Intralayer versus interlayer pairing in the copper oxide superconductors: The three- and four-layer problems

We have analyzed the possible superconducting states of a model layered superconductor with N conducting layers in a unit cell, for N=3 and 4, in the presence of both intralayer and interlayer pairing interactions. In case all layers are identical, the system has the same behavior as one-layer and two-layer systems, namely that pure intralayer pairing interaction stabilizes an s-wave state with equal and isotropic gaps in all N quasiparticle bands, while interlayer pairing produces both singlet and triplet paired states with anisotropic gap functions. The singlet state is always energetically more stable than the triplet state. The analysis can be easily extended to nonidentical layers. As one example, a model for ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$ consisting of two ${\mathrm{CuO}}_2$ layers in a unit cell separated by a layer of CuO chains is discussed. It is shown that the system reduces to an effective two-layer model, due to the large difference in electronic energy states between the planes and the chains. The chains are most likely nonsuperconducting. In another example, we show that ${\mathrm{Bi}}_2$ ${\mathrm{Sr}}_2$ ${\mathrm{CaCu}}_2$ ${\mathrm{O}}_8$ can be reduced to an effective three-layer system. Recent tunneling data can be understood on the basis of this model.