Phosphorus nanoclusters and insight into the formation of phosphorus allotropes

Elemental phosphorus has a striking variety of allotropes, which we analyze by looking at stable phosphorus clusters. We determine the ground-state structures of P clusters in a wide range of compositions (n = 2-50) using density functional calculations and global optimization techniques. We explain why the high-energy white phosphorus is so easily formed, compared to the much more stable allotropes - the tetrahedral P₄ cluster is so much more stable than nearby compositions that only by increasing the size to P₁₀ one can get a more stable non-P₄-based structure. Starting from 17 atoms, phosphorus clusters have a single-stranded structure, consisting of a set of well-resolved structural units connected by P₂ linking fragments. The investigation of relative stability has revealed even-odd alternations and structural magic numbers. The former are caused by the higher stability of clusters with even numbers of atoms due to closed electronic shells. The structural magic numbers are associated with the presence of particular stable structural units and lead to enhanced stability of P₁₈₊₁₂ (k = 0, 1, 2) clusters. We also compare the energies of the obtained ground-state structures with clusters of different phosphorus allotropes. Clusters of fibrous phosphorus are energetically the closest to the ground states, white phosphorus clusters are found to be less stable, and the least stable allotrope at the nanocluster scale is black phosphorene.