Elasticity of two-dimensional ferroelectrics across their paraelectric phase transformation

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