  We performed density functional theory calculations with self-consistent van
der Waals corrected exchange-correlation (XC) functionals to capture the
structure of black phosphorus and twelve monochalcogenide monolayers and find
the following results: (a) The in-plane unit cell changes its area in going
from the bulk to a monolayer. The change of in-plane distances implies that
bonds weaker than covalent or ionic ones are at work within the monolayers
themselves. This observation is relevant for the prediction of the critical
temperature $T_c$. (b) There is a hierarchy of independent parameters that
uniquely define a ground state ferroelectric unit cell (and square and
rectangular paraelectric unit cells as well): only 5 optimizable parameters are
needed to establish the unit cell vectors and the four basis vectors of the
ferroelectric ground state unit cell, while square and rectangular paraelectric
structures are defined by only 3 or 2 independent optimizable variables,
respectively. (c) The reduced number of independent structural variables
correlates with larger elastic energy barriers on a rectangular paraelectric
unit cell when compared to the elastic energy barrier of a square paraelectric
structure. This implies that $T_c$ obtained on a structure that keeps the
lattice parameters fixed (for example, using an NVT ensemble) should be larger
than the transition temperature on a structure that is allowed to change
in-plane lattice vectors (for example, using the NPT ensemble). (d) The
dissociation energy (bulk cleavage energy) of these materials is similar to the
energy required to exfoliate graphite and MoS$_2$. (e) There exists a linear
relation among the square paraelectric unit cell lattice parameter and the
lattice parameters of the rectangular ferroelectric ground state unit cell.
These results highlight the subtle atomistic structure of these novel 2D
ferroelectrics.
