Superconducting transition temperatures and coherence length in non-$s$-wave pairing materials correlated with spin-fluctuation mediated interaction

Following earlier work on electron or hole liquids flowing through assemblies with magnetic fluctuations, we have recently exposed a marked correlation of the superconducting temperature $T_c,$ for non-$s$-wave pairing materials, with coherence length $\xi$ and effective mass $m^{*}.$ The very recent study of Abanov et al. [Europhys. Lett. 54, 488 (2001)] and the prior investigation of Monthoux and Lonzarich [Phys. Rev. B 59, 14 598 (1999)] have each focused on the concept of a spin-fluctuation temperature $T_{\mathrm{sf}},$ which again is intimately related to $T_c.$ For the d-wave pairing via antiferromagnetic spin fluctuations in the cuprates, these studies are brought into close contact with our own work, and the result is that $k_{\mathrm{B}}{T}_{\mathrm{sf}}\sim {ħ}^2{/m}^{*}{\xi }^2.$ This demonstrates that $\xi$ is also determined by such antiferromagnetic spin-fluctuation mediated pair interaction. The coherence length in units of the lattice spacing is then essentially given in the cuprates as the square root of the ratio of two characteristic energies, namely, the kinetic energy of localization of a charge carrier of mass $m^{*}$ in a specified magnetic correlation length to the hopping energy. The quasi-two-dimensional ruthenate ${\mathrm{Sr}}_2{\mathrm{RuO}}_4,$ with $T_c\sim 1.3\hspace{0.5em}\mathrm{K},$ has p-wave spin-triplet pairing and so is also briefly discussed here.