What Determines the Structures of the Group 15 Elements?

The structures of black phosphorus (orthorhombic), As, Sb, and Bi (rhombohedral) have been traditionally interpreted as resulting from a Peierls distortion from an ideal simple cubic structure. We examine this idea in detail by calculating the Fermi surfaces of simple cubic phosphorus with the extended Hückel tight-binding method and by looking at some simpler models to understand the trends. The calculated Fermi surfaces for cubic P are not nested, which argues strongly that the structural distortion in black phosphorus has little to do with Peierls instability. Within the (simple) Hückel approximation for p-orbital interactions, the Fermi surface nesting is still perfect, even when interchain (π-type) interactions are included. Next-nearest-neighbor interaction and s-p mixing completely destroy the Fermi surface nesting. The observed strong s-p mixing implies that it is just this mixing that causes the deformation to the highly stable black phosphorus structure, forming lone pairs in the process. We suggest that in order to understand the trend in the magnitude of distortions in the structures of the group 15 elements (P>As>Sb>Bi) s-p mixing should not be neglected even in the heavier elements.