Anomalous vortex dynamics in

We report on measurements of current–voltage (I–V) characteristics for YNi₂B₂C single crystals with weak pinning in various fields at 7.6 K. We find nonmonotonic, N-shaped I–V curves in a certain field region deep in the vortex solid phase. This behavior is anomalous, since there exists an intermediate I region where flow voltage V shows a decrease with increasing I (a driving force). While the exact nature remains unknown, this phenomenon suggests vortex motion (driving I) induced pinning.     

Absence of an elastic vortex glass in disordered HTS

Analyses of standard current–voltage (I–V) characteristics of disordered high-temperature superconductors (HTS) indicate that the vortex phase at high magnetic fields H should be an elastic vortex glass, where the vortex pinning barriers diverge at low current densities, whereas dc magnetization relaxation measurements reveal the presence of nondiverging (plastic) pinning barriers in a wide H–T domain. We show that the different conclusions concerning the nature of the vortex phase at high H in disordered HTS seem to be due to the ordering effect of the driving force existing in various experiments.     

Pinning in a porous high-temperature superconductor Bi2223

The current-voltage characteristics of a porous superconductor Bi₂Sr₂Ca₂Cu₃O _x (Bi2223) have been measured at temperatures in the range from 10 to 90 K in magnetic fields of 0–80 kOe. The experimental dependences have been analyzed within the model allowing for pinning by clusters of a normal phase with fractal boundaries, as well as the model taking into account phase transformations of vortex matter. It has been found that the electrical resistance of the superconductor material significantly increases at temperatures of 60–70 K over the entire range of magnetic fields under consideration without changing in the sign of the curvature of the dependence R(I). It has been assumed that this behavior is associated with the specific feature of the pinning in a highly porous high-temperature superconductor, which lies in the fractality of the distribution of pinning centers in a wide range of self-similarity scales. The studied material at the aforementioned temperatures is characterized by the melting of the vortex structure.     

Collective multivortex states in periodic arrays of traps

We examine the vortex states in a 2D superconductor interacting with a square array of pinning sites. As a function of increasing pinning size or strength we find a series of novel phases including multivortex and composite superlattice states such as aligned dimer and trimer configurations at individual pinning sites. Interactions of the vortices give rise to an orientational ordering of the internal vortex structures in each pinning site. We also show that these vortex states can give rise to a multistage melting behavior.     

Interaction and pinning of plane vortices in a three-dimensional Josephson medium and possible distances between two isolated vortices

Within a continuous vortex model, exact expressions are obtained for the Josephson and magnetic energies of plane (laminar) vortices, as well as for the energy and force of pinning by cells in a three-dimensional Josephson medium. If the porosity of the medium is taken into account, the Josephson and magnetic energies of the vortex differ from those for the continuum case. The contributions to the pinning energy from the Josephson and magnetic energies have opposite signs. An algorithm for numerically solving a system of difference equations is proposed in order to find the shape and the energy of the vortex in its stable and unstable states. The continuous vortex model is shown to fail in predicting correct values of the Josephson and magnetic energy of the vortex, as well as of the pinning energy components. Expressions for the least possible distances between two isolated vortices are obtained for a small pinning parameter. Analytical results are in close agreement with computer simulation. An algorithm for numerically solving a system of difference equations is proposed in order to find the least possible distances between two isolated vortices when the pinning parameter I is not small. The minimal value of I at which the center-to-center distance N of the vortices equals three cells is 1.428; for N=2, I _min=1.947. At I>2.907, the vortices can be centered in adjacent cells.     

Excess current in point contacts on two-band superconductor MgB2 in magnetic field

A series of I(V) characteristics and bias-dependent differential resistance dV/dI(V) curves of point contacts made between a single crystal of two-band superconductor MgB₂ and Cu were measured in magnetic fields up to 9 T. The magnetic field dependences of the excess current in the I(V) curves were obtained and analyzed using Koshelev and Golubov's [Phys. Rev. Lett. 90, 177002 (2003)] theoretical results for the mixed state of a dirty two-band superconductor. Introducing a simple model for the excess current in the point contact in the mixed state, our data can be qualitatively described using the theoretical magnetic field dependence of the superconducting order parameter of the σ and π-bands and the averaged electronic density of states in MgB₂.     

Comparison on plasticity of current-driven vortex lattices in as-grown and heavy ion irradiated microbridge Bi2Sr2Ca1Cu2O8+δ single crystals

From the I-V characteristics for as-grown and irradiated Bi₂Sr₂CaCu₂O_8+δ single crystals at T=5 K in a magnetic field applied parallel to the c axis, we have seen two types of vortex dynamics near the depinning threshold. For the as-grown sample, at low field, the I-V curves show steps that clearly indicate a “fingerprint phenomenon” since they reflect the current dependence of differential resistance R_d=dV/dI. This can be ascribed to vortices flow through uncorrelated channels for the highly defective lattice. As fields sufficiently increase, these peaks merge, giving broader ones, indicating a crossover from filamentary strings to braid river. In contrast, in the irradiated sample, the pinning is found to be individual at low magnetic fields and collective when the vortex-vortex interactions are involved. Our result suggests a dynamic nature of the peak effect, in agreement with recent numerical simulations and experimental works.     

Different methods of calculating the pinning energy of plane vortices in a 3D Josephson medium: A comparative study

The pinning energy of plane (laminar) vortices in a 3D Josephson medium is calculated within a continuous vortex model considering functions of two types: V=1−cosϕ and V= 2/π⁴ϕ²(2π−ϕ)². The shape and energy of the stable and unstable vortices are found with an algorithm for the exact numerical solution of a set of difference equations. The vortex magnetic and Josephson energies diverge. The magnetic and Josephson components of the pinning energy are close in magnitude but differ in sign; as a result, the total pinning energy is smaller than its components by one order of magnitude. This result is confirmed analytically. An analytical computing method within the continuous vortex model is suggested. This method preserves the difference terms in the energy expression. The magnetic energy found by this method differs from the Josephson energy in magnitude, and the magnetic component of the pinning energy is opposite in sign to the Josephson component. Comparative analysis of the approximate approaches to energy calculation within the continuous vortex model when the difference terms are retained and when they are replaced by derivatives is performed. It is shown that the continuous vortex model gives incorrect values of the Josephson and magnetic components of the pinning energy. The actual values are several tens or several hundreds of times higher than those obtained with the continuous vortex model. Yet, since the Josephson and magnetic components of the pinning energy have different signs, the exact value of the total pinning energy and the approximate value obtained within the continuous vortex model differ insignificantly.     

Equilibrium states of planar vortices in a three-dimensional josephson medium and meaning of the pinning-energy concept

Various ways of specifying the pinning-energy concept for planar vortices in a three-dimensional cellular Josephson medium are analyzed. It is shown that, for values of the pinning parameter I that are not small, a universal characteristic of vortex interaction with the lattice cannot be found, since the displacement of a vortex distorts its shape. At small values of I, the maximum pinning force can be chosen for such a characteristic. Two equilibrium states of a vortex are analyzed for stability. It is revealed that the state of higher energy is not inevitably unstable. A correct analysis of stability must be based on exploring a quadratic form that describes the energy of a current configuration. Such an investigation is performed for the equilibrium state of a vortex. At small values of the pinning parameter, the vortex state of higher energy is quasistable.     

Dynamic magnetic permeability of a thin, high-T c superconducting wafer

The field dependence of the vibrational contribution to the dynamic magnetic permeability μ _V(H) is calculated for a thin (of thickness d∼λ) high-T _c superconducting wafer in a magnetic field parallel to the surface. The resulting curves are plotted on the basis of an exact numerical analysis of the vortex structures both for the thermodynamic-equilibrium vortex lattice and in the presence of pinning forces and the Bean-Livingston surface barrier. It is shown that the μ _V(H) curves are highly sensitive to the size factor (d/λ) and exhibit abrupt changes corresponding to a change in the number of vortex rows. The equilibrium μ _V(H) curve is found to be similar in its general behavior and absolute value (obtained with allowance for the distribution of grain sizes and with appropriate values of λ and ϰ) to the experimental μ _V(H) curve plotted at nitrogen temperature for fine-grained YBa₂Cu₃O_x with grain diameters 〈D〉∼λ in an increasing magnetic field. It is established that the main cause of the experimentally observed irreversible behavior of the μ _V(H) curves during cyclic variation of the applied magnetic field is the existence of a surface barrier to the exit of vortices from the superconductor. The lower limit H _min(B) of stability of the mixed state in the presence of an ideal surface barrier in a thin, high-T _c superconducting wafer (d∼λ) is determined, along with the range of the vortex state (H _max-H _min) for a fixed number of vortices in micrometer-size grains of the investigated YBaCuO samples. Fiz. Tverd. Tela (St. Petersburg) 39, 1943–1947 (November 1997)     

Pinning of pancake vortices in a three-dimensional Josephson medium and structure of the vortex lattice in the “critical” state

A system of pancake vortices formed near the boundary of a sample in a monotonically increasing external magnetic field is calculated with allowance for pinning due to the cellular structure of the medium for various values of the pinning parameter I, which is proportional to the critical current of the junction and the cell diameter. The shortest distance from the outermost vortex to the nearest neighbor is proportional to I ⁻¹¹. It is shown that the pinning parameter has a critical value I _c separating two regimes with different types of critical states. For I<I _c the external magnetic field has a threshold value H _t(I), above which the field immediately penetrates the interior of the junction to an infinite distance. For I>I _c the magnetic field decays linearly from the boundary into the interior of the junction. The value obtained in the study, I _c=3.369, differs from the value of 0.9716 postulated by other authors. The dependence of the slope of the magnetic field profile near the boundary on I is determined. It is shown that the slope is independent of I in intervals 2πk<I<2πk+π. Fiz. Tverd. Tela (St. Petersburg) 39, 1958–1963 (November 1997)     

Vortex lattice matching effects in a washboard pinning potential induced by Co nanostripe arrays

An advanced mask-less nanofabrication technique, focused electron beam-induced deposition (FEBID), has been employed on epitaxial Nb thin films to prepare ferromagnetic decorations in the form of an array of Co stripes. These substantially modify the non-patterned films’ superconducting properties, providing a washboard-like pinning potential landscape for the vortex motion. At small magnetic fields B ? 0.1 T, vortex lattice matching effects have been investigated by magneto-transport measurements. Step-like drops in the field dependencies of the films resistivity ρ(B) have been observed in particular for the vortex motion perpendicular to the Co stripes. The field values, corresponding to the middle points of these drops in ρ(B), meet the vortex lattice parameter matching the pinning structure’s period. These disagree with the results of Jaque et al. (2002) [11], who observed matching effects corresponding to the stripe width in Nb films grown on periodically distributed submicrometric lines of Ni.     

Vortex lattice ordering in the flux flow state of Nb thin films

We measure current–voltage characteristics at high driving currents for different magnetic fields and temperatures in Nb thin films of rather strong pinning. In a definite range of the B–T phase diagram we find that a current induced transition occurs in the flux flow motion of the vortex lattice, namely a dynamic ordering (DO). Contrary to the case of weaker pinning materials, DO is observed only at low fields, due to the stronger intrinsic disorder that can deform plastically the moving vortex lattice even for small applied fields.     

Two-Dimensional Coulomb Glass as a Model for Vortex Pinning in Superconducting Films

A glass model of vortex pinning in highly disordered thin superconducting films in magnetic fields B ≪ H_c2 at low temperatures is proposed. Strong collective pinning of a vortex system realized in disordered superconductors that are close to the quantum phase transition to the insulating phase, such as InO_x, NbN, TiN, MoGe, and nanogranular aluminum, is considered theoretically for the first time. Utilizing the replica trick developed for the spin glass theory, we demonstrate that such vortex system is in non-ergodic state of glass type with a large kinetic inductance per square L_K. The distribution function of local pinning energies is calculated, and it is shown that it possesses a wide gap; i.e., the probability to find a weakly pinned vortex is extremely low.

     

Ordering of the vortex lattice in Mo-Re films

We investigated the vortex lattice ordering induced by an applied current in thick (700-900 nm) Mo-Re films with strong pinning. Measurements of I-V characteristics as a function of field and at different temperatures were carried out. We found that, as in the case of weakly pinning amorphous samples, dynamic ordering can occur only if the size of the vortex correlated region is at least two times the intervortex distance. Received 20 July 2001     

Line vortices in a three-dimensional ordered Josephson medium

Two equilibrium configurations of a line vortex in a three-dimensional ordered Josephson medium are considered: (i) the vortex core is at the center of a cell and (ii) the vortex core is on a contact. Infinite systems of equations describing these configurations are derived. In going to a finite system, the currents far away from the center are neglected. A new technique for solving the finite system of equations is suggested. It does not require smallness of phase discontinuities at all vortex cells and, therefore, can be applied for any values of pinning parameter I down to zero. The structures and energies of both equilibrium states for isolated line vortices are calculated for any I from the range considered. For I >0.3, a vortex can be thought of as fitting a square of 5×5 cells. For lower I, the vortex energy can be expressed as a sum of the energies of the small discrete core and the quasi-continuous outside. The core energy is comparable to the energy of the outside and is a major contributor to the vortex energy when I is not too small. For any I, the energy of the vortex centered on the contact is higher than the energy of the configuration centered at the center of the cell.     

Angular dependence of vortex matter phase transition in MgB2 single crystal

The vortex matter phase transitions and intrinsic pinning effect were investigated in an MgB₂ single crystal using the torque magnetometry. For the field directions apart from the ab plane, we succeed in the observation of the vortex lattice melting transitions, which are transformed from the order-disorder transitions at low temperatures. Both transition fields with field directions can be describe by the GL effective mass model. For the field direction along the ab plane, these transitions become unobservable. Instead, the sudden increase in the hysteresis of magnetization curve occurs, indicating the existence of the intrinsic pinning coming from the layer structure.     

磁通密度对第Ⅱ类超导体磁通动力学的影响

计算了二维无序钉扎系统中磁通运动的平均速度、微分电阻、纵向电压噪声和静态结构因子.通过在不同磁通密度下的磁通运动形式,给出了磁通运动的动力学相图.研究表明,磁通晶格存在钉扎相、塑性流相、近晶流相,和运动玻璃相.在运动玻璃相中,随着驱动力的进一步增加,横向玻璃态和运动Bragg玻璃态相继出现.磁通密度增大有利于有序相的出现.当磁通密度增大到一定程度时,近晶流动相会消失.磁通运动随着外加驱动电流增大发生从塑性流动相到运动玻璃相的转变 关键词： Ⅱ类超导体 磁通动力学 运动玻璃     

Scanning tunneling microscopy study of pinning-induced vortex lattice distortion in ion-irradiated NbSe2

We observe vortex pinning in 2.2 GeV Au-ion irradiated NbSe₂ by scanning tunneling microscopy (STM) at 3K. The ion irradiation generates columnar defects which act as pinning sites. At various external magnetic fields the vortex arrangement is clearly resolved but shows strong distortion. The location of individual defects is extracted from STM data and compared to the vortex arrangement.     

Turbulent vortex rings in weakly turbulent and in turbulent ambient flow

Turbulent vortex rings were investigated in weakly turbulent flow and in three different grid generated turbulent flows to clarify the reciprocal action of the vortex ring with defined external turbulence. Assuming self-similarity and turbulent viscosity as proportional to V₀D₀ the equations for the ring diameter D(t) and the velocity of propagation V(t) were derived. The time difference Δt between the virtual origins of 1/V(t) respectively D²(t) led to an invariant term. The equation of momentum is fulfilled. – Position and diameter of the vortex rings were determined till their decay by means of an optical system, which did not disturb the vortex rings. The experimental results in weakly turbulent ambient flow obtained by the author and by others confirm the theory very well. The ambient turbulence was nearly constant in the measuring region; its effect could be described by simply adding its viscosity to the vortex ring’s internal turbulent viscosity. The results could be represented in unified non-dimensional diagrams. Moreover, an explanation was found as to why the mean internal turbulent viscosity is constant.