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Two-dimensional disorder in black phosphorus and monochalcogenide monolayers

Ridged, orthorhombic two-dimensional atomic crystals with a bulk {\em Pnma} structure such as black phosphorus and monochalcogenide monolayers are an exciting and novel material platform for a host of applications. Key to their crystallinity, monolayers of these materials have a four-fold degenerate structural ground state, and a single energy scale $E_C$ (representing the elastic energy required to switch the longer lattice vector along the $x-$ or $y-$direction) determines how disordered these monolayers are at finite temperature. Disorder arises when nearest neighboring atoms become gently reassigned as the system is thermally excited beyond a critical temperature $T_c$ that is proportional to $E_C/k_B$. $E_C$ is tunable by chemical composition and it leads to a classification of these materials into two categories: (i) Those for which $E_C\ge k_BT_m$, and (ii) those having $k_BT_m>E_C\ge 0$, where $T_m$ is a given material's melting temperature. Black phosphorus and SiS monolayers belong to category (i): these materials do not display an intermediate order-disorder transition and melt directly. All other monochalcogenide monolayers with $E_C>0$ belonging to class (ii) will undergo a two-dimensional transition prior to melting. $E_C/k_B$ is slightly larger than room temperature for GeS and GeSe, and smaller than 300 K for SnS and SnSe monolayers, so that these materials transition near room temperature. The onset of this generic atomistic phenomena is captured by a planar Potts model up to the order-disorder transition. The order-disorder phase transition in two dimensions described here is at the origin of the {\em Cmcm} phase being discussed within the context of bulk layered SnSe.