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VOLUME 77, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 9 DECEMBER 1996
s Martyrs,
, Russia
4
Reentrance of the Metallic Conductance in a Mesoscopic Proximity Superconductor
P. Charlat,1 H. Courtois,1 Ph. Gandit,1 D. Mailly,2 A. F. Volkov,3 and B. Pannetier11Centre de Recherches sur les Trs Basses TempraturesC.N.R.S. associ lUniversit Joseph Fourier, 25 Av. de
38042 Grenoble, France2Laboratoire de Microstructures et de MicrolectroniqueC.N.R.S., 196 Av. H. Ravera, 92220 Bagneux, France
3Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Mokhovaya St. 11, 103907 Moscow(Received 14 May 1996)
We present an experimental study of the diffusive transport in a normal metal near a superconductinginterface, showing the reentrance of the metallic conductance at very low temperature. This newmesoscopic regime comes in when the thermal coherence length of the electron pairs exceeds thesample size. The reentrance is suppressed by a bias voltage given by the Thouless energy and canbe strongly enhanced by an AharonovBohm flux. Experimental results are well described by thelinearized quasiclassical theory. [S00319007(96)018170]
PACS numbers: 74.50.+r, 73.20.Fz, 73.50.Jt, 74.80.Fpeetr
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omingingtion.outdsDuring the past few years, the proximity effect betwea superconductor (S) and a normal (N) metal has mnoticeable revival, thanks to spectacular progress infabrication of samples of mesoscopic size [1]. Expemental study of the transport near a SN interfaceshown that the proximity effect strongly affects electrtransport in mesoscopic SN systems: The deviationDGof the conductance from its normalstate value depestrongly on temperatureT and oscillates in an appliemagnetic fieldH if a N loop is present [24]. Varioustheoretical approaches were suggested to explain thishavior. A scattering matrix method based upon the Ldauer formula [5] as well as a numerical solution of tBogolubovde Gennes equations [6] were used. Thstudies demonstrated that superconductivity does nofect the charge transfer in the N metal if the temperaturTand the voltageV are zero, i.e.,DG is zero at zero energyA more powerful method based on the equations forquasiclassical Greens functions [710] was used totain the dependence ofDG on T andV. It has been established [9] that atV 0 the deviation of the conductancDG increases from zero atT 0 (if electronelectron interaction in N is negligible) with increasingT, reachesa maximum at approximately the Thouless temperatecykB hDykBL2, and decreases to zero atT ecykB.This constitutes the reentrance effect for the metallic cductance of the N metal. Similar dependence ofDGsV dat T 0 has been found in [10] both in a numerical slution of the Bogoliubovde Gennes equations and inanalytical solution of the equations for the quasiclassGreens functions.
The physics behind this reentrance effect involvnonequilibrium effects between quasiparticles injectedthe N reservoirs and electron pairs leaking from S.the NS interface, and incident electron is reflected inthole of the same energye compared to the Fermi leveEF , but with a slight change in wave vectordk due tothe branch crossing:dkykF eyEF , kF being the Fermi950 00319007y96y77(24)y4950(4)$10.00nt aheiasn
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wave vector. The phase conjugation between the elecand the hole results in a finite pair amplitude involvinstatesskF 1 eyhyF , 2kF 1 eyhyFd, yF being the Fermivelocity. Such a pair maintains coherence in N up toenergydependent diffusion lengthLe
phDye [7,11]
which coincides with the wellknown thermal lengLT
phDykBT at e 2pkBT .
In the hightemperature regimeLT , L or, equivalently, ec , kBT , it is well known that the proximity effect results in the subtraction of a lengthLT of N metalfrom the resistance of a SN junction. In the low temperatureLT . L or ec . kBT and low voltageeV , ec,electron pairs are coherent over the whole sample.proximity effect on the N metal resistance is still prdicted to be zero. In this Letter, we report the expemental realization of both limits (L , LT and L . LT )and the observation of the reentrance of the metallic cductance in a mesoscopic proximity superconductor. Tlowtemperature reentrant regime is destroyed by increing the temperature [9] or the voltage [10]. As will bdiscussed below, an AharonovBohm flux modifies thefective length of the sample and therefore shifts the enecrossover of the reentrant regime.
Figure 1 shows a micrograph of the sample made osquare copper (Cu) loop in contact with a single alumin(Al) island. The loop, although not essential for thoccurrence of the reentrance effect, allows one to conboundary conditions for the pair amplitude. The Cu wwidth is 150 nm and its thickness is about 40 nm. Tdistance between the Cu loop and the Al island is ab100 nm, whereas the perimeter of the loop is 2mm.One should note that the sample geometry differs frall previous sample geometries with two superconductcontacts [24] in that there is a single superconductphase and therefore no possible Josephson contribuTwo voltage probes measure the distribution at theflows of the reservoirs, which are the wide contact paat both ends of the Cu wire. The Cu surface isin situ 1996 The American Physical Society
VOLUME 77, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 9 DECEMBER 1996
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iset:FIG. 1. Micrograph of the sample made of a continuous( N) loop in contact with a single Al ( S) island. Inset:simple model. At halfinteger magnetic flux, a conditionF 0is enforced at the pointK at L0 1 mm from S. The distancebetween S and the right and left N reservoirs are, respectivL 2 mm and 0.5mm
cleaned before Al deposition in order to ensure an optimtransparency of the CuyAl interface [4].
We performed transport measurements in ammetalshielded dilution refrigerator down to 20 mK. Frothe normalstate conductanceGN 0.0937 S we finda diffusion coefficientD 70 cm2ys, an elastic meanfree path le 13 nm, and a thermal coherence lengLT 92 nmy
pT . Aluminum islands become superco
ducting belowTc . 1.4 K. The behavior of the conductance in the hightemperature regimeLT , L (i.e.,above 500 mK) is very similar to the twoisland ca[4]. At lower temperatures, so thatLT . L, and zeromagnetic field, we observe adecreaseof the lowvoltageconductance [Fig. 2(inset)]. This occurs below 50 mat the temperature where a Josephson coupling wbe expected in a twoisland geometry. The voltagependence of the measured conductance shows thestriking behavior, i.e., anincrease of the conductancewhen the bias voltage isincreased(Fig. 2). This nonlinear behavior discards an interpretation in termsweak localization, which is known to be insensitito voltage. The conductance peak is observed abias voltage (about 1.7mV) of the order of the calculated Thouless energyec 1.1 mV related with asample lengthL 2 mm. In the Fig. 2 inset, the peaposition is also consistent with the Thouless tempeture ecykB 13 mK. One can note that the discussenergies are much smaller than Al energy gapD.
Let us now analyze the effect of the magnetic fieFigure 3 shows oscillations of the magnetoconductawith a periodicity of one flux quantumf0 hy2e inthe loop area. Here the reentrance effect can bevery clearly atf f0y2, 3f0y2, and at higher field. Aspreviously observed in twoisland samples, the oscillatamplitude decays slowly with a 1yT power law down to200 mK [Fig. 3(inset)]. Figure 4 shows the temperatdependence of the conductance for various values ofmagnetic flux in the loop. On this scale, the reentrancu
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FIG. 2. Nonmonotonous voltage dependence of the samconductance atT 22 mK; the normalstate conductanceGNis 0.0937 S. Inset: temperature dependence. Measurecurrent of 70 nA in both curves.
zero field is hardly distinguishable [12]. Atf f0y2,the conductance maximum is near 500 mK and the retrance has a much larger amplitude. Atf f0, thecurve is close to the zerofield case, and, atf 3f0y2,close to thef0y2 case. Increasing further the magnefield [Fig. 4(left part)], the conductance peak is displacto higher temperature and thef0 periodic modulation issuppressed. Figure 5 shows the conductance peaksgies obtained from both the currentvoltage characterisand temperature dependence of the conductance as ation of the magnetic flux. Hence the magnetic field htwo effects on the reentrance: (i) a largef0 oscillation ofthe peak position at low field; (ii) a monotonous shifthigher fields.
Most of our observations can be analyzed in the framwork of the quasiclassical theory for inhomogenous supconductors [7,911]. We present here a simplified versthat, however, keeps the essential physical features.consider the mesoscopic regime where the inelastic stering length is larger than the sample lengthL betweenthe reservoirs. The flow of electrons at a particular eergy e is then uniform over the sample, and one hasconsider transport through independent channels at theergy e. In a perfect reservoir, electrons follow the Ferm
FIG. 3. Magnetoconductance oscillations showingF0 hy2e flux periodicity at T 50 mK (solid line) andT 500 mK (dashed line). Reentrance of the resistancevisible at halfinteger magnetic flux and at high field. Insoscillations amplitude together with a 1yT power law fit.4951
VOLUME 77, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 9 DECEMBER 1996
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enFIG. 4. Left: measured temperature dependence of theductance at different values of the magnetic fluxF in unitsof the flux quantum:FyF0 0; 1y2; 1; 3y2; 2; 3.8; 5.6; 7.4.Measurement current is 200 nA. Right: calculated conductain the linear approximation for the sample model of the Figinset and at the same values of the flux. The only adjustparameter is the effective width of the wires.
equilibrium distribution at temperatureT and chemical potential m. Charges are injected in the system from oreservoir atm eV and transferred to the other, so ththe current is carried by electrons within an energy wdow f0, m eV g with a thermal broadeningkBT . Let usassume that the proximity effect can be accounted forconductivity enhancementdsse, xd depending on both thenergye and the distancex from the S interface. From thbehavior ofdsse, xd it is then straightforward to calculatthe excess conductancedgsed for the precise geometry othe sample. The excess conductancedGsV , T d at voltageV and temperatureT writes as
dGsV , T d Z `
2`dgsedPsV 2 ed de , (1)
wherePsed f4kBTcosh2sey2kBTdg21 is a thermal kernel which reduces to the Dirac function atT 0. Hence,the lowtemperature differential conductancedIydV GN 1 dG probes the proximityinduced excess condtancedgsed at energye eV with a thermal broadening
FIG. 5. Energy of the conductance maximum as a funcof the magnetic fluxf in units of F0. The black squaredots are obtained from the voltageconductancedIydV sV dcharacteristics atT 100 mK, except forFyF0 0 and 1(T 22 and 45 mK). The white circles are obtained froFig. 4. The discrepancies in between reflect the imperfectof the reservoirs at low energies.4952n
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kBT . Independent measurements of the excess contance as a function ofT andV agree with Eq. (1) for nottoo low temperatures and voltages, provided the chempotential in the reservoirs is taken as equal to the msured voltage times a geometrical factor of 1.05. This crespondence, however, fails for energies less than a5 meV: we believe that, at these energies, the inelacollision rate is too small to ensure a good thermalizatin the reservoirs [13].
In N and at zero magnetic field,Fse, xd follows theUsadel equation that for smallF can be linearized as
hD2xF 1
2ie 2
hDL2w
!F 0 . (2)
The linearization used above is a very crude appromation, since near the interface, which is believed toclean, the pair amplitude should be large [in this appromation, we haveFse D, 0d 2ipy2 for a perfectlytransparent interface [13]]. At the contact with a N resvoir, F is assumed to be zero. However, this simple fmulation enables a straightforward understanding ofphysical root of the conductance enhancement in a prmity system. Indeed, Eq. (2) features simply a diffusiequation for the pair amplitudeF at the energye with adecay lengthLe and a cutoff atLw . At a particular energy e, the real part of the pair amplitude is zero at theinterface, maximum at a distanceLe if Le Lw , L, andthen decays in an oscillating way. The pair amplitudeF isresponsible for the local enhancement of the conductidsse, xd sN sssRefFse, xdgddd2 for small F, sN being thenormalstate conductivity [7,911]. It involves two cotributions: a positive and dominant one which is similarthe MakiThompson fluctuation term in superconductaboveTc [14] and a negative one related to the decreof the density of states. The two contributions cancel eother at zero energy [15].
We model the sample as two independent SN circin series as shown in the inset of Fig. 1. Thisour main approximation. It describes the main physof our particular geometry and illustrates more genesituations. Both circuits consist of a N wire betwea superconductor S and a normal reservoir N. Alothe wire, the excess conductivitydsse, xd at a givenenergy e increases from zero at the NS interfacea maximum of about0.3sN at a distanceLe from theinterface (ifL Lw) and then decays exponentially witx. The integrated excess conductancedgsed of the wholesample rises from zero with ane2 law at low energy,reaches a maximum of0.15GN at about5ec, and goesback to zero at higher energy with a1y
pe law. This
behavior is indeed confirmed in the experimental resin Fig. 2. We observe a conductance peak as a funcof both a temperature and voltage. The conductais maximum for a temperature (50 mK) close to tcalculated crossover temperatures5ecykB 65 mKd, seethe Fig. 2 inset. Only qualitative agreement betwe
VOLUME 77, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 9 DECEMBER 1996
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ndthe observed (1.7mV) and calculated energys5ec 5.5 mVd is obtained. This discrepancy is believedbe due to insufficient energy relaxation efficiency at loenergy in the Cu reservoirs.
Because of the loop geometry, a magnetic field induan AharonovBohm flux, which changes the boundconditions on the pair amplitudeFsed. At zero magneticflux, Fsed is zero only at contact with the normareservoirs. At halfmagnetic flux, destructive interferenof the pair functions in the two branches enforces a zin Fsed at the node K [see Fig. 1(inset)]. Consequenthe pair amplitude is also zero between the loop and threservoir. Half a flux quantum then r...