Schön, Gerd

Wilhelm, F.K.

Zaikin, Andrei D.

Wilhelm, F. K.

KITopen

Supercurrent in a mesoscopic proximity wireFrank K. Wilhelm, Andrei D. Zaikin, and Gerd SchönInstitut für Theoretische Festkörperphysik, Universität Karlsruhe (TH), D-76128Karlsruhe, GermanyRecent experiments on the proximity induced supercurrent in mesoscopicnormal wires revealed a surprising temperature dependence. They suggestclean-limit behavior although the wires are strongly disordered. We demon-strate that this unexpected scaling is actually contained in the conventionaldescription of diusive superconductors and nd excellent agreement withthe experimental results. In addition we propose a SQUID-like proximitystructure for further experimental investigations of the eects in question.PACS: 73.23.Ps, 74.50.+r, 74.80.Fp1. IntroductionA normal metal in direct contact with a superconductor acquires su-perconducting properties.1,2Although this proximity eect has been dis-cussed for a long time, it has recently attracted new attention because ofthe dramatic progress in nanotechnology which allows the fabrication andstudy of metallic structures in the mesoscopic regime. Due to the prox-imity eect a supercurrent can ow through a normal metal between twosuperconductors.3{5Recently, this current has been detected by Courtois etal.6in a thin normal wire in the diusive regime with superconducting stripsdeposited on its top (see Fig. 1).The supercurrent is determined by an overlap of Cooper-pair wave func-tions penetrating into the normal wire from the superconducting strips.Since in the diusive limit the eective penetration length is of order1,2N=pD=2T one should expect that the supercurrent in the system isproportional to4,5Ic(T ) / e pT=T0. In contrast, the experimental data ofRef.6showed a much better t to the dependence Ic(T ) / e T=T1, which istypical for ballistic systems.3Furthermore, the tting parameter T1deviatedSuperconducting properties of thin proximity wiresProximity-induced superconductivityS S SIN N enFig. 1. Experimental geometryfrom standard clean-limit results by more than one order of magnitude. Thishas raised the question, whether the usual criterion distinguishing betweenballistic and diusive limits is correct when applied to proximity inducedsuperconductivity in the normal metal, or whether the above phenomenonhas to be attributed to quantum eects not contained in the quasiclassicaltheory of superconductivity.Below we will demonstrate that neither of these conjectures is true.We employ a quasiclassical calculation of the supercurrent in the proxim-ity structures of Fig. 1 and show the corresponding results agree well withthe experimental data.6Furthermore, we will also propose a new, equiva-lent experiment, where superconducting material is deposited onto a normalring. In this 'proximity SQUID' the magnetic ux through the ring is theequivalent of the phase dierence across the normal wire.2. The Model and the FormalismSince the critical current of the structure of Fig.1 is determined by thelongest SNS-cell which serves as a \bottleneck" it is sucient to study asingle cell. We denote the distance between these two adjacent supercon-ducting strips as d and consider the case where d is much larger than thestrips' thickness.We will use the standard formalism of quasiclassical Green's functions7in the diusive limit8For a thin normal wire with a thickness  N(thiscondition is well justied in the experiment6) the Green functions in thewire directly below superconducting strips are equal to those of a supercon-ductor for all relevant energies.9In what follows we further assume thatthe superconductors have bulk properties, neglecting any suppression of thesuperconducting gap  in the strips in the vicinity of SN boundaries.3. Solution and ResultsDetails of the formalsim and the calculation will be published in a futurepublication, so we will only present the key results here.A characteristic energy scale is provided by the Thouless energy Ed=Superconducting properties of thin proximity wiresD=d2, where D is the diusion constant. For high temperatures, T  Ed(equivalent to the geometrical condition d  N), the mutual inuence be-tween the superconductors can be neglected. In this approximation, we getthe current-phase relation I = Icsin(') with critical currentIc=643 + 2p2TeRNdNexp dN/ Tqexp  sTT0!(1)with 2T0= Ed, q = 3=2 and RNis the normal state resistance of the normalpart. For comparison, the expression following from the Ginzburg-Landauanalysis2has the same T -dependence but with q = 1=2. On the other hand,in the clean limit one expects a temperature dependence Ic/ e T=T1.At this stage we note a remarkable mathematical artefact caused by theexponent q = 3=2. The logarithmical derivative of (1)dIc(T )Ic(T )dT=32T 12pTT0has a minimum at T = 36T0and varies very slowly at higher temperatures,so log Icis almost linear in T . As a good approximation for the slope in alogplot, we can take the logarithmical derivative in the minimum and getIc/ e T=Twhere T= 24T0. This implies that within a considerable tem-perature interval a \quasi-clean" scaling is found also in dirty SNS systems,which explains the behavior found in the experiments of Courtois et al.6In the low temperature limit T = 0 another simple estimate for thecritical current can be derived.Ic=RNearctanEd2Ed !D2RNed2: (2)This demonstrates the importance of the Thouless energy as a relevant en-ergy scale in SNS-junctions at Ed . The Thouless energy also determinesthe proximity induced eective gap in the quasiparticle spectrum of the N-metal (see Ref.10and further references therein). We further note that (2)resembles the well-known Ambegaokar-Barato formula for the critical cur-rent of a Josephson tunnel junction at T = 0 if we substitute D=d2by thegap  . Thus our results emphasize that the Thouless energy in a diu-sive proximity coupled normal wire plays the same role as the gap  in a\strong" superconductor and does not only determine the density of states,but also the critical current.A solution of the full problem as obtained numerically is shown in g2. The result matches quantitatively to the experiments, if d is chosenslightly larger than the average cell length of the experiment. This reectsthe fact, that the critical current of the whole chain is determined by theSuperconductingpropertiesofthinproximitywires0.0200.0400.0600.0800.0T [mK]0 0 1 10Ic [µA]Theoretical curveExperimental dataFig.2.Criticalcurrentfortheexperimentalparameters[6],experimentaldataaresuppliedbyH.CourtoislongestSNS-cellwiththelowestIc.Theadjustmentlieswithintherangeofexperimentalaccuracy.Ingeneralthecurrent-phaserelationI(')maydeviatefromsinusodiality.4,5Weinvestigatedalsothisproperty,butfoundintheSNSgeometrynearlynodeviationdowntoverylowtemperatures(seeFig.3).4.TheProximitySQUIDTheproximityinducedsupercurrentshouldleadtoaninterestingux-periodicbehaviorofthe\proximityloop"structureshowninFig.4.AnormalringiscontactedoverarangeoflentghdSbysuperconductingma-terial.Iftheringisnarrowitcanbemappedontothelinearsystemdis-cussedabovebyabsorbingthevectorpotentialinthegaugeinvariantphase'(r)='0(r)+2eRr0dr0A(r0).I.e.theanomalousGreen'sfunctionFcarriesthephasefactorexp[i'0(r)],but(r)and'(r)satisfytheUsadelequationwithappropriateboundaryconditionsandthesubstitutiond=L dS.SincetheGreen'sfunctionsmustbesingle-valuedateverypointofthering,the\real"phase'0canchangeonlybymultiplesof2whencirclingaroundtheSuperconducting properties of thin proximity wires0.0 0.5 1.0φ0.01.02.03.0I c(µA) T=2 mKT=5 mKT=10 mKT=20 mKT=50 mKT=100 mKT=200 mKT=500 mKFig. 3. Current phase relation, parameters as in the experiment, see Fig. 2ring. For this reason, at x = 0 the gauge-invariant phase drops by = '(0+)   '(0 ) = 2eIdr A(r) =2eh;where  is the magnetic ux through the ring.Taking into account the self-inductance, this system is completely equiv-alent to a standard SQUID formed by a superconducting loop interruptedby a weak link. The conguration considered here is complementary: a nor-mal loop is interrupted by a narrow superconducting strip, but due to theproximity eect it shows the same properties as the standard SQUID.At relatively high temperatures the supercurrent in the normal prox-imity loop is exponentially small. However, as the temperature is loweredbelow the corresponding Thouless energy the supercurrent becomes large(cf. (2)) and can be easily detected experimentally. Also the current-phaserelation of a diusive SNS structure can be easily studied.5. ConclusionsMaking use of a standard quasiclassical formalism of the superconduc-tivity theory we evaluated the supercurrent in mesoscopic proximity wires.Our results match quantitatively with the experimental data6thus demon-strating that the quasiclassical theory of superconductivity is sucient forSuperconducting properties of thin proximity wiresSNN−φ/2NΦ0j jjjGauge transformation-NjS00Sjφ/2−φ/2φ/2Fig. 4. Proximity SQUIDthe description of these systems. The deviations of the current-phase rela-tion from sinusodiality are found to be small even at very low temperatures.Finally we argue that a normal-metall loop interrupted by a narrow super-conducting strip has the same properties as a standard SQUID and can beused for further experimental investigations of the proximity eect in meso-scopic systems.We would like to thank C. Bruder, H. Courtois and B. Pannetier foruseful discussions. This work was supported by the Deutsche Forschungs-gemeinschaft through Sonderforschungsbereich 195. One of us (G.S.) ac-knowledges the A.v. Humboldt award of the Academy of Finland. and thehospitality of the Helsinki University of Technology.REFERENCES1. P.G. de Gennes, Rev. Mod. Phys. 36, 225 (1964).2. G. Deutscher and P.G. de Gennes in Parks, \Superconductivity", Vol. 1 (1969).3. I.O.Kulik, Zh. Eksp. Theor. Phys., 57, 1745 (1969) [Sov. Phys. JETP 30, 944(1970)].4. K.K. Likharev, Rev. Mod. Phys. 51, 101 (1979).5. A.D. Zaikin, and G.F. Zharkov, Fiz. Nizk. Temp. 7, 375 (1981)[Sov. J. LowTemp. Phys. 7, 184 (1981)]; A.D. Zaikin, in: Nonequilibrium Superconductivity,ed. V.L. Ginzburg (Nova Science, N.Y., 1986), p. 57.6. H. Courtois, Ph. Gandit, and B. Pannetier, Phys. Rev. B 52 1162 (1995).7. G. Eilenberger, Z. Physik 214, 195 (1968).8. K.D. Usadel, Phys. Rev. Lett. 25, 507 (1970).9. M.Yu. Kupriyanov, Sov. J. Superconductivity 2, 5 (1989).10. A.A. Golubov, F. Wilhelm, and A.D. Zaikin, submitted to Phys. Rev. B.

Supercurrent in a mesoscopic proximity wire

https://publikationen.bibliothek.kit.edu/61497/2149

Supercurrent in a mesoscopic proximity wire

Abstract

Similar works

Full text

Available Versions

KITopen