  In the last 60 years conventional solid and electrolyte gating allowed
sizable modulations of the surface carrier concentration in metallic
superconductors resulting in tuning their conductivity and changing their
critical temperature. Recent conventional gating experiments on superconducting
metal nano-structures showed full suppression of the critical current without
variations of the normal state resistance and the critical temperature. These
results still miss a microscopic explanation. In this article, we show a
complete set of gating experiments on Ti-based superconducting Dayem bridges
and a suggested classical thermodynamic model which seems to account for
several of our experimental findings. In particular, zero-bias resistance and
critical current IC measurements highlight the following: the suppression of IC
with both polarities of gate voltage, the surface nature of the effect, the
critical temperature independence from the electric field and the gate-induced
growth of a sub-gap dissipative component. In addition, the temperature
dependence of the Josephson critical current seems to show the transition from
the ballistic Kulik-Omelyanchuck behavior to the Ambegaokar-Baratoff
tunnel-like characteristic by increasing the electric field. Furthermore, the
IC suppression persists in the presence of sizeable perpendicular-to-plane
magnetic fields. We propose a classical thermodynamic model able to describe
some of the experimental observations of the present and previous works. Above
all, the model grabs the bipolar electric field induced suppression of IC and
the emergence of a sub-gap dissipative component near full suppression of the
supercurrent. Finally, applications employing the discussed effect are
proposed.
