  The 2H, 1T, and their distorted structures are known as prototype structures
of $AB_2$ monolayers. Here, we study a puckered structure that is truncated
from the (110) surface of fluorite-type materials. 53 fluorite-type materials
are investigated based on first-principles approach. The formation energy
calculations indicate that seven systems form the puckered structure in the
monolayer limit, while other systems form either 1T, 2H, or distorted 1T
structures. The puckered structures of PbF$_2$, PRh$_2$, and Ga$_2$Au exhibit
negative Poisson's ratio (NPR) in the out-of-plane direction. An analytical
model for the NPR is derived. The surface energy calculations predict the
appearance of NPR.
