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 the 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 in both the normal and the superconducting states. Using the solution of the coupled equations, we find an onset of d-wave pairing at T-c similar to 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, Delta((k) over right arrow (F), omega(n)), we derive and solve the nonlinear gap equation together with the modified equation for the bosonic self-energy which below T-c also depends on Delta((k) over right arrow (F), omega(n)). We find that Delta((k) over right arrow (F), omega(n)) is a nonmonotonic function of momentum along the Fermi surface, with its node along the zone diagonal and its maximum some distance away from it. We obtain 2 Delta(max)(T -> 0)/T-c similar to 4. We argue that the value of T-c, the nonmonotonicity of the gap, and the 2 Delta(max)/T-c ratio are all in good agreement with the experimental electron-doped cuprates.