Magneto-optical images of flux dendrites

Magneto-opitcal studies of a c-oriented MgB2 film show that below 10 K the global penetration of vortices is dominated by complex dendritic structures abruptly entering the film. This behavior contrasts the gradual uniform penetration usually found in superconducting films. Figure shows magneto-optical images of flux penetration (image brightness represents flux density) into the virgin state at 5 K. The respective images were taken at applied fields (perpendicular to the film) of 3.4, 8.5, 17, 60, 21, and 0 mT. This complex flux dynamics must be responsible for suppression and noisy behavior of magnetization: the graph is copied from Zhao et al, Phys. Rev. B 65, 064512 (2002).

Theory explaining dendritic patterns

Bulks:  Phys. Rev. B 2004   PDF Films:  Phys. Rev. B 2006   PDF A linear analysis of thermal diffusion and Maxwell equations shows that a thermo-magnetic instability can lead to formation of finger-like distributions of magnetic field and temperature. The fingering instability emerges when the background electric field is larger than a threshold field Ec, and the applied magnetic field exceeds a threshold value H(E), see the phase diagram. We derive the criterion for the instability, and estimate its build-up time and characteristic finger width. Numerical simulations support the analytical results, and allow us to follow the development of the fingering instability beyond the linear regime. Thin films are shown to be more unstable than bulk superconductors and have a stronger tendency to form fingering (dendritic) flux patterns.

Quantitative comparison of Theory and Experiment

Lower threshold field

Phys. Rev. Lett. 2006   PDF The work presents a detailed comparison of experimental data and theoretical predictions for the dendritic flux instability. It is shown that a thermo-magnetic model published very recently [Phys. Rev. B 73, 014512 (2006)] gives an excellent quantitative description of key features like the instability onset (first dendrite appearance) magnetic field, and how the onset field depends on both temperature and sample size. The measurements were made using magneto-optical imaging on a series of different strip-shaped samples of MgB2. Excellent agreement is also obtained by reanalyzing data previously published for Nb.

Upper threshold field

Phys. Rev. B 2007   PDF We propose a mechanism responsible for the abrupt vanishing of the dendritic flux instability when an increasing magnetic field is applied. The onset of flux avalanches and the subsequent reentrance of stability in NbN films was investigated using magneto-optical imaging, and the threshold fields were measured as functions of critical current density, jc. The results are explained with excellent quantitative agreement by a thermomagnetic model published in [Phys. Rev. B 73, 014512 (2006)], showing that the reentrant stability is a direct consequence of a monotonously decreasing jc versus field.

Movies of dendritic flux penetration

small size	AVI, 90 Kb	MPEG, 240 Kb	QuickTime, 1 Mb	Animated Gif, 1 Mb
normal size	AVI, 280 Kb	MPEG, 800 Kb	QuickTime, 3 Mb	Animated Gif, 2 Mb
large size	AVI, 800 Kb	MPEG, 2 Mb	QuickTime, 9 Mb	Animated Gif, 7 Mb

The movie is composed of 101 images taken at 3 K as the applied field increases from zero up to 35 mT and then decreases back to zero.

Temperature dependence

The flux pattern is strongly temperature dependent Figures (a-c) show flux distribution at T = 3.3, 9.9 and 10.5 K at applied fields of 13, 17 and 19 mT, respectively.

We suggest that the observed behavior is due to a thermo-magnetic instability, which is supported by vortex dynamics simulations. (d-f): Flux densities obtained by vortex dynamics simulations at T1 < T2 < T3, respectively, reproducing the morphology of the patterns in (a-c).

Remanent state

Remanent state of a highly branching dendritic structure at 9.9K. The dendrite was formed at applied of field 17mT, and then the field was decreased back to zero. The flux distribution along the film edge shows the conventional pattern with black annihilation zone (zero flux density), while the dendrite was not affected by the applied field reversal.

Dramatic role of critical current anisotropy on dendritic avalanches

Phys. Rev. Lett. 2007   PDF Anisotropic penetration of magnetic flux in MgB2 films grown on vicinal sapphire substrates is investigated using magneto-optical imaging. Regular penetration above 10 K proceeds more easily along the substrate surface steps, anisotropy of the critical current being 6%. At lower temperatures the penetration occurs via abrupt dendritic avalanches that preferentially propagate perpendicular to the surface steps. This inverse anisotropy in the penetration pattern becomes dramatic very close to 10 K where all flux avalanches propagate in the strongest-pinning direction. The observed behavior is fully explained using a thermomagnetic model of the dendritic instability.

Simultaneous Penetration of Flux and Antiflux Dendrites in MgB2 rings

Phys. Rev. B 2006   PDF Flux dendrites with opposite polarities simultaneously penetrate superconducting, ring-shaped MgB2 films. By applying a perpendicular magnetic field, branching dendritic structures nucleate at the outer edge and abruptly propagate deep into the rings. When these structures reach close to the inner edge, where flux with opposite polarity has penetrated the superconductor, they occasionally trigger anti-flux dendrites. These anti-dendrites do not branch, but instead trace the triggering dendrite in the backward direction. Two trigger mechanisms, a non-local magnetic and a local thermal, are considered as possible explanations for this unexpected behaviour. Increasing the applied field further, the rings are perforated by dendrites which carry flux to the center hole. Repeated perforations lead to a reversed field profile and new features of dendrite activity when the applied field is subsequently reduced.

Coexistense of small and dendritic jumps

More detailed studies show that in addition to the big dendritic jumps, small flux jumps also take place everywhere along the flux front. On the Figure: The first jump was detected at 4 mT and was as small as 20 vortices. At larger fields more jumps were found with average size gradually increasing to 10000 vortices. At 15 mT the first dendritic jump occured which consisted of millions of vortices. The observed jump size distribution and its field dependence is explained by our Adiabatic theory Phys.Rev.B 2005 PDF (Theory+Experiment) Physica C 2004 PDF (Experiment)

Local threshold field for the instability

As the applied field increases from 2.3 to 7.4 mT, more and more dendrites penetrate the film. The area with non-zero flux density thus becomes larger, however, one does not find areas with B>12 mT. This value, 12 mT, gives the flux density in the cores of all branches of all dendritic structures. We conclude that 12 mT is local threshold field for the instability. Whenever the local flux density exceeds 12mT, an avalanche is nucleated there, and the flux invasion proceeds as long as there exist regions where B>12mT. Phys. Rev. B 2003, PDF

Fractal analysis of dendritic patterns

Appl. Phys. Lett. 2005, PDF Fractal analysis of the first dendrites entering into a virgin NbN film shows that dendrites formed at higher T have larger fractal dimension. They are also formed at higher applied field and have total larger area. For T=4.8K the fractal dimension is found to be D=1.77, see the picture.

Interplay of dendritic avalanches and gradual flux penetration

Simultaneously with dendritic avalanches, gradual flux penetration also takes place and eventually covers the dendrites. A detailed analysis of this process shows an anomalously high gradient of the flux density across a dendrite branch, and a peak value that decreases as the applied field goes up. This unexpected behaviour is reproduced by flux creep simulations based on the non-local field-current relation in the perpendicular geometry. The simulations also provide indirect evidence that ux dendrites are formed at an elevated local temperature, consistent with a thermo-magnetic mechanism of the instability. Superc.Sci.Techn. 2003, PDF

Dendrites avoid crossing

Before

After

Difference

MO images taken before and after invasion of a dendrite. The new dendrite had to turn the growth direction several times (indicated by arrows) to avoid crossing the existing dendrites. The last image is obtained by subtraction of the first two. The grown dendrite is seen white, while the black regions indicate branches of existing dendrites affected by appearing the new one.

Supercond. Sci. Technol. 2001, PDF.

Dendrites induced by transport current

The dendrites can be induced by a pulse of transport current if the film is initially in the critical state created by applying a magnetic field. The pulse duration was a few milliseconds, the temperature range of the instability is up to 19K. The dendrites always develop from the edge where the transport and the shielding currents add up constructively (their directions are shown by black and white arrows, respectively). Appl. Phys. Lett. 2002, PDF.

The instability is suppressed by an Al foil or deposited Au layer

When the superconducting film is partially covered by a thermally conducting Al foil, or an Au layer is deposited on top of MgB2, the dendrites are formed only in the areas lacking the thermal conductor. When dendrites develop in the uncovered part they never invade into the covered region. The results suggest that the dendritic instability is thermal in origin. Physica C 2002, PDF Appl.Phys.Lett. 2005, PDF

Magneto-optical images

Simulations Flux and temperature distribution produced by vortex dynamics simulations White dots: individual vortices; Red: regions of enhanced temperature due to vortex motion Green: traces of recently moving vortices