Magnetic flux can penetrate a type-II superconductor in the form of Abrikosov vortices (also called flux lines, flux tubes, or fluxons) each carrying a quantum of magnetic flux phi 0=h/2e. These tiny vortices of supercurrent tend to arrange themselves in a triangular flux-line lattice (FLL), which is more or less perturbed by material inhomogeneities that pin the flux lines, and in high-Tc superconductors (HTSCs) also by thermal fluctuations. Many properties of the FLL are well described by the phenomenological Ginzburg-Landau theory or by the electromagnetic London theory, which treats the vortex core as a singularity. In Nb alloys and HTSCs the FLL is very soft mainly because of the large magnetic penetration depth lambda . The shear modulus of the FLL is c66~1/ lambda 2, and the tilt modulus c44(k)~(1+k2 lambda 2)-1 is dispersive and becomes very small for short distortion wavelengths 2 pi /k<< lambda . This softness is enhanced further by the pronounced anisotropy and layered structure of HTSCs, which strongly increases the penetration depth for currents along the c axis of these (nearly uniaxial) crystals and may even cause a decoupling of two-dimensional vortex lattices in the Cu-O layers. Thermal fluctuations and softening may `melt` the FLL and cause thermally activated depinning of the flux lines or ofthe two-dimensional `pancake vortices` in the layers. Various phase transitions are predicted for the FLL in layered HTSCs. Although large pinning forces and high critical currents have been achieved, the small depinning energy so far prevents the application of HTSCs as conductors at high temperatures except in cases when the applied current and the surrounding magnetic field are small.

https://iopscience.iop.org/article/10.1088/0034-4885/58/11/003/pdf

The flux-line lattice in superconductors

The authors present a review of recent muon spin rotation (\ensuremath{\mu}SR) studies of the vortex state in type-II superconductors. There are significant gaps in our understanding of this unusual phase of matter, especially in unconventional superconductors, for which the description of the vortex structure is a subject of great controversy. The \ensuremath{\mu}SR technique provides a sensitive local probe of the spatially inhomogeneous magnetic field associated with the vortex state. For the case of a regular vortex lattice, the magnetic penetration depth \ensuremath{\lambda} and the coherence length \ensuremath{\xi} can be simultaneously extracted from the measured internal field distribution. The penetration depth is directly related to the density of superconducting carriers in the material, and measurements of its variation with temperature, magnetic field, and impurities can provide essential information on the symmetry of the order parameter. The coherence length measured with \ensuremath{\mu}SR is the length scale for spatial variations of the order parameter within a vortex core. A primary goal of this review article is to show that measurements of these fundamental length scales are fairly robust with respect to the details of how the field distribution is modeled. The reliability of the results is demonstrated by a comparison of the \ensuremath{\mu}SR experiments with relevant theories and with other experimental techniques. The authors also review \ensuremath{\mu}SR measurements that have focused on the study of pinning-induced spatial disorder and vortex fluctuation phenomena. The \ensuremath{\mu}SR technique has proven to be a powerful tool for investigating exotic vortex phases, where vortex transitions are directly observable from changes in the \ensuremath{\mu}SR line shape. Particular emphasis is given to \ensuremath{\mu}SR experiments performed on high-temperature superconductors since high-quality single crystals have become available.

μSR studies of the vortex state in type-II superconductors

A set of highly sensitive cavity perturbation techniques, quite different from the ones most commonly used to measure thin films, has been developed for measuring the surface impedance of small crystals of high-${\mathit{T}}_{\mathit{c}}$ superconductors. Since the results of these studies differ markedly from most surface-impedance measurements of thin films of high-${\mathit{T}}_{\mathit{c}}$ superconductors, we describe the techniques in detail, including recent improvements. Measurements of the surface resistance and penetration depth of a twin-free crystal of ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{6.95}$ indicate two striking differences from the surface impedance of conventional superconductors. Both quantities exhibit linear temperature dependence at low temperatures rather than the exponentially activated behavior of an s-wave BCS superconductor. Also, a broad peak in the temperature dependence of the surface resistance indicates that the large transport scattering rate in the normal state collapses below ${\mathit{T}}_{\mathit{c}}$ as the holes condense into the superconducting condensate. Systematic studies of the influence of Zn and Ni impurities on the surface impedance indicate that these properties are very sensitive to defects and this sensitivity provides further insight into the pairing state. Although both types of impurities provide strong elastic scattering that limits the collapse of the scattering rate and suppresses the peak in the surface resistance, only Zn has a strong pair-breaking effect. As little as 0.31% Zn substitution makes ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{6.95}$ gapless, but Ni, which is a magnetic impurity has no such pair-breaking effect up to 0.75% substitution. This difference is the opposite of that expected for an s-wave BCS superconductor.

Comparison of the influence of Ni and Zn impurities on the electromagnetic properties of YBa2Cu3O6.95.

Magnetism in two-dimensional (2D) van der Waals (vdW) materials has recently emerged as one of the most promising areas in condensed matter research, with many exciting emerging properties and significant potential for applications ranging from topological magnonics to low-power spintronics, quantum computing, and optical communications. In the brief time after their discovery, 2D magnets have blossomed into a rich area for investigation, where fundamental concepts in magnetism are challenged by the behavior of spins that can develop at the single layer limit. However, much effort is still needed in multiple fronts before 2D magnets can be routinely used for practical implementations. In this comprehensive review, prominent authors with expertise in complementary fields of 2D magnetism (i.e., synthesis, device engineering, magneto-optics, imaging, transport, mechanics, spin excitations, and theory and simulations) have joined together to provide a genome of current knowledge and a guideline for future developments in 2D magnetic materials research.

/pdf/the-magnetic-genome-of-two-dimensional-van-der-waals-1hvraszg.pdf

The Magnetic Genome of Two-Dimensional van der Waals Materials

Muon-spin-rotation (\ensuremath{\mu}SR) experiments on a high-quality sintered ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{\mathit{x}}$ sample [x=6.970(1)] were performed, in order to obtain an accurate knowledge of the magnitude and the temperature dependence of the magnetic penetration depth in this copper oxide superconductor. Special attention was given to the data analysis. In particular, the systematic errors introduced by different types of analyses were estimated. Our results show that the temperature dependence of the effective penetration depth ${\ensuremath{\lambda}}_{\mathrm{eff}}$ into the sintered sample is well described by the two-fluid model, with ${\ensuremath{\lambda}}_{\mathrm{eff}}$(0)=155(10) nm. This behavior of ${\ensuremath{\lambda}}_{\mathrm{eff}}$(T) is consistent with conventional s-wave pairing. With the anisotropy ratio \ensuremath{\gamma}=${\ensuremath{\lambda}}_{\mathit{c}}$/${\ensuremath{\lambda}}_{\mathit{a}\mathit{b}}$=5(1) measured in a previous \ensuremath{\mu}SR experiment, the penetration depths ${\ensuremath{\lambda}}_{\mathit{a}\mathit{b}}$(0)=130(10) nm and ${\ensuremath{\lambda}}_{\mathit{c}}$(0)=500--800 nm (parallel and perpendicular to the ${\mathrm{CuO}}_{2}$ planes, respectively) were extracted. Our results are compared with those obtained by other experimental techniques and theoretical predictions.

Muon-spin-rotation measurements of the London penetration depths in YBa2Cu3O6.97.

To get accurate values of the London penetration depths in YBa2Cu3O x , μSR measurements were performed on a high quality, sintered sample and a c-axis-oriented polycrystal. For the sintered sample the temperature dependence of the effective penetration depth λeff is well described by the two-fluid model, with λeff(0) = 155(10) nm. This behavior of λeff(T) is consistent with conventional s-wave pairing. The anisotropy ratio λ c λ ab ⋍ 5(1) was determined from measurements on the polycrystal. These results were used to calculate λab(0) = 130(10) nm and λ c (0) ⋍ 500 – 800 nm .

Measurement of the London penetration depths in YBa2Cu3Ox by means of muon spin rotation (μSR) experiments

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Muon-spin-rotation measurements of the London penetration depths in YBa2Cu3O6.97.

Measurement of the London penetration depths in YBa2Cu3Ox by means of muon spin rotation (μSR) experiments