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We generalize the theory of Cooper pairing by spin excitations in themetallic antiferromagnetic state to include situations with electron and/orhole pockets. We show that Cooper pairing arises from transverse spin waves andfrom gapped longitudinal spin fluctuations of comparable strength. However,each of these interactions, projected on a particular symmetry of thesuperconducting gap, acts primarily within one type of pocket. We find anodeless $d_{x^2-y^2}$-wave state is supported primarily by the longitudinalfluctuations on the electron pockets, and both transverse and longitudinalfluctuations support nodeless odd-parity spin singlet $p-$wave symmetry on thehole pockets. Our results may be relevant to the asymmetry of the AF/SCcoexistence state in the cuprate phase diagram, as well as for the "nodal gap"observed recently for strongly underdoped cuprates.

Doping asymmetry of superconductivity coexisting with antiferromagnetism in spin fluctuation theory