Insights from Angle-Resolved Photoemission Spectroscopy of an Undoped Four-Layered Two-Gap High-$T_c$ Superconductor

An undoped cuprate with apical fluorine and inner ($i$) and outer ($o$) ${\mathrm{CuO}}_2$ layers is a 60 K superconductor whose Fermi surface has large $n$- and $p$-doped sheets with the superconducting gap on the $n$ sheet twice that on the $p$ sheet. The Fermi surface is not reproduced by the local density approximation, but the screening must be substantially reduced due to electronic correlations, and oxygen in the $o$ layers must be allowed to dimple outwards. This charges the $i$ layers by $0.01|e|$, causes a 0.4 eV Madelung-potential difference between the $i$ and $o$ layers, quenches the $i\mathrm{\text{−}}o$ hopping, and localizes the $n$ sheets onto the $i$ layers, thus protecting their $d$-wave pairs from being broken by scattering on impurities in the BaF layers. The correlation-reduced screening strengthens the coupling to $z$-axis phonons.

Figure 1

FS for F0234. Gray: ARPES [1]. The width reflects the experimental error bar. The smaller (larger) sheet is the $n$ ($p$) sheet. Open dots: Calculations. Top right: LDA. Remaining three quarters: $⟨\mathrm{LDA}+U⟩$. Bottom right: Flat ${\mathrm{CuO}}_2$ layers and 2% F-O exchange ($q=0.02$) treated in the VCA. Indistinguishable from this FS are the ones calculated with no F-O exchange, but with oxygen in the $o$ or $i$ layers dimpled outwards by 0.07 Å. See Fig. 2. The $n$ ($p$) sheet resides on the $i$ ($o$) layers. ${\epsilon }_F$ is adjusted such the FS fits along the nodal line, whereby $n=0.19$ and $p=0.12$. Bottom left: Dimpling as before, but with the vertical Ba and F positions adjusted such as to give the best agreement with the ARPES FS: $z_{\mathrm{Ba}}-z_{{\mathrm{Cu}}_o}$ is decreased from 2.0 to 1.7 Å and $z_F-z_{{\mathrm{Cu}}_o}$ is decreased from 2.4 to 2.1 Å. ${\epsilon }_F$ is adjusted as before, whereby $n=0.18$ and $p=0.21$. Top left: As before, but with O in the $o$ (or $i$) layers dimpled inwards by 0.10 Å. Now, the $n$ ($p$) sheet resides on the $o$ ($i$) layers. $z_{\mathrm{Ba}}-z_{{\mathrm{Cu}}_o}=1.6\AA$ and $z_F-z_{{\mathrm{Cu}}_o}=2.0\AA$. ${\epsilon }_F$ is adjusted as before, whereby $n=0.15$ and $p=0.18$. Note that intersheet splittings in $\mathbf{k}$ space are enhanced near the saddle points in the band structure.

Figure 2

Layer-averaged potentials calculated with $⟨\mathrm{LDA}+U⟩$ for dimplings of $\pm 0.1\AA$ of the inner (left) and outer (right) layer oxygens. This potential is the calculated Cu and O core levels, lined up on the layer which is not dimpled, and normalized to zero at the Ca mirror plane ($m$).

Figure 3

Total energy as a function of the Raman-active dimpling displacements calculated with $\mathrm{LDA}+U$. See Fig. 2.