Dynamic transition in current-driven disordered flux-line lattice in single-crystal of Bi-2212

Physica C 506 (2014) 47–52

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Dynamic transition in current-driven disordered flux-line lattice in single-crystal of Bi-2212 L. Ammor ⇑, A. Ruyter GREMAN-CNRS, Université de Tours, UMR 7347, Parc de Grandmont, 37200 Tours, France

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Article history: Received 4 April 2014 Received in revised form 10 July 2014 Accepted 29 August 2014 Available online 11 September 2014 Keywords: Cuprates superconductors Vortex dynamics Dynamical phase Vortex pinning Critical current

a b s t r a c t We have measured the current–voltage characteristics for both as-grown and irradiated Bi2Sr2CaCu2O8+d single crystals at T = 5 K in a magnetic field applied parallel to c axis. The results show a variety of dynamical behavior above the depinning threshold, depending on the vortex–vortex interaction (kab/a0) strength and the nature of the quenched disorder (point-like or columnar defects). When the flux lattice is soft, our experimental measurements in both samples have been attributed to plastic flow, including strong metastability and history dependence of the depinning process. The vortex motion in this regime is thought of relatively weakly pinned vortices past more strongly pinned neighbors. A power-law scaling, fit between voltage and applied current can be obtained for the onset of motion in both samples, with different apparent critical exponent depending on defect nature and the strength of interactions. In the plastic regime, the usual scaling ansatz associated with dynamic critical phenomena V scales as (I Ic)b, where b  2.2 ± 0.1 and 1.22 ± 0.021 for as-grown and b  1.49 ± 0.07 for irradiated samples, respectively. Finally, in both cases of defects, with increasing the strength of vortex–vortex interaction a dynamical transition is observed as confirmed by the discontinuity in the vortex–vortex interactions dependence of the critical exponent b. More, our results confirm the important role of the system dimensionnality on vortex dynamics. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction A wide variety of systems such as charge transport in disordered metallic dot arrays [1], charge-density waves (CDW) pinned by impurities [2,3], Wigner crystals [4], vortex matter in disordered superconductors [5], and colloidal crystals [6] present a generic property is elasticity. In these systems the competition between elasticity and disorder, caused by impurities inevitably present even in small quantities, gives rise to complex phenomena of pinning, which is a characteristic manifestation of the depinning transition in the presence of an external force. The dynamics of these systems under the action of an external force, in the vicinity of the threshold depinning, has strongly non-linear effects. This issue has been addressed in the theoretical work of Fisher which predicted that for many systems the elastic depinning would show criticality in the vicinity of the depinning threshold Fc at zero temperature and that velocity-force dependence curve would scale as vd  (F Fc)b, where f is the dc driving force, vd is the drift velocity and b < 1 is the dynamical critical exponent [7].

⇑ Corresponding author. http://dx.doi.org/10.1016/j.physc.2014.08.011 0921-4534/Ó 2014 Elsevier B.V. All rights reserved.

With an increasing strength of the disorder, a crossover to plastic depinning may occur. Such crossover has been observed in colloid [6] and superconducting vortex systems [8]. A plastic flow is observed in a large number of experiments and numerical simulations studies of particle motion in a random pinning disorder [6,9,10,11–16]. The largest numbers of results from the numerical simulations are the evolution of current–voltage characteristics and dynamic phase diagram. Several parameters may be involved in these studies as the defects density, the traps profile and vortex–vortex interactions. An examplecal advance was made on the basis of numerical simulations on studies in driven vortices as a function of lattices softness and driving current, who argued that there could be three distinct current regimes [11,12]. A pinned phase for applied currents below the threshold I < IC; a plastic flow for IC < I < IP and for current well above IP, they found a regime where the moving lattice was quite well ordered. In the plastic flow regime, depinning is observed to occur through plastic channels between pinned regions [12]. This type of plastic flow is observed in experiments [17] numerical simulations studies [18,19]. Other numerical simulations studies of driven vortices in three-dimensional layered superconductors with strong pinning, show that at zero temperature there are two additional dynamical

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2. Experiment Bi-2212 single crystals with a typical in-plane penetration length kab (T = 0)  1700 Å [22] were grown by a self-flux technique as described elsewhere [23]. Their high quality had been confirmed previously by X-ray diffraction measurements. Two samples, A and B, were extracted from the same single crystal and the thick microbridges were created by using laser. The advantage of this technique is to make it possible to reach low temperatures and to apply large current densities with a good homogeneity of the current flowing in the sample. The resulting sample was post-annealed in oxygen atmosphere. Low resistance of the electrical contacts (61X) was made by bonding gold wires with silver paste. The zero-field Ohmic resistance R (T) as a function of the temperature did not exhibit any anomaly (Fig. 1). We determined Tc as the average of the two that can be obtained from the maxima of dR(T)/dT. The crystals used in our experiment exhibit almost the same superconducting transition temperature Tc = 79.5 K with a transition width DTc of about 2 K. These types of measurements (i.e. local and overall) confirm the good homogeneity of the samples.

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regimes appear when increasing current above the depinning threshold Ic: a moving smectic for IP < I < It and a transverse solid for It < I < Is [20]. In this study, for fixed pinning density, the characteristics force IC, IP, and It depend on the disorder strength and vortex density. In the plastic flow regime, the power-law scaling has been also shown from both experimental and numerical simulation works for colloids [6,21], charge transport in metallic dots [13] and in driven vortices in type-II superconductors or Josephson junctions [9,13,14,16], with a wide spectrum of scaling exponent b in the range from 1.2 to 2.3. This exponent has not been explained theoretically. The question of whether there is a universal exponent for plastic depinning remains open. Finally, the dynamical phase diagrams for driven lattices as a function of disorder strength and driving current have been investigated theoretically by Giarmarchi et al. [10]. They predict the following dynamical phases upon increasing disorder strength: the moving Bragg glass where vortices move elastically-coupled along static channels; the moving transverse glass (or a moving smectic) and plastic where the filamentary motion of vortices proceeds via plastic channels between pinned regions. The vortex lattices in type II superconductors are model systems to test theories and numerical simulations where the notions of competition between the order within an elastic system and disorder (point defects, artificial defects) are prevailing. They are, because of their experimental accessibility, an ideal setting to study the elastic or plastic dynamics of disordered systems. Indeed, it is a system where the vortex density, their interactions and, under certain conditions, the disorder which they are subjected can be fixed in a controlled manner. I–V characteristics vs. temperature (T) and magnetic field (B) have been used just to investigate the nature of the depinning in the low-Tc materials. However, in high-Tc superconductors, those measurements were technically impossible to perform, due to the problem of contacts heating and the limitation of the applied current. To avoid that, one must apply only a very small current. However, in this case, it is hard to detect a signal. To overcome that, we have used the microbridge technique to be able to measure I–V characteristics at very low temperatures (i.e. T/TC ? 0). In this paper we will focus on the pinning–depinning mechanism and the dynamics of vortices in Bi2Sr2Ca1C2O8+d (Bi2212) single crystals with two different pinning centers (0D and 1D) and vortex–vortex interaction strengths (by changing the external magnetic field).

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T (K) Fig. 1. (a) The temperature dependence of Ohmic resistance R at zero Fields in an extended range of temperatures for the sample A. Solid line: linear fit to the normal state resistance. Inset: microbridge (200  400  100 lm3) of Bi-2212 single crystal. (b) R(T) curves measured at various magnetic fields for the sample A.

The sample B was irradiated with a beam of 5.8 Gev Pb ions (which traversed over the entire specimen) at the heavy-ion facility GANIL (Caen, France). The density of irradiated defects corresponds to a matching field B/ of 1 T, i.e. to an average distance d/  450 Å between the tracks. Using a standard dc four-probe method, I–V characteristics were obtained with a voltage resolution of 1 nV and a temperature stability better than 5 mK. The magnetic field was aligned with the ion tracks using a well-known dip feature occurring in dissipation for the field parallel to the columnar defects.

3. Results and discussion In order to investigate the influence of the history of current and field cycling on the pinning of the flux-lattice, we have performed current–voltage measurements through tuning the strength of vortex–vortex interaction (by changing the external magnetic field) for both samples in field cooling (FC) process at a fixed temperature (T = 5 K; i.e. T/TC ? 0). Typical results are shown in from Fig. 2. Concerning the unirradiated sample, as seen from Fig. 2(a), we present the typical V–I curves for different values of the strength of vortex–vortex interaction. We find a dramatic change in the gross features of V(I) curves as (kab/a0) is varied. For 3.5 6 (kab/a0) 6 10.43, we have measured the same dissipation in both ways either when the applied current is increased or decreased. For 0.35 6 (kab/a0) < 3.5, when the magnetic field is decreased, different behaviors were observed in a restricted region of the phase diagram. These curves exhibit pronounced history dependence (see Fig. 3(a)). This hysteresis disappeared at high currents. When the vortex lattice appeared under a low magnetic field, the V–I curves have an S-shape with a high threshold current I⁄ (1: open square), but only for the first increase of the current. After this initial ramp, a reproducible Ic < I⁄ can always be detected (2: closed square and 3: filled square). For (kab/a0) lower than 0.35, we do not observe any hysteresis in V–I curves. The two thresholds, Ic and I⁄, identify two distinct states of the FLL, with one which is more strongly pinned than the other. Another remarkable behavior in Fig. 3(a) is the multiple steps in the I–V curves measured upon increasing or decreasing bias current. For kab/a0 larger than 3.5, we do not observe any ‘‘step’’ in the V–I curve. One can see that the difference in V–I characteristics of irradiated sample in Fig 2(b) shows the V–I curve measured at fixed magnetic field and temperature. These curves present the usual form

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I (mA) Fig. 2. Plot of current–voltage curves near threshold after a FC recorded at T = 5 K for (a) as-grown: from the right to left the data were obtained at (kab/a0) as of 1.35, 1.56, 1.90, 2.46, 3.48, 2, 4.92, 7.78, and 10.43 and (b) heavy ion irradiated samples; from the right to the left the data were obtained at fields as of 0.35, 0.92, 1.10, 1.56, 2.46, 2.69, 3.48 and 7.78. All curves are concave upward.

V  (I IC)b for a current that is slightly larger than the critical current IC. The value of the critical current IC was determined by considering the non-linear V(I) curves and using a criterion of V = 1 nV (i.e. E = 100 nV/cm). One should notice that both increasing and decreasing the current give the same reversible dissipation for high kab/a0 (3.5 6 (kab/a0) 6 10.43) and for low kab/a0 (kab/a0 6 0.65). In contrast, when a magnetic field is increased, different behaviors could be observed in a restricted region of the phase diagram (0.65 6 (kab/a0) 6 0.85). One can see that in general, at a low field, a higher value of the critical current could be obtained upon an increase of the applied current (see Fig. 3(b)). Now let us turn back to the vicinity of IC. Many theoretical and experimental studies have been devoted to the study of the dynamics of vortex lattices near the depinning threshold. The subject is particularly interesting and rich insofar depinning mechanisms and theories that attempt to describe are common to a wide variety of physical system. By focusing on the nonlinear dynamics above the onset of motion, Fisher theoretically predicted that elastic depinning would show criticality and that velocity vs. force curves in the vicinity of a threshold force Fc would scale as V  (F Fc)b, where b is a critical exponent [7]. This scaling law has been intensively studied in 2D charge density waves (CDW) systems where b < 1 [7,21,24]. All reported values in the literature,

Fig. 3. History dependence of the depinning process in the plastic regime: (a) V(I) curve for (kab/a0) = 1.35: arrows indicate increasing and decreasing applied current. The first increase in current defines I⁄ (1: filled circle), the following decrease in the current defines Ic (2: open square), and the second increase in current (3: filled square). Inset: reversible I–V curves measured at (kab/a0) = 7.8 after both increasing (filled circle) and decreasing currents (open circle). (b) Irradiated: Log–Lin plot of V(I) curves measured for (kab/a0) = 2.2 with both increasing (filled square), and decreasing (open diamond) the current. Inset: Lin–Lin plot of the same curves.

in the case of an elastic depinning shows that b < 1. With an increasing strength of disorder, a crossover to plastic depinning may occur [6,8]. It is however, not known whether the b exponent occurs in other systems undergoing elastic flow or not. The question now is whether the law predicted for an elastic flow is also valid in the case of a depinning flowed by plastic flow? What is the influence of disorder on the b exponent? It therefore appears that the main way to characterize this dynamic is through the determination of the dynamic exponent b. From Fig. 4, one can see the typical plots for the V–I characteristics before heavy-ion irradiation. Broadly, three current regimes can be identified: a pinned state for I < IC and when increasing I above the critical depinning current two fluid states for IC < I < IP and for I > IP. In Ref. [25] the dynamical regime between IC and IP is identified as a disordered flow where some vortices remain immobile and large transverse excursions of the active channels are present. A large variety for the values, extracted from experimental and numerical studies, for the b exponent for the plastic depinning can be found in the literature (see Table 1). The set of values of the exponent b determined from studies by numerical simulations on periodic systems (vortex lattices, colloids and Wigner crystals) and experiments suggest that in the regimes studied plastic flow, the dynamic exponent is greater than unity. Several theoretical studies focused on the plastic of elastic objects in disordered media in the presence of a driving force. Among the models that provide an assessment of the critical exponent b include especially those proposed by Watson and Fisher [15] and Kawaguchi [26] of

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[6,29]. In numerical studies of colloids in disordered media, the exponent b has been determined in the case of plastic flow b = 1.94 ± 0.03 when the disorder dominates the elasticity [6]. Further work on 2D vortex lattice dynamic at zero temperature driven over a random disorder, offer two categories of depinning depending on the disorder strength [29,30]. For weak disorder (elastic depinning) the dynamics exponent b = 0.27 ± 0.04 (b < 1) [30] and b = 1.3 ± 0.1 (b > 1) in the case of strong disorder (plastic depinning) [29]. In the strong pinning case, the detailed study of the

Fig. 4. Typical scaling plots for the I–V characteristics obtained for as-grown sample at: (a) (k/a0) = 0.78 and (b) (k/a0) = 7.8. Three distinct current regimes are identified 1: a pinned state for I < IC; 2: a disordered flow state between IC < I < IP and 3: smectic flow state or ordered flow state.

interacting particles in a two-dimensional random medium and plastic depinning in one-dimensional random field XY model, respectively. In both models, numerical simulations of these models yield values of b exponent as b  1.53 ± 0.03 and b  1.67 ± 0.01, respectively. The experimental evolution of the current–voltage characteristics with increasing l0H remind us about the reports on layered low-Tc superconductor NbSe2 and a high critical temperature superconductor MgB2 single crystals describing the nonlinear transport properties of the FLL in this system [14,27]. They used magnetic field dependence of the V–I curves and differential resistance to investigate the dynamic nature of the vortex state in vicinity of ‘‘peek effect’’: below the onset of the peak (l0H)on; between (l0H)on and (l0H)peak, and above (l0H)peak. It has been reported than in these three characteristic regions, that below (l0H)on, a ordered vortex phase exists but above (l0H)peak, a disordered phase appears. Between (l0H)on and (l0H)peak, they expect, that both an ordered vortex phase and the disordered phase coexist. In addition, they observe the scaling relation only below (l0H)on and above (l0H)peak. It is found that, above (l0H)peak, the dimensionality effects are more pronounced in the disorder-dominated regime ‘‘plastic flow’’ where b  1.3 (T = 4.2 K; l0H = 2.1T) [28] and 1.75 (T = 4.2 K; l0H = 6.6T) [27] for 2D and 3D case, respectively. In this regime, where the disorder dominates, b is associated to a non-uniform filamentary motion. For a comparison, b  1.2 (T = 4.2 K; l0H = 1.5T and l0H = 4.0T) in both driven 2D and 3D vortex matter below (l0H)on where elastic flow is expected to occur [27]. The scaling relation V  (I Ic)b both below (l0H)on above (l0H)p was also observed in superconducting MgB2 single crystals [14]. The exponent b is 1.2 ± 0.1 (T = 20.5 K; l0H = 2.2T) below (l0H)peak while it is 1.5 ± 0.1 (T = 20.5 K; l0H = 3.75T) above (l0H)peak. The largest number of works concerned with the nature of the dynamics and in particular the determination of the dynamic exponent b is numerical simulations studies of two-dimensional elastic system. For vortices in type II and similar periodic systems such as colloids, b values could be determined in the plastic flow regime

Fig. 5. Scaling behavior of I–V curves obtained for both as-grown and irradiated samples and measured in the vicinity and above the hysteretic region.

Fig. 6. Field dependence of the exponent b measured in the vicinity and above the hysteretic region before (filled square) and (open square) after heavy-ion irradiation. The exponents in these two regimes are extracted from I–V characteristics with decreasing current after field cooling. Four regimes are distinguished by different values of b.

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L. Ammor, A. Ruyter / Physica C 506 (2014) 47–52 Table 1 Exponent b of power-law scaling observed in various simulations and experiments on numerous periodic system for the plastic depinning transition. Compounds

dim

Method

Exponent b

2H-NbSe2 2H-NbSe2 MgB2 Bi-2212 Vortex lattices Vortex lattices Vortex lattices and colloids Colloids Colloids Colloids Josephson junction arrays Metallic dots arrays Metallic dots arrays Metal nanocrystal arrays Metal nanocrystal arrays Wigner crystals CDW NbSe3

3 2 3 3 2 3 2 2 2 2 2 2 2 3 2 2 3

Transport measurements Transport measurements Transport measurements Transport measurements Simulations 3D XY model Model Simulations Confocal microscopy Video microscopy Simulations Simulations Simulations Transport measurements Transport measurements Simulations Transport measurements

2 1.3 1.51 ± 0.12 2.2 ± 0.1 1.3 ± 0.1 2.25 ± 0.02 1.53 ± 0.03 1.91 ± 0.03 1.5 2.2 2.22 ± 0.20 1.94 ± 0.15 2.0 2.25 ± 0.1 2.01 ± 0.04 1.61 ± 0.1 1.23 ± 0.07

vortex trajectories and structure factor identified four dynamical regimes: single-particle, disordered chaotic, smectic chaotic, and decoupled channels. The value of b = 1.3 ± 0.1 was obtained in the disordered chaotic regime [5,29]. Other simulation studies in the plastic flow regime found b  2.2 for vortex flow in strongly disordered Josephson-junction arrays [13], b = 2.0 for electron flow in metallic dots [31] and b = 1.61 ± 0.1 for Wigner Crystals interacting with quenched disorder from charged impurities [4]. Recently, experiments on colloids driven by an electric field and interacting with disordered substrate [24]. They observe plastic depinning with filamentary or river-like flow of colloids and velocity–force curve scaling with b  2.2. Fig. 5 show typical log–log plots of these fits close to the critical current for the as-grown sample and irradiated sample. The exponent b was determined by using the linear part of the V–I curves at low [(I IC)/IC] values. When varying the value of Ic around the previously determined values (i.e. in the confidence interval), the value of b changes in a way to fit the data near the onset of vortex motion and, thus, it allows us to estimate the error of b which is determined to be about 7%. For Ic < I < Ip regime, when increasing the strength of vortex–vortex interaction, we find a crossover in the b exponent. We show typical different dynamic exponent respectively in the low (kab/a0 < 2.5) in Fig. 5(a), and in high (kab/a0 > 2.5) with respect to b  2.2 ± 0.1 and b  1.22 ± 0.02 in Fig. 5(b). For the irradiated sample, similar plots are shown in Fig. 5(c) and (d) in order to compare. We show different power law scaling for (kab/a0) < 2.5 (Fig. 5(c)) and for (kab/a0) > 2.5 (Fig. 5(d)) with b  1.49 ± 0.07 and b  1.02 ± 0.02, respectively. More, it must be noticed that, for (kab/a0) > 2.5, the b = 1.0 is usually attributed to flux flow motion and, then, in favor of a collective élastic depinning. Fig. 6 shows the vortex–vortex interactions dependence of the exponent b measured in the vicinity and above the hysteretic region before and after heavy-ion irradiation. The exponents in these two regimes are extracted from V–I characteristics with decreasing current after field cooling. Note that exponents in these two regimes represent different physical phenomena. A crossover in b value is observed for both irradiated and unirradiated samples. This crossover appears at kab/a0  kab/dU for the irradiated sample confirming the wellknown change of vortex dynamics at the matching field BU which can be interpreted as a crossover from a plastic depinning for B < BU to a elastic one when B > BU. For the as-grown sample, the crossover is located at (kab/a0)  2.5 which correspond to a magnetic field of 0.2–0.3T. Such a magnetic field value as been attributed to the wellknown Fishtail effect observed for overdoped Bi2212 samples using magnetization measurements [37–39]. This effect has often been

Ref. 1.75

1.5 ± 0.02

1.58 ± 0.04 1.71 ± 0.1

[16,28] [27] [14] Our result [29] [32] [15] [6] [33] [21] 13 [1] [31] [34] [35] [4] [36]

associated to a change from single vortex creep to collective vortex creep with increasing field or a 3D/2D transition of the vortex lattice [39]. Our b values (2.2 and 1.2 at low and high magnetic field, respectively) confirm this last interpretation of the vortex dynamic change (see Table 1). Nevertheless, it is quite possible that a change of vortex dynamics in the vicinity of IC could be attributed, according to b values, to a transition from a strongly disordered phase (i.e. Vortex glass) to a possibly Bragg glass phase which could be understood as an direct effect of the increase of vortex–vortex interaction strength (i.e. the system becomes more and more elastic). But our results more strongly suggest a change in the vortex lattice dimensionnality (from 3D to 2D). 4. Conclusion Current–voltage characteristics over a wide range of vortex– vortex interactions strength are used to investigate the vortex dynamics for both as-grown and irradiated Bi2Sr2CaCu2O8+d single crystals at very low temperatures (i.e. T/TC ? 0) in a magnetic field applied parallel to c axis. Our results show a variety of dynamical behavior above the depinning threshold, depending on the vortex–vortex interaction (kab/a0) strength and the nature of the quenched disorder (low or strong pinning). Additionally, we have shown that, above the threshold current, a power law scaling law exists between the current and the voltage. Fit between voltage and applied current can be obtained for the onset of motion in both samples, with different apparent critical exponent depending on defect nature and strength of interactions: b  2.2 ± 0.1 and 1.22 ± 0.02 for point-like defects and b  1.49 ± 0.07 and 1.02 ± 0.02 for columnar defects, respectively. Finally, in both cases of defects, with increasing the strength of vortex–vortex interaction a dynamical transition is observed as confirmed by the crossover in kab/a0 dependence of the critical exponent b. This crossover appears at the matching field BU for the irradiated sample confirming the wellknown change of vortex dynamics. Our results are in favor of a crossover from a plastic depinning for B < BU to a elastic one when B > BU. For the as-grown sample, the crossover is attributed to a 3D/2D transition of the vortex lattice. Our transport measurements studies are in good agreement with previous results (measurements and simulations) and confirm the important role of the system dimensionality on vortex dynamics. Acknowledgments We gratefully acknowledge helpful discussions with A. Pautrat and Ch. Simon.

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