We present the results of a study of vortex arrangements in the peak-effect
regime of 2H-NbSe_2 by scanning tunneling microscopy. By slowly increasing the
temperature in a constant magnetic field, we observed a sharp transition from
collective vortex motion to positional fluctuations of individual vortices at
the temperature which coincides with the onset of the peak effect in
ac-susceptibility. We conclude that the peak effect is a disorder driven
transition, with the pinning energy winning from the elastic energy.Comment: 4 pages, 4 figures included Manuscript has been submitte

A. M. Troyanovski

A. B. Pippard

A. I. Larkin

B. L. T. Plourde

J. Aarts

M. van Hecke

M. A. R. Le Blanc

N. Saha

P. H. Kes

P. E. Goa

S. Bhattacharya

X. S. Ling

English

arXiv

Crossref

STM Imaging of Flux Line Arrangements in the Peak Effect Regime

Troyanovski, A.M.

Hecke, M.L. van

Saha, N.

Aarts, J.

Kes, P.H.

Leiden University Scholary Publications

VOLUME 89, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 30 SEPTEMBER 2002STM Imaging of Flux Line Arrangements in the Peak Effect RegimeA. M. Troyanovski,* M. van Hecke, N. Saha, J. Aarts, and P. H. KesKamerlingh Onnes Laboratory, Universiteit Leiden, P.O. Box 9504, 2300 RA Leiden, The Netherlands(Received 4 June 2002; published 17 September 2002)147006-1We present the results of a study of vortex arrangements in the peak-effect regime of 2H-NbSe2 byscanning tunneling microscopy. By slowly increasing the temperature in a constant magnetic field, weobserved a sharp transition from collective vortex motion to positional fluctuations of individual vorticesat the temperature which coincides with the onset of the peak effect in ac susceptibility. We conclude thatthe peak effect is a disorder driven transition, with the pinning energy winning against the elastic energy.DOI: 10.1103/PhysRevLett.89.147006 PACS numbers: 74.60.Ge, 64.60.CnFIG. 1. Phase diagram of NbSe2 obtained from susceptibilitymeasurements on a single crystal, showing Hc2T (solid line/circles), the peak of critical current HpT (solid line/plusses),and the onset of the peak Hp1T (dashed line/triangles). Theinset shows the peak effect in the bulk critical current densitypresent STM data of vortex structures obtained on a puresingle crystal of 2H-NbSe2 at temperatures around 4.5 Kusing the method due to Clem and co-workers [15,16] for a flatsample in perpendicular field vs temperature at 0H1:75 T.It is well known that the critical current density jc inweakly disordered type II superconductors shows a suddenincrease when the applied magnetic field H approaches theupper critical field Hc2 [1]. An explanation for this intrigu-ing phenomenon, known as the peak effect, was suggestedby Pippard [2] already in 1969. It is based on the competi-tion between the collective work done by the disorder (thepinning centers) on the vortex lattice (VL) and the elasticenergy stored in vortex lattice deformations due to thepinning [3]. The work depends linearly on Hc2 H,while the elastic energy essentially is proportional to theshear modulus c66 of the VL, which behaves as Hc2 H2. Therefore, at some field close to Hc2, the pinningwill exceed the elasticity and the VL will accommodate tothe random pinning potential, leading to the suddenincrease in jc observed experimentally. The discovery ofthe high temperature superconductors made it relevant toalso consider thermal fluctuations as a third energy scale,which lead to melting by suppression of c66. This wouldthen yield a peak effect as a precursor of the meltingtransition [4–6].Experimentally, it is hard to probe which mechanism ispredominant, because one needs information about struc-tural changes of the vortex lattice positional order as afunction of time. Small angle neutron scattering (SANS)yields structural information along the field direction aver-aged over the sample volume, i.e., about the amount ofentanglement of the VL. SANS has been successfully usedin the peak-effect regime [7–9], but it is resolution limitedin the transverse direction. Lorentz microscopy [10], scan-ning Hall probe [11], scanning SQUID [12,13], and, re-cently, magneto-optics experiments [14] yield positionalinformation on the scale of individual vortices, but sensethe magnetic field distribution, and work only in the lowflux density regime, usually far below Hc2, where vorticesare well separated and the vortex-vortex interaction is veryweak. In this Letter we show that information on the time-dependent positional order can be provided by scanningtunneling microscopy (STM), which is uniquely able toaccess the necessary length scales and flux densities. We0031-9007=02=89(14)=147006(4)$20.00(far below Tc  7:1 K) in the regime of the peak effect. Weobserved a sharp transition from collective vortex motionto positional fluctuations of individual vortices at a tem-perature which coincides with the onset of the peak effectin ac-susceptibility measurements. The state above theonset temperature is characterized by the fact that the shearmodulus of the VL does not play a role and shows theproperties of a pinned liquid.Figure 1 shows part of the H; T-phase diagram of one ofour crystals as obtained from magnetization measurementswith the field along the c axis (perpendicular to the cleavedsurface). Also shown (inset) is the critical current densityjc as function of temperature T in a constant field of1:75 T, obtained from the maximum of the out-of-phasecomponent of the ac susceptibility [15,16]. With thismethod the same pinning conditions hold for the entiresample, and geometrical effects are taken into account. Thepeak effect sets in at Tp1 ’ 4:5 K, it reaches a maximum atTp ’ 4:6 K, and superconductivity disappears at TcH ’4:7 K. It should be noticed that in dc-magnetization 2002 The American Physical Society 147006-1FIG. 2. Raw and processed data from one sequence of STMimages acquired at 1.75 T for three different temperatures. Theframe size is about 410 460 nm. (a)–(c) Original images (thedark elongated stain in the upper-right corner is caused by asurface defect); (d)–(f) result of convolution with the shape of‘‘single vortex’’; and (g)–(i) result of convolution with ‘‘perfect’’lattice cell of 3 3 units.VOLUME 89, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 30 SEPTEMBER 2002(SQUID) measurements no peak effect was observed dueto flux creep. This means that even in the peak-effectregime nonuniform flux distributions decay within a fewminutes to a uniform flux density.The STM setup was the same as in a previous report[17], with the sample immersed in liquid helium. Afterdecreasing the magnetic field from above Hc2 at 4:27 K toa preset value (e.g., 1.75 T) in persistent mode (relativedecay less than 5 108 s1), we waited about 10 minbefore starting the STM imaging. By closing the evapora-tion valve during imaging a slow temperature rise wasestablished at a rate of 5 105 K=s to a value of 4.9 K.The temperature in the STM, TSTM, was determined fromthe pressure with a precision of 2 mK, but we found thatTSTM and the SQUID temperature, TSQ, showed a smallsystematic difference, TSTM  TSQ  60 mK, due to thenonuniform temperature in the liquid. We corrected for thisdifference and estimate the residual uncertainty in thetemperature determination to be about 20 mK. The biasvoltage was set inside the superconducting gap (0.2–0.8 mV) and the tunneling current was about 8–15 pA.The signal-to-noise (S=N) ratio goes down when approach-ing TcH; e.g., the relative conductance modulation be-tween vortex core and vortex cell edge decreases from 0.25(far below Tp1) to 0.02 (at Tp). Therefore we had to use alow scanning speed of about 20–70 s per image to acquireimages of a typical size of 450 450 nm, with a resolutionof 512 512 pixels. At a field of 1.75 T we could probe thespatial arrangement of about 150 vortices. The imagesshow a small constant spatial distortion due to a slightasymmetry in the piezoscanner. By heating up in constantfield the vortex structure is very weakly perturbed by thealmost negligible change of the equilibrium magnetization.The effect of the tunneling current is also expected to benegligible because it is very small ( < 15 pA) and it isdirected parallel to the vortices. Moreover, there was noinfluence on the vortex arrangement when we changedthe tunnel current from 7 to 25 pA. In this respect, ourexperiments differ essentially from dc and/or ac magnet-ization or transport measurements which do affect thevortex configurations.Results of experiments at 1.75 T are shown in Figs. 2(a)–2(c). We selected three typical temperatures: (a) 4.3 K(well below Tp1), (b) 4.44 K (just below Tp1), and(c) 4.53 K (between Tp1 and Tp). The first image showsan almost perfect vortex arrangement without large visibledistortions. With increasing temperature we see an increas-ing shear distortion of the VL and the appearance of VLdefects. With a further increase of T the signal from thevortices gradually decreases to the noise level. Severalimage-processing procedures were used to extract the po-sitional information. The S=N ratio could be considerablyimproved by convoluting the original data with the imagesof perfect vortex lattice cells acquired at low T (directintegrated multiplication, no Fourier transforms), usingcell sizes up to 10 10 vortices. In Figs. 2(d)–2(f) theresults are shown of convolution with one vortex147006-2(‘‘1 1’’), in Figs. 2(g)–2(i) of convolution with a 3 3cell, both with respect to the corresponding data inFigs. 2(a)–2(c). Using the 1 1 convolutions, the displace-ments of individual vortices could be followed up to 4.56 K;see below. The low contrast in Figs. 2(e) and 2(f) is mostlikely due to noise, but may also denote local vortexdisplacements.With the knowledge of the vortex positions, furtheranalysis can be performed using the vortex motion.Inspecting consecutive frames in the temperature intervalbetween 4.3 and 4.6 K in a field of 1.75 T, we notice asudden change in the vortex mobility at 4.48 K. Below4.48 K we find collective jumps of large portions of the VL;above 4.48 K, we see only oscillations of individual vorti-ces around fixed positions. An example of a collectivejump is given in Fig. 3(a), which displays the differencebetween two consecutive frames of raw data at 4.31 K. Thescan direction is horizontal going from bottom to top.In the lower part of the figure the VL did not move sub-stantially, and the difference reflects the noise level. At thescan position denoted by the arrow, the entire VL movedover a considerable fraction of the VL parameter a0. Thenext difference picture (not shown) has a lattice below andnoise above the position of the arrow. The scan speed(40 ms=line) and the number of lines involved in thejump yield an upper limit for the jump time of 1 s. Wecould see no net motion of the VL; it is jumping betweenmetastable positions. Above 4.48 K, no collective jumps147006-2(a)(b)FIG. 4. (a) Order parameter Kn for the correlated displacementof vortices from frame to frame as a function of temperature at amagnetic field of 1.75 T. (b) Amplitude of vortex displacementn averaged over the total number of vortices in a frame in thesame sequences of images. Symbols (+) mark the value~~~~~~~~2hu2ip=a0 (the value of n for fluctuations of individual vorti-ces) calculated from the average of four selected vortices (seeFig. 3 and Table I).FIG. 3. (a) Difference between two consecutive images at4.31 K in a field of 1.75 T, to demonstrate coherent motion ofvortices. When scanning the second image from bottom to top,the VL moved at the line indicated by the arrow. (b) Average ofthe sum of 60 consecutive images taken with increasing tem-perature above Tp1. The brightness is a measure for the proba-bility of finding a vortex at a certain position. The numbersindicate individual vortices discussed in the text.VOLUME 89, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 30 SEPTEMBER 2002were observed; in the 60 images in which T slowly rose to4.57 K (total observation time about 70 min) we did notdetect any net motion of individual vortices in the image,although some vortices hopped back and forth between twometastable positions. A movie of the change in behavior,made of two-cell filtered frames for noise reduction, can beviewed as an EPAPS document [18].We can characterize this transition by introducing thefollowing two order parameters, based on the displacement~di;n of vortex i going from frame n to frame n 1, ~di;n ~ri;n1  ~ri;n: (i) n, in order to quantify the amount ofmotion of the vortices, we definen :1NXi~d2i;n1=2=a0; (1)where the sum runs over the N displacement vectors;(ii) Kn, in order to quantify the amount of correlatedmotion of the vortices, we defineKn :1NXhi;kidi;n ~~dk;nj ~di;nj  j ~dk;nj; (2)where the sum runs over the displacements on neighboringvortices i and k in a range 1:5a0. For both parameters, onlycontributions were retained for which ~di;n was two or morepixels.Clearly, when vortices move coherently, both n and Knwill be large, while for uncorrelated, random motion, Kn issmall but n remains finite. Both Kn and n are plottedversus T in Figs. 4(a) and 4(b). For both, the behaviorabruptly changes at TSTM4:48 K, which is essentiallyTp1 (see Fig. 1). Below Tp1 the behavior of Kn exhibitsthe collective vortex lattice jumps between metastablepositions in the collective pinning potential. At Tp1, thepicture suddenly changes to a situation in which eachvortex independently fluctuates within its own pin poten-tial. This transition from collective to single vortex behav-147006-3ior is reflected in the sudden decrease of fluctuations in Kndown to the noise level. From the plot of n versus T itfollows that the collective jumps span a distance of about0:1–0:25a0, which changes at Tp1 to individual jumpswith a much smaller amplitude spread. The distance ofcollective jumps reflects the typical length scale separatingindependent local minima in the pinning potential in thefield regime close to Hc2 (at low fields where the vortexcores are well separated, this scale would be the Ginzburg-Landau coherence length ). The jumps occur more fre-quently with increasing temperature.From the original STM pictures it can also be concludedthat the jumps are directed along the principal axes of theVL. In the regime above Tp1 the mean-squared displace-ment gradually grows with temperature to a valuehu2i1=2  0:1a0. This number is close to the Lindemanncriterion for melting, hu2i1=2  cLa0 with cL  0:15–0:25.However, we do not observe diffusive motion of vorticesover more than a0. This is illustrated in Fig. 3(b) where60 images in the temperature regime above Tp1 are aver-aged. Since the noise level for all vortices is supposedly thesame, the brightness of the spots represent the probabilityamplitude for finding a vortex at a certain position. Thebrightest spots display vortices with an almost fixed posi-tion. One of them is marked No. 1; other vortices (markedNos. 2–4) are less bright, indicating larger individual147006-3TABLE I. Mean square displacement hu2i1=2=a0 at threetemperatures between Tp1 and Tp for the four vortices indicatedin Fig. 3(b).Vortex number No. 1 No. 2 No. 3 No. 44.51 K 0.045 0.095 0.055 0.0774.53 K 0.055 0.080 0.13 0.0714.55 K 0.070 0.10 0.21 0.15VOLUME 89, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 30 SEPTEMBER 2002positional oscillations most likely caused by thermalfluctuations.In Table I the values are given of hu2i1=2=a0 for thesefour vortices determined from 20 images around the de-noted temperatures. At 4.51 K, just above Tp1, the ampli-tudes remain far below the Lindemann criterion.Approaching Tp, the amplitudes for vortices Nos. 3 and 4increase to values close to the Lindemann criterion.However, the estimated average of hu2i1=2=a0 for all de-tectable vortices coincides with the data in Fig. 4(b) andremains well below the criterion for melting. Therefore,and because of the absence of large scale vortex diffusionwithin the 70 min observation time, the change at Tp1signifies a disorder driven transition from collective pin-ning to single vortex pinning. The consequence is a suddenchange in the temperature and/or field dependence of thecritical current which causes the peak effect.Following this main result we add a few quantitativeremarks about the observed disorder and the relation toother results. From the convoluted images we can extractinformation about the size of the Larkin domains in thetemperature range below the peak effect (although thelimited number of vortices in the images does not allowmuch statistics). The positional order in an area of 10 10vortices disappears at TSTM  4:35 K, while for an area of6 6 vortices positional order is seen up to Tp1. Thecomputed correlation lengths from the Fourier transformof the images have similar values: 4–6a0 at the onset ofthe peak effect and 1–2a0 above the onset. It should benoted this correlation length is about twice as large as thetransverse size of the Larkin domain Rc. At the onset of thepeak effect the value of Rc=a0 drops from about 3 to 1. Thisis in reasonable agreement with the values we obtained in aprevious analysis of critical current data on a similarcrystal [16]. Based on SANS experiments on a Nb singlecrystal, Ling et al. [8] found evidence for supercooling andsuperheating of the vortex matter disorder around the peak-effect line in the phase diagram and concluded that this isdirect structural evidence for a first-order vortex solid-liquid transition. Very recent SANS work of Forgan et al.[9] demonstrated that the vortex solid in very pure Nb isstable to very close to the Hc2T line. This shows that thehistory dependence observed by Ling et al. is disorderdriven rather than of thermal origin, and this is in agree-ment with the disorder driven transition seen in our STM147006-4experiment. Our setup does not allow slow cool down toinvestigate the reversibility question.In conclusion, the mechanism for the peak effect is theconquest of the pinning energy over the elastic energy, justas was first suggested by Pippard [2], and later in moredetail described by Larkin and Ovchinnikov [19]. The hightemperature phase is characterized by the fact that theshear energy has become irrelevant. In this phase, thevortices are not free to move around as one would expectto occur in a liquid, but they remain trapped in their ownpin potential which may differ for each vortex. In terms ofthe collective pinning theory this would be the amorphouslimit, but it would be equally correct to say that we aredealing with a pinned liquid.This work was supported by the ‘‘Stichting FOM,’’ bythe Dutch-Russian science collaboration financed byNWO, and by the ESF network ‘‘Vortex.’’ We thankR. Drost for assistance in the early experiments.*Permanent address: Institute for High Pressure Physics,Russian Academy of Science, Troitsk, 142092, Russia.[1] M. A. R. Le Blanc and W. A. Little, in Proceedings of theVII International Conference on Low TemperaturePhysics, 1960 (University of Toronto Press, Toronto,1960), p. 198.[2] A. B. Pippard, Philos. Mag. 19, 217 (1969).[3] G. Blatter et al., Rev. Mod. Phys. 66, 1125 (1994).[4] P. Berghuis and P. H. Kes, Phys. Rev. B 47, 262 (1993).[5] S. Bhattacharya and M. J. Higgins, Physica (Amsterdam)257C, 232 (1996).[6] X. S. Ling, J. E. Berger, and D. E. Prober, Phys. Rev. B 57,R3249 (1998).[7] P. L. Gammel et al., Phys. Rev. Lett. 80, 833 (1998).[8] X. S. Ling et al., Phys. Rev. Lett. 86, 712 (2001).[9] E. M. Forgan et al., Phys. Rev. Lett. 88, 167003 (2002).[10] A. Tonomura et al., Nature (London) 412, 620 (2001).[11] A. Oral et al., Phys. Rev. Lett. 80, 3610 (1998).[12] B. L. T. Plourde et al., Physica (Amsterdam) 341C, 1023(2000).[13] J. R. Kirtley et al., Phys. Rev. Lett. 76, 1336 (1996).[14] P. E. Goa et al., Supercond. Sci. Technol. 14, 729 (2001).[15] J. R. Clem and A. Sanchez, Phys. Rev. B 50, 9355 (1994).[16] L. A. Angurel et al., Phys. Rev. B 56, 3425 (1997).[17] A. M. Troyanovski, J. Aarts, and P. H. Kes, Nature(London) 399, 665 (1999).[18] See EPAPS Document No. E-PRLTAO-89-060236 for amovie showing 180 consecutive images of a vortex lattice,measured on a sample of NbSe2 in a field of 1.75 T andwith slowly increasing temperature. A direct link to thisdocument may be found in the online article’s HTMLreference section. The document may also be reached viathe EPAPS homepage (http://www.aip.org/pubservs/epaps.html) or from ftp.aip.org in the directory /epaps/.See the EPAPS homepage for more information.[19] A. I. Larkin and Yu. N. Ovchinnikov, J. Low Temp. Phys.34, 409 (1979).147006-4

STM imaging of flux line arrangements in the peak effect regime

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STM Imaging of Flux Line Arrangements in the Peak Effect Regime

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