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We consider normal state properties, the pairing instability temperature, andthe structure of the pairing gap in electron-doped cuprates. We assume that thepairing is mediated by collective spin excitations, with antiferromagnetismemerging with the appearance of hot spots. We use a low-energy spin-fermionmodel and Eliashberg theory up to two-loop order. We justify ignoring vertexcorrections by extending the model to N >>1 fermionic flavors, with 1/N playingthe role of a small Eliashberg parameter. We argue, however, that it is stillnecessary to solve coupled integral equations for the frequency dependentfermionic and bosonic self-energies, both in the normal and superconductingstate. Using the solution of the coupled equations, we find an onset of d-wavepairing at Tc ~ 30 K, roughly three times larger than the one obtainedpreviously [P. Krotkov and A. Chubukov, Phys. Rev. B 74, 014509 (2006)], whereit was assumed that the equations for fermionic and bosonic self-energiesdecouple in the normal state. To obtain the momentum and frequency dependentd-wave superconducting gap D(k,w), we derive and solve the non-linear gapequation together with the modified equation for the bosonic self energy whichbelow Tc depends on the pairing gap. We find that the gap is a non-monotonicfunction of momentum along the Fermi surface, with its node along the zonediagonal and its maximum some distance away from it. We obtain 2D/Tc ~ 4. Weargue that the value of Tc, the non-monotonicity of the gap, and 2D/Tc ratioare all in good agreement with the experimental data on electron-dopedcuprates.

Self-consistent Eliashberg theory, Tc, and the gap function in electron-doped cuprates