Three-dimensional Ginzburg-Landau simulation of a vortex line displaced by a zigzag of pinning spheres

A vortex line is shaped by a zigzag of pinning centers and we study here how far the stretched vortex line is able to follow this path. The pinning center is described by an insulating sphere of coherence length size such that in its surface the de Gennes boundary condition applies. We calculate the free energy density of this system in the framework of the Ginzburg-Landau theory and study the critical displacement beyond which the vortex line is detached from the pinning center.     

Vortex pinning in thin superfluid films

We have formulated the theory of vortex pinning for thin film superfluids. The pinning site are closed regions where the superfluid density vanishes locally. Vortex pinning occurs due to vortex creation and annihilation at the boundaries of these regions and dominates pair processes in thicker films because of the cost of creating longer lengths of vortex line.     

Line vortices in a three-dimensional ordered Josephson medium at a small pinning parameter

The structure and energy of a line vortex whose axis is aligned with the symmetry axis of a finite-thickness slab indefinitely long in two directions is calculated by solving a set of linear finite-difference equations. Fluxoid quantization conditions in cells near the center of the vortex serve as boundary conditions. An exact solution is approached by iterations in phase stepwise discontinuities that cannot be considered small. A close similarity between the configuration under study and a periodic sequence (chain) of vortices makes it possible to allow for the effect of the domain boundary on the structure and energy of the vortex. It is shown that, at any width of the slab, one can find a pinning parameter value so small that the vortex cannot be viewed as solitary and contributions from other vortices should be taken into account in calculation. Proceeding in this way, one can find the structure and energy of the vortex however small the pinning parameter is. The total energy of the vortex is its intrinsic energy plus the sum of its energies of interaction with other members of the chain. In turn, the intrinsic energy is the sum of the energies of the small discrete core and quasi-continuous outer shell. It is demonstrated that the energy of the core is a linear function of the pinning parameter and is comparable to the energy of the shell.     

The volume pinning force for periodical interaction forces between defects and the flux line lattice

The volume pinning force for some forms of the interaction potential defect-flux line is calculated without restrictions on the vortex lattice distance and the interaction range of defects. It is shown that for larger maximum elementary interaction forces, the direct summation of pinning forces is realistic. However, if the interaction range of the defects is smaller than the vortex lattice distance, one obtains a region (about one order of magnitude) in which Labusch's quadratic dependence of the volume pinning force on the elementary interaction force is valid. In the region where the direct summation of pinning forces occur, the volume pinning force is proportional the vortex lattice distance and one obtains an additional magnetic field dependence of the volume pinning force.     

Effects of kinked linear defects on planar flux line arrays

In the hard core limit, interacting vortices in planar type II superconductors can be modeled as non-interacting one dimensional fermions propagating in imaginary time. We use this analogy to derive analytical expressions for the probability density and imaginary current of vortex lines interacting with an isolated bent line defect and to understand the pinning properties of such systems. When there is an abrupt change of the direction of the pinning defect, we find a sinusoidal modulation of the vortex density in directions both parallel and perpendicular to the defect.     

Shifting and pinning of a magnetic vortex core in a permalloy dot by a magnetic field

Magnetic pinning in thin films seems to be a major research subject in the near future, as it is involved in all switching processes which include a movement of a domain wall or a magnetic vortex. We used Lorentz transmission electron microscopy and vortex pinning at artificial pinning sites to investigate the pinning behavior of magnetic vortices for the first time with high spatial resolution.     

Critical vortex line length near a zigzag of pinning centers

A vortex line passes through as many pinning centers as possible on its way from one extremety of the superconductor to the other at the expense of increasing its self-energy. In the framework of the Ginzburg-Landau theory we study the relative growth in length, with respect to the straight line, of a vortex near a zigzag of defects. The defects are insulating pinning spheres that form a three-dimensional cubic array embedded in the superconductor. We determine the depinning transition beyond which the vortex line no longer follows the critical zigzag path of defects.Received: 23 July 2004, Published online: 26 November 2004PACS: 74.80.-g Spatially inhomogeneous structures - 74.25.-q General properties; correlations between physical properties in normal and superconducting states - 74.20.De Phenomenological theories (two-fluid, Ginzburg-Landau, etc.)     

Ordered states and structural transitions in a system of Abrikosov vortices with periodic pinning

A system of Abrikosov vortices in a quasi-two-dimensional HTSC plate is considered for various periodic lattices of pinning centers. The magnetization and equilibrium configurations of the vortex density for various values of external magnetic field and temperature are calculated using the Monte Carlo method. It is found that the interaction of the vortex system with the periodic lattice of pinning centers leads to the formation of various ordered vortex states through which the vortex system passes upon an increase or a decrease in the magnetic field. It is shown that ordered vortex states, as well as magnetic field screening processes, are responsible for the emergence of clearly manifested peaks on the magnetization curves. Extended pinning centers and the effect of multiple trapping of vortices on the behavior of magnetization are considered. Melting and crystallization of the vortex system under the periodic pinning conditions are investigated. It is found that the vortex system can crystallize upon heating in the case of periodic pinning.     

Influence of the pinning of Abrikosov vortices on the propagation of surface magnetostatic waves in a ferromagnet-superconductor structure

The influence of surface-layer vortex pinning in a type-II superconductor on the propagation of surface magnetostatic waves in a ferromagnet-superconductor structure is analyzed. The pinning is assumed to be strong enough to prevent vortex displacement under the influence of the Lorentz force generated by the surface magnetostatic waves, so that the ground state of the superconductor is determined by the elastic properties of the vortex lattice and by pinning. In the given model the problem reduces to the analysis of the wave spectrum in the scattered field created by the disordered vortex surface layer. A calculation shows that the influence of this field on the surface magnetostatic-wave spectrum is slight and, hence, degradation of the shielding properties of the superconductor does not take place in the presence of strong vortex pinning (as opposed to the ferromagnet-ideal superconductor structure). Fiz. Tverd. Tela (St. Petersburg) 40, 32–35 (January 1998)     

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.     

Dynamical phase transition of a driven vortex lattice with disordered pinning

We have numerically solved the overdamped equation of vortex motion in a two-dimensional driven vortex lattice with disordered pinning, in which the driving Lorentz force, the pinning force due to point defects, the intervortex interacting force, and the thermal fluctuation force are taken into account. It is found that the vortex density and pinning strength are two important factors of affecting the melting transition of a vortex lattice. At low magnetic fields, there exist hysteresis loops of the average vortex velocity and the average pinning force vs. the driving force, from which the feature of a first-order melting transition of the vortex motion can be clearly seen. As the magnetic field is increased beyond a critical value, the hysteresis loops disappear and the melting transition is replaced by a second-order glass transition. We have also studied the influence of intervortex interactions on the vortex melting transition by comparing several forms of repulsive forces between the vortices.     

Dynamics of two-dimensional vortex system in a strong square pinning array at the second matching field

We study the dynamics of a two-dimensional vortex system in a strong square pinning array at the second matching field. Two kinds of depinning behaviors, a continuous depinning transition at weak pinning and a discontinuous one at strong pinning, are found. We show that the two different kinds of vortex depinning transitions can be identified in transport as a function of the pinning strength and temperature. Moreover, interstitial vortex state can be probed from the transport properties of vortices.     

Rectified vortex motion in an Nb film with a spacing-graded array of holes

Superconducting Nb thin films with a spacing-graded array of holes were prepared by electron beam lithography. Two films with different hole gradients were fabricated. The ac-driven vortices were investigated in Nb superconductors with a spacing-graded array of holes. The measurements revealed pronounced rectified voltage when the vortex lattice is driven by an ac injected current. The rectified voltage is mainly caused by the strength of the vortex–vortex interaction. The rectified motion of a vortex is affected by the pinning potential of the spacing-graded array and the applied magnetic field. The vortex–vortex interaction strength changes the effective pinning landscape of the vortices and an asymmetric potential is formed. Vortices depin easily from high concentration to low concentration of pinning sites. In both samples, the ac-driven vortices exhibit a variety of dynamical responses and the rectified voltage is tunable with the applied magnetic field.     

Dynamics of a pinned magnetic vortex

We observe the dynamics of a single magnetic vortex pinned by a defect in a ferromagnetic film. At low excitation amplitudes, the vortex core gyrates about its equilibrium position with a frequency that is characteristic of a single pinning site. At high amplitudes, the frequency of gyration is determined by the magnetostatic energy of the entire vortex, which is confined in a micron-scale disk. We observe a sharp transition between these two amplitude regimes that is due to depinning of the vortex core from a local defect. The distribution of pinning sites is determined by mapping fluctuations in the frequency as the vortex core is displaced by a static in-plane magnetic field.     

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.     

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.     

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.     

Weak to strong pinning crossover

Material defects in hard type II superconductors pin the flux lines and thus establish the dissipation-free current transport in the presence of a finite magnetic field. Depending on the density and pinning force of the defects and the vortex density, pinning is either weak collective or strong. We analyze the weak to strong pinning crossover of vortex matter in disordered superconductors and discuss the peak effect appearing naturally in this context.     

Vortex mass in a superfluid at low frequencies

An inertial mass of a vortex can be calculated by driving it around in a circle with a steadily revolving pinning potential. We show that in the low-frequency limit this gives precisely the same formula that was used by Baym and Chandler, but find that the result is not unique and depends on the force field used to cause the acceleration. We apply this method to the Gross-Pitaevskii model, and derive a simple formula for the vortex mass. We study both the long-range and short-range properties of the solution. We agree with earlier results that the nonzero compressibility leads to a divergent mass. From the short-range behavior of the solution we find that the mass is sensitive to the form of the pinning potential, and diverges logarithmically when the radius of this potential tends to zero.     

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.