Superconductivity on a Y${\mathrm{Ba}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_7$ lattice

The interaction between two electrons near a conducting and polarizable plane is found to be attractive at large separations if the polarizability is large enough. Assuming such an interaction, a variational Bardeen-Cooper-Schrieffer superconducting state is constructed based upon the Y${\mathrm{Ba}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_7$ tight-binding electronic structure. There is no charge-density-wave instability, nor $s$-wave superconducting instability. However, allowing the phase of the order parameter to vary along the Fermi lines separates the paired electrons sufficiently to sample only the attractive electron-electron interaction. As the polarizability is increased from zero over a reasonable range, 90-K superconductivity is first obtained in the Cu${\mathrm{O}}_2$ planes. The corresponding Cooper-pair electrons are related by a reflection symmetry of the Brillouin zone rather than the usual $\mathbf{k}$ to $-\mathbf{k}$ inversion. The paired electrons move along third-neighbor copper rows. This particular $l=3\mathrm{}$ state would have triplet pairing but neighboring $l=2\mathrm{}$ and $l=4\mathrm{}$ superconducting states have singlet pairing. Superconductivity, with a smaller gap, simultaneously arises on the Cu${\mathrm{O}}_3$ chains with Cooper-pair electrons on second-neighbor chains. The ratio $\frac{2{\Delta }_0}{k_B{T}_c}$ is found to be 5.08 for the planes. Predicted values for a range of superconducting properties of the 90-K state are in reasonable accord with experiment.