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The knowledge of vortex nucleation barriers is crucial for applications ofsuperconductors, such as single-photon detectors and superconductor-basedqubits. Contrarily to the problem of finding energy minima and critical fields,there are no controllable methods to explore the energy landscape, identifysaddle points, and compute associated barriers. Similar problems exist inhigh-energy physics where the saddle-point configurations are calledsphalerons. Here, we present a generalization of the string method to gaugefield theories, which allows the calculation of energy barriers insuperconductors. We solve the problem of vortex nucleation, assessing theeffects of the nonlinearity of the model, complicated geometry, surfaceroughness, and pinning.

physical review. b./physical review. b

Vortex nucleation barrier in superconductors beyond the Bean-Livingston  approximation: A numerical approach for the sphaleron problem in a gauge  theory