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The effect of a magnetic field component parallel to the superconducting layers on longitudinal Josephson plasma oscillations in the layered high-temperature superconductor ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ is shown to depend on the thermodynamic state of the underlying vortex lattice. Whereas the parallel magnetic field component depresses the Josephson plasma resonance (JPR) frequency in the vortex solid phase, it may enhance it in the vortex liquid. There is a close correlation between the behavior of microwave absorption near the JPR frequency and the effectiveness of pancake vortex pinning, with the enhancement of the plasma resonance frequency occurring in the absence of pinning, at high temperature close to the vortex melting line. An interpretation is proposed in terms of the attraction between pancake vortices and Josephson vortices, apparently also present in the vortex liquid state.

light on this enigmatic feature. In the H CuO2 geometry, where the magnetic field initiates Josephson vortices, we observed a strong mode in the far-infrared which hardens with increasing field. The field dependence of the low-frequency resonance behavior is contrasted to that of two other cuprate materials: La2�xSrCuO4 compounds that we have investigated earlier, and Bi2Sr2CaCu2O8�. Specifically, there exist disparities in the number and field dependence of longitudinal modes measured for each system. Many of these differences can be explained through a new numerical solution of the interlayer phase equations which includes effects of both in-plane and c-axis dissipation parameters. Support for this approach is given by calculations of the Josephson vortex lattice ground state configuration, and further insight is gained through the phenomenological framework of the transverse JPR model, as well as a classical model of vortex dynamics.

Dynamic ordering of driven vortex matter has been investigated in the peak effect regime of both amorphous MoGe films and 2H-NbSe2 crystals by mode locking (ML) and dc transport measurements. ML features allow us to trace how the shear rigidity of driven vortices evolves with the average velocity. Determining the onset of ML resonance in different magnetic fields and/or temperatures, we find that the dynamic ordering frequency (velocity) exhibits a striking divergence in the higher part of the peak effect regime. Interestingly, this phenomenon is accompanied by a pronounced peak of dynamic critical current. Mapping out field-temperature phase diagrams, we find that divergent points follow well the thermodynamic melting curve of the ideal vortex lattice over wide field and/or temperature ranges. These findings provide a link between the dynamic and static melting phenomena which can be distinguished from the disorder induced peak effect.

The theory presented is based on a simple Hamiltonian for a vortex lattice in a weak impurity background which includes linear elasticity and plasticity, the latter in the form of integer valued fields accounting for defects. Within a quadratic approximation in the impurity potential, we find a first-order Bragg-glass, vortex-glass transition line showing a reentrant behavior for superconductors with a melting line near $ H_{c2} $. Going beyond the quadratic approximation by using the variational approach of M\'ezard and Parisi established for random manifolds, we obtain a phase diagram containing a third-order glass transition line. The glass transition line separates the vortex glass and the vortex liquid. Furthermore, we find a unified first-order line consisting of the melting transition between the Bragg glass and the vortex liquid phase as well as a disorder induced first-order line between the Bragg glass and the vortex glass phase. The reentrant behavior of this line within the quadratic approach mentioned above vanished. We calculate the entropy and magnetic induction jumps over the first-order line., Comment: 26 pages, 6 figures, typos corrected, An error in Section VI C. (replica symmetric solution in the fluid phase) is corrected leading to changes in the phase diagram in Section IX. Erratum sent to Phys. Rev. B

We have measured ac magnetic susceptibility (chi=chi(')+i chi '') of the electron-doped superconducting Pr0.88LaCe0.12CuO4-delta with applied magnetic field (H) either parallel or perpendicular to the CuO2 plane (H parallel to ab-plane or H parallel to c-axis). For H parallel to ab-plane, a peak in the temperature dependence of the screening current is revealed as a dip in the real part of the ac susceptibility chi. The temperature at which this peak anomaly occurs decreases with increasing field and lies well below the irreversibility line in the H-T phase diagram. This peak effect may arise from the phase transition of Josephson vortices from a quasiordered vortex lattice to a disordered glass phase.

The stability of the superconducting dissipationless and resistive states in single-crystalline NbSe2 nanobelts is characterized by transport measurements in an external magnetic field (H). Current-driven electrical measurements show voltage steps, indicating the nucleation of phase-slip structures. Well below the critical temperature, the position of the voltage steps exhibits a sharp, periodic dependence as a function of H. This phenomenon is discussed in the context of two possible mechanisms: the interference of the order parameter and the periodic rearrangement of the vortex lattice within the nanobelt., Comment: 4 figures

We derive the generalized Ginzburg-Landau free energy functional for conventional and unconventional singlet superconductors in the presence of paramagnetic, orbital and impurity effects. Within the mean field theory, we determine the criterion for appearence of the non uniform (Fulde-Ferrell-Larkin-Ovchinnikov) superconducting state, with vortex lattice structure and additional modulation along the magnetic field. We also discuss the possible change of the order of transition from normal to superconducting state. We find that the superconducting phase diagram is very sensitive to geometrical effects such as the nature of the order parameter and the shape of the Fermi surface. In particular, we obtain the qualitative phase diagrams for three-dimensional isotropic s-wave superconductors and in quasi two-dimensional d-wave superconductors under magnetic field perpendicular to the conducting layers. In addition, we determine the criterion for instability toward non uniform superconducting state in s-wave superconductors in the dirty limit., 15 pages, 4 figures

We measured the Josephson vortex flow resistance as a function of magnetic field applied parallel to the $ab$ planes using annealed $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{x}$ intrinsic Josephson junctions having high anisotropy $(\ensuremath{\sim}40)$ by oxygen content reduction. Periodic oscillations were observed in magnetic fields above $45--58\phantom{\rule{0.3em}{0ex}}\mathrm{kOe}$, corresponding to dense-dilute boundary for Josephson vortex lattice. The observed period of oscillations, which agrees well with the increase of one fluxon per two junctions $({H}_{p}={\ensuremath{\Phi}}_{0}∕2Ls)$, may correspond to formation of a triangular lattice of Josephson vortices as has been reported by Ooi et al. for highly anisotropic $(\ensuremath{\geqslant}200)$ Bi-2212 intrinsic Josephson junctions.

The ground state of interlayer Josephson vortex systems is investigated on the basis of a simplified Lawrence-Doniach model in which spatial dependence of the gauge field and the amplitude of superconducting order parameter is not taken into account. Energy landscape is drawn with respect to the in-plane field, the period of insulating layers including Josephson vortices, and the shift from the aligned vortex lattice. The energy landscape has a multi-valley structure and ground-state configurations correspond to bifurcation points of the valleys. In the high-field region, the shear modulus becomes independent of field and its anisotropy dependence is given by $c_{66}\propto \gamma^{-4}$., Comment: 4 pages, 7 figures (2 gif figures not in the text); RevTeX

The Bragg-chain model of the two-dimensional (2D) vortex state at high magnetic field [V. N. Zhuravlev and T. Maniv, Phys. Rev. B 60, 4277 (1999)] is extended to an array of coupled superconducting (SC) layers. Application to ${\mathrm{MgB}}_{2}$ and ${\mathrm{YNi}}_{2}{\mathrm{B}}_{2}\mathrm{C}$ yields good quantitative agreement with high-field magnetization measurements, indicating that the smeared transitions observed in these materials are, at least in great part, due to SC fluctuations. Similar to the situation in a 2D system, the melting of the vortex lattice in strongly coupled SC layers is predicted to occur well below the mean field ${H}_{c2}$.

Magnetic field and angle dependences of the critical current density ${J}_{c}(H,\ensuremath{\theta})$ in epitaxial $c$-oriented $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ thin films are measured by the four-probe transport current technique, low-frequency ac magnetic susceptibility, and superconducting quantum interference device magnetometry. The films under study are deposited by off-axis dc magnetron sputtering onto $r$-cut sapphire substrates buffered with a $\mathrm{Ce}{\mathrm{O}}_{2}$ layer. A consistent model of vortex pining and supercurrent limitation is developed and discussed. Rows of growth-induced out-of-plane edge dislocations forming low-angle boundaries (LAB's) are shown to play a key role in achievement of the highest critical current density ${J}_{c}\ensuremath{\geqslant}2\ifmmode\times\else\texttimes\fi{}{10}^{6}\phantom{\rule{0.3em}{0ex}}\mathrm{A}∕{\mathrm{cm}}^{2}$ at $77\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. The model takes into account the transparency of LAB's for supercurrent as well as the pinning of vortex lattice on a network of LAB's. Principal statistical parameters of the film defect structure, such as the domain size distribution and mean misorientation angle, are extracted from ${J}_{c}(H)$ curves measured in a magnetic field $H$ applied parallel to the $c$ axis and from x-ray diffraction data. An evolution of angle dependences ${J}_{c}(\ensuremath{\theta})$ with $H$ is shown to be consistent with the model supposing dominant pinning on edge dislocations. Strongly pinned vortices parallel to the $c$ axis appear to exist in tilted low magnetic fields up to a characteristic threshold field, below which the magnetic induction within the film obeys a simple relation $B=H\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\ensuremath{\theta}$. This feature is shown to explain the absence of the expected maximum of ${J}_{c}(\ensuremath{\theta})$ at $H\ensuremath{\parallel}c$ in a low applied field. A peak of ${J}_{c}(H)$ and an angular hysteresis of ${J}_{c}(\ensuremath{\theta})$, which have been observed in an intermediate-field range, are discussed in terms of film thickness, surface quality, and orientation of the applied field. The observed effects are found to be consistent with the developed model.

The vortex ratchet effect has been studied in Al films patterned with square arrays of submicron antidots. We have investigated the transport properties of two sets of samples: (i) asymmetrical antidots where vortices are driven by an unbiased ac current and (ii) symmetrical antidots where in addition to the ac drive a dc bias was used. For each sample, the rectified dc voltage is measured as a function of drive amplitude and frequency, magnetic field, and temperature. As unambiguously shown by our data, the voltage rectification in the asymmetric antidots is induced by the intrinsic asymmetry in the pinning potential created by the antidots, whereas the rectification in the symmetric antidots is induced by the dc bias. In addition, the experiments reveal interesting collective phenomena in the vortex ratchet effect. At fields below the first matching field (H-1), the dc-voltage-ac-drive characteristics present two rectification peaks, which is interpreted as an interplay between the one-dimensional motion of weakly pinned incommensurate vortex rows and the two-dimensional motion of the whole vortex lattice. We also discuss the different dynamical regimes controlling the motion of interstitial and trapped vortices at fields higher than H-1 and their implications for the vortex ratchet effect. 10 pages, 8 figures ispartof: Physical Review B: Condensed Matter and Materials Physics vol:73 issue:1 pages:1-10 status: published

We reanalyze published magnetization data and demonstrate that the conclusion of the original authors, claiming enhanced two-dimensional properties of the cuprate superconductor $\mathrm{Tl}{\mathrm{Ba}}_{2}{\mathrm{Ca}}_{3}{\mathrm{Cu}}_{4}{\mathrm{O}}_{y}$, is not supported by the experimental results. Our analysis shows that the magnetic field dependence of the mixed-state magnetization for this particular sample is amazingly close to the results of numerical calculations by Brandt for an ideal vortex lattice without fluctuations. This good agreement between experiment and theory allows for the evaluation of the absolute values of the upper critical field ${H}_{c2}(T)$.

We have investigated the magnetic penetration depth lambda and the superconducting coherence length xi in the vortex lattice of the filled skutterudite compound PrRu4Sb12 (T-c similar to 0.97 K) using transverse-field muon-spin rotation measurements. Zero-field and longitudinal-field studies were also carried out to investigate the time-reversal symmetry of the superconducting state. We found lambda=3650(20) angstrom and xi=345(5) angstrom at 0.05 K and using these values of lambda and xi, together with the Sommerfeld constant we have calculated an effective mass of the quasiparticles m(*)approximate to 10m(e) and superfluid carrier density n(s)approximate to 4x10(27) carriers/m(3). The temperature dependence of the vortex state muon-spin relaxation rate sigma(s)(T) is consistent with the phenomenological two-fluid model. Further, the zero-field and longitudinal field measurements do not reveal any signature of a spontaneous internal magnetic field below the superconducting transition temperature, indicating the preservation of time-reversal symmetry in the superconducting state of PrRu4Sb12, unlike the broken time-reversal symmetry of the superconducting state of the heavy-fermion superconductor PrOs4Sb12.

We numerically investigate Josephson vortex flux flow states in stacked Josephson junctions, motivated by recent experiments trying to observe the vortices in a square vortex lattice when a magnetic field is applied to layered high-${T}_{c}$ superconductors of the ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{x}$ type. By extensive numerical simulations, we are able to clearly distinguish between triangular and square vortex lattices and to identify the parameters leading to an in-phase vortex configuration.

Nonlocal response in Hall-shaped superconductors is studied using the time-dependent Ginzburg-Landau equations. Applying current in one pair of contacts leads to a voltage drop in another pair of contacts situated at a distance much larger than the coherence length. This effect is a consequence of the long range correlations in a one-dimensional vortex lattice squeezed in a narrow channel by screening currents. The discrete change in the number of vortices in the channel with applied magnetic field leads to a nonlocal response which is a nonmonotonous function of the magnetic field. For specific configurations of the Hall-shaped superconductor we found a rectifying effect.

The magneto-thermal conductivity kappa is calculated for the vortex state of UPd_2Al_3 by assuming horizontal gap nodes. The Green's function method we employed takes into account the effects of supercurrent flow and Andreev scattering on the quasiparticles due to Abrikosov's vortex lattice order parameter. The calculated angular dependence of kappa_{yy} for field rotation theta_0 in the ac-plane depends strongly on field strength H, impurity scattering, anisotropy of the Fermi velocity, and temperature. For finite temperatures and the clean unitary scattering limit we get qualitative agreement with recent experiments for all four proposed gap functions having horizontal line nodes at ck_z = 0, pi/4, and pi/2.

The use of artificial defects is known to enhance the superconducting critical parameters of thin films. In the case of conventional superconductors, regular arrays of submicron holes (antidots) substantially increase the critical temperature Tc(H) and critical current Ic(H) for all fields. Using electrical transport measurements, we study the effect of placing an additional small antidot in the unit cell of the array. This composite antidot lattice consists of two interpenetrating antidot square arrays with a different antidot size and the same lattice period. The smaller antidots are located exactly at the centers of the cells of the array of large antidots. We show that the composite antidot lattice can trap a higher number of flux quanta per unit cell inside the antidots, compared to a reference antidot film without the additional small antidots in the center of the cells. As a consequence, the field range in which an enhanced critical current is observed is considerably expanded. Finally, the possible stable vortex lattice patterns at several matching fields are determined by molecular dynamics simulations.

We have studied bulk vortex properties in ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ (BSCCO) using transport measurements with a unique Corbino disk contact geometry. This Corbino disk contact geometry allows us to measure true bulk vortex properties free from surface barrier effects. The investigated vortex properties include current-induced vortex dissipation in the vortex liquid and vortex solid phases, vortex matter phase transition (vortex lattice melting transition), and vortex correlation along the $c$ axis of BSCCO. No discrepancy is found between our measurements and others measurements using the conventional four-probe contact geometry, which renders vortex transport properties susceptible to the Bean-Livingston surface barrier effect. We conclude that the sample bulk vortex matter properties determine the vortex transport behaviors in BSCCO.

We have studied the field-orientational dependence of the zero-energy density of states (FODOS) for a series of systems with different Fermi surface and superconducting gap structures. Instead of the phenomenological Doppler-shift method, we use an approximate analytical solution of the Eilenberger equation together with self-consistent determination of the order parameter and a variational treatment of the vortex lattice. First, we compare the zero-energy density of states when a magnetic field is applied in the nodal direction $[{\ensuremath{\nu}}_{\mathit{node}}(0)]$ and in the antinodal direction $[{\ensuremath{\nu}}_{\mathit{anti}}(0)]$, by taking account of the field-angle dependence of the order parameter. As a result, we found that there exists a crossover magnetic field $H*$ so that ${\ensuremath{\nu}}_{\mathit{anti}}(0)g{\ensuremath{\nu}}_{\mathit{node}}(0)$ for $HlH*$, while ${\ensuremath{\nu}}_{\mathit{node}}(0)g{\ensuremath{\nu}}_{\mathit{anti}}(0)$ for $HgH*$, consistent with our previous analyses. Next, we showed that $H*$ and the shape of the FODOS are determined by a contribution from the small part of the Fermi surface where the Fermi velocity is parallel to the field-rotational plane. In particular, we found that $H*$ is lowered and the FODOS has broader minima when a superconducting gap has point nodes, in contrast to the result of the Doppler-shift method. We also studied the effects of in-plane anisotropy of the Fermi surface. We found that the in-plane anisotropy of a quasi-two-dimensional Fermi surface sometimes becomes larger than the effects of the Doppler shift and can destroy the Doppler-shift-predominant region. In particular, this tendency is strong in a multiband system where superconducting coherence lengths are isotropic. Finally, we addressed the problem of the cusplike singularity in $\mathrm{Y}{\mathrm{Ni}}_{2}{\mathrm{B}}_{2}\mathrm{C}$ and present a possible explanation of this phenomenon.

Heavy-ion irradiation experiments have been carried out for single crystals of YBa{sub 2}Cu{sub 4}O{sub 8} in order to compare the magnetic-field dependence of the flux pinning for damage channels with the intrinsic {open_quote}{open_quote}fishtail{close_quote}{close_quote} pinning that occurs in this material before irradiation. The two pinning effects seem to be additive in that irradiation increases the hysteresis at all fields and the fishtail hump persists in the irradiated samples with roughly the same magnitude as before the damage channels are produced. For irradiated samples, there are two regions where pinning rises with increasing field, one near zero field due to a matching effect of the defect density with the vortex lattice spacing and a second at the same field as the original fishtail. Both effects appear in the same sample in different magnetic-field ranges. {copyright} {ital 1996 The American Physical Society.}

Measurements of magnetization of ${\mathrm{N}}_{2}$-annealed ${\mathrm{Tl}}_{2}{\mathrm{Ba}}_{2}\mathrm{Cu}{\mathrm{O}}_{6}$ were carried out by using a micro-Hall sensor while applying a low-ac transverse magnetic field ${H}_{\mathit{ac}}$ along the $ab$ plane in addition to a dc magnetic field along the $c$ axis. A systematic reduction of dc magnetization was observed, and further, an anomalous second magnetization peak evolved with increasing ${H}_{\mathit{ac}}$. We explained the observed reduction of magnetization due to a dc electric field generated by an ac transverse field, which leads to the decay of the magnetization and then interpreted the development of the second peak at finite ${H}_{\mathit{ac}}$ due to the destruction of the vortex lattice by strong point pinning and the subsequent optimal redistribution of two-dimensional vortices to the pinning centers.

We have studied the influence of a square array of pinning centers on the dynamics of vortex avalanches in $\mathrm{Pb}$ thin films by means of $\mathrm{ac}\text{\ensuremath{-}}$ and $\mathrm{dc}\text{\ensuremath{-}}$magnetization measurements. Close to the superconducting transition ${T}_{c}$, the commensurability between the vortex lattice and the pinning array leads to the well known local increments of the critical current. As temperature $T$ decreases, matching features progressively fade out and eventually disappear. Further down in temperature, vortex avalanches develop and dominate the magnetic response. These avalanches manifest themselves as jumps in the $\mathrm{dc}$ magnetization and produce a lower $\mathrm{ac}$ shielding, giving rise to a paramagnetic reentrance in the $\mathrm{ac}$ screening ${\ensuremath{\chi}}^{\ensuremath{'}}(T)$. Within the flux-jump regime, two subregimes can be identified. Close to the boundary where vortex avalanches develop, the field separation between consecutive jumps follows the periodicity of the pinning array and a field- and temperature-dependent screening is observed. In this regime, the response also depends on frequency $f$ in agreement with theoretical models for magnetothermal instabilities. At low enough temperatures and fields, the screening saturates to a constant value independent of $T$, $H$, and $f$, where jumps are randomly distributed. We have also found that vortex instabilities occupy a larger portion of the $H\text{\ensuremath{-}}T$ diagram in patterned samples than in films without nanoengineered pinning sites. Finally, we discuss the possible origin of the vortex avalanches and compare our results with previous experimental and theoretical studies.

Muon spin rotation $({\ensuremath{\mu}}^{+}\mathrm{SR})$ measurements were conducted on a single crystal of ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7}$ with a superconducting transition temperature of ${T}_{c}\ensuremath{\approx}91.3\mathrm{K}$ and a transition width of $\ensuremath{\Delta}{T}_{c}l0.5\mathrm{K}$ in zero applied field. Data were taken at applied magnetic fields along the c axis of 0.05, 1.0, 3.0, and 6.0 T. We found, by taking into account the expected field-dependent and temperature-activated flux-line disorder, that our results were in fact consistent with a nodeless (s-wave) superconducting order parameter and that they appeared to be inconsistent with order parameters possessing nodes, such as those having ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ symmetry. This result is consistent with early \ensuremath{\mu}SR measurements on sintered samples in which (we believe) strong pinning eliminated the temperature and field dependence of the vortex lattice disorder. These data (including their observed dependences on magnetic field) are, however, completely consistent with s-wave (or extended s-wave) pairing, provided that field-dependent and temperature-activated vortex depinning is also accounted for. Our results (i) confirm the s-wave superconductivity character originally observed in 1989, and (ii) show that the features of \ensuremath{\mu}SR (and microwave) data claimed by other authors to be evidence for d-wave superconductivity are instead symptomatic of temperature-dependent depinning of vortices, which results in long-ranged distortion of the flux lattice. Indeed, the probability that any published d-wave model gives a better fit than the two-fluid model is less than $4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}.$

We report on the effects of neutron irradiation in superconducting ${\mathrm{MgB}}_{2}$ single crystals. Transmission electron microscopy investigations reveal large radiation-induced defects with dimensions comparable to the superconducting coherence length. Accordingly, measurements of the magnetic moment show pronounced modifications of the reversible and irreversible properties after irradiation. In particular, we concentrate on the emergence of a second peak (fishtail), situated near ${H}_{c2}$ at low neutron fluences, but extending over a large part of the superconducting phase diagram at high doses. The history dependence of the critical current density on the low-field side of the fishtail region is studied by measuring minor hysteresis loops and interpreted in terms of an order-disorder transition of the vortex lattice.

Superconductivity of a two-dimensional electron gas under a very high magnetic field is studied by using the Bogoliubov--nde Gennes equations. It is shown that in the quantum limit, where all of the carriers are confined into the lowest Landau level, the excitation spectrum is gapless for any distribution of zeros of the order parameter. Therefore, it is concluded that the presence of gapless excitations, which has been first demonstrated for ordered vortex lattice, is in fact a topological property of the system. The model describes the physics of type-II superconductors where the vortices are pinned by lattice imperfections.

The influence of finite dimensions of metallic multilayers on the superconducting phase nucleation and vortex mobility is studied. Resistive characteristics are sensitive to the geometrical symmetry of the samples. For multilayers with the symmetry plane in the center of superconducting layer the resistive transitions are widely spread for temperatures above the temperature of two- to three-dimensional crossover with respect to the samples with the symmetry plane in the center of the normal layer. Applying the Ginzburg-Landau theory, we correctly found the upper critical magnetic fields and explained the observed broadening by considering the common action of the Lorentz and pinning forces on the nascent vortex lattice.

The effect of a periodic pinning array on the vortex state in a two-dimensional superconductor at low temperatures is studied within the framework of the Ginzburg-Landau approach. It is shown that an attractive interaction of vortex cores to a commensurate pin lattice stabilizes vortex solid phases with long-range positional order against violent shear fluctuations. Exploiting a simple analytical method, based on the Landau orbital description, we derive a rather detailed picture of the low-temperature vortex-state phase diagram for the first matching magnetic field. It is predicted that for sufficiently clean samples application of an artificial periodic pinning array would enable one to directly detect the intrinsic shear stiffness anisotropy characterizing the ideal vortex lattice.

We have numerically investigated the dynamics of vortices in a clean layered superconductor placed in a perpendicular magnetic field. We describe the energetics using a Ginzburg-Landau free energy functional in the lowest Landau level approximation. The dynamics are determined using the time-dependent Ginzburg-Landau approximation, and thermal fluctuations are incorporated via a Langevin term. The c-axis conductivity at nonzero frequencies, as calculated from the Kubo formalism, shows a strong but not divergent increase as the melting temperature $T_M$ is approached from above, followed by an apparently discontinuous drop at the vortex lattice freezing temperature. The discontinuity is consistent with the occurrence of a first-order freezing. The calculated equilibrium properties agree with previous Monte Carlo studies using the same Hamiltonian. We briefly discuss the possibility of detecting this fluctuation conductivity experimentally., Comment: To be published in Phys. Rev. B

The vortex-dominated ac response of a superconducting film with a periodic array of pinning centers is studied in the linear regime for vortex densities greater than the saturation number. A simple model is introduced to describe small oscillations of the vortex lattice, which is considered to be composed of two distinct lattices formed by trapped vortices and interstitial vortices. The frequency-dependent complex resistivity is calculated as a function of frequency and magnetic induction for vortex densities between the first and the second matching fields. It is shown that the absorption spectra has two peaks corresponding to two characteristic pinning frequencies: one due to the periodic pinning array and another due to vortex-cage pinning. The range of validity of the model is determined from direct numerical simulations.

We have performed scanning tunneling spectroscopy imaging of vortices in ${\mathrm{MgB}}_{2}$ with both the tunneling current and the applied field perpendicular to the crystalline c axis. This revealed a well ordered vortex lattice with a relatively low anisotropy of $1.19\ifmmode\pm\else\textpm\fi{}0.04.$ This is significantly below the upper-critical-field anisotropy, but in good agreement with theoretical predictions for the penetration depth anisotropy, which at low fields is expected to be dominated by the $\ensuremath{\pi}$ band. The value of the anisotropy is smaller than estimated from small-angle neutron-scattering measurement.

We have considered vortices in p-wave superconductors with a nonvanishing order parameter in the vortex core for the arbitrary ratio $\ensuremath{\kappa}=\ensuremath{\lambda}/\ensuremath{\xi}$ (here $\ensuremath{\lambda}$ and $\ensuremath{\xi}$ are the magnetic length and the coherence length, correspondently). Vector vortices are characterized by an orientation in the order-parameter space, however, spatial profile of both the order parameter and magnetic field of the vortex demonstrate an essential difference of the vortex structure for different $\ensuremath{\kappa}$ parameters and from those calculated for the vortex in liquid ${}^{3}\mathrm{He}.$ We calculate lower critical field ${H}_{c1}$ and find that critical $\ensuremath{\kappa}={\ensuremath{\kappa}}_{c}=1/\sqrt{2}$ is independent of the detailed form of interactions. Strong asymmetry of some of the components of the order parameter disappears when $\ensuremath{\kappa}$ approaches ${\ensuremath{\kappa}}_{c}.$ The vortex lattice in the multicomponent superconductor resembles a spin system in which the orientation in the order-parameter space plays the role of ``spin.'' Interactions between vortices lead to long-range order. An effective theory for this system is proposed.

Magneto-optical snapshots of the induction distribution across a Bi 2 Sr 2 CaCu 2 O 8 + δ crystal subjected to ramped magnetic fields reveal strong effects of transient disordered vortex states near the disorder-driven vortex phase transition. These transient states, predominantly generated by edge contamination during field ascending and by supercooling of the disordered vortex lattice during field descending, cause smearing of the transition and shift of its onset towards lower inductions. With decreasing temperature, these effects are more pronounced as the lifetime of the transient vortex states increases. Similar effects are observed with increasing field sweep rates, as transient states with shorter lifetime play a role. These observations clarify previously reported puzzles, e.g., the absence of the second magnetization peak at low temperatures, its appearance at longer times, and the shift of its onset to lower inductions for higher field sweeping rates.

The specific-heat coefficient y,(H) and the electronic thermal conductivity κ e s (H) are calculated for Abri-kosov's vortex lattice by taking into account the effects of supercurrent flow and Andreev scattering. First we solve the gap equation for the entire range of magnetic fields. We take into account vertex corrections due to impurity scattering calculated in the Born approximation. The function γ s (H)/γ n increases from zero and becomes approximately linear above H/H c 2 ∼0.1. The dependence on impurity scattering is substantially reduced by the vertex corrections. The upward curvature of κ e s (H)/κ e n , which is caused by decreasing Andreev scattering for increasing field, is reduced for increasing impurity scattering. We also calculate the temperature dependence of the scattering rates 1/τ p s (H) of a phonon and 1/τ e s (H) of a quasiparticle due to quasiparticle and phonon scattering, respectively. At low temperatures the ratio τ p n /τ p s (H) increases rapidly to one as H tends to H c 2 which yields a rapid drop in the phononic thermal conductivity κ p h . Our results are in qualitative agreement with the experiments on the two-gap superconductor MgB 2 .

We report the observation of superconducting vortices in pure and lightly Al doped ${\mathrm{MgB}}_{2}$ single crystals. Low field experiments allow for the estimation of the absolute value of the London penetration depth, $\ensuremath{\lambda}=1900\ifmmode\pm\else\textpm\fi{}600\mathrm{\AA{}}$ for $T\ensuremath{\sim}6\mathrm{K}.$ Experiments in higher fields (e.g., 200 Oe) clearly show a triangular vortex lattice in both real space $(13\ifmmode\times\else\texttimes\fi{}13\ensuremath{\mu}{m}^{2}$ Bitter decoration image of over 1000 vortices) and reciprocal space.

Magneto-optical observations of distribution of vortices and vortex-lattice melting transition fields are used to characterize the inhomogeneities in high-quality ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+y}$ single crystals grown by the floating-zone method. Two kinds of characteristic features are found; one with expansion of vortex liquid phase forming arclike regions, and another with linear structures which are observed only in the vortex solid phase. The former is found to be related to the compositional inhomogeneities of the crystal, while the latter is related to the structural defects, most probably small angle grain boundaries. The origin of different evolutions of the vortex liquid region upon increase in the applied field is explained based on the sample-size dependent field profile in the crystal.

There has been considerable recent interest in the vortex state in highly anisotropic high-temperature superconductors in which the orthogonal pancake chain and Josephson vortices interpenetrate to form a crossed lattice. This paper reports magnetization measurements on thin films of conventional superconductors in which the field is applied at a small angle to the film plane. The experimental results demonstrate that the equilibrium vortex lattice structure under this condition consists of a similar interpenetrating lattice of orthogonal Abrikosov vortices and the crossed lattice is calculated to be the lowest-energy configuration.

We report measurements of the low-field structure of the magnetic vortex lattice in an untwinned YBCO single-crystal platelet. Measurements were carried out using an atomic beam magnetic-resonance (ABMR) technique. For a 10.7 G field applied parallel to the c axis of the sample, we find a triangular lattice with orientational order extending across the entire sample. We find the triangular lattice to be weakly distorted by the $a\ensuremath{-}b$ anisotropy of the material and measure a distortion factor, $f=1.16.$ Model-experiment comparisons determine a penetration depth, ${\ensuremath{\lambda}}_{\mathrm{ab}}=140(\ifmmode\pm\else\textpm\fi{}20)\mathrm{nm}.$ The paper includes a detailed description of the ABMR technique. We discuss both technical details of the experiment and modeling used to interpret the measurements.

The vortex phase diagram of optimally doped untwinned YBa 2 Cu 3 O y is studied. We find a first-order transition T L (H) in the vortex liquid above the terminal point H m c p (≃7 T) of both the vortex glass line T g (H) and the field-driven disordering transition line H*(T). The obtained small entropy change (∼0.02 k B /vortex/layer) and the critical endpoint H c e p (≃11 T) of the T L (H) line indicate that the vortex liquid undergoes the vortex slush regime before the solidification into the vortex glass phase. Below H m c p , the vortex liquid phase shows the first-order melting transition into the Bragg glass phase. We also study the oxygen content y dependence of the vortex phase diagram and find that the vortex slush regime is located in the borderline (i.e., 6.90≤y≤6.92) below which the vortex lattice melting transition disappears. The result indicates that the point disorder with the intermediate strength plays an important role in the vortex slush regime.

Angular dependence measurements of local magnetization in Bi 2 Sr 2 CaCu 2 O 8 + y revealed the existence of multiple peak effects peculiar to the hysteresis loops in tilted fields. We discuss the origins of these peak effects in relation to changes of underlying vortex states, and propose a phase diagram for two-component vortex matter composed of pancake and Josephson vortices. Furthermore, systematic studies with changing temperature and oxygen concentration clarified the nature of the breakpoint of linear suppression of the first-order vortex-lattice melting transition, and the second peak field at low temperatures. It is also demonstrated that the two-component vortex matter shows two kinds of penetration field for pancake vortices.

We report observation of the first-order phase transition of vortex lattice in Hg(Re)Ba 2 Ca 2 Cu 3 O y with critical temperature (T c ) over 100 K from in-plane magnetoresistance. The behaviors of H p t for high-T c superconductors could be universally understood in terms of a thermally induced sublimation model in wide ranges of the anisotropy factor (γ 2 ) and T c , and in multilayered systems including triple CuO 2 layers. The estimated value of γ 2 was extremely small (∼500) in spite of possessing large superconducting layer spacing, which is consistent with the resistivity anisotropy (ρ c /ρ a b ) and is comparable with that in nearly optimally doped (La 1 - x Sr x ) 2 CuO 4 .

The magnetic relaxation measurements on a heavily Pb-doped Bi 2 Sr 2 CaCu 2 O 8 + Φ single crystal were carried out in the vicinity of both the anomalous magnetization peak H 2 [ and its onset field H o n at different temperatures. The results show that both the normalized magnetic relaxation rate S(=-d ln M/d In t) and the pinning potential U 0 exhibit remarkable changes near H o n and H 2 [ in the temperature region where H o n (T) follows the relation of the disorder induced vortex lattice transition. H o n (T)=H o n (0)[1-(T/T c ) 4 ] 3 / 2 . For both H H 2 p , S increases and U 0 decreases, respectively, with increasing magnetic fields. A negative power law of U 0 (H)H was observed for both H H 2 p . There exists a minimum in the magnetic-field dependence of S(H) and a maximum in the pinning potential U 0 (H) in the field range between H o n and H 2 p . At lower temperature region where H o n (T) deviates from the above relation, U 0 (H) behaves differently. The results are discussed in terms of both collective flux creep model and plastic flux creep model.

Commensurability effects between vortex distance and crystal lattice constant are investigated by angular dependent magnetization in very pure twinned and twin-free ${\mathrm{NdBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{y}$ single crystals. With increasing temperature the incommensurate states split up and become finally commensurate with half the crystal unit cell. These additional commensurate states are related to a second intrinsic pinning potential within the unit cell or alternatively to a superstructure from distortion of the vortex lattice with respect to the crystal lattice. The role of twins which favor and/or stabilizes the double periodicity is discussed furthermore.

We report muon spin rotation $(\ensuremath{\mu}\mathrm{SR})$ measurements on the moving vortex lattice (VL) in the type-II superconductor Pb-In, backed up by small-angle neutron scattering (SANS) observations on the same sample. We observe a motional narrowing of the $\ensuremath{\mu}\mathrm{SR}$ lineshape $p(B)$ and by SANS, alignment of the VL to the direction of vortex motion. We have calculated the $\ensuremath{\mu}\mathrm{SR}$ lineshape expected with a range of orientations of the moving VL. We demonstrate how the new $\ensuremath{\mu}\mathrm{SR}$ results give information on the moving VL which is complementary and consistent with the SANS data.

We extend the application of the recently developed Landau orbital approach to the fluctuating vortex lattice in two-dimensional (2D) extremely type-II superconductors at high magnetic fields to the region above the melting point. We find that the abrupt incomplete melting of the quasicrystalline vortex phase at T=T m «T c is followed by a broad crossover into an intermediate state with a nematic liquid-crystalline-like order without 2D positional order. A special type of amplitude fluctuations, which can be described as classical transverse vibrations of vortex chains (while vortices are accumulated and depleted alternately in individual chains), controls this crossover. Due to the 1D nature of these vibrations, the structure factor suffers thermal broadening of the diffraction peaks, which remains effective at temperatures well below the mean field T c . The complete melting into an isotropic vortex fluid is shown to occur only far above T m , since the energy cost of the fluctuations which break the chain structure is of the order of the superconducting condensation energy.

We report on magnetization measurements in $({\mathrm{La}}_{0.937}{\mathrm{Sr}}_{0.063}{)}_{2}{\mathrm{CuO}}_{4}$ crystals with the field either parallel or perpendicular to the c axis. A second magnetization peak (``fishtail''), interpreted as indicating a vortex order-disorder transition, is observed in both directions. Differences in the details of the fishtail anomaly in the two directions (width, temperature and time dependence, and history effects) are attributed to the anisotropy and the twin boundaries in the ab plane. The transition fields in both directions, although different in magnitude, exhibit similar qualitative behavior, namely, a strong decrease with temperature in the entire measured range. This unique behavior is explained postulating that both thermally and disordered induced fluctuations contribute to the destruction of the vortex lattice. The resulting ``pinned liquid'' disordered state is characterized by large thermal fluctuations and irreversible magnetic behavior.

The thermal diffusivity of a Bi-2212 polycrystalline sample has been measured under a 1 T magnetic field applied perpendicularly to the heat flux. The magnetic contribution to the heat carrier mean free path has been extracted and is found to behave as a simple power law. This behavior can be attributed to a percolation process of electrons in the vortex lattice created by the magnetic field.

The induction of spin-density wave (SDW) and charge-density wave (CDW) orderings in the mixed state of high-${T}_{c}$ superconductors (HTS) is investigated by using the self-consistent Bogoliubov-de Gennes equations based upon an effective model Hamiltonian with competing SDW and d-wave superconductivity interactions. For optimized doping sample, the modulation of the induced SDW and its associated CDW is determined by the vortex lattice and their patterns obey the fourfold symmetry. By deceasing doping level, both SDW and CDW show quasi-one-dimensional-like behavior, and the CDW has a period just half of that of the SDW along one direction. From the calculation of the local density of states, we found that the majority of the quasi-particles inside the vortex core are localized. All these results are consistent with several recent experiments on HTS.

We report the angle-resolved magnetotransport measurements on ${\mathrm{MgB}}_{2}$ single crystals that exhibit moderate anisotropy (\ensuremath{\gamma}) in upper critical fields with $\ensuremath{\gamma}=2.6\ifmmode\pm\else\textpm\fi{}0.1.$ Unusual ``kink'' features in resistivity are observed, which appear most clearly for field parallel to the c axis. We discuss the origin of the ``kink'' features in relation with the vortex-lattice melting and the recently proposed model of two-gap superconductivity. The influences of anisotropy on superconducting properties including the kink features are also demonstrated.

The electrostatic potential in a superconductor is studied. To this end Bardeen's extension of the Ginzburg-Landau theory to low temperatures is used to derive three Ginzburg-Landau equations - the Maxwell equation for the vector potential, the Schroedinger equation for the wave function and the Poisson equation for the electrostatic potential. The electrostatic and the thermodynamic potential compensate each other to a great extent resulting into an effective potential acting on the superconducting condensate. For the Abrikosov vortex lattice in Niobium, numerical solutions are presented and the different contributions to the electrostatic potential and the related charge distribution are discussed., Comment: 19 pages, 11 figures