1. Folgefonna Glacier, Norway

2. Finger patterns produced by flux jumps in superconductors 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)

3. Linearly polarized light Faraday-active crystal Magnetic field H  (H)(H) F Magneto-optical Imaging

4. t BaBa Magneto-optical movie YBCO film Conventional flux penetration 1 mm Flux density  Brightness BaBa

5. t BaBa 250 sec50 sec 68 mT Flux pattern produced by instability T = 4 K 1 mm MgB 2 film [Sung-Ik Lee, POSTECH, Korea] J c ~ 10 7 A/cm 2 Magneto-optical movie

6. t=0 t=43.3ns final state How fast dendrites propagate ? Setup for ultrafast MO imaging P. Leiderer, University of Konstanz

7. sound velocity ~10 km/s << Fermi velocity ~500km/s << Dendrite propagation velocity Time (ns) Distance (  m) ~100 km/s v (km/s) H (mT)

8. Dendritic patterns in other MgB 2 films Screen printing, Al 2 O 3 substrate 3000 nm, T c =35K G. Gritzner, Univ. of Linz, Austria Pulse Laser Deposition on 1102 Al 2 O 3 substrate 400nm, T c =39K S.I. Lee, Pohang Univ., Korea PLD, SrTiO 3 substrate, 250nm, T c =28K S.X. Dou, Wollongong, Australia

9. Nb : C.A. Duran et al. PRB 52, 75 (1995) YBaCuO : only induced by laser pulse P. Leiderer et al. PRL 71, 2646 (1993) Nb 3 Sn : 150 nm, 3.8 K Oslo, cond-mat/0211349 NbN 150 nm, 4.2 K Oslo 2003 Dendritic patterns in other materials What is the role of sample inhomogeneities ?

10. 3 identical experiments: field ramp from 0 to 13.6mT for 10sec the nucleation place: well reproduced the exact flux pattern: never reproduced Irreproducibility

11. Sample Inhomogeneities OR Self-organization T=8.0K B a = 51 mT The exact pattern is every time different Irreproducibility at high fields

12. Theory  T 0   J c     Q   T >  T 0 1) Flux motion releases heat 2) T rise weakens flux pinning Thermal runaway Conventional flux jump mechanism: We look for spatially-nonuniform solutions

13. y z H 0 penetrated by flux no flux x l j,E Maxwell and thermal diffusion Linear Analysis unstable if Re > 0 non-uniform if k y  0

14. Solution: (k x,k y )

15. the local J remains constant fastest possible growth, maximal jE J the local J is reduced slowing down J Why narrow fingers ?

16. Slow Fast t h e r m a l d i f f u s i o n Contour plot of the instability increment Re (k x,k y ) Finite k y => Fastest growth  = 0.01  > 1

17. 0369 -0.5 0.0 0.5 k y Re B a The instability increment

18. Numerical Solution TemperatureElectric field T(t=0)=0.0001 * ``white noise” to introduce all k y y sample edge Finger pattern with some characteristic k y is formed in a self-organized way

19. Numerical Solution, finger propagation B E T Increasing applied magnetic field Beyond the linear regime a few strongest fingers survive and propagate into the flux free area..... in agreement with experiment linearized j(E) full non-linear j(E)

20. H(E) phase diagram Fingering is not sensitive to initial T(x,y), E(x,y) boundary conditions J c (B) dependence  11/n

21. Important estimates

22. Large electric field needed, E > 0.1 V/m Ramping magnetic field: E ~ w*dH/dt Flux jumping in thin films R.G.Mints and E.H. Brandt, PRB 1996 d=1  m, dH/dt=100T/s H=1T our experiment: 0.001 T/s E(r,t)  one needs: dH/dt ~ 100 T/s

23. increasing applied field real-time 25  m High resoltuion magneto-optical movie edge non-thermal vortex avalanches superconductor NbSe crystal from P. Gammel

24. Detecting vortex jumps  (r) B a =4G Subtract subsequent images:  B(r) vortex arrived vortex left 90 % no motion 1010 4040 1010 B (r)B (r)

25. Counting vortices  B (r) 150  0 B (r)B (r) 5 vortices has moved 11 vortices has entered from the edge dH/dt : =2.4 10 -10 V/m local  B : local E ~ 10 -7 V/m local E ~ 400 perhaps much more: 1  s <<  t << 1/24s

26. Effect of sample shape Bulk Film Linear theory Fingering (this presentation) Fingering (work in progress) Experiment few studies… Fingering + Branching Simulations No branching Branching

27. Conclusions Fingering instability is observed experimentally A linear theory based on the Maxwell and thermal diffusion equations is proposed. It predicts fingering for E>E c, H>H f (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

28. ФНТ, Шкловский, 1995-6 Shklovskij разогрев

29. A very recent preprint by Aranson et al., cond-mat/0407490 confirms the presence of fingering + branching in films

30. Modulation of magnetic field - minimize gap - sample with small - high sensitivity MO films Signal loss in optical system - optimize optics for polarization contrast Mechanical noise - reduce vibrations from cryo- system and other sources MO image FGF superconductor L Custom MO microscope Resolution: 3  m single vortex Abrikosov lattice in NbSe 2

31. t BaBa 200 sec 17mT movie MO movie MgB 2 film T=3.6K Movie of flux penetration: edge region 5 mm

32. typical size ~10-20  m number of flux quanta 50 - 10000  0 abrupt (  t<0.1s,  B a <0.02mT) Analyzing difference images 7.15 mT 7.40 mT linear ramp of B a 15 MO images T=3.6K black - flux left ??? = MO image (7.165mT) — MO image (7.150mT) white - flux arrived gray - no changes  B a = 0.015mT,  t=2.5 sec local increase of flux density - flux jump 2300  0 1100  0 250  0

33. jump size,  0 100 1,000,000 1000 dE/dj >  Q M >  Q  QMQM QTQT Number of jumps  B (r) 10  m 2 mm 200  m

34. 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 t i m e

35. 0369 -0.5 0.0 0.5 k y Re B a The instability increment

36. T i m e Simulations Temperature Electric field

37. Temperature distrbution Simulations based on Maxwell & thermal diffusion equations Aranson et al., Phys.Rev.Lett. 2001 Pattern not as in experiment No results for flux distribution

38. Irregular flux patterns can be due to sample inhomogeneities : What’s the role of inhomogeneities in formation of the dendritic flux patterns?

39. Molecular Dynamics Simulations: Force F M from Meissner current intervortex forces ~ 1/r 2 position-independent pinning, F pin (T) ~ 1 - T/T c moving vortices leave a trail of heated area T riri  v i = F M (r i ) +  j F(r i,r j ) + F pin Europhys.Lett. 59, 599 (2002) experiment simulations The simulations reproduce: channels branching irregular Possible to reproduce dendritic pattern?

40. Dendrites can damage material Brull et al, Annalen der Physik 1992, v.1, p.243 YBCO Dendrite was here

41. In MgB 2 dendrites only reduce Jc Dendritic instability reduces J c by a factor of 2

42. Nb disk, Goodman et al., Phys. Lett. 18, 236 (1965) Favors uniform jumps, k y =0 Thermal diffusion