A concept of the upper threshold field for dendritic vortex avalanches in superconductors.

Dendritic vortex avalanches in superconductors (SC), associated with thermo-magnetic instabilities, are known to destroy a metastable critical state and severely detriment performance of SC applications. Several thermo-magnetic models have been proposed to determine conditions under which a superconductor is to undergo a dendritic magnetic avalanche. However, there remains an omnipresent experimental fact that has not been given a theoretical explanation: instabilities always disappear above some upper threshold field. Our recently developed model [Phys. Rev. Lett. 97, 077002 (2006)] predicts divergence of the threshold field at low values of the critical current j$_{c}$. In its turn, at increasing magnetic field j$_{c}$ becomes strongly suppressed by the field. We explain disappearance of the dendritic avalanches in terms of the divergence of the upper threshold field due to a strong dependence of the critical current on magnetic field j$_{c}$(B) and support it by the results of our recent magneto-optical investigations of dendritic flux avalanches in NbN thin films. We then verify the model in a range of controllably varied j$_{c}$ values.