Tuning the Influence of Microscopic Decoherence on the Superconducting ProximityEffect in a Graphene Andreev Interferometer

Fabio Deon, Sandra Šopić, and Alberto F. Morpurgo*

Département de Physique de la Matière Condensée (DPMC) and Group of Applied Physics (GAP),University of Geneva, 24 Quai Ernest Ansermet, 1211 Genève 4, Switzerland

(Received 24 April 2013; revised manuscript received 30 September 2013; published 26 March 2014)

We discuss transport measurements through graphene Andreev interferometers exhibiting reentrance ofthe superconducting proximity effect. We observe that at high gate voltage (VBG) the energy dependence ofthe Andreev conductance oscillations exhibits a scaling in agreement with theoretical expectations, whichbreaks down at low VBG, when the Fermi energy approaches the charge neutrality point. The phenomenonis a manifestation of single particle dephasing that increasingly limits the propagation of superconductingcorrelations away from the superconductor-graphene interface. Our work addresses the interplay betweenmicroscopic decoherence and superconductivity, and shows that graphene provides a useful experimentalplatform to investigate unexplored regimes and phenomena in the superconducting proximity effect.

DOI: 10.1103/PhysRevLett.112.126803 PACS numbers: 73.23.−b, 74.45.+c, 85.25.Dq

Superconductivity originates from microscopic correla-tions between electron and hole excitations in the vicinity ofthe Fermi level of a normal metal, corresponding to theformation of Cooper pairs [1]. The characteristic length scaleof these correlations—i.e., the size of a Cooper pair—is thesuperconducting coherence length ξ, which in conventionalmetals at low temperature is usually much shorter than thephase coherence length of individual electrons Lϕ. In thiscase, dephasing does not pose limits to the formation ofthe superconducting correlations, and superconductivity isonly suppressed by thermal excitations at finite temperature.In the opposite case, when Lϕ becomes comparable to ξ,dephasing competes with superconductivity and, if Lϕ

becomes shorter than ξ, it suppresses superconductingcorrelations. In conventional bulk superconductors thisregime is realized only rarely. One example is providedby s-wave superconductors with spin impurities, where—upon increasing the impurity concentration—dephasing firstresults in gapless superconductivity, and eventually in thecomplete destruction of the superconducting state [2]. Giventhe relevance of the problem, it is worth finding new exper-imental systems where the competition between supercon-ductivity and dephasing can be investigated under controlledconditions. To this end, induced superconductivity and prox-imity systems provide a significantly wider experimentalflexibility [3].Here we present an experimental investigation of induced

superconductivity, in nanoelectronic devices where theeffective strength of dephasing can be tuned. Specifically,we investigate the superconducting proximity effect (PE)through measurements of phase-modulated transport in adiffusive Andreev interferometer [4] whose normal region isa T-shaped graphene ribbon. We exploit the possibility toelectrostatically tune the carrier density in the ribbon over avery broad range. We find that at large densities induced

superconductivity is suppressed by thermal fluctuations, aspredicted by the conventional theory. As the carrier densityis lowered, however, a crossover occurs to a regime in whichthe behavior of the PE is determined by dephasing, when Lϕ

becomes much shorter than the lateral dimension of ourinterferometer. Next to illustrating the competition betweensuperconducting correlations and dephasing, our experi-ments show how graphene allows studying unexploredregimes of the superconducting proximity effect.Our investigations focus on a specific manifestation of

the PE, the so-called reentrance effect (RE) [5]. Thiscounterintuitive phenomenon consists in the nonmonotonicenergy dependence of the conductance of diffusive N=Sjunctions with highly transmissive interfaces. When thetemperature T is lowered from just above the criticaltemperature TC, the conductance first increases, and thenunexpectedly decreases so that, in the ideal case, at T ¼ 0 itreturns to the normal-state value GN , as if the PE wascompletely absent. A similar nonmonotonic trend is alsoobserved at low temperature, when measuring the differ-ential resistance as a function of applied bias V (i.e.,decreasing V from V > Δ=e to 0; Δ is the superconductinggap), and in Andreev interferometers, when looking at theamplitude of the conductance oscillations as a function ofbias or temperature. Although not yet observed experi-mentally, theoretical work indicates that the RE shouldoccur also in graphene-based systems [6].Theory treating the normal conductor in the diffusive

limit relates the energy dependence of the conductancechange δGðEÞ to the Thouless energy ET ¼ ℏD=L2 (L isthe length of the N region, and ET ≪ Δ) and to the normalstate resistance RN ¼ 1=GN [7]. The hallmark of the theoryis the universal scaling of the phenomenon in terms ofreduced variables, i.e., when RNδGðEÞ is plotted as afunction of E=ET [7]. Pioneering experiments have

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observed the RE in systems where the normal conductorwas either a thin metal film [8,9] or a two-dimensionalelectron gas (2DEG) [10,11], and found excellent qualita-tive agreement of the experiments with theoretical predic-tions (quantitative deviations, especially in the case of2DEGs, were attributed to device nonidealities). However,the universality of the scaling between RNδGðEÞ and E=EThas never been verified experimentally.Compared to devices used in the past, graphene Andreev

interferometers provide a key experimental advantage, asthey enable a stable and reproducible electrostatic tuningover a broad range of densities. This allows us to studyexperimentally the dependence of the RE on the transportparameters of the normal conductor. Figure 1(a) shows aSEM micrograph of one of our Andreev interferometers. Asingle-layer graphene flake is patterned into a T-shapedribbon, connected to a superconducting loop, to controlthe relative phase of the superconducting order parameterδφ ¼ 2πΦ=Φ0 (Φ is the magnetic flux threading the loopand Φ0 ¼ h=2e). The device is fabricated on grapheneexfoliated onto a degenerately doped Si wafer (coated witha 285 nm thick SiO2 layer) acting as a gate electrode. Twoprobes are placed on the graphene region below the T. Thesuperconducting loop and these electrodes consist of atrilayer of Ti=V=Au (layer thicknesses are 5=17=5 nm).Measurements (see the Supplemental Material [12]) onS=graphene=S microjunctions fabricated with the sameprocess used to realize the Andreev interferometers indicatethat the S=graphene contacts have very high transparencyfor all values of VBG. The superconducting gap extracted

from the subharmonic structure of these microjunctions isΔ ∼ 500 μeV, which matches the TC ¼ 3.5 K of the film.Experiments were performed in a filtered 3He system,

down to 250 mK. The single-layer nature of the grapheneflake was confirmed by two-probe quantum Hall measure-ments between contacts 3 and 4 [13]. The resistance acrosstheribbon,measuredatB ¼ 0asafunctionofVBG [Fig.1(b)],raises by 2 orders of magnitude near the charge neutralitypoint. The strong suppression of the low-bias conductancenear charge neutrality is typical of etched graphene nano-ribbons, and is related to disorder and electron-electroninteractions (EEI): scattering by edge-disorder increasesthe tendency of electrons toward (Anderson) localization,enhancing the effect of EEI. Indeed, in sufficiently narrowribbons this leads to the formation of a full transport gaporiginating from Coulomb blockade [14]. Finally, Fig. 1(c)shows the conductance oscillations induced by a smallmagnetic field that modulates the superconducting phase,originating from quantum interference of holes Andreevreflected at the two different superconducting contacts.As the device transport properties are determined by

quantum interference, we need to distinguish betweenensemble-averaged (EA) and sample-specific (SS) contri-butions in the measured quantities. The SS component ofthe Andreev oscillations, which consists in phase-coherentconductance fluctuations modulated by the superconduct-ing phase [10], is dominant at small energies or at largemagnetic fields, where the EA component is suppressed,respectively, by the RE or by the breaking of time reversalsymmetry (when the magnetic flux threading the grapheneT exceeds one flux quantum h=2e). The mean peak-to-peakamplitude of the SS oscillations, measured in the fieldrange 200 G < B < 500 G, is shown in Fig. 2(c) as afunction of VBG. In order to minimize the SS componentand to isolate the EA contribution, for each quantity ofinterest we averaged measurements for 25 different valuesof VBG, stepping the gate voltage just enough to cause achange in EF larger than the correlation energy π2ET [15].A plot of ET ¼ ℏD=L2 for positive VBG is shown inFig. 2(d). The diffusion constant D is estimated from thezero-bias conductivity, assuming a linear dispersion forgraphene [16] (we take L ¼ 1 μm, corresponding to thedistance between the bottom of the T and the super-conducting contacts; note how D decreases as the Fermilevel approaches charge neutrality, consistently with theexpected tendency toward localization).Figures 2(a) and 2(d) show the results of the ensemble

averaging (performed around VBG ¼ 50 and 60 V, respec-tively) for the magnetic field dependence of the linearconductance and for the bias-dependent differential con-ductance. Around B ¼ 0 the amplitude of the conductanceoscillations measured at VSD ¼ 40 μeV is not muchaffected by the averaging process, because at this biasthe EA contribution is larger than the SS one. On thecontrary, at higher magnetic field (B > 15 mT), where the

FIG. 1 (color online). (a)Falsecolorscanningelectronmicrographof one of our devices; graphene and the superconducting electrodesare colored in yellow and purple, respectively (contacts 1 and 2 arejoined by a ≃12 μm2 superconducting loop. The scale bar corre-sponds to 1 μm). (b) Resistance per square of the T-shaped grapheneribbon (outlined by dashes) versusVBG. (c) Four-probe conductanceG3;2j4;1, periodically modulated by the flux threading the loop(measured at T ¼ 250 mK with an applied bias VSD ¼ 40 μV).

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EA contribution is suppressed, averaging over N tracessuppresses the amplitude proportionally to N−1=2 [seeFig. 2(e)]. Similarly, individual dI=dV curves are asym-metric and exhibit random bias dependent features,whereas the EA curve is symmetric [thick versus thin linesin Fig. 2(d)].Having established the averaging procedure, we look in

detail at the EA phase-modulated oscillations at largecharge density (VBG ¼ 60 V). The low-field conductanceoscillations are plotted in Fig. 3(a) at T ¼ 250 mK, for VSDbetween 0 and 0.55 mV. Similar measurements have beendone as a function of temperature, for VSD ¼ 0. The VSDand T dependence of the peak-to-peak amplitude of thefirst and second harmonic are shown in Figs. 3(b) and 3(c).The first harmonic exhibits reentrance in the bias and in thetemperature dependence, with the oscillation amplitudehaving a maximum at an energy (i.e., either bias ortemperature) comparable to the Thouless energy[ET ≈ 45 μeV at VBG ¼ 60 V, see Fig. 2(d)]. The secondharmonic, on the contrary, shows no reentrance. This isexpected, because the trajectories causing conductanceoscillations with twice the frequency have to Andreevreflect at both superconducting electrodes, and are therefore

longer (by approximately twice the distance between theS contacts). The effective Thouless energy associated tothese trajectories is therefore significantly smaller than ET ,so that the energy at which reentrance would occur for thesecond harmonic is smaller than the lowest temperaturereached in the experiment [17].We now analyze the evolution of the energy dependence

of the oscillations as a function of VBG. Figure 4(a) showsthe bias dependence of the EA oscillation amplitude (firstharmonic) for seven different values of VBG between 60and 12.5 V. Upon lowering VBG, the maximum oscillationamplitude decreases, qualitatively in line with the theory,because RN increases [see Fig. 1(c)]. The value of VSD forwhich the maximum oscillation amplitude occurs, however,remains essentially unchanged. Within the existing theory,this finding is inconsistent with the value of the Thoulessenergy ET, which changes from ∼45 to ∼10 μeV as VBG islowered [Fig. 2(d)]. For a more quantitative analysis, welook at the data in terms of normalized quantities, i.e.,plotting RNδG versus eVSD=ET [Fig. 4(b)]. At largedensities, for VBG ¼ 60, 50, and 40 V, the rescaled curvesfall on top of each other, as expected. When lowering VBGbelow 30 V, however, deviations from perfect scalingbecome progressively larger: the maximum relative oscil-lation amplitude decreases, and shifts to larger eVSD=ETratios. Both trends observed in the range 12.5V < VBG <30 V are in conflict with what is predicted by existingtheory.The suppression in the amplitude of the Andreev

oscillations and the shift of their maximum toward higherenergy can be explained by a progressive shortening of Lϕ.

FIG. 2 (color online). (a) Magnetoconductance (backgroundsubtracted), averaged over N ¼ 1, 5, and 25 VBG values around50 V (VSD ¼ 40 μV; the curves are offset for clarity). (b) Thou-less energy ET extracted from the VBG dependent resistanceof the device. (c) Amplitude of the large-field (200 G <B < 500 G) sample-specific conductance oscillations. (d) Ensem-ble-averaged (thick line) versus individual dI=dV curves (thinlines) measured around VBG ¼ 60 Vat T ¼ 250 mK. (e) Sample-specific oscillation amplitude versus the number of averagedexperimental curves.

FIG. 3 (color online). (a) EA conductance oscillations at VBG ¼60 V and T ¼ 250 mK, for VSD varying from 0 (bottom curve) to0.5 mV (top; curves offset for clarity). (b) Bias dependence of theamplitude of the first and second harmonics of the oscillationsshown in (a). (c) Temperature dependence of the harmonics of thezero-bias EA oscillations.

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A finite value of the phase-breaking length, which is takento be infinite in the simplest theory, introduces a cutoff forthe penetration of the pair amplitude in N. When Lϕ dropsbelow the size of the device L, Eϕ ¼ ℏD=L2

ϕ takes the roleof ET in determining the energy-scale of the reentrance.Having a new energy scale (next to ET) explains thedeviations from scaling on the energy axis. Dephasingobviously also explains why the amplitude of the proximityeffect decreases, since trajectories with a length larger thanLϕ cannot contribute to phase coherent effects. For a veryclosely related problem, namely how a reduction in Lϕ

affects the RE in the conductance of a single NS junction, aquantitative analysis has been performed by Charlat et al.[8], using linearized Usadel equations in which a finitevalue of Lϕ was introduced phenomenologically. The resultis illustrated in the inset of Fig. 4(b), which shows the bias-dependent linear conductance for five different values ofthe phase coherence length. At a qualitative level theevolution of the energy dependence reproduces the behaviorof the Andreev conductance oscillations measured in ourexperiments for different values of VBG, supporting the ideathat the behavior of the Andreev oscillations at low carrierdensity is governed by the shortening of Lϕ. Our devices,therefore, allow the continuous tuning between two regimes,with induced superconductivity being suppressed by thermalfluctuations at high gate voltage, and by electron dephasing

at low gate voltage, with dephasing eventually completelysuppressing the superconducting correlations.As for the origin of dephasing, definite conclusions

cannot be drawn at the moment. The progressive increaseof the influence of EEI as VBG approaches the charge-neutrality point, is a possible explanation. Evidence forsuch a scenario is provided by the evolution as a functionof VBG of the bias-dependent differential conductancedI=dVðVSDÞ, shown in Figs. 5(a)–5(c). At VBG ¼ 60 V(top panel) Andreev reflection results in a clear conduct-ance increase. At higher temperature (3.5 K) the conduct-ance enhancement extends to all subgap voltages while atlow T a conductance dip appears at low bias (i.e., thephenomenology of the RE). Upon lowering VBG to 20 V[Fig. 5(b)], the conductance enhancement at subgap voltagebecomes significantly less pronounced; eventually, for VBGsufficiently close to charge neutrality [VBG ¼ 12.5 V,Fig. 5(c)] no enhancement of dI=dVðVSDÞ is observed,and only a suppression persists, which occurs on an energyscale larger than the superconducting gap. This suppressionis what is typically seen in low-dimensional systems wheredynamical Coulomb blockade becomes relevant [18].Dephasing, however, may also have a different origin.Indeed, earlier weak-localization measurements in largegraphene flakes [19] have revealed an unexpected satu-ration of the dephasing time τϕ at low temperatures, abehavior that is not yet understood and that appearsincompatible with the sole effect of EEI. More experiments

FIG. 4 (color online). (a) Bias-dependent amplitude of the EAoscillations (first harmonic) for −10 G < B < 10 G, measured atT ¼ 250 mK and 12.5 V < VBG < 60 V. (b) Same data as in (a),plotted in dimensionless units: the curves measured forVBG ¼ 60, 50, and 40 V exhibit a perfect scaling, which breaksdown starting from VBG ¼ 30 V. Inset: calculated conductanceof a diffusive NS junction for decreasing values of Lϕ=L (∞,black curve, 0.6, 0.4, 0.3, 0.2, orange curve), using the linearizedUsadel equations, as in Ref. [8].

FIG. 5 (color online). (a)–(c) Ensemble-averaged dI=dV curvesbetween 265 mK (blue) and 3.5 K (red), for different values ofVBG (60, 20, and 12.5 V, respectively). With approaching chargeneutrality, the conductance enhancement due to Andreev reflec-tion visible in (a) is suppressed (b), and eventually completelydisappears (c).

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are needed, and possibly, the study of the PE may provide auseful tool to understand what limits τϕ in graphene.In conclusion, we have used graphene-based Andreev

interferometers to investigate and control the influence ofmicroscopic dephasing on induced superconductivity. Ourresults show how the possibility to gate-tune normal transportin graphene is particularly effective to investigate unexploredregimes of the superconducting proximity effect [3,20].

We thank A. Ferreira for technical support. Financialsupport from the ESF ENTS project, SNF, and NCCR-QSIT is gratefully acknowledged.

*Alberto.Morpurgo@unige.ch[1] P. G. de Gennes, Superconductivity of Metals and Alloys

edited by D. Pines (W. A. Benjamin, New York, 1966).[2] K. Maki, in Superconductivity, edited by R. D. Parks

(Dekker, New York, 1969) Chap. 18, pp. 1035–1105.[3] A. Allain , Z. Han, and V. Bouchiat, Nat. Mater. 11, 590

(2012); L. Serrier-Garcia, J. C. Cuevas, T. Cren, C. Brun,V. Cherkez, F. Debontridder, D. Fokin, F. S. Bergeret, andD. Roditchev, Phys. Rev. Lett. 110, 157003 (2013).

[4] H. Pothier, S. Guéron, D. Esteve, and M. H. Devoret, Phys.Rev. Lett. 73, 2488 (1994); A. Dimoulas, J. P. Heida, B. J.van Wees, T. M. Klapwijk, and W. van de Graaf, Phys. Rev.Lett. 74, 602 (1995); V. T. Petrashov, V. N. Antonov, P.Delsing, and T. Claeson, Phys. Rev. Lett. 74, 5268 (1995).

[5] S. N. Artemenko, A. F. Volkov, and A. V. Zaitsev, SolidState Commun. 30, 771 (1979); C. W. J. Beenakker, Phys.Rev. B 46, 12841 (1992); C. J. Lambert and R. Raimondi, J.Phys. Condens. Matter 10, 901 (1998).

[6] A. Ossipov, M. Titov, and C.W. J. Beenakker, Phys. Rev. B75, 241401 (2007).

[7] Y. V. Nazarov and T. H. Stoof, Phys. Rev. Lett. 76, 823(1996); T. H. Stoof and Y. V. Nazarov, Phys. Rev. B 53,14496 (1996); A. Volkov, N. Allsopp, and C. J. Lambert, J.Phys. Condens. Matter 8, L45 (1996).

[8] P. Charlat, H. Courtois, Ph. Gandit, D. Mailly, A. F. Volkov,and B. Pannetier, Phys. Rev. Lett. 77, 4950 (1996); H.Courtois, P. Charlat, Ph. Gandit, D. Mailly, and B. Pannetier,J. Low Temp. Phys. 116, 187 (1999).

[9] C. J. Chien and V. Chandrasekhar, Phys. Rev. B 60, 15356(1999); V. T. Petrashov, R. Sh. Shaikhaidarov, P. Delsing,and T. Claeson, JETP Lett. 67, 513 (1998).

[10] S. G. den Hartog, C. M. A. Kapteyn, B. J. vanWees, T. M.Klapwijk, and G. Borghs, Phys. Rev. Lett. 77, 4954 (1996);S. G. den Hartog, C. M. A. Kapteyn, B. J. vanWees, T. M.Klapwijk, and G. Borghs, Phys. Rev. Lett. 76, 4592 (1996);S. G. den Hartog, B. J. van Wees, T. M. Klapwijk, Y. V.Nazarov, and G. BorghsPhys. Rev. B 56, 13738 (1997).

[11] E. Toyoda, H. Takayanagi, and H. Nakano, Phys. Rev. B 59,R11653 (1999).

[12] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.112.126803 for repre-sentative experimental data. Reference [21] is also included.

[13] Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature(London) 438, 201 (2005); K. S. Novoselov, A. K. Geim,S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva,S. V. Dubonos, and A. A. Firsov, Nature (London) 438, 197(2005).

[14] M. Y. Han, B. Özyilmaz, Y. Zhang, and P. Kim, Phys. Rev.Lett. 98, 206805 (2007); X. Liu, J. B. Oostinga, A. F.Morpurgo, and L. M. K. Vandersypen, Phys. Rev. B 80,121407 (2009); F. Molitor, A. Jacobsen, C. Stampfer, J.Güttinger, T. Ihn, and K. Ensslin, Phys. Rev. B 79, 075426(2009); M. Y. Han, J. C. Brant, and P. Kim, Phys. Rev. Lett.104, 056801 (2010); J. B. Oostinga, B. Sacépé, M. F.Craciun, and A. F. Morpurgo, Phys. Rev. B 81, 193408(2010); P. Gallagher, K. Todd, and D. Goldhaber-Gordon,Phys. Rev. B 81, 115409 (2010).

[15] P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622(1985).

[16] N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, andB. J. van Wees, Nature (London) 448, 571 (2007); S. Russo,J. B. Oostinga, D. Wehenkel, H. B. Heersche, S. S. Sobhani,L. M. K. Vandersypen, and A. F. Morpurgo, Phys. Rev. B77, 085413 (2008).

[17] Our measurements far from charge neutrality reproduce whatwas found in 2DEG-based interferometers [10], havingdimensions, a diffusion constant, and carrier density com-parable to those of our graphene devices atVBG ¼ 60 V. Thisobservation indicates that the Dirac nature of electrons doesnot play an important role in disordered graphene on SiO2

(see also X. Du et al. in [20]). Indeed, possibly the onlytheoretically predicted manifestation of the Dirac nature ofcarriers on proximity effect in graphene is the so-calledspecular Andreev reflection [C.W. J. Beenakker, Phys. Rev.Lett. 97, 067007 (2006)], which requires extremely cleansamples, to be able to gate tune the Fermi energy near thecharge neutrality point on a scale of the superconducting gap(≃0.5 meV). This regime is not relevant here, since forgraphene on SiO2 potential fluctuations are approximately 2orders of magnitude larger.

[18] See, e.g., R. Egger and A. O. Gogolin, Phys. Rev. Lett. 87,066401 (2001); A. Bachtold, M. de Jonge, K. Grove-Rasmussen, P. L. McEuen, M. Buitelaar, and C. Schönen-berger, Phys. Rev. Lett. 87, 166801 (2001); A. Kanda, K.Tsukagoshi, Y. Aoyagi, and Y. Ootuka, Phys. Rev. Lett. 92,036801 (2004); Y. V. Nazarov and Y.M. Blanter, QuantumTransport (Cambridge University Press, 2009).

[19] F. V. Tikhonenko, D. W. Horsell, R. V. Gorbachev,and A. K. Savchenko, Phys. Rev. Lett. 100, 056802(2008).

[20] H. B. Heersche, P. Jarillo-Herrero, J. B. Oostinga, L. M. K.Vandersypen, and A. F. Morpurgo, Nature (London) 446, 56(2007); X. Du, I. Skachko, and E. Y. Andrei, Phys. Rev. B77, 184507 (2008); T. Dirks, T. L. Hughes, S. Lal, B. Uchoa,Y.-F. Chen, C. Chialvo, P. M. Goldbart, and N. Mason, Nat.Phys. 7, 386 (2011).

[21] K. Flensberg, J. Bindslev Hansen, and M. Octavio, Phys.Rev. B 38, 8707 (1988).

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