Current reversal in collective ratchets induced by lattice instability
L. Dinis, E.M. Gonzalez, J.V. Anguita, J.M.R. Parrondo, J.L. Vicent
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Current reversal in collective ratchets induced by lattice instability
L. Dinis, E.M. Gonz´alez, J.V. Anguita, J.M.R. Parrondo, and J.L. Vicent Departamento de F´ısica At´omica, Molecular y Nuclear and GISC,Universidad Complutense de Madrid, 28040-Madrid, Spain Departamento de F´ısica de Materiales, Universidad Complutense de Madrid, 28040-Madrid, Spain Instituto de Microelectr´onica de Madrid, Consejo Superior de Investigaciones Cient´ıficas, Tres Cantos, 28670-Madrid, Spain (Dated: October 30, 2018)A collective mechanism for current reversal in superconducting vortex ratchets is proposed. Themechanism is based on a two-dimensional instability of the ground state ( T = 0) of the system.We illustrate our results with numerical simulations and experiments in Nb superconducting filmsfabricated on top of Si substrates with artificially induced asymmetric pinning centers. PACS numbers: 05.40.-a, 02.30.Yy, 74.25.Qt, 85.25.-j
I. INTRODUCTION
Rectification of motion and fluctuations in thenanoscale is becoming a major field of research . Rec-tifying mechanisms or ratchets have been used to ex-plain how protein motors work and to design newseparation techniques or synthetical chemical motors .Superconductors have become a powerful tool to studyratchet mechanisms . Recently, a superconducting vor-tex ratchet device has been reported by Villegas et al . Inthat experiment, the rocking ratchet mechanism is due toa superconducting film patterned with a lattice of asym-metric potentials acting as pinning centers. An input accurrent yields an output dc voltage in the superconduct-ing film, revealing a rich phenomenology of single andmultiple current reversals , very sensitive to the un-derlying vortex dynamics. The interest of vortex ratch-ets is then twofold: it reveals new collective rectificationmechanisms and sheds light on the physics of vortices insuperconductors.Current reversals in vortex ratchets have been ex-plained, in the framework of one-dimensional models, bythe coexistence of pinned and interstitial vortices mov-ing in opposite directions or by the interaction betweenvortices within the pinning centers in one-dimensionalchannels . Olson and Reichhardt have studied numeri-cally a two-dimensional model, obtaining current reversalonly when interstitial vortices are present in the groundstate. They provide intuitive explanations of differentrectification mechanisms based on local interactions be-tween vortices and pinning centers.In this Letter we experimentally show that current re-versal can also occur when interstitial vortices are absentin the ground state (T=0). Remarkably, current rever-sal disappears increasing either the pinning strength orthe temperature. We also present numerical simulationsof vortices as interacting Brownian particles in two di-mensions, indicating that this current reversal is due toa new collective effect: an instability of the ground state,selective to the sign of the applied force. The influence oflattice instabilities on rectification has been also recentlyanalyzed by Lu et al. , for two dimensional vortices in a substrate with a one-dimensional modulation. II. EXPERIMENTAL RESULTS
For our experiments, two types of asymmetric pinningcenters have been fabricated: magnetic (Ni) nanotrian-gles and non-magnetic (Cu) nanotriangles. The vortexpinning force is enhanced by magnetic centers in com-parison with non-magnetic centers . The nanotriangleswere fabricated using e-beam lithography techniques andSi (100) wafers as substrates. The Ni or Cu arrays ofnanotriangles, on top of the substrate, are covered witha sputtered Nb thin film. Ni or Cu thickness (trianglesheight) is 40 nm and Nb film is 100 nm thick. Fur-ther details on this fabrication technique can be foundelsewhere . For the present work, we have fabricatedarrays with the same nanotriangle dimensions and arrayperiodicity than those in Ref. 8.We have measured magnetotransport in these films us-ing a commercial He cryostat. The variable tempera-ture insert allows controlling temperature with stabilityof 1 mK. For these experiments, samples were patternedwith a cross-shaped measuring bridge , by using opticallithography and ion-etching. This patterned bridge al-lows us to control the Lorentz force on the vortices in themixed state: taking into account ~F L = ~J × ~nφ (with φ = 2 . × − Wb and ~n a unitary vector parallel tothe applied magnetic field). On the other hand, from theexpression for the electric field ~E = ~B × ~v , where ~B isthe applied magnetic field and ~v the vortex-lattice veloc-ity, we can calculate this velocity v = V / ( dB ) from themeasured voltage drops V ( d being the distance betweencontacts). See Ref. 8 for more experimental details.The dc magnetoresistance in the mixed state of sam-ples with periodic arrays of pinning centers exhibits well-known commensurability phenomena , in which min-ima develop as a consequence of geometrical matchingbetween the vortex-lattice and the underlying periodicstructure. These minima are equally spaced (two neigh-bor minima are always separated by the same magneticfield value). For example, in the case of square arraysof nanostructured pinning centers, minima appear at ap-plied magnetic fields H m = n ( φ /a ), where a is the lat-tice parameter of the square array. Hence, the number ofvortices n per array unit cell can be known by simple in-spection of the dc magnetoresistance R ( H ) curves. More-over, for non-magnetic pinning centers Mkrtchyan andShmidt have given a rough estimation of the maximumnumber of vortices that can be pinned in each center,which could confirm the matching field minima values.This filling factor can be calculated as the ratio betweenhalf the dimension of the pinning center (the triangle sideis around 650 nm) and two times the superconduct-ing coherence length (around 60 nm for these samplesand temperatures close to the critical temperature) .In our samples this rough estimation gives us approxi-mately three vortices per triangle, in agreement with thematching field data (see also 8). Therefore, we know, forselected values of the applied magnetic field, how manyvortices there are per unit cell and where they are, this is,if they are interstitial vortices or vortices in the pinningcenters.We want to underline that collective behavior of vor-tices in films with periodic pinning is crucial to under-stand vortex lattice reconfiguration effects or vortexchanneling . Also, collective effects in ratchets havebeen shown to yield new interesting phenomena .Measurements of the vortex lattice average velocity asa function of the applied force are depicted in Fig. 1(a)for sample (A), Ni triangles, and three vortex per triangle( n = 3). There is no current reversal in the rectifiedsignal in agreement with the explanation given in Ref. 8(see below). Experiments for sample (B), Cu triangles,also for n = 3 (Fig. 1(b)) show a similar behavior for T = 8 . T = 0 . T c ), but a current reversal appearsdecreasing the temperature, despite the fact that thereare still three vortices per triangle.In the one-dimensional approach reported in Ref. 8,the vortex lattice does not play any role, and the currentreversal was explained assuming that pinned and inter-stitial vortices are rectified in opposite directions. How-ever, this assumption fails to explain current reversal forthe matching field n = 3, since in this case there areno interstitial vortices in the ground state of the system.The simulations performed by Olson and Reichhardt do not exhibit any current reversal for n = 3 and therectification mechanisms that they propose do not applyfor this case either. Therefore, a new explanation of cur-rent reversal is required. We have found by numericalsimulations such an explanation based on the interplaybetween the vortex lattice and the geometry of the tri-angular defects. (a) (b) FIG. 1: Net velocity of vortices versus ac Lorentz force am-plitude ( ω = 10kHz). Array periodicity 770 nm, trianglebase 620 nm. (a) Sample (A): Nb film with Ni triangles T c = 8 . K , T /T c = 0 .
99 ( T = 8 . (cid:3) ), T /T c = 0 . T = 8 . ◦ ). (b) Sample (B): Nb film with Cu triangles T c = 8 . T = 8 . (cid:3) ), T = 8 . × ), T = 8 . ◦ ), T = 8 . ♦ ). The negative velocity part of each curveis shown in the inset. III. THEORETICAL MODELA. Simulations
The simulations have been performed by numericallysolving Langevin equations for the movement of the vor-tices η ˙ x = − ∂ x U vv + ∇ V p + F ext + ξ ( t ) . (1) F ext is the Lorentz force resulting from the appliedcurrent, U vv the usual vortex-vortex interaction (seeRef. ), V p the pinning potential, η the friction coeffi-cient, and ξ ( t ) a white gaussian thermal noise. The pin-ning force and vortex-vortex interaction must be in agree-ment with the experimental situation of three vortices perpinning site (triangle). The interaction of the vorticeswith the pinning defects is modelled by a potential V p in the shape of a triangle and with a hyperbolic tangentprofile. The depth of the potential is V p = 0 .
002 pN µ m, F(pN) -50050100150200 v(m/s) v(m/s) F(pN)
FIG. 2: Simulation results. Three vortices per triangle, pin-ning 0 .
002 pN µ m. Inset: pinning 0 . µ m. so that depinning forces have typical values of the orderof 10 − N/m. Experiments show that the system isadiabatic in the region of frequencies used. This allowsus to obtain the expected ac signal from the velocity-force response curves obtained in simulations, both forconstant positive and negative applied force.Results are shown in Fig. 2 where a window of down-ward rectification can be observed. Let us stress atthis point that the “natural” direction of rectificationfor pinned vortices is upwards. The reason is that thevortex feels a smaller force at the tip of the triangle thanat the triangle base. The force at the tip can be, for anequilateral triangle, as lower as half the force at the baseof the triangle.
B. Ground state instability
What is then the origin of the downward rectificationfor small forces? Our simulations indicate that the rec-tification is due to a two-dimensional instability of theground state under small downward forces. The groundstate of the system consists of vortices located in the cor-ners of each triangle, without interstitial vortices. How-ever, for finite temperature, there are some “defective tri-angles” in the actual configuration of vortices, i.e., sometriangles which have only two vortices inside. Accord-ingly, some interstitial vortices will be randomly spreadalong the sample. Fig. 3 shows one of these configura-tion with only one defective triangle out of 6 ×
6. For lowforces, motion is induced by the interstitial vortices both in the downward and upward direction. However, con-trary to the picture presented in previous works , there isnot a continuous motion of interstitial vortices along thespace between triangles: the interstitial vortex enters thenearest triangle expelling one of the three vortices inside.The aforementioned instability, selective to the sign ofthe external force, is shown in Fig. 3. We have chosen aninitial condition with only one interstitial vortex and one t =0 t τ = FF FIG. 3: Downward rectification mechanism at T = 0 K: snap-shots from simulations. Initial condition (left) and configu-rations after evolution ( τ = 6 . × − s) with positive force F = 0 . − F (right, down).Dashed circles show the defective triangle and the only inter-stitial present in the initial condition. Thick arrows point outthe only two columns presenting any motion when positiveforce is applied. Comparison of the right panels shows thatthe number of interstitial vortices ( × ) dramatically increaseswhen the external force points downwards. defective triangle, located at some distance. In the rightfigures we have plotted the configurations of vortices af-ter an upward and downward force has been applied for τ = 6 . × − seconds (long enough for a depinned vor-tex to cover the whole sample several times), respectively.This simulations evolved at zero temperature to show themechanism in a clearer manner. We see that, in the caseof the upward force, the interstitial vortex remains in onesingle column . The column with a defective triangle alsomoves, but again vortices remain in that column. As aconsequence, there is a positive current of vortices butthe motion is constrained to two columns. When the in-terstitial vortex enters a triangle, the top vortex in thetriangle moves upwards, out of it and into the followingone. In addition to that, in the defective triangle, oneof the two vortices always escapes through the tip, be-coming and interstitial vortex and triggering a processsimilar to that in the column with an extra vortex. Bothmotions propagate along the column without disturbingthe neighboring columns.On the other hand, under a downward force, the ini-tial defect in the ground state propagates along the wholesample, as it is clearly shown in Fig. 3. A more detailedanalysis of the simulations show that, when an interstitialvortex enters a triangle, one of the two bottom vorticesis expelled but now can move to a triangle in one of thenearest columns . It even happens frequently that the two vv − U r vv −7 m(10 ) J m − ( ) FIG. 4: Vortex-vortex interaction versus vortex-vortex dis-tance. Solid line: interaction used in simulations corre-sponding to current reversal (see Fig. 2, pinning V p =0 . µ m.). Dashed line: interaction used in simulationsshowing no current reversal, which are depicted in the inset(pinning V p = 0 . µ m). vortices in the base of the triangle are depinned. Con-sequently, the initial defect is then spread out along thehorizontal direction, yielding a considerable large fractionof depinned vortices which increase the overall motion inthe system, yielding a net downward rectification and thecorresponding current reversal.As was noted before, current inversion may disappearwhen the pinning potential is increased. This effect wasobserved in experiments where the Cu triangles were re-placed by Ni ones which have a higher pinning strength(see Fig. 1(a)), and it is also reproduced in our simula-tions. Results for pinning potential V p = 0 . µ mdo not show current reversal, as depicted in the insetin Fig. 2. The increased pinning strength implies thata larger force is needed to depin vortices when pushingdownwards. According to simulations, when the same force is applied upwards, the aforementioned columnarmovement starts, but also neighboring columns withoutdefective triangles or extra vortices depin, and eventuallymotion spreads all over the sample. In other words, forthese moderate values of the force, the instability appearsin both directions.Finally, current reversal in sample (B), Cu triangles,also vanishes when temperature is raised (see Fig. 1(b)).We can explain this behavior taking into account that thepenetration depth of the superconductor increases withtemperature. As a result, the vortex-vortex interactionstrength, U vv , decreases at short distance but its rangebecomes longer , as depicted in Fig. 4. In this case weobserve that the long range vortex lattice order precludesthe instability responsible for current reversal (see theinset in Fig. 4). IV. CONCLUSIONS
In summary, our simulations show that the asym-metric substrate induces an instability sensitive to thedirection of the external force, affecting rectification.This effect can explain the current reversal observed inour experiments for n = 3. Moreover, our work indi-cates that the lattice configuration, and consequentlyits dynamical properties, can be controlled by exter-nal forces. This interplay between rectification, drivingforces and lattice configuration can induce other inter-esting phenomena such as transitions between differentlattice configurations , and could help to design newrectifying devices, not only in superconducting films, butalso in other two dimensional collective systems, such asJosephson arrays or colloidal suspensions.We acknowledge support by Spanish Ministerio de Ed-ucaci´on y Ciencia (NAN04-09087, FIS05-07392, MAT05-23924E, MOSAICO), CAM (S-0505/ESP/0337) andUCM-Santander. Computer simulations were performedat “Cluster de c´alculo para T´ecnicas F´ısicas” of UCM,funded in part by UE-FEDER program and in part byUCM and in the “Aula Sun Microsystems” at UCM. P. Reimann, Phys. Rep. , 57-265 (2002); Special issueed. by H. Linke, Appl. Phys. A: Mater. Sci. Process. ,167 (2002). G. Bar-Nahum, V. Epshtein, A.E. Rukenstein, R. Rafikov,A. Mustaev and E. Nudler, Cell , 183-193 (2005). J.C.M. Gebhardt, A.E.M. Clemen, J. Jaud, and M. Rief,Proc. Nat. Ac. Sci. , 8680 (2006). J. Rousselet, L. Solome, A. Ajdari, and J. Prost, Nature , 446 (1994). V. Serreli, C.F. Lee, E.R. Kay, and D.A. Leigh, Nature , 523 (2007). T.J. Huang, B. Brough, C.M. Hoa, Y. Liu, A.H. Flood,P.A. Bonvallet, H.R. Tseng, J. F. Stoddarta, M. Baller,and S. Magonov, Appl. Phys. Lett. , 5391 (2004). I. Zapata, R. Bartussek, F. Sols, and P. H¨anggi, Phys. Rev.Lett. , 2292 (1996); C. S. Lee, B. Janko, L. Der´enyi, and A. L. Bar´abasi, Nature , 337 (1999); J. F. Wambaugh,C. Reichhardt, C. J. Olson, F. Marchesoni, and F. Nori,Phys. Rev. Lett. , 5106 (1999). J.E. Villegas, S. Savelev, F. Nori, E.M. Gonzalez, J.V.Anguita, R. Garcia, and J.L. Vicent, Science , 1188(2003). C. C. de Souza Silva, J. V. de Vondel, M. Morelle, and V.V. Moshchalkov, Nature , 651 (2006). L. Dinis, E.M. Gonzlez, J.V. Anguita, J.M.R. Parrondo,and J.L. Vicent, New J. Phys. , 366 (2007). C. J. Olson-Reichhardt, and C. Reichhardt, Physica C , 125 (2005). Q. Lu, C.J. Olson Reichhardt, and C. Reichhardt, Phys.Rev. B , 054502 (2007). Y. Jaccard, J. I. Martin, M. C. Cyrille, M. Velez, J. L.Vicent, and I. K. Schuller, Phys. Rev. B , 8232 (1998). J. I. Martin, Y. Jaccard, A. Hoffmann, J. Nogues, J. M.George, J. L. Vicent, and I. K. Schuller, J. Appl. Phys. ,411 (1998). O. Daldini, P. Martinoli, J.L. Olsen, G. Berner, Phys. Rev.Lett. , 218 (1974); A. Pruymboom, P.H. Kes, E. van derDrift, S. Radelaar, Phys. Rev. Lett. , 1430 (1988); Y.Otani , B. Pannetier, J.P. Nozi`eres, D. Givord, J. Magn.Magn. Mater. , 622 (1993); M. Baert, V. Metlushko,R. Jonckheere, V.V. Moshchalkov, and Y. Bruynseraede,Phys. Rev. Lett. , 3269 (1995); D.J. Morgan, J.B. Ket-terson, Phys. Rev. Lett. , 3614 (1998). J.I. Mart´ın, M. V´elez, J. Nogu´es, I.K. Schuller, Phys. Rev.Lett. , 1929 (1997). G.S. Mkrtchyan and V.V. Shmidt, Sov. Phys. JETP ,195 (1972). J.I. Mart´ın, M. V´elez, A. Hoffmann, Ivan K. Schuller, and J. L. Vicent, Phys. Rev. Lett. , 1022 (1999). M. V´elez, D. Jaque, J.I. Martin, M.I. Montero, I.K.Schuller, J.L. Vicent, Phys. Rev. B , 104511 (2002). F. J¨ulicher, A. Ajdari, and J. Prost, Rev. Mod. Phys. , 1269 (1997); P. Reimann, R. Kawai, C. Van denBroeck, and P. H¨anggi, Europhys. Lett. , 545 (1999);F.J. Cao, L. Dinis, and J.M.R. Parrondo, Phys. Rev. Lett. , 040603 (2004). M. Tinkham,
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