Vortices in superconductors with a columnar defect: Finite size effects
Introduction
The analysis of vortex motion in the presence of defects is one of the most important topic in the superconductivity research field. The relevance of this theme is based on the fact that the behavior of type-II superconductors in the presence of an applied magnetic field can be described by the motion and pinning of vortices in the material.
In the last decades a number of magnetic anomalies, such as the Wohlleben effect, fish tail anomaly and the jumps on the magnetic response of mesoscopic samples, were reported relating the occurrence of defects and the granular conformance of both bulk samples and thin films [1], [2]. Simultaneously, several initiatives using the most different theoretical approaches have been carried out to describe granular superconductors and their characteristic magnetic behavior [3], [4]. Today it is well known that all these features and anomalies are straight linked to the vortex dynamics.
Recently, the advances in the fabrication process related to nanotechnology made possible the production of mesoscopic rings [5] and superconducting arrays with columnar defects [6]. Experiments using these nanoscopic samples showed that the connectivity and vorticity are quite different than those for the macroscopic ones. The physics revealed by those experiments has contributed to a major understanding of type-II granular superconductors. This has motivated both experimental and theoretical physicists to investigate the magnetic properties of either small superconductors or bulk samples with nanosized defects. The properties of the vortex lattice, for instance, in this superconductors of confined geometries change radically with respect to the bulk superconductor ones.
One of the open questions within the framework of mesoscopic superconducting materials is the validity and complete understanding of the simple rule proposed by Mkrtchyan and Shmidt [7]. This rule indicates that the maximum number of vortices that enter a columnar defect is ns = r/2ξ, where r is the radius of the cylindrical cavity and ξ is the superconducting coherence length. Although this rule has been questioned, it is a reasonable estimate of the cavity occupation [8], [9].
In this paper the work of Refs. [7], [10] is extended for the case of a long superconducting wire, of arbitrary size, with a columnar defect. The applied magnetic field is parallel to the cylinder axis and the cylindrical cavity axis coincides with the direction of the applied field. First we determine the local magnetic field with appropriate boundary conditions and further, we calculate the Gibbs free energy of an ensemble of vortices near the cavity with n vortices inside it. On using the Gibbs free energy, the force acting on a vortex near the cavity and near to the external surface of the superconductor is calculated. From the expression of the force, we find a criterion for the maximum number of vortices allowed in the cavity. It is shown that if the size of the cavity is not too small, the saturation number differs substantially from the classical rule of Mkrtchyan and Shmidt [7]. The surface barrier field for flux penetration into the cavity is also determined.
Section snippets
The magnetic field
The starting point of our study is the London equation for the local magnetic field of a very strong type-II superconductor for which the Ginzburg–Landau parameter . For our purposes, this equation can be more conveniently written in cylindrical coordinates aswhere ρi is the position of the i-vortex line, Φ0 is the quantum flux, and N is the number of vortices outside the columnar defect.
Consider a long superconducting wire under an applied
The Gibbs free energy
In order to proceed we need to evaluate the Gibbs free energy (per unit length) which is given bywhere F is the London free energy (the Helmholtz free energy in the thermodynamic context) and B is the average induction; here A is the area of the cylinder cross section. In the London approximation, the free energy is given byBy using integration by parts, we can writeNext, we compute the
Force
We will be most interested in the particular case N = 1. From Eq. (20), we haveNow the Green’s function is given byand the magnetic field in the cavity isThe force acting on a single vortex can be found by taking the derivative of
Summary and conclusions
In summary, we have derived the local field and the free energy of an ensemble of vortices around a columnar defect in a superconducting wire for both the cavity and the superconductor of arbitrary sizes. We also evaluated the force near to the cavity and external surfaces of the superconductor. It has also been found that for a large superconductor, the cavity saturates at a larger number of vortices as its size increases. However, not linearly in r/2ξ as predicted in [7]. In fact, in [8], [9]
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