Spatially-resolved probing of a non-equilibrium superconductor

0.55 MB·English

by  K. Yu. Arutyunov

Spatially-resolvedprobing ofa non-equilibrium superconductor
K. Yu. Arutyunov,1,2, H.-P. Auraneva,1 and A. S. Vasenko3,4,5
∗
1UniversityofJyva¨skyla¨, DepartmentofPhysics, NanoScienceCentre, PB35, 40014Jyva¨skyla¨, Finland
2Nuclear Physics Institute, Moscow State University, 119992 Moscow, Russia
3Institut Laue-Langevin, 6 rue Jules Horowitz, BP 156, 38042 Grenoble, France
4LPMMC,Universite´ JosephFourierandCNRS,25AvenuedesMartyrs,BP166,38042Grenoble, France
5DonostiaInternational PhysicsCenter(DIPC),ManueldeLardizbal 5,E-20018SanSebastia´n, Spain
(Dated:February1,2011)
1
1 Spatiallyresolvedrelaxationofnon-equilibriumquasiparticlesinasuperconductoratultra-lowtemperatures
0 wasexperimentally studied. Itwas found that thequasiparticle injectionthrough atunnel junction resultsin
2 themodificationoftheshapeoftheI-Vcharacteristicofaremote‘detector’junction. Theeffectdependson
thetemperature, injection current and proximity tothe injector. Thephenomena can beunderstood interms
n
ofcreationofquasiparticlechargeandenergydisequilibriumcharacterizedbytwodifferentlengthscalesL
Ja 5m mandL T 40m m.Thefindingsareingoodagreementwithexistingphenomenologicalmodels,whiQle∗
∼ ∗ ∼
moreelaborated microscopictheoryismandatory fordetailedquantitativecomparison withexperiment. The
1
resultsareoffundamentalimportanceforunderstandingelectrontransportphenomenainvariousnanoelectronic
3
circuits.

]
n PACSnumbers:74.25.Fy,74.40.+k,74.50.+r,74.78.Na
o
c
- I. INTRODUCTION we describe our setup and the experimental conditions. In
r
p Sec.IIIwediscussourexperimentaldata. Wealsoformulate
u aphenomenologicalmodelwhichisinagoodagreementwith
Conversionof electric currentat an interface betweendif-
s ourfindings. Themodelenablesusto obtainthe energyand
ferent materials is a common process in any electric circuit.
.
t chargemodesrelaxationlengths. Finally, we summarize the
a At nanoscales the whole system might behave as an inter-
resultsinSec.IV.
m face if the dimension(s) are comparable to the characteristic
relaxation length. Of particular interest are boundaries with
-
d a superconductorwhereelectric currentconvertsfromsingle
n electronstoCooperpairs. Inasuperconductoratafinitetem- II. EXPERIMENT
o
perature there are always non-paired electrons called equi-
c
[ librium quasiparticles. In the presence of additional distur- Multi-terminalnanostructures(Fig. 1) were fabricated us-
bancetheirconcentrationcanbeincreasedbynon-equilibrium ing electron beam lithographyand UHV evaporationof alu-
2 quasiparticles. Iftheseexcitationsoriginatesolelyfrompair- minum(S)andcopper(N)separatedbynaturally-grownalu-
v
breakingtheyequallypopulateexcitationspectrumwithmo- minumoxide(I). The sample layoutis conceptuallyclose to
9
5 mentapQ>pF (electron-like)andpQ<pF (hole-like),where the layout, described in Refs. 5 and 14. Electrons were in-
2 pF is the Fermimomentum. Such a perturbationcontributes jectedintothesuperconductorthroughalarge-areaNISinjec-
2 tothelongitudinal(orenergy)mode(L)contrarytotransverse tor with typical tunnel resistance 3 kW . The induced dis-
8. (orchargeimbalance)mode(T)wherethetwobranchesofthe equilibriumwas measuredbyrem∼otedetectorjunctionswith
0 excitation spectrum are non-equally populated.1 Within the typicaltunnelresistance 50kW . Evenatthelowestvalues
0 Keldysh formalism one can define the non-equilibrium dis- oftheutilizedtunnelresis∼tancesonecanneglecttheproximity
1 tributionfunctions fL,T.2Theusualelectrondistributionfunc- effect.
v: tionis2f =1−fL−fT. The 3He4He dilution refrigerator was located inside the
i These phenomena have attracted attention in mid-70s re- electromagneticallyshieldedroomwithonlybattery-powered
X sulting in an impressive number of papers.3,4 Those early analogue pre-amplifiers inside. Typical measuring set-up
r experiments were mainly performed on sandwich-type flat (Fig.1, toppanel)consistedoftwo circuits: injectorandde-
a
structuresnotadequateforspatially-resolvedstudies. Agree- tector. Fordetectiontwocomplementaryconfigurationswere
mentbetweenexperimentandtheorywasestablishedreliably used: either a single NIS junction, or two connected in se-
mainly in the high temperature limit T Tc. The opposite ries NISIN junctions from both sides of the superconduc-
→
limit T Tc has been poorly investigated. Recently the in- tor. The latter was found to be less sensitive to dc offsets
terest to≪the problem increased,5–8 while the understanding presumablyoriginatingfromparasitic thermoelectriceffects.
is still far frombeingsatisfactory. In additionto the general Carewastakennottoincreasethetemperatureofthesystem
interest in physics of superconductivity, relaxation of quasi- by the Joule heating at the injector by making the electrode
particleexcitationsisabottleneckprocessinmanynanoelec- rathermassive(Fig.1, bottompanel). AnextraNISjunction
tronic applications: solid state coolers,9–11 cold/hot electron (marked ‘Thermometer’ in Fig. 1 and located 1 m m away
bolometers,12 and Cooper pair boxes.13 Experimental study fromthe injector junction)was used to measur≃e the electron
ofthecorrespondingphenomenaisthesubjectofthepaper. temperatureof the injector Ti(I) by fitting the experimental
e i
The paper is organized as follows. In the next Section, I (V ,I =const)dependencieswiththetext-bookexpression
T T i
2
I 1m Atheelectrontemperatureoftheinjectorincreasedby
i
d T≃i Ti T 100 mK. The electron-phononinteraction is
rathee≡rweea−katu≃ltra-lowtemperatures,9andthesuperconduct-
ing‘body’ofthesample,thermallyanchoredtothesubstrate,
has low thermal conductivity. Hence, the phonon tempera-
ture of the remote detectors should not deviate significantly
from the bath temperature T. On the contrary, the electron
temperature Td of the normal detectors, determined by fit-
e
tingthezero-injectionI (V ,I =0)dependency,wasalways
d d i
higher than the base temperature T. The effect originates
fromthe poorthermalcouplingtypicalforultra-lowtemper-
atures and from inevitable electron heating coming from the
noisy EM environment. At a base temperature T 20 mK
≃
and zero injection I = 0 the absolute value of the detector
i
electrontemperatureoffsetd Td Td T variedfrom 10 to
e ≡ e −
40 mK depending on the coupling of the particular junction
with the measuring circuit. As the detectors are decoupled
bythetwoNISjunctionsfromtheremote‘hot’injector,itis
reasonable to assume that for the given detector its electron
FIG.1: (Coloronline)Top: schematicsofthesampleandmeasure- temperatureTd shouldnotdependontheinjectioncurrentI:
e i
ments.Electrontemperatureofthenormalmetalinjectorismeasured Td(I >0,T) Td(I =0,T).
withtheNISjunction‘Thermometer’. Thequasiparticlerelaxation e i ≃ e i
The TS denote the superconductor electron temperature.
inthesuperconductinglongbarisdeterminedbythe‘Detectors’:ei- e
Formally,itshouldbedeterminedtakingintoaccountacom-
ther a single NIS junction, or a pair of connected in series NISIN
junctions from the opposite sides of the superconductor. Bottom: plicatedenergybalance.9 Fromtheverybasicconsiderations
scanningprobemicroscopeimageoftheSi/SiOxsubstratewithtwo onemayexpectthatatnon-zeroquasiparticleinjectionsIi>0
closestdetectorpairsandtheinjector(leftmostmassiveprobe).Typ- the actual(thermodynamic)temperatureofthe superconduc-
icalthicknessof thealuminum wire, copper detectors, andinjector tor TS, which enters into the non-equilibrium distribution
e
are25,40and80nm,andthewidth400,180and1600nm,respec- function fS(E,TS), should be higher than the bath temper-
T e
tively.Criticaltemperatureofthealuminumwireisabout1.35Kand ature T. However, within the phenomenological formalism
thelowtemperaturemeanfreepathℓ 20nm. employedinthepaper,TS cannotbedetermined. Moreelab-
≃ e
oratedmicroscopicanalysis(toourbestknowledge,currently
absent)shouldbeusedtodeterminethenon-equilibrium(and
forthetunnelcurrentofaNISjunction.HereandbelowIi(Vi), - essentially asymmetric with respect to the chemical poten-
Id(Vd), and IT(VT) are the I-V characteristicsof the injector, tial) distribution function. Otherwise, any arbitrary value of
detector,andthermometertunneljunctions,respectively(see TS substituted into a symmetric (e.g. equilibrium) distribu-
Fig.1,toppanel). OncetherelationTei(Ii,T =const)wases- tieonfunction fS(E,TS)givesexactlythesametunnelcurrent
T e
tablished, the corresponding ‘Thermometer’circuit was dis-
ofaNISjunction.
connectedfortherestoftheexperiments.
Finally,theT characterizesthereductionofthesupercon-
∗
ductinggapD . Whenthe superconductinggapis suppressed
by the quasiparticle injection one can use two complemen-
III. RESULTSANDDISCUSSION tary descriptions: either to deal with the bare experimen-
tal data D (I), or to convert the non-equilibrium energy gap
i
To avoid confusionit is instructiveto summarize the defi- intoaneffectivetemperatureT∗ usingconventional(equilib-
nitions of various parameters of the dimensionality ‘temper- rium)BCSexpression. InthelattercaseT∗ justmeans: what
ature’ hereafterused in the paper. The T standsfor the bath would be the equilibrium temperature of the superconductor
temperature of the system measured by the two RuOx resis- T =T∗ resulting in the corresponding reduction of the gap
torsthermallyanchoredto themixingchamberandthe mas- D (TeS,Ii>0)=D BCS(T∗,Ii=0). Itshouldbeemphasizedthat
sivecoppersampleholder.TheRuOx sensorswerecalibrated T∗ isonlyaconvenientparameterofdimensionality‘temper-
against the nuclear orientation thermometer, and during the ature’andhasnodirectrelationtothe(non-equilibrium)ther-
experimentstheirreadingsdifferlessthanbyfewmK. modynamictemperatureTS. As it will be shown below, the
e
TheTiandTd denotetheelectrontemperaturesofthenor- impactofquasiparticleinjectiononasuperconductorisman-
e e
mal metal injector and detectors, respectively. The Ti and ifold. Inparticular,thelongitudinal(energy)modecannotbe
e
Td are determined by fitting the corresponding I(V) and describedsolelybythereductionofthegap,oralternatively-
e i i
Id(Vd)experimentaldependenciesbythewell-knownequilib- theeffectivetemperatureT∗.
riumexpressionforthetunnelcurrentofaNISjunction. The Injectionofnon-equilibriumquasiparticlesintoasupercon-
Ti depends on the injection current, Ti(I), due to the triv- ductorresultsinadeviationofitsdensityofstates(DOS)N
e e i S
ial Joule heating of the normalmetal injector. At the lowest anddistributionfunctions fS fromequilibrium. Undercer-
L,T
bathtemperatureT 20mKandthehighestinjectioncurrents tainassumptionsthedistributionfunctioncanbefoundbyde-
≃
3
FIG.3:(Coloronline)DeparingparameterG vs.injectionenergyfor
threedetectorsatL =0.8( ),2.8((cid:3))and7.8(⋆)m m. Leftinset:
id
sampleC2NISdetector lo△cated atLid =7.8 m m G (Ii)vs. D (Ii)at
various injection currents Ii at T =18 mK ( ) and T =200 mK
( ).Rightinset:sampleC2G vs.injectionen(cid:13)ergyatL =2.8m mat
id
△
T =17.5((cid:3)),200((cid:4))and400mK(⊞).
the transverse mode excitations. For the analysis of the ex-
perimentaldata we utilized the simplified phenomenological
approachwiththreefittingparameters: DOSbroadeningG ,16
superconductinggap D , and effective quasiparticle chemical
potentialm .17,18 Allparameterscanbedefinedbyfittingex-
∗
perimentalI (V )or/anddI /dV dependencies. Itshouldbe
d d d d
FIG.2:(Coloronline)(a)SampleC8Id(Vd)characteristicsatvarious noted that the utilized phenomenologicalanalysis, based on
injectionsforthedoublejunctionNISINdetectorlocatedatL =0.8 thepostulatedequilibriumfunctionalformofthedistribution
id
m m away from the injector. Inset: the corresponding dId/dVd de- function fS(E,TeS,m ∗),disablesdeterminationofthe(thermo-
pendencies. SlightasymmetryofthedId/dVd athighinjectionspre- dynamic)temperatureof the superconductorTeS. Moreelab-
sumably originatesfromthenot perfect equalityof theeachof the orated model (to our best knowledge - currently absent) is
NISjunctionsconnetced inseries. (b)SampleC2NISdetectorlo- mandatory to deduce the non-equilibrium f (E,TS,m ) and
catedatLid=2.8m mexperimentaldId/dVddependenciesatT=18 1 thecorrespondingTS. S e ∗
mK atthreeinjectionseVi/D i=0((cid:13)), 10(△) and20 ((cid:3)), and±at As the injected neon-equilibrium quasiparticles relax on a
T =200(⋆)andT =400(♦)mKatzeroinjections.Solidlinesare
certainlength,thefittingparametersshoulddependonthein-
the theoretical best fits with the fitting parameters indicated in the
jectionrate(energyeV orcurrentI =VGi )andthedistance
plot. i i i NN
betweenthedetectorandthe injectorL . TheI (V )depen-
id d d
dency of a NIS detector in the presence of non-equilibrium
convolutionofthetunnelingI-Vdependencies.15Inmoregen- injectionis
eral cases, self-consistent solution of Keldysh-Usadel equa- ¥
ttiaoknesthiserseiqmupirleifide.2dFaoprprtohaechintpeorpstrueltaattiinogn eoqfutihliebrrieusmultfsunwce- Id =(GdNN/e)Z0 NS(E,D ,G )(cid:2)fTN−fTS(cid:3)dE, (3)
tionalformsoftheDOSanddistributionfunction fS,butwith where Gd is the normal state conductivity of the junc-
parametersdeviatingfromtheirequilibriumvalues, tion, fN(NEN,V ,TN) = tanh((E +eV )/2k TN) tanh((E
eV )/2Tk TN)dtheeequilibrium valuedof f Binethe−normal d−e-
d B e T
NS(E,D ,G )= Re (E+iG /2)/ (E+iG /2)2 D 2 , tector, and fTS(E,Lid,TeS) the local non-equilibriumvalue of
| { q − }| f in the superconductor. The first term corresponds to the
(1) T
text-bookequilibriumexpressionforaNIStunnelcurrent,ex-
fS(E,TeS,m ∗)=1/ exp (E−m ∗)/kBTeS +1 . (2) plicitly depending on properties of the superconductor only
(cid:8) (cid:0) (cid:1) (cid:9) throughitsDOSN (E,D ,G ).
S
Deviationof fS from equilibriumformallycorrespondsto TheshapeoftheexperimentalI (V )characteristicdepends
L d d
an increase of TS above the bath temperature T, while de- onthequasiparticleinjection(Fig.2). ByfittingI (V )or/and
e d d
viation of fS - to a non-zero shift of chemical potential m . dI /dV dependenciesonefindsthatthe parametersG andD
T ∗ d d
Consideringtheaboveexpressionfor f withtwoparameters, dependontheinjectionrate,proximitytotheinjectorL ,and
S id
TS and m , one might account for both the longitudinaland temperature T (Figs. 3, 4). The smearing of the I (V ) de-
e ∗ d d
4
Keldysh-Usadel equations, is D =l ¥ fS(E,L )Re(F)dE,
0 L id
where l is the electron-phonon couRpling constant, F the
anomalous pair amplitude, and fS(E,L ) the local non-
L id
equilibrium value of f in the superconductor.2 At any en-
L
ergyE thefunction fS(E,L )issmallerthanitsequilibrium
L id
value tanh(E/2k T),22 reducing D (Fig. 4a). The effect is
B
smalleratthelowesttemperaturesandincreaseswithincreas-
ingbathtemperature(Fig.4a,inset).Utilizingthewell-known
BCStemperaturedependenceforthesuperconductinggapone
canassignaneffectivetemperatureT ,whichwouldprovide
∗
the same effect on the energygap as the quasiparticle injec-
tion: D (TS,I,L )=D (T ,I =0,L ) (Fig. 4b). Assum-
e i id BCS ∗ i id
inganexponentialdecayT =T (0)exp( L /L )onefinds
∗ ∗ id T
thecharacteristiclengthL =40 m m −20m m(∗Fig.4b,in-
T
set). The large uncertainty∗ comes fro±m the weak D (T)
BCS
dependence at low temperatures and the limited set of L
id
points (number of detectors). Once again we would like to
remind that T is a convenient parameter introduced to ac-
∗
countforthe reductionofthe superconductinggapD (I)and
i
has no direct relation to the thermodynamic temperature of
thesuperconductorTS whichshouldenterthecorresponding
e
non-equilibriumdistribution function. It might even happen
that in the limit of strong injection and weak quasiparticle-
quasiparticlescatteringTS cannotbedefinedatall,whilethe
e
non-equilibriumenergygapD (I),oralternativelyT (I),can
i ∗ i
bestilldeducedfromtheexperimentalI (V )characteristics.
d d
We believe that the reduction of the gap D (I) and the in-
i
FIG.4: (Coloronline)(a)SuperconductinggapD vs. injectionen- crease of G (I) are the two manifestations of the same phe-
ergyatL =0.8( ),2.8((cid:3)),7.8(⋆)and17.8(♦)m m.Inset:reduc- i
id nomenon - energy disbalance (longitudinalmode) accompa-
△
tionofthesuperconductinggapmeasuredbythesameNISdetector
nyinginjectionofthenon-equilibriumquasiparticles.Presum-
at Lid=7.8 m m at T =17.5 (open star), 200 (⋆) and 400 (crossed ably,inacomprehensivemicroscopicdescription(toourbest
star)mK.(b)EffectivetemperatureT vs. injectionenergyforsam-
∗
knowledge, currently absent) the longitudinal and the trans-
plesC3(18mK,opensymbols),C2(19mK,solidsymbols)andC8
(24mK,crossedsymbols)atLid=0.8( ),2.8((cid:3))and7.8(⋆)m m. versemodesmaybe entangledbothcontributingto the D (Ii)
Linesareguidestotheeye. Inset:spati△aldecayofT∗averagedover andG (Ii)dependencies.Withinthe‘orthodox’BCSmodelthe
several samplesateVi/D i=70((cid:3)), 16( )and 2.5( ). Linesare (temperature-dependent) smearing of a NIS detector Id(Vd)
fitsassumingexponentialdependence∼(cid:13)exp(−Lid/L T△∗). characteristic comes solely from the distribution function of
the normal electrode fN(E,V ,TN) and not from the DOS
d e
or distribution function of the superconductor. Temperature
pendence[roundingofthe‘corners’ateVd D (Ii)]canbeas- of the superconductor TeS enters solely through the temper-
signed to the parameter G (Ii) which enters≃the superconduc- ature dependence of the gap D (TeS). In a typical equilib-
tor DOS and is physically related to the finite quasiparticle rium situation T =TeN =TeS. Employed in the paper anal-
lifetime.16 For the same injection rate I the temperaturede- ysis deviates from that ‘orthodox’ model by introduction of
i
pendenceG (T, Ii=const, Lid =const)(rightinsetinFig. 3) the Dynes smearing G [Ii] entering the DOS of the supercon-
most probably originates from the more intensive relaxation ductorNS(E,D [Ii],G [Ii]), Eq. (1), at a given, but not defined,
ofthequasiparticles(i.e. fromthereductionofthequasipar- temperatureofthesuperconductorTeS,whichindeedleadsto
ticlelifetimeresponsibleforfiniteG )duetoinelasticscatter- theextra(non-temperaturedependent)smearingoftheId(Vd)
ing on phonons. The totalnumberof quasiparticlesentering dependence,Eq.(3).
the superconductorper unit time is proportionalto I, while The employed analysis of the experimentaldata based on
i
only the fraction exp( L /√Dt ) reaches the detector. the equilibrium expression for the tunnel current [first term
id R
At low temperatur∼es the−recombination time t can be es- inEq.(3)]withtheDOSdeviatingfromthe‘orthodox’BCS
R
timated from the relation, t /t (T/T)1/2exp( D /k T), model [Eqs. (1), (2)] enables separation of the two impacts
0 R c B
witht beingtheelectron-phonon∼scatteringtime.1−9,20 These D (I)andG (I).ThereductionofthegapD (I)(Fig.4)leadsto
0 i i i
simple arguments qualitatively explain the spatial, energy, a‘sharp’I (V )characteristic:asifthetemperatureisstilllow
d d
and temperature dependencies of G (Fig. 3), while a more TS T,buttheenergygapisreducedD (I >0)<D (I =0).
elaboratedmodel,presumablyincorporatingtheeffectofthe Tehe≃impact of the finite G (I) manifests iitself as the simear-
i
environment,21isrequiredforaquantitativeanalysis. ingoftheI (V )characteristicsatthe‘corners’eV D (I)
d d d i
Neglectingthesuperconductingphase,thegapequationfor whichcannotbeaccountedforanyreasonablevalue≃of±TeN.
thepairingpotentialD ,tobesolvedself-consistentlywiththe It is important to emphasize that the experimental non-
5
zero-injectionhigh-temperatureI (V ) characteristicscan be
d d
nicely fit by the equilibrium BCS expression for the tunnel
current[firstterminEq.(3)]assumingsmallintrinsicDynes
broadeningof the DOS, Eq. (1), with G (I =0)/D (I =0)
i i
≃
0.02 (Fig. 2b) in a reasonable agreement with existing data
onaluminum.21However,onerunsintoacontradictiontrying
tofittheexperimentalI (V ,I >0,T)characteristicsusinga
d d i
singlefittingparameterT andthefirsttermintheexpression
fit
(3) for the NIS tunnel current. To properly account for the
gap reduction D (I) one should set the best fit ‘equilibrium’
i
temperatureT =T T (compareFigs. 2band4b)being
fit ∗
≫
significantly higher than the temperature of ‘hottest’ part of
the system - the injector. While to accountfor the observed
broadening at the gap edge, the effective fitting temperature
T shouldbesetsignificantlylowerT Te T (Fig.2b).
fit fit≃ N≪ ∗
Summarizing,oneshouldconcludethatatnon-zeroinjections
the I (V ,I >0,T) characteristics cannot be described by a
d d i
single ‘equilibrium’ temperature T . The two independent
fit
parameters,D (I)andG (I),arenecessary. Thedependencies
i i
D (I) and G (I) on the injection rate are different. At ultra-
i i
low temperatures T T and modest injections eV/D .10
c i 0
≪
the suppression of the gap is rather small (Fig. 4a), while
theincreaseoftheDOSbroadeningparameterG isquitepro-
nounced(Fig. 3). Atelevatedtemperaturesthedependenceof
G on the injection rate is rather weak (insets in Fig. 3). We
wouldliketoemphasizethatthenecessitytointroducethetwo
independentparametersD (I)andG (I)istheresultoftheem-
i i
ployedphenomenologicalapproachbasedonEqs.(1)-(3). We
wouldwishtobelievethatinanelaboratedmicroscopicmodel
dealingwiththeessentiallynon-equilibriumdistributionfunc-
tion f (E,TS) a single parameter - the truly thermodynamic
S e
temperatureofthesuperconductorTS -wouldbesufficientto
e
accountforthewholevarietyoftheexperimentaldata.
A remarkable feature of the tunnel current expression is
theexistenceofthenon-zeroexcesscurrentIex=I (V =0)
d d d
originating from the independence of fS(E,L ,TS) on the
T id e
bias volatage V [second term in Eq. (3)]. The effect has
d
been observed in NISIS’ flat sandwiches23,24 and recently
in multiterminalNISstructures.5 Within ourphenomenolog-
FIG.5: (Color online) SampleC3NISdetection. (a) Zoomof the ical approach, the experimentally measured excess current
sub-gapregionoftheId(Vd)characteristicatdifferentinjectionvolt- is linked to the effective quasiparticle chemical potential,24
tae3iVg.o6ein/,ssD5Vo.i2if=arat0en,dvdi3ef,rf6se5.er4,ep1nmo0tmldaarinfsiotdtyar.n1tc5h(ebems)dELVeix.tdecL.ceitsLnoserinssceulcsororcarearrneteetsdpfiIdeotaxsntvduLstti.oliiz=ntihjnee2gc.s8tLia,omQ7n∗.e8e=,inne1j3r7eg.c.6y8-, gcIdeuaxprr=eInd(cid:0)t(GVindNd)cNr/eceah(cid:1)saemrsa∗c.wteAirtihsttyiicpniccirsaelassehexoawomfnptlhieneoFifniagje.Nc5tIiaSo.nderTatehtecetoreerxvsceuerbsss--
id ing its sign with the change of the injection polarity. The
and37.8m m, respectively.Inset:spatialdecayoftheexcesscurrent
measured at eVi/D i =33.7( ), 6.7((cid:3))and4.5( ). (c)Normal- effect decreases with increasing distance to the injector Lid
izedchemica|lpotentia|lm vs(cid:13). injectionenergyme△asuredatvarious (Fig. 5b) and the bath temperature (Fig. 5c). Slight asym-
temperaturesbythesame∗detectoratLid=7.8m m. metryofIdex withrespectto zeroinjection(Fig.5a)probably
originatesfroma parasitic residualthermo-electricpotential.
It should be noted that in the NISIN detector geometry the
excesscurrentis absentsince the contributionsfromthetwo
equilibriumId(Vd,Ii>0,T)dependenciescannotbedescribed NISjunctionsconnectedinseriesandlocatedatthesamedis-
bythe‘orthodox’BCSmodeljustassumingthatthetemper-
tanceL (fromtheoppositesidesofthesuperconductingbar,
id
ature of the system has increased above the bath tempera-
see Fig. 1), cancel each other. The double NISIN junctions
ture: T →TeN,TeS ≫T. At T ≪Tc the shape of the exper- were used only as a complementaryconfigurationin the ex-
imentalId(Vd,Ii>0,T)characteristicsisqualitativelydiffer- perimentsontheenergymoderelaxation(Fig.2a).
ent from the reference equilibrium characteristics I (V ,I =
d d i
0,T )takenatelevatedtemperaturesT >T (Fig. 2b). Those Thesolutionofthediffusionequationforthechargeimbal-
′ ′
6
anceQ givestheexcesscurrent Ref. 25, (T T)/T (t T) 2/3, where t is the energy
∗ c c E c − E
− ≪
relaxation time (measured in 1/K). Analysis of this expres-
Iex=I F∗L Q∗GdNN exp( L /L ), (4) sionleadstotheconclusionthattheonlyphysicallimitwhen
d i2e2N(0)Ds − id Q∗ alwaysL Q L E isatT Tc,whereL Q diverges.Formally
∗≫ → ∗
itmeansthatinthenormalstatethereisnobranchrelaxation
where N(0)=1.08 1047 1/J m3 is the normal state DOS mechanism. Ourexperimentsweremadeatverylowtemper-
× ×
foraluminiumattheFermilevel,andF isasmoothfunction aturesmuchsmallerthenT . Atsuchlowtemperatures,toour
∗ c
varyingfrom0atnoinjectionto1ateV >>D .24Substituting bestknowledge,thereisnogeneralcriteriontoapriorirelate
i i
themeanfreepathℓ 20nm,FermivelocityvF =1.36 106 L Q withL T .
m/s, diffusivityD=≃1/3vFℓ andcrosssection s =25n×m ∗ ∗ IV. CONCLUSION
×
400 nm one gets an acceptable fit for the dependence of the
excesscurrenton theinjectionenergy(Fig.5b). The best-fit
In conclusion, we performed spatially resolved measure-
chargeimbalancelengthL = Dt variesfrom3.5to6.5
Q∗ Q∗ ments of the non-equilibrium quasiparticle relaxation in su-
m mbeinginareasonableagreempentwithrecentfindings.5,8
perconducting aluminum at ultra-low temperatures T T.
c
Weaktemperaturedependenceofthechargeimbalancere- ≪
The analysis employs the phenomenological interpretation
laxationQ Iex m (Fig.5c) atultra-lowtemperaturesis
∗ ∼ d ∼ ∗ postulatingequilibriumfunctionaldependenciesofDOSand
expected:intheabsenceofaneffectiveelectron-phononinter-
distribution function of the superconductor, while assuming
actiontheremaining(weak)mechanismistheelasticscatter- thatthebroadeningparameterG ,effectivechemicalpotential
ing dueto the gapinhomogeneity,or/andanisotropyor even m , and superconductinggap D depend on the rate of quasi-
just a finite supercurrent.24 Contrary to the transverse mode ∗
particleinjectionandtheproximitytotheinjector.Twoscales
relaxation,therelaxationofthelongitudinal(energy)modeal- L = 40 20 m m and L =5 1.5 m m are responsible
T Q
waysrequiresthepresenceofinelasticmechanism(s). Atthe ∗ ± ∗ ±
forrelaxationof the energy(longitudinal)and charge(trans-
ultra-low temperaturesof the experimentk T D there are
B verse) modes, respectively. Both quantities were measured
≪
very few equilibrium phonons capable to provide the quasi-
simultaneously in the same samples using the same experi-
particleenergyrelaxation. Theonly(potential)sourceofthe
mentaltechniqueeliminatingsample/measurementartifacts.
phonons with energies above the bath temperature T is the
Thisisthecentralresultofthepaper.Itshouldbeemphasized
Jouleheatingintheinjector.However,howithasbeenalready
thatboththeenergyandthechargedisequilibriumareuniver-
discussed, even in the worst-case scenario of the strongest
sal phenomena which should be taken into consideration in
injection the diffusion of heat from the ‘hot’ injector into a broad class of systems involving electron,9–13 spin and/or
the superconductor is not sufficient to account for our find- coherentnon-local26transport. Despitethereasonableagree-
ings. The maximum rise of the electron temperature in the
ment with the phenomenological description, application of
injector d Ti 100 mK. Even assuming a perfect electron-
e ≃ therelaxationapproximationfortheessentiallyspatiallyinho-
phononcouplingintheinjector,theenergyofthecorrespond-
mogeneousproblemisnotfullyjustifiedatlowtemperatures
ing phonons is still much smaller than the energy 2D re-
T T .Amoreelaboratedmicroscopicmodelisrequiredfor
≥ c
quiredfor a pair of hole-likeand electron-likequasiparticles ≪
aquantitativeanalysis. Wehopethatourfindingswilltrigger
to forma Cooperpair. However,as the quasiparticlerecom-
thecorrespondingresearchactivity.
binationfinally does happenon ‘astronomically’large (fora
superconductor)scalesL 40m m,someweakmechanism
T
∗ ≃
should exist. For example, emission / re-absorption of non-
equilibrium phononsor photons, might provide the required
Acknowledgments
energy relaxation. Certainly, these mechanisms should also
contribute to the relaxation of the transverse mode. The ob-
servationL <L supportsourconclusionthatatultra-low The authors would like to acknowledge support of
Q T
∗ ∗
temperaturestheelasticscatteringisanimportantchannelof TEKES project DEMAPP, FCPK grant 2010-1.5-508-
thechargeimbalancerelaxation. 005-037, NanoSCIERA project Nanofridge and ANR-
SubstantialquantitativedifferencebetweenL andL at DYCOSMA. We thank A. Julukian and L. Leino for AFM
T Q
∗ ∗
ultra-lowtemperatures,T T ,isinaqualitativeagreement analysis, and F. Hekking for discussions. A.S.V. acknowl-
c
with previous calculations≪.19,20 The explicit condition when edgethehospitalityofDonostiaInternationalPhysicsCenter
one can neglect the elastic scattering has been obtained in (DIPS)duringhisstayinSanSebastia´n,Spain.
Electronicaddress:[email protected].fi (PlenumPress,NewYork,1981).
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