Role of the Apical Oxygen in Cuprate High-Temperature Superconductors

Scanning tunneling microscopy measurements exploiting the natural superstructure modulation of the cuprate superconductor Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} (Bi-2212) have revealed a possible correlation between the Cu-apical-O distance δ_{api} and the superconducting order parameter m_{SC}, as reported recently by O'Mahony et al. [On the electron pairing mechanism of copper-oxide high temperature superconductivity, Proc. Natl. Acad. Sci. U.S.A. 119, e2207449119 (2022)PNASA60027-842410.1073/pnas.2207449119]. These observations were interpreted as evidence for a direct link between superconductivity and the charge-transfer gap, and more broadly revived the long-standing question of the role of apical oxygens in cuprate superconductivity. Using a combination of density-functional theory and cluster dynamical mean-field theory, we compute from first principles the variations of m_{SC} induced solely by apical oxygen displacement in Bi_{2}Sr_{2}CuO_{6+δ}, Bi-2212, and HgBa_{2}CuO_{4+δ}. The quantitative agreement between our calculations and experiments allows us to unambiguously attribute the observed variations of m_{SC} to changes in δ_{api}. We demonstrate, however, that these variations of m_{SC} originate predominantly from changes in the effective hole-doping of the CuO_{2} planes, with negligible effect on the charge-transfer gap. The modest magnitude of the m_{SC} modulation induced by apical-oxygen displacement alone warrants caution in interpreting correlations between T_{c} and δ_{api} inferred from comparisons across different cuprate compounds. Our work demonstrates that the present ab initio framework can quantitatively resolve the influence of specific structural degrees of freedom on superconductivity in correlated oxides.