We study a model for a pair of copper-oxide planes, described by the t-J model with a small interplane exchange term ${\mathit{J}}_{\mathrm{\perp }}$${\mathbf{S}}_{\mathit{i}}^{\mathrm{}\left(1\right)\mathrm{}}$⋅${\mathbf{S}}_{\mathit{i}}^{\mathrm{}\left(2\right)\mathrm{}}$. We show that this interplane interaction can be responsible for some of the unusual properties of multilayer high-${\mathit{T}}_{\mathit{c}}$ cuprates. An important role is played by antiferromagnetic spin correlations, causing strong peaks in the susceptibility ${\mathrm{\chi }}^{\mathrm{RPA}}$(q,${\mathit{q}}_{\mathit{z}}$) at the incommensurate nesting vectors ${\mathbf{Q}}_{\mathrm{AF}}$≃(π,π±2x). Consequently the effective coupling constants ${\mathit{J}}_{\mathrm{\parallel }}^{\mathrm{eff}}$(q) and ${\mathit{J}}_{\mathrm{\perp }}^{\mathrm{eff}}$(q) are also peaked at q=${\mathbf{Q}}_{\mathrm{AF}}$. We show that this leads to strongly enhanced interplane pairing between fermions on adjacent ${\mathrm{CuO}}_2$ planes, characterized by a pairing order parameter ${\mathrm{\Delta }}_{\mathrm{\perp }}$(${\mathbf{r}}_{\mathit{i}\mathit{j}}$)=〈${\mathit{f}}_{\mathit{i}\mathrm{\uparrow }}^{\mathrm{}\left(1\right)\mathrm{}}$${\mathit{f}}_{\mathit{j}\mathrm{\downarrow }}^{\mathrm{}\left(2\right)\mathrm{}}$-${\mathit{f}}_{\mathit{i}}$ ${}_{\mathrm{\downarrow }}{}^{\mathrm{}\left(1\right)\mathrm{}}{}_{}{}^{}$${\mathit{f}}_{\mathit{j}\mathrm{\uparrow }}^{\mathrm{}\left(2\right)\mathrm{}}$〉 which extends over several lattice spacings. We find that ${\mathrm{\Delta }}_{\mathrm{\perp }}$(k) has an (extended) s-wave symmetry and is peaked at the corners of the Fermi surface. We give qualitative arguments why the gauge field, which at low doping is very effective in destroying the in-plane pairing ${\mathrm{\Delta }}_{\mathrm{\parallel }}$, is less effective in destroying the interplane pairing.

This leads to the conclusion that ${\mathrm{\Delta }}_{\mathrm{\perp }}$(k) can be responsible for the observed spin-gap phase in bilayer cuprates. We use this model to calculate the NMR-relaxation rate, the echo-decay rate, and the Knight shift. Our numerical results are in qualitative agreement with the experimental data on ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6.6}$. We also suggest that the superconducting state in bilayer materials is a combination of d-wave pairing in the plane and s-wave pairing between planes. In heavily underdoped materials where the spin gap is large, the interplane pairing may dominate and a nodeless gap structure is predicted.

• Received 16 December 1993

DOI:https://doi.org/10.1103/PhysRevB.50.438