Structural similarity between the magnetic-flux profile in superconductors and the surface of a 2d rice pile

– We analyze the magnetic-flux profile in the critical state of an YBa2Cu3O7−x thin film. It is found that the profile forms a self-affine structure in both space and time. Furthermore, the surface of a pile of rice is studied, which also shows a self-affine structure. These experimental results are compared to simulations of a two-dimensional version of the Oslo model for sandpiles. The exponents characterizing the roughness and dynamics of the profiles for both experimental systems agree with each other, the values being α = 0.39(3) for the roughness exponents and β = 0.50(3) for the growth exponents. These values are reproduced in the simulations as well. This shows that a simple sandpile model can be used to describe the shape and dynamics of the vortex landscape in the critical state. Conversely, superconductors may be used as a model granular system. The connection between sandpiles and the critical state in superconductors goes back to de Gennes [1], more than thirty years ago. He invoked the sandpile as a mechanical model for the occurrence of the critical state. In this analogon, the vortices of quantized magnetic flux in the superconductor take the role of sand grains. The friction forces between the grains are mapped onto pinning forces due to impurities and gravity takes the form of a “magnetic pressure” arising from the repulsive inter-vortex interactions [2]. Given these identifications, the building up of the critical state in a superconductor is similar to that of a sandpile, with its self-organizing slope. In recent years, the image of a sandpile has become a paradigm for the complex behaviour of simple dynamical systems. This is mainly due to the concept of self-organized criticality (SOC) and its sandpile cellular automaton [3]. However, experiments on real sand and its dynamics have been inconclusive for a long time as to whether SOC does in fact apply to them [4]. Recent experiments on the behaviour of one-dimensional (1d) piles of rice have however shown the features of a critical system, such as finite-size scaling [5]. Further investigations into rice piles have also shown that the surface of such a rice pile is a self-affine structure [6, 7] giving further insights into the dynamics of experimentally realized SOC systems. Extending these studies to the structure of a 2d rice pile opens up a way for a proper comparison of the critical state in superconductors with a granular pile. The corresponding flux profile,