Symmetry-broken crystal structure of elemental boron at low temperature

Abstract

The crystal structure of boron is unique among chemical elements, highly complex, and imperfectly known. Experimentalists report that the $\ensuremath{\beta}$-rhombohedral (black) form is stable over all temperatures from absolute zero to melting. However, early calculations found its energy to be greater than the energy of the $\ensuremath{\alpha}$-rhombohedral (red) form, implying that the $\ensuremath{\beta}$ phase cannot be stable at low temperatures. Furthermore, the $\ensuremath{\beta}$ form exhibits partially occupied sites, seemingly in conflict with the thermodynamic requirement that entropy vanish at low temperature. Using electronic density functional theory methods and an extensive search of the configuration space we find a unique, energy-minimizing pattern of occupied and vacant sites that can be stable at low temperatures but that breaks the $\ensuremath{\beta}$-rhombohedral symmetry. Even lower energies occur within larger unit cells. Alternative configurations lie nearby in energy, allowing the entropy of partial occupancy to stabilize the $\ensuremath{\beta}$-rhombohedral structure through a phase transition at moderate temperature.