  Dark matter (DM) occupies the majority of matter content in the universe and
is probably cold (CDM). However, modifications to the standard CDM model may be
required by the small-scale observations, and DM may be self-interacting (SIDM)
or warm (WDM). Here we show that the diffractive lensing of gravitational waves
(GWs) from binary black hole mergers by small halos ($\sim10^3-10^6M_\odot$;
mini-halos) may serve as a clean probe to the nature of DM, free from the
contamination of baryonic processes in the DM studies based on dwarf/satellite
galaxies. The expected lensed GW signals and event rates resulting from CDM,
WDM, and SIDM models are significantly different from each other, because of
the differences in halo density profiles and abundances predicted by these
models. We estimate the detection rates of such lensed GW events for a number
of current and future GW detectors, such as the Laser Interferometer
Gravitational Observatories (LIGO), the Einstein Telescope (ET), the Cosmic
Explorer (CE), Gravitational-wave Lunar Observatory for Cosmology (GLOC), the
Deci-Hertz Interferometer Gravitational Wave Observatory (DECIGO), and the Big
Bang Observer (BBO). We find that GLOC may detect one such events per year
assuming the CDM model, DECIGO (BBO) may detect more than several (hundreds of)
such events per year, by assuming the CDM, WDM (with mass $>30$\,keV) or SIDM
model, suggesting that the DM nature may be strongly constrained by DECIGO and
BBO via the detection of diffractive lensed GW events by mini-halos. Other GW
detectors are unlikely to detect a significant number of such events within a
limited observational time period. However, if the inner slope of the mini-halo
density profile is sufficiently steeper than the Navarro-Frenk-White (NFW)
profile, e.g., the pseudo-Jaffe profile, one may be able to detect one to more
than hundred such GW events by ET and CE.
