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The currents and field distributions of a vortex in a thin superconducting strip of a width $W$ is considered. It is shown that unlike infinite films where the vortex field crosses the film only in one direction (say, from the half-space under the film to the half-space above it), in strips (as well as in finite size film samples of any shape) the field lines go back to the lower half-space not only out of the sample but also through certain strip parts. The strip patches where the magnetic lines ``dive" back to the space under the strip are situated mostly near the strip edges at the strip and out of it. The magnetic flux through the strip associated with the vortex is shown to be less than the flux quantum and depends on the vortex position. The suppression of the vortex flux is strong in narrow strips with $W<<\Lambda=2\lambda^2/d$ where $\lambda$ is the bulk London penetration depth and $d$ is the film thickness. The vortex energy scales roughly with $W/\Lambda$ and depends on vortex position., Comment: 7 pages, 8 figures

A recent theory of the disorder-dependent slope of the upper critical field, $H_{c2}$, at the superconducting transition temperature, $T_{c}$, is extended to multiband superconductors aiming at iron-based superconductors, considering two constant gaps of different magnitude and, potentially, different signs. The result shows that there is only a narrow domain inside the $s_{\pm}$ pairing state where the slope increases with the increase of transport (non-magnetic) scattering rate, $P$. In most phase space, the slope should decrease in an $s_{\pm}$ state and increase in the $s_{++}$ pairing state. The experiment shows that in an archetypal iron-based superconductor, $\textrm{Ba}_{1-x}\textrm{K}_{x}\textrm{Fe}_{2}\textrm{As}_{2}$ (BaK122), non-magnetic disorder induced by electron irradiation increases the slope $S$ across the superconducting ``dome,'' at different $x$. This implies that $\textrm{Ba}_{1-x}\textrm{K}_{x}\textrm{Fe}_{2}\textrm{As}_{2}$ is likely an $s_{++}$ superconductor with two (or more) gaps of different magnitudes. This work reopens a decade-long discussion of the nature of the superconducting order parameter in iron pnictides., Comment: v2 updated with the new data (Fig.10)

We study the slopes of the upper critical field $\partial_{T}H_{c2}|_{T_{c}}\equiv\partial H_{_{c2}}/\partial T$ at $T_{c}$ in anisotropic superconductors with transport (non-magnetic) scattering employing the Ginzburg-Landau theory, developed for this situation by S. Pokrovsky and V. Pokrovsky, Phys. Rev. B 54, 13275 (1996). We found unexpected behavior of the slopes for a $d-$wave superconductor and in a more general case of materials with line nodes in the order parameter. Specifically, the presence of line nodes causes $\partial_{T}H_{c2}|_{T_{c}}$ to decrease with increasing non-magnetic scattering parameter $P$, unlike the nodeless case where the slope increases. In a pure $d-$wave case, the slope $\partial H_{c2}|_{T_{c}}$ changes from decreasing to increasing when scattering parameter approaches $P\approx0.91\,P_{{\rm crit}}$, where $P_{{\rm crit}}\approx0.2807$ at which $T_{c}\to0$ that implies the the existence of a gapless state in $d-$wave superconductors with transport scattering in the interval, $0.91\,P_{{\rm {crit}}}

The problem of a thin superconducting film in a parallel magnetic field, first discussed in the classical paper by Ginzburg and Landau for temperatures close to the critical, is revisited with the help of the microscopic BCS theory for arbitrary temperatures taking pair-breaking and transport scattering into account. While confirming experimental findings of the $T_c$ enhancement by the magnetic field, we find that the transport scattering pushes the phase transition curve to higher fields and higher temperatures for nearly all practical scattering rates. Still, the $T_c$ enhancement disappears in the dirty limit. We also consider intriguing changes, such as re-entrant superconductivity, caused to the phase boundary by pair-breaking magnetic ions spread on one of the film faces. These features await experimental verification.

The upper critical field $H_{c2} $, the field $H_{c3}$ for nucleation of the surface superconductivity, and the thermodynamic $H_c $ are evaluated within the weak-coupling theory for the isotropic s-wave case and arbitrary transport and pair-breaking scattering. We find that for the standard geometry of a half-space sample in a magnetic field parallel to the surface, the ratio ${\cal R}=H_{c3}/H_{c2}$ is within the window $1.55\lesssim {\cal R}\lesssim 2.34$, regardless of temperature, magnetic or non-magnetic scattering. While the non-magnetic impurities tend to flatten the ${\cal R}\left(T\right)$ variation, the magnetic scattering merely shifts the maximum of ${\cal R}\left(T\right)$ to lower temperatures. Surprisingly, while reducing the transition temperature, magnetic scattering has a milder impact on ${\cal R}$ than the non-magnetic scattering. The surface superconductivity is quite robust; in fact, the ratio ${\cal R}\approx 1.7$ even in the gapless state. We used Eilenberger's energy functional to evaluate the condensation energy $F_c$ and the thermodynamic critical field $H_c$ for any temperature and scattering parameters. By comparing $H_{c2} $ and $H_{c}$, we find that unlike the transport scattering, the pair-breaking pushes materials toward type-I behavior. We find a peculiar behavior of $F_c$ as a function of the pair-breaking scattering parameter at the low-$T$ transition from gapped to gapless phases, which has recently been associated with the topological transition in the superconducting density of states.

Members of the CaK(Fe$_{1-x}$Mn$_{x}$)$_{4}$As$_{4}$ series have been synthesized in single crystalline form and characterized by elemental analysis, thermodynamic and transport measurements. These measurements show that the superconducting transition temperature decreases monotonically and is finally suppressed below 1.8 K. For $x$-values greater than 0.016, signatures of a magnetic transition can be detected in both thermodynamic and transport measurements in which kink-like features allow for the determination of the transition temperature, $T^*$, that increases as Mn substitution increases. A temperature-composition ($T$-$x$) phase diagram is constructed, revealing a half-dome of superconductivity with the magnetic transition temperature, $T^*$, appearing near 26 K for $x$ $\sim$ 0.017 and rising slowly up to 33 K for $x$ $\sim$ 0.036. Specific heat data are used to track the jump in specific heat at $T_c$; The CaK(Fe$_{1-x}$Mn$_x$)$_4$As$_4$ data does not follow the scaling of $\Delta$$C_{p}$ with $T_{c}^3$ as many of the other Fe-based superconducting systems do. Elastoresistivity coefficients, $2m_{66}$ and $m_{11}-m_{12}$, as a function of temperature are also measured. $2m_{66}$ and $m_{11}-m_{12}$ are qualitatively similar to CaK(Fe$_{1-x}$Ni$_x$)$_4$As$_4$. This may indicate that the magnetic order in Mn substituted system may be still the same as CaK(Fe$_{1-x}$Ni$_x$)$_4$As$_4$. A clear change in $H^\prime_{c2}$($T$)/$T_c$, where $H^\prime_{c2}$($T$) is d$H_{c2}$($T$)/d$T$, at $x$ $\sim$ 0.015 is observed and probably is related to change of the Fermi surface due to magnetic order. Coherence lengths and the London penetration depths are also calculated based on $H_{c1}$ and $H_{c2}$ data. Coherence lengths as the function of $x$ also shows the changes near $x$ = 0.015., Comment: Fig. 16 a and b were corrected

Moving vortices in thin superconducting films are considered within the time-dependent London description. The dissipation due to out-of-core normal excitations for two vortices moving together turns out to have a minimum for the separation vector $\bm a$ parallel to the velocity and equal to $a_m \approx 2.2\, \Lambda$, where $\Lambda $ is the Pearl length. The minimum entropy production suggests that moving vortices should have a tendency to form chains along the velocity with a period of the order $a_m$., Comment: 4 pages, 4 figures

The London penetration depth, $\lambda(T)$, was measured in a single crystal V$_{3}$Si. The superfluid density obtained from this measurement shows a distinct signature of two almost decoupled superconducting gaps. This alone is insufficient to distinguish between $s_{\pm}$ and $s_{++}$ pairing states, but it can be achieved by studying the effect of a controlled non-magnetic disorder on the superconducting transition temperature, $T_{c}$. For this purpose, the same $\text{V}_{3}\text{Si}$ crystal was sequentially irradiated by 2.5 MeV electrons three times, repeating the measurement between the irradiation runs. A total dose of 10 C/cm$^{2}$ ($6.24\times10^{19}$ electrons/$\textrm{cm}^{2}$) was accumulated, for which $T_{c}$ has changed from 16.4 K in a pristine state to 14.7 K (9.3 $\%$). This substantial suppression is impossible for a single isotropic gap, but also it is not large enough for a sign-changing $s_{\pm}$ pairing state. Our electronic band-structure calculations show how five bands crossing the Fermi energy can be naturally grouped to support two effective gaps, not dissimilar from the iron pnictides physics. We analyze the results using two-gap models for both, $\lambda(T)$ and $T_{c}$, which describe the data very well. Thus, the experimental results and theoretical analysis provide strong support for an $s_{++}$ superconductivity with two unequal gaps, $\Delta_{1}\left(0\right)\approx2.53\;\textrm{meV}$ and $\Delta_{2}\left(0\right)\approx1.42\;\textrm{meV}$, and a very weak inter-band coupling in $\text{V}_{3}\text{Si}$ superconductor.

The magnetic field of vortices in anisotropic superconducting films is considered in the framework of anisotropic London approach. It is found that at distances large relative to the core size, the magnetic field normal to the film surface may change sign. We find that the magnetic field attenuates at large distances as $1/r^3$ as it does in isotropic films, but the anisotropy induces an angular dependence to the supercurrents which causes the sign of the field to change for anisotropy parameters $\gamma=\lambda_2/\lambda_1>\sqrt{2}$ in some parts of the $(x,y)$ plane., Comment: 8 pages, 12 figures

The magnetic field of moving vortices in anisotropic superconductors is considered in the framework of time-dependent London approach. It is found that at distances large relative to the core size, the field may change sign that alludes to a non-trivial intervortex interaction which depends on the crystal anisotropy and on the speed and direction of motion. These effects are caused to the electric fields and corresponding normal currents which appear due to the moving vortex magnetic structure. We find that the motion related part of the magnetic field attenuates at large distances as $1/r^3$ unlike the exponential decay of the static vortex field. The electric field induced by the vortex motion decreases as $1/r^2$., Comment: 9 pages, 10 figures

We take account of normal currents that emerge when vortices move. Moving Abrikosov vortices in the bulk and Pearl vortices in thin films are considered. Velocity dependent distributions of both normal and persistent currents are studied in the frame of time-dependent London equations. In thin films near the Pearl vortex core, these distributions are intriguing in particular., Comment: 8 pages,10 figures. arXiv admin note: text overlap with arXiv:2102.00073

The magnetic field $h_z$ of a moving Pearl vortex in a superconducting thin-film in $(x,y)$ plane is studied with the help of time-dependent London equation. It is found that for a vortex at the origin moving in $+x$ direction, $h_z(x,y)$ is suppressed in front of the vortex, $x>0$, and enhanced behind ($x<0$). The distribution asymmetry is proportional to the velocity and to the conductivity of normal quasiparticles. The vortex self-energy and the interaction of two moving vortices are evaluated., Comment: 6 pages, 4 figures. arXiv admin note: text overlap with arXiv:2004.03663

We study the effects of anisotropic order parameters on the temperature dependence of London penetration depth anisotropy $\gamma_\lambda(T)$. After MgB$_2$, this dependence is commonly attributed to distinct gaps on multi-band Fermi surfaces in superconductors. We have found, however, that the anisotropy parameter may depend on temperature also in one-band materials with anisotropic order parameters $\Delta(T,k_F)$, a few such examples are given. We have found also that for different order parameters, the temperature dependence of $\Delta(T)/\Delta(0)$ can be represented with good accuracy by the interpolation suggested by D. Einzel, J. Low Temp. Phys, {\bf 131}, 1 (2003), which simplifies considerably the evaluation of $\gamma_\lambda(T)$. Of particular interest is mixed order parameters of two symmetries for which $\gamma_\lambda(T)$ may go through a maximum for a certain relative weight of two phases. Also, for this case, we find that the ratio $\Delta_{max}(0)/T_c$ may exceed substantially the weak coupling limit of 1.76. It, however, does not imply a strong coupling, rather it is due to significantly anisotropic angular variation of $\Delta$.

The anisotropic London equations taking into account the normal currents are derived and applied to the problem of the surface impedance in the Meisner state of anisotropic materials. It is shown that the complex susceptibility of anisotropic slab depends on the orientation of the applied microwave field relative to the crystal axes. In particular, the anisotropic sample in the microwave field is subject to a torque, unless the field is directed along with one of the crystal principle axes.

The self-energy of a moving vortex is shown do decrease with increasing velocity. The interaction energy of two parallel slowly moving vortices differs from the static case by a small term $\propto v^2$; the "slow" motion is defined as having the velocity $vv_c(T)$, the interaction energy of two vortices situated along the velocity direction is enhanced and along the perpendicular direction is suppressed compared to the static case.

We study vortex current distributions in narrow thin-film superconducting strips. If one defines the vortex core ``boundary" as a curve where the current reaches the depairing value, intriguing features emerge. Our conclusions based on the London approach have only qualitative relevance since the approach breaks down near the core. Still, the main observation which might be useful is that the core size near the strip edges is smaller than in the rest of the strip. If so, the Bardeen-Stephen flux-flow resistivity should be reduced near the edges. Moreover, at elevated temperatures, when the depairing current is small, the vortex core may extend to the whole strip width, thus turning into an edge-to-edge phase-slip line.

A simple relation is established between the zero-$T$ penetration depth $\lambda (0)$ and the slope of $\lambda^{-2}(T)$ near $T_c$, similar to Helfand-Werthamer's relation for $H_{c2}(0)$ and the slope of $H_{c2}(T)$ at $T_c$ for the isotropic s-wave case with non-magnetic scattering.\cite{HW} When the scattering parameter $\rho=\hbar v/2\pi T_c\ell$ ($v$ is the Fermi velocity and $\ell$ is the mean-free path) varies from 1 to 10, the coefficient of proportionality between $\lambda^{-2}(0)$ and $T_c (d\lambda^{-2}/dT)_{T_c}$ changes from 0.43 to 0.38. Combining this relation with the Rutgers thermodynamic identity, one can express $\lambda (0)$ in terms of the slope $(dH_{c2}/dT)_{T_c}$ and the density of states.

Strains in superconductors due to moving vortices and vortex lattices are discussed. It is shown that the energy stored in elastic strains increases with vortex velocity. For moving vortex lattices, the elastic energy depends on velocity orientation relative to the lattice having minimum for the velocity directed along one of the unit cell vectors of the moving lattice. It is shown that for supersonic motion, the vortex induced stress field has a shape similar to supersonic shock waves., Comment: 6 pages, 2 figures

We show on a few examples of one-band materials with spheroidal Fermi surfaces and anisotropic order parameters that anisotropies $\gamma_H$ of the upper critical field and $\gamma_\lambda$ of the London penetration depth depend on temperature, the feature commonly attributed to multi-band superconductors. The parameters $\gamma_H$ and $\gamma_\lambda$ may have opposite temperature dependencies or may change in the same direction depending on Fermi surface shape and on character of the gap nodes. For two-band systems, the behavior of anisotropies is affected by the ratios of bands densities of states, Fermi velocities, anisotropies, and order parameters. We investigate in detail the conditions determining the directions of temperature dependences of the two anisotropy factors.

Enhanced microwave absorption, larger than that in the normal state, is observed in fine grains of type-II superconductors (MgB$_2$ and K$_3$C$_{60}$) for magnetic fields as small as a few $\%$ of the upper critical field. The effect is predicted by the theory of vortex motion in type-II superconductors, however its direct observation has been elusive due to skin-depth limitations; conventional microwave absorption studies employ larger samples where the microwave magnetic field exclusion significantly lowers the absorption. We show that the enhancement is observable in grains smaller than the penetration depth. A quantitative analysis on K$_3$C$_{60}$ in the framework of the Coffey--Clem (CC) theory explains well the temperature dependence of the microwave absorption and also allows to determine the vortex pinning force constant.

The dissipative currents due to normal excitations are included in the London description. The resulting time dependent London equations are solved for a moving vortex and a moving vortex lattice. It is shown that the field distribution of a moving vortex looses it cylindrical symmetry, it experiences contraction which is stronger in the direction of the motion, than in the direction normal to the velocity $\bm v$. The London contribution of normal currents to dissipation is small relative to the Bardeen-Stephen core dissipation at small velocities, but approaches the latter at high velocities, where this contribution is no longer proportional to $v^2$. To minimize the London contribution to dissipation, the vortex lattice orients as to have one of the unit cell vectors along the velocity, the effect seen in experiments and predicted within the time-dependent Ginzburg-Landau theory., Comment: 6 pages, 5 figures

Effective demagnetizing factors that connect the sample magnetic moment with the applied magnetic field are calculated numerically for perfectly diamagnetic samples of various non-ellipsoidal shapes. The procedure is based on calculating total magnetic moment by integrating the magnetic induction obtained from a full three dimensional solution of the Maxwell equations using adaptive mesh. The results are relevant for superconductors (and conductors in AC fields) when the London penetration depth (or the skin depth) is much smaller than the sample size. Simple but reasonably accurate approximate formulas are given for practical shapes including rectangular cuboids, finite cylinders in axial and transverse field as well as infinite rectangular and elliptical cross-section strips., Comment: updated version, corrected typos etc

It is of interest to determine the exit angle of a vortex from a superconducting surface, since this affects the intervortex interactions and their consequences. Two ways to determine this angle are to image the vortex magnetic fields above the surface, or the vortex core shape at the surface. In this work we evaluate the field h(x, y, z) above a flat superconducting surface x, y and the currents J(x,y) at that surface for a straight vortex tilted relative to the normal to the surface, for both the isotropic and anisotropic cases. In principle, these results can be used to determine the vortex exit tilt angle from analyses of magnetic field imaging or density of states data., Comment: 5 pages, 4 figures

We study vortex lattices (VLs) in superconducting weak-pinning platelet-like crystals of $\beta$-Bi$_{2}$Pd in tilted fields with a Scanning Tunneling Microscope. We show that vortices exit the sample perpendicular to the surface and are thus bent beneath the surface. The structure and orientation of tilted VL in the bulk are, for large tilt angles, strongly affected by Coulomb-type intervortex repulsion at the surface due to stray fields.

A model of a clean two-band s-wave superconductor with cylindrical Fermi surfaces, different Fermi velocities v_{1,2}, and a general 2x2 coupling matrix V_{alpha beta} is used to study the order parameter distribution in vortex lattices. The Eilenberger weak coupling formalism is used to calculate numerically the spatial distributions of the pairing amplitudes Delta_1_ and Delta_2_ of the two bands for vortices parallel to the Fermi cylinders. For generic values of the interband coupling V_{12}, it is shown that, independently of the couplings V_{alpha beta}, of the ratio v_1 /v_2, of the temperature, and the applied field, the length scales of spatial variation of Delta_1 and of Delta_2 are the same within the accuracy of our calculations. The only exception from this single length-scale behavior is found for V_{12} --> 0, i.e., for nearly decoupled bands., Comment: 7 pages, 8 figures

We study the superconducting density of states and vortex lattice of single crystals of CaKFe$_4$As$_4$ using a scanning tunneling microscope (STM). This material has a critical temperature of $T_c= 35\,$ K, which is one of the highest among stoichiometric iron based superconductors (FeBSC) and is comparable to $T_c$ found near optimal doping in other FeBSC. Using quasi-particle interference we identify the hole sheets around the zone center and find that two superconducting gaps open in these sheets. The scattering centers are small defects that can be localized in the surface topography and just produce quasiparticle interference, without suppressing the superconducting order parameter. This shows that sign inversion is not within hole bands, but between hole and the electron bands. Vortex core bound states show electron-hole asymmetric bound states due to proximity of the top of one of the hole bands to the Fermi level $E_F$. This places CaKFe$_4$As$_4$ in a similar situation as FeSe or related materials, with a superconducting gap $\Delta$ just a few times smaller than $E_F$. On the other hand, we also identify locations showing strong suppression of the superconducting order parameter. Their size is of order of the vortex core size and vortices are pinned at these locations, leading to a disordered vortex lattice., Comment: 7 figures

We study the spatial distribution of the density of states (DOS) at zero bias N(r) in the mixed state of single and multigap superconductors. We provide an analytic expression for N(r) based on deGennes' relation between DOS and the order parameter that reproduces well Scanning Tunneling Microscopy (STM) data in several superconducting materials. In the single gap superconductor $\beta$-Bi$_2$Pd, we find that N(r) is governed by a length scale $\xi_H=\sqrt{\phi_0/2\pi H}$, which decreases in rising fields. The vortex core size $\cal C$, defined via the slope of the order parameter at the vortex center, $\cal C$ $\propto (d \Delta /dr |_{r \to 0})^{-1}$, differs from $\xi_H$ by a material dependent numerical factor. For two gap superconductors 2H-NbSe$_{1.8}$S$_{0.2}$ and 2H-NbS$_2$, we find that $\cal C$ is field independent and has the same value for both bands. We conclude that, independently of the magnetic field induced variation of the order parameter values in both bands, the spatial variation of the order parameter close to the vortex core is the same for all bands., Comment: 8 pages, 9 figures

A two-band model with repulsive interband coupling and interband {\it transport} (potential) scattering is considered to elucidate their effects on material properties. In agreement with previous work, we find that the bands order parameters $\Delta_{1,2}$ differ and the large is at the band with a smaller normal density of states (DOS), $N_{n2}

It is shown that the order parameter $\Delta$ induced in the normal part of superconductor-normal-superconductor proximity system is modulated in the magnetic field differently from vortices in bulk superconductors. Whereas $\Delta$ turns zero at vortex centers, the magnetic structure of these vortices differs from that of Abrikosov's., Comment: 5 pages, 7 figures

Although accumulation of molecular damage is suggested to be an important molecular mechanism of aging, a quantitative link between the dynamics of damage accumulation and mortality of species has so far remained elusive. To address this question, we examine stability properties of a generic gene regulatory network (GRN) and demonstrate that many characteristics of aging and the associated population mortality rate emerge as inherent properties of the critical dynamics of gene regulation and metabolic levels. Based on the analysis of age-dependent changes in gene-expression and metabolic profiles in Drosophila melanogaster, we explicitly show that the underlying GRNs are nearly critical and inherently unstable. This instability manifests itself as aging in the form of distortion of gene expression and metabolic profiles with age, and causes the characteristic increase in mortality rate with age as described by a form of the Gompertz law. In addition, we explain late-life mortality deceleration observed at very late ages for large populations. We show that aging contains a stochastic component, related to accumulation of regulatory errors in transcription/translation/metabolic pathways due to imperfection of signaling cascades in the network and of responses to environmental factors. We also establish that there is a strong deterministic component, suggesting genetic control. Since mortality in humans, where it is characterized best, is strongly associated with the incidence of age-related diseases, our findings support the idea that aging is the driving force behind the development of chronic human diseases., Comment: 22 pages, 9 figures

By evaluating the upper and thermodynamic critical fields, $H_{c2}$ and $H_c$, and their ratio $H_{c2}/H_c $ at arbitrary temperatures, we argue that situations are possible when a type-II material is transformed to type-I by adding magnetic impurities.

New thin-film Josephson junctions have recently been tested in which the current injected into one of the junction banks governs Josephson phenomena. One thus can continuously manage the phase distribution at the junction by changing the injected current. A method of calculating the distribution of injected currents is proposed for a half-infinite thin-film strip with source-sink points at arbitrary positions at the film edges. The strip width $W$ is assumed small relative to $\Lambda=2\lambda^2/d$, $\lambda$ is the bulk London penetration depth of the film material, $d$ is the film thickness., Comment: 5 pages, 7 figures

The classical criterion for classification of superconductors as type-I or type-II based on the isotropic Ginzburg-Landau theory is generalized to arbitrary temperatures for materials with anisotropic Fermi surfaces and order parameters. We argue that the relevant quantity for this classification is the ratio of the upper and thermodynamic critical fields, $H_{c2}/H_c$, rather than the traditional ratio of the penetration depth and the coherence length, $\lambda/\xi$. Even in the isotropic case, $H_{c2}/H_c$ coincides with $\sqrt{2}\lambda/\xi$ only at the critical temperature $T_c$ and they differ as $T$ decreases, the long known fact. Anisotropies of Fermi surfaces and order parameters may amplify this difference and render false the criterion based on the value of $\kappa=\lambda/\xi$.

The upper critical field $H_{c2}$ and its anisotropy are calculated for order parameters with line nodes at equators, $k_z=0$, of the Fermi surface of uniaxial superconductors. It is shown that characteristic features found in Fe-based materials -- a nearly linear $H_{c2}(T)$ in a broad $T$ domain, a low and increasing on warming anisotropy $\gamma_H= H_{c2,ab}/ H_{c2,c}\, $ -- can be caused by competing effects of the equatorial nodes and of the Fermi surface anisotropy. For certain material parameters, $\gamma_H(T)-1$ may change sign on warming in agreement with recorded behavior of FeTeS system. It is also shown that the anisotropy of the penetration depth $\gamma_\lambda= \lambda_c/\lambda_{ab} $ decreases on warming to reach $\gamma_H$ at $T_c$ in agreement with data available. For some materials $\gamma_\lambda(T)$ may change on warming from $\gamma_\lambda>1$ at low $T$s to $\gamma_\lambda<1$ at high $T$s.

It is shown that a vortex trapped in one of the banks of a planar edge-type Josephson junction in a narrow thin-film superconducting strip can change drastically the field dependence of the junction critical current $I_c(H)$. When the vortex is trapped at certain positions in the strip middle, the pattern $I_c(H)$ has zero at $H=0$ instead of the traditional maximum of '0-type' junctions. The number of these positions is equal to the number of vortices trapped at the same location. When the junction-vortex separation exceeds approximately $2W$, $I_c(H)$ is no longer sensitive to the vortex presence.

Large high-quality single crystals of hole-doped iron-based superconductor (Ba$_{1-x}$K$_x$)Fe$_2$As$_2$ were grown over a broad composition range $0.22 \leq x \leq 1$ by inverted temperature gradient method. We found that high soaking temperature, fast cooling rate, and adjusted temperature window of the growth are necessary to obtain single crystals of heavily K doped crystals (0.65$\leq x \leq$ 0.92) with narrow compositional distributions as revealed by sharp superconducting transitions in magnetization measurements and close to 100% superconducting volume fraction. The crystals were extensively characterized by x-ray and compositional analysis, revealing monotonic evolution of the $c$-axis crystal lattice parameter with K substitution. Quantitative measurements of the temperature-dependent in-plane resistivity, $\rho(T)$ found doping-independent, constant within error bars, resistivity at room temperature, $\rho(300K)$, in sharp contrast with significant doping dependence in electron and isovalent substituted BaFe$_2$As$_2$ based compositions. The shape of the temperature dependent resistivity, $\rho(T)$, shows systematic doping-evolution, being close to $T^2$ in overdoped and revealing significant contribution of the $T$-linear component at optimum doping. The slope of the upper critical field, $d H_{c2}/dT$, scales linearly with $T_c$ for both $H\parallel c$, $ H_{c2,c}$, and $H \parallel ab$, $H_{c2,ab}$. The anisotropy of the upper critical field, $\gamma \equiv H_{c2,ab} / H_{c2,c}$ determined near zero-field $T_c$ increases from $\sim$2 to 4-5 with increasing K doping level from optimal $x \sim$0.4 to strongly overdoped $x$=1., Comment: 16 figures and 1 table

The phase difference between the banks of an edge-type planar Josephson junction crossing the narrow thin-film strip depends on wether or not vortices are present in the junction banks. For a vortex close to the junction this effect has been seen by Golod, Rydh, and Krasnov, \prl {\bf 104}, 227003 (2010), who showed that the vortex may turn the junction into $\pi$-type. It is shown here that even if the vortex is far away from the junction, it still changes the 0-junction to $\pi$-junction when situated close to the strip edges. Within the approximation used, the latter effect is independent of the vortex-junction separation, a manifestation of topology of the vortex phase which extends to macroscopic distances of superconducting coherence., Comment: 4 pages, 3 figures

The London penetration depth is evaluated for isotropic materials for any transport and pair-breaking Born scattering rates. Besides known results, a number of new features are found. The slope $|d\rho/d\theta |$ of the normalized superfluid density $\rho=\lambda^2(0)/\lambda^2(\theta)$ at the transition $\theta=T/T_c=1$ has a minimum near the value of the pair-breaking parameter separating gapped and gapless states. The low-$T$ exponentially flat part of $\rho$ for the s-wave materials is suppressed by increasing pair breaking. For strong $T_c $ suppression by magnetic impurities the "Homes scaling" $\lambda^{-2}(0) \propto \sigma T_c$ with $\sigma$ being the normal conductivity gives way to $\lambda^{-2}(0) \propto \sigma T_c^2$. For the d-wave order parameter, the transport and spin-flip Born scattering rates enter the theory only as a sum, in particular, they affect the $T_c$ depression in the same manner. We confirm that the linear low temperature behavior of $\rho$ in a broad range of the combined scattering parameter turns to the $T^2$ behavior only when the critical temperature is suppressed at least by a factor of 3 relative to the clean limit $T_{c0}$. Moreover, in this range, $\rho (\theta)$ is only weakly dependent on the scattering parameter, i.e. it is nearly universal.

The stress caused by vortices in tetragonal superconductors contributes to the intervortex interaction which depends on vortex orientation within the crystal, on elastic moduli, and is attractive within certain angular regions even in fields along the $c$ crystal axis. For sufficiently strong stress dependence of the critical temperature, this contribution may result in distortions of the hexagonal vortex lattice for $\bm H||\bm c$. In small fields it leads to formation of a square vortex lattice with a fixed $H$ independent spacing. This should be seen in the magnetization $M(H)$ as a discontinuous jump of magnetization at the transition from the Meissner to mixed states., Comment: 5 pages, 2 figures

It is shown that the thermodynamic Rutgers relation for the second order phase transitions can be used for the analysis of the superfluid density data irrespective of complexities of the Fermi surface, structure of the superconducting gap, pairing strength, or scattering. The only limitation is that critical fluctuations should be weak, so that the mean field theory of the second order phase transitions is applicable. By using the Rutgers relation, the zero temperature value of the London penetration depth, $\lambda(0)$, is related to the specific heat jump $\Delta C$ and the slope of upper critical field $dH_{c2}/dT$ at the transition temperature $T_c$, provided the data on $\Delta\lambda=\lambda(T)-\lambda(0)$ are available in a broad temperature domain. We then provide a new way of determination of $\lambda(0)$, the quantity difficult to determine within many techniques.

The orbital upper critical field $H_{c2} $ is evaluated for isotropic materials with arbitrary transport and pair-breaking scattering rates. It is shown that unlike transport scattering which enhances $H_{c2} $, the pair breaking suppresses the upper critical field and reduces the dimensionless ratio $h^*(0)=H_{c2}(0)/T_c(dH_{c2}/dT)_{T_c} $ from the Helfand-Werthamer value of $\approx 0.7$ to 0.5 for a strong pair-breaking. $h^*(T)$ is evaluated for arbitrary transport and pair-breaking scattering. A phenomenological model for the pair-breaking suppression by magnetic fields is introduced. It shows qualitative features such as a positive curvature of $H_{c2}(T) $ and the low temperature upturn usually associated with multi-band superconductivity.

It is argued on the basis of the BCS theory that the zero-$T$ penetration depth satisfies $ \lambda^{-2}(0)\propto\sigma T_c $ ($\sigma$ is the normal state dc conductivity) not only in the extreme dirty limit $\xi_0/\ell \gg 1$, but in a broad range of scattering parameters down to $\xi_0/\ell \sim 1$ ($\xi_0$ is the zero-$T$ BCS coherence length and $\ell$ is the mean-free path). Hence, the scaling $ \lambda^{-2}(0)\propto\sigma T_c $, suggested as a new universal property of superconductors,\cite{Homes} finds a natural explanation within the BCS theory., Comment: 2 pages, 2 figures

Measurements of temperature-dependent in-plane resistivity, $\rho(T)$, were used to determine the upper critical field and its anisotropy in high quality single crystals of stoichiometric iron arsenide superconductor KFe$_2$As$_2$. The crystals were characterized by residual resistivity ratio, $\rho(300K)/\rho(0)$ up to 3000 and resistive transition midpoint temperature, $T_c$=3.8 K, significantly higher than in previous studies on the same material. We find increased $H_{c2}(T)$ for both directions of the magnetic field, which scale with the increased $T_c$. This unusual linear $H_{c2}(T_c)$ scaling is not expected for orbital limiting mechanism of the upper critical field in clean materials.

In-plane resistivity measurements as a function of temperature, magnetic field and its orientation with respect to the crystallographic $ab-$plane were used to study the upper critical field, $H_{c2}$, of two overdoped compositions of the iron-based superconductor Ba(Fe$_{1-x}$Ni$_x$)$_2$As$_2$, $x=$0.054 and $x$=0.072. Measurements were performed using precise alignment (with accuracy less than 0.1$^o$) of magnetic field with respect to the Fe-As-plane. The dependence of the $H_{c2}$ on angle $\theta$ between the field and the $ab$-plane was measured in isothermal conditions in a broad temperature range. We found that the shape of $H_{c2}(\theta)$, clearly deviates from Ginzburg-Landau functional form.

It is argued that the stress caused by vortex cores in the mixed state of superconductors may result in a field dependent contribution to the free energy and magnetization of measurable levels. For sufficiently strong stress dependence of the critical temperature, $\partial T_c/\partial p$, this contribution may result in the so-called "second peak" in the field dependence of the reversible magnetization, the effect often masked by vortex pinning and creep., Comment: 5 pages, 2 figures

We use both Eilenberger-Usadel and Ginzburg-Landau (GL) theory to calculate the superfluid's temperature-dependent kinetic inductance for all currents up to the depairing current in thin and narrow superconducting films. The calculations apply to BCS weak-coupling superconductors with isotropic gaps and transport mean-free paths much less than the BCS coherence length. The kinetic inductance is calculated for the response to a small alternating current when the film is carrying a dc bias current. In the slow-experiment/fast-relaxation limit, in which the superconducting order parameter quasistatically follows the time-dependent current, the kinetic inductance diverges as the bias current approaches the depairing value. However, in the fast-experiment/slow-relaxiation limit, in which the the superconducting order parameter remains fixed at a value corresponding to the dc bias current, the kinetic inductance rises to a finite value at the depairing current. We then use time-dependent GL theory to calculate the kinetic impedance of the superfluid, which includes not only the kinetic reactance but also the kinetic resistance of the superfluid arising from dissipation due to order-parameter relaxation. The kinetic resistance is largest for angular frequencies $\omega$ obeying $\omega \tau_s > 1$, where $\tau_s$ is the order-parameter relaxation time, and for bias currents close to the depairing current. We also include the normal fluid's contribution to dissipation in deriving an expression for the total kinetic impedance. The Appendices contain many details about the temperature-dependent behavior of superconductors carrying current up to the depairing value., Comment: 17 pages, 14 figures, many changes made following referee suggestions, new material added regarding normal-fluid dissipation