In order to theoretically identify the factors governing superconductivity in multilayer cuprates, a three-layer Hubbard model is studied with the two-particle self-consistent (TPSC) approach so as to incorporate electron correlations. The linearized Eliashberg equation is then solved for the gap function in a matrix form to resolve the role of outer ${\mathrm{CuO}}_2$ planes (OPs) and inner plane (IP). We show that OPs dominate IP in the $d_{x^2-y^2}$-wave superconductivity, while IP dominates in the antiferromagnetism. This comes from an electron correlation effect in that the correlation makes the doping rates different between OPs and IP (i.e., a self-doping effect), which occurs in intermediate and strong correlation regimes. Namely, the antiferromagnetic fluctuations in IP are stronger due to a stronger electron correlation, which simultaneously reduces the quasiparticle density of states in IP with a suppressed $d_{x^2-y^2}$-wave superconductivity. Intriguingly, while the off-diagonal (interlayer) elements in the gap function matrix are tiny, interlayer pair scattering processes are in fact at work in enhancing the superconducting transition temperature $T_{\text{c}}$ through the interlayer Green's functions. This actually causes the trilayer system to have higher $T_{\text{c}}$ than the single-layer in a weak- and intermediate-coupling regimes. This picture holds for a range of values of the on-site Hubbard repulsion $U$ that contains those estimated for the cuprates. The present result is qualitatively consistent with nuclear magnetic resonance experiments in multilayer cuprate superconductors.

• Received 12 June 2018
• Revised 23 October 2018

DOI:https://doi.org/10.1103/PhysRevB.98.174508