Tuning energy barriers by doping 2D group-IV monochalcogenides

Structural degeneracies underpin the ferroic behavior of next-generation two-dimensional materials, and lead to peculiar two-dimensional structural transformations under external fields, charge doping and/or temperature. The most direct indicator of the ease of these transformations is an {\em elastic energy barrier}, defined as the energy difference between the (degenerate) structural ground state unit cell, and a unit cell with an increased structural symmetry. Proximity of a two-dimensional material to a bulk substrate can affect the magnitude of the critical fields and/or temperature at which these transformations occur, with the first effect being a relative charge transfer, which could trigger a structural quantum phase transition. With this physical picture in mind, we report the effect of modest charge doping (within $-0.2$ and $+0.2$ electrons per unit cell) on the elastic energy barrier of ferroelastic black phosphorene and nine ferroelectric monochalcogenide monolayers. The elastic energy barrier $J_s$ is the energy needed to create a $Pnm2_1\to P4/nmm$ two-dimensional structural transformation. Similar to the effect on the elastic energy barrier of ferroelastic SnO monolayers, group-IV monochalcogenide monolayers show a tunable elastic energy barrier for similar amounts of doping: a decrease (increase) of $J_s$ can be engineered under a modest hole (electron) doping of no more than one tenth of an electron or a hole per atom.