Coincidence of Checkerboard Charge Order and Antinodal State Decoherence in Strongly Underdoped Superconducting ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$

The doping dependence of nanoscale electronic structure in superconducting ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ is studied by scanning tunneling microscopy. At all dopings, the low energy density-of-states modulations are analyzed according to a simple model of quasiparticle interference and found to be consistent with Fermi-arc superconductivity. The superconducting coherence peaks, ubiquitous in near-optimal tunneling spectra, are destroyed with strong underdoping and a new spectral type appears. Exclusively in regions exhibiting this new spectrum, we find local “checkerboard” charge ordering of high energy states, with a wave vector of $\overrightarrow{Q}=(\pm 2\pi /4.5a_0,0)$; $(0,\pm 2\pi /4.5a_0)\pm 15\%$. Surprisingly, this spatial ordering of high energy states coexists harmoniously with the low energy Bogoliubov quasiparticle states.

Figure 1

(a)–(d) Measured $\Delta (\overrightarrow{r})$, gap maps, for the four different hole-doping levels listed in . Color scales identical.

Figure 2

(a) The average spectrum, $g(E)$, associated with each gap value in a given FOV (field of view) from 1. These were extracted from 1b but the equivalent analysis for $g(\overrightarrow{r},V)$ at all dopings yields results which are indistinguishable. The coherence peaks are seen spectra 1–4. (b) Characteristic spectra from the two regions $\Delta <65$ (red) and $\Delta \nless 65$ (black). It is not the absolute scale of spectra, but changes in their shape, which is our focus.

Figure 3

(a) A schematic representation of the 1st Brillouin zone and Fermi-arc location of Bi-2212. The flat-band regions near the zone face are shaded in green. The eight locations which determine the scattering within the octet model [17] (for one sub gap energy) are shown as red circles and the scattering vectors which connect these locations are show as arrows labeled by the designation of each scattering vector. (b) Measured dispersions of the LDOS modulations ${\overrightarrow{q}}_1$, ${\overrightarrow{q}}_5$, and ${\overrightarrow{q}}_7$ for the three dopings whose gap maps are shown in Figs. 1a, 1c, 1d. (c) Calculated loci of scattering, ${\overrightarrow{k}}_s$ for all three dopings. For the lowest doping the internal consistency of Eqs. (1, 2) is worse than for optimal doping.

Figure 4

(a) FT of $g(\overrightarrow{r},V){|}_{\Delta \nless 65}$ integrated from $V=-65\mathrm{meV}$ to $V=-150\mathrm{meV}$ [taken from Fig. 1d]. (b) FT of the complementary, integrated $g(\overrightarrow{r},V){|}_{\Delta <65}$. (c) Dispersion of ${\overrightarrow{q}}_1$ in regions with dSC coherence peaked spectra $\Delta <65\mathrm{meV}$ (red circles) and in regions with ZTPG spectra for $E<36\mathrm{meV}$ (black squares). At low energies the dispersive LDOS modulation ${\overrightarrow{q}}_1$ are identical in the two regions. For $E>65\mathrm{meV}$, a new wave vector, ${\overrightarrow{q}}^{*}$, modulation is found only in the ZTPG regions (black). To within our uncertainty they do not disperse and ${\overrightarrow{q}}^{*}=(\pm 2\pi /4.5a_0,0)$, $(0,\pm 2\pi /4.5a_0)\pm 15\%$. (d) The magnitude of the integrated $g(\overrightarrow{q},V)$ along the $\overrightarrow{q}\parallel (\pi ,0)$ direction [lines in 4a, 4b] for $\Delta \nless 65\mathrm{meV}$ ($\Delta <65\mathrm{meV}$) black (red). (e) The magnitude of the FT of the masked topographic image along the $\overrightarrow{q}\parallel (\pi ,0)$ direction for $\Delta \nless 65\mathrm{meV}$ and ($\Delta <65\mathrm{meV}$) black (red). The solid lines are guides to the eye. (f) A plot of the amplitude of the ${\overrightarrow{q}}_1$ LDOS modulation as a function of $|{\overrightarrow{q}}_1|$ for the same data set yielding Fig. 1d. The maximum intensity of the modulations in the ZTPG regions occurs at $|{\overrightarrow{q}}_1|=2\pi /4.5a_0\pm 10\%$. No enhanced scattering of the quasiparticles in the dSC regions (red) is seen.