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A self-doping effect between outer and inner CuO$_2$ planes (OPs and IPs) inmulti-layer cuprate superconductors is studied. When one considers athree-layer tight-binding model of the Hg-based three-layer cuprate derivedfrom the first principle calculations, the electron concentration gets to belarge in the OP compared to IP. This is inconsistent with the experimental factthat more hole carriers tend to be introduced into the OP than IP.Weinvestigate a three-layer Hubbard model with the two-particle self-consistentapproach for multi-layer systems to incorporate electron correlations. Weobserve that the double occupancy (antiferromagnetic instability) in the IPdecreases (increases) more than the OP, and also reveal that more electronstend to be introduced into the IP than OP to obtain the energy gain from theon-site Hubbard interaction. These results are consistent with the experimentalfacts, and this electron distribution between the OP and IP can be interpretedas a self-doping effect arising from strong electron correlations.

Self-doping effect arising from electron correlations in multi-layer cuprates