Supermodulation in the atomic structure of the superconductor ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+x}$ from ab initio calculations

We present results of density functional theory (DFT) calculation of the structural supermodulation in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+x}$ structure, and show that the supermodulation is indeed a spontaneous symmetry breaking of the nominal crystal symmetry, rather than a phenomenon driven by interstitial O dopants. The structure obtained is in excellent quantitative agreement with recent x-ray studies, and reproduces several qualitative aspects of scanning tunneling microscopy (STM) experiments as well. The primary structural modulation affecting the $\mathrm{Cu}{\mathrm{O}}_2$ plane is found to be a buckling wave of tilted $\mathrm{Cu}{\mathrm{O}}_5$ half-octahedra, with maximum tilt angle near the phase of the supermodulation where recent STM experiments have discovered an enhancement of the superconducting gap. We argue that the tilting of the half-octahedra and concomitant planar buckling are directly modulating the superconducting pair interaction.

Figure 1

(Color online) Left: Upper half of a nominal Bi-2212 unit cell. Lattice constants shown are between cations in each layer, values taken from Levin et al. (Ref. 16) (from this work). Right: Side view of cation distribution for a layer of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+x}$ one unit cell thick for (top) DFT bulk calculation, and (bottom) x-ray diffraction experiment (Ref. 16).

Figure 2

(Color online) Displacements in the BiO layer projected into the $x\text{−}y$ plane for (a) DFT bulk calculation and (b) x-ray (Ref. 16). Dashed lines are mirror planes, and X’s mark points of stability found for O dopants by DFT. Small circles without arrows (dark green) mark positions deduced by x-ray experiment (c) Average $z$ displacements of Bi for DFT (solid line) and x ray (dashed line). (d)–(f) Same as (a)–(c), but for the SrO layer. All arrow lengths were multiplied by 2 for clarity.

Figure 3

(Color online) (a) $z$ displacements of Bi atoms in DFT surface calculation (solid line) and STM measurement (Ref. 1) (dashed line); (b) modulation of measured spectral gap for overdoped BSCCO sample vs supermodulation phase angle ${\varphi }_{\mathit{SM}}$ from Ref. 1; (c) schematic tilting of $\mathrm{Cu}{\mathrm{O}}_5$ half-octahedra and its dependence on ${\varphi }_{\mathit{SM}}$. Tilt angles have been exaggerated for clarity.