Critical state controlled by microscopic flux jumps in superconductors Daniel Shantsev Physics Department, University of Oslo, Norway in collaboration with Vitali Yurchenko, Alexander Bobyl, Yuri Galperin, Tom Johansen Eun-Mi Choi, Sung-Ik Lee Pohang University of Science and Technology, Korea

a superconductor can carry? What determines the maximal current a superconductor can carry?

the critical magnetic field 1. Solsby Rule H Magnetic field created by current, should not exceed the critical magnetic field H = I / 2p R < Hc R Jc(1) = 2Hc / R I

I J < Jc(2)  Hc / l 2. Depairing current density Ginsburg-Landau equations have a solution only if R J < Jc(2)  Hc / l For J>Jc the kinetic energy of Cooper pairs exceeds the superconducting energy gap I

x~10 Å J l  B dA = h/2e = 0 Meissner effect normal core Flux quantum: x~10 Å l J B(r) normal core Vortex lattice

Lorentz force F = j F0 current Ba J Vortices are driven by Lorentz force and their motion creates electric field E ~ dB/dt Lorentz force F = j F0 Ba J pinning force Lorentz force Vortices get pinned by tiny defects and start moving only if Lorentz force > Pinning force current

J < Jc(3) = U / F0 Jc(3) ~ Hc / l U(r) 3. Depinning current density Superconductor remains in the non-resistive state only if Lorentz force < Pinning force, i.e. if U(r) J < Jc(3) = U / F0 Ideal pinning center is a non-SC column of radius ~ x so that U ~ Hc2x2 and similar to the depairing Jc Jc(3) ~ Hc / l

positive feedback J*E velocity current +kT E ~ dB/dt Vortex motion dissipates energy, J*E Local Temperature Increases velocity +kT It is easier for vortices to overcome pinning barriers positive feedback current Vortices move faster

x Hfj  Hfjslab (d/w)1/2 x H Jc(4) = (2C Jc(3) [d Jc(3) /dT]-1)1/2/2w Thermal instability criterion ~ Swartz &Bean, JAP 1968 dQM > dQT - instability starts dQT = C(T) dT dQM = Jc(T) dF = H2/2Jc dJc/dT dT H > Hfj = (2C Jc [dJc/dT]-1)1/2 Jc(4) = (2C Jc(3) [d Jc(3) /dT]-1)1/2/2w j H dF x Hfj  Hfjslab (d/w)1/2 x H j 2w d<<w dF D. S. et al. PRB 2005

List of current-limiting mechanisms Solsby, Jc ~ Hc/R Depairing current Jc ~ Hc / l Depinning current, Jc (U) Thermal instability current, Jc(C,..) Jc(3) < Jc(4) < Jc(1) < Jc(2) We need to know which Jc is the most important i.e. the smallest! Achieved

J >Jc(3) a small finite resistance appears How to distinguish between Jc’s J >Jc(3) a small finite resistance appears J >Jc(4) a catastrophic flux jump occurs (T rises to ~Tc or higher) Brull et al, Annalen der Physik 1992, v.1, p.243 Gaevski et al, APL 1997

Critical state is destroyed Global flux jumps M(H) loop DM ~ M Critical state is destroyed Muller & Andrikidis, PRB-94

Critical state is destroyed locally Dendritic flux jumps MgB2 film DM ~ 0.01 M Critical state is destroyed locally Europhys. Lett. 59, 599-605 (2002) Magneto-optical imaging Zhao et al, PRB 2002

Microscopic flux jumps 5 mm MgB2 film fabricated by S.I. Lee (Pohang, Korea) MgB2 film 100 mm Magneto-optical movie shows that flux penetration proceeds via small jumps

2300F0 1100F0 250F0 Analyzing difference images flux jump 7.15 mT 7.40 mT linear ramp of Ba 15 MO images T=3.6K = MO image (7.165mT) — MO image (7.150mT) local increase of flux density - flux jump 2300F0 1100F0 250F0

Jc(3) OR Jc(4) ? The problem with microscopic jumps 31,000F0 7,500F0 Too small, DM ~ 10-5 M : invisible in M(H) Critical state is not destroyed B-distribution looks as usual x From the standard measurements one can not tell what limits Jc: vortex pinning OR thermal instabilities Jc(3) OR Jc(4) ? edge Flux profiles before and after a flux jump have similar shapes

What can be done One should measure dynamics of flux penetration and look for jumps If any, compare their statistics, B-profiles etc with thermal instability theories If they fit, then Jc=Jc(4) , determined by instability; actions – improve C, heat removal conditions etc, if not, then Jc=Jc(3), determined by pinning; actions – create better pinning centers Jump size (F0) Number of jumps power-law Altshuler et al. PRB 2005 peak (thermal mechanism)

Two Jc’s in one sample 300 mm 70 mm Jcleft  2 Jcright Jc(3) Jc(4)

Dendritic instability can be suppressed by a contact with normal metal Baziljevich et al 2002

Jc(3) Jc(4) MgB2 300 mm 70 mm Two Jc’s in one sample Au 9 mm w 3 mm Au suppresses jumps, Jc is determined by pinning Jc is determined by jumps Jc(3) Jc(4)

A graphical way to determine Jc’s: d-lines

MgB2 3 mm Au Jc1 Jc2 ?

α

α ≈ π/3 β α α ! jc1 ≈ 2jc2 !

Conclusions Thermal avalanches can be truly microscopic as observed by MOI and described by a proposed adiabatic model These avalanches can not be detected either in M(H) loops or in static MO images => “What determines Jc?” - is an open question MO images of MgB2 films partly covered with Au show two distinct Jc’s: - Jc determined by stability with respect to thermal avalanches - a higher Jc determined by pinning http://www.fys.uio.no/super/

Evolution of local flux density 5x5 mm2 Local B grows by small and repeated steps 7.9mT 7.4mT 7mT linear ramp 6 mT/s local flux density calculated from local intensity of MO image; each point on the curve corresponds to one MO image

Jc is determined by Jc depends on stability with respect to thermal avalanches Jc depends on thermal coupling to environment, specific heat, sample dimensions But we need to prove that the observed microscopic avalanches are indeed of thermal origin

Adiabatic critical state for a thin strip In the spirit of Swartz &Bean in 1968 Adiabatic : All energy released by flux motion is absorbed Critical state Biot-Savart for thin film Flux that has passed through “x” during avalanche

Flux jump size Thermal origin of avalanches T=0.1Tc We fit Bfj ~ 2 mT Tth ~ 13 K F(Ba) dependence using only one parameter: Thermal origin of avalanches

Irreproducibility B(r) DB(r) The final pattern is the same T=3.6K Ba = 13.6 mT B(r) the flux pattern almost repeats itself MOI(8.7mT) - MOI(8.5mT) DB(r) DB(r) is irreproducible! The final pattern is the same but the sequences of avalanches are different

Magneto-optical Imaging polarizer P A mirror MO indicator image large small Faraday rotation S N light source Linearly polarized light Faraday-active crystal Magnetic field H q (H) F Square YBaCuO film