Dependence of the superconducting transition temperature of organic molecular crystals on intrinsically nonmagnetic disorder: A signature of either unconventional superconductivity or the atypical formation of magnetic moments

We give a theoretical analysis of published experimental studies of the effects of impurities and disorder on the superconducting transition temperature $T_c$ of the organic molecular crystals $\kappa -(\mathrm{BEDT}-\mathrm{TTF}{)}_{2}X$ (where $X=\mathrm{Cu}\left[\mathrm{N}\right(\mathrm{CN}{)}_2]\mathrm{Br}$ and $\mathrm{Cu}(\mathrm{NCS}{)}_2$ and BEDT-TTF is bis(ethylenedithio)tetrathiafulvalene) and $\beta -(\mathrm{BEDT}-\mathrm{TTF}{)}_{2}X$ (for ${X=I}_3$ and ${\mathrm{IBr}}_2).\mathrm{}$ The Abrikosov-Gorkov (AG) formula describes the suppression of $T_c$ both by magnetic impurities in singlet superconductors, including s-wave superconductors and by nonmagnetic impurities in a non-$s$-wave superconductor. We show that various sources of disorder (alloying anions, fast electron irradiation, disorder accidentally produced during fabrication, and cooling rate induced disorder) lead to the suppression of $T_c$ as described by the AG formula. This is confirmed by the excellent fit to the data, the fact that these materials are in the clean limit and the excellent agreement between the value of the interlayer hopping integral $t_{\perp }$ calculated from this fit and the value of $t_{\perp }$ found from angular-dependent magnetoresistance and quantum oscillation experiments. There are only two scenarios consistent with the current state of experimental knowledge. If the disorder induced by all of the four methods considered in this paper is, as seems most likely, nonmagnetic then the pairing state cannot be s wave. We show that published measurements of the cooling rate dependence of the magnetization are inconsistent with paramagnetic impurities. Triplet pairing is ruled out by NMR and upper critical field experiments. Thus if the disorder is nonmagnetic then this implies that $l>~2,$ in which case Occam’s razor suggests that d-wave pairing is realized in both $\beta -(\mathrm{BEDT}-\mathrm{TTF}{)}_{2}X$ and $\kappa -(\mathrm{BEDT}-\mathrm{TTF}{)}_{2}X.\mathrm{}$ However, particularly given the proximity of these materials to an antiferromagnetic Mott transition, it is possible that the disorder leads to the formation of local magnetic moments via some atypical mechanism. Thus we conclude that either $\beta -(\mathrm{BEDT}-\mathrm{TTF}{)}_{2}X$ and $\kappa -(\mathrm{BEDT}-\mathrm{TTF}{)}_{2}X$ are d-wave superconductors or else they display an atypical mechanism for the formation of localized moments, possibly related to the competition between the antiferromagnetic and superconducting grounds states. We suggest systematic experiments to differentiate between these two scenarios.