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Simulation Of Vortex Pinning in Two-BandSuperconductors
Chad Sockwell
Florida State University
February 5, 2016
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About Me
Chad Sockwell
Undergrad in Physics and S.C. at FSU (Honors Thesis)
Modeling Superconductivity With Max Gunzburger and JanetPeterson
Currently Master’s student in S.C.
Applying to DOE Fellowship
Possible PhD in Physics or SC
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Outline
Background and Motivation
Simulation of Superconductivity
Challenges and Future Work
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Background
What is a Superconductor(SC)?
Zero Electrical Resistance
No Waste Heat
Normal metals are penetratedby Mag. Fields
Some SC are not (MeissnerEffect)
Some are penetrated only bytubes of flux (Vortices)
Figure : 3D SC with Vortices
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Why Vortex Dynamics are Important
Moving Vortices (Flux flow)creates Resistance
f x = J y× B z
E y = B z× u x
Vortices (B) + Current (J)=Flux flow
Flux Flow induces ElectricField (E) and Voltage (V)
Resistance Now Exists (VI =R)
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Vortex Pinning Comes to the Rescue
Immobilizing the Vortices IsCrucial
Non Superconducting Metal=Normal Metal= Pinning Sites(Outlined in Black)
Vortices ”Stick” To Impurities
Limited Increase In Jc
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Applications of Large Scale Simulations
Large Scale Simulations Could Improve Technology:
Efficient Current Carriers
Powerful Magnets (bymagnetization)
MRI
Efficient Mag Lev
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Modeling Vortex Pinning with Normal Inclusions
TD-Ginzburg-Landau model ( Coupled System of NL PDES)
ψ ∈ C, Like complex phase field order parameter
A ∈ R2 magnetic vector potential
Γ(∂ψ
∂t− i
κ
σJy) + (|ψ|2 − τ)ψ + (
i
κ∇− A)2ψ = 0 in Ω
σ∂A
∂t− J +∇×∇× A +
i
2κ(ψ∗∇ψ − ψ∇ψ∗) + |ψ|2A = ∇×H in Ω
∇× A× n = (H− J z(x − L/2))× n on ∂Ω
A · n = 0 on ∂Ω
∇ψ · n = 0 on ∂Ω
ψ(t = 0) = ψ0 ; A(t = 0) = A0 ; ∇ · A(t = 0) = 0 in Ω
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Model To Simulation
Simulation of Magnesium Diboride (Two Band Model)
Need Material Parameters: κ, σ, Γ
Various Inputs: Field (H), Applied Current (J), Temperature (τ)
Change Material Parameters in Normal Metals (Impurities)
Γ(∂ψ
∂t− i
κ
σJy) + (|ψ|2 − τ)ψ + (
i
κ∇− A)2ψ = 0 in Ω
σ∂A
∂t− J +∇×∇× A +
i
2κ(ψ∗∇ψ − ψ∇ψ∗) + |ψ|2A = ∇×H in Ω
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Numerical Methods
FEM →Triangular Piecewise Parabolic Elements & Gauss Quadrature
Newton’s Method, FullJacobian
Sparse Storage (CRS)
Adaptive Backward Euler
Parallel Solver (SUPERLU)(9/10 NL Time Cost)
These methods were ”GoodEnough for Now”
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Passing a Resistance Free Current
Metal - SuperconductorInterface
ψ →0 in Vortices and Metals
Flux Flow produces Resistive(or Normal) Current
Impurities Outline In Black
Normal Current→Resistance
Did Pinning Prevent FluxFlow?
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Computational Challenges For Practical Simulation
Large Spatial and Time Scale
Large Storage Costs→ Distributed memory
Long Solve Times → Parallel Iterative Solvers & Preconditioners
Trilinos Distributed Environment & Solver’sML & Hypre AMG Preconditioners?Jacobian Free Newton-Kylov Methods?
3D-Modeling to Infinity
BEM for the Exterior?
Finding Maximum Current
Optimization, Continuation?
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Need Lots of Vortices
Figure : A general function f (x)
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