Impact of border defects on the magnetic flux penetration in superconducting films

TL;DR

This paper reviews how border defects modify magnetic flux penetration in superconducting films by contrasting London/GL vortex-entry pictures with continuous-media descriptions. It shows that defects create current crowding, lower edge barriers, and long-range perturbations in current flow, reducing the penetration field $H_p$ and altering vortex-entry conditions. Through critical-state and hodograph-based analyses, it connects defect geometry to observable $d$-lines and flux patterns, while GL calculations refine the quantitative predictions near surfaces. The work also discusses how border defects influence thermomagnetic instabilities, highlighting conditions under which edge features promote or suppress flux avalanches, with implications for devices like resonators, detectors, and cavities.

Abstract

Defects in superconducting systems are ubiquitous and nearly unavoidable. They can vary in nature, geometry, and size, ranging from microscopic-size defects such as dislocations, grain boundaries, twin planes, and oxygen vacancies, to macroscopic-size defects such as segregations, indentations, contamination, cracks, or voids. Irrespective of their type, defects perturb the otherwise laminar flow of electric current, forcing it to deviate from its path. In the best-case scenario, the associated perturbation can be damped within a distance of the order of the size of the defect if the rigidity of the superconducting state, characterized by the creep exponent $n$, is low. In most cases, however, this perturbation spans macroscopic distances covering the entire superconducting sample and thus dramatically influences the response of the system. In this work, we review the current state of theoretical understanding and experimental evidence on the modification of magnetic flux patterns in superconductors by border defects, including the influence of their geometry, temperature, and applied magnetic field. We scrutinize and contrast the picture emerging from a continuous media standpoint, i.e. ignoring the granularity imposed by the vortex quantization, with that provided by a phenomenological approach dictated by the vortex dynamics. In addition, we discuss the influence of border indentations on the nucleation of thermomagnetic instabilities. Assessing the impact of surface and border defects is of utmost importance for all superconducting technologies, including superconducting resonators, superconducting single-photon detectors, superconducting radio-frequency cavities and accelerators, superconducting cables, superconducting metamaterials, superconducting diodes, and many others.

Figures (24)

• Figure 1: (a) current streamlines in a conducting plane with a central hole of radius $R$. (b) The dimensionless perturbation of the norm of the current density, $\delta j / j_0$, as a function of $x/R$, the dimensionless distance from the center of the circular defect. $j_0$ is the value of the uniform current density in a plain sample, without the circular defect. Inset: The decay exponent, $p$, as a function of the $E$-$j$ exponent, $n$ (${\bf E} \sim j^{n-1} \bf{j}$). The exponent $p$ corresponds to the best-fit power-law ($A.x^p$, $A$ being a constant) of the graph $\delta j / j_0$ vs. $x/R$.
• Figure 2: A brick wall distorted by a single book by artist Jorge Méndez Blake. At the base of the wall is Frank Kafka’s The Castle (Das Schloss). This art piece illustrates the fact that small perturbations in a rigid medium propagate long distances from the source.
• Figure 3: (a) Superheating field $H_p$ as a function of the Ginzburg-Landau parameter $\kappa$ using analytical approximations corresponding to the regimes of one-dimensional (long wavelength of the instability of the Meissner state) critical perturbations ($\kappa < 1.1495$, blue curve) and two-dimensional (short wavelength) critical perturbations ($\kappa > 1.1495$, black curve). Adapted from Ref.Transtrum. (b) Superheating field as a function of the reduced temperature $T/T_c$ normalized by the critical field at zero temperature (main panel) and by $H_c(T)$ (inset). Adapted from Ref.Catelani-2008
• Figure 4: Local magnetization loops $B-H_a$ as a function of applied magnetic field $H_a$ in BSCCO crystals of platelet (a) and prism (b) shapes at $T = 80$ K. The platelet crystal shows hysteretic magnetization below the irreversibility field $H_{IL}$. In the prism sample, the geometrical barrier is strongly suppressed, and a fully reversible magnetization is obtained at temperatures above 76 K. The vortex melting transition $H_m$ is observed as a sharp thermodynamic step in the local magnetization. A cross-section of the experimental setup is shown schematically in the insets (not to scale). The two-dimensional electron gas active layer of the sensors resides about 0.1 µm below the surface. The BSCCO crystals are in contact with the GaAs surface, and the local vertical component of the magnetic field $B$ is measured directly. The external field $H_a$, is applied parallel to the crystalline $c$-axis. Reproduced from Ref.Majer1995.
• Figure 5: (a) Magneto-optical image showing the vortex penetration in zero-field cooling condition in a 100 µm thick NbSe$_2$ single crystal, at an applied magnetic field of 0.2 mT. The vortex-free region is seen between the edge (bright vertical band on the left) and the vortex-filled region. The scale bar corresponds to 10 µm. Reroduced from Ref.Olsen. Scanning SQUID microscopy covering a surface of 12 µm $\times$ 12 µm obtained in a Pb film with a 5.7 µm-wide central constriction at 4.2 K and after field-cooling in fields of (b) 2.7 mT, (c) 5.4 mT, and (d) 12 mT. The vortex-free region is seen to shrink as the magnetic field increases. Reproduced from Ref.Embon.