  Superconducting nanowires fabricated via carbon-nanotube-templating can be
used to realize and study quasi-one-dimensional superconductors. However,
measurement of the linear resistance of these nanowires have been inconclusive
in determining the low-temperature behavior of phase-slip fluctuations, both
quantal and thermal. Thus, we are motivated to study the nonlinear
current-voltage characteristics in current-biased nanowires and the stochastic
dynamics of superconductive-resistive switching, as a way of probing phase-slip
events. In particular, we address the question: Can a single phase-slip event
occurring somewhere along the wire--during which the order-parameter fluctuates
to zero--induce switching, via the local heating it causes? We explore this and
related issues by constructing a stochastic model for the time-evolution of the
temperature in a nanowire whose ends are maintained at a fixed temperature. We
derive the corresponding master equation as tool for evaluating and analyzing
the mean switching time at a given value of current. The model indicates that
although, in general, several phase-slip events are necessary to induce
switching via a thermal runaway, there is indeed a regime of temperatures and
currents in which a single event is sufficient. We carry out a detailed
comparison of the results of the model with experimental measurements of the
distribution of switching currents, and provide an explanation for the
counter-intuitive broadening of the distribution width that is observed upon
lowering the temperature. Moreover, we identify a regime in which the
experiments are probing individual phase-slip events, and thus offer a way for
exploring the physics of nanoscale quantum tunneling of the superconducting
order parameter.
