Folgefonna Glacier , Norway
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Folgefonna Glacier , Norway Finger patterns produced by flux jumps in superconductors Daniel Shantsev Department of Physics, University of Oslo, Norway in collaboration with A.L. Rakhmanov, Inst. Th&Appl. Electrodyn., Moscow, Russia Y. M. Galperin, T. H. Johansen, University of Oslo, Norway • Experiment (2001-2004) • Theory (2004) Magneto-optical Imaging image q F(H) Faraday-active crystal A small Linearly polarized light large Faraday rotation polarizer P H Magnetic field light source MO indicator mirror N S small Conventional flux penetration Flux density Brightness Ba Magneto-optical movie Ba 1 mm t YBCO film Flux pattern produced by instability MgB2 film [Sung-Ik Lee, POSTECH, Korea] Jc ~ 107 A/cm2 T=4K Magneto-optical movie Ba 68 mT t 250 sec 50 sec 1 mm How fast dendrites propagate ? Setup for ultrafast MO imaging P. Leiderer , University of Konstanz t=0 beam splitter 0.5 mm Bext || c-axis analyzer Ti:sapphire laser 150 fs, 800nm lens YBCO-film with YIG variable delay line polarizer camera delay line for double exposures t=43.3ns final state v (km/s) Distance (mm) Dendrite propagation velocity Time (ns) sound velocity ~10 km/s << ~100 km/s << H (mT) Fermi velocity ~500km/s Dendritic patterns in other MgB2 films Pulse Laser Deposition on 1102 Al2O3 substrate 400nm, Tc=39K S.I. Lee, Pohang Univ., Korea Screen printing, Al2O3 substrate 3000 nm, Tc=35K G. Gritzner, Univ. of Linz, Austria PLD, SrTiO3 substrate, 250nm, Tc=28K S.X. Dou, Wollongong, Australia Dendritic patterns in other materials Nb: Nb3Sn: 150 nm, 3.8 K C.A. Duran et al. PRB 52, 75 (1995) NbN 150 nm, 4.2 K Oslo 2003 Oslo, cond-mat/0211349 YBaCuO: only induced by laser pulse P. Leiderer et al. PRL 71, 2646 (1993) What is the role of sample inhomogeneities ? Irreproducibility 3 identical experiments: field ramp from 0 to 13.6mT for 10sec the nucleation place: the exact flux pattern: well reproduced never reproduced Irreproducibility at high fields Ba = 51 mT T=8.0K The exact pattern is every time different Sample Inhomogeneities OR Self-organization Theory Conventional flux jump mechanism: 1) Flux motion releases heat 2) T rise weakens flux pinning T0 Jc Q T > T0 Thermal runaway We look for spatially-nonuniform solutions Maxwell and thermal diffusion z j,E y H 0 x l penetrated by flux no flux Linear Analysis unstable if Re > 0 non-uniform if ky0 Solution: (kx,ky) Why narrow fingers ? the local J is reduced J slowing down the local J remains constant J fastest possible growth, maximal jE Contour plot of the instability increment t = 0.01 Re (kx,ky) Slow Fast thermal diffusion Finite ky => Fastest growth t>1 The instability increment 0.0 Re 0.5 -0.5 Ba -1.0 0 3 ky 6 9 Numerical Solution Temperature Electric field y T(t=0)=0.0001 * ``white noise” to introduce all ky sample edge Finger pattern with some characteristic ky is formed in a self-organized way Numerical Solution, finger propagation Increasing applied magnetic field B linearized j(E) full non-linear j(E) E T Beyond the linear regime • a few strongest fingers survive • and propagate into the flux free area . . . . . in agreement with experiment H(E) phase diagram H Fingering is not sensitive to • initial T(x,y), E(x,y) • boundary conditions • Jc(B) dependence Uniform jumps dynamic criterion adiabatic criterion Hadiab S t Fingering t Ec/n 1 a b l Ec 1/n e E Important estimates Large electric field needed, Ramping magnetic field: E ~ w*dH/dt one needs: E > 0.1 V/m dH/dt ~ 100 T/s our experiment: Flux jumping in thin films R.G.Mints and E.H. Brandt, PRB 1996 d=1mm, dH/dt=100T/s H=1T E(r,t) <E> 0.001 T/s High resoltuion magneto-optical movie 25 mm increasing applied field real-time superconductor non-thermal vortex avalanches edge NbSe crystal from P. Gammel Detecting vortex jumps B (r) Ba=4G Subtract subsequent images: B(r) vortex arrived B (r) vortex left 10 90 % no motion 40 10 Counting vortices B (r) B (r) 1500 5 vortices has moved 11 vortices has entered from the edge dH/dt local E ~ 400 <E> perhaps much more: 1ms << t << 1/24s : <E>=2.4 10-10 V/m local B: local E ~ 10-7 V/m Effect of sample shape Bulk Linear theory Experiment Fingering Fingering (this presentation) (work in progress) few studies… No branching Simulations Film Fingering + Branching Branching Conclusions A linear theory based on the Maxwell and thermal diffusion equations is proposed. It predicts fingering for E>Ec, H>Hf(E) Fingering instability is observed experimentally H Uniform jumps dynamic criterion Fingering adiabatic criterion Hadiab S t Ec/n a b l e Ec E Simulations support the theory, show how the instability evolves beyond the linear regime More info: http://www.fys.uio.no/super, cond-mat/0405446 ФНТ, Шкловский, 1995-6 Shklovskij разогрев A very recent preprint by Aranson et al., cond-mat/0407490 confirms the presence of fingering + branching in films Custom MO microscope 3 mm Abrikosov lattice in NbSe2 Resolution: single vortex MO image • Modulation of magnetic field - minimize gap - sample with small - high sensitivity MO films FGF FGF FGF FGF superconductor L • Signal loss in optical system - optimize optics for polarization contrast • Mechanical noise - reduce vibrations from cryosystem and other sources Movie of flux penetration: edge region Ba movie 17mT 5 mm t 200 sec T=3.6K MO movie MgB2 film Analyzing difference images 7.15 mT = MO image (7.165mT) — MO image (7.150mT) Ba= 0.015mT, t=2.5 sec white - flux arrived gray - no changes linear ramp of Ba 15 MO images black - flux left ??? local increase of flux density - 23000 flux jump • typical size ~10-20mm T=3.6K 11000 7.40 mT 2500 • number of flux quanta 50 - 10000 0 • abrupt (t<0.1s, Ba<0.02mT) QM Number of jumps QT dE/dj > k 100 QM > QT 1000 1,000,000 jump size, 0 B (r) 200 mm 10 mm 2 mm Conclusion from Simulations In the linear regime • fingers are formed in agreement with the theory Beyond the linear regime • one finger will eventually dominate over others • and propagate into the flux free area in agreement with experiment time time The instability increment 0.0 Re 0.5 -0.5 Ba -1.0 0 3 ky 6 9 Simulations Temperature Electric field Time Aranson et al., Phys.Rev.Lett. 2001 Temperature distrbution Simulations based on Maxwell & thermal diffusion equations c H c z [ J ( J , T )], J Hzˆ t 4 T h C div (kT ) (T T0 ) J 2 (T , J ) t d • Pattern not as in experiment • No results for flux distribution Irregular flux patterns can be due to sample inhomogeneities : What’s the role of inhomogeneities in formation of the dendritic flux patterns? Possible to reproduce dendritic pattern? Europhys.Lett. 59, 599 (2002) Molecular Dynamics Simulations: vi = FM(ri) + jF(ri ,rj) + Fpin • Force FM from Meissner current • intervortex forces ~ 1/r2 T ri • position-independent pinning, Fpin(T) ~ 1 - T/Tc • moving vortices leave a trail of heated area experiment simulations The simulations reproduce: channels branching irregular Dendrites can damage material YBCO Brull et al, Annalen der Physik 1992, v.1, p.243 In MgB2 dendrites only reduce Jc from MO images 2 JC (MA/cm ) 10 8 6 4 from M(H) 2 0 0 10 T (K) 20 30 Dendritic instability reduces Jc by a factor of 2 Thermal diffusion Nb disk, Goodman et al., Phys. Lett. 18, 236 (1965) Favors uniform jumps, ky=0
