Strong electronic distortions are typically accompanied by structural distortions, and vice versa. Determining the relationship between these orders can be complicated, but a clue comes from their (co-)dependence on other parameters. In a cuprate superconductor, for example, both superconductivity and the pseudogap are highly dependent on doping, temperature, and magnetic field. Here, we investigate whether the structural symmetry in BSCCO is similarly dependent on these parameters, or whether it is an omnipresent background within which the electronic states evolve.Structural symmetries are traditionally measured by scattering experiments, such as xray or neutron scattering to determine bulk symmetries, or low energy electron diffraction (LEED) to determine surface symmetries. The structure of double layer Bi 2 Sr 2 CaCu 2 O 8+x (Bi-2212) is sketched in Fig. 1a. Although nearly tetragonal, a ∼0.5% difference between a and b axes 14 makes the true structure orthorhombic. However, despite numerous scattering experiments on BSCCO spanning two decades, the more detailed structure has remained enigmatic, due in part to an incommensurate structural "supermodulation" which pervades the bulk of these materials 14 , and to dopant disorder which leads Bi atoms to stochastically occupy inequivalent sites in different unit cells 15 . [18][19][20][21][22][23] . Thus, to investigate the role of structure in these broken symmetry electronic states, it is imperative to make atomic scale measurements of the structural symmetry.To undertake this investigation, we use three different home-built scanning tunneling microscopes. In each case, a sample is cleaved at low temperature in cryogenic ultra-high vacuum, and immediately inserted into the scanning head. BSCCO typically cleaves between two BiO mirror planes (Fig. 1a). Data was acquired at T=6K unless otherwise noted. The tip is rastered across the sample surface, while a feedback loop adjusts its height to maintain a constant tip-sample tunneling current. This results in a topographic image of the BiO surface.Temperature drift (typically < 10 mK), piezo hysteresis, and piezo nonlinearity, can lead to small but problematic warping of topographic images. Recently, Lawler et al introduced a ground-breaking algorithm to correct these picometer-scale drifts 7 . We show that Lawler's algorithm can also be used to remove subtle periodic noise (see Supplementary that the pseudogap is characterized by intra-unit-cell inversion symmetry breaking? To investigate this, we characterize the dependence of the structural distortion on parameters which are known to heavily influence electronic ordered states: doping, temperature, and magnetic field. Fig. 3a locates in a three-dimensional phase diagram the 21 datasets in which we measured the structure. The key results are summarized in Figs. 3b-d. We do not find a dependence of the structural distortion on doping, temperature, or field, across a wide range of values. We have measured the distortion both inside and outside the superconducting...

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Charge modulations have been widely observed in cuprates, suggesting their centrality for understanding the high-Tc superconductivity in these materials. However, the dimensionality of these modulations remains controversial, including whether their wavevector is unidirectional or bidirectional, and also whether they extend seamlessly from the surface of the material into the bulk. Material disorder presents severe challenges to understanding the charge modulations through bulk scattering techniques. We use a local technique, scanning tunneling microscopy, to image the static charge modulations on Bi2−zPbzSr2−yLayCuO6+x. By comparing the phase correlation length ξCDW with the orientation correlation length ξorient, we show that the charge modulations are more consistent with an underlying unidirectional wave vector. Using cluster techniques, we show that these locally 1D charge modulations are actually a bulk effect resulting from 3D criticality throughout the entire superconducting doping range.

Charge modulations have been widely observed in cuprates, suggesting their centrality for understanding the high-Tc superconductivity in these materials. However, the dimensionality of these modulations remains controversial, including whether their wavevector is unidirectional or bidirectional, and also whether they extend seamlessly from the surface of the material into the bulk. Material disorder presents severe challenges to understanding the charge modulations through bulk scattering techniques. We use a local technique, scanning tunneling microscopy, to image the static charge modulations on Bi2−zPbzSr2−yLayCuO6+x. The ratio of the phase correlation length ξCDW to the orientation correlation length ξorient points to unidirectional charge modulations. By computing new critical exponents at free surfaces including that of the pair connectivity correlation function, we show that these locally 1D charge modulations are actually a bulk effect resulting from classical 3D criticality of the random field Ising model throughout the entire superconducting doping range.