency,
By choosingan appropriatevoltagedependence of the ϕ
fitting parameter τ /τ and L /L, we find that the au-
ϕ ν ϕ 4
tocorrelation scales with Lϕ(V) as follows: Lϕ ∼ Dτ | → Lϕ = D ¯hg. (11)
L ϕ eVLϕ/L L L2 eV
∆V τ L 3 q (cid:18) (cid:19)
ϕ ϕ
K (V,∆V)≈ F , , , (10)
g V τ L where g = h¯σ/(e2L) in 1−d. This would give a decay
ν=XC,D " c ν (cid:18) (cid:19) # law for the coherence length Lϕ ∼ V−1/4, which is not
with Dτ = L2 and τ = τ , V = h¯/τ . The explicit clearly recognizable in our experiment.
ϕ ϕ C D c ϕ
form of the function F will be given elsewhere41. Its
limiting form for ∆V →0 gives Eq.(8). This scaling law
VI. DISCUSSION
is exploited to plot the curves of Fig(10b).
The scaling among the blue, red and cyan curves re-
producesreasonablywellthe experimentalpattern. This Werecollectherethemainexperimentalfactsthatcan
indicates that the V dependence of eq.(10) is well ac- beextractedfromourdataregardingquantumtransport
counted for by simply reducing L with increasing V. in a GB YBCO JJ .
ϕ
On the contrary, the black curve at the lowest voltage Themagneticdependenceofthemaximumcriticalcur-
V ∼ 7.5mV requires adjusting the prefactor after the rent suggests an active transport channel of the order
scaling, to make the central peak higher and narrower. of 50-100 nm. Uniformity of the critical current is on
Thiscouldbe ahinttothe factthat,whenthe voltageis scales larger than about 20 nm. The Thouless energy
rather low, the superconducting correlations may be rel- E =h¯D/L2 turns outto be the relevantenergy scale in
c
evant and should be included in deriving the functional thiscase. ThenormalresistanceR oftheHTSjunction
N
form of the F function. Needless to say, an increase of is of the orderof 200Ω,increasingup to 480Ωwith time,
the τ /τ parameter implies a reduction of the value of due to aging of the sample. The zero field Josephson
ϕ ν
var[g]. critical current appears to satisfy I R ∼ E /e. This
C N c
In writing Eq.(10), extra contributions arising from product is definitely much smaller than ∆/e, where the
the non linear response have been neglected. Actually, nominal superconducting gap is ∆= 20meV. This rep-
10
resents additional evidence that the proximity effect in- by the experiment, non-equilibrium cannot be ignored.
ducedinthebridgeintheabsenceofappliedvoltageisof Our derivation extends the calculation of Ref.(43,44).
mesoscopic origin. The superconductive pair coherence We give a simple estimate of the conductance autocor-
length ξs < Lϕ <∼ L, as opposed to the classical regime, relation to fit our experiments. We invoke the sim-
L<ξ , when the tailof the orderparameterenters both plest non-equilibriumdistribution for diffusing quasipar-
s
superconductorsofthejunction47. Wespeculatethatthe ticles, that is the collisionless limit51, by lumping the
oscillationsintheresistanceasafunctionofV, shownin relaxation processes in the damping parameter of the
the inset of Fig.(4), could be due to this mesoscopic ori- Cooperon/Diffuson propagators τ . The dependence
C/D
gin. on the applied voltage is introduced by tuning τ . We
ϕ
Remarkableconductancefluctuationshavebeenfound obtainoscillationsinthenegativetailoftheautocorrela-
inavoltagerangeupto20Ec inthemagneticfieldrange tion, Kg(V,0,∆V) (see Fig.(10)) and our scaling proce-
ofH ∈(−100,100)Gfortemperatures below3 K.Inthe dure fulfills the relation Dτϕ = L2ϕ. Extra contributions
explored window, we do not measure an halving of the that are specific of the non equilibrium theory and are
variance with increasing field H48. The crossover field known to be responsible first for a linear increase of the
H at which the Cooperon contribution to the variance autocorrelationfunction with V andsubsequently for its
o
is expected to disappear, as givenby Eq.(4) is estimated power-law decay are not included here.
ofthe orderoffew Teslas48. Decoherenceinducedbythe
Zeeman energy splitting requires even larger fields.
Transport has been measured in highly non-
equilibrium conditions. Hence the temperature depen-
VII. CONCLUSIONS
dence is quite weak up to T ∼ 1.5K. We have concen-
trated our analysis in the voltage range V ∈(7,30)mV,
where the conductance fluctuations reach a steady value We have reported about transport measurements of
for the variance at T below 1 Kelvin, var[g](V) <∼ 1. high quality biepitaxial Grain Boundary YBCO Joseph-
Thesepropertiesconfirmthatthefluctuationsaredueto sonJunctionattemperaturesbelow 1Kelvin, performed
quantum coherence at a mesoscopic scale. over a time period of about 18 months. A global
The PSD of the autocorrelation of the conductance view on the data offers a consistent picture, pointing
at different fields ∆H allows to identify H ≈ 10G as to transport across a single SNS(superconductor nor-
c
the field scale for the mesoscopic correlations, weakly mal superconductor)-like diffusive conduction channel
dependent on V. This value leads to a phase coher- of mesoscopic size L <∼ 0.1 µm. We have mostly ex-
ent length Lϕ <∼ 1 µm. We have plotted the coher- plored the magnetoconductance fluctuations in the volt-
ence length extracted from the autocorrelation PSD vs agerangeeV >>Ec >>kBT,wheretheThoulessenergy
V in Fig.9 and compared it with Lϕ(V) ∼ (V/Vo)−s Ec ∼1meV. The Thouless energy, 2−3 orders of mag-
with s ∼ 0.2544. A similar exponent has been found nitude larger than the one usually experienced in nor-
in a limited range of voltage bias V > Ec/e in gold mal mesoscopic or low-Tc superconducting samples, de-
samples49, in which UCF (universal conductance fluctu- termines qualitatively the quantum coherentdiffusion in
ations)andAharonov-Bohmoscillationswerefound. De- the channel. We believethatthe oscillationsintheresis-
phasing mechanisms are low frequency electron-electron tance that canbe seenin Fig.(4) canbe due to quantum
interaction, magnetic impurity-mediated interaction50 diffusion.
and non equilibrium quasiparticle distribution51. Ac- The mesoscopic correlations are found to be quite ro-
cording to Fig.9, the comparison is not conclusive. As bust in our GB narrow channel, even at large voltages.
a matter of fact all data of conductance autocorrelation This could not occur if the lifetime of the carriers were
at finite voltage reportedfor normalwires22,49 identify a strongly cut by non-equilibrium relaxation. We con-
Thouless energy Ec ∼ 1µeV, three orders of magnitude clude that mesoscopic effects deeply involve supercon-
smallerthaninourHTSdeviceandrefertoappliedvolt- ducting electron-electron correlations, which persist at
ages not larger than mV’s. Still, mesoscopic coherence larger voltages. Transport features due supercurrents
persists in our sample, up to voltages much larger than and quasiparticles at finite voltages cannot be disentan-
the Thouless energy. We do not find any linear increase gledinthepatternoftheconductance,norinitsvariance.
of the conductance autocorrelationwith voltage at large This consideration has led us to approach the problem
voltages43. with a non-equilibrium model calculation for generic co-
Our model calculation appears to reproduce the gross herent transport, which highlights the role of the phase
features in the dependence of the conductance autocor- breakingtimeτ ,withoutincludingsuperconductingcor-
ϕ
relation Kg(V,∆H,∆V) on ∆H as well as on ∆V. In relationexplicitly. Fig.10showsthe comparisonbetween
the case of Kg(V,∆H,0) we limit ourselves to the linear our model results and the autocorrelation experimental
responsetermonly andthe effect ofthe voltagebiasjust data,whichisencouraging. Theremarkablylonglifetime
appeared as a small reduction of Lϕ(V) with increasing of the carriers, which we find, appears to be a generic
V. propertyinhigh-T YBCOjunctionsasprovedbyoptical
c
To model the trend of K (V,0,∆V) vs ∆V given measurements32andMacroscopicQuantumTunneling13.
g
