Dynamics of superconducting vortices driven by oscillatory forces in the plastic flow regime
aa r X i v : . [ c ond - m a t . s up r- c on ] J u l Dynamics of superconducting vortices driven by oscillatory forces in the plastic flowregime
D. P´erez Daroca, ∗ G. Pasquini, G. S. Lozano, and V. Bekeris
Departamento de F´ısica, FCEyN, Universidad de Buenos Aires and IFIBA,CONICET; Pabellon 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina
We study experimentally and theoretically, the reorganization of superconducting vortices drivenby oscillatory forces near the plastic depinning transition. We show that the system can be taken toconfigurations that are tagged by the shaking parameters but keep no trace of the initial conditions.In experiments performed in
NbSe crystals, the periodic drive is induced by ac magnetic shakingfields and the overall order of the resulting configuration is determined by non invasive ac susceptibil-ity measurements. With a model of interacting particles driven over random landscapes, we performmolecular dynamics simulations that reveal the nature of the shaking dynamics as fluctuating statessimilar to those predicted for other interacting particle systems. PACS numbers: 74.25.Uv, 74.25.Wx, 74.62.En
Vortices in type II superconductors , charge den-sity waves , colloidal particles , Wigner crystals and domain walls are only few of the many examplesof physical systems belonging to the category of drivenelastic manifolds over random landscapes. In all thesesystems, competing interactions give rise to a complexdynamics characterized by a region of plastic motion: un-der the action of an external driving force, strong enoughto depin the system, a non-linear response develops, someparticles are mobile while others remain pinned. Thisplastic depinning and its relation with the proliferationof topological defects has been object of study for severalyears. Recently, it has been claimed that the plasticdepinning of a colloidal system with random quencheddisorder under the action of a dc force can be describedin terms of a non equilibrium phase transition with adivergent transient time and it can be related to the”absorbing transition” from a random self-organizationto a drive-dependent fluctuating steady state (FSS) ob-served in experiments on periodically sheared particlesuspensions .Vortex matter offers an ideal playground to studythis phenomenology both numerically and experimen-tally. Dynamic phase transition and plastic flow havebeen theoretically described and measured in the neigh-borhood of the order-disorder (O-D) transition, wherethe system evolves from a quasi-ordered Bragg glass(BG) to a disordered phase with proliferation of topo-logical defects. The experimental fingerprint of the O-Dtransition is the anomalous non-monotonous dependenceof the critical current density J C with both temperatureand magnetic field, known as Peak Effect (PE) . There,the response at a fixed temperature and field can de-pend on thermal, magnetic and dynamical history thatcan modify the topology of the vortex lattice configura-tions (VLCs) by creation or annihilation of disclinations .In transport experiments vortex instability phenomenaand the smearing of the PE have been claimed to orig-inate on a competition between the injection of a disor-dered vortex phase at the sample edges, and the dynamic annealing of this metastable disorder by the transportcurrent . However, using an ultrafast transport tech-nique that avoids current-induced vortex lattice (VL) re-organization, Xiao et al. have shown the existence ofan enlarged crossing phase boundary between orderedand disordered phases in N bSe crystals arising from thebulk VL response, that has been corroborated in a re-cent work by our group using non invasive ac susceptibil-ity measurements in the linear regime . In experiments,a shaking ac field is often applied to order the VL andthe most accepted picture is that the shaking field assiststhe system in an equilibration process, from a disorderedmetastable configuration to the equilibrium BG phase.However, in our experiments, we have shown that in thetransition region a shaking ac field can either order ordisorder the VL, driving the system to VLCs with inter-mediate degree of disorder that are independent of theinitial VLC. Very recently, noise transport experiments(in a-Mo x Ge − x ) in this intermediate region have shownevidence of a depinning transition with critical behaviorsimilar to the absorbing transition .In this work we study experimentally and numericallythe reorganization of a vortex system in the plastic flowregion under the action of ac shaking magnetic fields. Wehave done experiments changing the frequency and thewaveform of the shaking fields and measuring the ac re-sponse of the final VLC in each case with very low acamplitude, avoiding modifications in the final VLC. Wehave also performed numerical calculations that mimicour experiment. We have first used the Brandt method to determine the induced current dependence on the ex-ternal magnetic field and as a second step we have usedthis result as input for a molecular dynamics simulation.Although the model is rather simple, we obtain resultswhich are qualitatively consistent with the experimen-tal results and show evidence of ac driven vortex latticereorganization.Experimental results shown in this work correspondto a N bSe single crystal of approximate dimensions(0 . × . × . mm , with T c = 7 . K (defined as -0,8-0,7-0,6-0,5-0,4 ’ T (K) ’’’
T (K)
FIG. 1. (Inset) Typical χ ,, and χ , FCC and FCW curves inthe linear regime measured at f = 30 kHz . Arrows indicatethe direction of temperature variation. (Main panel) Zoomof the region delimited in the inset. Gray triangles (blackcircles) correspond to shaking frequency f sh = 100 kHz ( f sh = 10 Hz ). Open (full) symbols for warming (cooling)experiment. In darkgray small circles a FCC curve is shown. the mid point of the ac susceptibility linear transitionat H = 0) and ∆ T c = ± . K . A neutron study in samples synthesized with the same technique as ours,shows an excellent agreement with the VL structure pre-dicted for randomly distributed point defects . The acsusceptibility has been measured with a home made sus-ceptometer based in the mutual inductance technique,where both the ac and dc fields are parallel to the c axisof the sample. To study the VL response without dis-turbances, the measurements have been restricted to thelinear Campbell regime , where a very small ac field h a superimposed to the dc field H is applied, forcing vorticesto perform small (harmonic) oscillations inside their ef-fective pinning potential wells. In this regime, the acsusceptibility is independent of amplitude and frequencyand the inductive component of the ac susceptibility χ , isdetermined by the experimental geometry and the curva-ture of the effective pinning potential well α L and a lowersusceptibility χ , (closer to -1) can be directly associatedto a more pinned VL (see Ref. 24). Therefore, at fixed T and H , different χ , (i. e. different degree of effectivepinning) can be associated to the existence of VLCs withdifferent degree of disorder.Figure 1 shows χ , ( T ) measured in the linear regime( h a = 0 . Oe , f = 30 kHz ), in a dc field H = 320 Oe after different thermal and dynamical histories. In theInset, both components χ , and χ ,, in field cool cool-ing (FCC) and field cool warming (FCW) processes areshown in the full temperature range, and the region ofthe PE (main panel) is indicated. The curve with smalldarkgray circles in the main panel was obtained in a FCCprocess, where the VL remains trapped in a metastable -45 -30 -15 0 15 -0,75-0,70-0,65-0,60-0,55-0,50 -0,62-0,60-0,58-0,56-0,5410 -0,66-0,63-0,60-0,57-0,54-0,51 FCC , t-t (s)T = 7.06 K (a)
100 kHz10 Hz (b)
T=7.06 K ’ frequency (Hz) T=7.05 K ’ (c) FIG. 2. (a) χ , versus time before and after shaking the sys-tem (the shaking time is not in scale and it is indicated bya gap without data). Gray (black) symbol correspond to f sh = 100 kHz ( f sh = 10 Hz ). (b) χ , values versus theshaking frequency (log scale). Arrows point from the valuesof χ , before shaking to the values obtained after shaking. (c) χ , values versus the fundamental shaking frequency for sinu-soidal (open dots) and square (black squares) shaking wave-forms. disordered and strong pinned configuration, and the PEdisappears. The other experimental points have beenobtained by measuring the system after shaking the VLat different frequencies. In all those cases, the follow-ing procedure has been performed: The system has beenstabilized at a shaking temperature T sh and measuredin the linear regime, recording the χ , value correspond-ing to the starting configuration. Then the measurementwas interrupted and N sh cycles of a sinusoidal shakingac field ( h sh = 3 . Oe ) were applied. After switching offthe shaking field, χ , corresponding to the final value wasrecorded in the linear regime; then the system was drivento the next shaking temperature. Full and open sym-bols correspond to procedures increasing and decreasing T sh . For clarity, in this figure only the final values areshown. Gray triangles indicate the response after shak-ing the system with N sh = 100 cycles at a frequency f sh = 100 kHz , whereas black circles show the responseafter a similar procedure at f sh = 10 Hz . One remark-able feature published in our previous work is that thefinal VLCs do not show appreciable relaxation (withinour experimental time window) and are independent onthe initial condition. Surprisingly, we find that these fi-nal states are not unique: they depend on the shakingfrequency, but however the independence of initial con-ditions persists.In Fig. 2(a), the response at fixed temperature, beforeand after applying shaking protocols, is shown for differ-ent starting VLCs (same frequencies as in Fig. 1). It canbe seen that the shaking field can order or disorder theVL. The various curves χ , ( t ) obtained at different initialconfigurations collapse in an unique curve after shakingthe system at a given frequency. Figure 2(b) shows the re-sponse after applying sinusoidal shaking fields at variousfrequencies. Arrows point from each starting VLC to thecorresponding final configuration after shaking. In all thecases the final response is independent of the initial one,for shaking amplitudes greater than 1.8 Oe and above anumber of applied cycles (in our system N sh ' .In that scenario, the frequency dependence would bedue mainly to the different vortex velocities (i. e. forces)induced by the different shaking ac fields. Additionalevidence supporting this fact arises from the responseof the system to shaking fields with a square waveform.In this case it is expected that vortices move at highervelocities during short time intervals triggered and de-termined by the rising and falling edges, whereas theshaking frequency only modifies the waiting time be-tween these short intervals. In Fig 2(c), it is seen that forsquare shaking fields, the VLCs are maximally ordered.On the other hand, frequency independence shows thatrelaxation processes are not relevant at least in this timewindow. As another evidence, we have confirmed thattriangular and sinusoidal shaking fields (that at the samefundamental frequency induce similar vortex velocities)produce similar results.The simulated system describes a very small regionof the real macroscopic sample, where the mean vor-tex density and mean current density can be assumedto be homogeneous. In the experiment, the shaking acfield induces macroscopic currents J sh ( t ) that move vor-tices, that have been introduced in our simulations bya Lorentz force term F L = φ J sh × b z , where b z is theversor parallel to the vortex direction. The explicit func-tional relation between J sh ( t ) and the shaking h ac ( t ) isnot straightforward. In order to gain some intuition, wehave applied the method developed in Ref. 18 to obtainthe distribution of J ( t ) for a certain geometry, assum-ing a non linear electric field-current constitutive relation E ( J ) = E c ( J/J c ) n . We have chosen a simple geome-try (a disk) and we have estimated n by comparing nonlinear susceptibility measurements and polar plots of χ ,, vs χ , . Under this non linear regime, given an external h sh ( t ) = h sh cos( ωt ) the current throughout most of thesample can be described by a square waveform of fre-quency ω and normalized amplitude J ( ω ) /J c & J sh ( t ) = J ( ω )Sign(sin( ωt )).In our simulations, we consider N v rigid vorticeswith coordinates r i in a two-dimensional rectangle ofsize L x × L y that evolve according to the dynam-ics η v i = F vvi + F vpi + F L , where v i its velocity, η the Bardeen-Stephen viscosity coefficient, F vvi = P N v j = i f f vv K ( | r i − r j | λ )ˆ r ij is the vortex-vortex interaction -0.018-0.012-0.006 0 0.006 0.012 0.018 2 2.5 3 3.5 4 v t/P(b) f L = 0.012f ω = 0.0188 ω ω = 0.0019 ω -1 0 1 J / J c (a) FIG. 3. (a) Theoretical current density
J/J c at a fixed po-sition, in a superconducting disk of radius a and a perpen-dicular sinusoidal magnetic field, as a function of t/P for twodifferent frequencies, 1˜ ω , solid line and 100˜ ω , dashed line,˜ ω = πE c µ aJ c . Throughout most of the sample, the current canbe described by a square waveform. (b) Calculated vortexmean velocity v, obtained from molecular dynamic simula-tions, as a function of t/P . and F vpi = − P N p k =1 F pk e − ( | ri − Rk | rp ) ( r i − R k ) is the pinningattraction. Here, φ is the quantum of magnetic flux, λ isthe London penetration length, K is the special Besselfunction and f vv is a dimensionless parameter that can berelated to the stiffness of the vortex lattice. The N p pin-ning centers are located at random positions R k , withstrength and range F pk and r p .We measure lengths inunits of λ , forces (per unit length) in units of f = φ π λ ,time in units of t = ηλ/f . We will consider N v = 1600, L x = 40 λ , L y = √ L x / N p = 25 N v , r p = 0 . λ and F pk chosen from a Gaussian distribution with mean value F p = 0 . . F p . The equa-tions of motion are integrated using a standard Euler al-gorithm with step h = 0 . t , and a hard cut-off Λ = 4 λ for the vortex-vortex force. The mean vortex velocityis defined as v( t ) = N v P N v i v i ( t ), where v i ( t ) is the in-stantaneous velocity of the i-vortex in the force direction.The observable used to characterize the degree of order isthe proportion of vortices with 6 neighbors P = 1 − n d ,where n d is the density of disclinations. P is calculatedat the end of each cycle.We assume that the main effect of temperature nearthe PE is the increase of the ratio f vp /f vv , causing aspontaneous disordering of the VL. In the simulations,starting with a perfect ordered lattice (i. e. P = 1) andleading the system to evolve without any applied externalforce, there is a spontaneous creation of disclinations, anda decrease in P . In the inset of Fig. 4, the spontaneous P starting from an ordered configuration is plotted asa function of f vp /f vv . We identify the parameters corre-sponding to the experimental temperature region shownin Fig. 1 as the range where the VL spontaneously dis- P t/P(a) ω = 0.0188 ω ω = 0.0019 ω f L =0.012f f vp /f vv = 0.25 1 41 81 121t/P(b) f L = 0.012f f L = 0.014f ω = 0.0188 ω P f vp /f vv FIG. 4. P versus t/P starting from an ordered and a disor-dered configuration. (a) For two different ω and fixed f L . (b)For two different f L and fixed ω . (Inset) P versus f vp /f vv ,showing a spontaneous creation of disclinations, and a de-crease in P at around f vp /f vv = 0 . orders. Therefore, we set the ratio f vp /f vv = 0 .
25 tosimulate the experimental system at a fixed temperature(experiments of Fig. 2). The dc depinning force per unitlength F dcc = φ J c ≈ . f was obtained by increasinga dc F L until the mean velocity v exceeds a ”criterionvoltage”. The shaking frequencies have been selected inthe low frequency regime ( ω << ω c ∼ α L η ∼ . ,where pinning forces dominate over viscous drags, be-cause the experimental shaking frequencies are well be-low f c = ω c / π > M Hz
Figure 3(b) shows the calculated mean velocity as afunction of time during two shaking cycles for two differ-ent frequencies. After switching the force direction, v de-cays towards a steady regime characteristic of a dc driv-ing force, similar to that reported in Ref. 14 that couldbe associated to a FSS. The transient time τ and the finalmean velocity depend on the shaking force. At very lowfrequencies, the system reaches the steady regime insideeach cycle (black curves in Fig. 3(b)) , and the responseis essentially dc. This is not the case at higher frequencies(gray curve), were we observe that both τ and the finalmean velocity slightly vary while the system evolves froma cycle to the next one. This fact is more clear in Fig. 4,that shows P as a function of the number of cycles of theshaking force, starting from different initial conditions, ata fixed amplitude and two different frequencies (4(a)) anda fixed frequency and two amplitudes slightly above thedepinning force (4(b)). After a new transient time τ ac ,that can involve many shaking cycles, the disclinationdensity remains fluctuating around a steady value, in adynamic state independent of the initial condition, thatdepends on the amplitude of the shaking force. When the force is very near the dc depinning force, a clear fre-quency dependence is also observed (Fig. 4(a)). It be-comes clear that the higher the shaking frequency andamplitude of the shaking force (both effects expected byincreasing, in the experiment, the frequency of a shakingfield), the more ordered the resulting VLC, in agreementwith that observed in our experiments. In this process,there are two different transient times that involve thecreation or annihilation of disclinations: the characteris-tic dc time to decay to the FSS inside a cycle τ ( prob-ably related with the experimental transient time τ inRef. 2 ) and the transient time τ ac that can take manyshaking cycles and could be related with the transientresponse observed in ac transport experiments . Thissecond process is inherent of an oscillatory drive, and dy-namically reorganizes the system in a VLC independentof the initial conditions.In conclusion, a dynamic reorganization in the plas-tic regime that keeps memory of the frequency drive hasbeen observed experimentally and numerically. Through-out the order disorder crossing phase boundary in N bSe there is a large collection of VLCs, that can be accessedafter the application of ac shaking fields of different fre-quency, either ordering or disordering the initial state.The order of the VLCs is determined by non invasive acsusceptibility measurements, showing that higher shak-ing frequencies lead to more ordered VLCs. The systemforgets the initial condition and but keeps memory of theshaking frequency, suggesting that the nature of the fi-nal states is more complex than previously conjectured .Molecular dynamics simulations performed for a compat-ible set of interaction parameters in the plastic regimehave reproduced qualitatively the salient features of ex-perimental shaking protocols, revealing a plausible na-ture of the experimental VLCs. The dynamic steadystates are reminiscent of FSS observed in other systems,as VL disclinations and mean vortex velocity fluctuatearound a steady value. 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50 has been set for
J < J cc