Figure 1

(Color online) [(a) and (b)] DFT electronic band structure of ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_8$ shown for different energy ranges. (c) Band structure of the O-doped ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$ supercell in the same energy window as (b). The weight of the $\text{Cu}3d_{x^2-y^2}$ character in the eigenvalues is proportional to the circles’ size.Reuse & Permissions

Figure 2

(Color online) Plot of the DFT calculated $\text{Cu}3d_{x^2-y^2}$ bands (dots) and the TB1 Hamiltonian spectrum (lines) for ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_8$ and ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$: (a) comparison of the ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_8$ electronic structure to the $\text{TB}{1}_{\text{undoped}}$ model; (b) comparison of the ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$ electronic structure to the $\text{TB}{1}_{\text{undoped}}$ model plotted in the folded Brillouin zone; and (c) comparison of the ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$ electronic structure to the $\text{TB}{1}_{\text{loc}\text{.}\text{doped}}$ model (see Fig. 4). High symmetry points are given by $\left(k_x,k_y\right)$ only, $k_z=0$; thus $\left(0,0\right)=\Gamma$, $\left(\pi ,0\right)=X$, and $\left(\pi ,\pi \right)=T$.Reuse & Permissions

Figure 3

(Color online) Plot of DFT calculated $\text{Cu}3d_{x^2-y^2}$ bands (dots) and the TB2 Hamiltonian spectrum (line) for ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_8$ and ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$: (a) comparison of the ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_8$ electronic structure to the $\text{TB}{2}_{\text{undoped}}$ model; (b) comparison of the ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$ electronic structure to the $\text{TB}{2}_{\text{undoped}}$ model plotted in the folded Brillouin zone; and (c) comparison of the ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$ electronic structure to the homogeneous $\text{TB}{2}_{\text{hom}\text{.}\text{doped}}$ model. (Table second row).Reuse & Permissions

Figure 4

(Color online) The schematic lattice of dopant-displaced Cu atoms in the oxygen-doped supercell of ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$. The smallest circle represents the interstitial oxygen atom and the larger circles stand for Cu atoms. Darker color is used for Cu atoms in the ${\text{CuO}}_2$ layer closest to the interstitial oxygen. The Cu-Cu bonds that correspond to the hopping integrals of the $t_{100}$ type are represented by solid lines, the hopping integrals of the $t_{110}$ type by dashed lines, and the hopping integrals of the $t_{200}$ type by dash-dotted lines. The numbers over the bonds stand for the optimized values of corresponding hopping integrals of the doped supercell TB model based upon the nearest-neighbors parent compound model (see text). The optimized value of the on-site energy of this model for the Cu in the next-nearest ${\text{CuO}}_2$ layer (the light Cu atom symbols) is $\mu =0.4445\text{eV}$. The six optimized on-site energy values for Cu in the nearest ${\text{CuO}}_2$ layer (the dark Cu atom symbols) are $\mu =0.5757$, 0.5057, 0.5341, 0.5151, 0.4930, and 0.5186 eV. $\mu =0.5757\text{eV}$ corresponds to the Cu atom which is displaced most by the dopant and is marked with an arrow. $m$ labels the mirror plane.Reuse & Permissions

Figure 5

(Color online) The static spin susceptibility of (a) the two parent compound TB models $\text{TB}{1}_{\text{undoped}}$ and $\text{TB}{2}_{\text{undoped}}$ and the homogeneous doped supercell model $\text{TB}{2}_{\text{hom}\text{.}\text{doped}}$, plotted in the full Brillouin zone, and (b) the two parent compound TB models $\text{TB}{1}_{\text{undoped}}$ and $\text{TB}{2}_{\text{undoped}}$ and the inhomogeneous $\text{TB}{1}_{\text{loc}\text{.}\text{doped}}$ and homogeneous $\text{TB}{2}_{\text{hom}\text{.}\text{doped}}$ doped supercell models, plotted in the folded Brillouin zone.Reuse & Permissions

Figure 6

(Color online) (a) The pairing strength $\lambda$ for the undoped (triangles) and doped (circles) ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$ TB models as a function of Coulomb repulsion $U$. (b) The superconducting gap function $g\left(\mathbf{k}\right)$ on the $k$-point mesh at the Fermi surface of the undoped ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_8$ unit cell for the undoped (triangles) and doped (circles) models. Positive $g\left(\mathbf{k}\right)$ values are mostly above the $(0,0)-(\pi ,\pi )$ line (red), negative values mostly below (blue). The intensity of the color is proportional to the absolute value of $g\left(\mathbf{k}\right)$. Half the nesting vector $q_{\text{nest}}$ is shown by an arrow.Reuse & Permissions