## Abstract

The flux avalanche in type-II superconducting thin film is numerically simulated in this paper. We mainly consider the effect of non-uniform critical current density on the thermomagnetic stability. The nonlinear electromagnetic constitutive relation of the superconductor is adopted. Then, Maxwell's equations and heat diffusion equation are numerically solved by the fast Fourier transform technique. We find that the non-uniform critical current density can remarkably affect the behaviour of the flux avalanche. The external magnetic field ramp rate and the environmental temperature have been taken into account. The results are compared with a film with uniform critical current density. The flux avalanche first appears at the interface where the critical current density is discontinuous. Under the same environmental temperature or magnetic field, the flux avalanche occurs more easily for the film with the non-uniform critical current density. The avalanche structure is a finger-like pattern rather than a dendritic structure at low environmental temperatures.

## 1. Introduction

Type-II superconductors have great potential in many fields, such as superconducting cables [[1],[2]], nuclear magnetic resonance magnets [[3],[4]] and so on. The distributions of the magnetic field and current density in superconductors have received much attention. Researchers have carried out a lot of work on electromagnetic behaviour in superconductors based on different theoretical models [[5]–[8]]. When a type-II superconductor is under an external magnetic field, the magnetic field will penetrate into the superconductor in the form of flux vortex lines [[9]]. If there are some defects at the edges of the superconductor or there is a temperature perturbation, the magnetic flux lines may suddenly enter the superconductor from the edges and a flux avalanche will happen [[10]]. The magnetic flux lines in type-II superconductors are subject to the Lorentz force and flux pinning force. When the Lorentz force is equal to the flux pinning force, the flux lines will not move. With increasing applied field, the Lorentz force will exceed the flux pinning force, and the flux lines will move [[11]]. Moving flux lines can lead to energy dissipation. The local temperature will rise due to the generated heat in the superconductor. Then, the flux pinning force will decrease, which makes the flux lines move more easily. Such a positive feedback leads to a number of magnetic flux lines penetrating into the superconductor, i.e. a flux avalanche [[12]]. Thus, it is important to study this behaviour and to avoid a flux avalanche. Researchers have done much work on different superconducting materials to understand the characteristics of flux avalanche. Flux avalanche is closely related to the external magnetic field [[13],[14]], the geometry of the superconducting material [[15],[16]], the field ramp rate [[17]] and the environmental temperature [[18]]. Baziljevich *et al*. [[19]] studied the dendritic instability in a YBa_{2}Cu_{3}O_{7–d} film at very high magnetic ramp rates. Through the magneto-optical imaging technique, the flux avalanche and evolution process [[18],[20]–[22]] were observed. The propagation velocity of the magnetic avalanche is very high, and it can reach hundreds of kilometres per second [[23],[24]]. Moreover, a large amount of heat is generated during the flux avalanche. Thus, the local temperature will rise sharply and even exceed the critical temperature, which can lead to a large temperature gradient and quench. There is high thermal strain in the superconductor induced by the temperature change. Wang *et al.* [[25]] proposed a method to detect the quench in superconductors via experimental observation of the thermal strain. In addition, the electromagnetic response in superconductors has been studied widely. Brandt [[26]–[28]] introduced a local magnetization *g* and studied the flux penetration in different superconductors. Mints & Brandt [[29]] studied the flux jumping behaviour in a thin superconducting film with the bean model. They found that the threshold field of flux jump is related to the external magnetic field ramp rate and the thermal resistivity. Aranson *et al*. [[30]] analysed the flux avalanche in a thin superconducting film by combining Maxwell's equations with the heat diffusion equation numerically. The positive feedback between non-local flux diffusion or flux movement and the Joule heat leads to the flux avalanche and its propagation. Considering the temperature-dependent critical current, Vestgarden *et al*. [[14],[31]–[34]] solved Maxwell's equations coupled to the thermal diffusion equation using the fast Fourier transform (FFT) technique. They pointed out that, at low environmental temperatures, the first avalanche would occur in the middle of the long edges. At a higher environmental temperature, the location of the first avalanche is related to the fluctuations due to a random disturbance. Jing *et al*. [[17],[35],[36]] studied the flux avalanche and the thermal strain in type-II superconductors with different applied magnetic field ramp rates. During the fabrication of a thin film, the flux pinning is enhanced locally with graded pinning landscapes [[37]]. Thus, the critical current density may be non-homogeneous in the film. Schuster *et al*. [[38]] studied the flux penetrating behaviours of non-uniform thin type II superconductor films. The flux avalanche in the sample with a non-uniform critical current density was investigated experimentally [[39],[40]]. In this paper, we study the flux avalanche in a non-uniform thin superconducting film. The effect of the non-uniform critical current density on the flux avalanche behaviour is mainly studied. The external magnetic field ramp rate and the environmental temperature are taken into account. We also calculate the flux avalanche in a non-uniform film with a slit. The results in the non-uniform film are compared with those in a uniform thin film.

## 2. Model and basic equations

Consider a thin square superconducting film with length 2*a* and thickness of *d*. It is assumed that the thickness is much smaller than the length, i.e. *L* and thickness *d _{s}*.

The electromagnetic behaviour of a thin superconducting film in an external magnetic field is determined by Maxwell's equations
** B**,

**,**

*E***and**

*H***are the magnetic flux density, electric field, magnetic field and sheet current density of the thin superconducting film, respectively.**

*J**δ*(

*z*) is the Dirac delta function. The sheet current density satisfies the continuity equation

*c*,

*κ*and

*h*are the specific heat, the thermal diffusivity and the heat transfer coefficient of the superconductor, respectively. The temperature of the superconductor is

*T*, the environmental temperature is

*T*

_{0}and the electrical conductivity is

*σ*.

The nonlinear electromagnetic constitutive relation of the superconducting thin film is the *E* − *J* power law [[28]]:
*ρ*(*J*) is the resistivity of the superconductor and *ρ*_{0} is the resistivity of the normal state. *J*_{c} is the critical sheet current density and *T*_{c} is the critical temperature of the superconductor. The critical sheet current density *J*_{c} and the flux creep exponent *n* are temperature dependent [[31]]: *n*_{0} and *J*_{c0} are constants. We adopt the local magnetization *g* = *g*(*x*,*y*) [[14],[27]], and the current density can be expressed as
*F*^{−1} and *F* are the forward and inverse Fourier transform, respectively. The inverse transformation of the above formula is

## 3. Results and discussions

The width of the square superconducting film is 2*a* = 2 × 2.2 mm and the thickness is *d* = 0.5 µm (figure 1). Our numerical simulation region is 2 *L* × 2 *L*, where *L* = 1.3*a* [[17]]. The simulation region is discretized with 512 × 512 equidistant grids. For the thin film with uniform critical current density, we use the parameters of MgB_{2} [[41]], where *T*_{c} = 39 K, *n*_{0} = 19. For the uniform film, *J*_{0} = 50 KA m^{−1}. For a non-uniform thin film, the critical current density is defined as follows:
*B _{a}* = 12.6 mT, and the ramp rates are d

*B*/d

*t*= 500, 1000, 1500 T s

^{−1}, respectively. For the uniform film, there is no dendritic structure at a field ramp rate of 500 T s

^{−1}. However, at the same field ramp rate, the flux avalanche appears in the non-uniform film and the dendritic structure locates at the interface where the current density is discontinuous. For a larger field ramp rate, the dendritic pattern exists in both the uniform and non-uniform films. Comparing two different films, one can find that the locations for the first avalanche are also different. For the uniform film, the flux avalanche appears at the centre-right position of the edge first. However, the initial avalanche appears at the interface where the critical current density is discontinuous for the non-uniform film. This is due to the fact that the electric field is higher at the interface [[29]]. The dendritic pattern at the higher field ramp rate is denser than that at the lower field ramp rate. Comparing the dendritic patterns in the uniform and non-uniform films, we can also find that the avalanche is denser in the non-uniform film. At a field ramp rate of 1500 T s

^{−1}, the size of the single dendritic pattern becomes smaller. These results mean that the flux avalanche occurs more easily in the non-uniform film.

Figure 3 shows the average magnetization curve in the uniform and non-uniform films during the flux avalanche, where *m*_{0} = *a*^{3}*J*_{0} [[42]]. For the uniform film, there is no flux jump at the magnetic field ramp rate of 500 T s^{−1}, which indicates that the flux avalanche does not happen. However, the flux avalanche occurs at the same field ramp rate in the non-uniform film. This result is in agreement with the result in figure 2. We can see that the frequency of the jump increases with the magnetic field ramp rate. However, the magnitude in the uniform film is greater than that in the non-uniform film. Figure 4 gives the maximum temperature in two kinds of films. The maximum temperature shows the same trend with the magnetization curve. This means that, as the field ramp rate increases, more heat is generated during flux movement. As expected, the jump frequency in the non-uniform film is larger than that in the uniform film. The peak value of the maximum temperature is about 1.5 *T*_{c}. It can be seen that there is no magnetization and the temperature jumps at lower ramping rates. The stability in the thin film can be enhanced by lowering the ramping rate.

Figure 5 shows the flux avalanche in the non-uniform film at different environmental temperatures. The external magnetic field increases from zero to 6, 9, 11 mT, with a field ramp rate of 500 T s^{−1}. The environmental temperature is *T*_{0} = 0.5 *T*_{c} (figure 5*a*) and *T*_{0} = 0.25 *T*_{c} (figure 5*b*). It can be seen that there is no flux avalanche at a field of 6 mT. When the magnetic field reaches 9 and 11 mT, the dendritic flux avalanche appears in the film and the dendritic patterns are different. It is interesting to find that the dendritic flux avalanche begins to occur at the interface where the current density is discontinuous. As the external field is equal to 9 mT, the avalanche is located at the left interface. This result means that the field at the interface region is higher than that in surrounding area. The heat generated in the film will not dissipate in time, so the temperature will continue to rise and cause more flux avalanches at the interface. Then, the flux avalanche will also appear at the right interface between the two regions which have a higher critical current density for the field of 11 mT. As a comparison, the results at *T*_{0} = 0.25 *T*_{c} are shown in figure 5*b*. It can be seen that, when the field increases to 6 mT, there are some dendritic structures at the edge of the film. When the magnetic field reaches 9 and 11 mT, the results show that most of the geometry patterns during the flux avalanche are finger-like patterns rather than dendritic patterns. These finger-like patterns were also reported in [[43]].

Figure 6 shows the magnetization curve during the flux avalanche at different environmental temperatures. The magnetic field reaches *B _{a}* = 12.6 mT with a field ramp rate of 500 T s

^{−1}, and the environmental temperatures are

*T*

_{0}= 0.25, 0.35, 0.45, 0.5

*T*

_{c}. For the same magnetic field ramp rate, the lower environmental temperature leads to a higher jump frequency of magnetization. In addition, a lower environmental temperature also leads to a smaller jump amplitude. This indicates that the temperature has different effects on the frequency and jump amplitude.

The maximum temperatures in the non-uniform film during the flux avalanche at different environmental temperatures are plotted in figure 7. The external magnetic field is 12.6 mT and the magnetic field ramp rate is 500 T s^{−1}. The field of the temperature jump increases with the environmental temperature. At a lower environment temperature, the jump frequency of the temperature is higher. This trend is similar to the magnetization jump. The peak value of the local temperature can exceed 100 K at an environmental temperature of 0.25 *T*_{c}, and this temperature is higher than the critical temperature of MgB_{2}. If the heat cannot be removed rapidly quench may occur, which could cause the failure of the superconductor.

In figure 2, we can see that the flux avalanche appears at the interface where the critical current density is discontinuous in the non-uniform film. When the magnetic field ramp rate increases, the flux avalanche can also appear in other regions. In addition, it was reported that flux avalanche will also occur first in the vicinity of the slit as the superconducting film has a slit [[17]]. In order to compare the effect of the slit and the discontinuous interface on the flux avalanche, we introduced two slits on the right and left sides of the non-uniform film. The length of the slits is *L _{s}* = 0.4

*L*and the width is

*W*= 0.01

_{s}*L*. The distributions of the magnetic field in the non-uniform thin film with two slits are plotted in figure 8. The magnetic field is increased from zero to

*B*= 5.0, 7.5, 10.0, 12.5 mT, respectively, with a field ramp rate of 500 T s

_{a}^{−1}. The environmental temperature is

*T*

_{0}= 0.5

*T*

_{c}. With the increasing magnetic field, the dendritic pattern appears at the tip of the slits first and then becomes larger. When the external applied magnetic field is small (

*B*< 7.5 mT), the dendritic structure appears only around the slits. However, when the magnetic field is more than 10 mT, the flux avalanche also occurs at the discontinuous interface. Interestingly, the sequence of flux avalanche at the interface is different from that in the non-uniform film without slits. From figure 8, we can see that the magnetic field at the crack tip is higher than that in the other regions. The dendritic structure first appears around the crack tip. The magnetization curves in the non-uniform thin film with and without slits are given in figure 9

_{a}*a*. The jump frequency in the film with two slits is higher. It can be expected that the high magnetization jump frequency corresponds to the lower jump magnitude (figure 9

*a*). The maximum temperature during the flux avalanche is given in figure 9

*b*. The peak value of the temperature in the film with slits is higher than that without slits. Thus, the effect of the slits on the flux avalanche is more obvious than the effect of the discontinuous interface.

## 4. Conclusion

In this paper, we numerically simulate the flux avalanche in a type-II superconducting film. Based on the FFT, the coupled Maxwell equation and the heat diffusion equation are numerically solved. The effect of a non-uniform critical current density on the flux avalanche is mainly considered. We present the results for the different magnetic field ramp rates and environmental temperatures. Finally, the non-uniform film with slits is taken into account. The results for the uniform film and non-uniform film are compared. It should be noted that the external magnetic field ramp rates are quite low in the experiments for the MgB_{2} specimen. However, we select a relatively large ramp rate as a qualitative simulation. It can be found that for a certain magnetic field ramp rate, with increasing external magnetic field, the flux avalanche first appears at the interface where the current density is discontinuous. The dendritic structure will also appear in other regions when the magnetic field ramp rate is increased. The larger the magnetic field ramp rate, the smaller the dendritic structures. The jumping frequencies of magnetization and maximum temperature in the non-uniform film are higher than those in the uniform film. The flux avalanche structure has a finger-like pattern instead of a dendritic pattern at a temperature of 0.25 *T*_{c}. For the non-uniform film with slits, the flux avalanche first takes place at the tip of the slits. With increasing external magnetic field, the flux avalanche can also appear at the discontinuous interfaces.

## 5. Computational solution

The numerical solution was performed on the 512 × 512 mesh using the FFT technique in MATLAB. The time step was 10^{−10} and the number of iterations was 2.5 × 10^{5}. A heat pulse was applied to trigger the flux avalanche.

## Authors' contributions

Y.L., Z.J., H.Y. and Y.Z. conceived the mathematical models, interpreted the computational results, and Y.L. wrote the paper. All authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

This work was supported by the National Natural Science Foundation of China (grant nos. 11202087, 11472120, 11421062 and 11602185), the National Key Project of Magneto-Constrained Fusion Energy Development Program (grant no. 2013GB110002), New Century Excellent Talents at the University of the Ministry of Education of China (NCET-13-0266) and the National Key Project of Scientific Instrument and Equipment Development (11327802).

- Received June 11, 2016.
- Accepted September 22, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.