Fourier transform spectroscopy of $d$-wave quasiparticles in the presence of atomic scale pairing disorder

The local density of states power spectrum of optimally doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+x}$ (BSCCO) has been interpreted in terms of quasiparticle interference peaks corresponding to an “octet” of scattering wave vectors connecting $\mathbf{k}$ points where the density of states is maximal. Until now, theoretical treatments have not been able to reproduce the experimentally observed weights and widths of these octet peaks; in particular, the predominance of the dispersing “${\mathbf{q}}_1$” peak parallel to the Cu-O bond directions has remained a mystery. In addition, such theories predict “background” features which are not observed experimentally. Here, we show that most of the discrepancies can be resolved when a realistic model for the out-of-plane disorder in BSCCO is used. Weak extended potential scatterers, which are assumed to represent cation disorder, suppress large-momentum features and broaden the low-energy “${\mathbf{q}}_7$” peaks, whereas scattering at order parameter variations, possibly caused by a dopant-modulated pair interaction around interstitial oxygens, strongly enhances the dispersing ${\mathbf{q}}_1$ peaks.

Figure 1

(Color online). Schematic depiction of the first Brillouin zone in cuprate superconductors. Green (dark gray) and blue (light gray) regions represent ${\Delta }_{\mathbf{k}}>0$ and ${\Delta }_{\mathbf{k}}<0$, respectively. Red (gray) dots are intersections of constant quasiparticle energy contours with the Fermi surface. The fraction of octet vectors, ${\mathbf{q}}_1$, ${\mathbf{q}}_4$, ${\mathbf{q}}_5$ which should in principle be visible in the case of a “pointlike” (four bond) order parameter modulation (${\tau }_1$ scatterer) are denoted with solid lines. The remaining octet vectors are indicated with dashed lines.

Figure 2

Effect of out-of-plane scattering potentials of different type on the power spectrum. The power spectrum is shown for: first column: pointlike ${\tau }_3$ scatterer with strength $V_s=120\mathrm{meV}$, second column: extended ${\tau }_3$ scatterer with $V_0=40\mathrm{meV}$, $\lambda =r_z=1.5$, third column: pointlike ${\tau }_1$ scatterer with $\delta \Delta ={\Delta }_0$ on the four central bonds, fourth column: slightly extended ${\tau }_1$ scatterer with $\delta \Delta ={\Delta }_0$, $\lambda =r_z=1.2$, fifth column: combination of 7.5% extended ${\tau }_1$ scatterers ($\delta \Delta ={\Delta }_0$, $\lambda =r_z=1.2$) and 3% extended ${\tau }_3$ scatterers ($V_0=40\mathrm{meV}$, $\lambda =r_z=1.5$), sixth column: same as fifth column plus 0.2% unitary $(V_s=3.9\mathrm{meV})$ pointlike scatterers, seventh column: experimental power spectrum (Refs. 12, 13).

Figure 3

Cut along the diagonal of the Brillouin zone $(k_x=k_y)$ through the power spectrum of a strong pointlike ${\tau }_3$ scatterer with $V_s=3.9\mathrm{eV}$ (solid line), a weak extended ${\tau }_3$ scatterer with $V_0=40\mathrm{meV}$, $\lambda =r_z=1.5$ (dashed line) and a pointlike ${\tau }_1$ scatterer with $\delta \Delta ={\Delta }_0$ (dotted line) for an energy of (a) $\omega =14\mathrm{meV}$ and (b) $\omega =30\mathrm{meV}$.