Vortex configurations in high-{Tc} superconducting films (original) (raw)
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Nonlinearity, 1997
The time-dependent Ginzburg-Landau equations (TDGL) model a thin-film superconductor of finite size placed under a magnetic field. For numerical computation, we use a staggered grid discretization, a technique well known in numerical fluid mechanics. Some properties of the solutions are established. An efficient explicit-implicit method based on the forward Euler method is developed. In our simulations, we impose natural boundary conditions at the edge of the superconductor. With suitable choices of parameters (corresponding to physical superconductors of type II) and the strength of the external magnetic field, the steady-state solutions exhibit vortices. When a variable strength magnetic field, simulating a transient current, is introduced, we observe motion of the vortices in a periodic pattern.
Vortex-vortex interaction in bulk superconductors: Ginzburg-Landau theory
Physical Review B, 2011
The vortex-vortex interaction potential in bulk superconductors is calculated within the Ginzburg-Landau (GL) theory and is obtained from a numerical solution of a set of two coupled non-linear GL differential equations for the vector potential and the superconducting order parameter, where the merger of vortices into a giant vortex is allowed. Further, the interaction potentials between a vortex and a giant vortex and between a vortex and an antivortex are obtained for both type-I and type-II superconductors. Our numerical results agree asymptotically with the analytical expressions for large inter-vortex separations which are available in the literature. We propose new empirical expressions valid over the full interaction range, which are fitted to our numerical data for different values of the GL parameter.
Physical Review B, 2006
Using the non-linear Ginzburg-Landau (GL) theory, we obtain the possible vortex configurations in superconducting thin films containing a square lattice of antidots. The equilibrium structural phase diagram is constructed which gives the different ground-state vortex configurations as function of the size and periodicity of the antidots for a given effective GL parameter κ *. Giant-vortex states, combination of giant-and multi-vortex states, as well as symmetry imposed vortex-antivortex states are found to be the ground state for particular geometrical parameters of the sample. The antidot occupation number no is calculated as a function of related parameters and comparison with existing expressions for the saturation number ns and with experimental results is given. For a small radius of antidots a triangular vortex lattice is obtained, where some of the vortices are pinned by the antidots and some of them are located between them. Transition between the square pinned and triangular vortex lattices is given for different values of the applied field. The enhanced critical current at integer and rational matching fields is found, where the level of enhancement at given magnetic field directly depends on the vortex-occupation number of the antidots. For certain parameters of the antidot lattice and/or temperature the critical current is found to be larger for higher magnetic fields. Superconducting/normal H − T phase boundary exhibits different regimes as antidots are made larger, and we transit from a plain superconducting film to a thin-wire superconducting network. Presented results are in good agreement with available experiments and suggest possible new experiments.
(Giant) Vortex-(anti) vortex interaction in bulk superconductors: The Ginzburg-Landau theory
Arxiv preprint arXiv: …, 2010
The vortex-vortex interaction potential in bulk superconductors is calculated within the Ginzburg-Landau (GL) theory and is obtained from a numerical solution of a set of two coupled non-linear GL differential equations for the vector potential and the superconducting order parameter, where the merger of vortices into a giant vortex is allowed. Further, the interaction potentials between a vortex and a giant vortex and between a vortex and an antivortex are obtained for both type-I and type-II superconductors. Our numerical results agree asymptotically with the analytical expressions for large inter-vortex separations which are available in the literature. We propose new empirical expressions valid over the full interaction range, which are fitted to our numerical data for different values of the GL parameter.
Numerical Simulation of Vortex Dynamics in Type-II Superconductors
Journal of Computational Physics, 1996
This article describes the results of several numerical simulations of vortex dynamics in type-II superconductors. The underlying mathematical model is the time-dependent Ginzburg-Landau model. The simulations concern vortex penetration in the presence of twin boundaries, interface patterns between regions of opposite vortex orientation, and magnetic-ux entry patterns in superconducting samples. r r A = ? 4 c 1 c @A @t + r + 4 c J s :
Use of the modified Ginzburg–Landau equations in high temperature superconductors
physica status solidi (b), 2005
The modified Ginzburg–Landau equations are used to study some selected problems for the high temperature superconductors. The derivations of modified Ginzburg–Landau equations are briefly demonstrated by taking into consideration the layered features of the sample according to the relation between the coherence length and the separation distance of the layers. The domain wall energy, the maximum supercurrent and the parallel critical field of high temperature superconducting thin film were evaluated. Comparison with previous works is given. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
On Ginzburg–Landau Vortices of Superconducting Thin Films
Acta Mathematica Sinica, English Series, 2006
In this paper, we discuss the vortex structure of the superconducting thin films placed in a magnetic field. We show that the global minimizer of the functional modelling the superconducting thin films has a bounded number of vortices when the applied magnetic field h ex < H c 1 + K log | log ε| where H c 1 is the lower critical field of the film obtained by Ding and Du in SIAM J. Math. Anal., 2002. The locations of the vortices are also given.
Journal of Physics-condensed Matter, 1998
We extend our previous one-dimensional Ginzburg-Landau calculations of the pinning energy of vortices to two dimensions, in order to achieve an understanding of the pinning forces exerted on vortices by defects. By minimizing the free energy using a relaxation scheme, we obtain the spatial variation of the order parameter and supercurrents for a vortex in the vicinity of a cylindrical defect in an extreme type-II superconductor. The resulting twodimensional field distributions provide a direct mapping of the spatial dependence of the vortexdefect pinning potential, thereby yielding the pinning force and depinning current as a function of the defect size and magnetic field. We also use periodic boundary conditions in the twodimensional Ginzburg-Landau equations to solve for the known vortex-vortex interaction, in order to verify the resolution and accuracy of our approach for extreme type-II superconductors. Our direct numerical derivation of the pinning force per vortex is shown to be applicable to a wide range of magnetic fields and columnar-defect densities, and the calculated results are consistent with experimental observation.
Ginzburg-Landau theory of type II superconductors in magnetic field
Reviews of modern physics, 2010
Thermodynamics of type II superconductors in electromagnetic field based on the Ginzburg -Landau theory is presented. The Abrikosov flux lattice solution is derived using an expansion in a parameter characterizing the "distance" to the superconductor -normal phase transition line. The expansion allows a systematic improvement of the solution. The phase diagram of the vortex matter in magnetic field is determined in detail. In the presence of significant thermal fluctuations on the mesoscopic scale (for example in high Tc materials) the vortex crystal melts into a vortex liquid. A quantitative theory of thermal fluctuations using the lowest Landau level approximation is given. It allows to determine the melting line and discontinuities at melt, as well as important characteristics of the vortex liquid state. In the presence of quenched disorder (pinning) the vortex matter acquires certain "glassy" properties. The irreversibility line and static properties of the vortex glass state are studied using the "replica" method. Most of the analytical methods are introduced and presented in some detail. Various quantitative and qualitative features are compared to experiments in type II superconductors, although the use of a rather universal Ginzburg -Landau theory is not restricted to superconductivity and can be applied with certain adjustments to other physical systems, for example rotating Bose -Einstein condensate. A. GL equations. 49 B. Theory of thermal fluctuations in GL model 50 C. The effects of quenched disorder 52 D. Other fields of physics 53 E. Acknowledgments 53 VI. Appendices 53 A. Integrals of products of the quasimomentum eigenfunctions 53 1. Rhombic lattice quasimomentum functions 53 2. The basic Fourier transform formulas 53 3. Calculation of the β k , γ k functions and their small momentum expansion 54 B. Parisi algebra for hierarchial matrices 56
Physica C: Superconductivity, 2000
The first integrals of the Ginzburg-Landau equations for a vortex-free state of superconductors with different mixed symmetries of the order parameter are found. The general boundary conditions for the order parameter at the ideal interface between the superconductor and vacuum are derived. Based on these integrals and boundary conditions, we analyze the stability criteria for vortex-free state in unconventional superconductors. The threshold field s H above which the Abrikosov vortices can enter the superconductor is found to be higher or equal to the thermodynamic critical field for all states under study.