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We demonstrate that superconducting aluminium nano-bridges can be driven intoa state with complete suppression of the critical supercurrent viaelectrostatic gating. Probing both in- and out-of-plane magnetic fieldresponses in the presence of electrostatic gating can unveil the mechanismsthat primarily cause the superconducting electric field effects. Remarkably, wefind that a magnetic field, independently of its orientation, has only a weakinfluence on the critical electric field that identifies the transition fromthe superconducting state to a phase with vanishing critical supercurrent. Thisobservation points to the absence of a direct coupling between the electricfield and the amplitude of the superconducting order parameter or $2\pi$-phaseslips via vortex generation. The magnetic field effect observed in the presenceof electrostatic gating is described within a microscopic model where aspatially uniform inter-band $\pi$-phase is stabilized by the electric field.Such an intrinsic superconducting phase rearrangement can account for thesuppression of the supercurrent, as well as for the weak dependence of thecritical magnetic fields on the electric field.

Unveiling Mechanisms of Electric Field Effects on Superconductors by Magnetic Field Response