We report a new nonlocal effect in vortex matter, where an electric current
confined to a small region of a long and sufficiently narrow superconducting
wire causes vortex flow at distances hundreds of inter-vortex separations away.
The observed remote traffic of vortices is attributed to a very efficient
transfer of a local strain through the one-dimensional vortex lattice, even in
the presence of disorder. We also observe mesoscopic fluctuations in the
nonlocal vortex flow, which arise due to "traffic jams" when vortex
arrangements do not match a local geometry of a superconducting channel.Comment: a slightly longer version of a tentatively accepted PR

A. K. Geim

A. I. Larkin

A. M. Campbell

D. Y. Vodolazov

E. H. Brandt

F. M. Peeters

H. R. Kerchner

I. V. Grigorieva

J. Osquiguil

J. W. Ekin

K. S. Novoselov

L. J. van der Pauw

M. Hesselberth

N. Toyota

P. H. Kes

S. V. Dubonos

English

arXiv

A nonlocal effect in vortex matter, where an electric current confined to a small region of a long and sufficiently narrow superconducting wire causes vortex flow at distance hundreds of intervortex separation away, was discussed. The remote traffic of vortices was attributed to a very efficient transfer of a local strain through the one-dimensional vortex lattice (VL). The nonlocal vortex flow was observed at distances up to ≈ 5μm. It was found that the mesoscopic fluctuations in the nonlocal vortex flow, which arise due to traffic disorder

Grigorieva, I. V.

Geim, A. K.

Dubonos, S. V.

Novoselov, K. S.

Vodolazov, D. Y.

Peeters, F. M.

Kes, P. H.

Hesselberth, M.

The University of Manchester - Institutional Repository

Long-range nonlocal flow of vortices in narrow superconducting channels

Grigorieva, I.V.

Geim, A.K.

Dubonos, S.V.

Novoselov, K.S.

Vodolazov, D.Y.

Peeters, F.M.

Kes, P.H.

Hesselberth, M.B.S.

Leiden University Scholary Publications

P H Y S I C A L R E V I E W L E T T E R S week ending11 JUNE 2004VOLUME 92, NUMBER 23Long-Range Nonlocal Flow of Vortices in Narrow Superconducting ChannelsI.V. Grigorieva, A. K. Geim, S.V. Dubonos, and K. S. NovoselovDepartment of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United KingdomD.Y. Vodolazov and F. M. PeetersDepartement Natuurkunde, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, BelgiumP. H. Kes and M. HesselberthKamerlingh Onnes Laboratorium, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands(Received 1 December 2003; published 8 June 2004)237001-1We report a new nonlocal effect in vortex matter, where an electric current confined to a small regionof a long and sufficiently narrow superconducting wire causes vortex flow at distances hundreds ofintervortex separations away. The observed remote traffic of vortices is attributed to a very efficienttransfer of a local strain through the one-dimensional vortex lattice (VL), even in the presence ofdisorder.We also observe mesoscopic fluctuations in the nonlocal vortex flow, which arise due to ‘‘trafficjams’’ when vortex arrangements do not match a local geometry of a superconducting channel.DOI: 10.1103/PhysRevLett.92.237001 PACS numbers: 74.25.Qt, 73.23.–b, 74.20.–z, 74.78.–wkind, which arises in the absence of a driving current due probe geometry.Phenomena associated with vortex motion in super-conductors have been subject to intense interest formany decades, as they are important both for applicationsand in terms of interesting, complex physics involved.Vortices start moving when the Lorentz force fL actingon them exceeds pinning forces arising from always-present defects. The force is determined by the localcurrent density j and, hence, the resulting vortex motionis confined essentially to the region where the appliedcurrent flows [1,2]. There are only a few cases knownwhere vortex flow becomes nonlocal (i.e., not limited tothe current region), most notably in Giaever’s flux trans-former [3] and in layered superconductors [4]. In theformer case, fL is applied to vortices in one of the super-conducting films comprising the transformer, while thevoltage is generated in the second film, due to electro-magnetic coupling between vortices in the two films [3,5].In layered superconductors, a drag effect (somewhat simi-lar to that in Giaever’s transformer) is observed due tocoupling between pancake vortices in different layers.Both nonlocal effects occur along vortices and are basi-cally due to their finite rigidity. A high viscosity of avortex matter can also lead to a nonlocal response in thedirection perpendicular to vortices [6–9]. In this case,local vortex displacements induced by j create secondaryforces on their neighbors pushing them along. Suchnonlocal correlations were observed in the vicinity ofthe melting transition in high-temperature superconduc-tors [8,9]. This is a dynamic effect where VL’s regions—generally moving at different speeds due to differentabove-critical currents—suddenly become locked in along-range collective motion. In the absence of a drivingcurrent, such viscosity-induced nonlocality is expected todie off at a few vortex separations [6,7].In this Letter, we report a nonlocal effect of a different0031-9007=04=92(23)=237001(4)$22.50 to a long-range collective response of a rigid one-dimen-sional (1D) VL and survives at strikingly long distances,corresponding to several hundred vortex spacings.Nonlocal vortex flow in our experiments is observed atdistances up to  5 m, provided a superconductingchannel contains only one or two vortex rows. To thebest of our knowledge, such nonlocality has neitherbeen observed nor considered theoretically.Our starting samples were thin films of amorphoussuperconductor MoGe (  60) with various thicknessesd from 50 to 200 nm. We have chosen amorphous filmsbecause they are known for their quality and very lowpinning and have been extensively studied in the past interms of pinning and vortex flow (see, e.g., [10,11]). Thesharp superconducting transitions ( < 0:1 K) measuredon mm-sized samples of our films indicate their highquality and homogeneity. The critical current jC in inter-mediate fields b  H=Hc2  0:3–0:6 was measured to be 102 A=cm2 (at 5 K), where H is the applied field andHc2 the upper critical field.jC increased several times atlower temperatures. The MoGe films were patterned intomultiterminal submicron wires of various widths w (be-tween 70 nm and 2 m) and lengths L (between 0.5 and12 m) using e-beam lithography and dry etching (seeFig. 1). Electrical measurements were carried out usingthe standard low-frequency (3 to 300 Hz) lock-in tech-nique at temperatures T down to 0.3 K. The results wereindependent of frequency, which proves that the measuredac signals are just the same as if one were using a dcmeasurement technique, provided the latter could allowthe same sensitivity ( < 1 nV). The external field H wasapplied perpendicular to the structured films. For brevity,we focus below on the results obtained in the nonlocalgeometry and omit discussions of the complementarymeasurements carried out in the standard (local) four-2004 The American Physical Society 237001-1FIG. 2. Dependence of RNL on length L and width w of thecentral wire. (a) Nonlocal resistance at T  6:0 K for differentwires (their L and w values are shown on the graph). Curves areshifted vertically for clarity. (b) Nonlocal resistance at itsmaximum value as a function of L (w  150 nm). The signalat 6.0 K is also representative of the behavior observed at lowerT. The dashed line is a guide to the eye. The inset showstemperature dependence of the field corresponding to the dis-appearance of R NL (solid circles). The solid line is Hc2Tmeasured on macroscopic films.FIG. 1. Nonlocal resistance RNL as a function of applied fieldH measured on a 150 nm wide wire at a distance of 1 mbetween the current and voltage leads. Different curves areshifted vertically for clarity (RNL is always zero in the normalstate). The inset shows an AFM image of the studied sample.The vertical wire in this image is referred to as central wire.Scale bar, 1 m.P H Y S I C A L R E V I E W L E T T E R S week ending11 JUNE 2004VOLUME 92, NUMBER 23The nonlocal geometry is explained in Fig. 1. Here, theelectric current is passed through leads marked I andI and voltage is measured at terminals V and V. Inthis geometry, the portion of applied current I thatgoes sideways along the central wire (see Fig. 1) andreaches the area between the voltage probes is negligiblysmall. Indeed, in both normal and superconducting states[12], the current along the central wire decays as / I expx=w, which means that the current density re-duces by a factor of 10 already at distances x  w and,typically, by 1010 in the nonlocal region (x  L) in ourexperiments. This also means that all vortices in thecentral wire, except for one or two nearest to the cur-rent-carrying wire, experience the current density manyorders of magnitude below the critical value. Therefore,no voltage can be expected to be observable in the non-local geometry. In stark contrast, our measurementsrevealed a pronounced nonlocal voltage VNL, whichemerged just below [13] the critical temperature TC andpersisted deep into the superconducting state (Fig. 1).The signal appeared above a certain value of H 0:2Hc2, reached its maximum at 0:5–0:7 Hc2 andthen gradually disappeared as H approached Hc2. VNLwas found to depend linearly on I that was variedbetween 0.2 and 5 A. At lower I, VNL became so small( < 100 pV) that it disappeared under noise, while highercurrents led to heating effects. The linear dependenceallows us to present the results in terms of resistanceRNL  VNL=I. With increasing L, RNL was found todecay relatively slowly (for L 	 4 m) and quickly dis-appeared for longer wires as well as for the wide ones(w  0:5 m) (Fig. 2). The general shape of RNLHcurves was identical for all samples but fluctuations(sharp peaks) seen in Fig. 1 varied from sample to237001-2sample. A closer inspection of the fluctuations for differ-ent samples shows that they have the same characteristicinterval of magnetic field over which RNL changes rapidly.This correlation field BC corresponds to the entry of oneflux quantum 0 into the area L  w between the currentand voltage leads, so that BC  0=L  w.To understand the nonlocal signal, we note that withinthe accessible range of I, its density inside the current-carrying wire was in the range of  103 to 105 A=cm2(i.e.,  jC) and, accordingly, caused a vortex flowthrough this wire. Indeed, whenever VNL was observed,measurements in the local geometry showed the behaviortypical for the flux flow regime. This indicates that thenonlocal resistance is related to the vortex flow in thecurrent-carrying part of the structures, which then some-how propagates along the central wire to the regionbetween V and V terminals, where no electric currentis applied. The mechanism of the propagation can beunderstood as follows. The Lorentz force —acting on237001-2FIG. 3. (a) Comparison of RNLH observed experimentally(lower curve; data of Fig. 1 at 6 K) with the nonlocal signalexpected in the proposed 1D model (upper curve) and withresults of our numerical analysis (middle curve). (b),(c)Snapshots of vortex configurations corresponding to pro-nounced changes in mobility of 1D vortex matter. Dashedlines indicate the average orientation of vortex rows in thecross areas.P H Y S I C A L R E V I E W L E T T E R S week ending11 JUNE 2004VOLUME 92, NUMBER 23vortices located at the intersection between current-carrying and central wires—pushes/pulls them alongthe central wire. In the absence of edge defects alongthis wire, the surface barrier prevents these vortices fromleaving a superconductor [14] and, hence, the local dis-tortion of the VL can be expected to propagate along thecentral wire, away from the current-carrying region. Ifthe vortex motion reaches the remote intersection be-tween the central and voltage wires, a voltage is generatedby vortices passing through this region. For an infinitelyrigid VL, such a local distortion would propagate anydistance. However, for a soft VL and in the presence ofdisorder, the lattice can be compressed and vortices be-come jammed at pinning sites. The softer the lattice, theshorter the distance over which the distortion is damp-ened. Note that, as we discuss a dc phenomenon, thereshould exist a constant flow of vortices through thesample. We believe that this is ensured by large contactregions that act as vortex reservoirs.The interplay between pinning and VL’s elasticity isimportant in many vortex phenomena, and the spatialscale, over which a VL behaves as almost rigid (respondscollectively), is usually determined by the correlationlength RC [15,16]. This concept had been successfullyused in the past to explain the behavior of jC in macro-scopic thin films, where the only relevant elastic modulusdefining RC is the shear modulus C66 [17,18]. For ourparticular films, the maximum value of RC can be esti-mated as  20a0(reached at  0:3Hc2) and then RCgradually reduces to  a0 as H approaches Hc2 (here,a0  0=B1=2 is the VL period and B the magneticinduction) [10,19]. This length scale is in agreementwith predictions [6,7] and clearly too short to explainthe observed RNL. For example, at 4.5 K, RNL was de-tected at distances up to 5 m and in fields up to 3.5 T.This means that the entire vortex ensemble between thecurrent and voltage wires, which is over 200 vorticeslong, is set in motion by a localized current.To explain these unexpectedly long-range correlations,we argue that the VL in mesoscopic wires is much morerigid than in macroscopic films due to its 1D characterand the presence of the edge confinement that preventstransverse vortex displacements. Indeed, if there are onlya few vortex rows in a narrow channel, the only possibledeformation of the lattice is via uniaxial compression.This deformation is described by compressional modulusC11  C66. In this case, the characteristic length, overwhich one should expect collective response, is muchlonger and given by another correlation length C C11=L1=2, where L  Fp=rp is a characteristic ofthe pinning strength, Fp  jC  B the bulk pinning force,and rp the pinning range (rp  a0=2 for b > 0:2)[2,20,21].To calculate CH we used the expression C11  0 B=2 0  2  a0  k expected for a 1D channel [22].Here,  is the field- and temperature-dependent penetra-237001-3tion depth [22,23] and k the wave vector of VL deforma-tion. Our numerical simulations show that the mostrelevant k is given by VL’s distortion in the cross-shapedregions [see Fig. 3(b)] and, accordingly, we assume k 1=w. The estimated C in intermediate fields at T  6 Kis  3–10 m, in agreement with our experiment. Theabove model also describes well the observed field de-pendences of RNL. The theory curve in Fig. 3(a) takes intoaccount that the nonlocal signal should decay as RNL /expL= C where C  0:w1=2=20:jC1=2 andthat, for narrow wires, it is thermodynamically unfavor-able for vortices to penetrate the narrow wires until Hreaches a critical value HS  0=!w. The latter effectis modeled by pinning at the surface barrier, which re-sults in an additional part in jC / expH=HS. Thedisappearance of RNL below 4 K is attributed to higherjC at lower T. Note that the exponential dependenceimplies that changes in jC by a factor of 4, which occurbelow 5 K, result in a rapid suppression of RNL.It is clear that the above description applies only towires that accommodate just a few vortex rows. As thenumber of rows increases, the VL gains an additional237001-3P H Y S I C A L R E V I E W L E T T E R S week ending11 JUNE 2004VOLUME 92, NUMBER 23(lateral) degree of freedom and a local compression be-comes dampened by both longitudinal and lateral defor-mations. One eventually expects a transition to the 2Dcase described by the shear modulus C66 and a muchshorter correlation length RC. In addition, as more vortexrows are added, elastic correlations are expected to be-come less relevant, as vortex dynamics becomes domi-nated by VL’s plastic deformation [22]. The latter isforbidden in a 1D case but in wider channels it canbecome a dominant mechanism for dampening of collec-tive flow. This qualitatively explains the disappearance ofRNL in wider wires.To support the discussed model further, we carried outdirect numerical simulations of nonlocal vortex traffic,using time-dependent GL equations. The middle curve inFig. 3(a) plots a typical example of the obtained fielddependence of RNL for the geometry shown in Fig. 3(b).One can see that the GL simulations reproduce the overallshape of RNLH observed experimentally. Furthermore,the numerical analysis allowed us to clarify the origin offluctuations inRNLH: they appear due to sudden changesin vortex configurations. Figure 3(b) shows such changesfor the field marked by the arrow in Fig. 3(a), where asharp fall in RNL is observed. Here, approximately twoadditional vortices enter the central wire, which resultsin a transition from an easy-flow vortex configuration(H  0:52Hc2) to a blocked one (0:56 Hc2). In the lattercase, vortices in the cross-shaped regions are distributedrather randomly and break down the continuity of thevortex rows formed in the central wire. This leads toblockage of collective vortex motion. For H  0:52 Hc2,vortices in the cross’ areas are more equally spaced, andthe corresponding vortex rows make a shallower anglewith rows in the central wire [Fig. 3(b)]. In this case, thereis less impediment to vortex motion through the crossregions which leads to a larger nonlocal voltage.The mechanism of the sudden blocking/unblocking ofvortex flow at different H becomes even clearer if oneconsiders an imaginary configuration containing just afew vortices—see Fig. 3(c). Here, we find a sharp fall inRNL when the number of vortices changes from 9 to 11(N  10 is a thermodynamically unstable state for thisgeometry). For N  9, the vortex row passes continu-ously through the whole central wire, allowing its motionas a whole when pushed or pulled along by a localizedcurrent. In contrast, for N  11, there is a vortex pair ineach of the crosses which prevents such vortex motion.In conclusion, we have observed pronounced flux flowat distances corresponding to hundreds of VL periodsfrom the region where applied current flows. We attributethe observed behavior to an enhanced rigidity of the237001-4vortex lattice confined in narrow channels and providea theoretical model for this.We thank M. Blamire, M. Moore, and V. Falko forhelpful discussions.[1] G. Blatter et al., Rev. Mod. Phys. 66, 1125 (1994).[2] E. H. Brandt, Rep. Prog. Phys. 58, 1465 (1995).[3] I. Giaever, Phys. Rev. Lett. 15, 825 (1965).[4] R. Busch et al., Phys. Rev. Lett. 69, 522 (1992); H. Safaret al., Phys. Rev. B 46, 14 238 (1992).[5] J.W. Ekin, B. Serin, and J. R. Clem, Phys. Rev. B 9, 912(1974).[6] M. C. Marchetti and D. R. Nelson, Phys. Rev. B 42, 9938(1990).[7] R. Wortis and D. A. Huse, Phys. Rev. B 54, 12 413 (1996);S. J. Phillipson, M. A. Moore, and T. Blum, Phys. Rev. B57, 5512 (1998).[8] D. López et al., Phys. Rev. Lett. 82, 1277 (1999).[9] Yu. Eltsev et al., Physica (Amsterdam) 341C–348C, 1107(2000); J. H. S. Torres et al., Solid State Commun. 125, 11(2003).[10] P. H. Kes and C. C. Tsuei, Phys. Rev. B 28, 5126 (1983);R. Wördenweber and P. H. Kes, ibid. 34, 494 (1986).[11] A. Pruymboom et al., Phys. Rev. Lett. 60, 1430 (1988);N. Kokubo et al., Phys. Rev. Lett. 88, 247004 (2002).[12] The dependence expx=w follows directly from theformalism introduced by L. J. van der Pauw, Philips Tech.Rev. 20, 220 (1958). We have also validated this formulafor the zero resistance state through numerical simula-tions using GL equations.[13] Very close to TC, we observed a nonlocal effect ofanother origin. This signal exhibits a different shapeand different L and w dependences (e.g., it could bedetected for any w but only for L 	 2 m). We attributethe near-TC signal to quantum interference corrections toconductivity [L. I. Glazman et al., Phys. Rev. B 46, 9074(1992)].[14] L. Burlachkov, Phys. Rev. B 47, 8056 (1993).[15] A. I. Larkin and Yu. N. Ovchinnikov, J. Low Temp. Phys.34, 409 (1979).[16] H. R. Kerchner, J. Low Temp. Phys. 50, 337 (1983).[17] P. H. Kes and C. C. Tsuei, Phys. Rev. Lett. 47, 1930 (1981).[18] E. H. Brandt, Phys. Rev. Lett. 50, 1599 (1983); J. LowTemp. Phys. 53, 71 (1983).[19] N. Toyota et al., J. Low Temp. Phys. 55, 393 (1984);J. Osquiguil,V. L. P. Frank, and F. de La Cruz, Solid StateCommun. 55, 222 (1985).[20] A. M. Campbell, J. Phys. C 2, 1492 (1969); 4, 3186 (1971).[21] E. H. Brandt, Phys. Rev. Lett. 67, 2219 (1991).[22] R. Besseling et al., Europhys. Lett. 62, 419 (2003).[23] E. H. Brandt, J. Low Temp. Phys. 26, 709 (1977); 26, 735(1977).237001-4

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Long-Range Nonlocal Flow of Vortices in Narrow Superconducting Channels

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Institutional Repository Universiteit Antwerpen

Long-range nonlocal flow of vortices in narrow superconducting channels

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The University of Manchester - Institutional Repository

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Institutional Repository Universiteit Antwerpen