Electronic structure and two-band superconductivity in unconventional high-$T_c$ cuprates ${\mathrm{Ba}}_2 {\mathrm{CuO}}_{3+\delta }$

The recently discovered cuprate superconductor ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{O}}_{3+\delta }$ exhibits a high $T_c\simeq 73\mathrm{K}$ at $\delta \simeq 0.2$. The polycrystal grown under high pressure has a structure similar to ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$ but with dramatically different lattice parameters due to the $\mathrm{Cu}{\mathrm{O}}_6$ octahedron compression. The crystal field in the compressed ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{O}}_4$ leads to an inverted Cu $3d{e}_g$ complex with the $d_{x^2-y^2}$ orbital sitting below the $d_{3z^2-r^2}$ and an electronic structure highly unusual compared to the conventional cuprates. We construct a two-orbital Hubbard model for the Cu $d^9$ state at hole doping $x=2\delta$ and study the orbital-dependent strong correlation and superconductivity. For the undoped case at $x=0$, we found that strong correlation drives an orbital-polarized Mott-insulating state with the spin-$1/2$ moment of the localized $d_{3z^2-r^2}$ orbital. In contrast to the single-band cuprates where superconductivity is suppressed in the overdoped regime, hole doping the two-orbital Mott insulator leads to orbital-dependent correlations and the robust spin and orbital exchange interactions produce a high-$T_c$ antiphase $d$-wave superconductor even in the heavily doped regime at $x=0.4$. We conjecture that ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{O}}_{3+\delta }$ realizes mixtures of such heavily hole-doped superconducting ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{O}}_4$ and disordered ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{O}}_3$ chains in a single-layer or predominately separated bilayer structure. Our findings suggest that unconventional cuprates with liberated orbitals as doped two-band Mott insulators can be a direction for realizing high-$T_c$ superconductivity with enhanced transition temperature $T_c$.

Figure 1

Mixed structures of (a) Ba214 regions with $\mathrm{Cu}{\mathrm{O}}_2$ plane and Ba213 regions with short Cu-O chains and substantial oxygen vacancies within a single layer and (d) bilayer of a mostly Ba214 with $\mathrm{Cu}{\mathrm{O}}_2$ plane and a disordered Ba213 with short Cu-O chains and substantial oxygen vacancies. [(b) and (e)] Atomic structures of stoichiometric Ba214 and Ba214/Ba213 bilayer corresponding to (a) and (d). [(c) and (f)] DFT band structures of compressed Ba214 corresponding to (b) and (e) using structural parameters measured by neutron scattering. Zero marks the Fermi level corresponding to Cu $3d^9$ configuration.

Figure 2

(a) Band dispersion in the TB model with color-coded orbital content: $d_{x^2}$ orbital (red and 1) and $d_{z^2}$ orbital (blue and 0). The Fermi levels are marked by black ($x=0$) and blue ($x=0.4$) dashed lines. (b) The hopping renormalization factor $g_{\alpha \beta }$ as a function of $U$ at $x=0$, showing a Brinkman-Rice transition at $U_c\simeq 4.4$ eV. (c) Correlated band dispersion at $x=0$ and $U=4.4$ eV. (d) Correlated band structure at $x=0.4$ and $U=7$ eV.

Figure 3

(a) Normal state FSs at $x=0.4$. (b) Anisotropic pairing energy gaps as a function of angle $\theta$ depicted in (a) along the two FSs around $\mathrm{\Gamma }$ (blue) and $M$ (red), showing two antiphase $d$-wave gap functions. (c) Total (red line) and $d_{z^2}$ orbital contribution (black dashed line) to local tunneling density of states, showing two $d$-wave gaps with coherent peaks. A thermal broadening of 0.5 meV is used. (d) Variations of the larger $d$-wave gap as a function of $U$ and $K$.

Figure 4

(a) The two FSs (solid squares) inside the first BZ (dashed square). Circles and triangles denote the eight patches, each representing one flat section of the FS. The inter-FS coupling $g_{u\parallel }$ (green arrows) describes the pair scattering between two same colored patches (circles and triangles), while $g_{u\perp }$ (orange arrows) describes the pair scattering between two different colored patches. (b) The intra-FS coupling $g_1$ describes the backward scattering of electrons between solid and dashed patches. The red line indicates the $(Q_1,0)$ nesting vector. (b) $g_2$ describes the forward scattering of electrons on the solid and dashed patches. The red line indicates the the (0,$Q_1$) nesting vector. (c) $g_3$ describes the scattering of electrons between black and blue patches.

Figure 5

The RG flow of the effective couplings ($\mathrm{\Gamma }$'s) normalized by a scaling factor $g_0$. (a) Comparison of the RG flow of antiphase $d$-wave superconductivity (${\mathrm{\Gamma }}_{d\mathrm{SC}}^{\pm }$) (red) to those associated with density waves: the SDW [${\mathrm{\Gamma }}_{\mathrm{SDW}}^{(Q_1,0)}$] (blue) and CDW [${\mathrm{\Gamma }}_{\mathrm{CDW}}^{(Q_1,0)}$] (black). (b) Comparison of the RG flow of antiphase $d$-wave superconductivity (${\mathrm{\Gamma }}_{d\mathrm{SC}}^{\pm }$) (red) to other superconductivity channels: In-phase $d$-wave superconductivity (${\mathrm{\Gamma }}_{d\mathrm{SC}}^{++}$) (green), antiphase $s$-wave superconductivity (${\mathrm{\Gamma }}_{s\mathrm{SC}}^{\pm }$) (orange), and in-phase $s$-wave superconductivity (${\mathrm{\Gamma }}_{s\mathrm{SC}}^{++}$) (dark green). The initial dimensionless coupling constants are $g_1=g_2=1.0,g_3=1.0,g_{u\parallel }=1.3,g_{u\perp }=1.0$, and $\alpha =0.8$. Panels (a) and (b) show that the antiphase $d$-wave pairing is the leading instability.

Figure 6

(a) Crystal structure of ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{O}}_4$. (b) Effective molecular orbital through hybridization between the Cu $d_{z^2}$ orbital and apical oxygen $p_z$ orbital. Right side is the schematic molecular orbital ${\varphi }_1$. (c) Orbital-resolved DFT band structure for compressed Ba214 calculated using the structural parameters in Table 1.

Figure 7

(a) Crystal structure of ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{O}}_3$ with Cu-O chains along the $x$ direction. (b) The Fermi surface of ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{O}}_3$. (c) Orbital-resolved DFT band dispersions.

Figure 8

(a) Crystal structure of Ba213/Ba214 bilayers corresponding to ${\mathrm{Ba}}_2\mathrm{Cu}{\mathrm{O}}_{3.5}$. (b) Orbital-resolved band dispersions in the $\mathrm{Cu}{\mathrm{O}}_2$ plane (Ba214). (c) Orbital-resolved band dispersions in the chain plane with Cu-O chains along the $x$ direction (Ba213).