Hole Dispersion and Symmetry of the Superconducting Order Parameter for Underdoped CuO₂ Bilayers and 3D Antiferromagnets

We calculate the dispersion of a hole in ${\rm CuO_2}$ bilayers and 3D antiferromagnets, using the self-consistent Born approximation (SCBA) 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 (AF) fluctuations, as described in recent studies for the single layer cuprates. The well-known pairing in the ${ d_{x^2-y^2}}$ 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 a ``s-wave'' state is observed with an order parameter mixture of ${ d_{3z^2-r^2}}$ and ${ s_{x^2+y^2 + z^2}}$. For an isotropic 3D antiferromagnet the $d_{x^2 -y^2}$ and $d_{3z^2 - r^2}$ channels are degenerate.