Dendritic Flux Avalanches and Nonlocal Electrodynamics in Thin Superconducting Films

We report a mechanism of nonisothermal dendritic flux penetration in superconducting films. Our numerical and analytical analysis of coupled nonlinear Maxwell and thermal diffusion equations shows that dendritic flux pattern formation results from spontaneous branching of propagating flux filaments due to nonlocal magnetic flux diffusion and positive feedback between flux motion and Joule heating. The branching is triggered by a thermomagnetic edge instability, which causes stratification of the critical state. The resulting distribution of thermomagnetic microavalanches is not universal, because it depends on a spatial distribution of defects. Our results are in good agreement with experiments on Nb films.

Figure 1

Temperature distributions in a film with no defects for ${\dot{H}}_0=5J_1/t_h$, $\tau =0.0025$, $\alpha =0.008$, system size $600L_h\times 150L_h$ (only a quarter part is shown), and $j_0=20$ at (a) $t=16.75t_h$ and (b) $t=17.75t_h$; (c) magnetic flux pattern for $t=20t_h$.

Figure 2

Flux penetration in a film of $320L_h\times 40L_h$ with 30 random edge defects with ${\xi }_0=0.25L_h$, and amplitudes $q_i$ uniformly distributed between $0$ and $0.4$, for ${\dot{H}}_0=3J_1/t_h$, $\alpha =0.08$, $\tau =0.0025$, and $t=25t_h$ for (a) $j_0=20$ and (b) $j_0=80$. Black and white correspond to the Meissner and vortex phases, respectively. (c) Wide film of $600L_h\times 150L_h$ with 20 edge defects for ${\dot{H}}_0=5J_1/t_h$, $j_0=60$. The giant avalanche develops in a defect free region marked by the arrow. (d) Same wide film with 500 randomly distributed bulk defects at $j_0=80$. See also movies 2–6 in 16.

Figure 3

MOI of flux branching in Nb films of Ref. 7 at (a) $T=6.2\mathrm{K}$ and $B_0=31.2\mathrm{m}\mathrm{T}$ and (b) $T=4.7\mathrm{K}$ and $B_0=36.5\mathrm{m}\mathrm{T}$. (c) “Giant flux avalanche” at 4.5 K in the Nb film of 23. See also movies 7–8 in 16.