The role of the apical oxygen in cuprate high-temperature superconductors

TL;DR

This work investigates whether the apical-oxygen distance $δ_{\mathrm{api}}$ controls superconductivity in cuprates. Using a first-principles DFT+CDMFT approach on Bi-2201, Bi-2212, and Hg-1201 with controlled $δ_{\mathrm{api}}$ variations, the authors reproduce the STM-observed modulations of the superconducting order parameter $m_{\rm SC}$ and show that the effect is a modest modulation driven mainly by changes in the effective hole-doping of the CuO$_2$ planes, not by the charge-transfer gap (CTG). A downfolded three-band analysis reveals that CTG increases with $δ_{\mathrm{api}}$, contradicting the STM-based interpretation and supporting a doping-driven mechanism. The results emphasize material-dependent pathways for charge transfer and call for caution when inferring $T_c$–$δ_{\mathrm{api}}$ correlations across cuprate families, providing a quantitative link between apical-oxygen displacement and superconductivity while highlighting the central role of effective hole-doping.

Abstract

Scanning tunneling microscopy measurements exploiting the natural superstructure modulation of the cuprate superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ (Bi-2212) have revealed a possible correlation between the Cu-apical-O distance $δ_{\mathrm{api}}$ and the superconducting order parameter $m_{\mathrm{SC}}$, as reported recently by O'Mahony et al. (Proc. Natl. Acad. Sci. 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 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 $m_{\mathrm{SC}}$ to changes in $δ_{\mathrm{api}}$. We demonstrate, however, that these variations of $m_{\mathrm{SC}}$ 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 $m_{\mathrm{SC}}$ modulation induced by apical-oxygen displacement alone therefore warrants caution in interpreting correlations between $T_c$ and $δ_{\mathrm{api}}$ inferred from comparisons across different cuprate compounds.

Figures (8)

• Figure 1: Half simplified ($P4/mmm$) unit cells of (a) Bi-2201 and (b) Bi-2212. $\delta_{\rm api}$, $c/2$ and $a$ are displayed (See SM SupMat for details).
• Figure 2: Computed superconducting order parameter $m_{\rm SC}$ vs $\delta_{\rm api}$ for Bi-2201 and Bi-2212 (top). Note the scaling of the $y$ axis. Relative variations of the computed and measured superfluid density $|m_{\rm SC}|^2/\overline{|m_{\rm SC}|^2}$ vs $\delta_{\rm api}$ (bottom). The experimental data are digitized from Ref. omahony_electron_2022.
• Figure 3: (a) $t_{pd}$ and $t_{pp}$ (Wannier), (b) $\left(\varepsilon^{DFT}_p-\varepsilon^{DFT}_d\right)$ (Wannier), (c) $U$ (constrained DFT) and (d) $J$ (Cu$_2$O$_{25}$ clusters) vs $\delta_{\rm api}$. All parameters were computed for Bi-2201.
• Figure 4: (a) Electron occupation variations $\Delta n_{\rm CuO_2}$ of the in-plane Cu-$d_{x^2-y^2}$ and O-$p_x/p_y$ orbitals vs $\delta_{\rm api}/{\rm min}\left[\delta_{\rm api}\right]$. (b) Relative variations of the superfluid density $|m_{\rm SC}|^2/\overline{|m_{\rm SC}|^2}$ vs $\Delta n_{\rm CuO_2}$. (c) Representative sketch of the cuprates phase diagram. $m_{SC}$ is an increasing function of $\delta_{\rm api}$ in all compounds since the effective hole doping increases in underdoped Hg-1201 (green), while it decreases in overdoped Bi-2201 and Bi-2212 (blue).
• Figure 5: Non-interacting DOS of the low-energy effective model of (a1) Bi-2201 and (b1) Bi-2212 for all values of $\delta_{\rm api}$. (b) Variations of occupation $\Delta n_{\rm CuO_2}$ estimated from the non-interacting effective models.