Technical Comments

Comment on “Broken translational and rotational symmetry via charge stripe order in underdoped YBa2Cu3O6+y

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Science  15 Jan 2016:
Vol. 351, Issue 6270, pp. 235
DOI: 10.1126/science.aac4454


Comin et al. (Reports, 20 March 2015, p. 1335) have interpreted their resonant x-ray scattering experiment as indicating that charge inhomogeneities in the family of high-temperature superconductors YBa2Cu3O6+y (YBCO) have the character of one-dimensional stripes rather than two-dimensional checkerboards. The present Comment shows that one cannot distinguish between stripes and checkerboards on the basis of the above experiment.

Comin et al. (1) conducted a resonant x-ray scattering (RXS) experiment for three different compounds belonging to the family of high-temperature superconductors YBa2Cu3O6+y (YBCO). The experiment focused on the accurate measurements of the shapes of four charge-ordering peaks appearing in the RXS structure factor Embedded Image at positions Embedded Image and Embedded Image, where Embedded Image in the units of inverse lattice period. These peaks can originate from either two-dimensional (2D) checkerboard-like modulations of charge density or 1D stripe-like modulations. The stripe interpretation implies that the sample can be fully partitioned into regions of mutually orthogonal 1D modulations. Each such region would generate only two dominant peaks at either Embedded Image or Embedded Image. The checkerboard interpretation implies that, under any partition, there will be regions generating all four peaks with comparable intensity. Distinguishing between stripes and checkerboards was the goal of Comin et al. In other families of high-temperature superconductors, this goal proved to be notoriously difficult to achieve [see, for example, (25)]. In particular, the experimental effort of (3) and the discussion given in (4) are very reminiscent of the present case.

Comin et al. have found that the shapes of measured charge peaks are elongated in the direction perpendicular to the wave vectors defining the centers of the peaks. In their analysis, the above authors associated the finite width of the peaks with the finite size of either stripe or checkerboard domains and then, in the main text of the Report, they pointed out that, in the case of checkerboards, the shapes of individual peaks should either have the fourfold symmetry—i.e., not be elongated—or the orientation of the elongation should be different. At the same time, stripe domains could reproduce the observed elongated peak shapes. The supplementary materials, however, indicated that the difference was not so clear-cut, because “canted” checkerboard domains would reproduce the observed elongation of the charge peaks as well. Comin et al. then introduced a quantitative constraint on the canted checkerboard scenario (equation S16 of their supplementary materials) and, in supplementary table S3, showed that their experimental results violate this constraint. Finally, Comin et al. concluded that their experimental observations are incompatible with checkerboard modulations and hence indicate stripe-like modulations.

The goal of this Comment is to raise the objection to the above conclusion. The problem with the reasoning of Comin et al. is that, instead of doing the Fourier analysis directly, they adopted various oversimplifying assumptions involving rigid domains of perfectly periodic structures for both stripes and checkerboards. Adopting a domain picture amounts to an implicit assumption of a certain kind of phase coherence between different Fourier components of the charge modulation, for which, to the best of this author’s knowledge, there is no experimental evidence. At the same time, in the opposite limit of no coherence between different modulation harmonics, sufficiently narrow charge peaks, such as those observed by Comin et al., are consistent with a rather routine-looking checkerboard modulation irrespective of the shape of the peaks.

To demonstrate the above statement, let us assume Gaussian peak shapes (which will not be essential for the conclusions) and represent the four peaks in the structure factor asEmbedded Image(1)where i is the index of the peaks; Embedded Image are the positions of the centers of the peaks admitting values Embedded Image, Embedded Image, Embedded Image, and Embedded Image; Embedded Image, Embedded Image are the peak widths in the directions indicated by the subscripts; and Embedded Image are the normalization constants.

Let us choose Embedded Image and then Embedded Image, Embedded Image for peaks centered at Embedded Image and Embedded Image, and Embedded Image, Embedded Image for peaks centered at Embedded Image and Embedded Image. The resulting structure factor Embedded Image is plotted in Fig. 1A. Such a choice of parameters violates the constraint [equation S16 of (1)] on the canted checkerboards by a factor of about 2. In the framework of the assumptions adopted in (1), these four peaks cannot correspond to a checkerboard.

Fig. 1 Fluctuating checkerboard modulation of charge density.

(A) Four charge modulation peaks in the structure factor Embedded Image with parameters given in the text. (B) Density-density correlation function Embedded Image obtained as the 2D Fourier transform of Embedded Image given in (A). (C) Example of density modulation Embedded Image corresponding to Embedded Image given in (A).

Figure 1B demonstrates the 2D Fourier transforms of the above structure factor, which gives the correlation function Embedded Image, where Embedded Image describes the fluctuation of the charge density with respect to the average value, and 〈…〉 (x0, y0) denotes averaging over x0 and y0. Independently of the shape of the four narrow peaks, Embedded Image is bound to show strong checkerboard correlations of the kind appearing in Fig. 1B.

Now, to generate a possible pattern of 2D density fluctuations Embedded Image underlying the peaks in Embedded Image, let us recall that the Embedded Image also represents the square of the amplitude of a harmonic with wave numbers Embedded Image. Therefore, for the sake of producing an example, let us resolve the spectral peaks around Embedded Image and Embedded Image into two 41 x 41 grids of discrete Fourier components uniformly spanning the ranges Embedded Image and Embedded Image around the peak centers in the x and y directions, respectively. [The apparent character of the resulting function Embedded Image does not change, if a denser Embedded Image grid is used.] The density modulations in the real space are then obtained asEmbedded Image(2)where index m labels all Embedded Image discrete Fourier components participating in the expansion, Embedded Image the corresponding amplitude (up to a normalization constant), and Embedded Image is the random phase of each component.

The numerically generated function Embedded Image for one possible set of random phases Embedded Image is shown in Fig. 1C. It conveys a clear impression of a fluctuating checkerboard, in fact not much different from the results of the scanning tunneling microscopy experiments (6, 7) for other cuprate compounds. As seen in Fig. 1C, the randomness of the phases Embedded Image implies that the correlation length controlling the width of the charge modulation peaks originates from the distortions of mostly continuous superstructures rather than from the domains of perfectly periodic modulations.

To conclude, the experimental results reported in (1) represent new valuable microscopic information that has implications for both the stripe and the checkerboard scenarios of charge modulations in cuprates. As such, however, the above results do not rule out the checkerboard modulations.

References and Notes

  1. Acknowledgments: I am grateful to V. K. Bhartiya for drawing my attention to the subject of this Comment.
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