Dendritic flux avalanches in superconducting films

Thermomagnetic instability in general, and dendritic flux avalanches in particular, have attracted considerable attention of both scientists and engineers working on superconductor applications. Though being harmful for the performance of many superconducting devices, the avalanches provide a fruitf...

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Datum:	2009
Hauptverfasser:	Yurchenko, V.V., Johansen, T.H., Galperin, Y.M.
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Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2009
Schriftenreihe:	Физика низких температур
Schlagworte:	Электронные свойства проводящих систем
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spelling	irk-123456789-1173052017-05-22T03:02:48Z Dendritic flux avalanches in superconducting films Yurchenko, V.V. Johansen, T.H. Galperin, Y.M. Электронные свойства проводящих систем Thermomagnetic instability in general, and dendritic flux avalanches in particular, have attracted considerable attention of both scientists and engineers working on superconductor applications. Though being harmful for the performance of many superconducting devices, the avalanches provide a fruitful playground for experimental and theoretical studies of complex dynamics of the vortex matter. In this paper, we report on the progress in understanding the mechanisms responsible for the development of the giant magnetic avalanches. We review recent results on magneto-optical imaging of the fingering instability in superconducting films and analyze them basing on the recent theoretical model that establishes criteria for onset of the dendritic avalanches. 2009 Article Dendritic flux avalanches in superconducting films / V.V. Yurchenko, T.H. Johansen, Y.M. Galperin // Физика низких температур. — 2009. — Т. 35, № 8-9. — С. 789-797. — Бібліогр.: 36 назв. — англ. 0132-6414 PACS: 74.25.Qt, 74.25.Ha, 68.60.Dv http://dspace.nbuv.gov.ua/handle/123456789/117305 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Электронные свойства проводящих систем Электронные свойства проводящих систем
spellingShingle	Электронные свойства проводящих систем Электронные свойства проводящих систем Yurchenko, V.V. Johansen, T.H. Galperin, Y.M. Dendritic flux avalanches in superconducting films Физика низких температур
description	Thermomagnetic instability in general, and dendritic flux avalanches in particular, have attracted considerable attention of both scientists and engineers working on superconductor applications. Though being harmful for the performance of many superconducting devices, the avalanches provide a fruitful playground for experimental and theoretical studies of complex dynamics of the vortex matter. In this paper, we report on the progress in understanding the mechanisms responsible for the development of the giant magnetic avalanches. We review recent results on magneto-optical imaging of the fingering instability in superconducting films and analyze them basing on the recent theoretical model that establishes criteria for onset of the dendritic avalanches.
format	Article
author	Yurchenko, V.V. Johansen, T.H. Galperin, Y.M.
author_facet	Yurchenko, V.V. Johansen, T.H. Galperin, Y.M.
author_sort	Yurchenko, V.V.
title	Dendritic flux avalanches in superconducting films
title_short	Dendritic flux avalanches in superconducting films
title_full	Dendritic flux avalanches in superconducting films
title_fullStr	Dendritic flux avalanches in superconducting films
title_full_unstemmed	Dendritic flux avalanches in superconducting films
title_sort	dendritic flux avalanches in superconducting films
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2009
topic_facet	Электронные свойства проводящих систем
url	http://dspace.nbuv.gov.ua/handle/123456789/117305
citation_txt	Dendritic flux avalanches in superconducting films / V.V. Yurchenko, T.H. Johansen, Y.M. Galperin // Физика низких температур. — 2009. — Т. 35, № 8-9. — С. 789-797. — Бібліогр.: 36 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT yurchenkovv dendriticfluxavalanchesinsuperconductingfilms AT johansenth dendriticfluxavalanchesinsuperconductingfilms AT galperinym dendriticfluxavalanchesinsuperconductingfilms
first_indexed	2025-07-08T12:00:35Z
last_indexed	2025-07-08T12:00:35Z
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fulltext	Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9, p. 789–797 Dendritic flux avalanches in superconducting films V.V. Yurchenko1, T.H. Johansen1,2, and Y.M. Galperin1,3 1 Department of Physics and Centre for Materials Science and Nanotechnology, University of Oslo P.O. Box 1048, Blindern, 0316 Oslo, Norway E-mail: Vitaliy.Yurchenko@smn.uio.no 2 Institute for Superconducting and Electronic Materials, University of Wollongong, NSW, Australia 3 A.F. Ioffe Physico-Technical Institute, St Petersburg 194021, Russia Received February 13, 2009 Thermomagnetic instability in general, and dendritic flux avalanches in particular, have attracted con- siderable attention of both scientists and engineers working on superconductor applications. Though being harmful for the performance of many superconducting devices, the avalanches provide a fruitful playground for experimental and theoretical studies of complex dynamics of the vortex matter. In this paper, we report on the progress in understanding the mechanisms responsible for the development of the giant magnetic ava- lanches. We review recent results on magneto-optical imaging of the fingering instability in superconduct- ing films and analyze them basing on the recent theoretical model that establishes criteria for onset of the dendritic avalanches. PACS: 74.25.Qt Vortex lattices, flux pinning, flux creep; 74.25.Ha Magnetic properties; 68.60.Dv Thermal stability; thermal effects. Keywords: superconducting films, flux avalanche, vortices in superconductor. 1. Introduction 1.1. Vortex pinning and motion One of the characteristic features of superconductors is perfect diamagnetism or expulsion of a weak external magnetic field, H, from their interior, the so-called Meiss- ner and Ochsenfeld effect. In type-II superconductors, the perfect diamagnetism exists for applied fields below a lower critical field, Hc1, and there is a broad domain of magnetic fields, H H Hc c1 2� � , where the field pene- trates into the sample in a form of quantized flux lines. In a perfect sample these lines-vortices form an Abrikosov lattice [1]. Above the upper critical field, Hc2 , the bulk superconductivity seizes to exist. An isolated vortex con- sists of a core where the superconducting order parameter is suppressed, while the magnetic field reaches a local maximum. The radius of the core is of the order of the co- herence length, � . The magnetic field outside the core de- cays exponentially over a distance of a magnetic penetra- tion depth � , where also electrical current circulates around the core. Each vortex carries one flux quantum �0 152 2 07 10� � � �h e/ . Wb. The magnetic field is sup- ported by circular currents around the core. The flux lines repel each other, an interaction that can be understood ap- plying the Ampere’s law to the circular currents. This re- pulsion leads to a formation of a flux line lattice, most typically a hexagonal and in some cases square. A number of phases and dynamic effects in the flux line lattice was reviewed in Refs. 2, 3. The vortices interact with a transport current via the Lorentz force per unit length f j n� �0 [ ] , where j is the current density and n is the unit vector along the flux line. Since the vortex motion implies mov- ing of normal cores it is accompanied with dissipation, which is proportional to the velocity. This dissipation can be described by an effective viscosity, and the velocity is determined by the balance between the Lorentz force and the viscous force. Therefore a free vortex lattice would move as a whole with a constant velocity resulting in a fi- nite resistance of the sample. Fortunately, in all real superconductors the flux lines interact with material defects that act as pinning sites and © V.V. Yurchenko, T.H. Johansen, and Y.M. Galperin, 2009 hamper their motion. Besides the pinning barriers that arise from rather inevitable structural irregularities such as vacancies, dislocations, grain boundaries, etc., there exists a rich zoo of artificially introduced pinning sites like magnetic inclusions, phases of weaker (stronger) su- perconductivity, lithographically patterned «antidots», magnetic dots, etc. According to the particular nature and dimensionality of the defects the pinning potential has different spatial extent, magnetic field and temperature dependencies, see Ref. 4 for a review. When a supercon- ductor is exposed to an increasing magnetic field (or self field of a transport current), vortices are formed at the edges and then propagate inwards. The presence of pin- ning leads to formation of an inhomogeneous distribution of the magnetic flux. According to the critical state model [5] the stationary distribution can be found from Am- pere’s law with the condition that the current density at each point is equal to its local critical value, j Tc ( , )B , i.e., �B j� 0 , \| \| ( , )j B� j Tc . The case where jc is independent on magnetic field is called the Bean model [5]. The critical current density is in practice a key measure of the applicability of supercon- ductors since jc is the maximum current density in a ma- terial for transport without energy loss. Microscopic eva- luation of the critical current requires direct summation of vortex–vortex interactions and all elementary pinning forces, which all together is a rather unfeasible task. Therefore the critical state model has become a major pa- radigm in the studies of electromagnetic properties of type-II superconductors. The critical state emerging from the balance between the Lorentz and the pinning forces is metastable. Even a slow increase in the external magnetic field may lead to total collapse of the critical state and sudden large-scale redistribution of the flux. Such dramatic events observed experimentally as abrupt drops in the magnetization are ascribed to a thermomagnetic instability where the local heat dissipation associated with vortex motion reduces the pinning, which in turn facilitates further vortex mo- tion. With this positive feedback, a small perturbation can quickly evolve into a flux avalanche [6–19]. 1.2. Experimental methods Experimental techniques used to investigate the ava- lanches in the vortex matter are conventionally subdi- vided in two groups: integral and spatially resolved. The former one includes all types of magnetometry: inductive coils, vibrating sample magnetometry, SQUID mag- netometry [20]. These methods are sensitive to global re- distributions of the flux, current flow, and the resulting change in the total magnetic moment caused by an ava- lanche in the sample. A drawback of the integral methods is a lack of detailed information about the jumps, their lo- cation in the sample, their morphology, etc. Moreover, the relatively low sampling rate in these methods prevents identification of jumps occurring with short time inter- vals. Obviously there is no chance to separate simul- taneous avalanches. Thus, measurement of the volume averaged magnetization excludes a possibility to study detailed statistics of flux jumps. Yet another disadvantage is that it is impossible also to discriminate between small jumps and instrument noise. These problems are partly solved in locally resolved magnetometry, an overview of available methods can be found elsewhere [21]. One of the most powerful tools is magneto-optical imaging (MOI), which combines high magneto-spatial resolution and short acquisition times (restrained only by the exposure time of a camera). This makes MOI the most suitable real-time technique for investigation of flux avalanches. In our experiments we use MOI based on the Faraday effect where the sensing element is an in-plane magne- tized ferrite garnet film with strong Faraday rotation. The film, which is grown separately on a transparent gadolin- ium gallium garnet substrate, is placed directly on the top of the sample under investigation. Stray fields from the magnetic flux pattern of the sample locally remagnetize the indicator, and when shining polarized light through the film, the out-of-plane component of magnetization in- duces Faraday rotation. The effect is doubled after the re- flected light (either from the mirror layer or, in more rare cases, from the sample surface) passes through the garnet film again. Figure 1 illustrates the principal experimental scheme. An analyzer set at 90° relative to the polarizer fil- ters out the nonrotated light. The rotation angle, �, is a monotonous function of the local magnetic field, B. The transmitted light is projected on the matrix of a CCD camera, its intensity, I, being related to the rotation angle according to Malus’ law I I Bo� sin [ ( ) ]2 � � . Here � is an offset angle (equal to 0 when the polarizer and the analyzer are crossed). The dependence �( )B can be determined by a proper calibration of the indicator film. Using this dependence together with Malus’ law one can recalculate the distribution of the magnetic field in the sample from the intensity map of the projected light. Since MOI was invented in the 1950s several materials have been used as indicator films [22]. The MOI results reported in the present paper were obtained utilizing in-plane magnetized garnet films (Lu,Bi)3(Fe,Ga)5O12, grown as a few micron thick epitaxial layer on gadolin- ium gallium garnet (transparent) substrates. One limita- tion of these indicator films is a relatively small magnetic 790 Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 V.V. Yurchenko, T.H. Johansen, and Y.M. Galperin field range: above � 100 mT the films reach saturation, i.e., they become magnetized totally out-of-plane. 2. Dendritic flux avalanches Thermomagnetic instabilities have been in focus of many experimental and theoretical studies for decades [11,23]. However, only with the use of MOI it has been discovered that in most superconducting films, such as MgB2, Nb, Pb, Nb3Sn, NbN, YBa Cu O2 3 x , and YNi2B2C, the avalanches create dendritic flux patterns (see Fig. 2) [6,11,12,17]. From the massive experimental data collected on the subject one can identify some com- mon features of the dendritic instability: (i) It occurs below a certain temperature Tth where T Tcth � . (ii) It occurs in a limited range of applied fields: H H H 1 2 th th� � , where H 1 th and H 2 th are the so-called lower and upper threshold fields, respectively. Shown in Fig. 3 is a set of images recorded during a slow ramping up of the applied magnetic field in a thin film of NbN after the sample was initially zero-field cooled to T � 4 K. In the image taken at H � 8 Oe, there is only a shallow penetration, with a flux front that gradu- ally advanced during the field increase. This smooth mode of penetration ended abruptly when reaching H � = 12 Oe, where a flux avalanche suddenly occurred. As the applied field increased further, more and more ava- Dendritic flux avalanches in superconducting films Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 791 CCD camera Analyzer Beam splitter Coil Light source MO film Mirror Sample Cold finger Polarizer Fig. 1. Schematic draft of a MOI setup. A sample is mounted on a massive cold finger of a He-flow cryostat. Resistive coils are used as a source of an external magnetic field. The light from a mercury lamp shines through a polarizer and is guided onto an indicator film, where it experiences the Faraday rota- tion. The light becomes reflected by a thin mirror and pro- jected through an analyzer on a CCD matrix of a computer-op- erated camera. Fig. 2. Dendritic flux avalanche observed in MgB2 film by MOI. From Ref. 6. 8 Oe 42.5 Oe 400 Oe 12 Oe 357 Oe 425 Oe H Th 1 H Th 2 Fig. 3. Magneto-optical images of a NbN film at 4 K in in- creasing applied field illustrating development of the dendritic flux avalanches. From Ref. 24. lanches took place, creating a complex pattern of flux dendrites covering most of the sample area, Fig. 3 (H � = 42.5 Oe). Then, as the field reached H � 357Oe, the av- alanche activity stopped entirely although the field con- tinued to increase. The advancing flux front now erased the previously formed dendritic structures, see images taken at H � 400 and 425 Oe, and the critical state mode of flux penetration was restored. The bright fan-like features in the lower part of the images are due to tiny film defects, and are not related to the flux instability. (iii) The formation of the dendritic instability is a sto- chastic process. Usually indentations on the sample edges serve as the most probable origins of the avalanches, as they give a significant local magnetic field amplification and in- crease in the electric field [26]. Nevertheless, the exact nucleation place of the next dendrite, field interval be- tween two consecutive events, and the final shape of the dendritic structure are nonreproducible. To illustrate this, Fig. 4 shows three magneto-optical images of flux pene- tration in a MgB2 film taken after three repeated expe- rimental runs with initial zero-field-cooling the sample 9.2 K and applying a field of 20 mT. Image (b) was ob- tained by superimposing the three original images co- lored red, green, and blue, respectively. In those parts of the final image where they overlap, i.e., the flux front’s behavior is reproducible, one observes a gradation of white. Different colors show the places where there is no overlap, or the overlap is only partial. Strong irrep- roducibility is seen in the dendrite shapes, while the pene- tration near the edge and along static defects is repeat- able. Randomness in the behavior of the vortex matter also explains a large statistical variation of the measured H 1 2, th values, which will be discussed later. (iv) The degree of branching of the dendrites, some- times represented by their fractal dimensionality, and size vary with temperature and the field. Figure 5 illustrates typical behaviors of the dendrites in a NbN film at different temperatures. At T � 4 K the number of the dendritic avalanches per interval of the field was higher compared to T � 6 K. The size of the den- drites shows opposite trend — it increases when the tem- perature approached Tth . Regarding the dependences on magnetic field, at lower (slowly increasing) fields the avalanches are smaller in size, but appear in larger num- bers per field interval. At larger fields, the dendrites are less frequent, but larger in size. Images in Fig. 3 taken at H �12 and 357 Oe illustrate this property. 3. Theory of the dendritic flux jumps There exist numerous experimental indications that the instability is a rather unavoidable generic feature of the vortex matter in superconductor thin films. The con- ventional theory of the thermomagnetic instability con- sidered only «uniform» flux jumps, where the flux front is smooth and essentially straight [23,27]. The problem of the flux pattern formation was examined in the slab ge- ometry [28] and more recently in thin films placed in per- pendicular magnetic field [29,30]. The criteria for the in- stability onset were obtained from the linear analysis of small coordinate-dependent perturbations in the electric field, E, and temperature, T [25,31]. The problem was solved for a thin (thickness d is much less than the 792 Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 V.V. Yurchenko, T.H. Johansen, and Y.M. Galperin a b 500 m� Fig. 4. MOI demonstrating irreproducibility of the flux distri- bution that results from the dendritic avalanches. (a) Three im- ages were taken under identical experimental conditions. (b) Image obtained by superimposing the three images above. Repeatable parts of the flux front appear as gradations of white. From Ref. 25. a b Fig. 5. Magneto-optical images of dendritic flux avalanches in a NbN film taken at a) T � 4 K and b) T = 6 K. width 2w) superconducting strip infinite in y direction, placed in a transverse magnetic field, H, and thermally coupled to the substrate (Fig. 6). 3.1. Model and basic equations The thermomagnetic instability can be described by the set including the Maxwell equations and the equation for thermal conduction: curl B j�� 0 , (1) curl E B � � � �t , (2) C T t T � � � �� 2 jE . (3) In the above equations, j is the current density, E is the electric field, B is the magnetic induction, B H�� 0 ; C is the specific heat, � is the thermal conductivity. The heat release into substrate is taken into account by applying the boundary condition ��nT h T T� � �0 0( ) (4) to both film surfaces. Here T0 is temperature of the sub- strate, while h0 characterizes heat transfer through the in- terface between the sample and the substrate. The above set should be supplemented by a relation between the cur- rent density, and the electric field, j j E B T� ( , , ). This rela- tion is chosen as j E� j T g E Ec ( ) ( ) ( / ) . (5) Here g E( ) represents a very steep E j( ) curve having large logarithmic derivative, n E E j j Ec( ) ln / ln /� � � � �� 1, (6) where � is the differential electrical conductivity, �( ) /E j E� � � . Parameter n stands for the exponent in the frequently used power-law relation E jn� . A strong nonlinearity of the function g E( ) leads to a formation of a quasistatic critical state with j j Tc� ( ), where jc is the critical current density. Note that in this model B-depen- dence of jc is neglected, as in the Bean’s model [5]. We will come back to this assumption later. The key dimensionless parameter of the model is the ratio of the thermal and magnetic diffusion coefficients [23], � � �� 0 / C . (7) The smaller is �, the slower heat diffuses from the pertur- bation region into the surrounding areas. Hence, one can expect that for smaller � the superconductor is more un- stable, and the formation of instability-induced nonuni- form structures is more likely. Solutions of Eqs. (1)–(5) were sought within linear perturbation analysis in the form T T x y z t � ( , , , ), E E � ( , , , )x y z t , j j � ( , , , )x y z t with perturbations � �T Ex y, , , and �jx y, determined via corresponding dimensionless Fourier amplitudes �, �, and i: � � � � �T T t t ik ikx y� * exp ( / )0 , � � � � �E E t t ik ikx y x y x y, , exp ( / )� 0 , � � � �j j i t t ik ikx y c x y x y, , exp ( / )� 0 . (8) These components can be regarded as z-independent for a thick slab. For the case of a thin film they should be prop- erly averaged over z, see Ref. 31 for details. 3.2. Criteria for dendritic instability in thin films Here we will focus on the analysis of the main results of Ref. 31. Provided there exists a set of parameters that satisfies Re ( , )� k kx y � 0, the superconductor becomes unstable. The wave vectors kx and k y characterize the extent of the flux front perturbations in corresponding directions. Instability would develop to a fingering pat- tern if k kx y�� 0. The critical k y * and kx * for the finger- ing instability can be found from the requirement max{Re ( )}� k y � 0 for k y � 0 in the limit d l/ 2 1�� (nota- tions follow from Fig. 6): k n nh n x * � �1 � � , k nh n nh n y * / /( ) ( ) � � � � � 1 1 11 4 1 2 , h h C j d j Tc c � � � 2 0 0 2� � ( ln / ) . (9) Dendritic flux avalanches in superconducting films Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 793 z l no flux j, E d/2 –d/2 H 0 x y flux substrate w m id dl e of th e st rip Fig. 6. One half of a superconductor strip on a substrate. The dark gray area is the flux-penetrated region (Ref. 31). Since the front penetration depth, l, is related to the ap- plied field according to Ref. 32, the quantity k lx * /�1 de- termines the applied magnetic field at which the instabi- lity first takes place. The quantity k y * determines spatial scale of the fingers. Dispersion relations Re ( , )� k kx y for arbitrary k y are presented in Fig. 7. For small � and large kx , i.e., small magnetic field, Re � is negative for all k y (Fig. 7,a), meaning the superconductor is stable. At small kx , Re � becomes positive in some finite range of k y , which implies that perturbations with finger patterns of elevated T and E will start growing perpendicular to the flux front, giving rise to the dendritic instability. For large �, i.e., large heat diffusion, at small kx the maximum of Re � corresponds to k y � 0, and the uniform perturba- tion will be dominant (Fig. 7,b). One of the most important results that can be directly tested in experiments is the threshold flux penetration depth l* , when the superconductor becomes unstable [25]. Putting in Eqs. (9) l k y * /�1 one obtains: l j E h nd j Ec c \| \| \| \| � � � � � � � � � ! ! � " � 2 1 2 0 1 . (10) Here �jc is the temperature derivative of the critical current density. By combining Eq. (10) with the Bean model expression for the flux penetration depth of a thin superconducting strip in transverse field [32] one imme- diately arrives at the expression for the threshold field: H j d w w l c th arccosh� � � � �� !!" * . (11) Equations (10), (11) set the most practically important relations between the external parameters, such as the field and temperature, on one hand, and the sample prop- erties (�, h0 , jc , w) on the other. In order to check the va- lidity of the model, the threshold field was measured first in a series of MgB2 samples of different width. A set of eight thin film samples was deposited in one run to allow 794 Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 V.V. Yurchenko, T.H. Johansen, and Y.M. Galperin � = 0.01 � = 7 H 0.50.5 0 –0.5 –1.0 k <x k* x k >x k* x 0 –0.2 –0.4 k* y ky ky kx k* x a b R e � R e � Fig. 7. Dispersion relations Re ( , )� k kx y for small and large �, for d l/ .2 0 001� and n � 20. From Ref. 31. 1.6 mm 0.2 mm Fig. 8. MOI of flux distribution in MgB2 samples of different width at 4 K and magnetic field of 15 mT. From Ref. 25. for a direct comparative analysis and shaped later by photolithography into 3 mm long rectangles having dif- ferent widths ranging from 0.2 to 1.6 mm. In Fig. 8 pre- senting the results of MOI experiments, the dendrites pro- nouncedly invade the wider strips, while there are fewer of them in the samples with smaller widths, and no den- drites at all in the one with w � 02. mm. The quantitative results for H wth ( ) are summarized in Fig. 9. The fitted data reproduce the experimentally observed tendency of the threshold field to diverge as the strip becomes nar- rower. Rephrasing the latter statement, the narrower is the superconductor the more thermomagnetically stable it is. A good agreement between the theory and experiment was also obtained for the threshold field as a function of temperature. To fit the H Tth ( ), observed in Nb and MgB2 thin films (Fig. 10), some assumptions for temperature dependent model parameters had to be made. The as- sumed cubic dependencies for the thermal conductivity, � �� ~( / )T Tc 3 , and the heat transfer coefficient, h0 � � ~ ( / )h T Tc0 3 , were based on the low-temperature data for MgB2 [33] and the acoustic mismatch model [34], respec- tively. The critical current density was assumed to depend linearly on temperature: j j T Tc c c� �0 1( / ). Given a pin- ning potential, U T Tc� �( / )1 , the exponent n U kT~ / be- comes T-dependent, n n� ~ ( / )T Tc �1 . The model clearly reproduces the property (i) mentioned above — existence of a threshold temperature Tth such that for T T� th the in- stability is absent, and a steep increase of the threshold field H th when T approaches Tth . 4. Reentrant stability of superconducting films Although the existence of the lower threshold mag- netic field, H 1 th , was explained in Sec. 3, the origin of the upper threshold field, H 2 th , above which the flux distribu- tions are stable, was not discussed. Recently, we managed to put the existence of an upper threshold into the frame- work of the model described in the previous section [24]. As it follows from Eqs. (10) and (11), the threshold field depends nonmonotonously on the critical current density, as shown in Fig. 11. For intermediate jc the threshold field is nearly constant, and increases slowly as jc becomes larger giving eventually a sublinear asymp- totic dependence, H jcth ~ /3 4 . More importantly, when jc decreases the threshold field diverges at some finite jc , Dendritic flux avalanches in superconducting films Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 795 0 5 10 15 20 25 w, mm 82 m� 0.2 0.4 0.6 0.8 Dendritic instability No dendrities � 0 th H , m T Fig. 9. Threshold magnetic field for onset of the dendritic in- stability in MgB2 strips of different width. Experimental data (symbols) are in a good agreement with a fitted theoretical curve (full line), for jc � �0 9 1011. A/m 2 . From Ref. 25. 0 2 4 6 8 10 5 10 15 20 25 � 0 th H , m T Nb MgB2 T, K Fig. 10. Temperature dependence of the threshold magnetic field (from Ref. 25). Experimental data obtained for the 5 mm wide MgB2 sample (�) and for a 1.8 mm wide Nb film (�) [13]. The full lines are theoretical fits. The dashed lines show the limiting temperature above which the instability vanishes. j (B)c sm la ler j c(B ) – al a w ys stab el H2 th H1 th Instability region j , arb. unitsc H , ar b . u n it s H (j )th c Fig. 11. Schematic plot illustrating the existence of two thresh- old fields for the dendritic instability adapted from Ref. 24. The two main curves represent dependencies of the threshold filed on the critical current density H jcth ( ) and a typical mo- notonous field dependence of jc. The intersection of the two curves defines the two thresholds H 1 th and H 2 th for the onset and vanishing of the instability. corresponding to l jc * ( ) approaching w. Taking into ac- count that the critical current density most commonly de- creases with the field, it follows that the field range with unstable behavior can indeed have both a lower and upper limit, H 1 th and H 2 th , as indicated in the figure. Evidently, it is essential how fast jc is decreasing with the field. In par- ticular, along the Bean model (constant jc), or with a jc having only weak field dependence, the conditions for having an upper threshold will never be met. Note also that if jc is sufficiently small, e.g., because of a larger temperature, the thermomagnetic avalanches will not occur at any magnetic field. This idea was supported by MOI experiments on a NbN superconductor thin film. To measure both jc and the upper and lower instability threshold fields the sample was field-cooled to 4 K in various constant magnetic fields, H fc . The field-dependent critical current density j Hc ( )fc was determined from the depth of the penetra- tion front in the middle section of the rectangular sample [32,35]. A nearly exponential decay was found, with jc decreasing from 76 1010. � A/m2 at zero field to almost one half at H fc � 300 Oe, which agrees well with the results obtained earlier from ac susceptibility measurements [36] on the same type of films. The threshold fields were deter- mined by slowly ramping up the additional field after an initial field-cooling in extern field H fc . Figures 12 and 13 present the experimentally obtained functions j Hc ( )fc and H H 1 2, ( )th fc , which were used to make the parametric plot of the instability onset field #H � � �H H 1 th fc shown in Fig. 14. Note that under field-cool- ing the vortices are distributed homogeneously inside this superconductor where pinning is fairly strong. Application of additional field leads to a build-up of the critical state, which collapses due to the thermomag- netic instability as soon as the flux front advances over the threshold length l* corresponding to applying the ad- ditional field #H. Figure 14 demonstrates an excellent quantitative agreement between the experimental data for NbN (symbols) and theoretical curve obtained from Eqs. (10) and (11), using only two fitting parameters, �T Eth / � 500 mA and h T nE0 5700th / � A/m (these pa- rameter values can be achieved, e.g., with the combi- nation � � 25 mW/(K·m), h0 1424� W/(K·m2), n � 5, E � 055. V/m, Tth �11 K). As mentioned above, the den- drite formation process is stochastic, which is reflected in a noticeable dispersion of the measured threshold fields and the error bars in the figures, calculated as a standard deviation for repeated experiments. 796 Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 V.V. Yurchenko, T.H. Johansen, and Y.M. Galperin 7.5 7.0 6.5 6.0 5.0 5.5 4.5 4.0 0 50 100 150 200 250 300 j , 1 0 A /m c 1 0 2 H , Oefc Fig. 12. Field dependence of the critical current density at T � 4 K measured using MOI in NbN thin film (Ref. 24). H2 th H1 th #H Instability 400 300 200 100 0 Hfc Hfc, Oe 0 100 200 300 H , O e th Fig. 13. Thermomagnetic instability diagram of NbN at T � 4 K (Ref. 24): the lower and upper threshold fields H 1 th and H 2 th , measured at different H fc, limit the region where the dendritic avalanches were observed. 4 5 6 7 8 0 10 20 30 40 50 60 70 80 90 # H , O e j , 10 A/mc 10 2 Fig. 14. Instability onset field #H as a function of the critical current at 4 K in NbN thin film. From Ref. 24. 5. Summary and conclusions In this work, we review both well established facts and the results of very recent experiments on dendritic insta- bilities in superconductor films. The most common prop- erties of the dendritic instability are summarized in Sec. 2. In Sec. 3 we introduced the recent theoretical model allowing to understand key features of the instabil- ity and link them to the samples’ parameters. These links are provided by Eqs. (10) and (11) for the threshold field H th as a function of temperature, the critical current den- sity jc , the sample width w, the thermal conductivity �. 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Dendritic flux avalanches in superconducting films

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