  We investigate the ultrafast carrier and phonon dynamics in the ferrimagnetic
semiconductor Mn$_3$Si$_2$Te$_6$ using time-resolved optical pump-probe
spectroscopy. Our results reveal that the electron-phonon thermalization
process with a subpicosecond timescale is prolonged by the hot-phonon
bottleneck effect. We identify the subsequent relaxation processes associated
with two non-radiative recombination mechanisms, i.e., phonon-assisted
electron-hole recombination and defect-related Shockley-Read-Hall
recombination. Temperature-dependent measurements indicate that all three
relaxation components show large variation around 175 and 78 K, which is
related to the initiation of spin fluctuation and ferrimagnetic order in
Mn$_3$Si$_2$Te$_6$. In addition, two pronounced coherent optical phonons are
observed, in which the phonon with a frequency of 3.7 THz is attributed to the
$A_{1g}$ mode of Te precipitates. Applying the strain pulse propagation model
to the coherent acoustic phonons yields a penetration depth of 506 nm and a
sound speed of 2.42 km/s in Mn$_3$Si$_2$Te$_6$. Our results develop
understanding of the nonequilibrium properties of the ferrimagnetic
semiconductor Mn$_3$Si$_2$Te$_6$, and also shed light on its potential
applications in optoelectronic and spintronic devices.
