Two-dimensional disorder in black phosphorus and monochalcogenide monolayers

Ridged, orthorhombic two-dimensional atomic crystals 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 long 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 $T_c\simeq 1.1 E_C/k_B$; an atomistic phenomena that is captured by a planar Potts model. $E_C$ is tunable by chemical composition, and while $T_c>5,000$ K for a freestanding black phosphorus monolayer --thus implying its structural order at room temperature-- $T_c$ is slightly larger than room temperature for GeS and GeSe, and smaller than 300 K for SnS and SnSe monolayers: These materials display a phase transition in two dimensions.