Influence of surface defects on the vortex penetration and arrangement at mesoscopic superconducting samples

All superconductor applications lie on carry dissipationless current; however, in the presence of external magnetic fields, including the self-field, vortices penetrate the sample, and their dissipative motion generates resistive states. Thus, once the superconducting state survives for higher magnetic fields due to the presence of vortices, those specimens cannot move to increase the material’s critical current density; thus, in this work, we studied the influence of surface defects on the vortex penetration at square mesoscopic superconducting materials using the time-dependent Guinzburg-Landau framework. The lateral size of the samples was 12 times the coherence length at zero Kelvin, with defects distributed in two opposite borders. The main result showed that the currents crowd around the surface defects are responsible for vortex penetration at 60% of critical temperature.


Introduction
With the advance of nanotechnology and intensification of nanomaterials manufacture [1,2], there is enormous interest in studying superconductivity on this scale, i.e., the mesoscopic scale.At such scale, the vortex matter in mesoscopic superconductors (SC) presents different interactions compared with macroscopic samples, such as giant vortices and non-Abrikosov lattices(i.e., the vortex lattice depends on the samples' geometry) [3].
Additionally, the boundary between mesoscopic and macroscopic scales depends on the sample size and the bath temperature.Thus, in [4,5] showed a crossover in the vortex lattice in such a threshold, i.e., the Abrikosov lattice is present in the macroscopic superconductor until the transition to the normal state; however, a crossover between the Abrikosiv lattice and a square lattice takes place at mesoscopic scales [4].Such a behavior means that the vortices begin to be influenced by the surfaces.Other characteristics of mesoscopic superconductors can be found in [6][7][8][9][10][11][12][13], and references therein.
Another widespread effect is the crowding-current (CC), i.e., an accumulation of current lines in a specific region where they need to change their path abruptly.This effect can cause a drop in the system's critical current density.In addition, these regions can become paths for vortices to enter the sample [14,15].So, it can be demonstrated that the CC effect is not a rule to guide or facilitate the vortex penetration by studying samples with surface defects and using the time-dependent Ginzburg-Landau framework.

Theoretical Formalism
The time-dependent Ginzburg-Landau (TDGL) formalism relates the superconducting order parameter ψ, the vector potential A, and the scalar electric potential Φ, as can be seen in the Equation (1) and Equation (2) [16,17].
Where, D is the diffusion coefficient, is the Planck constant, m is the electron's mass, i the imaginary number, e is the electron charge, c is the light speed, σ is the electrical conductivity a(T ) and b are phenomenological constants, ψ is the order parameter, where its modulus squared represents the density of Cooper's pairs.J s is the superconducting current density and the applied magnetic field H = ∇ × A. However, for convenience the Equation (1) and Equation ( 2) are normalized, so the lengths are in units of ξ(0) the coherence length, the fields in units of H(0) c2 the second critical field, currents in units of J c , temperature in units of the critical temperature T c , the square modulus of ψ in terms of |ψ 0 | 2 = a(T )/b and time in units of the characteristic time t 0 = π /(8k B T c ), where k B is the Boltzmann constant.
To carry on the simulations, samples with lateral size L = 12ξ(0) were considered with temperatures of T = 0.6T c and the Ginzburg-Landau parameter κ = 5.The defects were 0.250ξ(0) thickness; they were arranged on two sides of the sample and in three different configurations: (i) spaced by 2ξ(0) (S2.0), Figure 1(a), (ii) a random distribution (Random), Figure 1(b), and (iii) a single defect (Single) on each side of the sample with a distance of 4ξ(0) to the nearest border, Figure 1(c).

Results and discussion
Figure 2 shows the superconducting currents' distribution at different regions of the samples (see Figure 2(d)) and on their Meissner state; the peaks are due to the crowding of the currents.From Figure 2(a) to Figure 2(c), can be noted that local J s is only higher than in the smooth borders when it crowds around the defects.For the systems shown in Figure 1, the first vortex penetrates by the defect region.Therefore, Figure 3 shows the first vortex penetration for these samples where, due to the CC effect in the defects' vertices, the vortex penetration occurs by the defected region.However, in the pristine system, the vortex penetrates the upper and bottom faces.
The second penetration was also studied, as shown in Figure 4; where, can be noticed the absence of giant vortices.However, there is a formation of a vortex lattice.

Conclusion
In this work, it was possible to observe the dynamics of vortices in mesoscopic superconductors with different distributions of surface defects; as seen, the currents crowd around the defects, promoting peaks in the |J| profile.additionally, the formation of giant vortices is rare in those kinds of samples.

Figure 1 .
Figure 1.Simulated superconducting systems with defects on both sides of the sample.(a) Defects spaced by 2ξ(0) from each other (sample 2.0), (b) randomly distributed (Random), (c), and a single defect (Single).

Figure 2 .
Figure 2. Superconducting current density (J s ) distribution at T = 0.6 T c for samples (a) 2.0, (b) Random, and (c) Single.(d) The arrows show the places where J s was analyzed.m

Figure 3 .
Figure 3. First vortex penetration for T = 0.6 T c (a) 2.0 system; (b) Random system; (c) Single system and (d) pristine system.Insets show the stationary state.

Figure 4 .
Figure 4. First vortex penetration for T = 0.6 T c (a) 2.0 system; (b) Random system; (c) Single system and (d) pristine system.Insets show the stationary state.