Low-energy tetrahedral networks for carbon and silicon from (2+1)-regular bipartite-like graphs.

Post-graphite in cold-compression [Mao et al. (2003). Science 302, 425-427] has attracted widespread research interest in condensed matter physics. Subsequently, many low-energy carbon phases, such as M-carbon, W-carbon and Z-carbon, have been proposed as structural candidates. Based on graph theory, we found that these 4-coordinated post-graphite candidates can not only be decomposed into 3-regular graphitic graphs but also yield (2+1)-regular graphs in a non-graphitic manner from different decomposition directions. This inspires a general idea of generating 3D 4-coordinate networks based on 2D (2+1)-regular bipartite-like graphs, which can be efficiently generated by a random method combined with group and graph theory (RG2). Associated with such a general graph-based method, a large number of 4-coordinate networks have been discovered and investigated by first-principles calculations as potential carbon/silicon phases. Most are confirmed as low-energy carbon/silicon phases and identified as direct or quasi-direct band gap semiconductors. Two complex configurations, Pbam48 and Pbam40, show energetic stabilities exceeding or comparable to the previously proposed Pbam24. They are further confirmed to be dynamically and mechanically stable phases as carbon/silicon. As carbon phases, Pbam48 and Pbam40 are superhard insulators with quasi-direct band gaps of 5.622 and 5.890 eV, and hardness values of 85.352 and 85.558 GPa, respectively. Their X-ray diffraction (XRD) results can largely explain the experimental XRD patterns of cold-compressed graphite. As silicon allotropes, Pbam48 and Pbam40 have quasi-direct band gaps of 1.386 and 1.451 eV, respectively, making them potential solar cell absorber materials.