Traditionally, all superhard carbon phases including diamond are electric insulators and all conductive carbon phases including graphite are mechanically soft. Based on first-principles calculation results, we report a superhard but conductive carbon phase C21-sc which can be obtained through increasing the sp3 bonds in the previously proposed soft and conductive phase C20-sc (Phys. Rev. B 74, 172101 2006). We also show that further increase of sp3 bonds in C21-sc results in a superhard and insulating phase C22-sc with sp3 bonds only. With C20-sc, C21-sc, C22-sc and graphite, the X-ray diffraction peaks from the unidentified carbon material synthesized by compressing the mixture of tetracyanoethylene and carbon black (Carbon, 41, 1309, 2003) can be understood. In view of its positive stability, superhard and conductive features, and the strong possibility of existence in previous experiments, C21-sc is a promising multi-functional material with potential applications in extreme conditions.
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