Approximate Ginzburg-Landau solution for the regular flux-line lattice: Circular cell method

Abstract

A variational model is proposed to describe the magnetic properties of type-II superconductors in the entire field range between Hc1 and Hc2 for any values of the Ginzburg-Landau parameter κ > 1/ √ 2. The hexagonal unit cell of the triangular flux-line lattice is replaced by a circle of the same area, and the periodic solutions to the Ginzburg-Landau equations within this cell are approximated by rotationally symmetric solutions. The Ginzburg-Landau equations are solved by a trial function for the order parameter. The calculated spatial distributions of the order parameter and the magnetic field are compared with the corresponding distributions obtained by numerical solution of the Ginzburg-Landau equations. The comparison reveals good agreement with an accuracy of a few percent for all κ values exceeding κ ≈ 1. The model can be extended to anisotropic superconductors when the vortices are directed along one of the principal axes. The reversible magnetization curve is calculated and an analytical formula for the magnetization is proposed. At low fields, the theory reduces to the London approach at κ ≫ 1, provided that the exact value of Hc1 is used. At high fields, our model reproduces the main features of the well-known Abrikosov theory. The magnetic field dependences of the reversible magnetization found numerically and by our variational method practically coincide. The model also refines the limits of some approximations which have been widely used. The calculated magnetization curves are in a good agreement with experimental data on high-Tc superconductors.

Accurate Symbolic Solution of Ginzburg-Landau Equations in the Circular Cell Approximation by Variational Method: Magnetization of Ideal Type II Superconductor

“…On the other hand, Equation (20) of [9] is a substantial improvement of corresponding Equation (14) of [6] as it is explained in [8] [9] [10] (note that the corresponding Equation (11) [8] contains a typing error, the correct equations are: (2.16) in [10], (20) in [9] and Equation (16) …”

“…It should be noted that this correction of the magnetisation uses the symbolic form of the theoretical result [7] at 1 b → and therefore the solution obtained here remains self-sufficient (even though it now uses two solutions of GL equations: [8] [9] [10] and [7]). So far I did not use any numerical results (the fact that calculated numerically magnetisation m 2 also agree with the conditions of Equations (27), (28) only means that these conditions are just).…”

“…At medium magnetic fields a symbolic approximation considering vortex lattice of circular unit cells is developed [5] [6] [8] [10]. The drawbacks of [5] [6] are dealt with in [8] [9] [10] and will not be discussed here. As stated in [8] [9] [10], their method approximately describes magnetic properties of type II superconductors with periodic lattice over the entire range of magnetic fields and for any value of the GL parameter with λ, ξ London penetration depth and the coherence length respectively.…”

“…Clearly, the accuracy of the obtained solution of GL equations is vital: a symbolic one only has added value when its accuracy is comparable to that of the numerics. In this paper I will access the degree of approximation of the method [8] [9] [10], refine it in a clear, concise and rational way, identify and eliminate the errors. Furthermore, I will improve accuracy of the method by providing for the first time accurate dependencies of the variational parameters; correct and calculate magnetisation, compare it with the one calculated numerically and conclude they agree for any k within 98.5% or better at magnetic field …”

Vortex phases in mesoscopic cylinders with suppressed surface superconductivity

“…Instead of the straightforward integration of this equation, it is possible to use trial functions for the coordinate dependence f Lj (r). Notice that different variational procedures allowing one to solve approximately the Ginzburg-Landau equations were used in numerous papers for mesoscopic [7,10,15], bulk [28][29][30][31], and different-shaped [32] superconductors. One can easily show that if Eq.…”