Hole dispersion and symmetry of the superconducting order parameter for underdoped CuO2 bilayers and the three-dimensional antiferromagnets

We calculate the dispersion of a hole in CuO2 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 d(x2-y2) 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 d(3z2-r2) and s(x2+y2+z2). For an isotropic 3D antiferromagnet the d(x2-y2) and d(3z2-r2) channels are degenerate. Our results are compared with other predictions in the literature.