  Hysteretic I-V (current-voltage) is studied in narrow Al nanowires. The
nanowires have a cross section as small as 50 nm^2. We focus on the retapping
current in a down-sweep of the current, at which a nanowire re-enters the
superconducting state from a normal state. The retrapping current is found to
be significantly smaller than the switching current at which the nanowire
switches into the normal state from a superconducting state during a current
up-sweep. For wires of different lengths, we analyze the heat removal due to
various processes, including electronic and phonon processes. For a short wires
1.5 um in length, electronic thermal conduction is effective; for longer wires
10um in length, phonon conduction becomes important. We demonstrate that the
measured retrapping current as a function of temperature can be quantitatively
accounted for by the selfheating occurring in the normal portions of the
nanowires to better than 20 % accuracy. For the phonon processes, the extracted
thermal conduction parameters support the notion of a reduced phase-space below
3-dimensions, consistent with the phonon thermal wavelength having exceeded the
lateral dimensions at temperatures below ~ 1.3K. Nevertheless, surprisingly the
best fit was achieved with a functional form corresponding to 3-dimensional
phonons, albeit requiring parameters far exceeding known values in the
literature.
