Coincidence of Checkerboard Charge Order and Antinodal State Decoherence in Strongly Underdoped Superconducting<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>Bi</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>Sr</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mi>Ca</mml:mi><mml:msub><mml:mi>Cu</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mrow><mml:mn>8</mml:mn><mml:mo>+</mml:mo><mml:mi>δ</mml:mi></mml:mrow></mml:msub></mml:math>

The doping dependence of nanoscale electronic structure in superconducting ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ 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 $\stackrel{\ensuremath{\rightarrow}}{Q}=(\ifmmode\pm\else\textpm\fi{}2\ensuremath{\pi}/4.5{a}_{0},0)$; $(0,\ifmmode\pm\else\textpm\fi{}2\ensuremath{\pi}/4.5{a}_{0})\ifmmode\pm\else\textpm\fi{}15%$. Surprisingly, this spatial ordering of high energy states coexists harmoniously with the low energy Bogoliubov quasiparticle states.

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