Dynamic roughening of the magnetic flux landscape in
Introduction
When a type II superconductor is put in a slowly ramped external magnetic field, vortices start to penetrate the sample from its edges. These vortices get pinned by dislocations or other crystallographic defects, leading to the build-up of a flux gradient, which is only marginally stable, as is the slope of a slowly grown pile of sand [1]. Thus, it can happen that small changes in the applied field can lead to large rearrangements of flux in the sample, known as flux avalanches [2]. Due to the analogy of the flux landscape with a pile of sand, the properties of flux avalanches were previously studied in the context of self-organized criticality (SOC) [3], which predicts that many slowly driven non-equilibrium systems have avalanches which are distributed according to a power-law [4]. As a matter of fact, vortex avalanches in superconductors are thought of as an ideal experimental system in which to study SOC, due to the over-damped dynamics of the vortices [2], [5]. In the past, power-law distributed avalanches have been observed in a number of controlled experiments, which were ascribed to SOC [6]. Furthermore, the microscopic dynamics of the particles (vortices) is well known [7] and the collective dynamics can then be for instance studied in detail using molecular dynamics simulations [8]. In more detail however, SOC predicts that a system not only shows power-law behaviour, but that it organizes into a critical state, and should thus show finite-size scaling in the distribution of avalanches as well [9]. This has now also recently been shown for the flux-avalanches in a thin film of (YBCO) [10], where also the shape of the flux avalanches and their fractal dimension was characterized. The shape and distribution of avalanches moreover strongly influence the shape of the magnetic flux landscape, leading to a rough, self-affine surface [11]. The characteristic exponents of this surface can then be obtained quantitatively from the avalanche properties via a set of scaling relations derived by Paczuski et al. for many SOC models [12]. The (numerical) values obtained can be compared with a direct measurement of the growth and roughness exponents.
Here, we study the roughening properties of the magnetic flux landscape in a thin film of YBCO in two dimensions (2D). While dynamic roughening has been experimentally studied in many 1D systems [13], [14], [15], among which there was also a study of the front of penetrating flux in YBCO [16], [17], 2D characterization of roughening systems are rare in the experimental literature [18]. A full 2D characterization of the roughness properties makes it possible to compare properties of the avalanches with those of the surface, in order to have a stringent test for SOC in the flux avalanches in YBCO [10]. Furthermore, we compare the roughness results with numerical integrations of the Edwards–Wilkinson (EW) equation [19] in the presence of quenched disorder in order to have a comparison with the static pinning in the experiment.
Section 2 describes the experimental setup in detail, while Section 3 introduces the analysis methods for a 2D surface, which are applied to the data. In Section 4, we present the results of the numerical studies of the quenched EW equation, before turning to the roughness of the flux landscape in Section 5. These roughness results will finally be compared with the avalanche properties, determined elsewhere [10], in Section 6.
Section snippets
Experimental setup
The magnetic flux density just above the YBCO thin film is measured by means of the Faraday-effect [20] in an advanced magneto-optical microscope [21]. In this setup, the polarization rotation angle is measured directly by means of a lock-in technique. The YBCO films studied here were grown on a NdGaO3 substrate to a thickness of 80 nm using pulsed laser ablation [22]. Pinning sites in the sample consist mostly of screw dislocations and are distributed uniformly over the sample, acting as
Analysis methods
As can be seen from Fig. 1, the magnetic flux surface, , has an average profile, , with high values near the sample edge and zero magnetic flux inside the sample, where denotes an average over the subscript s. In order to only study the properties of the fluctuations in the surface, this average profile is subtracted from the data, such that the properties of are studied in the following. Furthermore, the time evolution of the flux landscape is
Numerical simulations
In order to study the influence of quenched disorder, analogous to the static point pins in the superconductor, on the roughening properties of the flux surface, we also carried out numerical simulations of the two dimensional EW equation in the presence of static disorder. The EW equation describes the height of an interface, , as a function of time, which is pulled though a disordered medium with a speed v. In the experimental case, the driving can be seen as the increasing applied
Flux landscape
Given the rough landscape shown in Fig. 1, we determine the self-affine properties of the fluctuations around the mean surface as discussed in Section 3 above. In order to have a reasonable size for the 2D area used in the analysis, we analyse images starting from an applied field of 5 mT and determine the spatial and temporal correlation functions for the subsequent 128 images. Averaged over these 128 images and over all experiments, the full 2D correlation function, , is shown as a
Comparison with avalanche properties
It has been noted that the roughness of a surface of a SOC system is created by the shape and distribution of the avalanches in the system [11]. Thus, there should be a connection between these two phenomena, which can be tested [12]. Such a quantitative connection has been previously shown in the properties of the avalanches and the surface roughness of a three dimensional pile of rice [18]. From a theoretical point of view, the scaling relations have been derived in the context of models of
Conclusions
In conclusion, we have shown that the two dimensional flux surface of an YBCO thin film in the mixed state is self-affine and shows power-law scaling in its roughness and growth, given by and . This behaviour is in good agreement with the expectations from a simple roughening system (the EW equation) in the presence of quenched disorder. In addition, the surface roughness is connected to the avalanche properties of the magnetic flux jumps, as is expected for a SOC system. The
Acknowledgements
We would like to thank Jan Rector for providing the sample. This work is supported by FOM (Stichting voor Fundamenteel Onderzoek der Materie), which is financially supported by NWO (Nederlandse Organisatie voor Wetenschappelijk Onderzoek).
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