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. Pannetier1

1Centre de Recherches sur les Très Basses Températures-C.N.R.S. associé à l’Université Joseph Fourier, 25 Av. de38042 Grenoble, France

2Laboratoire de Microstructures et de Microélectronique-C.N.R.S., 196 Av. H. Ravera, 92220 Bagneux, France3Institute 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 Aharonov-Bohm flux. Experimental results are well described by thelinearized quasiclassical theory. [S0031-9007(96)01817-0]

PACS numbers: 74.50.+r, 73.20.Fz, 73.50.Jt, 74.80.Fp

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During 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 S-N interfaceshown that the proximity effect strongly affects electrtransport in mesoscopic S-N systems: The deviationDGof the conductance from its normal-state value depestrongly on temperatureT and oscillates in an appliemagnetic fieldH if a N loop is present [2–4]. Varioustheoretical approaches were suggested to explain thishavior. A scattering matrix method based upon the Ldauer formula [5] as well as a numerical solution of tBogolubov–de 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 Green’s functions [7–10] 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 electron-electron in-teraction in N is negligible) with increasingT, reachesa maximum at approximately the Thouless temperatecykB ­ h̄DykBL2, 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 Bogoliubov–de Gennes equations and inanalytical solution of the equations for the quasiclassGreen’s functions.

The physics behind this reentrance effect involvnonequilibrium effects between quasiparticles injectedthe N reservoirs and electron pairs leaking from S.the N-S 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 Fermi

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wave vector. The phase conjugation between the elecand the hole results in a finite pair amplitude involvinstatesskF 1 eyh̄yF , 2kF 1 eyh̄yFd, yF being the Fermivelocity. Such a pair maintains coherence in N up toenergy-dependent diffusion lengthLe ­

ph̄Dye [7,11]

which coincides with the well-known thermal lengLT ­

ph̄DykBT at e ­ 2pkBT .

In the high-temperature regimeLT , L or, equiva-lently, ec , kBT , it is well known that the proximity ef-fect results in the subtraction of a lengthLT of N metalfrom the resistance of a S-N 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. Tlow-temperature reentrant regime is destroyed by increing the temperature [9] or the voltage [10]. As will bdiscussed below, an Aharonov-Bohm 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 [2–4] 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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FIG. 1. Micrograph of the sample made of a continuous(­ N) loop in contact with a single Al (­ S) island. Inset:simple model. At half-integer 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 am-metal-shielded dilution refrigerator down to 20 mK. Frothe normal-state 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 high-temperature regimeLT , L (i.e.,above 500 mK) is very similar to the two-island ca[4]. At lower temperatures, so thatLT . L, and zeromagnetic field, we observe adecreaseof the low-voltageconductance [Fig. 2(inset)]. This occurs below 50 mat the temperature where a Josephson coupling wbe expected in a two-island geometry. The voltagependence of the measured conductance shows thestriking behavior, i.e., anincrease of the conductancewhen the bias voltage isincreased(Fig. 2). This non-linear 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 cal-culated 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 two-island 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 reentranc

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FIG. 2. Non-monotonous voltage dependence of the samconductance atT ­ 22 mK; the normal-state 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 zero-field 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 current-voltage 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 [7,9–11]. 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 half-integer magnetic flux and at high field. Insoscillations amplitude together with a 1yT power law fit.

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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

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FIG. 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 po-tential 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 ker-nel which reduces to the Dirac function atT ­ 0. Hence,the low-temperature differential conductancedIydV ­GN 1 dG probes the proximity-induced 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 voltage-conductancedIydV 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.

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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

h̄D≠2xF 1

√2ie 2

h̄DL2

w

!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 ø 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 en-ergy e, the real part of the pair amplitude is zero at theinterface, maximum at a distanceLe if Le ø 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 thenormal-state conductivity [7,9–11]. It involves two cotributions: a positive and dominant one which is similarthe Maki-Thompson 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 S-N 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 N-S interfacea maximum of about0.3sN at a distanceLe from theinterface (ifL ø 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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the 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 Aharonov-Bohm flux, which changes the boundconditions on the pair amplitudeFsed. At zero magneticflux, Fsed is zero only at contact with the normareservoirs. At half-magnetic 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 reduces the effectsample size to the lengthL0 between the S interfacand the pointK. In the intermediate temperature regimkBT . ec (or LT , L), this modulates the conductancwith a relative amplitude of the order ofecykBT [13], inqualitative agreement with the experiment.

As an additional effect of the magnetic fieldH, thephase-memory lengthLw is renormalized due to the finitwidth w of the Cu wire [16]:

L22w sHd ­ L22

w s0d 1p2

3H2w2

F20

. (3)

When smaller than the sample lengthL, the phase-memory lengthLwsHd plays the role of an effectivelength for the sample. As a result, the conductance pis shifted to higher temperatures and energies whenmagnetic field increases, see Figs. 4 and 5. At hmagnetic field, the position of the conductance maxidoes not increase as rapidly as would be expecbecause of the field-induced depletion of the gapD.In the right part of Fig. 4, we show the calculateconductance using Eqs. (1)–(3) in the modeled geomof the Fig. 1 inset in the case of a fully transpareinterface. The only free parameter is the width of twires which has been adjusted so that the experimedamping of the amplitude of the magnetoconductaoscillations by the magnetic field, see Fig. 3, is wdescribed by the calculation. The discrepancy betwthe fitted value w ­ 65 nm and the measured valuis attributed to deviations of sample geometry froour simple model. Our calculation accounts for bothe global shape and amplitude of the curves, andtheir behavior as a function of the magnetic flux. This particularly remarkable with respect to the stroassumptions of the model. One should note that the qutative shape and amplitude of the curves are conseif nonlinearized Usadel equations or slightly differegeometrical parameters are used.

In conclusion, we have measured the energy depdence of the proximity effect on the conductance neaN-S junction. As predicted in recent works [9,10], whave observed the reentrance of the metallic conducta

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when all energies involved are below the Thouless eneof the sample. In contrast with a very recent similar oservation [17], the energy crossover has been trackedfunction of temperature, voltage, and magnetic field. Oexperimental results are well described by the linearizUsadel equations from the quasiclassical theory.

We thank P. Butaud, M. Dévoret, D. Estève, B. Spvak, T. Stoof, A. Zaikin, and F. Zhou for stimulating discussions. A. F. V. thanks P. Monceau for hospitality, tRussian Fund for Fundamental Research, and the cooration program between the Ecole Normale Superiede Paris and the Landau Institute for Theoretical Physfor support. We also acknowledge financial support froRégion-Rhône-Alpes and D. R. E. T.

[1] See references in “Mesoscopic Superconductivity”, eby F. Hekking, G. Schön, and D. Averin, Physic(Amsterdam)203B (1994).

[2] V. T. Petrashov, V. N. Antonov, P. Delsing, anT. Claeson, Phys. Rev. Lett.70, 347 (1993); Phys. RevLett. 74, 5268 (1995).

[3] A. Dimoulas, J. P. Heida, B. J. van Wees, T. M. KlapwijW. v. d. Graaf, and G. Borghs, Phys. Rev. Lett.74, 602(1995).

[4] H. Courtois, Ph. Gandit, D. Mailly, and B. PannetiePhys. Rev. Lett.76, 130 (1996).

[5] C. W. J. Beenakker, Phys. Rev. B46, 12 841 (1992).[6] C. J. Lambert, J. Phys. Condens. Matter3, 6579 (1991).[7] A. F. Volkov, A. V. Zaitsev, and T. M. Klapwijk, Physica

(Amsterdam)210C, 21 (1993); A. F. Volkov and A. V.Zaitsev, Phys. Rev. B53, 9267 (1996); A. V. Zaitsev,JETP Lett.51, 35 (1990); Phys. Lett. A194, 315 (1994).

[8] A. A. Golubov, F. Wilhelm, and A. D. Zaikin (to bepublished).

[9] Y. V. Nazarov and T. H. Stoof, Phys. Rev. Lett.76, 823(1996).

[10] A. F. Volkov, N. Allsopp, and C. J. Lambert, J. PhyCondens. Matter8, L45 (1996).

[11] F. Zhou, B. Spivak, and A. Zyuzin, Phys. Rev. B52, 4467(1995).

[12] We observed recently a reentrance effect with a mularger amplitude in smaller samples; see P. CharH. Courtois, Ph. Gandit, D. Mailly, A. Volkov, andB. Pannetier, Czech. J. Phys.46, S6 (1996).

[13] P. Charlat, H. Courtois, Ph. Gandit, D. Mailly, anB. Pannetier (unpublished).

[14] B. R. Patton,Proceedings of the 13th International Conference on Low Temp. Phys., 1972edited by W. S.O’Sullivan (Plenum, New York, 1974).

[15] A. F. Volkov and V. V. Pavlovsky (to be published).[16] B. Pannetier, J. Chaussy, and R. Rammal, Phys. Scr

T13, 245–251 (1986).[17] S. G. den Hartog, C. M. A. Kapteyn, B. J. van Wees, a

T. M. Klapwijk, Phys. Rev. Lett.76, 4592 (1996).

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