Influence of Lattice Defect to Vortex Flow on Type II Superconductor

Adding defect on type II superconductor can influence to vortex dynamic. It declares vortex movement, so speed of vortex is more stable. In this research, we applied lattice defect. In case to study dynamics of the vortex in type II superconductor material, this study was based on numerical solution of the Time-Dependent Ginzburg Landau (TDGL) equations by means of finite difference method with Forward Time Centred Space (FTCS) scheme. Applied external current density Je and external magnetic field He to superconductor material generate a vortex flow from higher magnetic field to lower magnetic field. Furthermore, applied external current density Je without external magnetic field He generates vortex flow between lattice defects.


Introduction
Superconductivity is the properties of which has resistivity approach to zero. Generally, a material which is under critical temperature has superconductivity properties. Superconductor can be categorized into two types are type I and type II. Type I has one critical magnetic field, whereas type II has two critical magnetic fields. In type II, the external magnetic field may penetrate to superconductor material. It enters into superconductor material in the form of vortex [1].
In recent years, research about vortex dynamic is more attention. It can obtain important properties of type II superconductor, such as plastic properties of vortex which this property affects on critical current density in superconductor [2]. The vortex dynamic can be studied by time-dependent Ginzburg-Landau equation which is a non-linear differential equation. Thus it can be solved by computational methods [1,3,4]. One of the computational methods is UΨ [5][6][7][8] This method has an advantage namely converging property for the high magnetic field [8][9][10].
In previous research about vortex dynamics with a single defect on the material has been successfully studied. From that research, we know that the influence of applied external current on vortex dynamics can be measured by the potential difference on the side of the material. In the presence of a defect, potential difference changes slower than on clean superconductor [11][12][13]. On the other research, when current density applied with a different value, it can change the vortex flow [14]. From the previous research, vortex dynamics could not show stability. One of the solution to get the stability is watched the dynamics of the vortex in the presence of defects and clean superconductor closely.

Methods
In this research, the influence of lattice defect on superconductor type II was studied by numerical methods. The Time-Dependent Ginzburg-Landau (TDGL) equation is used on this research. This equation has been successfully explaining superconductivity phenomenon for any sample which consists of the normal and superconducting part. Based on a modified TDGL equation, vortex dynamic has been successfully studied by [12,[15][16][17]. It shows that this model has the potential for explaining how the characteristics of vortex when there is a defect on type II superconductor. The normalized TDGL equations with potential gauge can be expressed by [18].
Consider a system of type II superconductor with computational grid size is Nx× Ny, and the size of typical grid size is hx× hy = 0.2ξ0 × 0.2ξ0, like on Figure 1. External current density Je applied to this material. A computational grid is obtained by divide Lx with Nx, and every part of the typical grid can be written as hx = Lx/Nx. And also divided Ly with Ny and every part of the typical grid can be written as hy= Ly/Ny. From this way, we can produce a homogency computational grid, shown on

Results and Discussion
In this research, vortex dynamic with adding lattice defect has been simulated. The size of superconductor is Lx × Ly = 50ξ x 50ξ, made from niobium = κ =1,3, hx = hy = 0,5ξ, and ∆t = 0,001ξ 2 /D. The superconducting material was on 0 Kelvin condition. When external current density flow on the material, the magnetic induction will appear of 3 external magnetic field Heks, so there is resultant between the induction magnetic field and external magnetic field, namely vortex. Vortex enters into superconductor material because of interaction between external current density and potential vector. In the equilibrium state, the induction magnetic field is equal to the external magnetic field, Bz = Hz. If the external current density flows in the xdirection, then In the last term of equation (4) In external current density form, the Lorentz force can be written by  Figure 2 shows the dynamics of vortex when external current density Je = 0.016 applied to the materials. In figure 2(a) is the condition when the vortex does not penetrates to the material yet. In Figure 2(b) show that vortex has penetrating to the material. From 2(b), we can see that the defect is attracting the vortex one by one from it upper side. When the vortex have filling the upper defect, the other vortex will flow between the defect and it will be attracting again with the empty defect and it will happen continuously, like in Figure 2 [19]. From that equation, the high value of the potential difference is implicating faster movement of the vortex.
Red line a curve in Figure 3 indicate the vortex dynamics in a clean superconductor. Value of potential difference increase indicates that there are many vortexes penetrate to the material and implicate that vortex inside the material move faster. When the value of potential difference decrease, it indicates that penetrating of vortex decrease and also the velocity of the vortex movement is slower than before. When the value of potential difference on equilibrium state, t = 10000 to t = 20000, it indicates that there are no more penetrate of the vortex, motionless of the vortex, and there is no vortex left the material.
Black line a curve in Figure 3 indicate that vortex dynamics in the material with the existence of lattice defect. From that line, we can see that the curve has more stability than the curve for a clean superconductor. The sharp peak on that line indicates when the defect will pin vortex. The small peaks indicate when vortex penetrate to the material. The value of the potential difference is lower than on clean superconductor, because vortex could not move faster in the presence of a defect. Furthermore, the vortex will get it equilibrium state slower than on clean superconductor.
Comparing to the previous research, Figure 4 below shows the influence of defect on the dynamics of the vortex on type II superconductor. IFigure (4) shows that the defect influences characteristics V-I curve. Figure 4(a) shows the dotted line and black line for the evolution of vortex dynamics for clean superconductor and with a single defect, respectively. For clean superconductor, peak P1 and P2 show there is a dissipation of energy when vortex penetrates to the materials. Moreover, peak P3 and P4 show when vortex left the material from the other side [11]. For the material with a single defect on the center of materials, peak D1 show when the first vortex penetrate to the material. We can see from figure 2(a) that peak D2 is sharper than peak D1. This peak is total of raising of first vortex velocity around defect before pinning and second vortex penetration. Peak D3 show the third vortex penetrate to the material. Peak D4 show the raising of second vortex velocity around the defect before pinning. In this case, we can see that there are two vortexes inside the defect. After that, the sharp peaks will not exist anymore on the curve because there is no vortex movement on the materials [11].
The presence of defect on the material will change characteristic curve of V-I, in Figure 4(b). Figure  4(b) explains that the current I will change in range 0.22 ≤ I ≤ 0.025 and not giving an influence of potential difference V changing. That means, in this range, the defect can pin vortex and stopping the movement of the vortex on materials. When current I increase to 0,026, value of potential difference will rise sharply. In this condition happen when vortex released from defect and move faster avoid the defect [11].

Conclusion
Based on the discussion above, it can be concluded that the existence of lattice deffect in superconductor type II influences on vortex dynamics showing the slower flow of the vortex. It indicate the paining vortex flow of the system. Furthermore, the defects have role as pinning vortex and increase the superconductivity of material. In another word, the defects also have as essential role to increase the stability of vortex flow. Interestingly, the stability of vortex flow is able to enhance the critical current of the superconductor.