  The next generation of space-borne gravitational wave detectors may detect
gravitational waves from extreme mass-ratio inspirals with primordial black
holes. To produce primordial black holes which contribute a non-negligible
abundance of dark matter and are consistent with the observations, a large
enhancement in the primordial curvature power spectrum is needed. For a single
field slow-roll inflation, the enhancement requires a very flat potential for
the inflaton, and this will increase the number of $e$-folds. To avoid the
problem, an ultra-slow-roll inflation at the near inflection point is required.
We elaborate the conditions to successfully produce primordial black hole dark
matter from single field inflation and propose a toy model with polynomial
potential to realize the big enhancement of the curvature power spectrum at
small scales while maintaining the consistency with the observations at large
scales. The power spectrum for the second order gravitational waves generated
by the large density perturbations at small scales is consistent with the
current pulsar timing array observations.
