Global and local flux jumps in MgB2 films: Magneto-optical imaging and theory Daniel Shantsev, Yuri Galperin, Alexaner Bobyl, Tom Johansen Physics Department, University of Oslo, Norway Sung-Ik Lee Pohang University of Science and Technology, Korea

What determines the critical current density J c ? Thermal Vortex Avalanches at least for MgB 2 films for T<15 K stable critical state described by critical current J c thermo-magnetic instability (flux jumps) OR AND J c due to depinning of vortices J c due to thermal vortex avalanches >

 T 0   J c     Q   T >  T 0  Flux motion releases heat  Temperature rise weakens flux pinning Mechanism of Thermo-Magnetic Instability Positive feedback loop

Catastrophic flux jumps   M ~ M  Critical state is destroyed  Temperature rises to ~ T c Muller & Andrikidis, PRB-94 M(H) loop

Dendritic flux jumps Zhao et al, PRB 2002   M ~ 0.01 M  Critical state is destroyed locally  Temperature rises to ~ T c locally Europhys. Lett. 59, (2002) Magneto-optical imaging MgB 2 film

Microscopic flux jumps 5 mm MgB 2 film 100  m MgB 2 film fabricated by S.I. Lee (Pohang, Korea) Magneto-optical movie shows that flux penetration proceeds via small avalanches

number of vortices Analyzing difference images 7.15 mT 7.40 mT linear ramp of B a 15 MO images T=3.6K = MO image (7.165mT) — MO image (7.150mT) local increase of flux density - avalanche 2300    0

local flux density calculated from local intensity of MO image; each point on the curve corresponds to one MO image 5 x 5  m 2 linear ramp 6  T/s Evolution of local flux density 7mT7.4mT7.9mT Local B grows by small and repeated steps

x Flux density profiles film edge 31,000  0 7,500  0 Microscopic flux jumps do not destroy the critical state Flux profiles before and after a flux jump have similar shapes

Catastrophic jumps Dendritic jumps Microscopic jumps   M ~ M  Critical state is destroyed   M ~ 0.01 M : noisy M(H)  Critical state is destroyed locally  Global J c is suppressed   M ~ M : invisible in M(H)  Critical state is preserved What determines J c ?

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

Avalanche size distribution The distribution has a peak at some typical size (self-organized criticality suggests a power-law) hints to the thermal mechanism

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

x B, T - profiles film edge 31,000  0 7,500  0

We fit B fj ~ 2 mT T th ~ 13 K  (B a ) dependence using only one parameter: T=0.1T c 0.3T c Thermal origin of avalanches Flux jump size

Materials Dendritic avalanches seen by magneto-optics – all kinds of MgB 2 films (T<10K), C-doped MgB 2 Nb, NbN, Nb 3 Sn, YBaCuO, Pb, YB 2 C 2 O Peaked size distribution of avalanches measured by Hall probes Nb, Pb

Results Small flux avalanches (~1000  0 ) are observed in MgB 2 films using magneto-optical imaging for T<15 K Adiabatic model for the size of flux avalanches in a thin film is developed Good agreement suggests the thermal mechanism of avalanches  Thermal avalanches can be microscopic  These avalanches can control formation of the critical state without destroying it  J c is then determined by stability with respect to these thermal avalanches rather than by pinning  The avalanches are too small to be detected in M(H) loops Phys. Rev. B 72, (2005) Conclusions

temperature T th the instability field for a thin strip: Microscopic flux jumps Dendritic jumps H - T phase diagram

What determines the critical current density J c ? J c due to depairing of Cooper pairs J c due to depinning of vortices J c due to thermal vortex avalanches at least for MgB 2 films for T<15 K a new type of critical state with a new J c thermal avalanches (flux jumps) Breakdown of critical state <<< usually Vortex Pinning

Critical state Vortices : driven inside due to applied field get pinned by tiny inhomogeneities => Metastable critical state

Critical state in a superconductor YBaCuO film, picture from R.Wijngarden Distribution of flux density Sandpile Critical currentCritical angle picture from E.Altshuler

Pierre G. de Gennes comments in his classic 1966 book Superconductivity of Metals and Alloys: ‘‘We can get some physical feeling for this critical state by thinking of a sand hill. If the slope of the sand hill exceeds some critical value, the sand starts flowing downwards (avalanche). The analogy is, in fact, rather good since it has been shown (by careful experiments with pickup coils) that, when the system becomes overcritical, the lines do not move as single units, but rather in the form of avalanches including typically 50 lines or more’’

Motivation to study vortex avalanches To understand something about vortices To understand something about self-organization (local interactions between vortices lead to long-scales correlations) To enhance J c, i.e. the slope of the vortex pile (for various applications of superconductors) JcJc Trapped field magnets up to 17 Tesla

Statistics of vortex avalanches From E. Altshuler and T. H. Johansen, Reviews of Modern Physics, 76, 471 (2004)

Using Magneto-optical Imaging to position the Hall probes

Magneto-optical imagin to measure avalanches 5 mm MgB 2 film 100  m

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

 T 0   J c     Q   T >  T 0 1) Flux motion releases heat 2) T rise weakens flux pinning Thermal effects The thermal instability is usually associated with catastrophic avalanches

Nb disk, Goodman et al., Phys. Lett. 18, 236 (1965) Sometimes thermal avalanches are not complete, but they are limited only by sample dimensions, and obviously destroy the critical state  M ~ 0.2 M

Are there small thermal avalanches ? Are there thermal avalanches that do not destroy the critical state? Can thermal avalanches stop before reaching the sample dimensions? Can we calculate the size of a thermal avalanche?

C dT = jE dt = j c d  All energy released by flux motion is absorbed the amount of flux that has passed through the given point during an avalanche Adiabatic energy balance

Biot-Savart Adiabatic critical state for a thin strip is given by a set of equations: Critical state

Number of avalanches avalanche size,  0 1,000, mm “Shape” model E > E c dend large E dH/dt~1G/s too small  B (r) 10  m 200  m “Size” model 20  m

Vortex Avalanches in MgB 2 films Small: ,000 vortices Round shape Big: ~5,000,000 vortices Dendritic shape “Shape” Model Maxwell + Thermal diffusion “uniform” shape dendritic shape “Size” Model adiabatic critical state Thermal effects control dendritic avalanches micro-avalanches down to 50 vortices 20  m 1 mm Criterion H(E,h 0 ) Dendrite width Build-up time More info: Conclusions

Linearly polarized light Faraday-active crystal Magnetic field H  (H)(H) F Magneto-optical Imaging Square YBaCuO film

 B dA = h/2e =  0 Flux quantum:  J B(r) normal core The vortex core interacts with tiny inhomogeneities (  nanometers) => vortices get pinned (don’t want to move)

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)

We want to understand how the critical state is formed because: it determines the critical current density J c – the key parameter for most applications of superconductors (high-current cables, trapped-field magnets) to test models, e.g. self-organized criticality, for applicability to vortices (that move in a disordered landscape and don’t have inertia)

local flux density calculated from local intensity of MO image; each point on the curve corresponds to one MO image No long-range correlation between the jumps Frequent jumps at the same place 5 x 5  m 2 linear ramp 6  T/s Evolution of local flux density 7mT7.4mT7.9mT

Why small and big jumps ? Nb films: also 2 types of jumps, big and small: James et al., Phys.C 2000 Nowak et al, PRB 1997 Both types of jumps have the same threshold T=10K the same mechanism

 all jumps  i  final  initial = ? < 100% Fraction of flux arrived via jumps: Distribution functions of jump sizes Dendritic 10% 50% 90% Some flux penetrates into the sample via very small jumps or without jumps at all resolution limit