Imperfect Crystal and Unusual Semiconductor: Boron, a Frustrated Element

All elements, except for helium, appear to solidify into crystalline forms at zero temperature, and it is generally assumed that the introduction of lattice defects results in an increase in internal energy. beta-Rhombohedral boron, a thermodynamically stable form of elemental boron at high temperature, is known to have a large amount of partial occupied sites, seemingly in conflict with our common knowledge. By using lattice Monte Carlo techniques combined with ab initio calculations, we find that the beta-phase is stabilized by a macroscopic amount of intrinsic defects that are responsible not only for entropic effects but also for a reduction in internal energy. These defects enable the conversion of two-center to three-center bonds and are accompanied by the presence of localized, nonconductive electronic states in the optical gap. In addition we find that the ab initio Ising model describing the partial occupancy of beta-boron has macroscopic residual entropy, suggesting that boron is a frustrated system analogous to ice and spin ice.