Numerical simulation of vortex arrays in thin superconducting films

Abstract

Numerical simulations of the evolution of the order parameter and the vector potential in thin type-II superconducting films are reported. The theoretical framework is provided by the well-known time-dependent Ginzburg-Landau (TDGL) equations coupled with the Maxwell equations. The external field is applied parallel to the surfaces. Several maxima appear in the magnetization curve, a phenomenon that has been observed in experiments and up to now only explained using a London approach. It is proved that these maxima are indeed predicted by the fu11 TDGL approach, and are not necessarily linked with structural changes in the vortex lattice. A mechanism for the appearance of magnetization maxima in finite samples is identified, based on the behavior of surface supercurrents.The magnetization of a thin superconducting film as a function of the applied field parallel to its faces is considered. In the last few years this process has attracted attention, ' mainly focused on the oscillations exhibited by the magnetization. Some models, based on a London-type approach, have been able to explain why magnetization peaks appear at certain fields. ' Though the experiments were carried out in layered superconductors and in oriented YBaCuO films, the models make no use of the internal periodic structure of the sample and nevertheless arrive at satisfactory predictions. It is thus interesting to analyze this problem using a fully coupled Ginzburg-Landau approach, by numerically solving the time-dependent Ginzburg-Landau (TDGL) equations coupled with the Maxwell equations in a homogeneous, isotropic, type-II superconducting thin film. Numerical simulations using the same approach have already proved useful in the modeling of other superconductivity phenomena.Our results can be sunitnarized as the following: (i) It is confirmed that the full TDGL approach predicts a series of maxima in the magnetization of a homogeneous type-II film with the applied field parallel to its faces. (ii)It is proved that these maxima are not necessarily linked with structural changes in the vortex lattice. (iii) A mechanism for the appearance of such maxima is identified, strongly linked with the behavior of surface supercurrents. This mechanism is independent of that proposed in Refs. 1 and 2. Let us brieAy describe the mathematical model and the numerical method. We use the time-dependent GinzburgLandau (TDGL) equations coupled with Maxwell equations, leading to the following mathematical problem for the order parameter P and the vector potential A (the scalar potential is eliminated through an appropriate choice of gauge): BA =(1 -T)Re[&/i*( -iV -A)p] -~2 Vx VxA. (2) Bt Lengths have been scaled in units of $(0), time in units of to= mfa/(96k~T, ), A in units of H, z(0)g(0) and temperatures in units of T, . It has been assumed that ((T) = $(0)(1 -T/T, ) ', where T is the temperature and T, the. critical temperature, and that the Ginzburg-Landau parameter is independent of temperature. In (1), r/ is proportional to the ratio of characteristic times f...

ac magnetic response of mesoscopic type-II superconductors

“…Each metastable state becomes unstable at the i-th penetration field H pi in which vortices enter the sample and the magnetization has a discontinuous jump. These jumps have been observed numerically 16,20,29 and experimentally in mesoscopic Al disks. 20 In the last years numerical simulations of the time dependent Ginzburg Landau (TDGL) equations have been an important tool in the study of the static and dynamic properties of superconductors.…”

“…16,29 In mesoscopic samples each vortex entrance event produces discontinuos jumps in the magnetization curve at certain H dc values. Each discontinuous jump in M (H) corresponds to a sudden increase in the number of vortices.…”

“…In this case, it is possible to use phenomenological models to describe the macroscopic behavior of the superconducting samples. 4,5,6,7 Recently there has been an increasing interest in the study of vortex physics on a mesoscopic scale, 15,16,17,18,19,20,21,22,23,24,25,26,27,28,29 a regime in which a small number of vortices are confined in a small sample. The behavior of mesoscopic superconductors is different from the behavior of bulk samples.…”

“…29 The magnetic properties strongly depend on the sample sizes and geometry. 22,23,29 The most studied geometries are the the mesoscopic disk 15,18,19,20,21,22,23,24,25,26 the mesoscopic slab 16,17 and the mesoscopic square. 27,28,29 In particular the mesoscopic samples can develop Abrikosov multivortex states 21 and depending on the size of the sample it is possible to observe first or second order transitions.…”

“…20 In the last years numerical simulations of the time dependent Ginzburg Landau (TDGL) equations have been an important tool in the study of the static and dynamic properties of superconductors. 16,31,32,33,34,35,36,37,38,39,40 The TDGL model has been used to study the flux growth dynamics 31,32 , the magnetic response 16,33 and the I-V characteristics of superconducting samples. 34 In particular, Enomoto et al 38 using the TDGL equations studied the temperature dependence of the ac susceptibility at a fixed frequency in the absence of a dc magnetic field for a large sample.…”