Elasticity of two-dimensional ferroelectrics across their paraelectric phase transformation

The mechanical behavior of two-dimensional (2D) materials across 2D phase\nchanges is unknown, and the finite temperature ($T$) elasticity of paradigmatic\nSnSe monolayers -- ferroelectric 2D materials turning paraelectric as their\nunit cell (u.c.) turns from a rectangle onto a square -- is described here in a\nprogressive manner. To begin with, their zero$-T$ {\\em elastic energy\nlandscape} gives way to (Boltzmann-like) averages from which the elastic\nbehavior is determined. These estimates are complemented with results from the\nstrain-fluctuation method, which employs the energy landscape or {\\em ab\ninitio} molecular dynamics (MD) data. Both approaches capture the coalescence\nof elastic moduli $\\langle C_{11}(T)\\rangle=\\langle C_{22}(T)\\rangle$ due to\nthe structural transformation. The broad evolution and sudden changes of\nelastic parameters $\\langle C_{11}(T)\\rangle$, $\\langle C_{22}(T)\\rangle$, and\n$\\langle C_{12}(T)\\rangle$ of these atomically-thin phase-change membranes\nestablishes a heretofore overlooked connection among 2D materials and soft\nmatter.\n