  In this paper, we propose a novel mechanism in $\mathbb{T}^2$-inflation to
enhance the power spectrum large enough to seed primordial black holes (PBHs)
formation. To accomplish this, we consider the coupling function between the
inflaton field and $\mathbb{T}^2= T_{\mu \nu}T^{\mu \nu}$ term. PBHs formed
within this scenario can contribute partially or entirely to dark matter (DM)
abundance. Furthermore, the amplification in the scalar power spectrum will
concurrently produce significant scalar-induced gravitational waves (SIGWs) as
a second-order effect. In addition, the energy spectrum associated with SIGWs
can be compatible with the recent NANOGrav 15-year stochastic gravitational
wave detection and fall into the sensitivity range of other forthcoming GW
observatories.
