TDGL Simulation on Dynamics of Helical Vortices in Thin Superconducting Wires in the Force-Free Configuration

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5th International Workshop on Numerical Modelling of High-Temperature Superconductors, 6/15-17/2016, Bologna, Italy TDGL Simulation on Dynamics

Science and Technology (AIST) Collaborators: T. Matsuno, Ariake National Collage of Technology; M. Masuda and E. S.

Vortex Dynamics and Electromagnetic Response 4. Longitudinal Field Effect in the Force-Free Configuration 5.

Mawatari, 160615 HTS Modelling 2016, TDGL-Force-Free 1 / 17 Modelling of Superconductors Macroscopic Modelling (> mm) l modelling of wires,

phenomena and electronics devices l Ginzburg-Landau equation, London model, Josephson equation l length scales: coherence length ξ ~ 1nm,

1 5th International Workshop on Numerical Modelling of High-Temperature Superconductors, 6/15-17/2016, Bologna, Italy TDGL Simulation on Dynamics of Helical Vortices in Thin Superconducting Wires in the Force-Free Configuration Yasunori Mawatari, National Institute of Advanced Industrial Science and Technology (AIST) Collaborators: T. Matsuno, Ariake National Collage of Technology; M. Masuda and E. S. Otabe, Kyusyu Institute of Technology 1. Time-Dependent Ginzburg-Landau (TDGL) Equations 2. 2D Simulation by TDGL equations 3. Vortex Dynamics and Electromagnetic Response 4. Longitudinal Field Effect in the Force-Free Configuration 5. 3D Simulation: Transverse Configuration 6. 3D Simulation: Longitudinal (Force-Free) Configuration 7. Summary Y. Mawatari, HTS Modelling 2016, TDGL-Force-Free 1 / 17 Modelling of Superconductors Macroscopic Modelling (> mm) l modelling of wires, conductors, and power devices l critical state model, power-law model E ~ J n Nexans Mesoscopic Modelling (nm ~ µm) l modelling of nanoscale phenomena and electronics devices l Ginzburg-Landau equation, London model, Josephson equation l length scales: coherence length ξ ~ 1nm, penetration depth λ ~ 0.1µm vortex structure M. Ejrnaes, SPIE Newsroom Microscopic Theory (< nm) l theory for electronic state and superconducting mechanism l BCS theory, first-principle calculation University of Birmingham Y. Mawatari, HTS Modelling 2016, TDGL-Force-Free 2 / 17

Time-Dependent Ginzburg-Landau Equation Time-dependent Ginzburg-Landau (TDGL) equation: l order

state at zero field) ) + * +, + τ 0! t + i 2π $ # Φ " φ & ψ = ξ!

# 2π ϕ A % ', & " A j n = σ n E = σ n $ # t + Φ % ' & A. Schmid, Phys. Kondens.

Mawatari, 160615 HTS Modelling 2016, TDGL-Force-Free 3 / 17 $ &ψ % è fluxoid quantization l upper

0 λ 2 ξ l τ 0 : GL relaxation time l ξ 0 : coherence length l λ 0 : penetration depth l Φ : scalar

5K 39K 95K τ 0 0.3ps 0.08ps 0.03ps ξ 0 4nm 5nm 0.

2 Time-Dependent Ginzburg-Landau Equation Time-dependent Ginzburg-Landau (TDGL) equation: l order parameter ψ = ψ exp(iϕ ); superfluid density ψ 2, phase ϕ l ψ normalized ( ψ = 1 for the equilibrium state at zero field) ) + * +, + τ 0! t + i 2π $ # Φ " φ & ψ = ξ! 2 0 i 2π $ # A 0 % " φ & 0 % Current density, rot rot A /µ 0 = j s + j n j s = ψ 2 " φ 2 $ 0 µ 0 λ 0 # 2π ϕ A % ', & " A j n = σ n E = σ n $ # t + Φ % ' & A. Schmid, Phys. Kondens. Materie 5, 302 (1966). 2 ψ +! # 1 ψ 2 " Y. Mawatari, HTS Modelling 2016, TDGL-Force-Free 3 / 17 $ &ψ % è fluxoid quantization l upper critical field B c2 = φ 0 2 π ξ 2 l depairing current density J d = 2 2 B c 3 3 µ 0 λ = φ π µ 0 λ 2 ξ l τ 0 : GL relaxation time l ξ 0 : coherence length l λ 0 : penetration depth l Φ : scalar potential l A : vector potential l φ 0 : flux quantum NbTi MgB 2 YBa 2 Cu 3 O y T c 9.5K 39K 95K τ 0 0.3ps 0.08ps 0.03ps ξ 0 4nm 5nm 0.4 3nm λ 0 300nm 140nm nm 2D TDGL Simulation for SC Disks SC disk in a perpendicular magnetic field (2D) 2R 0 = 20 ξ 0 l vortex penetration SC disk magnetic field ψ 2 vortex B a = 0.16 B c2 B a = 0.2 B c2 B a = 0.4 B c2 Y. Mawatari, HTS Modelling 2016, TDGL-Force-Free 4 / 17

2D TDGL Simulation for Current Carrying SC

current J t SC strip w = 50 ξ 0 ψ 2 vortex

9 J d similar to the weak link (Josephson

vortex and antivortex nucleation l vortex

current J t photon SC strip w = 50 ξ 0 ψ 2

3 2D TDGL Simulation for Current Carrying SC Strips current-carrying SC strip with an edge defect (2D) l vortex nucleation at the edge defect l vortex flow transport current J t SC strip w = 50 ξ 0 ψ 2 vortex 0.6 J d 0.8 J d 0.9 J d similar to the weak link (Josephson junction) Y. Mawatari, HTS Modelling 2016, TDGL-Force-Free 5 / 17 2D TDGL Simulation for SC Strip Photon Detectors SC strip photon detector (2D) simulation based on the TDGL and heat diffusion equations l hot spot due to the photon incidence l vortex and antivortex nucleation l vortex flow l beltlike normal domain transport current J t photon SC strip w = 50 ξ 0 ψ 2 antivortex normal domain vortex J t t = 11 J t t = 24 J t t = 35 Y. Mawatari, HTS Modelling 2016, TDGL-Force-Free 6 / 17

Vortex Dynamics and Electromagnetic Response Vortex Dynamics in the Transverse (B J) Configuration: l driving (Lorentz) force on vortices, F L = J B ~ current density, J l vortex

.. = 0 (F p : pinning force and F v : viscous drag force) ~ E-J characteristics, critical state model ~ electromagnetic response of superconductors B transverse, B J B vortex J B

Mawatari, 160615 HTS Modelling 2016, TDGL-Force-Free 7 / 17 Longitudinal Field Effect Vortex Dynamics in the Longitudinal (B //J) Configuration: l F L = J B ~ 0 (force free?) l v?

4 Vortex Dynamics and Electromagnetic Response Vortex Dynamics in the Transverse (B J) Configuration: l driving (Lorentz) force on vortices, F L = J B ~ current density, J l vortex velocity, v ~ electric field, E = B v J l force balance equation for vortices, F L + F p + F v +... = 0 (F p : pinning force and F v : viscous drag force) ~ E-J characteristics, critical state model ~ electromagnetic response of superconductors B transverse, B J B vortex J B vortex E = B v F L = J B v Y. Mawatari, HTS Modelling 2016, TDGL-Force-Free 7 / 17 Longitudinal Field Effect Vortex Dynamics in the Longitudinal (B //J) Configuration: l F L = J B ~ 0 (force free?) l v? ~ E B v? l force balance equation? ~ E-J characteristics?? (double critical state model,...) ~ electromagnetic response??? (e.g., large I c, paramagnetic effect,... ) J B longitudinal, B //J right-handed helical vortex B a applied magnetic field J B j self field due to the transport current left-handed right-handed Lehninger et al. Principles of Biochemistry Y. Mawatari, HTS Modelling 2016, TDGL-Force-Free 8 / 17

TDGL Simulation on Thin Superconducting Wires l radius R 0 = 10 ξ 0, length l applied magnetic field B a = 0.2 0.4 B c2 l transport current density 0.3 0.

5 TDGL Simulation on Thin Superconducting Wires l radius R 0 = 10 ξ 0, length l applied magnetic field B a = B c2 l transport current density J d TDGL equation for thin superconducting wires: l self field neglected, A = B a r/2 τ 0 t + i 2π Φ φ ψ = ξ 2 0 i 2π 2 A 0 φ 0 ( j s + j n ) = 0 l boundary conditions n ( (2πi φ 0 )A)ψ = 0 n Φ = 0 z Φ = J t σ n at the surface, ψ + 1 ψ 2 at the side surface, ψ, J t at the ends (normal current injected) Parameters for YBa 2 Cu 3 O 7 at T = 0K l τ 0 : GL relaxation time ~0.03ps l ξ 0 : coherence length ~1nm l λ 0 : penetration depth ~100nm l B c2 : upper critical field ~100T 2R 0 = 20 ξ 0 l J d : depairing current density ~10 12 A/m 2 Y. Mawatari, HTS Modelling 2016, TDGL-Force-Free 9 / 17 FEM Calculation by COMSOL Multiphysics Numerical TDGL-FEM simulation has been done by using the commercial software, COMSOL Multiphysics ( Coefficient Form PDE l order parameter ψ (x, y, z, t) l scalar potential Φ (x, y, z, t) 2R 0 = 20 ξ 0 Mesh: l Element size parameter (in units of ξ 0 ): max. element size 2.1, min. element size 0.09 l Complete mesh consists of: 33,317 domain elements, 2,576 boundary elements, and 180 edge elements Calculation: l Number of degrees of freedom solved for 188,608 (plus 22,072 internal DOFs) l Calculation time ~15 hours for 0 < t < 1,000 Y. Mawatari, HTS Modelling 2016, TDGL-Force-Free 10 / 17

Simulation Results (I): Transverse Field l

15 B c2 l transport current density 0.26 J d 0.

15 B 2R c2 0 = 20 ξ 0 vortex flow ψ 2 J t Y.

TDGL-Force-Free 11 / 17 Simulation Results (II):

6 Simulation Results (I): Transverse Field l transverse magnetic field B a = 0.15 B c2 l transport current density 0.26 J d 0.26 J d B a = 0.15 B 2R c2 0 = 20 ξ 0 vortex flow ψ 2 J t Y. Mawatari, HTS Modelling 2016, TDGL-Force-Free 11 / 17 Simulation Results (II): Transverse Field l transverse magnetic field B a = 0.15 B c2 l transport current density 0.26 J d 0.26 J d B a = 0.15 B 2R c2 0 = 20 ξ 0 vortex flow vortex flow ψ 2 J t Y. Mawatari, HTS Modelling 2016, TDGL-Force-Free 12 / 17

7 Simulation Results (III): Longitudinal Field l longitudinal magnetic field B a = 0.2 B c2 l transport current density 0.5 J d 0.5 J d B a = 0.2 B 2R c2 0 = 20 ξ 0 J t l right-handed helical vortices Y. Mawatari, HTS Modelling 2016, TDGL-Force-Free 13 / 17 Simulation Results (IV): Longitudinal Field l longitudinal magnetic field B a = 0.2 B c2 l transport current density 0.57 J d 0.57 J d B a = 0.2 B 2R c2 0 = 20 ξ 0 vortex cutting è voltage pulse t = 860 t = 900 t = 970 Y. Mawatari, HTS Modelling 2016, TDGL-Force-Free 14 / 17

Mawatari, 160615 HTS Modelling 2016, TDGL-Force-Free 15 / 17 Simulation Results (VI): Longitudinal Field l longitudinal magnetic field B a = 0.

8 Simulation Results (V): Longitudinal Field l longitudinal magnetic field B a = 0.4 B c2 l transport current density 0.35 J d l propagation of vortex cutting 0.35 J d L 0 = 40 ξ 0 B a = 0.4 B 2R c2 0 = 20 ξ 0 t = 570 t = 586 t = 596 Y. Mawatari, HTS Modelling 2016, TDGL-Force-Free 15 / 17 Simulation Results (VI): Longitudinal Field l longitudinal magnetic field B a = 0.4 B c2 l transport current density 0.52 J d 0.52 J d L 0 = 20 ξ 0 B a = 0.4 B 2R c2 0 = 10 ξ 0 Right-handed cutting left-handed straight t = 195 t = 215 t = 230 t = 275 Y. Mawatari, HTS Modelling 2016, TDGL-Force-Free 16 / 17

9 Summary We have numerically investigated three-dimensional dynamics of vortices in thin superconducting (SC) cylinders exposed to transport currents and longitudinal magnetic fields. l The time-dependent Ginzburg-Landau (TDGL) equations are useful for modelling of nanoscale (nm µm) SC phenomena (e.g., vortex dynamics, SC electronics devices, ). l Helical vortex structure in current carrying SC wires in longitudinal magnetic fields l Complicated vortex dynamics with cutting for large current: n propagation of vortex cutting in many vortices n right-handed helices è cutting è left-handed helices è Future issues: l Further investigation (by changing applied field, current, and sample size, ) to clarify the complicated vortex dynamics l Large scale simulation for bulk behavior l Pinning, anisotropy, thermal fluctuation, Y. Mawatari, HTS Modelling 2016, TDGL-Force-Free 17 / 17

Ginzburg-Landau. Time-Dependent Ginzburg-Landau Eqs.

Ginzburg-Landau. Time-Dependent Ginzburg-Landau Eqs. ,, 9/13/15 Ginzburg-Landau The Time-Dependent Ginzburg-Landau Theory 1. Time-dependent Ginzburg-Landau (TDGL) Equations. Energy Balane 3. Vortex Flow 4. Marosopi Modelling Simulation Y. Mawatari, 15913,