Theory of antiferromagnetic superconductors

In this paper a theory is presented of antiferromagnetic superconductors in which a spin-density wave (SDW) ordering with a wave vector $\overrightarrow{\mathrm{Q}}$ may coexist with superconductivity. The effect of the antiferromagnetic molecular field $h_Q\mathrm{}\left(T\right)\mathrm{}$ on the Cooper pairing is studied, and it is shown that, below the magnetic transition temperature $T_N$, the Bardeen-Cooper-Schrieffer coupling parameter is reduced by a factor [$1-\frac{\mathrm{const}\left|h_Q\mathrm{}\right(T\left)\right|}{{\epsilon }_F}$] due to the formation of energy gaps of SDW on the Fermi surface along $\overrightarrow{\mathrm{Q}}$ and this reduction can explain the anomaly in the upper critical field $H_{c2\mathrm{}}$ just below $T_N$ as observed in $R{\mathrm{Mo}}_6{\mathrm{S}}_8$ ($T=\mathrm{G}\mathrm{d},\mathrm{T}\mathrm{b},\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{D}\mathrm{y}$). Taking account of both the spin-orbit scattering and spin-fluctuation effect near $T_N$ in addition to the effect of $h_Q\mathrm{}\left(T\right)\mathrm{}$, a theoretical calculation of the superconducting transition temperature $T_c$ and $H_{c2\mathrm{}}\mathrm{}\left(T\right)\mathrm{}$ is performed. Detailed quantitative comparisons between theory and experiments on $H_{c2\mathrm{}}\mathrm{}\left(T\right)\mathrm{}$ are made with fairly good accord for the above three compounds. Some speculation is devoted to the remaining discrepancies between theory and experiments and certain phenomena not yet explained by the theory.