Toward a thermodynamic stability order of the phosphorus allotropes

In this work, we investigate through quantum–mechanical calculations the relative stability of white-γ, white-β, fibrous red, violet, and orthorhombic black phosphorus allotropes, a longstanding yet challenging problem. We performed DFT-D3 calculations with the CRYSTAL code as well as periodic local second-order Møller–Plesset perturbation theory (p-LMP2) calculations. DFT and Spin-Component Scaled p-LMP2 place violet phosphorus as the thermodynamically most stable allotrope both at 0 K and at 298 K, yet within a tiny margin from the black phosphorus. Pure p-LMP2 suggests that black phosphorus is the most stable, although its accuracy may be affected by the narrow band gap in this material.