Fingering, Fronts, and Patterns in Superconductors

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Fingering, Fronts, and Patterns in Superconductors

• Alan Dorsey
• University of Florida
• Collaborators
• Ray Goldstein (U Arizona)
• John DiBartolo (Brooklyn Poly)
• Salman Ullah (Microsoft)

Support from the NSF
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Welcome to Florida!
Gainesville
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UF Lightning Research
International Center for Lightning Research and
Testing (ICLRT)
Prof. Martin Uman
Prof. Vladimir Rakov
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Outline

• Interface motion in superconductors
• Interfacial instabilities
• Analogies with dendritic growth
• Propagating fronts
• Modulated phases and the intermediate state of
  type-I superconductors
• Nonequilibrium vortex patterns and thermal
  instabilities

http//www.fys.uio.no/super/dend/
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Free boundary model for the moving
superconductor/normal interface

• Normal regions moving interface generates eddy
  currents (Amperes Law plus Ohms Law)

• In the superconducting region
• the magnetic field is zero.

• At the interface we have the
• boundary condition

• For a flat interface the field at the
• interface is the critical field for a
• curved interface

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Interfacial (Mullins-Sekerka) instability

• is largest near the bump

• Since the normal
• velocity is largest near the
• bump, so bumps grow faster!

• A linear stability analysis
• shows that the growth rate is
• positive at long wavelengths.
• Surface tension stabilizes the
• growth at short wavelengths.

• A similar instability occurs in
• the dendritic growth of solids.

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Flux expulsion/dendritic growth analogy

• A piece of solid grows into its
• supercooled liquid phase. This
• releases a latent heat L that
• must diffuse away from the
• interface for the solid to grow.

• At the interface the rate of heat
• production is equal to the rate
• at which heat flows into the solid
• and liquid.

• The Gibbs-Thomson condition

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Modeling time dependent Ginzburg Landau theory

• Coupled nonlinear PDEs for the
• order parameter and the vector
• potential

• Solve numerically using lattice
• gauge theory methods (Frahm,
• Ullah, Dorsey (1991).

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Propagating front solutions

• DiBartolo and Dorsey (1996) special one
  dimensional solutions of TDGL equations for an
  interface.
• Exact solution for special parameter values.
• Matched asymptotics and marginal stability
  analysis.
• Pulled vs. pushed fronts (Ebert and van
  Saarloos).

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Competing interactions

• Long range repulsive force uniform phase
• Short range attractive force compact structures
• Competition between forces?inhomogeneous (meso)
  phase
• Ferromagnetic films, ferrofluids, type-I
  superconductors, block copolymers

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Ferrofluid in a Hele-Shaw cell

• Ferrofluid colloid of 1 micron spheres. Fluid
  becomes magnetized in an applied field.
• Hele-Shaw cell ferrofluid between two glass
  plates

Surface tension competes with dipole-dipole
interaction
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Results courtesy of Ken Cooper
ferromovie.mov
http//www.its.caltech.edu/jpelab/Ken_web_page/fe
rrofluid.html
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Modulated phases
Langmuir monolayer (phospholipid and cholesterol)
Intermediate state of type-I superconductor
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The intermediate state

• For thin films complete flux explusion is
  energetically unfavorable.
• The sample breaks up into normal and
  superconducting regions that coexist.
• The domain size is set by a competition between
• Demagnetizing energy (favors finely divided
  structure).
• Surface energy (favors a coarse structure).
• Laminar model developed by Landau in 1937.

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Current loop model

• Supercurrents circulate on the normal/superconduct
  or boundries.
• There is a long range Biot-Savart interaction
  that causes branching.
• The instability is regulated on short scales by
  surface tension.
• Overdamped dynamics proposed by Dorsey and
  Goldstein (1998).

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Experiments
C. R. Reisen and S. G. Lipson, Phys. Rev. B
(2000). Pb-In sample, 3mm diameter, 0.14 mm thick
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Nonequilibrium vortex patterns

• Vortex entry in type-II superconductors often
  results in dendrites.
• Subtle interplay of geometry, thermal effects,
  and nonlinear IV characteristics.
• Recent theoretical work by I. S. Aranson et al.,
  Physical Review Letters (2005).

Simulations of Aranson et al.
Experiments magnetooptics images Of Niobium films
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Summary

• Fingering dynamical instabilities during
  magnetic flux entry (free boundary problem,
  Mullins-Sekerka instability).
• Fronts novel propagating front (interface)
  solutions in time-dependent GL theory.
• Patterns
• Competing interactions attractive short range
  and repulsive long range lead to mesoscale
  patterns.
• Intermediate state patterns in type-I
  superconductors.
• Nonequilibrium vortex patterns during field entry
  and exit.