Intralayer and interlayer spin-singlet pairing and energy gap functions with different possible symmetries in high-${\mathrm{T}}_{\mathrm{c}}$ layered superconductors

Anisotropy and the wave-vector dependence of the energy gap function determine many important properties of a superconductor. Starting from first principles, we present here a complete analysis of possible symmetries of the superconducting gap function ${\mathrm{E}}_{\mathrm{g}}$(k) at the Fermi surface in high-${\mathrm{T}}_{\mathrm{c}}$ layered superconductors with either a simple orthorhombic or a tetragonal unit cell. This is done within the framework of Gorkov's mean-field theory of superconductivity in the so-called 'layer representation' introduced by us earlier. For N conducting cuprate layers, J=1,2,…,N, in each unit cell, the spin-singlet order parameters ${\mathrm{\Delta }}_{\mathrm{JJ}\prime }$(k) can be expanded in terms of possible basis functions of all the irreducible representations relevant to layered crystals, which are obtained here. In layered materials, the symmetry is restricted to the translational lattice periodicity in the direction perpendicular to the layers and the residual point group and translational symmetries for the two-dimensional unit cell in each layer of the three-dimensional unit cell. We derive an exact general relation to determine different branches of the energy gap function ${\mathrm{E}}_{\mathrm{g}}$(k) at the Fermi surface in terms of ${\mathrm{\Delta }}_{\mathrm{JJ}\prime }$(k), which include both intralayer and interlayer order parameters. For N=2, we also obtain an exact expression for quasiparticle energies ${\mathrm{E}}_{\mathrm{p}}$(k), p=1,2, in the superconducting state in the presence of intralayer and complex interlayer order parameters as well as complex tunneling matrix elements between the two layers in the unit cell, which need not be equivalent. The form of the possible basis functions are also listed in terms of cylindrical coordinates ${\mathrm{k}}_{\mathrm{t}}$,φ,${\mathrm{k}}_{\mathrm{z}}$ to take advantage of the orthogonality of functions with respect to φ integrations. In layered materials, with open Fermi surfaces in the ${\mathrm{k}}_{\mathrm{z}}$ direction, there is orthogonality of basis functions with respect to ${\mathrm{k}}_{\mathrm{z}}$ also (-π⩽${\mathrm{k}}_{\mathrm{z}}$d⩽π).mOur results show that in orthorhombic systems, planar ${\mathrm{d}}_{{\mathrm{k}}_{\mathrm{x}}^2\mathrm{-}{\mathrm{k}}_{\mathrm{y}}^2}$-like (${\mathrm{B}}_{1\mathrm{g}}$) and ${\mathrm{d}}_{\mathrm{kx}}$${\mathrm{k}}_{\mathrm{y}}$-like (${\mathrm{B}}_{2\mathrm{g}}$) symmetries are always mixed, respectively, with the planar s-wave-like (${\mathrm{A}}_{1\mathrm{g}}$) and ${\mathrm{A}}_{2\mathrm{g}}$-like symmetries of the corresponding tetragonal system. There is also the possibility of a weak modulation of ${\mathrm{E}}_{\mathrm{g}}$(k) as a function of ${\mathrm{k}}_{\mathrm{z}}$(∼cos ${\mathrm{k}}_{\mathrm{z}}$d). In addition, in the presence of interlayer pairings which may or may not have the same symmetry as the intralayer order parameters, even in tetragonal systems the nodes of the ${\mathrm{d}}_{{\mathrm{k}}_{\mathrm{x}}^2\mathrm{-}{\mathrm{k}}_{\mathrm{y}}^2}$-like intralayer gap function will be shifted. In view of this, some suggestions for analyzing experimental data are also presented.