TDGL Simulation on the Motion of Flux
Lines in a Thin Superconducting Wire
in a Transverse Magnetic Field
T. Yoshihara, K. Tanimura, E. S. Otabe, M. Kiuchi(Kyushu Inst. of Tech.)Y. Mawatari(AIST)
T. Matsuno(Ariake Natl. Coll. of Tech.)

The 65th JSAP Spring Meeting
March 17, WASEDA University, Nishi-Waseda Campus
1. Background
2. In present study
3. TDGL equation
4. Simulation model
5. 𝐸-𝐽 property
6. 𝐽c -𝐵 property
7. Conclusion

𝐽 : current density
𝐽d : destruction current density
𝐽c : critical current density

1. Background
𝐽𝐽d 0𝐽c 0

On normal conductor(pin)…
density
shape / size
arrangement
𝛹 = 0,𝛼 > 0 𝛽 > 0

𝐵 : magnetic field𝐵c2 : critical magnetic field𝛹 : order parameter

𝐽𝐽c 0

Prediction ofchanges in 𝐽c

Differencesin conditions

𝐵c2

𝐵c2

𝐵

𝐵

Which condition of the pin affects 𝐽c ?
Improvement of computer processing
performance in recent years

It has become easier to investigate the relationshipby solving TDGL simulation
22. In present study
On normal conductor
spheredensity
For example
shape / size
arrangementboundary condition

cylinderplane

Transverse
magnetic field
spherecylinderplane𝑑Investigate 𝐸-𝐽 and 𝐽c -𝐵 properties.
3TDGL equation• Ginzburg-Landau(GL) equation
・ Phenomenological theory to explain superconductivity
・ Calculate order parameter 𝛹・ 𝛹 2 :Superconducting electron density
1−iℏ𝜵 + 2𝑒𝐴 2 𝛹 + 𝛼𝛹 + 𝛽 𝛹 2 𝛹 = 0∗2𝑚2iℏ𝑒 ∗4𝑒𝒋 = ∗ 𝛹 𝜵𝛹 − 𝛹𝜵𝛹 ∗ − ∗ 𝛹 2 𝑨𝑚𝑚• Time Dependent GL(TDGL) equation
・ GL equations with time dependency𝜕𝛹
1+ 2i𝑒𝑉𝛹 +−𝑖ℏ𝜵 + 2𝑒𝑨 2 𝛹 + α𝛹 + 𝛽 𝛹 2 𝛹 = 0∗𝜕𝑡2𝑚
𝜕𝑨
1iℏ𝑒 ∗4𝑒 2∗
𝜈+ 𝛁𝑉 + 𝜵 × 𝜵 × 𝑨 − ∗ 𝛹 𝜵𝛹 − 𝛹𝜵𝛹 + ∗ 𝛹 𝟐 𝑨 = 0𝜕𝑡𝜇0𝑚𝑚

𝛾43. Normalize TDGL equation• Equations
𝜕𝛹+ i𝑉𝛹 + −i𝛻 − 𝑨 2 𝛹 − 𝛹 + 𝛹 2 𝛹 = 0𝜕𝑡i2𝜎𝛻 𝑉 = 𝛹 ∗ 𝛻 2 𝛹 − 𝛹𝛻 2 𝛹 ∗ − 𝛻 ∙ ( 𝛹 2 𝑨)2
𝛹=0(1)(2)(3)

1 𝛹 in superconducting region
2 Scalar potential 𝑉(3) 𝛹 in normal region54. Simulation model・Assuming in vacuum
・Consider a cube space(side length:10𝜉)・Mesh size：50 × 50 × 50

6

Conditions (thin wire)𝑦𝐽𝐵𝑧𝑥■Current density 𝐽, Magnetic flux density 𝐵
・𝑱 = 𝐽𝑦 𝒊𝑦 , 𝑩 = 𝐵𝑧 𝒊𝑧 ∴ 𝑱 ⊥ 𝑩
・𝑨 = 𝐵𝑥 𝒊𝑦(𝒊𝑥 , 𝒊𝑦 , 𝒊𝑧 : Unit vector of each axis)■ Initial conditions
・𝛹 𝑡 = 0 = cos𝜃 + i sin𝜃・𝑉 𝑡 = 0 = −𝐽𝑦/𝜎■ Boundary conditions・𝒏 ∙ 𝛻𝛹 + i𝑨𝛹 = 0・𝛻𝑉 = −𝑱/𝜎※𝒏：A unit vector perpendicular to the plane78𝜉

5. 𝐸-𝐽 property

Critical current density 𝐽c is definedusing a resistance standard
as indicated by a red straight linein the 𝐸-𝐽 property4𝜉𝛹=0𝑬 = −𝛁𝑉𝐵→𝐵
𝐵𝑐2

Normalized magnetic field 𝐵2e𝛾𝛼𝜉field 𝐸 field
electricelectric
Normalized
𝐸E→ENormalized

0.2

B=0.1
B=0.2
B=0.3
B=0.4
B=0.5
B=0.6

0.1

00

0.1

0.2

0.3
3221 J
Normalized current density
𝐽→𝐽Normalized current density 𝐽Super ：𝜕𝛹𝜕𝑡+ i𝑉𝛹 + −i𝛻 − 𝑨 2 𝛹 − 𝛹 + 𝛹 2 𝛹 = 0Normal：𝛹 = 032 𝐽dIn each 𝐸-𝐽 property, it is possible toconfirm the rise of the electric field.
9𝐽c -𝐵 property
0.385

Normalized critical current density J

2 2 1𝐽→𝐽
32 𝐽dNormalized critical current density 𝐽

3・ Each 𝐽c -𝐵 property has a different
0.3

Sphere 4Pin(d=4.0)Cylinder 2Pin(d=4.0)Plane 1Pin

tendency
・ It is confirmed that the peak
appears at “Cylinder 2Pin”.
・ 𝐽c monotonically decreases as

0.2

𝐵 increases at “Plane 1Pin”
0.1000.5

1𝐵→B
𝐵c2

Normalized
magneticmagnetic
field 𝐵
Normarized
field

B
Super ：𝜕𝛹𝜕𝑡+ i𝑉𝛹 + −i𝛻 − 𝑨 2 𝛹 − 𝛹 + 𝛹 2 𝛹 = 0

Normal：𝛹 = 010

𝜉

𝐽c -𝐵 property
𝑑0.385・ It is confirmed that the peak

32 2 132 𝐽ddensity
critical current
Normalized
𝐽𝐽 → 𝐽J
Normalized
criticalcurrent
density

Pin shape(𝑑)appears in the 𝐽c -𝐵 property.

Cylinder Pin(d = 4.0)Cylinder Pin(d = 4.6)

0.3※Peak effectflux line

0.2
Peak appears

𝑎f =0.1

2𝜙03𝐵12∝1𝐵

Dr. B. Obst in Researarch Center in Karlsruhe

𝑎f000.51𝑑+ i𝑉𝛹 + −i𝛻 − 𝑨 2 𝛹 − 𝛹 + 𝛹 2 𝛹 = 0𝑎f𝐵𝐵c2

Normalized
magnetic
field 𝐵 field→ BBNormarizedmagnetic

Super ：

𝜕𝛹𝜕𝑡Normal：𝛹 = 0𝑑𝐵1

𝐵2 > 𝐵111

Scaling law of pin force density

Fp / Fpmax

1𝐹p / 𝐹pmax = (𝐵 /𝐵c2 ) γ (1−(𝐵 /𝐵c2 ))δ

Plane𝛾-> 1.12, 𝛿-> 0.99
Cylinder𝛾->0.92, 𝛿->0.98
Sphere𝛾-> 1.10, 𝛿-> 1.75

0.5

Sphere 4Pin(d=4.0)Cylinder 2Pin(d=4.0)Plane 1Pin

000.5
Normarized magnetic field B17. Conclusions
・The 𝐽c -𝐵 property was clarified by solving thesimplified three - dimensional TDGL equation bysimulation.
・Significant peak effect was confirmed in cylindricalpin.
・Plane pin shows largest pinning effect among of all.

・ It may be possible in the future to discussmacroscopic features such as the scale rule of 𝐽c by
performing large scale calculations.
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