Advances in Condensed Matter Physics

Volume 2018, Article ID 7615862, 7 pages

https://doi.org/10.1155/2018/7615862

## Effect of Background Magnetic Field on Type-II Superconductor under Oscillating Magnetic Field Simulated Using Ginzburg-Landau Model

^{1}Department of Physics, University of Science and Technology Beijing, Beijing 100083, China^{2}ASIC, China Center for Information Industry Development, Beijing 100083, China^{3}Advanced Research Institute of Multidisciplinary Science, Beijing Institute of Technology, Beijing 100081, China

Correspondence should be addressed to Xingqiao Ma; nc.ude.btsu.sas@amqx

Received 8 August 2018; Accepted 27 September 2018; Published 12 November 2018

Academic Editor: Joseph S. Poon

Copyright © 2018 Hasnain Mehdi Jafri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Cubic superconducting sample was simulated using time-dependent Ginzburg-Landau model under oscillating magnetic field with and without additional background static magnetic field. Vortex dynamics including entrance and exit from the sample was simulated. Magnetization and carrier concentration densities of the sample were studied as a function of external magnetic field variations. Anomalies in carrier concentration density were observed at certain values of the magnetic field which were correlated with the entrance and exit processes of vortices. Area swept by superconductor magnetization with magnetic field was observed to have a hysteresis-like behavior where area representing energy dissipated per cycle. This energy accumulation was suggested to cause instability in superconductor over the number of cycles and may result in thermal quenching. Temporal distribution of energy components showed consistency with the pattern observed for carrier concentration and magnetization under oscillating magnetic field. Rapid phase changes with magnetic oscillations resulted in oscillations in energy components, and irregular peaks and ripples in superconducting energy represent the situation of exit and entry of vortices. While the rise in interaction energy with cycles is referred to vortex relaxation time in a cycle, this energy is expected to accumulate and take other forms (e.g., heat) and is predicted to cause thermal quenching. In the presence of background static magnetic field, this energy dissipation was calculated to increase significantly while superconductor is subjected to oscillating magnetic field.

#### 1. Introduction

In general, the behavior of superconductor in applied electric and magnetic field is attributed to the vortex dynamics in it, which generally results in energy dissipation (e.g., heat) and resulting thermal quenching of superconductivity. According to Bardeen-Cooper-Schrieffer (BCS) theory, spin singlet state (opposite spin electron pair), bound by phonon interaction, constitutes the ground state of condensate [1]. The proposed macroscopic quantum theory by Ginzburg and Landau (named Ginzburg-Landau (GL) theory) used Higgs mechanism of spontaneous symmetry breaking and quartic potential [2–5] to generate a local mass term of vector potential. Development of homogeneous ferromagnetic spins destroys the superconductivity if it exceeds critical magnetic field of the superconductor. This model successfully describes the Meissner-Ochsenfeld effect [4, 6–8]. The prediction of penetrating strong magnetic fields in type-II superconductors by Abrikosov [9] gave further credibility to GL model which was later verified experimentally [10–12]. Over the past 70 years, not only macroscopic properties of superconductors (such as categorizing superconductors in type-I and type-II, and description of vortex state in type-II superconductors [9]) but also mesoscopic superconducting samples were successfully described using this theory. GL model is probably the most accurate phenomenological model to describe macroscopic properties of superconductors [13, 14]. Gorkov [15] proved that close to critical temperatures microscopic BCS theory also reduces to GL theory. Various numerical methods, including finite difference [16–18], finite element [19–21], and spectral method [22], have been developed for the solution of this model. In this report, we used the semi-implicit finite difference method with staggered grid scheme.

Generally, high-temperature superconductor (HTS) based magnetic Levitation (maglev) vehicle systems consist of HTS on a permanent magnet (PM) guideway [23–25]. In general, uniform magnetic field is considered for PM guideways but practically this is not the case; there are situations of magnetic field variations (e.g., cracks, magnetic contacts, and structural and magnetic defects) resulting in a variable magnetic field. The frequency of this variation depends on the speed of the HTS on PM guideway (i.e., speed of the vehicle). The magnetic field in such a case is not exactly oscillating (which was reported earlier [18]), but it is (practically) a constant magnetic field having small jitters (or oscillations) in its amplitude appearing due to imperfections in PM guideway. In this work, we studied such a magnetic field and compared the results with results reported earlier for perfectly oscillating magnetic fields [18]. To investigate this type of energy loss, we studied the behavior of type-II superconductor exposed to an oscillating magnetic field simulated using time-dependent Ginzburg-Landau (TDGL) equations [15, 26, 27] near critical temperature with and without an additional static background magnetic field. The dynamics of Abrikosov vortices in oscillating magnetic field is studied. A hysteresis-like behavior of sample magnetization was observed under oscillating magnetic fields, while background static magnetic field plays a vital role in thermal quenching of superconductor. The behavior of different energy components is also discussed in detail.

#### 2. Ginzburg-Landau Theory

Primary variables in GL model are order parameter,* ψ*, and magnetic vector potential,

*A*, occupying a superconductor sample in a three-dimensional region

*Ω*having boundary

*Γ*. In nondimensional state, these primary variables are linked to physical quantities, given as follows:

*Density of superconducting charge carriers,*

*Induced magnetic field,*

*Current density,*.

Ginzburg-Landau free energy is taken from Gorkov and Eliashberg [28] given as follows:where is charge and is effective mass of superconducting charge carriers, is external magnetic field, and are reduced Plank's constant (i.e., h/2*π*), and , while *α* and *β* are phenomenological parameters depending on environmental factors (e.g., temperature).

We introduce London penetration depth () and coherence length () as length scale parameters. We introduce a length scale and nondimensionalized physical quantities, special and time coordinates () and* t*, order parameter* ψ*, magnetic vector potential , and magnetic field according to the following transformations:where

*γ*is relaxation parameter. For simplicity, we choose length scale in terms of coherence length (i.e.

*l*=

*ξ*). Ginzburg-Landau parameter

*is defined as .*

*κ**defines the type of superconductor, for type-I and for type-II superconductors. Minimizer of Ginzburg-Landau energy functional (1) satisfies Euler-Lagrange equations, generally known as Ginzburg-Landau (GL) equations:*

*κ**is electrical conductivity defined as , while is normal conductance. External magnetic field is assumed to be uniformly distributed over the sample surface. GL equations were solved using natural boundary conditions, i.e., supercurrent across the boundaries is zero and magnetic field at the boundaries is given as follows:where is unit normal vector on the boundary*

*σ**Γ*of the region

*Ω*.

TDGL model is gauge invariant under transformation:where gauge *χ* is a function of space and time. For the present study, we choose zero electric potential gauge () known as Coulomb Gauge. So (3) and (4) reduce toEnergy density* H*_{tot}*=H*_{tot}*(x,y,z,t)* in superconductor is distributed into three parts: superconducting energy density* H*_{sup}*=H*_{sup}*(x,y,z,t),* interaction energy density* H*_{int}*=H*_{int}*(x,y,z,t),* and magnetic energy density* H*_{mag}*=H*_{mag}*(x,y,z,t)* given as[18, 29] where length scales are taken in terms of London penetration depth (i.e.* l= λ*). Total energy (

*H*

_{tot}) density is the sum of these three energy densities,

*H*

_{tot}

*= H*

_{sup}

*+ H*

_{int}

*+ H*

_{mag}; total energy is the integral of total energy density over the region

*Ω*, .

Gauge invariant discretization [17, 30–33] is the most popular and widely used approach to solve TDGL equations, which is first-order accurate in time and second-order accurate in space, and other finite element [19, 20], finite difference [34, 35], and spectral method [22] have also been developed. Link variable schemes [16, 17, 36] are the most commonly used schemes to solve coupled GL equations. In the present work, in order to investigate carrier concentration, magnetization, and vortex dynamics in superconductor, we used coupled nonlinear TDGL equations (8) and (9) solved by finite difference scheme using link variables, described by Winiecki and Adams [16], implemented in three-dimensional code we have developed [18]. Equations (8) and (9) along with boundary condition equations (5) and (6) form the basis of the present work. The present work was performed close to critical temperature allowing the assumption of thermal suppression of surface barrier [37–39]. The frequency of oscillating magnetic field was set to* 0.00278/t* in nondimensionalized coordinates for the present work.

#### 3. Results and Discussion

Three-dimensional, cubic superconductor domain of size , periodic along z-axis (along the direction of applied magnetic field), was discretized with grid size* 0.5 ξ* in each direction, and magnetic field was applied in z-axis direction with Ginzburg-Landau parameter

*. Initially, a constant magnetic field was applied, and on reaching equilibrium state, it was allowed to oscillate as cosine wave. Cubic superconductor under two different types of magnetic field schemes, (A) and (B) (where*

*κ*=4*), was simulated to investigate the effect of static background magnetic field on the superconductor in the presence of an oscillating magnetic field. The system was initialized with the completely superconducting sample () and no magnetic vector potential (*

*θ*=*ω*t=2*π*ft*A=0*). The system was relaxed to minimize Gibbs free energy. During this energy minimization process, vortices entered the sample and relaxed their positions to minimum energy. Magnetic field oscillations were introduced as the sample reached minimum energy state, resulting in a new series of entrance and leaving of vortices, corresponding to different positions of magnetic field wave. Figure 1 shows a variation of average carrier concentration of the sample with magnetic field. The pattern of carrier concentration is symmetric for positive and negative half cycles but nonsymmetric for increasing and decreasing parts of magnetic field oscillation, shown in Figure 1 (for magnetic field of type (B)), indicating a lag in vortex dynamics behind the oscillating magnetic field. For the situation of magnetic field of type (A), the system is initially subjected to , and after sufficient time, on reaching equilibrium state, a small oscillation with amplitude 0.15 in magnetic field was introduced.