Systematic search for low-enthalpysp3carbon allotropes using evolutionary metadynamics

We present a systematic search for low-energy metastable superhard carbon allotropes by using the recently developed evolutionary metadynamics technique. It is known that cold compression of graphite produces an allotrope at 15--20 GPa. Here we look for all low-enthalpy structures accessible from graphite. Starting from 2H- or 3R-graphite and applying a pressure of 20 GPa, a large variety of intermediate $s{p}^{3}$ carbon allotropes were observed in evolutionary metadynamics simulation. Our calculation not only found all the previous proposed candidates for ``superhard graphite,'' but also predicted two allotropes ($X$-carbon and $Y$-carbon) showing unusual types of 5+7 and 4+8 topologies. These superhard carbon allotropes can be classified into five families based on 6 (diamond/lonsdaleite), 5+7 ($M$- and $W$-carbon), 5+7 ($X$-carbon), 4+8 (bct-C${}_{4}$), and 4+8 ($Y$-carbon) topologies. This study shows that evolutionary metadynamics is a powerful approach both to find the global minima and systematically search for low-energy metastable phases reachable from given starting materials.