Two-Dimensional Disorder in Black Phosphorus and Monochalcogenide Monolayers

Ridged, orthorhombic two-dimensional
atomic crystals with a bulk 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 4-fold degenerate structural
ground state, and a single energy scale EC (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 Tc that is proportional to EC/kB. EC is tunable by chemical composition and it leads to a classification
of these materials into two categories: (i) Those for which EC ≥ kBTm, and (ii) those having kBTm > EC ≥ 0, where Tm 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 EC > 0 belonging to
class (ii) will undergo a two-dimensional transition prior to melting. EC/kB 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 Cmcm phase
being discussed within the context of bulk layered SnSe.