Impurity and spin-fluctuation effects in antiferromagnetic superconductors

We derive the superconducting density of states $N_s\left(\omega \right)$ of an antiferromagnetic superconductor in the presence of impurities and spin fluctuations. We use a self-consistent ansatz for the Green's function $G$ and our previous mean-field-theory results. $N_s\left(\omega \right)$ depends on four functions of frequency: the renormalized frequency $\tilde{\omega }$, gap $\tilde{\Delta }$, molecular field ${\tilde{H}}_Q$, and "pseudogap" $\tilde{\Omega }$, Equations for these four parameters are set up in general and solved for the case of elastically scattering impurities and spin fluctuations. In the absence of any disorder the density of states has structure at the gap frequency $\Delta$ and at $\omega =|\Delta \pm H_Q|$, where $H_Q$ is the (screened) molecular field. Most of our numerical work for dirty superconductors is based on a "quasi-three-dimensional" approximation which appears to be reasonable, although it underestimates gaplessness and overestimates the strength of the subsidiary peaks at $|\Delta \pm H_Q|$. We find in this approximation that magnetic impurities behave as might be expected but that nonmagnetic impurities can make an antiferromagnetic (AF) superconductor gapless at moderate concentrations. As the concentration is increased still further, the nonmagnetic impurities "screen" out the molecular field, a gap reopens, and $N_s\left(\omega \right)$ obtains its BCS value. In the process of solving for $N_s\left(\omega \right)$, we have simultaneously solved the self-consistent equations for $\Delta$ (as well as $H_Q$) as a function of temperature $T$ for dirty superconductors. We find that both $\Delta$ and the thermodynamic critical field $H_c$ exhibit structure at the magnetic ordering temperature $T_M$. As in the case of our previous mean-field-theory results, the behavior of $H_c$ is similar to that of $H_{c2\mathrm{}}$, observed in the ternary compounds. When spin-fluctuation effects are included, $N_s\left(\omega \right)$ and its derivative are found to exhibit structure at the spin-wave frequencies. In analogy with phonons in strong-coupling superconductors, this observation suggests that a measurement of $N_s\left(\omega \right)$ in AF superconductors can provide detailed information about the dynamic structure factor for the localized spins.