Magnetic interactions in a single-band model for the cuprates and ruthenates

Abstract

We consider the mean-field theory of superconductivity in quasi-two-dimensions assuming that a single tight-binding energy band is dominant. The effective interaction between quasiparticles is assumed to be of magnetic origin and built out of the Lindhard function of the single band together with a molecular field parameter, which is determined self-consistently. The tight-binding hopping matrix elements are chosen to be appropriate for the ${d}_{xy}$ band of ${\mathrm{Sr}}_{2}\mathrm{Ru}{\mathrm{O}}_{4}$, which is believed to be responsible for spin-triplet $p$-wave superconductivity below $1.5\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. As the band filling varies, the Lindhard function and hence the magnetic correlations display a surprisingly rich behavior even for our chosen simple tight-binding band. In particular, the Lindhard function displays both high-wave-vector and low-wave-vector peaks, which lead, respectively, to incommensurate antiferromagnetic and incommensurate ferromagnetic correlations. Near half filling the Fermi surface has strong high-wave-vector nesting features that produce dominant antiferromagnetic correlations. When sufficiently enhanced by the molecular field, these correlations lead to robust spin-singlet $d$-wave pairing. As the band filling is increased, the Fermi surface expands and touches the zone boundary at the Van Hove singularity point. In this case, ferromagnetic correlations are dominant and even when enhanced strongly, only lead to surprisingly weak $p$-wave pairing, with a maximum superconducting transition temperature ${T}_{c}$ more than two orders of magnitude lower than for $d$-wave pairing near half filling for the chosen model parameters. This $p$-wave pairing depends on a number of rather subtle aspects of the electronic structure and magnetic interaction and its weakness can be understood as the result of a near cancellation of large effects. Our findings can account in a natural way for the difficulty of finding superconducting analogs of spin-triplet $p$-wave superfluidity in $^{3}\mathrm{He}$ such as ${\mathrm{Sr}}_{2}\mathrm{Ru}{\mathrm{O}}_{4}$. With the help of a symmetry transformation, the simple chosen tight-binding band can also describe systems with less than a half-filled band with Fermi surfaces similar to those of the cuprates. Our results therefore suggest that the dramatically different superconducting properties of the cuprates and of strontium ruthenate may be qualitatively understood, and naturally so, within the same framework.