  In this work, we propose a new subclass of extreme-mass-ratio-inspirals
(EMRIs): mass-gap EMRIs, consisting of a compact object in the lower mass gap
$\sim (2.5-5) M_\odot$ and a massive black hole (MBH). The mass-gap object
(MGO) may be a primordial black hole or produced from a delayed supernova
explosion. We calculate the formation rate of mass-gap EMRIs in both the (dry)
loss-cone channel and the (wet) active galactic nucleus disk channel by solving
Fokker-Planck-type equations for the phase-space distribution. In the dry
channel, the mass-gap EMRI rate is strongly suppressed compared to the EMRI
rate of stellar-mass black holes (sBHs) as a result of mass segregation effect.
In the wet channel, the suppression is roughly equal to the mass ratio of sBHs
over MGOs, because the migration speed of a compact object in an active
galactic nucleus disk is proportional to its mass. We find that the wet channel
is much more promising to produce mass-gap EMRIs observable by spaceborne
gravitation wave detectors. (Non-)detection of mass-gap EMRIs may be used to
distinguish different supernova explosion mechanisms and constrainthe abundance
of primordial black holes around MBHs.
