Intralayer-versus-interlayer pairing in the copper oxide superconductors

We have investigated the competing roles of intralayer s-wave and interlayer BCS-like pairing in determining the c-axis versus ab-plane energy-gap anisotropy in layered superconductors with N=1,2 conducting planes per unit-cell edge s. For N=1, intralayer s-wave pairing leads to a conventional order parameter (OP) Δ₀ with a transition temperature (Tc) value Tc0 and isotropic energy gap 2{}Δ₀{}. For interlayer pairing, the four-vector gap function {Δ}̃ \~₃(kz) contains a singlet OP Δₛ and a vector triplet OP Δₜ, with corresponding normalized gap functions {}2 coskzs and {}2 sinkzs. Since Δₛ and Δₜ have identical Tc values (Tcs=Tct), the free energy is minimized when {}Δₛ{}={}Δₜ{}, and the resulting gap 2{}{Δ}{} is found to be isotropic. However, 2{}{Δ}{} is completely incompatible with 2{}Δ₀{}. For N=2, there are two intralayer pairing OP's Δ₀ and Δ₁, and two interlayer four-vector gap functions {Δ}̃ \~₃(kz) and {Δ}̃ \~₂(kz), all of which could contribute to measurable quantities in the Gaussian-fluctuation regime above the highest Tc value.However, interband pair breaking causes Δ₁ and {Δ}̃ \~₂ to vanish below the maximum Tc value. Of the two singlet OP's in {Δ}̃ \~₃(kz), the OP Δₛ has the higher Tc value, Tcs, and the second singlet OP can be neglected. The triplet OP Δₜ has the Tc value Tct{≤}Tcs. For Tc0>Tcs, the gap is just 2{}Δ₀{}. For Tc0{T}ct$, ${{{Δ}}}₀=0 and the gap 2Δ({k}z$,T){} is anisotropic. There are two regions of the parameters with different gap-anisotropy behavior. In region I, the gap is always nodeless, and is nearly pure singlet for weak coupling. In region II, the gap has a pair of nodes near to Tcs, but is nodeless at low T, as {}Δₜ{}{≠}0. For Tcs>Tc0>Tct, the gap 2{}{Δ}(kz,T){} is anisotropic, nodeless at low T, and could on rare occasions involve Δ₀{≠}0, in addition to Δₛ{≠}0. In any event, the gap is the same on both of the quasiparticle bands. Our results are discussed in terms of recent far-infrared-reflectance, Raman-scattering, and point-contact-tunneling experiments. Additional experiments to clarify the c-axis versus ab-plane gap anisotropy and the microscopic pairing mechanism are suggested.