Structure of intralayer and interlayer pairing interactions, the anisotropy of order parameters, and the transition temperature in layered superconductors

The structure of the general effective pairing interaction V(${\mathbf{r}}_1$,${\mathbf{r}}_2$;ω) for spin-singlet superconductivity in layered systems is examined in detail in a layer representation for the single-particle states. This interaction, averaged over the layer-plane momentum transfers near the Fermi surface [i.e., ${}_{}{}^{\mathrm{pp}}{}_{}{}^{}$V¯(q,q’;ω)], is expressed in terms of intracell-intralayer, intracell-interlayer, intercell-intralayer, and intercell-interlayer couplings, where q,q’ are wave vectors in the direction of the normal to the layers. In our general picture, because of the interaction process, a localized particle can scatter across from one layer to another in the same unit cell, as well as from one vertical unit cell to another cell. The dominant coupling terms are due to the interaction of two particles localized on layers within the same vertical unit cell, both before and after the scattering. The intercell coupling terms correspond to scattering processes in which at least one of the particles is localized in a different vertical unit cell either before or after the scattering. Whereas the ratio ${\mathrm{\delta }}_{\mathit{c}}$ between the intracell and nearest-neighbor intercell couplings is assumed to be small compared with unity in our analysis, no perturbative approach is used to distinguish the magnitudes of interlayer couplings from the corresponding intralayer couplings.

General results for ${\mathit{T}}_{\mathit{c}}$ and the superconducting-order-parameter matrix Δ, are discussed for the case of N layers per unit cell, both in the limit in which the off-diagonal tunneling terms, t(q), appearing in the inverse of the normal-state single-particle N×N Green-function matrix, ${\mathit{g}}^{\mathrm{-}1}$, in the layer representation, are either neglected or included to the lowest order in ‖t(q)‖/μ. Here, μ is the chemical potential. We find that the q,q’ dependence of ${}_{}{}^{\mathrm{pp}}{}_{}{}^{}$V arises only from intercell couplings which in turn implies that the q dependence and anisotropy in Δ are entirely due to intercell interactions. Explicit results are given for ${\mathit{T}}_{\mathit{c}}$ as well as the order parameters for the case of N=1 and 2, to leading orders in ${\mathrm{\delta }}_{\mathit{c}}$ and ‖t(q)‖/μ. We find that anisotropy in the gap parameter in the layer representation always arises due to the intercell interactions, and is present even in the N=1 case. It is also found that the presence of interlayer couplings tends to enhance ${\mathit{T}}_{\mathit{c}}$ in general. In a representation in which ${\mathit{g}}^{\mathrm{-}1}$ is diagonal, called the α-band representation here, we reformulate the problem in terms of new order parameters, ${\mathrm{\Delta }}^{\left(\mathrm{\alpha }\right)}$ and new effective interaction, ${}_{}{}^{\mathrm{pp}}{}_{}{}^{}$${\mathit{V}}^{\left(\mathrm{\alpha }\right)}$(qq’;ω). Although, in this representation there appear additional q,q’ dependences in ${}_{}{}^{\mathrm{pp}}{}_{}{}^{}$${\mathit{V}}^{\left(\mathrm{\alpha }\right)}$, these do not imply extra anisotropy in the corresponding gap parameter Δ in the layer representation. This is only an artifact of the use of the α representation as revealed by the relation between ${\mathrm{\Delta }}^{\left(\mathrm{\alpha }\right)}$ and Δ. For interband pairing in the α representation, we also present a complete calculation for the resulting order parameter.