Hole dispersion and symmetry of the superconducting order parameter for underdoped ${\mathrm{CuO}}_2$ bilayers and the three-dimensional antiferromagnets

We calculate the dispersion of a hole in ${\mathrm{CuO}}_2$ bilayers and three-dimensional (3D) antiferromagnets, using the self-consistent Born approximation to the t-J model. Superconductivity and the symmetry of its order parameter are studied introducing a nearest-neighbor density-density attraction induced by short range antiferromagnetic fluctuations, as described in recent studies for the single layer cuprates. The well-known pairing in the ${\mathit{d}}_{{\mathit{x}}^2\mathrm{-}{\mathit{y}}^2}$ channel observed for one plane remains robust when three-dimensional interactions are turned on. In bilayers, as the exchange along the direction perpendicular to the planes grows, eventually a transition to an ‘‘s-wave’’ state is observed with an order parameter mixture of ${\mathit{d}}_{3{\mathit{z}}^2\mathrm{-}{\mathit{r}}^2}$ and ${\mathit{s}}_{{\mathit{x}}^2+{\mathit{y}}^2+{\mathit{z}}^2}$. For an isotropic 3D antiferromagnet the ${\mathit{d}}_{{\mathit{x}}^2\mathrm{-}{\mathit{y}}^2}$ and ${\mathit{d}}_{3{\mathit{z}}^2\mathrm{-}{\mathit{r}}^2}$ channels are degenerate. Our results are compared with other predictions in the literature. © 1996 The American Physical Society.