d-wave superconducting condensation in the spin-density-wave background

It is shown within the random-phase approximation that through exchanging the amplitude fluctuations an attractive interaction is induced between the holes doped in the commensurate spin-density-wave background. Without including the polaronic effects, such an effective interaction is shown to lead to a $d$-wave superconducting condensation of the doped holes, instead of the $s$-wave pairing concluded by Schrieffer, Wen, and Zhang Phys. Rev. Lett. 60 944 1988 in a similar approach. The superconducting order parameter as a function of the doping concentration has been calculated also.