2017/1/31
量子化磁束動力学シミュレーション研究グループ第四回研究会

Investigation of the flux lines motion in superconductors
in a longitudinal magnetic field by the computer simulation
using the Time-Dependent Ginzburg-Landau equations*Kento Adachi，Yusuke Ichino，Yuji Tsuchiya，Yutaka Yoshida (Nagoya Univ.)

This work was partly supported by a Grants-in-aid for Scientific Research(23226014,15H04252, 15K14301, 15K14302 and JST-ALCA).

Introduction ~ Time-Dependent Ginzburg-Landau Equations ~Time-Dependent Ginzburg-Landau (TDGL) equations
 Description of the vortex dynamics in type-II superconductors★ Introduction of study about vortex dynamics by TDGL equations
 A. E. Koshelev [Argonne National Laboratory] (2016)[1] Visualizing the flux lines in nano-particles doped superconductor
 Investigation of optimal parameters (PC size and density) for the pinning to maximize
the critical current density Jc

Fig. Snapshot of the flux lines through the nanoparticles. (The particles are shown astransparent spheres, and the flux lines outside
particles are red and inside particles are blue.)Fig. The dependences of the criticalcurrent Jc on the particle volume
fraction f.

[1] A. E. Koshelev et al.: Phys. Rev. B. 93 (2016) 060508

Fig. Schematic view of the type IIsuperconductor in the LMF state. (B ⊥ I )Introduction ~ Longitudinal Magnetic Field Effect ~Longitudinal magnetic field (LMF) state
I Lorentz force
hardly works

B
Flux line
Fig. Schematic view of the type II superconductor in theLMF state. (B // I)

Fig. Magnetic field dependence of Jc in maximum force state
and the LMF state of the pure SmBa2Cu3Oy film andmultilayered SmBa2Cu3Oy film at 77K under B//ab [2],[3].

 Jc increases in a magnetic field more than the one at zero fieldObjective To visualize the quantized flux lines motion in the LMF state
the computer simulation solving the 3D-TDGL equations
 To investigate the effective APC shape in the LMF state[2] A. Tsuruta et al.: Jpn. J. Appl. Phys. 53 (2014) 078003[3] K. Sugihara et al.: Supercond. Sci. Technol. 28 (2015) 104004

TDGL Equations
TDGL equations[4] Calculation of time development of the order parameter and vector potential2𝜕∆ 𝒓, 𝑡𝜕𝑡1=−12𝛻
− 𝑨 𝒓, 𝑡𝑖

𝜕𝑨 𝒓, 𝑡𝜕𝑡

= 1 − 𝑇 Re ∆∗ 𝒓, 𝑡

∆ 𝒓, 𝑡 − (1 − 𝑇) 𝛼 + 𝛽 ∆ 𝒓, 𝑡2∆ 𝒓, 𝑡𝛻
− 𝑨 𝒓, 𝑡 ∆ 𝒓, 𝑡 − 𝜅 2 𝛻 × 𝛻 × 𝑨 𝒓, 𝑡𝑖∆ : order parameter, 𝑨 : vector potentialParameter[5]−1 ： superconductor
𝛼=𝜉0𝜉𝑛2= 𝛼𝑛 ： normal conductor

𝛽=1 ：superconductor

0 ：normal conductor

 Calculation of the motion of the flux lines with the progress at time
 Arranging artificial pinning centers (APCs) arbitrarily[4] R. Kato et al.: Phys. Rev. B. 47 (1993) 13[5] T. Matsushita et al.: Adv. Cryog. Eng. Mater. 36 (1990) 263

Condition of the Numerical Simulation
vacuumNeumannB

23ξ0superconductor
23ξ0𝒛𝒚60ξ0𝒙I

Table Conditions of this simulation.

Size

60ξ0×23ξ0×23ξ0GL parameterκ =λ/ξ = 3

Boundary condition

vacuumTemperature

0.5Tc

vacuum Applied magnetic field, B
Applied current, I

0.20Bc200.0025-0.015Jd0ξ02Fig. Simulation scale and external condition.

 Cylindrical shapesuperconductor20ξ0𝒛< View of result >Removing the cylindrical edge for ease of viewing.

β=0normal conductor

𝒚Fig. The model of cylindrical shape.

Fig. Simulation result.

Fig. Simulation result after removing
the cylindrical edge.

Simulation Result (Non-doped)
Trapped by APCs
Flux flow

Magnetic field ( Bc2)

0.4

Ic

0.2

Just before continue
the flux flow

0.0

Fig. Flux motion at LMF state. (I = 0.0075Jd0ξ02)

zx

2

468

1012Current (10 Jd00)-31416Fig. The relationship with current value and flux flow.

B’ is induced by the applied current
IB0B’

I : Current
B : External magnetic field
B’ : Induced magnetic field

The flux lines are leaned for direction of B’
Lorentz force works on the flux lines

yFig. Induced magnetic field image.

The flux lines are distorted by spiral shapeCondition of the Numerical Simulation with APCs
 APCs Shape & Size & Volume① Random nano-particles② Multi layers
3ξ03ξ03ξ0Fig. The snapshot of random nano-particles (3.0 vol.%).

Fig. The snapshot of multi layer (6.1 vol.%).

 Shape : ① Random nano-particles② Multi layers
 Size : ① The diameter is 3ξ0② The diameter & the height are 3ξ0 Volume : ① 3.0, 6.0, 9.0 vol.%② 6.1, 8.9, 11.9 vol.%3ξ03ξ0Thickness of superconductor
layer is 3ξ0Fig. The snapshot of multi layer.

Simulation Results (Random nano-particles)(a) 3.0 vol.%(b) 6.0 vol.%(c) 9.0 vol.% Removing the APCs for ease of viewing
in 6.0 and 9.0 vol.%Fig. Snapshot of the flux lines in superconductors with random nano-particles. (a)3.0 vo.l% (b) 6.0 vol.% (c) 9.0 vol.%Simulation Results (Random nano-particles)(b) 6.0 vol.%(a) 3.0 vol.%(c) 9.0 vol.%

The flux lines are trapped by APCs
In 9.0 vol.%, the flux lines keep on
moving through the dense APCs

Fig. Flux motion at LMF state in random nano-particles doped superconductor. (I = 0.0075Jd0ξ02) (a)3.0 vol.% (b) 6.0 vol.% (c) 9.0 vol.%Discussion (Random nano-particles) Ic and APC volume2Jd00 )1.0

Critical current (10-30.8

0.60.4

0.20.002468The Ic (0.009J0ξ02) increase from thepure system (0.006Jd0ξ02)
The Ic decrease at 9.0 vol.%10APC volume (vol.%)

Fig. The dependence of the Ic on the particle volume.

Mean distance between APCs( 0) Ic and mean distance between APCs (d) The Ic decrease when d < 2ξ(T) (9.0 vol.%)⇒ Flux channeling become easy

98d2(T)7(a) d > 2ξ(T) (2.83ξ0)6Trapped

54d

ξ(T)

|Ψ|2(b) d < 2ξ(T)dξ(T)Channeling

APCNormalstate3202468APC volume (vol.%)

10Fig. The relationship between d and 2ξ(T).

Fig. Image of the decaying cooper pair density in APCs.

The cooper pair density |Ψ|2 decay into APCs with ξ(T)Simulation Results (Multi Layers)
6.1 vol.%

1.02

The Ic decrease from the pure system

0.8

Critical current (10-3

11.9 vol.%Fig. Flux motion in multi layers (8.9 vol.%) superconductor.
(I = 0.006Jd0ξ02)

Jd00 )

Fig. Flux motion in multi layers (6.1 vol.%) superconductor.
(I = 0.006Jd0ξ02)

8.9 vol.%

Fig. Flux motion in multi layers (11.9 vol.%) superconductor.
(I = 0.006Jd0ξ02)

0.60.40.20.0024681012APC volume (vol.%)
Fig. The dependence of the Ic on the particle volume.

Discussion (Multi Layers)
 Definition of dx-y
Mean distance between APCs
in x-y plane Definition of dz

dz = 3ξ0Fig. The snapshot of multi layers.

Mean distance between APCs( 0) Ic and mean distance between APCs (dx-y, dz)
10dx-ydz
2(T)864206912

APC volume (vol.%)
Fig. The relationship between distance between APCs (dx-y, dz) and 2ξ(T).

 The dz is almost 2ξ(T)⇒ Flux channeling become easy in z plane in our simulation.
 The dx-y is shorter than 2ξ(T) at about 12.0 vol.%.
⇒ Flux channeling become easy toward the all direction at about 12.0 vol.%.

SummaryIn this study, we visualized flux motion and calculated Ic at LMF state insuperconductors by using TDGL equations. Moreover, we simulated flux
motion in APCs-doped superconductors as well, and compared Ic with puresystem. Consequently, we obtained following findings.
•In LMF state, the flux lines are distorted in a spiral shape by a current-induced
magnetic field.

• In the case of introducing random nano-particles as APCs, the Ic increased fromthe pure system.
• In the case of introducing multi layers as APCs, the Ic decreased from the puresystem, because of short inter-layer distance of APC doped layers.
• Each APC distance is important parameter to consider the optimal APC volume orshape, it need at least over 2ξ(T) in our simulation.