Pinning force in thin superconductors with small number of flux line rows

It is shown that the volume pinning force (and hence the critical current density) of thin superconducting slabs decreases strongly after the transition of the one-dimensional flux line lattice into the two-dimensional structure. The results are applied to superfine filamentary super-conductors (diameterd0·1 m) and they are in accordance with experiments on NbTi superfine filaments, as well as on weakly reacted Nb₃Sn multifilamentary superconductors. The maximum of the volume pinning force appears therefore at the transition fieldB ₁₂ from the one-dimensional to the two-dimensional flux line lattice. This field is much smaller than the fieldB _m at whichF _p is maximum for macroscopic superconductors. So, e.g. for filament diameterd=60 nm of NbTi superconductor,B ₁₂1·2 T, whereas theB _m value for macroscopic NbTi is about 5 T.     

Critical current density of a YBa2Cu3O7 film: Comparison between experiment and collective pinning theory

With the vibrating reed technique we have measured the critical current densityJ _c of a Y 123 200 nm thin film as a function of magnetic field perpendicular to the CuO₂ planes (O T<B8 T) and temperature (20KT60 K). At fieldsB<0.1 TJ _c is magnetic field independent and decreases at higher fields. A comparison with theory indicates that a crossover from a single pinning to a small bundle collective pinning regime may explain the observed behaviour. According to our estimate the main pinning centers are weak point pins due to oxygen vacancies. From the temperature dependence ofJ _c atB0 we obtain a temperature dependence of the thermodynamical critical fieldB _c (1–T/T _c )² forT20 K which agrees with the anomalous temperature dependence ofB _c2 observed recently in highly anisotropic high temperature superconductors.     

Flux diffusion in high-T c superconductors

In type-II superconductors in the flux flow (J J _c ), flux creep (J _c J _c ), and thermally activated flux flow (TAFF) (J J _c ) regimes the inductionB(r,t), averaged over several penetration depths , in general follows from a nonlinear equation of motion into which enter the nonlinear resistivities (B, J ,T) caused by flux motion and (B, J ,T) caused by other dissipative processes.J andJ are the current densities perpendicular and parallel toB,B=|B|, andT is the temperature. For flux flow and TAFF in isotropic superconductors with weak relative spatial variation ofB, this equation reduces to the diffusion equation plus a correction term which vanishes whenJ =0 (this means B××B=0) or when – = 0 (isotropic normal conductor). When this diffusion equation holds the material anisotropy may be accounted for by a tensorial . The response of a superconductor to an applied current or to a change of the applied magnetic field is considered for various geometries. Such perturbations affect only a surface layer of thickness where a shielding current flows which pulls at the flux lines; the resulting deformation of the vortex lattice diffuses into the interior until a new equilibrium or a new stationary state is reached. The a.c. response, in particular the frequency with maximum damping, depends thus on the geometry and size of the superconductor.     

Irreversibility line and flux pinning properties in iron-based high-Tc superconductor SmFeAsO0.85

The irreversibility line and flux pinning properties of high-T_c superconductor SmFeAsO_0.85 were studied using DC magnetization data. Polycrystalline SmFeAsO_0.85 was prepared in a high pressure synthesis apparatus under the pressure of 6 GPa. The results of DC susceptibility showed the superconducting transition at about 55 K. A critical current density J_c(B) was calculated using Bean’s critical state model. At low temperatures (20 K), J_c(B) showed a relatively high value with weak dependence on an applied magnetic field. At higher temperatures, a stronger dependence of the magnetic field was observed, which resulted from decrease in a critical current density probably due to the flux creep effect. The irreversibility line (IL) agreed well with the flux creep theory of Matsushita et al. A comparison of normalized pinning force density with the theoretical models showed that the irreversible behavior in SmFeAsO_0.85 is dominated mainly by normal point pinning (δT_c) and surface pinning mechanisms.     

Model temperature dependence of the imaginary ac susceptibility in simple and double superconducting structures

Closed expressions are derived for the temperature dependent imaginary part of the ac susceptibility of simple geometrical forms of superconductors by means of the relation between susceptibility and losses at ac field excitation. The magnetic field dependence of the critical current densityj _c is assumed to follow Bean's or Kim's model. The temperature dependence ofj _c is assumed to be linearly decreasing with a moderate nonlinear tail near the critical temperatureT _c . The Meissner as well as surface barrier shielding effect are also involved into consideration. Simple two component structures are entirely analysed regarding the variety of their relative critical current densities, critical temperatures and dimensions. Using the proposed procedure, it is very simple to derive similar as shown expressions for arbitrary parameters and to use them for getting insight into the inner structure of the investigate superconductors.     

The study of modified critical current distribution and its application to the resistive state of macroscopically inhomogeneous type II superconductors

A modification of the Gauss distribution function is outlined which can be useful mainly for probability distributions of such parameters whose most probable values are close to the maximum allowable values (restricted by intrinsic or extrinsic sources). The formulas for the mean value of the considered parameter, of its square, as well as for the mean square deviation are given.This modified distribution function is then applied to the calculation of the current-voltage characteristics in the resistive state of composites with type II superconductors under the assumption that the flux flow is the mean dissipation mechanism when the current flows through the superconductor.It is shown that the induced voltages are much smaller than those calculated by Evetts and Plummer assuming simple Gauss distribution function. The results show that the measured critical current densitiesJ _c (defined approximately by the 0.1 V/cm voltage criterion) in macroscopically inhomogeoeous type II superconductors are only about the dispersion value smaller than the mean critical current density¯J _c of the composite, in contradiction to the results of Evetts and Plummer, where much larger values for¯J _c were obtained than the actually measured values ofJ _c.Some improvements of the model are proposed, mainly by introducing the dissipation mechanism due to the flux creep and by including the current sharing in the normal matrix.     

Effect of grain-boundary flux pinning in MgB2 with columnar structure

We studied the flux pinning properties by grain boundaries in MgB₂ films prepared by using a hybrid physical chemical vapor deposition method on the c-axis oriented sapphire substrates. All the films we report here had the columnar grains with the growth direction perpendicular to the substrates and the grain sizes in the range of a few hundred nanometers. At very low magnetic fields, no discernable grain-boundary (GB) pinning effect was observed in all measuring temperatures, but above those fields, the effect of GB flux pinning was observed as enhanced critical current densities (J_cs) and reduced resistances when an external magnetic field (B) was aligned parallel to the c-axis. We interpret the B dependence of J_c in the terms of flux line lattice shear inside the columnar grains activated by dislocations of Frank–Read source while the flux lines pinned by GB act as anchors for dislocations. Magnetic field dependence of flux pinning force density for B parallel to the c-axis was reasonably explained by the above model.     

Point-contact measurements of CeB6 and CeCu6 in high magnetic fields

Epitaxial YBa₂Cu₄O₇-thin films with thec-axis oriented perpendicular to the film plane were prepared bydc-sputtering from a single stoichiometric target on (100) SrTiO₃-substrates. Typical values of the inductively measured superconducting transitions were about 90 K with a width less than 0.5 K. Critical current densities were measured on 5 to 10 m wide strips as function of magnetic field and temperature. The temperature dependences ofj _c follow a universal functionj _c(B,T)=j _c ^* (T=0,B)·(1–T/T _c (B)) with =1.5±0.1. ForB=0 andT=77 K we obtainedj _c =4·10⁶ A/cm². The field dependence of the resistive transitions was measured with the magnetic field parallel to thec-axis. The slope of the upper critical fieldB _c2 (T) was determined for different criteria. The carrier concentration evaluated from Hall-effect measurements was found to decrease linearly from one per unit cell at 240 K with decreasing temperature extrapolating nearly through zero forT=0. Highly resolved angular dependent measurements of the critical current density withB perpendicular to the current but tilted from thec-axis show a very strong and sharp enhancement ofj _c for the magnetic field parallel to the (CuO₂)-layers (Bc). Additionally to this phenomenon, which is caused by an intrinsic pinning mechanism due to the layered structure of high-T _c -superconductors the influence of the anisotropy of the upper critical field onj _c (B, T, ) is evident nearT _c .     

Monte Carlo study of entropy for face-centered cubic Ising antiferromagnets

A face-centered cubic Ising model with nearest neighbor antiferromagnetic exchangeJ _nn in the presence of a magnetic fieldH is investigated by Monte Carlo methods. Free energy and entropy of the model are obtained by integrating the equation of state along various paths, starting at suitable reference states. It is shown that at low temperatures first-order phase transitions can be located with very good precision. At the two critical fieldsH _c1/|J _nn |=4,H _c2/|J _nn |=12 a residual ground-state entropyS(0) is found, which is estimated as aboutS(0)/k _B (ln 2)/3 in both cases.In the presence of a ferromagnetic next-nearest neighbor exchange there is no longer a nonzero entropy at the critical fields, however. ForR+J _nnn /J _nn +–1 we find the same structure of the phase diagram as qualitatively predicted by Domany et al., where lines of 3-state and 4-state Potts model-like transitions meet at a multicritical point atH=0. Some consequences of our results for interpreting the ordering of face-centered cubic binary alloys are also discussed.     

Low field anomaly in magnetization curves

We present a calculation of the isothermal magnetization hysteresis curves appropriate to highT _csuperconductors. We discuss the nature of the low field anomaly as one goes from this strong pinning case to the weak pinning case. We show that the shape of the equilibrium (thermodynamic) magnetization curve is recovered in the limit ofJ _capproaching zero.     

Theory and experimental evidence for superconductors with short coherence length

We sketch the theory of London superconductors with a complex order parameter undergoing a phase transition dominated by thermal fluctuations. This theory is essentially that of superfluid helium, appropriately modified. We derive relations between the transition temperatureT _c , the London penetration depth, the phase correlation length, the fall ofT _c with reduced thickness in thin slabs and the associated appearance of Kosterlitz-Thouless behavior. AtH _c2 thermal fluctuations drive the transition first order. Accordingly, magnetic properties will exhibit hysteresis phenomena and glassy behavior. Our analysis of recent specific heat and SR measurements and of experiments on ultrathin slabs as well as the experimental evidence for hysteresis phenomena and glassy behavior in magnetic properties reveal that this theory describes a large class of superconductors that encompasses the cuprates, bismuthates and fullerenes.     

Transport critical-current and voltage-current characteristics of Bi(2223) silver-clamped thick films

The transport critical current properties of Bi(2223) silver-clamped thick films are studied by the measurement of its dependence on magnetic field and temperature close to T _c. It is found that the transport critical current follows a power law J _c(1–T/T _c)^3/2 for the sample with J _c>2.0×10⁴ A/cm² (77K, zero field) and that J _c(H) is basically reversible for increasing and decreasing magnetic field. After the transport current exceeds the critical current, the voltage-current (V-I) characteristics show a flux-creep-like behaviour until they smoothly join the flux-flow state. From the measurement of V-I curves, the pinning property of the sample may be estimated using the flux-creep mode. The flux-flow resistance is found to have a nonlinear magnetic field dependence.     

The flux distribution and hysteresis losses in highT c superconductors including viscous forces on flux lines

By comparing the contributions of pinning and viscous forces to the restoring force on flux ines in type II and highT _c superconductors, it is shown that the flux flow in highT _c superconducto rs should play an important role in determining the magnetic flux distribution and hence the hyste resis losses in a.c. fields. Both quantities are calculated in the extreme case of very large viscous forces with respect to the pinning force. The magnetic field and frequency dependence of the losses are changed with respect to the results obtained from the critical state model. The theoretical results are qualitatively confirmed by a.c. susceptibility measurements at different magnetic field amplitudes and frequencies. The quantitative differences indicate that the flux flow effects in highT _c superconductors are by far not so strong as expected and supposed by some theories. The reasons for these discrepancies are discussed.     

On the effect of a magnetic field on the yield point and kinetics of macroplasticity in LiF crystals

A strong effect of a static magnetic field B on active deformation kinetics ( =const) in LiF crystals is observed. This is a threshold effect with respect to B and ; it is observed only for B> B _c (B≃0.4 T) and . Magnetic sensitivity is exhibited by the yield stress τ _y , which decreases by approximately a factor of 1.5 for B=0.48 T, and by the stage-II and stage-III hardening coefficients θ_II and θ _III, the former decreasing and the latter increasing in a magnetic field. A physical interpretation is proposed for the observed behavior. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 6, 470–474 (25 March 1997)     

Influence of non-uniform J<Subscript>c</Subscript> distributions on flux jumps in high-temperature superconductors

The influence of non-uniform J_c distributions on flux jumps in high-temperature superconductors is investigated with the simplified models in the form of a composite superconducting slab consisting of different pinning regions. The magnetization loops and flux jumps for the simplified models are calculated with the modified recursion formulas which could predict the flux-jump fields. The valid ranges of the temperature and sweep rate for B_fj1 are specified and the fields of the subsequent jumps after the first flux jump of the inhomogeneous models are obviously lower than that of the homogeneous models. The results indicate that flux jump instability could be stimulated by non-uniform J_c distributions and with the increase of the discrepancy of J_c between different pinning regions the instability also increase.     

Superconductivity and magnetism and their interplay in quaternary borocarbides RNi2B2C

Since 1986, most of the interest in superconductivity became focused on high-T _c cuprates. The discovery of the superconducting quaternary borocarbide system Y–Ni–B–C with T _c as high as?～12?K inspired research into intermetallic superconductors (IMS) once again. Several reasons can be attributed to this revival of interest in IMS: (i) In the tetragonal quaternary magnetic superconductors RNi₂B₂C, superconductivity and magnetism occur with T _c and T _N?～?10?K, thereby allowing studies of exotic phenomena associated with, and arising from, the interplay of superconductivity and magnetism. (ii) High T _N's and a variety of commensurate and incommensurate magnetic structures in RNi₂B₂C (Fermi surface nesting playing a central role) strongly suggest that R-spins are coupled via the RKKY-exchange interaction. Hence, unlike in most other magnetic superconductors known so far, conduction electrons take part in superconductivity and magnetism. (iii) Quaternary borocarbides open up new pathways to try and synthesize multicomponent intermetallic superconductors. Their remarkable intrinsic superconducting and magnetic properties and the availability of high quality samples (bulk polycrystalline, large single crystals and thin films) make RNi₂B₂C particularly special to investigate. Several unusual phenomena have been reported, such as, to name a few, dramatic phonon mode softening at T _c, H _c2(T) exhibiting a positive curvature near T _c and a four-fold anisotropy in the basal plane; a variety of exceptional and fascinating flux line lattice (FLL) related effects — FLL-symmetry transformations and alignments with the underlying crystal lattice as a function of applied field (manifestation of nonlocal electrodynamics despite high κ?～?10, and thermal fluctuation effects even though T _c,?～?16?K, is not too high) and a four-fold symmetric star-shaped (in real space) vortex core. RNi₂B₂C are strong coupling s-wave BCS superconductors and, remarkably, have a superconducting gap with extreme anisotropy. Strong experimental evidence shows that the four-fold symmetric superconducting gap has point nodes along the 100 Carter, S, Batlogg, B, Cava, R, Krajewski, J, Peck, W Jr and Takagi, H. 1994. Phys. Rev. B, 50: 4216[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]- and 10 Baltensperger, W and Strässler, S. 1963. Phys. Kondens Materie, 1: 20 [Google Scholar]-directions, a feature that has been shown consistent with (s?+?g)-Cooper pairing. An energy gap with such strong anisotropy is unusual for an s-wave superconductor and, hence, calls for a pairing mechanism different from conventional electron–phonon coupling. Antiferromagnetic fluctuations possibly play an important role in the mechanism. Magnetic superconductors RNi₂B₂C (R?=?Dy, Ho, Er, Tm) reveal several phenomena, not observed earlier, associated with the interplay of superconductivity and magnetism. Microscopic evidence (via square FLL interacting with magnetism) of the coexistence of magnetism and superconductivity; intrinsic FLL-pinning by magnetic ions; weak ferromagnetism (local moment) coexisting with superconductivity (down to the lowest temperature) and the spontaneous vortex phase (ErNi₂B₂C); superconductivity setting in an already magnetically ordered lattice (DyNi₂B₂C) and pair-breaking by nonmagnetic ions in such materials; rich and complex magnetic structures and double (nearly) re-entrant superconductivity (HoNi₂B₂C) and changes in the FLL-symmetry in the vicinity of magnetic transition (TmNi₂B₂C) and 4f-quadrupole ordering (TmNi₂B₂C) are several exciting phenomena that magnetic superconductors RNi₂B₂C exhibit.

At the end of this review are indicated some possible further studies in quaternary borocarbide superconductors. These studies may turn out to be important not only with respect to borocarbides themselves but also from the standpoint of superconductivity in general.     

London field penetration in heavy fermion superconductors

We present results of a series of high resolution, low fielddc-magnetization measurements on the heavy fermion superconductors UPt₃, UBe₁₃, U_1–x Th _X Be₁₃ and CeCu₂Si₂, from which values of the magnetic penetration depth can be extracted. A study of the temperature variation reveals aT ² power law in all cases. This can not be reconciled with a BCS-like isotropic energy gap but may be explained by the presence of low energy quasiparticle states inside the gap. In the case of very pure superconductors, one such possibility is the assumption of point-nodes in the gap function. We argue, however, that an interpretation in terms of resonant impurity scattering in various anisotropic superconducting states is more likely to explain a broad range of experimental data. The results on differently oriented single crystals of UBe₁₃ and UPt₃ reveal no pronounced anisotropy related to the crystal lattice. Absolute values of atT=0 are deduced by a novel method in which the field is first screened out from the sample by means of an evaporated superconducting film (of lowerT _c ). Above this lowerT _c the sudden penetration of field into the sample can then be measured absolutely.     

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. I. General Theory and Square-Lattice Chromatic Polynomial

We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) P _G (q) for m×n rectangular subsets of the square lattice, with m8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer matrix in the Fortuin–Kasteleyn representation. In particular, we extract the limiting curves of partition-function zeros when n, which arise from the crossing in modulus of dominant eigenvalues (Beraha–Kahane–Weiss theorem). We also provide evidence that the Beraha numbers B ₂,B ₃,B ₄,B ₅ are limiting points of partition-function zeros as n whenever the strip width m is 7 (periodic transverse b.c.) or 8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B ₁₀) cannot be a chromatic root of any graph.     

Partition function zeros for the one-dimensional ordered plasma in Dirichlet boundary conditions

We consider the grand canonical partition function for the ordered one-dimensional, two-component plasma at fugacity in an applied electric fieldE with Dirichlet boundary conditions. The system has a phase transition from a low-coupling phase with equally spaced particles to a high-coupling phase with particles clustered into dipolar pairs. An exact expression for the partition function is developed. In zero applied field the zeros in the plane occupy the imaginary axis from –i to –i_c and i_c to i for some _c. They also occupy the diamond shape of four straight lines from ±i_c to _c and from ±i_c to –_c. The fugacity acts like a temperature or coupling variable. The symmetry-breaking field is the applied electric fieldE. A finite-size scaling representation for the partition in scaled coupling and scaled electric field is developed. It has standard mean field form. When the scaled coupling is real, the zeros in the scaled field lie on the imaginary axis and pinch the real scaled field axis as the scaled coupling increases. The scaled partition function considered as a function of two complex variables, scaled coupling and scaled field, has zeros on a two-dimensional surface in a domain of four real variables. A numerical discussion of some of the properties of this surface is presented.     

Finite-Size Scaling in the p-State Mean-Field Potts Glass: A Monte Carlo Investigation

The p-state mean-field Potts glass with bimodal bond distribution (±J) is studied by Monte Carlo simulations, both for p = 3 and p = 6 states, for system sizes from N = 5 to N = 120 spins, considering particularly the finite-size scaling behavior at the exactly known glass transition temperature T _c. It is shown that for p = 3 the moments q ^(k) of the spin-glass order parameter satisfy a simple scaling behavior, being the appropriate scaling function and T the temperature. Also the specific heat maxima have a similar behavior, , while moments of the magnetization scale as . The approach of the positions T _max of these specific heat maxima to T _c as N is nonmonotonic. For p = 6 the results are compatible with a first-order transition, q ^(k) (q _jump)^k as N but since the order parameter q _jump at T _c is rather small, a behavior q ^(k) N ^-k/3 as N also is compatible with the data. Thus no firm conclusions on the finite-size behavior of the order parameter can be drawn. The specific heat maxima c _V ^max behave qualitatively in the same way as for p = 3, consistent with the prediction that there is no latent heat. A speculative phenomenological discussion of finite-size scaling for such transitions is given. For small N (N 15 for p = 3, N 12 for p = 6) the Monte Carlo data are compared to exact partition function calculations, and excellent agreement is found. We also discuss ratios , for various quantities X, to test the possible lack of self-averaging at T _c.