Sensitivity of the thermomagnetic instability in superconducting film to magnetic perturbation for electromagnetic interference detection

Thermomagnetic instability is one of the crucial factors that limits the safe and stable operation of superconducting devices, which has been observed by magnetic-optical imaging (MOI) in the form of flux avalanches in superconducting films of different materials (e.g. MgB₂ [1–3], YBa₂Cu₃O$_{7-\delta}$ [4–7], Pb [8, 9], Nb [10–12], Nb₃Sn [13], NbN [14–16], a-MoSi [17], and V₃Si [18]). Numerous efforts have been devoted to understanding and controlling flux avalanches. The experimental results obtained by MOI suggest that there is a threshold field $H_\mathrm{th}$, above which the flux avalanches are triggered [19, 20]. Then, linear stability analysis and numerical simulations based on fast Fourier transform are utilized to explain the onset conditions and the evolution of flux avalanches [21–23]. The thermomagnetic instabilities are found to be suppressed by decreasing the sample dimensions [19], covering the superconducting films with a layer of normal metal [24–27], and guided by introducing the artificial antidot array [8, 28–30]. Moreover, it is commonly accepted that these abrupt events occur when numerous vortices are driven out from pinning centers and start to move, locally heating the film and facilitating more flux motion. Eventually, a positive feedback loop starting from a small initial fluctuation leads to large-scale thermomagnetic instability.

Besides the sample size, structure, and intrinsic properties of superconducting films mentioned above, the threshold field as well as the morphology of flux avalanches are found to be sensitive to external conditions. For various superconducting films exposed to an increasing transverse field, numerous experiments and simulations demonstrate that the threshold field $H_\mathrm{th}$ and the scale of flux avalanches can be increased by increasing the working temperature T₀, while the number of sequential avalanche events decreases [1, 22, 31, 32]. Similar results can also be observed by reducing the ramp rate of the applied field [32–34]. Besides the above DC cases, where the applied field increases linearly with a constant ramp rate, the electromagnetic responses (e.g. critical current, AC loss, and thermomagnetic instability) of superconductors exposed to the AC field have been widely investigated [35–45]. Motta et al experimentally observed the flux avalanches triggered by an AC field in a superconducting Nb film and revealed that the flux avalanches triggered at the increasing branch of the applied field can guide antiflux entry in the decreasing applied field [43, 44]. This feature of flux avalanches under an AC field was then reproduced by Jing and Ainslie using numerical simulations [45]. However, there is also another interesting and unresolved question on the sensitivity of thermomagnetic instability in the superconducting film to the magnetic perturbation: namely, how the applied magnetic field affects the electromagnetic breakdown event if superimposing a small AC magnetic oscillation on a linearly increasing DC field.

As the instability of the superconducting state shows extreme sensitivity to the external environment, the superconductor is commonly used for detection of single photons (such as superconducting quantum photodetectors developed for x-ray spectroscopy and modern nanowire superconducting single-photon detectors) [46–50]. The mechanism of the single-photon detection by the current-carrying superconducting film can be explained as follows: when a single photon is absorbed, a hot spot is created in the superconducting film, which can locally destroy the superconductivity and reduce the critical current $I_{\mathrm{c}}$. Currently, this single-photon detection system is the most promising detection technology in the infrared range, which can achieve the detection efficiency of 93% at 1550 nm [51]. However, to our knowledge, the detection of the lower-frequency electromagnetic perturbation by superconducting film has not been reported yet.

In this paper, we numerically investigate the sensitivity of the thermomagnetic instability of the superconducting film to AC magnetic perturbations, by simulating flux behavior in the film subjected to applied fields with AC magnetic oscillations of different amplitudes B₀ and frequencies f. An unexpected non-monotonicity effect is observed in the superconducting film, which depends on the working temperature and oscillation amplitude. The presented non-monotonicity is further proved to be attributed to the rapid increase of the maximum electric field in the superconducting film induced by the magnetic oscillations of a specific frequency range (of the order of a few MHz). Finally, a novel possible electromagnetic interference detection system is proposed to detect the target electromagnetic perturbation of the specific frequency in a complex electromagnetic environment, by analyzing jumps in magnetic moment curves. In section 2, we briefly introduce the thermomagnetic model used to numerically study the thermomagnetic instability of superconducting films. The sensitivity of flux avalanches in superconducting films to electromagnetic perturbations of different amplitudes and frequencies is presented and discussed in section 3. Finally, the conclusions of the most salient results are summarized in section 4.

For the model system, as shown in figure 1(a), we consider a square superconducting film of width 2a and thickness d, which is subjected to an oscillating magnetic field. The external magnetic disturbance is realized by artificially superposing an AC magnetic field $B_{0}\mathrm {sin}(2\pi ft)$ on the DC applied field $\dot{B}_{a}t$ (see figure 1(b)). The nonlocal electrodynamics of superconducting films are governed by Maxwell equations:

$$\begin{equation} {}\dot{\boldsymbol{B}}=-\nabla \times \boldsymbol{E},~{\nabla} \times \boldsymbol{H}=\boldsymbol{J}\delta(z) \; \text{and} \; {\nabla} \cdot \boldsymbol{B}=0, \end{equation} \tag{ 1 }$$

where B is the magnetic field, which is related to the magnetic field strength by $\boldsymbol{B} = \mu_{0}\boldsymbol{H}$, E is the electric field, and $\boldsymbol{J}\delta(z)$ is the current density, where J is the sheet current and $\delta(z)$ is the Dirac delta distribution.

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As we know, the occurrence of flux avalanches is a phenomenon that involves both the critical-state and ohmic properties [21, 22, 33, 52, 53]. The highly nonlinear material characteristics of superconductors causing thermomagnetic instability can be well described by the conventional power law [54]

$$\begin{align} \boldsymbol{E}=\rho(J)\boldsymbol{J}/d,\nonumber\\ \end{align}$$

$$\begin{align} \boldsymbol{\rho}=\left\{ \begin{array}{ll} \rho_{0}(J/J_{c})^{n-1}, & J\lt J_{c},~T \lt T_{c},\\ \rho_{n}, &\mathrm {otherwise}, \end{array} \right. \end{align} \tag{ 2 }$$

here, ρ₀ and ρ_n are a constant resistivity and the normal state resistivity, respectively, J_c is the critical sheet current density, and n is the flux creep exponent. The temperature dependencies are taken as $J_{c} = J_{c0}(1-T/T_{c})$ and $n = n_{0}T_{c}/T+1$, where $J_{c0}$ and n₀ are constants. The local temperature is found by solving the heat diffusion equation

$$\begin{equation} dc\dot{T}=d\kappa\nabla^2 T-h(T-T_{0})+\boldsymbol{J}\cdot\boldsymbol{E}, \end{equation} \tag{ 3 }$$

in which κ and c are the thermal conductivity and the specific heat of the superconductor, T₀ is the substrate temperature, which is fixed with a constant, and h is the heat transfer coefficient between the superconducting film and the substrate. The thermal parameters depend on the temperature, and are assumed as $\kappa = \kappa_{0}(T/T_{c})^{3}$, $c = c_{0}(T/T_{c})^{3}$, and $h = h_{0}(T/T_{c})^{3}$ [22]. In this thermomagnetic model, the flux avalanches are modeled by coupling the equations describing nonlocal and nonlinear electrodynamics (equations (1) and (2)) with an equation (equation (3)) for the increase and propagation of the heat. The key element in solving the local inhomogeneities is the $E(T,J)$ characteristic, which accounts for the resistive $\rho(x,y)$ at different positions for the flux creep regime at $J(x,y)\lt J_{c}$ and $T(x,y)\lt T_{c}$, the flux flow regime at $J(x,y)\gt J_{c}$, and the normal state at $T(x,y)\gt T_{c}$ [22, 53]. It is widely reported that the main characteristics of flux avalanches can be captured by this thermomagnetic model, which is used to understand the experiment results [23, 52, 53].

Numerical simulations are effectively performed using a Fourier real-space hybrid algorithm to confirm zero current outside the film (see the details in [22]). The width and the thickness of the film are a = 2.2 mm and $d = 0.5\,\mu$m, respectively. The material parameters used in the simulations are typical for MgB₂ films, given as $T_{c} = 39$ K, $J_{c0} = 50$ kA m⁻¹, $\rho_{0} = \rho_{n} = 7\,\mu\Omega$ cm, and $n_{0} = 19$, and limit n(T) to $n(T)\leqslant59$ at low temperature. The thermal parameters are given as $\kappa_{0} = 0.17$ kW Km⁻¹, $c_{0} = 35$ kJ Km⁻³, and $h_{0} = 220$ kW Km⁻².

Figures 2(a)–(c) show the flux distribution of the superconducting films subjected to an increasing applied field with an electromagnetic perturbation (i.e. a DC applied field of ramp rate $\dot{B}_{a} = 100$ T s⁻¹ superposed with an AC field of $B_{0} = 0.1$ mT and f = 0.1 MHz) at different working temperatures. The maximum positive magnetic field corresponds to the brightest intensity, whereas the negative local magnetic field is indicated by the dark one. In this paper, all simulations are implemented using a DC field of this ramp rate. Since the actual electric field can be much larger than that obtained from the simple theoretical calculation due to the strong nonuniformity of the flux penetration both in space and in time, the ramp rates that trigger the thermomagnetic instability are much smaller in experiments [22, 34, 55–57]. We also show the flux distribution of the films when the magnetic field increases linearly to the same field $B_{a} = 8$ mT for comparison (see figures 2(d–f)). It is found that the working temperature T₀ plays a crucial role in the thermomagnetic instability of superconducting films, even if in the applied field with the magnetic perturbation. The numerical results show a transition from many finger-like avalanches at low temperatures to a few branched dendrites at higher temperatures in both situations, which arises from a change in the mechanism of heat dissipation, as explained in [58, 59]. Note that flux avalanches in the film exposed to the increasing field superposed with a small AC field are more easily triggered with a smaller morphology and a higher frequency, indicating sensitivity of the thermomagnetic instability of the superconducting film to the artificially applied magnetic field perturbation. At low temperature ($T_{0} = 7$ K), the number of flux avalanches is slightly larger in the film subjected to the applied field with added magnetic perturbation, but the size is similar to that under the DC applied field. Remarkably, when increasing the temperature, this sensitivity becomes more pronounced at higher temperature, leading to a qualitative change in the thermomagnetic state at $T_{0} = 15$ K.

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The temperature dependence of the threshold magnetic field of the flux avalanche in the superconducting film for the two excitation modes mentioned above is shown in figure 3. At low temperature ($T_{0}\leqslant10$ K), the threshold field varies very slightly, which is also previously reported as a general characteristic of superconducting MgB₂ films [60] and tapes [3]. As the temperature increases beyond 11 K, one can find that the artificially superposed magnetic perturbation can significantly reduce the threshold field for the onset of the dendritic avalanche. Moreover, both threshold fields increase with the temperature, as well as the fact that the difference between them also shows an increasing trend, evidencing the temperature dependence of this magnetic perturbation sensitivity.

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To understand this sensitivity systematically, as shown in figure 4, we now study the electromagnetic response of the superconducting film to an AC perturbation of different amplitudes $B_{0} = 0.01$ mT, 0.1 mT, 1 mT, and 5 mT. For a low amplitude $B_{0} = 0.01$ mT, both the frequency and the size of the flux avalanches in figure 4(a) are the same as those of the flux avalanches in the increasing applied field without AC magnetic perturbation shown in figure 2(e), indicating that the influence of the AC perturbation on the thermomagnetic instability of the superconducting film can be ignored when the amplitude B₀ is low enough. However, when increasing the amplitude of the AC field B₀, the flux avalanches become smaller and the number of avalanches significantly increases. In other words, the sensitivity of the thermomagnetic instabilities in films to the AC electromagnetic perturbation becomes stronger with the increase in the amplitude. For the superposed AC field with a larger amplitude $B_{0} = 5$ mT, figure 4(d) shows a mixture of flux avalanches (bright) and antiflux avalanches (dark), similar to the flux behavior in superconducting films under an AC magnetic field of a triangular waveform reported in [43, 45]. Interestingly, the reversed flux penetrates the sample through the same tracks as the positive flux avalanches, but instead of completely overlapping the positive flux avalanches, it eventually forms smaller antiflux patterns.

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Let us now investigate the drastic process of thermomagnetic dynamics in an increasing field with the added AC magnetic field of a large amplitude $B_{0} = 5$ mT. In figure 5, we present the maximum temperature evolution and jumps in the magnetic moment normalized by a magnetization constant $M_{0} = a^{3}J_{c0}$, as reported in [22, 26, 61], of the superconducting film during the first period $\Gamma$ of the added AC field. The inset images in figure 5(b) correspond to the flux penetrations at $t = \Gamma/4, \Gamma/2, 3\Gamma/4, \Gamma$ respectively. It can be found that when the applied field reaches a threshold field B_th (${\thicksim}3.0$ mT), flux avalanches penetrate into the film abruptly in the form of finger-like dendrites, accompanied by the continuous increase in the maximum temperature. As the applied field decreases, the temperature returns back to the working temperature until the time close to $\Gamma/2$. At $t = \Gamma/2$, flux avalanches in the film are found to be composed of flux and antiflux. Upon further decreasing the applied field, more antiflux avalanches occur and overlap the former flux avalanches completely when the field drops to a minimum value. However, the size of the positive flux avalanches is much larger than that of the antiflux avalanches after several cycles of magnetic excitation, as shown in figure 4(d). It is worth noting that these periodic intermittent jumps in the maximum temperature and magnetic moment indicate that flux avalanches mainly occur at the increasing branch of the applied field, whereas flux penetrates the film smoothly at the decreasing branch in this case.

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Figure 6 shows the threshold magnetic fields for the onset of the flux avalanche as a function of the amplitude B₀ of the added AC magnetic perturbations with different frequencies. From the figure one sees that for the smaller oscillation frequencies (f = 0.05 MHz and 0.1 MHz), the threshold field $B_\mathrm{th}$ for very low B₀ is also similar to that for the DC applied field (∼6.5 mT), and then decreases with B₀ to varying degrees. Above a certain amplitude B₀, found to be close to 0.4 mT for f = 0.05 Hz, and 0.2 mT for f = 0.1 MHz, the threshold field becomes merely constant. Moreover, the threshold field for a high magnetic perturbation frequency f = 0.5 MHz is independent of B₀, and remains at a low level (∼2.6 mT). Thus, it seems that the sensitivity of flux avalanches to the magnetic perturbation can also be tuned by changing the oscillation frequency.

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Now we discuss the effect of the magnetic perturbation frequency f on the thermomagnetic instability in the superconducting film. As a first step, we focus on the flux profiles in superconducting films for different frequencies of the additional AC fields at $T_{0} = 16$ K (see figure 7). For the lower frequency (f = 0.02 MHz), one can find that the flux smoothly penetrates into the entire film without any flux avalanches. When the frequency f increases to 0.1 MHz, the smooth penetration is interrupted by a flux avalanche at 13 mT. Upon further increasing the frequency to 0.6 MHz, as shown in figure 7(c), numerous flux avalanches of smaller size are observed at the same applied field as in panel (b), indicating that the thermomagnetic instability seems to become more sensitive with the increase in the magnetic field perturbation frequency. However, for a higher frequency f = 4 MHz, there is only one flux avalanche observed at the same applied field $B_{a} = 13$ mT, which is similar to that in panel (b) for f = 0.1 MHz. The most interesting aspect of these results is that the influence of the frequency on the flux avalanches in superconducting films appears to be non-monotonic. More specifically, the number of flux avalanches in the superconducting films at the same magnetic field increases first and then decreases with increasing frequency. In fact, the actual critical current J_c is dependent on the frequency of the AC field, when the film is subjected to an AC applied field with/without a small DC transport current [35–39]. However, the rapidly rising DC magnetic field may play a major role in the applied field relative to the AC field in this work, because the amplitude of the AC magnetic field is quite small. It is suggested that the new-found non-monotonic characteristic presented here cannot be changed, even by considering the frequency-dependent critical current $J_{c}(T,f)$ of the film subjected to the applied field dominated by the AC field (see the appendix for a detailed discussion). In this case, the frequency dependence of J_c is neglected in this work.

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Since the sample is usually mounted on a metallic cold finger in the MOI experiment, the AC magnetic field of such frequencies in which a non-monotonic effect is expected (${\sim}0.1$ MHz), is probably attenuated due to the eddy current through the metallic parts of the cryostat at the low temperature, bringing the challenge to a similar MOI experiment. Hence, we also show the normalized magnetization of the superconducting film as a function of the DC magnetic field with $T_{0} = 16$ K, $B_{0} = 0.05$ mT in figure 8, which allows us to further verify this non-monotonic effect by the DC magnetometry, as reported in [2]. The insets are the flux distributions of the superconducting film at 15.5 mT, which are marked by blue asterisks. One finds that the number of flux avalanches shown in these insets is consistent with the number of jumps in the corresponding magnetization curves, indicating that each vertical step corresponds to the occurrence of flux avalanches. As shown in figure 8(a), for lower frequencies, the threshold field of the flux avalanches decreases with the increase of frequency, and then increases with the frequency when f > 0.6 MHz, as shown in panel 8(b). Moreover, the variation of the jump frequency of the magnetization curve also shows non-monotonic characteristics with the change in the magnetic oscillation frequency.

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To quantify this unexpected non-monotonic effect of the frequency f, we show in figure 9(a) the threshold field $B_\mathrm{th}$ for the onset of the first flux avalanche as a function of the frequency of the superposed AC field. The inset images present the flux distributions in the superconducting films subjected to a DC field, as well as the situations for large frequencies of the added applied field with different oscillation amplitudes at $T_{0} = 11$ K and 16 K. Surprisingly, the threshold field shows a significant non-monotonicity dependence on the frequency of the added AC field, i.e. the threshold field $B_\mathrm{th}$ significantly decreases when $f\leqslant0.1$ MHz, and then rises relatively gently with the frequency of the magnetic perturbation. At low temperature ($T_{0} = 11$ K), the first avalanche is triggered after linearly ramping the applied field to 6.4 mT, which is close to the threshold field of the flux avalanche in the film exposed to an applied field with low frequency perturbation (f = 0.01 MHz). Moreover, one also finds that both the threshold field and the flux morphology tend to be close to those under the DC applied field, as the frequency increases to 10 MHz. The same effect can be seen at the higher temperature $T_{0} = 16$ K, with the sole difference being that the flux has almost smoothly penetrated the entire sample and has generated an X-shaped discontinuity line (d-line) [62] under the applied fields with AC perturbation of very high (or low) frequency, as well as exposed to the DC applied field. Therefore, the non-monotonic effect of the frequency indicates that the magnetic oscillation of very high/low frequency is not the main factor triggering avalanches.

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To clarify the mechanism of this abnormal non-monotonicity, we calculate the maximum electric field in the superconducting films during the same magnetic field ramp time before the thermomagnetic instabilities occur. The associated flux avalanche patterns corresponding to figure 9(b) are shown in figures 7 (b)–(d). The pink curves are an eye guide to clearly show the variation trend of the maximum electric field. Significant oscillations of the maximum electric field are observed after the magnetic field ramps for some time (i.e. $t\geqslant 2\times10^5$ s). For the lower frequency (f = 0.1 MHz), the electric field stays at a low level at this stage, where the maximum applied magnetic field is far less than the threshold field of the avalanche. Upon increasing the frequency to 0.6 MHz, one can find that $E_\mathrm{max}$ increases exponentially with time, reaching the threshold conditions for flux avalanches, as expected when the sample locally enters the ohmic regime. Note that the electric field of the superconducting film in the critical state before the thermomagnetic instability is very small (around 10⁻²–10⁰ V m⁻¹), which is in agreement with the critical electric field E_c reported in [21, 55, 63], i.e. about 10⁻⁴ V m⁻¹ for a strip, 0.1 V m⁻¹ for a slab, and 0.5 V m⁻¹ for a film. Meanwhile, for the higher frequency f = 4 MHz, $E_\mathrm{max}$ increases at a relatively slow rate despite the appearance of a non-uniform oscillation, which reflects the insensitivity of the thermomagnetic instability to the magnetic field perturbation of very high frequency. As reported in [25, 64], there are several oscillatory regimes in the temperature and electric field before the appearance of avalanches, especially non-uniform oscillations, which can be regarded as the onset signal of thermomagnetic instability. Nevertheless, the oscillation of the maximum electric field here is induced by the artificial external magnetic perturbation, which is also observed in the case of essentially full flux penetration without any avalanches for very low/high f (not shown). These results can be well explained by a characteristic oscillation frequency corresponding to the lowest threshold field, with which maximum electric field oscillations dominate the thermomagnetic instability. More specifically, when the frequency of the external magnetic oscillation f is close to this characteristic frequency, the flux avalanche is more likely to be triggered at a lower threshold field. Direct evidence for the characteristic oscillation frequency determining the non-monotonic threshold field is presented in panel 9(a). The minimum threshold field for the onset of the flux avalanche corresponds to a lower frequency at the higher temperature $T_{0} = 16$ K than that at the lower temperature $T_{0} = 11$ K, signaling that the characteristic frequency decreases when increasing the temperature, which is fully consistent with the theoretical results in [64].

Since the non-monotonic effect of the frequency implies that the thermomagnetic instability can be affected only when the frequency of the magnetic field perturbation reaches a certain range, it is expected that the thermomagnetic instability can be utilized to detect the electromagnetic interference with such a range of frequency. For this purpose, we further study the electromagnetic response of the superconducting film subjected to a complex electromagnetic environment containing multiple magnetic field perturbations of different frequencies: $f_{1} = 0.1$ MHz, $f_{2} = 0.6$ MHz, and $f_{3} = 4$ MHz, at $T_{0} = 16$ K. Figures 10(a)–(d) show the distributions of B_z in the superconducting film when the applied field increases to 11 mT through different excitation processes. The corresponding excitation processes with different magnetic perturbations $B_{ac1}$-$B_{ac4}$ are shown in figure 10(e). From panel (a), we find that when the added AC magnetic oscillation consists of two magnetic perturbations at lower frequencies (i.e. $B_{ac1} = B_{0}(\mathrm{sin}(2\pi f_{1}t)+\mathrm{sin}(2\pi f_{2}t))$, where $f_{1} = 0.1$ MHz and $f_{2} = 0.6$ MHz), the number and morphology of the flux avalanches are close to those for a single magnetic perturbation with f = 0.6 MHz shown in figure 7(c). For the perturbation shown in panel 10(c), $B_{ac3}$ with two higher frequencies $f_{2} = 0.6$ MHz and $f_{3} = 4$ MHz, more flux avalanches of smaller size are triggered at the same applied field $B_{a} = 11$ mT, which are similar to the case when the added AC oscillation consists of all the magnetic perturbations mentioned above in figure 10(d). Moreover, this similarity is also reflected in the corresponding excitation process shown in figure 10(e). It is also worth pointing out that when we shield the magnetic perturbation of the frequency $f_{2} = 0.6$ MHz, the flux penetrates into the film smoothly without any flux avalanches (see figure 10(b)), confirming that the magnetic perturbation of the middle frequency $f_{2} = 0.6$ MHz is the main factor that causes the thermomagnetic instability in the superconducting film.

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Figure 10(f) displays the moment in units of $M_{0} = a^{3}J_{c0}$ as a function of the excitation time for four electromagnetic perturbations shown in panel (e). Each vertical jump corresponds to a flux avalanche. One can see that in such a complex magnetic field environment, the electromagnetic perturbation of the frequency $f_{2} = 0.6$ MHz can be easily detected by the magnetic moment measurements. Namely, the thermomagnetic instability events are noticeable in the magnetic moment curve and show several pronounced drops, which are the signature of the existence of the magnetic perturbation of such frequencies. The insets present the flux patterns corresponding to the first flux jump of each moment curve, and show that the flux avalanche of smaller size can be triggered at a lower threshold field when the applied field contains magnetic perturbations of two higher frequencies ($f_{1} = 0.6$ MHz and $f_{2} = 4$ MHz).

In summary, we utilize a thermomagnetic model to numerically study the sensitivity of thermomagnetic instability in superconducting films to superimposed sinusoidal magnetic perturbation. By comparing with the flux penetration in a superconducting film subjected to a DC applied field, we find that the added magnetic oscillation can reduce the threshold field for the onset of flux avalanches and increase the frequency of such events. In particular, at high temperatures, the stable state of the superconducting film exposed to a DC field can be interrupted by thermomagnetic instability by adding a small AC magnetic perturbation. Moreover, a varying magnetic perturbation amplitude or frequency leads to progressive changes in this sensitivity. It is shown that the influence of the artificial magnetic perturbation on the thermomagnetic instability becomes more pronounced with the increase of the amplitude, which can be ignored for a very low B₀.

The most intriguing part of this study is the unexpected non-monotonic effect of the oscillation frequency on the thermomagnetic behavior of superconducting films (i.e. both the threshold field for the onset of flux avalanches and the morphology of flux patterns decrease first with the frequency of the additional AC field and then increase relatively slowly), showing a dependence on the working temperature and oscillation amplitude. This non-monotonic effect can be well explained by the characteristic oscillation of the electric field determining the thermomagnetic instability, which cannot be aggravated by the AC magnetic perturbation with very low/high frequency. Based on these findings, we propose a novel method for the detection of electromagnetic interference by superconducting film in the complex electromagnetic environment, by analyzing the jumps in magnetic moment curves.

We acknowledge support from the National Natural Science Foundation of China (Grant No. 11972298).

All data that support the findings of this study are included within the article (and any supplementary files).

As reported in [35–39], the critical current J_c of the superconducting film is dependent on the frequency of the AC field, when the film is subjected to an AC applied field with/without a small DC transport current. In addition, such a frequency dependence of the critical current is given by several empirical relationships. However, the critical current frequency dependence of MgB₂ film exposed to the complex case of DC and AC magnetic fields, i.e. a linearly increasing magnetic field superposed with an AC magnetic field of small amplitude, has not been described by a mature formula. It is supposed that the result obtained by the E – J model of a single DC magnetic field is the lower limit case, while the full frequency dependence of J_c is the upper limit case. In this case, we numerically study the flux avalanches of the superconducting film (with $T_{0} = 16$ K, $B_{0} = 0.5$ mT) to verify the validity of the non-monotonic effect of the magnetic perturbation frequency, taking into account the frequency dependence of J_c [35]

$$\begin{equation} J_{c}(T,f)=J_{c0}(1-T/T_{c})\left(\frac{f}{f_{0}}\right)^{1/n}, \end{equation} \tag{ 4 }$$

where f₀ is a constant frequency and n is the creep exponent. As shown in figure 11, both the number of flux avalanches at the same magnetic field and the threshold field for the onset of the thermomagnetic instability show a non-monotonic change with the increase in frequency, which confirms that the new finding of the non-monotonic characteristic presented in this work cannot be changed, even by considering the frequency dependence of $J_{c}(T,f)$ in the film subjected to the applied field dominated by the AC field.

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