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

摘要

A variational model is proposed to describe the magnetic properties oftype-II superconductors in the entire field range between $H_{c1}$ and $H_{c2}$for any values of the Ginzburg-Landau parameter $\kappa>1/\sqrt{2}$. Thehexagonal unit cell of the triangular flux-line lattice is replaced by a circleof the same area, and the periodic solutions to the Ginzburg-Landau equationswithin this cell are approximated by rotationally symmetric solutions. TheGinzburg-Landau equations are solved by a trial function for the orderparameter. The calculated spatial distributions of the order parameter and themagnetic field are compared with the corresponding distributions obtained bynumerical solution of the Ginzburg-Landau equations. The comparison revealsgood agreement with an accuracy of a few percent for all $\kappa$ valuesexceeding $\kappa \approx 1$. The model can be extended to anisotropicsuperconductors when the vortices are directed along one of the principal axes.The reversible magnetization curve is calculated and an analytical formula forthe magnetization is proposed. At low fields, the theory reduces to the Londonapproach at $\kappa \gg 1$, provided that the exact value of $H_{c1}$ is used.At high fields, our model reproduces the main features of the well-knownAbrikosov theory. The magnetic field dependences of the reversiblemagnetization found numerically and by our variational method practicallycoincide. The model also refines the limits of some approximations which havebeen widely used. The calculated magnetization curves are in a good agreementwith experimental data on high-T$_c$ superconductors.