  Primordial black holes (PBHs) are dark matter candidates that span broad mass
ranges from $10^{-17}$ $M_\odot$ to $\sim 100$ $M_\odot$. We show that the
stochastic gravitational wave background can be a powerful window for the
detection of sub-solar mass PBHs and shed light on their formation channel via
third-generation gravitational wave detectors such as Cosmic Explorer and the
Einstein Telescope. By using the mass distribution of the compact objects and
the redshift evolution of the merger rates, we can distinguish astrophysical
sources from PBHs and will be able to constrain the fraction of sub-solar mass
PBHs $\leq 1$ $M_\odot$ in the form of dark matter $f_{PBH}\leq 1\%$ at $68\%$
C.L. even for a pessimistic value of a binary suppression factor. In the
absence of any suppression of the merger rate, constraints on $f_{PBH}$ will be
less than $0.001\%$. Furthermore, we will be able to measure the redshift
evolution of the PBH merger rate with about $1\%$ accuracy, making it possible
to uniquely distinguish between the Poisson and clustered PBH scenarios.
