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

We consider normal state properties, the pairing instability temperature, and the structure of the pairing gap in electron-doped cuprates. We assume that the pairing is mediated by collective spin excitations, with antiferromagnetism emerging with the appearance of hot spots. We use a low-energy spin-fermion model and Eliashberg theory up to two-loop order. We justify ignoring vertex corrections by extending the model to N >>1 fermionic flavors, with 1/N playing the role of a small Eliashberg parameter. We argue, however, that it is still necessary to solve coupled integral equations for the frequency dependent fermionic and bosonic self-energies, both in the normal and superconducting state. Using the solution of the coupled equations, we find an onset of d-wave pairing at Tc ~ 30 K, roughly three times larger than the one obtained previously [P. Krotkov and A. Chubukov, Phys. Rev. B 74, 014509 (2006)], where it was assumed that the equations for fermionic and bosonic self-energies decouple in the normal state. To obtain the momentum and frequency dependent d-wave superconducting gap D(k,w), we derive and solve the non-linear gap equation together with the modified equation for the bosonic self energy which below Tc depends on the pairing gap. We find that the gap is a non-monotonic function of momentum along the Fermi surface, with its node along the zone diagonal and its maximum some distance away from it. We obtain 2D/Tc ~ 4. We argue that the value of Tc, the non-monotonicity of the gap, and 2D/Tc ratio are all in good agreement with the experimental data on electron-doped cuprates.