Quasi-one-dimensional intermittent flux behavior in superconducting films

Intermittent ﬁlamentary dynamics of the vortex matter in superconductors is found in ﬁlms of YBa 2 Cu 3 O 7 - (cid:1) deposited on tilted substrates. Deposition of this material on such substrates creates parallel channels of easy ﬂux penetration when a magnetic ﬁeld is applied perpendicular to the ﬁlm. As the applied ﬁeld is gradually increased, magneto-optical imaging reveals that ﬂux penetrates via numerous quasi-one-dimensional jumps. The distribution of ﬂux avalanche sizes follows a power law, and data collapse is obtained by ﬁnite-size scaling, with the depth of the ﬂux front used as crossover length. The intermittent behavior shows no threshold value in the applied ﬁeld, in contrast to conventional ﬂux jumping. The results strongly suggest that the quasi-one-dimensional ﬂux jumps are of a different nature than the thermomagnetic dendritic (branching) avalanches that are commonly found in superconducting ﬁlms.

Since the prediction of a regular vortex lattice in type-II superconductors [1], a wide variety of vortex systems has been thoroughly investigated [2,3]. The pronounced dependencies of the range and strength of the vortex-vortex interaction on temperature and magnetic field make the vortex matter a unique, tunable model system for probing the statistical properties of interacting systems. The behavior of the vortex matter is to a large extent also determined by its interaction with quenched disorder in the material, i.e., with microscopic defects that pin the vortices and therefore serve as pinning centers in the superconductor. The interplay between the pinning forces and the driving Lorentz force leads to formation of a metastable critical state [4], where the current density has a critical magnitude, j c , which corresponds to the maximal nondissipative current density.
In films of YBa 2 Cu 3 O 7-(YBCO) grown epitaxially, i.e., with proper matching of crystal lattice parameters, on a substrate slightly tilted from a major crystal plane (vicinal films), self-organized arrays of planar defects (antiphase boundaries) are formed with a typical period of 2-5 nm [5]. The terrace structure of the surface is characterized by steps that are a few Å ngströms in height, and the steps provide well-oriented seeding sites for growth along the main axis (c axis) of the overlying YBCO film. Films grown on substrates with a tilt angle of close to 10 have better grain alignment and improved current-carrying ability [5,6]. There has been considerable interest in such films due to their potential application in Josephsonjunction design [7][8][9], in particular for highly sensitive detectors of electromagnetic radiation.
The growth of YBCO films on tilted substrates is also known to produce planar defects, which become channels for easy vortex motion [5,6,10]. These channels are separated by 5-10 m and run parallel to the terrace structure, causing considerable anisotropy in both the flux penetration and the effective critical current. At low temperatures, where the anisotropy is most pronounced, the vortices are essentially confined inside the channels, and the flux penetration has a strong quasi-one-dimensional (quasi-1D) character. In this paper, we show, using magneto-optical imaging (MOI), that the quasi-1D flux dynamics is highly intermittent at low temperatures in samples with moderate tilt angles.
Intermittent flux motion is very harmful for electronic devices, so the source of intermittency should be identified and the behavior characterized. Until now, intermittent flux behavior in superconducting films has been observed in Nb, Pb, Nb 3 Sn, NbN, MgB 2 , and YNi 2 B 2 C [11][12][13][14][15][16], where, during slow ramping of the applied field, the flux avalanches take the form of large dendritic (branching) structures. These avalanches are believed to be caused by a thermomagnetic instability [17][18][19][20]. We find that, in YBCO on tilted substrates, the flux jumps are largely Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. different from the thermomagnetic avalanches both in morphology and in size distribution, suggesting that the origin of the intermittency is of a different nature.
Superconducting films are prepared by laser ablation (vaporization by a laser beam and subsequent crystallization) of YBa 2 Cu 3 O 7-ceramic targets on NdGaO 3 substrates with tilt angles of up to 33 . (See Ref. [9] for sample preparation details.) The films we investigate in the present work have thickness of d ¼ 200 nm and a critical temperature of T c ¼ 88 K. They are shaped by optical lithography into long strips of width 2w ¼ 0:9 mm. Lowtemperature MOI is performed using an in-plane magnetized ferrite-garnet film as the Faraday rotating sensor [21,22]. The measurement configuration is illustrated in Fig. 1. Figure 2(a) shows a magneto-optical image of the flux penetration in a ¼ 14 film where the surface steps are aligned vertically in the picture. The sample is initially cooled to T ¼ 4 K at zero field, and then a perpendicular applied magnetic field is slowly increased at a rate of 0:2 mT=s. The image is recorded at B a ¼ 16:2 mT. The flux penetrates the sample from the two long edges, seen in the figure as bright horizontal lines, in a distinctly filamentary manner, where the filaments are aligned with the underlying terrace steps. From the short edge, however, the flux penetration is essentially blocked by the strong pinning associated with vortex motion transverse to the terrace structure. The full flux dynamics during such a field ramp is seen in Video 1.
From the real-time observation, it is evident that the penetration is strongly intermittent, where the flux enters from both long edges via discrete quasi-1D avalanches. This penetration is also documented in panel (b) of Fig  Analyzing the sequence of avalanches from such movies, we find no correlation between flux jumps in adjacent channels. Thus, the overall flux-penetration process can be considered as a sequence of independent avalanche events. More insight into the phenomenon is obtained by determining whether the avalanches are reproducible or not. The well-known dendritic flux avalanches caused by thermomagnetic instability have the characteristics of being irreproducible [15,24,25]. When a field sweep is repeated under the same experimental conditions, the flux patterns show essentially no overlap, suggesting that stochastic fluctuations play a main role. The present quasi-1D avalanches also display irreproducible behavior, although on a smaller scale. Figure 3 shows a superposition of three monochrome color-coded (red, green, and blue) differential images taken at the same applied field and temperature during three different consecutive experimental runs. In an image made in this way, repeatable features appear as a gradation of gray for repeatable features, whereas events that occur in only one run appear as red, green, or blue. The superposition is black in areas where no flux change occurs in any of the images. Evidently, there is no overlap in the three sets of flux avalanches [26]. The same irreproducibility has been found at all fields investigated.
All the quasi-1D avalanches are limited in length by the overall flux-penetration front as it advances toward the middle of the strip in response to the increasing applied magnetic field. The front is built up as an envelope of numerous flux needles formed by the avalanches penetrating the most deeply into the film. This result is strikingly different from the dendritic type of avalanches, where the size of the jumps can span the sample even with a small applied field, and the flux front in general attains a very complex shape. In our case, the flux front is far more well behaved.
To quantify the new type of intermittent flux dynamics, we record magneto-optical images while the applied field VIDEO 2. The differential video obtained by subtracting subsequent frames in Video 1.
FIG. 3. Superposition of three magneto-optical differential images of flux penetration in a YBCO film on a ¼ 14 -tilted substrate. The different monochrome image series are color coded in red, green, and blue, respectively. The presence of colored superpositions shows that the flux jumps are not reproducible. The sample temperature is T ¼ 4 K, the applied magnetic field is B a ¼ 16:7 mT, and the field step is ÁB a ¼ 42:5 T. The scale is the same as for Fig. 2(a). is slowly ramped from zero to B a ¼ 17 mT. The sample becomes nearly fully penetrated by flux. Images are recorded every 42:5 T. From all these images, a sequence of difference images is prepared, and, for every avalanche that occurs, its length ' is measured by using a patternrecognition program. Figure 4 shows how ' max , the size of the longest avalanche within each field interval, varies with the applied field. The maximum length follows a smooth curve, which corresponds very well to the advancement of the overall flux front, i.e., at all fields, there are avalanches starting at the edges and ending at the flux front. Actually, the ' max ðB a Þ behaves in full accord with the fluxpenetration depth for a long thin superconducting strip in the critical state. For a strip in the Bean critical state, the flux front reaches a depth of L given by [27] L=w ¼ 1 À cosh À1 ðB a =B d Þ; where B d ¼ 0 j c d=. In Fig. 4, LðB a Þ is plotted to fit the experimental data using a critical current density of j c ¼ 0:78 Â 10 11 Am À2 as the only fitting parameter. Based on the good agreement, we conclude that the overall flux front evolves with the applied field according to the conventional critical-state behavior, whereas the detailed dynamics is governed by quasi-1D flux avalanches of length up to the depth of the flux front. Figure 5 shows the size distribution of all the detected avalanches of more than 10 000, as a normalized probability for the occurrence of an avalanche of length '. The data are grouped into equal field intervals of 1.7 mT. From the log-log plot, it is evident that the vast majority of the avalanches are distributed as a power law Pð'Þ $ ' À . The avalanche distribution has a natural cutoff length at the overall flux-penetration depth LðB a Þ. A fit to the linear part of the data gives ¼ 1:3 for the exponent. Applying finite-size scaling according to the relation where f is a scaling function, the plot in Fig. 6 shows that a good data collapse is obtained with D ¼ 1.
To summarize the experimental results, we have found that, at small fields, the avalanches are also small, and, as the applied field becomes larger, the maximum avalanche size gradually increases. The envelope curve of the fluxpenetration front is slightly ramified (split into branches) by the avalanche activity, but it still behaves essentially in full accord with the critical-state model. Analyzing avalanches that occur within a small field interval as the applied field increases, we find that the distribution of avalanche sizes is a monotonously decreasing curve. All these characteristics are in stark contrast to the observations reported for flux avalanches caused by thermomagnetic runaways. Figure 7 illustrates the typical dendritic morphology of the thermomagnetic-runaway type of avalanches, here seen in a film of MgB 2 . A movie of the dendritic flux penetration into a film of MgB 2 is seen in Video 3. The avalanches protrude in highly nonregular shapes, deep into the Meissner-state part of the film, and histograms of the avalanche size show a strongly peaked distribution [28]. Moreover, in contrast to the dendritic avalanches, our observations of the present YBCO films do not detect any threshold field for onset of the avalanche activity. Thus, there are several significant differences between the two types of avalanches, which suggest that the quasi-1D flux jumps in YBCO on tilted substrates are not of a thermomagnetic origin. A more likely scenario is that the intermittent filamentary behavior is a result of vortex dynamics under conditions of channeling, which is clearly seen in the current MOI experiments. The film has a large number of linear defects which form parallel narrow channels with low pinning surrounded by areas of strong pinning, thus providing quasi-1D confinement. As flux enters these channels, some vortices will also penetrate into the strongpinning areas, thus forming a fence on both sides. These vortices, which sit at strong-pinning sites, certainly will have substantial positional disorder and hence add a considerable random component to the confinement. Thus, the channels will contain a sequence of random bottlenecks for the traffic of vortices. As the applied magnetic field increases, more and more vortices penetrate into the channels, and situations analogous to a traffic jam, or clogging, will occur. Whenever one of these blocked regions eventually breaks open, an avalanche of vortex motion takes place along the channel. Since these barriers can be arbitrarily small, there is no threshold applied field for the onset of this intermittent behavior. This picture is consistent with our experimental observations. Moreover, this mechanism allows avalanches to start not only from the sample edge, but also from inside the channels, which is also in accordance with our observations. Interestingly, this scenario resembles the flux motion found in artificially created mesoscopic flow channels. (See Ref. [29] and references therein.) It is also worth discussing our results within the framework of self-organized criticality (SOC). Although the first dynamical system shown to display SOC, the Bak-Tang-Wiesenfeld sand pile [30], does not exhibit SOC in one dimension, several similar algorithms with different toppling rules and intrinsic randomness do [31][32][33]. These models were invented to study numerically a mechanical quasi-1D dynamical system, the Oslo rice pile [34], which for certain types of rice grains displays SOC behavior. The numerical simulations give power-law avalanche-size distributions with exponents in the range of 1.3-1.6, depending on the details of the model. Our measured avalanche exponent falls within this range.
In the SOC literature, various definitions are used for the size of an avalanche: the number of sites being toppled, the amount of dissipated energy, the spatial extent of an avalanche, etc. In the present work, we have also performed an alternative statistical analysis where the avalanche size is taken to be the amount of the flux being abruptly displaced in each event, rather than the length of the filamentary jumps. This alternative analysis also leads to good scaling and a power-law exponent that agrees within 0.1 of that reported in this paper. Aegerter et al. [35] have also reported power-law statistics of flux avalanches in YBCO thin films deposited on plain substrates. Using MOI, they find the exponent close to ours. However, their observed avalanches are far from being one dimensional, having D ¼ 1:89. Closer-to-one-dimensional motion has been observed in small bundles of vortices in superconductor films of tungsten by use of scanning microscopy [36]. However, the data were accumulated over periods of many minutes, showing very different time characteristics compared to our observations. Note that, in contrast to the previous fluximaging experiments [35,36], the present MOI observations cover the avalanche activity of the entire flux pile, enabling us to perform finite-size scaling with the physical size of the pile L, instead of analyzing a small viewing window.
Worth noting in Fig. 6 is the slight overshoot in the curves near the cutoff length, L. Also, simulations VIDEO 3. For comparison, a typical set of flux avalanches in a MgB 2 film. Notice the different morphology between these avalanches spanning the sample in complex dendritic patterns, and those seen in Video 1. [31][32][33] show this feature, i.e., a small increase in the probability of having avalanches of maximum size, which is interpreted as a finite-size effect [31]. To our knowledge, this effect has not been observed experimentally before in vortex systems.
Since inertia can destroy SOC in mechanical systems [37], superconducting vortices, commonly considered as massless, indeed represent a very favorable candidate for SOC, and a lot of numerical simulation work has been reported on the topic [38][39][40][41]. In particular, the simulations for two-dimensional vortex motion in a slab [39] show SOC behavior with scaling and power-law distributions, including an overshoot similar to that in our data.
We also perform similar experiments on YBCO films using two other ramp rates: 0:12 mT=s and 0:02 mT=s. No significant qualitative or quantitative changes in the behavior are observed. Moreover, intermittent flux dynamics has also been observed in vicinal samples with tilt angles of ¼ 8 , 10 , and 12 . However, in films with ¼ 3 , 20 , 26 , and 33 , we find, not abrupt flux motion, but rather a smooth penetration into the channels. A reference sample with zero-degree tilt angle has shown neither flux jumps nor channeling. More work is required to clarify the relationship between the sample morphology and flux dynamics in superconductor films grown on vicinal-cut substrates.
In conclusion, we find that a gradual injection of the magnetic flux into a system of well-defined channels may trigger quasi-1D vortex avalanches, the sizes of which are distributed by a power law with the avalanche exponent ¼ 1:3. Introducing the depth of the flux front as a dynamic length scale makes the distributions invariant.