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A Little Bit of Carbon Can do a Lot for Superconductivity in H$_3$S

First-principles calculations were carried out to provide a chemical basis for proposed structures associated with the recently reported room-temperature superconductivity in a carbonaceous sulfur hydride material under pressure. Calculations were performed on supercells of H$_3$S doped with 1.85-25\% carbon, corresponding to SH$_3$~$\rightarrow$~CH$_3$ or SH$_3$~$\rightarrow$~CH$_4$ substitutions, primarily at pressures of 270 GPa where the maximum critical temperature, $T_c$, has been reported. In the first type of substitution, the carbon atoms can be six-fold coordinated, stabilizing a CH$_6$ configuration within the cubic H$_3$S framework structure that forms under pressure. In the second, the carbon can be four-fold coordinated as methane intercalated into the H-S lattice, with or without an additional hydrogen in the framework. The results indicate that unusual local bonding configurations with respect to carbon can be stabilized under pressure. The doping breaks degenerate bands, lowering the density of states at the Fermi level ($N_F$), and localizing electrons in C-H bonds. Low levels of CH$_4$ doping do not increase $N_F$ to values as high as those calculated for $Im\bar{3}m$ H$_3$S, but they do result in a larger logarithmic average phonon frequency, and an electron-phonon coupling parameter comparable to that of $R3m$ H$_3$S. The $T_c$s estimated for carbon doping levels ranging from 1.85-5.7\% are compatible with experimental measurements for the C-S-H superconductor.