Scanning tunneling microscopy measurements exploiting the natural superstructure modulation of the cuprate superconductor ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ (Bi-2212) have revealed a possible correlation between the Cu-apical-O distance ${\delta }_{\mathrm{api}}$ and the superconducting order parameter $m_{\mathrm{SC}}$, 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)]. 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 $m_{\mathrm{SC}}$ induced solely by apical oxygen displacement in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CuO}}_{6+\delta }$, Bi-2212, and ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+\delta }$. The quantitative agreement between our calculations and experiments allows us to unambiguously attribute the observed variations of $m_{\mathrm{SC}}$ to changes in ${\delta }_{\mathrm{api}}$. We demonstrate, however, that these variations of $m_{\mathrm{SC}}$ originate predominantly from changes in the effective hole-doping of the ${\mathrm{CuO}}_2$ planes, with negligible effect on the charge-transfer gap. The modest magnitude of the $m_{\mathrm{SC}}$ modulation induced by apical-oxygen displacement alone warrants caution in interpreting correlations between $T_c$ and ${\delta }_{\mathrm{api}}$ 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.

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• Superconducting order parameter
• Superconductivity
• Cuprates
• High-temperature superconductors
• Strongly correlated systems
• Density functional theory
• Dynamical mean field theory
• First-principles calculations