  The Josephson current-phase characteristics, that arise when a supercurrent
flows across two fermionic superfluids separated by a potential barrier, can be
controlled by varying either the inter-particle coupling or the temperature.
While the coupling dependence has been addressed in detail both theoretically
and experimentally for an attractive Fermi gas undergoing the BCS-BEC
crossover, a corresponding study of the temperature dependence of the Josephson
characteristics is still lacking in this context. Here, we investigate the
combined coupling and temperature dependence of the Josephson characteristics
in a systematic way for a wide set of barriers, within ranges of height and
width that can be experimentally explored. Our study smoothly connects the two
limiting cases, of non-overlapping composite bosons at low temperature
described by the Gross-Piatevskii equation, and of strongly overlapping Cooper
pairs near the critical temperature described by the Ginzburg-Landau equation.
In this way, we are able to explore several interesting effects related to how
the current-phase characteristics evolve along the BCS-BEC crossover as a
function of temperature and of barrier shape. These effects include the
coherence length outside the barrier and the pair penetration length inside the
barrier (which is related to the proximity effect), as well as the temperature
evolution of the Landau criterion in the limit of a vanishingly small barrier.
A comparison is also presented between the available experimental data for the
critical current and our theoretical results over a wide range of couplings
along the BCS-BEC crossover.
