11 May 2005 Lorentz Center Leiden 1

Fingering, Fronts, and Patterns in Superconductors

Alan DorseyUniversity 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

Prof. Martin UmanProf. Vladimir Rakov

International Center for Lightning Research and Testing (ICLRT)

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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

BDB Bt2

iBni nBDvB )/(

)1( 0 nci vdHB

• In the superconducting region the magnetic field is zero.

• Normal regions: moving interface generates eddy currents (Ampere’s Law plus Ohm’s Law):

• 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

nBvn /

B

• 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.

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

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Flux expulsion/dendritic growth analogy

TDT Tt2

])/()/([ ''liquidpTsolidpTn nTcDnTcDLv

)1( 0 nmi vdTT

• 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

222

||)/2(2

])/2([

baem

eit

A

i

)(4 sn JJA

• 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 forcesinhomogeneous (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

http://www.its.caltech.edu/~jpelab/Ken_web_page/ferrofluid.html

ferromovie.mov

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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/superconductor 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).

Experiments: magnetooptics imagesOf Niobium films

Simulations of Aranson et al.

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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.