deletions | additions       diff --git a/cuprates2b.tex b/cuprates2b.tex  index 2bea9e2..48979eb 100644  --- a/cuprates2b.tex  +++ b/cuprates2b.tex 
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Our goal is to tune the in-plane effective $U$, so the relevant layers to focus on are the BaO layers immediately adjacent to the CuO$_2$ plane. Due to their spatial proximity, the BaO layers tune the hoppings and interaction strengths of the in-plane Hamiltonian. Designing compounds with novel adjacent layers provides a mechanism for controlling superconductivity. However, robotically cycling through all roughly $100 \times 100$ elemental substitutions for BaO in the periodic table using structural prediction is clearly too naive and computationally expensive.  To select plausible compositions, we noted that the BaO layers form a rock salt structure. Using materials databases, we selected all naturally occurring rock salt compounds AX, composed of an cation A and an anion X, starting with 333 in total. We then quickly pre-screened candidates by discarding compositions with (1) large lattice mismatches relative to the in-plane Cu-Cu distance, which we took to be 3.82~\AA, and (2) anions less electronegative than Cu, as these anions would capture dopants intended for the superconducting plane, producing additional Fermi surfaces.  In Given  the Hg-cuprates, the HgO$_\delta$ layer harbors dopant atoms which tune the  chemical potential. The BaO layers immediately adjacent to the CuO$_2$ plane  spatially separate remaining list of $\sim$20 potential compositions, we screened for stable compounds by first point-substituting  the superconducting electrons from proposed elements into  the detrimental effects  of HBCO structure and checked for local stability via phonon calculations at  the disordered dopant layer. Additionally, $\Gamma$, $(\pi,0)$ and $(\pi,\pi)$ points. If  the highly ionic nature of composition was stable in  the BaO layer means they do not capture dopant electrons intended for HBCO structure, then we used USPEX to perform  the CuO$_2$  plane. The preference of Hg roughly week-long calculations necessary  to be dumbbell coordinated bonds determine whether  the entire HBCO  structure together without introducing structural distortions. Finally, is indeed  the highly ionic O-Hg-O dumbbells minimize $c$-axis hopping to maintain  2-dimensionality. preferred energetic minimum.  Due to their spatial proximity, the adjacent We found three  BaO layers tune substitutions to be stable in  the hoppings HBCO structure after phonon screening: CaS, ZrAs,  and interaction strengths of the in-plane Hamiltonian. Designing compounds with  novel adjacent layers provides a mechanism for controlling superconductivity  by, e.g., reducing the charge-transfer energy. We quickly realized the most  stringent constraint is structural stability, so we focused first on isolating YbS. USPEX found that only Hg(CaS)$_2$CuO$_2$ was  stable candidates, then subsequently investigating their electronic properties. in the layered cuprate structure.  To maximize the likelihood that a proposed composition is stable, we note that  the layers adjacent to the CuO$_2$ plane form a rock salt structure. Using  materials databases, we selected all naturally occurring rock salt compounds  AX, composed of an cation A and an anion X. The phase space is large and the  rate limiting step is structural prediction, so we quickly pre-screen  candidates by discarding compositions with (1) large lattice mismatches  relative to the in-plane Cu-Cu distance, which we took to be 3.82~\AA, and (2)  anions less electronegative than Cu, as these anions would capture dopants  intended for the superconducting plane, producing additional Fermi surfaces. \emph{Global stability} -- We construct the convex hull by computing the energies of all known compounds in the Hg-Ca-Cu-S-O system, which would produce a 4-dimensional tetrahedron. Realistic In reality,  synthesis is performed in an oxygen environment parameterized by the chemical potential $\mu(\text{O}_2)$. We find that for all values of $\mu(\text{O}_2)$, HCSCO is unstable, so we pick the value for which the compound is closest to the convex hull and plot the results in Fig.~\ref{fig:hcsco-hull}.