Hard and superconducting cubic boron phase via swarm-intelligence structural prediction driven by a machine-learning potential

Boron is an intriguing element due to its electron deficiency and the ability to form multicenter bonds in allotropes and borides, exhibiting diversified structures, unique chemical bonds, and interesting properties. Using swarm-intelligence structural prediction driven by a machine learning potential, we identified a boron phase with a 24-atom cubic unit cell, called $c\text{\ensuremath{-}}{\mathrm{B}}_{24}$, consisting of a ${\mathrm{B}}_{6}$ octahedron in addition to well-known ${\mathrm{B}}_{2}$ pairs and ${\mathrm{B}}_{12}$ icosahedra at ambient pressure. There appear unusual four-center-two-electron (4c-2e) bonds in the ${\mathrm{B}}_{12}$ icosahedron, originating from the peculiar bonding pattern between the ${\mathrm{B}}_{2}$ pair and ${\mathrm{B}}_{12}$ icosahedron, which is in sharp contrast with the 3c-2e and 2c-2e bonds in $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{B}}_{12}$. More interestingly, $c\text{\ensuremath{-}}{\mathrm{B}}_{24}$ is a metal with a superconducting critical temperature of 13.8 K at ambient pressure. The predicted Vickers hardness (23.1 GPa) indicates that $c\text{\ensuremath{-}}{\mathrm{B}}_{24}$ is a potential hard material. Notably, it also has a good shear/tensile resistance (48.9/29.3 GPa). Our work not only enriches the understanding of the chemical properties of boron, but also sparks efforts on trying to synthesize this particular compound, $c\text{\ensuremath{-}}{\mathrm{B}}_{24}$.