Tunable domino effect of thermomagnetic instabilities in superconducting films with multiply-connected topological structures

The MgB₂ thin films were prepared by hybrid physical-chemical vapor deposition (HPCVD) developed by Xi et al [32]. Details of the HPCVD system used in this work can be found in references [33, 34]. Briefly, Mg ingots ($> 99.5\%$ purity) were placed around a 5 × 5 × 0.3 mm MgO (111) substrate on a molybdenum susceptor, and heated up to 660 °C in 4 kPa ultrahigh pure hydrogen ($> 99.9999\%$ purity). The deposition started as 5% B₂H₆ balanced with hydrogen were slowly flowed into the vacuum chamber at a rate of 3 sccm. 50 nm-thick MgB₂ thin film were deposited on substrate after 4 min. The superconducting ring pattern schematically shown in figure 1(a) (with R_i1 = R_o1/2 = R_i2/3 = R_o2/4 = 0.5 mm) were obtained by photolithographic process by dry-etching with argon ion milling.

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The MO technique is employed to observe the flux penetration in the concentric rings, where a Bi-doped yttrium iron garnet with in-plane magnetic domains placed on top of the superconducting specimen is used as an indicator film based on Faraday effect [35]. The sample was first zero-field-cooled to 4.3 K, and then an increasing magnetic field was applied perpendicular to the sample.

For a very thin superconducting film, exposed to an applied magnetic field perpendicular to the film plane, the electrodynamics follows Maxwell's equations:

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

with B = μ₀ H and ∇ ⋅ J = 0, where J δ(z) is the current density with J the sheet current and δ(z) the delta distribution. The heat flow in the film is governed by the equation

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

where T is the local temperature in the superconductor, T₀ is the base temperature which is kept at a constant value, d is the sample thickness and J ⋅ E represents the Joule heating. The thermal conductivity and the specific heat of the superconductor are κ and c, and h is the coefficient of heat transfer between the superconducting film and the substrate. The temperature dependencies of the thermal parameters are assumed as $\kappa ={\kappa }_{0}{(T/{T}_{\mathrm{c}})}^{3}$, $c={c}_{0}{(T/{T}_{\mathrm{c}})}^{3}$ and $h={h}_{0}{(T/{T}_{\mathrm{c}})}^{3}$ [36].

The relationship between current and electric field is described by the commonly used phenomenological model [37]

$$\begin{equation}\boldsymbol{E}=\rho (J)\boldsymbol{J}/d,\end{equation} \tag{ 3 }$$

with a resistivity law

$$\begin{equation}\rho (J)=\begin{cases}{\rho }_{0}{(J/{J}_{\mathrm{c}})}^{n-1},\quad \hfill & J< {J}_{\mathrm{c}},\enspace T< {T}_{\mathrm{c}},\hfill \\ {\rho }_{\mathrm{n}},\quad \hfill & \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},\hfill \end{cases}\end{equation} \tag{ 4 }$$

where 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 so-called creep exponent. The E–J relationship is highly nonlinear and corresponds to Bean model when n → ∞. Here, the temperature dependence of J_c and n are taken as

$$\begin{equation}{J}_{\mathrm{c}}={J}_{\mathrm{c}\mathrm{0}}(1-T/{T}_{\mathrm{c}})\quad \mathrm{a}\mathrm{n}\mathrm{d}\quad n={n}_{0}{T}_{\mathrm{c}}/T+1,\end{equation} \tag{ 5 }$$

where J_c0 and n₀ are constants.

To solve the electromagnetic behavior, the sheet current is expressed through the local magnetization, g(x, y), as [38]

$$\begin{equation}{J}_{x}=\frac{\partial g}{\partial y},\qquad {J}_{y}=-\frac{\partial g}{\partial x},\end{equation} \tag{ 6 }$$

We can obtain the time derivative of g by the Biot–Savart law using the fast Fourier transform method as reported in reference [36]

$$\begin{equation}\dot{g}(\boldsymbol{r},t)={\mathcal{F}}^{-1}\left\{\frac{2}{k}\mathcal{F}[{\dot{B}}_{z}(\boldsymbol{r},t)/{\mu }_{0}-{\dot{H}}_{\mathrm{a}}(t)]\right\},\end{equation} \tag{ 7 }$$

where $\mathcal{F}$ is the two-dimensional spatial Fourier transform and k = | k |. This equation can be used to describe the dynamics of multiply-connected samples.

For a thin superconducting film as the one shown in figure 1, the xy plane is divided into three parts: the superconductor, the area in the hole and the area outside the ring. Inside the superconductor, ${\dot{B}}_{z}$ can be obtained by combining Faraday's law with the constitutive equations (3) and (4) of the superconductor, given by

$$\begin{equation}{\dot{B}}_{z}=\nabla \cdot (\rho \nabla g)/d.\end{equation} \tag{ 8 }$$

For the area outside the superconducting ring, ${\dot{B}}_{z}$ can be obtained by an iterative procedure that guarantees g = 0 outside the sample. For the area inside the hole, there are generally three different implementations to obtain ${\dot{B}}_{z}$ that satisfies the boundary conditions of the hole (g is kept constant in the hole). (i) g in the hole area is set to the lowest value of the inner edge of the ring [39, 40]. This method is simple and flexible to implement. However, unphysical net flux will be introduced into the hole in this way. (ii) The hole is considered as part of the superconducting sample, but its critical current is strongly reduced (or the resistance is increased) [41, 42]. This method is physically feasible, but the calculation efficiency is penalized due to the reduction of the time step. (iii) A modified iterative procedure is used to find ${\dot{B}}_{z}$ in the holes as reported by Vestgården et al [43, 44]. This approach does not require additional assumptions, but is more demanding from computational point of view. In what follows, we use the second method to model the holes. By testing different values of ρ, we find that ρ = ρ_n is large enough to make g = constant and J = 0 in the holes.

Numerical simulations are performed over a square of half-width L = 1.3R_o discretized on a 256 × 256 equidistant grid. The material parameters used in this calculation are typical for MgB₂ films, with T_c = 39 K, J_c0 = 50 kA m⁻¹, ρ₀ = ρ_n = 7 μΩ cm [36]. We use n₀ = 19 and limit n(T) to n(T) ⩽ 59 at low temperature. A weak quenched disorder is introduced by randomly decreasing J_c0 by 10% at 5% of the grid points. The thermal parameters are κ₀ = 0.17 kW K⁻¹ m⁻¹, c₀ = 35 kJ K⁻¹ m^{− 3} and h₀ = 220 kW K⁻¹ m^{− 2}. As we will be pointed out below, the numerically imposed ramp rate of the applied magnetic field is substantially larger than the ramp rate applied experimentally.

Figures 2(a)–(d) show MOI of concentric rings at different applied fields. The external field is ramped up in steps of 0.1 Oe, which is generated by a copper coil with resistance R = 22.1 Ω and inductance L = 26 mH. Since the corresponding time constant is τ = L/R ≈ 1 ms, a rate of about 100 Oe s⁻¹ = 0.01 T s⁻¹ is obtained. The brightness in these images represents the magnitude of the local magnetic flux density. In order to distinguish flux avalanches from the regular penetration, we display the differential images in figures 2(e)–(h), obtained by subtracting the corresponding previous images recorded at slightly lower magnetic fields from the upper panels, using the ImageJ software with the Image Calculator tool, as introduced in [25, 45–47]. The brightness here indicates an increase of the local magnetic flux with increasing applied field. At μ₀ H_a = 0.31 mT, several flux avalanches are triggered exclusively in the outer ring, including magnetic perforation accompanied by sudden flux increase in the outer hole between the two rings [panels (a) and (e)].

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When the applied field reaches μ₀ H_a = 0.49 mT, the first crossing dendrite appears in the inner ring as shown in panels (b) and (f). The increase in the flux field inside the central hole and the decrease of flux at the outer edge of the inner superconducting ring prove that a large amount of magnetic flux in the outer hole is injected into the central hole through this channel (i.e. magnetic perforation of the inner ring). With further increasing the applied field (e.g., μ₀ H_a = 1.57 mT and μ₀ H_a = 2.5 mT), several flux instabilities appear in the inner ring, including short-range flux avalanches and crossing dendrites [see panels (c), (d) and (g) and (h)]. Note that at μ₀ H_a = 1.57 mT flux avalanches are triggered simultaneously in both inner and outer superconducting rings as demonstrated in figure 2(g).

Remarkably, the smooth Bean-like magnetic profile has not yet fully penetrated the outer ring, when the first crossing dendrite in the superconducting inner ring is triggered in the vicinity of a flux avalanche in the outer ring, as shown in figure 2(b). This non-local correlation between avalanches in the outer and inner ring is suggestive of a domino effect.

Since the thermomagnetic breakdown of the inner ring seems to depend on the flux instability behaviors in the adjacent outer ring as suggested by MO images, it is instructive to start first by investigating numerically the flux avalanches and the magnetic perforation in a single ring. To the best of our knowledge, this analysis has not been done so far. As shown in figure 1(b), the inner radius, outer radius and thickness of the superconducting ring are R_i = 1.1 mm, R_o = 2.2 mm and d = 0.5 μm, respectively. Note that there is a small negative field at the inner edge of the ring resulting from the circulating supercurrent, which has also been reported and explained by Pannetier et al [48]. We will then extend the numerical simulations of the flux behavior for two concentric rings, and demonstrate the domino effect of thermomagnetic instabilities in which the flux avalanches in the inner ring can be triggered by the thermomagnetic instability at the outer ring. In this way we will be able to clarify the mechanism and triggering condition of thermomagnetic instability of the inner ring.

It is well known that the morphology of flux avalanches is strongly influenced by the temperature [1, 47]. Figures 3(a)–(d) show the magnetic flux distribution in the ring exposed to an increasing applied field with ${\mu }_{0}{\dot{H}}_{\mathrm{a}}=100$ T s⁻¹ at four different temperatures, where panels (a)–(c) correspond to the first magnetic perforation. At low working temperature (T₀ = 7 K), the first flux channel connecting the inner and outer edge of the ring appears at roughly μ₀ H_mp = 6.5 mT. As shown in figure 3(a), frequent finger-like flux avalanches in the ring appear before the formation of the flux channel, which is consistent with previous experimental reports [25]. In this case, the threshold field μ₀ H_th for the onset of the thermomagnetic instabilities is smaller than the threshold perforation field μ₀ H_mp for the appearance of the first crossing dendrite. With increasing temperature, the number of flux avalanches before the first magnetic perforation is found to decrease (not shown). At somewhat higher temperatures (e.g., T₀ = 11 K, 15 K), the magnetic perforation coincides with the onset of avalanche regime, which is consistent with the experimental results in reference [27]. Therefore, whether μ₀ H_mp is larger than μ₀ H_th or equal to can be controlled by varying the temperature. Furthermore, for the highest temperature of 20 K, the flux front almost reaches the inner edge of the ring without any thermomagnetic instability as seen in figure 3(d).

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With the occurrence of the magnetic perforation event, a large amount of flux is injected into the central hole through the flux channel. Figure 3(e) shows the corresponding evolution of average magnetic field in the central hole during the ramping up of the applied field. The dashed line in this figure indicates the applied magnetic field μ₀ H_a. At high temperature (T₀ = 20 K), there is no flux avalanches and hence no magnetic perforation event triggered. However, after the applied field reaches about 11 mT, the average magnetic field in the central hole increases linearly with the applied field due to the saturation of the current in the superconducting ring. As the working temperature decrease to a moderate value (e.g. T₀ = 11 K and 15 K), the average field in the central hole exhibits abrupt jumps up to the value of the applied field μ₀ H_a corresponding to the formation of the flux channel, in agreement with the experimental results shown in reference [27]. Moreover, this quasiperiodic perforation event increases with the decrease of working temperature. Interestingly, the average field in the hole no longer jumps to the value of the applied field at low temperature (e.g., T₀ = 7 K) but instead it remains below μ₀ H_a. This effect is ascribed to the shrinking of the heated flux channel with decreasing temperature, resulting in less flux being injected into the central hole of the superconducting ring.

In addition to the temperature, the ramping rate of applied field also plays an important role in the occurrence and morphology of the thermomagnetic instabilities [9, 19, 49–52]. In order to clarify the effect of the ramping rate on the flux avalanches and magnetic perforation, in figure 4 we show the magnetic field distribution in the rings (a)–(d) and average magnetic field in the central holes (e) exposed to increasing applied fields with ramping rates ranging from 25 T s⁻¹ to 200 T s⁻¹. For the lowest ramp rate ${\mu }_{0}{\dot{H}}_{\mathrm{a}}=25$ T s⁻¹, the ring remains stable in a state similar to that observed at high temperature as shown in figure 3(d). Remarkably, for the ramp rate of 50 T s⁻¹, a flux channel is formed after the first flux avalanche is triggered. Via this channel the central hole is filled with flux which manifests itself as a sudden increase of average field in the hole, and simultaneously nucleates several secondary avalanches from the inner edge of the ring. The behavior of flux avalanches in the ring for ${\mu }_{0}{\dot{H}}_{\mathrm{a}}=100$ T s⁻¹ is similar to that for ${\mu }_{0}{\dot{H}}_{\mathrm{a}}=50$ T s⁻¹, but with a lower perforation field and narrower flux channel. With further increasing the ramp rate, there are frequent and small flux avalanches triggered in the ring, but no flux is captured in the central hole until the first crossing dendrite is formed. In fact, avalanches continue to occur between the jumps of B_z shown in figure 4(e).

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As reported in reference [27], the threshold field μ₀ H_th in a ring-shaped superconductor is independent of the rim width. However, μ₀ H_th should be dependent on the sample size. Indeed, as shown in figure 5, the threshold field μ₀ H_th for rings with the same rim width (the symbols connected by solid lines) and that for rings with a fix R_i (the square symbols connected by a blue dotted line) both decrease with the radius of the ring at different temperatures. For the smallest R_i (R_i = 0.275 mm), there is no flux avalanche triggered in the ring at T₀ = 11 K. This can be explained by the fact that the local field at the border of the ring is increased by a large geometric factor ${B}_{\text{edge}}\sim {\mu }_{0}{H}_{\mathrm{a}}\sqrt{{R}_{\mathrm{o}}/d}$ [53–55], leading to a similar increase in the rate of local magnetic field change at the border. This argument leads as to conclude that μ₀ H_th should decrease with increasing R_o, as observed in figure 5. This is also consistent with the observations reported in reference [56]. Based on the same reasoning concerning the demagnetization factor, no dependence of μ₀ H_th on w (w = R_o − R_i) is to be expected, as indeed shown in figure 5, with symbols connected by a black dotted line.

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In addition, we also investigate the electrodynamic behavior of a single ring for decreasing applied field by applying a magnetic field of 10.0 mT after cooling down to 11 K, and subsequently reducing the field to −19.6 mT. For μ₀ H_a = 10.0 mT, which is larger than μ₀ H_th, there is a crossing dendrite in the ring, resulting in a jump of average field B_z in the central hole (see figures 6(a) and (e)). Figure 6(b) presents the flux penetration in the superconducting ring after the applied field is decreased from its maximum value to 0 mT. Bright and dark colors indicate positive high and negative local magnetic field, respectively. The field at the outer edge drops and negative flux (i.e., flux of opposite polarity to the applied field) appears, whereas the field at the inner edge increases. With further decreasing the applied field to −0.3 mT, a clear avalanche of negative flux is triggered retracing along the same path than the initial avalanche, which is consistent with the experimental results reported before [25, 57]. The negative flux outside the ring is injected into the central hole through the dark flux channel, corresponding to the abrupt drop of B_z in figure 6(e). After the first anti-flux perforation event, new events become frequent and several dark multi-branched dendrites appear in the superconducting ring, with more negative flux entering into the central hole and multiple jumps of average magnetic field in the central hole.

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We have shown that magnetic flux can reach the central hole of the single superconducting ring via a flux channel, confirming that the flux jump in the outer hole of concentric rings observed by MOI in figure 2 is caused by the magnetic perforation of the outer ring. Subsequently, a thermomagnetic instability in the inner superconducting ring shown in figure 2 occurs, as in the well-known 'domino effect'. Thus, we will further discuss whether the flux avalanches can continue occurring in the inner superconducting ring.

Figure 7 shows the magnetic field distribution of concentric superconducting rings (with R_i1 = R_o2/4, R_o1 = R_o2/2, R_i2 = 3R_o2/4, R_o2 = 2.2 mm) at different applied fields for T₀ = 9 K and ${\mu }_{0}{\dot{H}}_{\mathrm{a}}=100$ T s⁻¹. The simulated results reproduce the thermomagnetic instability process of the concentric rings observed by MO images presented in figure 2. For the lower applied field, several flux avalanches are triggered in the outer ring, while the inner ring is on a Bean's critical state. As the applied field increases, some flux channels are formed in the outer ring, injecting a large amount of flux into the outer hole as shown in panel (b). This process causes a stepwise increase in the magnetic field in the outer hole as demonstrated in the single ring case. When the field reaches μ₀ H_a = 7.5 mT, a thermomagnetic instability of the inner ring is triggered by the increase of the magnetic field in the outer hole. With further increasing applied field to 9.7 mT, the first magnetic perforation event appears in the inner ring accompanied by a jump of the magnetic field in the central hole, while magnetic flux has not fully penetrated the outer ring yet.

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It is important to note that the threshold field of avalanches and the magnetic field captured in the hole strongly depend on temperature as shown in figure 3. We present in figure 8 the influence of T₀ on the investigated superconducting concentric rings (R_i1 = R_o2/4, R_o1 = R_o2/2, R_i2 = 3R_o2/4, R_o2 = 2.2 mm) for a ramp rate ${\mu }_{0}{\dot{H}}_{\mathrm{a}}=100$ T s⁻¹. Panels on the left column show the field landscape just after the first channel in the inner superconducting ring has formed, and panels on the right column correspond to the average fields injected in each hole. For T₀ = 7 K (figures 8(a) and (b)), before the first magnetic perforation of the inner ring, there are frequent dendritic avalanches triggered in the concentric rings and several thermal channels formed in the outer ring, which is also confirmed by the quasiperiodic jumps of the average field in the outer hole. With increasing temperature, the number of avalanches before the first magnetic perforation of the inner ring decreases and the jump of average field in the outer hole becomes larger. For the highest temperature T₀ = 11 K, the average magnetic field in the outer hole jumps to the value of the applied field μ₀ H_a due to the appearance of a crossing dendrite in the outer ring. Interestingly, the threshold field for the onset of magnetic perforation in the inner ring μ₀ H_mp-inner does not increase monotonically with temperature, and the crossing dendrite in the inner ring is located near the flux channel of the outer ring, which is marked by the black ellipse. During this rapid process, the average magnetic field in the outer hole first jumps to a certain finite value and then is found to drop slightly. However, this decrease is much smaller than the increase of the average magnetic field in the central hole. These results suggest that the thermomagnetic instability at the inner ring can be triggered by the magnetic perforation in the outer ring, and therefore the mechanism of flux avalanches in this case is more complex.

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Figure 9 shows the magnetic field distribution for concentric superconducting rings (with R_i1 = R_o2/4, R_o1 = R_o2/2, R_i2 = 3R_o2/4, R_o2 = 2.2 mm) (a) and (b), and a single ring (with R_i = R_o/4, R_o = 2.2 mm) (c) and (d), for T₀ = 10 K and ${\mu }_{0}{\dot{H}}_{\mathrm{a}}=100$ T s⁻¹. As shown in panel (a), when the applied field reaches 7.1 mT, a flux channel is formed in the outer superconducting ring, through which the magnetic flux is injected into the outer hole and nucleates at multiple positions on the edge of the inner ring. Interestingly, the flux channel in the concentric superconducting rings terminates at the outer hole, and does not extend to the central hole. With further increasing the applied field to 10 mT, there is still only one flux avalanche triggered in the concentric rings as shown in figure 9(b). However, at the same applied field, several flux avalanches occur in the single ring and form two flux channels (figure 9(d)). Surprisingly, the outer hole of the concentric rings shows a stabilization effect, which is also observed in a circular hole [50, 58, 59]. This result indicates that despite the domino effect, the two-concentric-ring structure can not only reduce the scale and frequency of flux avalanches, but also effectively improves the thermomagnetic stability of the inner superconducting films, avoiding flux injection into the central hole of the system.

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In order to clarify the process and the mechanism of this cascade dynamics behavior, in figure 10 we compare the magnetic field distribution of the investigated superconducting rings in figure 8 with that of the single superconducting ring of the same size as the inner ring, as well as calculate the maximum electric field of each of them (panels on the right column). Figures 10(a) and (d) show the magnetic field landscape of concentric rings at μ₀ H_mp-outer = 8.6 mT and of single-layer ring at μ₀ H_mp = 18.5 mT for the same ${\mu }_{0}{\dot{H}}_{\mathrm{a}}$ and T₀ with that in figure 8(e), respectively. For the superconducting concentric rings in this case, once the flux channel is formed in the outer ring, the thermomagnetic instability of the inner ring is activated, and the nucleation loci of the first avalanches in the inner ring is near the channel [see supplemental material 1 at https://stacks.iop.org/NJP/24/083017/mmedia for flux penetration process]. Note that, the threshold field for the onset of avalanches in the inner ring is approximately equal to that of the outer ring (here H_th-outer = H_mp-outer), which is significantly lower than the threshold field of the single-layer of the same size as the inner ring. This observation confirms that the magnetic perforation in the outer ring of the superconducting concentric rings can trigger the flux avalanches in the inner ring. It is well known that with the formation of a magnetic flux channel in a superconducting ring, the average field of the central hole jumps to a certain value. To study the mechanism of the domino effect, we further compare the magnetic field distribution in the outer hole of the concentric rings (figure 10(b)) just after the formation of the flux channels with that outside the single ring (figure 10(e)) before the avalanches nucleate. Remarkably, for the concentric rings, a large amount of magnetic flux is injected into the outer hole through the flux channel, resulting in a rapid increase of local magnetic field in the vicinity, which is much larger than everywhere else in the hole. In contrast to that, for the single-layer superconducting ring, the magnetic field outside the superconductor is equal along the ring. According to the theoretical model developed in reference [60] ${H}_{\mathrm{t}\mathrm{h}}=d{j}_{\mathrm{c}}/\pi a\,\mathrm{tanh}(h{T}^{\ast }/nad{j}_{\mathrm{c}}{\mu }_{0}{\dot{H}}_{\mathrm{a}})$, where T* = |∂ln j_c/∂T|⁻¹, the threshold magnetic field for the onset of avalanches can be lowered by increasing the ramp rate of applied field. As a consequence, the fast rate of magnetic field variation in the hole near the flux channel favors the thermomagnetic instability nearby, triggering the flux avalanches in the inner ring.

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According to the linear analysis and numerical results in references [61, 62], avalanches are triggered once the local background electric field exceeds its critical value E_c, and keeps its maximum during the break-down event. This property is illustrated in figure 10(c) where the maximum electric field in the inner ring of concentric rings E_max-inner is shown and compared with that for the single-layer ring E_max (figure 10(f)). The main differences are that the maximum electric field for the inner ring of the concentric rings E_max-inner is lower before it abruptly jumps by almost ten orders of magnitude, and that the thermomagnetic instability of the inner ring lasts longer, which is due to the continuous penetration of the flux through the heated channel of the outer ring. For a single superconducting ring, with the nucleation and growth of avalanches, a large amount of heat is absorbed quickly and dissipated to stabilize the superconductor. However, for the superconducting concentric rings, it takes a longer time for the magnetic flux to flow into the central hole through the flux channel in the outer ring. During this period, the magnetic field distribution in the outer hole is not uniform and the ramp rate is not constant, which triggers the continuous flux avalanches of the inner ring at different points. These results are consistent with our suggestion that the domino effect forms the triggering mechanism of the avalanches in superconducting concentric rings.

Let us now investigate the influence of the gap separating the concentric rings on the flux avalanches by keeping the inner ring width constant and changing its radius. Figure 11 shows the magnetic field distribution and the maximum electric field of the superconducting concentric rings (with R_i1 = 3R_o2/8, R_o1 = 5R_o2/8, R_i2 = 3R_o2/4, R_o2 = 2.2 mm) and those of the single-layer ring of the same size as the inner ring. For the superconducting concentric rings, we can find that the threshold field of the inner ring μ₀ H_th-inner determined by the magnetic perforation event of the outer ring slightly increases when the gap is reduced by comparing figures 10(a) and 11(a). On the contrary, the threshold field of the single-layer superconducting ring μ₀ H_th decreases from 18.7 mT to 14.7 mT by increasing its outer and inner radius by 0.275 mm. These results seem to indicate that increasing the inner ring radius can increase the stability of the superconducting concentric rings. From panels (c) and (f), one can also see that the break-down event of the inner ring lasts longer than that of the single-layer ring. It is worth emphasizing that for the superconducting concentric rings with a larger inner ring, once a heated channel is formed in the outer ring, the magnetic perforation event in the inner ring is first triggered near the channel (i.e. μ₀ H_mp-inner = μ₀ H_th-inner) (see supplemental material 2 for magnetic perforation process in the concentric rings). The physical mechanism leading to this domino effect can be understood as follows: as the normal region (avalanche) propagates into the outer ring, supercurrents get compressed toward the inner rim of the outer ring thus generating a local higher magnetic field at the inner rim of the outer ring which in turn triggers the avalanche in the inner ring. As shown in figures 10(b) and 11(b), the local magnetic field at the border of the inner ring near the channel in the outer ring is indeed significantly larger than other region far away. Moreover, the domino effect is stronger when the gap between inner and outer rings is smaller.

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The discussion above suggests that the threshold magnetic fields of the inner ring μ₀ H_th-inner and for the magnetic perforation μ₀ H_mp-inner depend on the inner/outer radius of the inner ring (i.e. the gap of concentric rings). To further study the triggering condition of the break down event, we present in figure 12 the threshold field for the thermomagnetic instability μ₀ H_th-inner as a function of R_o1/R_o2 with R_o2 = 2.2 mm, and we also build up phase diagrams delineating the boundaries of regimes of smooth penetration, flux avalanches (μ₀ H_th-inner < μ₀ H_mp-inner), and magnetic perforation (μ₀ H_th-inner = μ₀ H_mp-inner) firstly triggered in the inner ring. The threshold field of the inner ring increases with the outer radius R_o1. Note that in this case, the flux channel is formed in the outer ring after the first thermomagnetic instability event occurs. The inner ring remains stable until the flux is injected into the outer hole through this channel, corresponding to the lower regime in figure 12. The left orange dots correspond to smaller R_o1 for which flux avalanches are triggered in the inner ring without the formation of flux channel connecting the outer hole and the central hole of the superconducting concentric rings. When the outer radius of the inner ring increases to a certain value (R_o1 ⩾ 13R_o2/24), a crossing dendrite is formed in the inner ring after the magnetic perforation event in the outer ring (i.e., μ₀ H_mp-inner = μ₀ H_th-inner = μ₀ H_mp-outer = μ₀ H_th-outer), and the flux is injected into the central hole of the superconductor, which is due to the decrease of gap between the rings. This is in accordance with the previous prediction based on figure 11 and the above explanation concerning the domino effect.

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