Metallic boron allotropes

Bulk boron phases are typically insulating due to sufficiently localized valence electrons, so that stable metallic boron phases at ambient conditions are rare. This situation is further exacerbated by the limited understanding of mechanism governing electronic properties of boron phases. Here, we report a series of stable metallic and insulating boron phases at ambient pressure, identified by an extensive structural search combining graph theory, group theory-based random strategies, and global optimization algorithms. Among these, an R3c-symmetry phase (${\mathrm{B}}_{\text{R}3\mathrm{c}}$), composed of six ${\mathrm{B}}_{9}$ structural units, exhibits exceptional thermal stability up to 1200 K, a Vickers hardness exceeding 40 GPa, and higher thermodynamic stability than the experimentally known \ensuremath{\epsilon}-B phase by 0.11 eV/atom. First-principles analysis reveals that the ${\mathrm{B}}_{\text{R}3\mathrm{c}}$ metallicity stems from delocalized electronic states by uniform B-B bond lengths and high-coordinate number. Motivated by this, we develop a physically interpretable one-dimensional descriptor based on the bond topology and electron localization parameters, which can correctly predict 95.1% of all reported metallic or insulating boron phases. These results constitute a predictive framework for designing boron and boron-based functional materials.