Scaling of the Equilibrium Magnetization in the Mixed State of Type-II Superconductors

No Heading We discuss the analysis of mixed-state magnetization data of type-II superconductors using a recently developed scaling procedure. It is based on the fact that, if the Ginzburg-Landau parameter does not depend on temperature, the magnetic susceptibility (H, T) is a universal function of H/H_c2(T), leading to a simple relation between magnetizations at different temperatures. Although this scaling procedure does not provide absolute values of the upper critical field H_c2(T), its temperature variation can be established rather accurately. This provides an opportunity to validate theoretical models that are usually employed for the evaluation of H_c2(T) from equilibrium magnetization data. In the second part of the paper we apply this scaling procedure for a discussion of the notorious first order phase transition in the mixed state of high-Tc superconductors. Our analysis, based on experimental magnetization data available in the literature, shows that the shift of the magnetization accross the transition may adopt either sign, depending on the particular chosen sample. We argue that this observation is inconsistent with the interpretation that this transition always represents the melting transition of the vortex lattice.PACS numbers: 74.60.-w, 74.60.Ec, 74.60.Ge, 74.72.-h     

Dynamical fluctuation of the order parameter and diamagnetism of superconductors. I. Dirty limit

Limiting ourselves to dirty superconductors, we study theoretically the fluctuation-induced diamagnetism above the superconducting transition. We find that the dynamical fluctuation is extremely important except in the immediate vicinity of the transition region [i.e.,TT _c(H) orHH _c2(T), whereH _c2 is the upper critical field] and that this contribution does account for a large discrepancy between previous calculations and the recent experiment by Gollub, Beasley, and Tinkham.     

Dimensional Crossover in the Fluctuation Magnetization in YBa2Cu3O7- x and its Evolution with the Content of Oxygen

We have studied the field-induced dimensional crossover in the fluctuation magnetization of three single crystals of YBa₂Cu₃O_7-x, with superconducting transition temperatures, T_c = 62.5, 52, and 41 K. The dimensional crossover is observed by studying the diamagnetic vortex fluctuations of the lowest-Landau-level type which occur in isochamps magnetization curves, MvsT, for temperatures close to the transition temperature T_c(H). The study was accomplished by obtaining isochamps magnetization curves as a function of temperature for fields in the range of 0.1–5 T. Magnetization curves for each sample when plotted together show two distinct well resolved crossing points, one formed by low field curves and located at a higher temperature than the other formed by high field curves. A lowest-Landau-level scaling analysis is applied to the curves forming the crossing points and it is verified that lower field curves obey the three-dimensional form of this scaling while the higher field curves obey the two-dimensional form. The results allow to observe the evolution of the dimensional crossover field H_cross, 3D–2D, with the content of oxygen in YBa₂Cu₃O_7-x. It is observed that the evolution of the field H_cross with the content of oxygen in each sample qualitatively agrees with theoretical predictions and allow us to estimate the ratio of the anisotropy among the studied samples.     

Fluctuation-induced diamagnetism in type II superconductors above H c2

The fluctuation-induced diamagnetism in strong magnetic fields in type II superconductors is studied theoretically. It is shown that the fluctuation-induced diamagnetism at low temperatures (i.e., TT _c0) is described in terms of a universal function of [B \s- H _c2(T)]/H_s(T) for both dirty and pure superconductors, where H _s(T), the scaling field, is proportional to (T/T _c0)H_c2(0).Work supported by the National Science Foundation under Grant DMR 76-21032.     

Superconducting gap function and upper critical field in the presence of inhomogeneous magnetic order

Rare earth ternary superconductors are known to exhibit oscillatory magnetic orders below their superconducting transition temperatures. The study of behaviour of superconducting electrons in an inhomogeneous magnetic field is therefore important for such systems. We report here the results of our theoretical study of superconducting gap function Δ(T) and upper critical fieldH _c2 (T). The results are applied to analyse and explain the variation of Δ(T) andH _c2 (T) in case of NdRh₄B₄.     

Upper Critical Field and Specific Heat of Cuprates

A representative set of magnetotransport measurements in novel superconductors is analyzed. The resistive upper critical field, H _c2 (T) of many cuprates, of superconducting spin-ladders, and organic (TMTSF)2X systems has a universal nonlinear temperature dependence H _c2 (T_c – T)^3/2 in a wide temperature interval near T _c, while its low-temperature behavior depends on the chemical formula and sample quality. The unusual H _c2(T) is described as the Bose–Einstein condensation field of preformed pairs. Its universal temperature dependence follows from the scaling arguments. Controversy in the determination of H _c2 (T) from the resistivity and specific heat measurements is resolved in the framework of the charged Bose-gas model with the impurity scattering. It is shown that specific heat shows two anomalies. The high-temperature anomaly is strong and shows only weak shift with applied field. The low-temperature anomaly corresponds to resistive transition and is very weak in agreement with the experiments. Both anomalies coincide at H = 0.     

Scaling of the Resistance and Magnetization for High Temperature Superconductors under Pressure

The scaling relation of single parameter scaling hypothesis is applied to the study of the scaling behavior of high temperature superconductors under pressure. The data of resistance and magnetization under various pressures are scaled onto a universal curve according to this scaling relation. The scaling parameters are pressure dependent while temperature independent. It is found that the controlling parameter B _i equals the relative critical temperature t _cP , which indicates that the superconducting energy gap at the zero temperature 2Δ _s0 is the controlling parameter in this scaling.      

Thermal fluctuation effects on the magnetization above and below the superconducting transition in Bi2Sr2CaCu2O8 and YBa2Cu3O7−δ crystals in the weak magnetic field limit

We present detailed experimental data of the magnetization, M_ab(T,H), of Bi₂Sr₂CaCu₂O₈ and YBa₂Cu₃O_7– crystals on both sides of the superconducting transition, for magnetic fields, H, applied perpendicularly to the ab (CuO₂) planes. The data are analyzed in terms of thermal fluctuations in the weak H limit: In the reversible mixed state below the transition, by taking into account the vortex fluctuations, as first proposed by Bulaevskii, Ledvij and Kogan, which are much more important in the Bi-based compounds. Above the transition, by taking into account the order parameter amplitude fluctuations (OPF), through a generalization to multilayered superconductors of the Schmidt-like approach.     

Critical Scaling of the Fluctuation Conductivity for ReFeAsO (Re=Nd,Pr) Superconductors

It is studied within the scaling analysis the fluctuation conductivity under the magnetic fields in ReFeAsO (Re=Nd, Pr) superconductors reported in literatures. A crossover is firstly observed in these iron-based superconductors from the Lowest Landau level (LLL) regime above the magnetic field H _LLL∼6–7 T to the critical fluctuation region at the low field. The static exponent ν of ReFeAsO superconductors is estimated to be 1.85±0.05, which is close to that of Bi₂Sr₂CaCu₂O₈, and the dynamic exponent z of ReFeAsO superconductors deviates from the theoretical prediction but is similar to that extracted from the resistivity experiment using the crossing-point technique.     

Equilibrium magnetization properties of irreversible superconducting niobium with peak effect

Measurements of the equilibrium magnetization curve of type II superconductors have usually only been possible on reversible specimens. In this paper, a method to measure the equilibrium magnetization curve of irreversible type II superconductors is described. Results obtained by this method from a hysteretic Nb specimen exhibiting the peak effect are reported. They show an irregularity in the shape of the equilibrium magnetization curve in the neighborhood ofH _c2 which corresponds to a minimum of the compressional modulusK of the flux line lattice. The minimum ofK also corresponds with the maximum of the critical current densityj _c, which strongly suggests that in this case the peak effect is originated by a “soft point” of the flux line lattice. The observed anomaly ofK is discussed in terms of a possible first-order phase transition in the flux line lattice.     

Complex borides with a ferromagnetic component: Development of the problem of magnetic superconductors

Superconductors with a ferromagnet as one of the principal components are considered. A group of new stable superconductors (of the Y-Ni-B system) containing a ferromagnetic component up to 50% at. is found out as well. The YNi_1.9 B_1.2 phase is revealed to have the highest critical parameters among the known superconductors with a ferromagnetic component (T_c∼13,5 K , ΔT_c∼ 0,6 K, J_c∼ 5 ^.10⁴ A/cm², H_c2(0) up to 30 kOe) . The model of its crystal structure is proposed. The YNi_1.9B_1.2 superconductor is characterized by an anomalous position of the H_c2(T) curve (the positive convexity within 9.5−13.5 K). In a normal state, it is characterised by an anomalous dependence of magnetic susceptibility on the temperature and magnetic field, i.e. magnetic susceptibilityχ shows no changes with increasing temperature (within 15–273 K) in a fixed magnetic field and decreases with increasing magnetic field from 0.214 to 4.28 kOe at a fixed temperature. The anomalous position of the curve can be described within the WHH theory in the Fulde and Maki approximation, the positive convexity of the H_c2(T) curve being attributed to anomalous magnetic properties of the YNi_1.9B_1.2 compound in a normal state. The coexistence of superconductivity with antiferromagnetism is assumed in the analyzed compound. Received: 9 November 2000 / Reviewed: 23 January 2001 / Accepted: 23 January 2001     

Magnetic Entropy Change by Mean-Field Theory for the Second-Order Phase Transition Manganite Nd<Subscript>0.6</Subscript>Sr<Subscript>0.3</Subscript>Ca<Subscript>0.1</Subscript>Mn<Subscript>0.975</Subscript>Fe<Subscript>0.025</Subscript>O<Subscript>3</Subscript>

In this work, we are going to show the method based on mean-field scaling for the Nd_0.6Sr_0.3Ca_0.1Mn_0.975Fe_0.025 O₃ sample, where from scaling of experimental magnetization data, the mean-field exchange parameter λ and the f function of the equation of state \(M(T,H)=B_{S} [\frac {\left ({H+H_{\text {ex}}} \right )}{T}]\) are directly determined. The scaling approach allows finding the dependence of H _ex on T or higher powers of M, which determine the order of the phase transition. Quantum spin number has been determined. In this study, we use \(\left | {\Delta S_{M} (T)} \right |\) obtained from isothermal magnetization measurements; we compare this result to mean-field theory fittings from a novel scaling method through the use of theoretical results S, g, and λ. The obtained results by mean-field theory are suitable and in good agreement with the classical Maxwell relation.     

A Comparative Study of High-Field Diamagnetic Fluctuations in Deoxygenated YBa2Cu3O7-x and Polycrystalline (Bi–Pb)2Sr2Ca2Cu3O10

We studied three single crystals of YBa₂Cu₃O_7-x, (Y123) with superconducting transition temperature, T_c=62.5, 52, and 41 K, and a highly textured polycrystalline specimen of (Bi–Pb)₂Sr₂Ca₂Cu₃O₁₀ (Bi2223), with T_c=108K. Isofield magnetization data were obtained as a function of temperature, with the magnetic field applied parallel to the c axis of each sample. The reversible magnetization data for all samples exhibited a rounded transition as magnetization tended toward zero. The reversible data were interpreted in terms of two-dimensional diamagnetic lowest-Landau-level (LLL) fluctuation theory. The LLL scaling analysis yielded consistent values of the superconducting transition temperatures T_c(H) for the various samples. The resulting scaling data were fit well by the two-dimensional LLL expression for magnetization obtained by Tesanovic and colaborators, producing reasonable values of κ but the fitting parameter ∂H_c2/∂T produced values that were larger than the experimentally determined ones. We performed simultaneous scaling of Y123 data and Bi2223, obtaining a single collapsed curve. The single curve was obtained after multiplying the x and y axis of each scaling curve by appropriate sample-dependent scaling factors. An expression for the two-dimensional x-axis LLL scaling was extracted from theory, allowing comparison of theoretical values of the x-axis scaling factors with the experimental values. The comparison between the values of the x-axis produced a deviation of 40% which suggests that the hypothesis of universality of the two-dimensional LLL fluctuations is not supported by the studied samples. We also observe that Y123 magnetization data for temperatures above T_c obbey a universal scaling obtained for the diamagnetic fluctuation magnetization from a theory considering non-local field effects. The same scaling was not obbeyed by the corresponding magnetization calculated from the two-dimensional LLL theory.     

Anomalous magnetization due to the vortex lattice melting transition in YBa2Cu3O y

Magnetization measurements in YBa₂Cu₃O_y single crystals are performed as a function of the temperature T and the magnetic field H. It is found that an anomalous magnetization step which provides the thermodynamic evidence of the first-order vortex lattice melting transition and the resistive kink with the hysteresis are observed in the same phase boundary in H-T plane. For samples with a small value of the irreversibility field H_irr(T) as compared with the vortex lattice melting line H_m(T), the vortex lattice melting is observed in the reversible magnetization region and the entropy change per vortex per CuO₂ layer at the transition is estimated to be s0.5–1.0 k _B . For samples with a slightly larger value of H_irr(T) comparable to H_m(T), on the other hand, the jump height and the value of s is enhanced. Thus, the value of s is affected significantly by the relation between H _irr (T) which varies from sample to sample and H_m(T) which is an universal line.     

Flux trapping and its effect on AC susceptibility of granular superconductors

Experimental data on the real (x′) and imaginary (x″) parts of AC magnetic susceptibility as a function of DC bias field (H) shows the effect of trapped flux in granular highT _c superconductors. The aim was to substantiate our recent theoretical findings on the basis of a two-component critical state model suitable for granular highT _c superconductors. Stress has been given to understanding the origin of hysteresis inx′(H) andx″(H). It was seen in the experimental data that above a certain value of DC field range irreversibility appears inx′(H) andx″(H) creating hysteresis like loops. Comparison of data with calculated loops shows good agreement.x′(H) andx″(H) curves show considerable asymmetry in presence of trapped flux.     

Surface magnetism of type II superconductors

The magnetization of a type II superconductor in the mixed state is a sum of two contributions, one from an array of flux lines and the other from shielding currents flowing along the surface. The former can be evaluated with the help of results of neutron diffraction experiments, using an extrapolation procedure. The observed magnetization curve thus enables us to estimate the surface contribution. This process is applied to the data on niobium obtained by Schelten, Ullmaier, and Lippmann. It is found that at low temperatures (T/T _c =0.161 and 0.457), the surface contribution may not be reduced atH _c2 as much as theories predict. At higher temperatures (T/T _c =0.717 and 0.891), it is likely to be paramagnetic in a region rather close toH _c1 . Attractive interaction between flux lines is suggested to be a mechanism of this anomaly. However, more experimental and theoretical work is needed to establish these findings to help us understand the surface magnetism of type II superconductors.     

The upper critical field of thin superconducting films with large resistance

The influence of the electron-electron interaction on the upper critical fieldH _c2 of thin superconducting films with large resistance and of layered superconductors is investigated theoretically. The orbital effect only is taken into account. It is shown that the electron-electron interaction in the diffusion and Cooper channels influencesH _c2 in different manners. The interaction in the diffusion channel (dynamically screened Coulomb interaction) leads to essential deviations from the standard BCS dirty limitH _c2 (T) curve at low temperatures and as a result upward curvature inH _c2 (T) is possible. The interaction in the Cooper channel is significant only at temperaturesT _c–TT_c and enlarges the slopedH _c2 (T)/dt atT=T _c.     

Vortex Pinning Energies and H(T) Characteristic Lines in the MoSr2YCu2O8−δ Superconductor

We investigated the vortex pinning energies and H(T) characteristic lines of samples with nominal composition MoSr₂YCu₂O_8?δ . The vortex pinning energies were obtained using two approximations of the resistive transition. The characteristic lines H _c2(T) and H _irr(T) were obtained using resistivity, susceptibility and magnetization measurements. The values of H _c2(0) and different irreversibility field values are discussed.     

The Activation Energy <Emphasis Type="Italic">U</Emphasis>(<Emphasis Type="Italic">T</Emphasis>,<Emphasis Type="Italic">H</Emphasis>) in Y-based Superconductors

Based on the Arrhenius equation, a method to calculate the activation energy from the resistance transition is proposed for high temperature superconductors. This method is applied to the Y-based superconductors. The activation energy is found to be U(T,H)∼(1−T/T _c )^4.8(H/H ₀)^−3.8 of YBCO crystal, and U(T,H)∼(1−T/T _c )^3.3(H/H ₀)^−2.2 of Er doped MTG YBCO crystal, respectively. With the obtained activation energy U(T,H), the lower part of the experimental curve ρ(T,H) and its derivative can be reproduced.