Doping asymmetry of superconductivity coexisting with antiferromagnetism in spin fluctuation theory

We generalize the theory of Cooper-pairing by spin excitations in the metallic antiferromagnetic state to include situations with electron and/or hole pockets. We show that Cooper-pairing arises from transverse spin waves and from gapped longitudinal spin fluctuations of comparable strength. However, each of these interactions, projected on a particular symmetry of the superconducting gap, acts primarily within one type of pocket. We find a nodeless d(x2-y2)-wave state is supported primarily by the longitudinal fluctuations on the electron pockets, and both transverse and longitudinal fluctuations support nodal d(x2-y2)-wave symmetry on the hole pockets. Our results may be relevant to the asymmetry of the AF/SC coexistence state in the cuprate phase diagram.