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) ${\mathrm{\Delta }}_0$ with a transition temperature (${\mathit{T}}_{\mathit{c}}$) value ${\mathit{T}}_{\mathit{c}0}$ and isotropic energy gap 2‖${\mathrm{\Delta }}_0$‖. For interlayer pairing, the four-vector gap function Δ${\mathrm{̃}}_3$(${\mathit{k}}_{\mathit{z}}$) contains a singlet OP ${\mathrm{\Delta }}_{\mathit{s}}$ and a vector triplet OP ${\mathrm{\Delta }}_{\mathit{t}}$, with corresponding normalized gap functions √2 cos${\mathit{k}}_{\mathit{z}}$s and √2 sin${\mathit{k}}_{\mathit{z}}$s. Since ${\mathrm{\Delta }}_{\mathit{s}}$ and ${\mathrm{\Delta }}_{\mathit{t}}$ have identical ${\mathit{T}}_{\mathit{c}}$ values (${\mathit{T}}_{\mathit{c}\mathit{s}}$=${\mathit{T}}_{\mathit{c}\mathit{t}}$), the free energy is minimized when ‖${\mathrm{\Delta }}_{\mathit{s}}$‖=‖${\mathrm{\Delta }}_{\mathit{t}}$‖, and the resulting gap 2‖Δ‖ is found to be isotropic. However, 2‖Δ‖ is completely incompatible with 2‖${\mathrm{\Delta }}_0$‖. For N=2, there are two intralayer pairing OP’s ${\mathrm{\Delta }}_0$ and ${\mathrm{\Delta }}_1$, and two interlayer four-vector gap functions Δ${\mathrm{̃}}_3$(${\mathit{k}}_{\mathit{z}}$) and Δ${\mathrm{̃}}_2$(${\mathit{k}}_{\mathit{z}}$), all of which could contribute to measurable quantities in the Gaussian-fluctuation regime above the highest ${\mathit{T}}_{\mathit{c}}$ value.