Fingering, Fronts, and Patterns in Superconductors Alan Dorsey University of Florida Collaborators: Ray Goldstein (U Arizona) John DiBartolo (Brooklyn Poly) Salman Ullah (Microsoft) Wexler and Dorsey, Phys Rev B 64, 115312 (2001) LR and ATD, PRL 88, 216802 (2002) Support from the NSF 11 May 2005 Lorentz Center Leiden

Welcome to Florida! Gainesville 11 May 2005 Lorentz Center Leiden

UF Lightning Research International Center for Lightning Research and Testing (ICLRT) Prof. Martin Uman Prof. Vladimir Rakov 11 May 2005 Lorentz Center Leiden

11 May 2005 Lorentz Center Leiden

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/ 11 May 2005 Lorentz Center Leiden

Free boundary model for the moving superconductor/normal interface Normal regions: moving interface generates eddy currents (Ampere’s Law plus Ohm’s 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: 11 May 2005 Lorentz Center Leiden

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. 11 May 2005 Lorentz Center Leiden

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: 11 May 2005 Lorentz Center Leiden

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). 11 May 2005 Lorentz Center Leiden

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). 11 May 2005 Lorentz Center Leiden

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 Type of pattern depends upon the area fraction. When the area fraction is close to ½, one observes stripes; when it is close to zero or one, one observes bubbles (which tend to form a hexagonal lattice). 11 May 2005 Lorentz Center Leiden

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 Ferrofluid is often mangenite (iron oxide) suspended in kerosene. Good reference is Rosensweig, Ferrohydrodynamics. Dimensionless number which depends the importance of surface tension is the Bond number. It is the ratio of the field energy to the surface energy. Dynamics is highly overdamped. See Langer, Goldstein, and Jackson (1992). Surface tension competes with dipole-dipole interaction… 11 May 2005 Lorentz Center Leiden

Results courtesy of Ken Cooper ferromovie.mov http://www.its.caltech.edu/~jpelab/Ken_web_page/ferrofluid.html 11 May 2005 Lorentz Center Leiden

Langmuir monolayer (phospholipid and cholesterol) Modulated phases Emphasize the common patterns. Langmuir monolayer pattern is taken from M. Seul and V. S. Chen, PRL 70, 1658 (1993). Note the cybotactic groups which have local layer order. Garnet film figure taken from Seul and Wolfe, PRA 46, 7519 (1992). Langmuir monolayer (phospholipid and cholesterol) Intermediate state of type-I superconductor 11 May 2005 Lorentz Center Leiden

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. 11 May 2005 Lorentz Center Leiden

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). 11 May 2005 Lorentz Center Leiden

Experiments C. R. Reisen and S. G. Lipson, Phys. Rev. B (2000). Pb-In sample, 3mm diameter, 0.14 mm thick 11 May 2005 Lorentz Center Leiden

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 11 May 2005 Lorentz Center Leiden

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. 11 May 2005 Lorentz Center Leiden