There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Quasiclassic Usadel equations for two-band superconductors in the dirty limit are derived with the account of both intraband and interband scattering by nonmagnetic impurities. From these equations, the Ginzburg-Landau equations, and the equations for the critical temperature ${T}_{c}$ and the upper critical field ${H}_{c2}$ are obtained. The equation for ${H}_{c2}$ determines both the temperature dependence of ${H}_{c2}(T)$ and the orientational dependence of ${H}_{c2}(\ensuremath{\theta})$ as a function of the angle $\ensuremath{\theta}$ between $\mathbf{H}$ and the c axis. It is shown that the shape of the ${H}_{c2}(T)$ curve essentially depends on the ratio of the intraband electron diffusivities ${D}_{1}$ and ${D}_{2},$ and can be very different from the one-gap dirty-limit theory. In particular, value ${H}_{c2}(0)$ can considerably exceed ${0.7T}_{c}{\mathrm{dH}}_{c2}{/dT}_{c},$ which can have important consequences for applications of ${\mathrm{MgB}}_{2}.$ Based on a general equation for ${H}_{c2}(T,\ensuremath{\theta}),$ a scaling relation, which enables one to get the angular dependence of ${H}_{c2}(\ensuremath{\theta})$ from the equation for ${H}_{c2}$ at $\mathbf{H}\ensuremath{\Vert}c,$ is proposed. The behavior of ${H}_{c2}(\ensuremath{\theta})$ and ${H}_{c1}(\ensuremath{\theta})$ can deviate markedly from the results of the anisotropic Ginzburg-Landau theory, and the ratio of the upper critical field ${H}_{c2}^{\ensuremath{\Vert}}{/H}_{c2}^{\ensuremath{\perp}}$ for $\mathbf{H}\ensuremath{\Vert}\mathrm{ab}$ and $\mathbf{H}\ensuremath{\perp}\mathrm{ab}$ can both increase and decrease with temperature, depending on the relation between ${D}_{1}$ and ${D}_{2}.$ Implications of the obtained results for ${\mathrm{MgB}}_{2}$ are discussed.

https://arxiv.org/pdf/cond-mat/0212129.pdf

Enhancement of the upper critical field by nonmagnetic impurities in dirty two-gap superconductors

Less than two years after the discovery of high temperature superconductivity in oxypnictide LaFeAs(O, F) several families of superconductors based on Fe layers (1111, 122, 11, 111) are available. They share several characteristics with cuprate superconductors that compromise easy applications, such as the layered structure, the small coherence length and unconventional pairing. On the other hand, the Fe-based superconductors have metallic parent compounds and their electronic anisotropy is generally smaller and does not strongly depend on the level of doping, and the supposed order parameter symmetry is s-wave, thus in principle not so detrimental to current transmission across grain boundaries. From the application point of view, the main efforts are still devoted to investigate the superconducting properties, to distinguish intrinsic from extrinsic behaviors and to compare the different families in order to identify which one is the fittest for the quest for better and more practical superconductors. The 1111 family shows the highest Tc, huge but also the most anisotropic upper critical field and in-field, fan-shaped resistive transitions reminiscent of those of cuprates. On the other hand, the 122 family is much less anisotropic with sharper resistive transitions as in low temperature superconductors, but with about half the Tc of the 1111 compounds. An overview of the main superconducting properties relevant to applications will be presented. Upper critical field, electronic anisotropy parameter, and intragranular and intergranular critical current density will be discussed and compared, where possible, across the Fe-based superconductor families.

/pdf/new-fe-based-superconductors-properties-relevant-for-2iik9h8fq2.pdf

New Fe-based superconductors: properties relevant for applications

This review focuses on the superconducting properties of MgB2 that are relevant for power applications. The reversible mixed state parameters are the most important, since they define the limiting conditions for loss-free currents: the transition temperature, the upper critical field and the depairing current. They also determine the flux pinning energy, the pinning force and the elastic properties of the flux line lattice and, therefore, strongly influence the critical current densities. The magnetic properties of magnesium diboride are anisotropic and influenced by the two different energy gaps of the σ- and π-bands. Whereas the transition temperature could not be enhanced significantly during the past five years, the upper critical field was considerably increased by impurity scattering or doping. Flux pinning is very weak in MgB2 single crystals and was only improved by irradiation techniques so far. In polycrystalline samples, grain boundary pinning seems to play the dominant role. High critical currents close to the theoretical limit were found in c-axis oriented thin films. The anisotropy of the upper critical field strongly reduces the critical currents in untextured MgB2 at high magnetic fields, where the supercurrents become highly percolative, since not all grains are superconducting anymore. The performance of polycrystalline wires and tapes was significantly improved during the past few years by increasing the upper critical field and by reducing its anisotropy. Pinning seems to be nearly optimized in many forms of this material, but the connectivity between the grains might be further improved.

https://iopscience.iop.org/article/10.1088/0953-2048/20/12/R01/pdf

Magnetic properties and critical currents of MgB2

Numerical modeling of superconductors is widely recognized as a powerful tool for interpreting experimental results, understanding physical mechanisms, and predicting the performance of high-temperature-superconductor (HTS) tapes, wires, and devices. This is particularly true for ac loss calculation since a sufficiently low ac loss value is imperative to make these materials attractive for commercialization. In recent years, a large variety of numerical models, which are based on different techniques and implementations, has been proposed by researchers around the world, with the purpose of being able to estimate ac losses in HTSs quickly and accurately. This paper presents a literature review of the methods for computing ac losses in HTS tapes, wires, and devices. Technical superconductors have a relatively complex geometry (filaments, which might be twisted or transposed, or layers) and consist of different materials. As a result, different loss contributions exist. In this paper, we describe the ways of computing such loss contributions, which include hysteresis losses, eddy-current losses, coupling losses, and losses in ferromagnetic materials. We also provide an estimation of the losses occurring in a variety of power applications.

Computation of Losses in HTS Under the Action of Varying Magnetic Fields and Currents

The influence of anisotropy on the transport current in ${\mathrm{M}\mathrm{g}\mathrm{B}}_{2}$ polycrystalline bulk samples and wires is discussed. A model for the critical current density is proposed, which is based on anisotropic London theory, grain boundary pinning, and percolation theory. The calculated currents agree convincingly with experimental data, and the fit parameters, especially the anisotropy, obtained from percolation theory agree with experiment or theoretical predictions.

Current percolation and anisotropy in polycrystalline MgB(2).

Two-band superconductivity has become an important topic over the past ten years Extensive experimental and theoretical studies started with MgB2 and are now focused on iron-based and other new superconductors In this review, I describe how important thermodynamic, reversible mixed-state, and other superconducting properties are changed by two-band superconductivity and, for comparison, by other effects such as anisotropy in a single-band material or an energy-gap structure different from the conventional s-wave symmetry The work consists of three main parts, in which I review (i) theoretical models and what they predict for experimentally accessible properties in the two-band and other scenarios, (ii) experimental methods applied for investigating superconducting properties and the results obtained in potential two-band materials, and (iii) materials, for which two- or multi-band superconductivity has been suggested It is shown that two-band effects appear in most of the analyzed properties and that they can be quite significant but usually fade away as interband interactions increase Anisotropy often leads to similar modifications in single-band superconductors, which is why the distinction of two-band and anisotropy effects is usually difficult, particularly when the temperature dependence of the quantities is examined In contrast, the field-dependent effects are more often different and thus more often allow a reliable distinction between the models

https://iopscience.iop.org/article/10.1088/0953-2048/26/4/043001/pdf

A review of two-band superconductivity: materials and effects on the thermodynamic and reversible mixed-state properties

Sintered samples of MgB2 were irradiated in a fission reactor. Defects in the bulk microstructure are produced during this process mainly by the 10B(n,α)7Li reaction while collisions of fast neutrons with the lattice atoms induce much less damage. Self-shielding effects turn out to be very important and lead to a highly inhomogeneous defect distribution in the irradiated samples. The resulting disorder enhances the normal state resistivity and the upper critical field. The irreversibility line shifts to higher fields at low temperatures and the measured critical current densities increase following irradiation.

https://iopscience.iop.org/article/10.1088/0953-2048/15/2/101/pdf

Neutron irradiation of MgB2 bulk superconductors

Sintered samples of MgB2 were irradiated in a fission reactor. Defects in the bulk microstructure are produced during this process mainly by the 10B(n,a)7Li reaction while collisions of fast neutrons with the lattice atoms induce much less damage. Self-shielding effects turn out to be very important and lead to a highly inhomogeneous defect distribution in the irradiated samples. The resulting disorder enhances the normal state resistivity and the upper critical field. The irreversibility line shifts to higher fields at low temperatures and the measured critical current densities increase following irradiation.

Neutron Irradiation of MgB2 Bulk Superconductors

We report on the effects of neutron irradiation in superconducting ${\mathrm{MgB}}_{2}$ single crystals. Transmission electron microscopy investigations reveal large radiation-induced defects with dimensions comparable to the superconducting coherence length. Accordingly, measurements of the magnetic moment show pronounced modifications of the reversible and irreversible properties after irradiation. In particular, we concentrate on the emergence of a second peak (fishtail), situated near ${H}_{c2}$ at low neutron fluences, but extending over a large part of the superconducting phase diagram at high doses. The history dependence of the critical current density on the low-field side of the fishtail region is studied by measuring minor hysteresis loops and interpreted in terms of an order-disorder transition of the vortex lattice.

M. Zehetmayer

Papers

Current percolation and anisotropy in polycrystalline MgB(2).

A review of two-band superconductivity: materials and effects on the thermodynamic and reversible mixed-state properties

Neutron irradiation of MgB2 bulk superconductors

Neutron Irradiation of MgB2 Bulk Superconductors

Fishtail effect in neutron-irradiated superconducting MgB 2 single crystals