Magnetic Field Penetration Depth Research Articles

Competing length scales and 2D versus 3D dimensionality in relatively thick superconducting NbN films

Magneto-transport characteristics of 2D and 3D superconducting layers, in particular, temperature and angular dependences of the upper critical field Hc2, are usually considered to be fundamentally different. In the work, using non-local resistance measurements at temperatures near the normal-to-superconducting transition, we probed an effective dimensionality of nm-thick NbN films. It was found that in relatively thick NbN layers, the thicknesses of which varied from 50 to 100 nm, the temperature effect on Hc2 certainly pointed to the three-dimensionality of the samples, while the angular dependence of Hc2 revealed behavior typical for 2D samples. The seeming contradiction is explained by an intriguing interplay of three length scales in the dimensionally confined superconducting films: the thickness, the Ginzburg–Landau coherence length, and the magnetic-field penetration depth. Our results provide new insights into the physics of superconducting films with an extremely large ratio of the London penetration depth to the Ginzburg–Landau coherence length exhibiting simultaneously 3D isotropic superconducting properties and the 2D transport regime.

Size-Dependent Superconducting Properties of In Nanowire Arrays.

Arrays of superconducting nanowires may be useful as elements of novel nanoelectronic devices. The superconducting properties of nanowires differ significantly from the properties of bulk structures. For instance, different vortex configurations of the magnetic field have previously been predicted for nanowires with different diameters. In the present study, arrays of parallel superconducting In nanowires with the diameters of 45 nm, 200 nm, and 550 nm-the same order of magnitude as coherence length ξ-were fabricated by templated electrodeposition. Values of magnetic moment M of the samples were measured as a function of magnetic field H and temperature T in axial and transverse fields. M(H) curves for the arrays of nanowires with 45 nm and 200 nm diameters are reversible, whereas magnetization curves for the array of nanowires with 550 nm diameter have several feature points and show a significant difference between increasing and decreasing field branches. Critical fields increase with a decrease in diameter, and the thinnest nanowires exceed bulk critical fields by 20 times. The qualitative change indicates that magnetic field configurations are different in the nanowires with different diameters. Variation of M(H) slope in small fields, heat capacity, and the magnetic field penetration depth with the temperature were measured. Superconductivity in In nanowires is proven to exist above the bulk critical temperature.

Two distinctive temperature dependences of the London penetration depth in high-Tc cuprate superconductors as support for the theory of Bose-liquid superconductivity

In the framework of the theory of Bose-liquid superconductivity, we show that in high-Tc cuprate superconductors the magnetic field penetration depth (known as the London penetration depth λL(T)) decreases first exponentially below the superconducting transition temperature Tc with decreasing the temperature T and then λL(T) decreases like a certain power of T at lower temperatures due to the vanishing of the energy gap Δg(T) in the excitation spectrum of a superfluid Bose-liquid at a characteristic temperature Tc⁎ lower than Tc. Below Tc, the London penetration depth λL(T) has an exponential temperature dependence down to Tc⁎, which is different from a BCS- like exponential temperature dependence, and then λL(T) will follow a power law temperature dependence down to T=0, which is also different from power-law temperature dependences predicted by some BCS-like (d-wave) pairing theories. In unconventional (bosonic) cuprate superconductors characterized by the strong interboson coupling constant (γB≳1), λL(T) exhibits the distinctly different temperature dependences in different temperature intervals 0≤T≤Tc⁎<<Tc and Tc⁎<T<Tc when the energy gap Δg(T) vanishes at a temperature Tc⁎ well below Tc. In the weak interboson coupling (γB<<1) the energy gap Δg(T) vanishes at a temperature Tc⁎ close to Tc and the reduced London penetration depth λL(T)/λL(0) has an unusual Tn power-law temperature dependence in a wide temperature range 0<T<Tc⁎ below Tc, i.e. the expression λL(T)/λL(0)≃[1−aL(T)(T/Tc)n]−1/2 (where n≳4 and aL(T)≲1) is valid only below Tc⁎ and is similar to the well-known empirical Gorter-Casimir law. Our theoretical predictions of the distinctive power-law and exponential temperature dependences of λL(T)/λL(0) are in good agreement with the experimental results reported for GdBa2Cu3O7−δ ceramics and YBa2Cu3O7−δ films. The peculiar temperature dependences of λL(T)/λL(0) observed in these high-Tc materials at low and high temperatures do not follow the predictions of the s- and d-wave BCS-like theories of Fermi-liquid superconductivity and give evidence for the predictions of the theory of Bose-liquid superconductivity.

Mutual and Self-Inductance in Planarized Multilayered Superconductor Integrated Circuits: Microstrips, Striplines, Bends, Meanders, Ground Plane Perforations

Data are presented on mutual and self-inductance of various inductors used in multilayered superconductor integrated circuits: microstrips and striplines with widths from 250 nm to a few micrometers, located on the same layer at various distances from each other, from 250 nm to a few micrometers, and/or on different layers spaced vertically; effect of long slits in the ground plane(s) along the inductors on their mutual inductance; inductance of right-angled bends; and inductance of meanders. Measurements were done using circuits fabricated in the SFQ5ee and the SC1 fully planarized fabrication processes with eight niobium layers and Nb/Al-AlO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</sub> /Nb Josephson junctions, developed at MIT Lincoln Laboratory for superconductor electronics. Simple analytical expressions for mutual and self-inductance of the basic inductors are given, describing experimental data with accuracy better than 2% over a very wide range of parameters. Mutual coupling between microstrips is long-ranged and decreases as a second power of distance between them, making microstrips unsuitable for very large scale integrated circuits. Mutual inductance of striplines decreases exponentially with distance between them on a scale of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$({\boldsymbol{H}} + 2{\boldsymbol{\lambda }})/{\boldsymbol{\pi }}$</tex-math></inline-formula> ∼ 0.3 μm, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\boldsymbol{H}}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\boldsymbol{\lambda }}$</tex-math></inline-formula> are, respectively, the distance between and magnetic field penetration depth in superconducting ground planes, whereas superconducting properties of the signal traces are practically irrelevant. This decay distance determines the scale of integration above which adjacent inductors in a circuit become strongly coupled. Dependence of mutual inductance on the linewidth of wires is weak. As a result, the area of flux transformers—an essential component of all superconductor digital circuits using ac power, qubits, and sensor arrays—scales poorly with the linewidth, potentially restricting scalability of the circuits.

The FFLO State in the Dimer Mott Organic Superconductor κ-(BEDT-TTF)2Cu[N(CN)2]Br

The superconducting phase diagram for a quasi-two-dimensional organic superconductor, κ-(BEDT-TTF)2Cu[N(CN)2]Br, was studied using pulsed magnetic field penetration depth measurements under rotating magnetic fields. At low temperatures, Hc2 was abruptly suppressed even by small tilts of the applied fields owing to the orbital pair-breaking effect. In magnetic fields parallel to the conducting plane, the temperature dependence of the upper critical field Hc2 exhibited an upturn and exceeded the Pauli limit field HP in the lower temperature region. Further analyses with the second derivative of the penetration depth showed an anomaly at 31–32 T, which roughly corresponded to HP. The origin of the anomaly should not be related to the orbital effect, but the paramagnetic effect, which is almost isotropic in organic salts, because it barely depends on the field angle. Based on these results, the observed anomaly is most likely due to the transition between the Bardeen-Cooper-Schrieffer (BCS) and the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states. Additionally, we discuss the phase diagram and physical parameters of the transition by comparing them with other FFLO candidates.

Electronic structure, electron-phonon coupling, and superconductivity in noncentrosymmetric ThCoC2 from ab initio calculations

Superconductors without inversion symmetry in their crystal structure are known to exhibit unconventional properties. Recently, based on the measured temperature dependence of the magnetic field penetration depth, superconductivity in noncentrosymmetric $\mathrm{ThCoC_2}$ was proposed to be a nodal $d$-wave and mediated by the spin fluctuations. Moreover, a non-BCS behavior of the temperature dependence of the electronic specific heat and the magnetic upper critical field were reported. In this work, the electronic structure, phonons and electron-phonon coupling are studied in $\mathrm{ThCoC_2}$ on the basis of \textit{ab initio} computations. The effect of the spin-orbit coupling on the electronic structure and electron-phonon interaction is analyzed, and a large splitting of the electronic band structure is found. The calculated electron-phonon coupling constant $\lambda = 0.59$ remains in decent agreement with the experimental estimates, suggesting that the electron-phonon interaction is strong enough to explain superconductivity with $T_c \simeq 2.5$~K. Nevertheless, we show that the conventional isotropic Eliashberg formalism is unable to describe the thermodynamic properties of the superconducting state, as calculated temperature dependence of the electronic specific heat and magnetic penetration depth deviate from experiments, which is likely driven by the strong spin-orbit coupling and inversion symmetry breaking. In addition, to shed more light on the pairing mechanism, we propose to measure the carbon isotope effect, as our calculations based on the electron-phonon coupling predict the observation of the isotope effect with an exponent $\alpha \simeq 0.15$.

Influence of spatial winding distribution of flexible eddy current sensor on quantitative monitoring of subsurface cracks

This paper proposes a flexible eddy current sensor based on TMR sensor. Firstly, the influence of three kinds of sensors with different spatial winding distributions on the penetration depth of magnetic field is analyzed through theoretical derivation. Then, the finite element model is established to analyze the eddy current distribution on the cross-section of the specimen under different excitation frequency and spatial winding distribution. The distribution of eddy current can be improved by increasing the spatial winding distribution. Finally, the corresponding sensors are prepared for subsurface crack monitoring. The experimental results show that the sensor with a spatial winding distribution of 10 mm can quantitatively monitor the cracks below 3 mm of material surface, and increasing the space winding distribution of the sensor to 20 mm, it can quantitatively monitor cracks below 5 mm of material surface, and the crack monitoring accuracy is consistent with the spacing between excitation lines.

Low kinetic inductance superconducting MgB2 nanowires with a 130 ps relaxation time for single-photon detection applications

Properties of superconducting nanowires set the performance level for superconducting nanowire single photon detectors (SNSPDs). Reset time in commonly employed large area SNSPDs, 1–10 ns, is known to be limited by the nanowire’s kinetic inductance to the load impedance ratio. On the other hand, reduction of the kinetic inductance in small area (waveguide integrated) SNSPDs prevents biasing them close to the critical current due to latching into a permanent resistive state. In order to reduce the reset time in SNSPDs, superconducting nanowires with both low kinetic inductance and fast electron energy relaxation are required. In this paper, we report on a study of kinetic inductance in narrow (15–100 nm) and long (up to 120 μm) superconducting MgB2 nanowires made from 5 nm thick films, offering such combination of properties. Such films were grown using hybrid physical chemical vapor deposition, resulting in a critical temperature of ∼32 K, and a switch current density of 5 × 107 A cm−2 (at 4.8 K). Using microwave reflectometry, we measured a kinetic inductance of L k0(4.8 K) = 1.3–1.6 pH/□ regardless of the nanowire width, which results in a magnetic field penetration depth of ∼90 nm. These values are very close to those in pristine MgB2. We showed that after excitations by a 50 fs pulsed laser the reset time in 35 nm × 120 μm MgB2 nanowires is 130 ps, which is more than a factor of 10 shorter than in NbN nanowires of similar length-to-width ratios. Depending on the bias current, such MgB2 nanowires function as single-, double, or triple-photon detectors for both visible (λ = 630 nm) and infrared (λ = 1550 nm) photons, with a dark count rate of <10 cps. Although the apparent photon detection efficiency seems so far to be low, further technological advances (uniform nanowire width, smaller thickness, increasing the switching current closer to the pair-breaking current) may improve this figure of merit.

Mechanism of the Generation of the Critical Current in High-Temperature Superconductors with Through Microdefects

The critical current of a model high-temperature superconductor (HTSC) with defects in the form of through holes (antidots) with characteristic sizes on the order of the magnetic-field penetration depth (or larger) is calculated. To this end, subprocesses equivalent to the magnetic-flux trapping by a hole and creation of a vortex near the hole edge are introduced into the model of a layered HTSC. It is shown that account for these subprocesses yields a physical mechanism, which makes it possible to describe correctly the nonmonotonic dependence of the critical current on the antidot characteristic size, which is similar to the experimental one. The calculations are carried out for a pure superconductor and a superconductor with nanoscale pinning centers. It is shown that the presence of nanoscale pinning centers along with antidots does not make qualitative changes in the relationship between the antidot radius and magnetic-flux pinning character and the critical-current behavior in an HTSC.

Suppression of the s-Wave Order Parameter Near the Surface of the Infinite-Layer Electron-Doped Cuprate Superconductor Sr0.9La0.1CuO2

The temperature dependencies of the in-plane (λab) and out-of-plane (λc) components of the magnetic field penetration depth were investigated near the surface and in the bulk of the electron-doped superconductor Sr0.9La0.1CuO2 by means of magnetization measurements. The measured λab(T) and λc(T) were analyzed in terms of a two-gap model with mixed s+d-wave symmetry of the order parameter. λab(T) is well described by an almost pure anisotropic d-wave symmetry component (≃96%), mainly reflecting the surface properties of the sample. In contrast, λc(T) exhibits a mixed s+d-wave order parameter with a substantial s-wave component of more than 50%. The comparison of λab−2(T) measured near the surface with that determined in the bulk by means of the muon-spin rotation/relaxation technique demonstrates that the suppression of the s-wave component of the order parameter near the surface is associated with a reduction of the superfluid density by more than a factor of two.

Simulation of dynamics of the order parameter in superconducting nanostructured materials: Effect of the magnetic field renormalization

The effect of the magnetic field renormalization in superconducting nanostructured materials is quantitatively evaluated. For demonstration purposes, three superconducting structures with various geometric shapes and dimensions and functioning in different resistive regimes are considered. Simulation is based on a set of equations including the time-dependent Ginzburg-Landau equation coupled with the Maxwell equations. An impact of the order parameter on the vector and scalar potentials is taken into account. It is shown that for Nb structures having thicknesses (∼200 nm) less than the magnetic field penetration depth (∼300 nm), the effect of the magnetic field renormalization equivocally affects the spatiotemporal distribution of superconducting vortices. For a slab with the thickness of ∼100 nm, the absolute value of the average voltage generated by moving vortices changes by less than 1%. With increasing the thickness of C-shaped structures up to 500 nm, the renormalization effect leads to the growth of the average voltage by up to 10%.

Self-Consistent Two-Gap Approach in Studying Multi-Band Superconductivity of NdFeAsO0.65F0.35

High-quality single crystals of NdFeAsO$_{0.65}$F$_{0.35}$ (the transition temperature $T_{\rm c} \simeq 30.6$~K) were studied in zero-field (ZF) and transverse-field (TF) muon-spin rotation/relaxation ($\mu$SR) experiments. An upturn in muon-spin depolarization rate at $T\lesssim 3$~K was observed in ZF-$\mu$SR measurements and it was associated with the onset of ordering of Nd electronic moments. Measurements of the magnetic field penetration depth ($\lambda$) were performed in the TF geometry. By applying the external magnetic field $B_{\rm ex}$ parallel to the crystallographic $c$-axis ($B_{\rm ex}\| c$) and parallel to the $ab$-plane ($B_{\rm ex}\| ab$), the temperature dependencies of the in-plane component ($\lambda_{ab}^{-2}$) and the combination of the in-plane and the out of plane components ($\lambda_{ab,c}^{-2}$) of the superfluid density were determined, respectively. The out-of-plane superfluid density component ($\lambda_{c}^{-2}$) was further obtained by combining the results of $B_{\rm ex} \| c$ and $B_{\rm ex} \| {ab}$ set of experiments. The temperature dependencies of $\lambda_{ab}^{-2}$, $\lambda_{ab,c}^{-2}$, and $\lambda_{c}^{-2}$ were analyzed within the framework of a self-consistent two-gap model despite of using the traditional $\alpha$-model. Interband coupling was taken into account, instead of assuming it to be zero as it stated in the $\alpha$-model. A relatively small value of the interband coupling constant $\Lambda_{12} \simeq 0.01$ was obtained, thus indicating that the energy bands in NdFeAsO$_{0.65}$F$_{0.35}$ are only weakly coupled. In spite of their small magnitude, the coupling between the bands leads to the single value of the superconducting transition temperature $T_{\rm c}$.

Enhanced Anharmonic Oscillations in Cu0.5Tl0.5Ba2(Ca2−yMgy)Cu3−xCdxO10−δ (y = 0, 1; x = 0, 1.5) Superconductors

Cu0.5Tl0.5Ba2Ca2Cu3O10−δ (undoped), Cu0.5Tl0.5Ba2(CaMg)Cu3O10−δ (Mg-doped), Cu0.5Tl0.5Ba2Ca2Cu1.5Cd1.5O10−δ (Cd-doped), and Cu0.5Tl0.5Ba2(CaMg)Cu1.5Cd1.5O10−δ (Cd/Mg-codoped) samples were synthesized by two-step solid-state reaction at 860°C to investigate the possible role of anharmonic oscillations of atoms in the mechanism of high critical temperature (Tc) superconductivity. The samples showed orthorhombic crystal structure with smaller and larger unit cell volume for the Mg- and Cd/Mg-codoped samples, respectively, compared with the undoped sample. Tc observed by resistivity measurements decreased in the Cd-doped samples, but the maximum suppression was observed in the Cd/Mg-codoped sample showing semiconductor behavior. Fourier-transform infrared (FTIR) absorption spectroscopy revealed hardening of apical oxygen modes on the incorporation of Mg atoms as compared with the undoped sample and similarly with the doping of Mg atoms in the Cd-doped sample, confirming the intrinsic inclusion of these elements in the final compound. Excess conductivity analysis revealed suppression in the coherence length (ξc(0)), interlayer coupling (J), phase relaxation time (τφ) of carriers, Fermi velocity (VF) of carriers, Ginzburg–Landau (GL) parameter κ, breaking energy (Ebreak), and magnetic field penetration depth (λp.d) in Cd-doped samples. The values of Bc0(T), Bc1(T), and Jc(0) increased in the Cd-doped samples, showing enhancement of flux-pinning characteristics. It is proposed that Cd doping at CuO2-planar site induces anharmonic oscillation in the solid-state medium, which suppresses the density of phonons and thereby the density of Cooper pairs and hence superconductivity. These studies emphasize the essential role of electron–phonon interactions in the mechanism of high-Tc superconductivity.

An-harmonic Oscillations and Superconductivity in (Cu0.5-xKxTl0.5)Ba2Ca2Cu3-y-z ZnyCdzO10-δ (x = 0, 0.25; y = z = 0, 0.25, 0.5, 0.75, 1) Samples

(Cu0.5-xKxTl0.5)Ba2Ca2Cu3-y-z ZnyCdzO10-δ (x = 0, 0.25; y = z = 0, 0.25, 0.5, 0.75, 1) samples are synthesized for the investigation of role of an-harmonic oscillations in the mechanism of high Tc superconductivity. In the conducting planes, the carrier density is enhanced by K doping at the charge reservoir layer (Cu0.5Tl0.5)Ba2O4-δ by synthesizing (Cu0.25K0.25Tl0.5)Ba2Ca2Cu3-y-zZnyCdzO10-δ (y = z = 0, 0.25, 0.5, 0.75, 1) samples. The (Cu0.5-xKxTl0.5)Ba2Ca2Cu3-y-z ZnyCdzO10-δ (x = 0, 0.25; y = z = 0, 0.25, 0.5, 0.75, 1) samples were prepared by employing two-step solid-state reaction method accomplished at 860 °C and understand their superconducting properties by resistivity measurements, X-ray diffraction analysis, and FTIR absorption measurements. The orthorhombic unit cell of our samples was confirmed via XRD analysis. The analysis also reflected the increase in c-axis length and the volume of the unit cell with the enhanced dopants (Cd and Zn) concentration. The suppression of Tc(onset) and Tc(R = 0) was found with the increasing concentration of Zn and Cd. An improvement in the Tc(onset) observed in samples doped by K that arises due to increase in the density of free carriers supplied by alkali atoms. With the increasing content of Cd and Zn, the unit cell exhibits the softening of Cu(1)-OA-Cu(2) and Tl-OA-Cu(2) apical oxygen mode and hardening of planar oxygen mode. From FIC analysis, the Tcmf and T* and various cross over temperature were shifted towards lower temperature with the enhanced Zn and Cd dopant concentration. The value of α determining the defect density also increases with the increasing dopant (Zn and Cd) concentration. The numerical values of different parameter like Bc0(T), Bc1(T), Jc(0), and Ebreak showing increasing trend but the phase relaxation time of the carriers τφ points out decreasing trend with the enhanced doping of Cd and Zn. Moreover, these parameters reflected inferior values in K-doped samples relative to K-undoped samples. The value of other parameter like coherence length ξc(0) along c-axis, interlayer coupling J, magnetic field penetration depth λp.d, the Ginzburg Landau (GL) parameter κ, and the Fermi velocity VF of the carriers decreases as doping of Zn and Cd enhanced. These parameters show superior values in K-doped samples relative to K-undoped samples due to increase in the density of free carriers supplied by K atoms. The negative values of real part of dielectric constant suppress in both K undoped and K doped samples with increasing concentration of Cd and Zn. The imaginary part of dielectric constant and ac conductivity showing suppression in their negative values in K undoped samples while showing increase in absolute values in K doped samples with increasing concentration of Cd and Zn.

Effect of Cd intercalation on the superconducting properties of (Cu0.5-yKyTl0.5)Ba2Ca2Cu3-xCdxO10-δ (y = 0, 0.25; x = 0, 0.5, 1.0, 1.5, 2.0) superconductors

(Cu0.5Tl0.5)Ba2Ca2 Cu3-xCdxO10-δ (x = 0, 0.5, 1.0, 1.5, 2.0) samples are synthesized for the investigation of role of an-harmonic oscillations in the mechanism of high Tc superconductivity. The density of carriers in the conducting planes is enhanced with the doping of K at the (Cu0.5Tl0.5)Ba2O4-δ by synthesizing (Cu0.25K0.25Tl0.5)Ba2Ca2Cu3-xCdxO10-δ (x = 0, 0.5, 1.0, 1.5, 2.0) samples. These samples were synthesized by two step solid state reaction method accomplished at 860 °C and studied their superconducting properties by resistivity measurements, X-ray diffraction analysis and FTIR absorption measurements. The planar reflections observed in the XRD analysis of the samples are best fitted to the orthorhombic crystal structure. The c-axis length and the volume of the unit cell increases with the enhanced doping of Cd in the final compound. The onset temperature of superconductivity Tc (onset) and the zero-resistivity critical temperature Tc (R = 0) suppresses with the increased doping of Cd in the final compound. An improvement in the Tc (onset) and Tc (R = 0) is observed in K-doped samples that arises due to increase in the density of free carriers supplied by alkali atoms. A softening of apical oxygen mode of type Cu(1)‒OA‒Cu(2)/Cd is observed with the increased doping of Cd at the CuO2 planar sites. The apical oxygen mode of type Tl‒OA‒Cu(2)/Cd is, however, did not change with the doping of Cd in the final compound. The planar oxygen mode (545 ‒ 570 cm−1) is hardened with the enhanced doping of Cd in the final compound. The excess conductivity analyses of conductivity data have shown a shift in the various cross-over temperatures, Tcmf and T* towards lower temperatures with the enhanced doping of Cd in the final compound. The value of α determining the defect density also increases with the increased doping of Cd in the final compound. The coherence length along the c-axis ξc(0), interlayer coupling J, magnetic field penetration depth λp.d and Ginzburg Landau (GL) parameter κ decreased with the increase doping of Cd. The phase relaxation time of the carriers Ꞇϕ, the Fermi velocity VF of the carriers decreased while Bc0(T), Bc1(T) and Jc (0) are enhanced with increase doing of Cd in the final compound.

Universal behavior of the intermediate mixed state domain formation in superconducting niobium

In the intermediate mixed state (IMS) of type-II/1 superconductors, vortex lattice (VL) and Meissner state domains coexist due to a partially attractive vortex interaction. Using a neutron-based multiscale approach combined with magnetization measurements, we study the continuous decomposition of a homogeneous VL into increasingly dense domains in the IMS in bulk niobium samples of varying purity. We find a universal temperature dependence of the vortex spacing, closely related to the London penetration depth and independent of the external magnetic field. The rearrangement of vortices occurs even in the presence of a flux freezing transition, i.e. pronounced pinning, indicating a breakdown of pinning at the onset of the vortex attraction.

Strong Spin–Charge Coupling and Its Manifestation in the Quasiparticle Structure, Cooper Instability, and Electromagnetic Properties of Cuprates

The Fermi excitation spectrum, the problem of Cooper instability, and the Londons magnetic field penetration depth in cuprate superconductors are considered using the unified conception based on accounting for the strong coupling between the spin of copper ions and holes at oxygen ions. This coupling leads to strong renormalization of the primary spectrum of oxygen holes with the formation of spin-polaron quasiparticles. Analysis of Cooper instability performed using the spin-polaron concept for different channels has shown that only the superconducting d-wave pairing occurs in the ensemble of spin-polaron quasiparticles, and there are no solutions corresponding to the s-wave pairing. It has been demonstrated that the superconducting d-wave pairing is not suppressed by the Coulomb repulsion of holes located at neighboring oxygen ions. This effect is due to peculiarities in the crystallographic structure of the CuO2 plane and the aforementioned strong spin–fermion coupling. As a result, such interaction of holes is omitted in the kernel of the integral equation for the superconducting order parameter with the d-wave symmetry. It has been shown the Hubbard repulsion of holes and their interaction for the second coordination sphere of the oxygen sublattice for actual intensities of the interaction do not suppress the d-wave type of superconductivity. For the spin-polaron ensemble, we have analyzed the dependence of the Londons magnetic field penetration depth on the temperature and hole concentration. It has been established that the peculiarities of this dependence are closely related to specific features of the spin-polaron spectrum.

Discussion on the Principle and Reliability Improvement of AC Magnetic Flux Leakage Detection of Steel Rod

In this paper, the principle of ac magnetic flux leakage (referred to as MFL) detection is presented by analyzing the formation of ac magnetic flux leakage and how it is collected. It is found that ac excitation frequency is a very important equipment parameter for ac magnetic flux leakage detection of steel rod. High-frequency current not only stabilizes the penetration depth of ac magnetic field under the rod surface and leakage magnetic field of defects, but also improves the ability to detect defects and adapts to higher detection speed. It is concluded that the lift-off effect is the fundamental reason affecting the reliability of MFL detection of steel rod, and there is an inherent signal amplitude deviation from the irremovable probe bouncing on the surface of steel rod. Then, the factors that aggravate the lift-off effect are analyzed synthetically, and the corresponding countermeasures are put forward. It provides an effective basis for improving the test reliability and reasonably controls the quality risk of the rod.

Tuning the in-plane Fulde-Ferrell-Larkin-Ovchinnikov state in a superconductor/ferromagnet/normal-metal hybrid structure by current or magnetic field

Temperature-induced transition of thin superconductor/ferromagnet/normal-metal (S/F/N) hybrid structure to an in-plane Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is accompanied by vanishing of effective inverse magnetic field penetration depth ${\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$ [S. V. Mironov et al., Phys. Rev. Lett. 121, 077002 (2018)]. Here we show that ${\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$ goes to zero only in the limit of zero magnetic field $H\ensuremath{\rightarrow}0$ and at any finite parallel $H$ or in-plane current $I$ it is finite and positive in the FFLO state, which implies a diamagnetic response. We demonstrate that ${\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$ has a nonmonotonic dependence on $H$ and $I$ not only in the parameter range corresponding to the FFLO phase domain but also in its vicinity. We find that for S/F/N/F/S structures with certain thicknesses of F layers there is a temperature-, current-, and magnetic-field-driven transition to and out of the FFLO phase with a simultaneous jump of ${\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$.

Nonlocal optical response of a layered high-temperature superconductor slab

We theoretically study the effect of the spatial dispersion on the optical response of a layered high-temperature superconductor slab. The nonlocality of the inherently-anisotropic layered superconductor comes from the wave vector dependence of its average permittivity tensor, and leads to the generation of additional electromagnetic modes just above the characteristic Josephson plasma frequency, that is in the terahertz range. We calculate p-polarization optical spectra for a Bi2Sr2CaCu2O8+δ (Bi2212) superconductor slab, which show very narrow resonances associated with the quantization of the wave vectors of both long-wavelength electromagnetic modes, having negative dispersion, and short-wavelength additional (nonlocal) modes of positive dispersion. The dependence of the frequency position and shape of the resonances on the nonlocality parameter, the slab thickness, and the components of the quasiparticle conductivity is analyzed. We have found that the quantized long-wavelength modes of negative dispersion, which can only be observed at relatively-large slab thicknesses, give rise to prominent resonances in the p-polarization reflectivity spectrum. On the other hand, the resonances associated with quantized additional short-wavelength electromagnetic modes are weak, but they can be clearly observed when the superconductor slab thickness is smaller than the smallest magnetic-field penetration depth.