  The monolithic integration of extended short-wave infrared (e-SWIR)
photodetectors (PDs) on silicon is highly sought-after to implement
manufacturable, cost-effective sensing and imaging technologies. With this
perspective, GeSn PIN PDs have been the subject of extensive investigations
because of their bandgap tunability and silicon compatibility. However, due to
growth defects, these PDs suffer a relatively high dark current density as
compared to commercial III-V PDs. Herein, we elucidate the mechanisms governing
the dark current in $2.6 \, \mu$m GeSn PDs at a Sn content of $10$ at.%. It was
found that in the temperature range of $293 \, $K -- $363 \,$K and at low bias,
the diffusion and Shockley-Read-Hall (SRH) leakage mechanisms dominate the dark
current in small diameter ($20 \, \mu$m) devices, while combined SRH and trap
assisted tunneling (TAT) leakage mechanisms are prominent in larger diameter
($160 \, \mu$m) devices. However, at high reverse bias, TAT leakage mechanism
becomes dominant regardless of the operating temperature and device size. The
effective non-radiative carrier lifetime in these devices was found to reach
$\sim 300$ -- $400$ ps at low bias. Owing to TAT leakage current, however, this
lifetime reduces progressively as the bias increases.
