# Structure of Charge Density Waves in La1.875Ba0.125CuO4

J. Sears  Y. Shen  Department of Condensed Matter Physics and Materials
Science, Brookhaven National Laboratory, Upton, New York 11973, USA  M. J.
Krogstad  Materials Science Division, Argonne National Laboratory, Lemont, IL
60439, USA  Advanced Photon Source, Argonne National Laboratory, Argonne,
Illinois 60439, USA  H. Miao  Department of Condensed Matter Physics and
Materials Science, Brookhaven National Laboratory, Upton, New York 11973, USA
Material Science and Technology Division, Oak Ridge National Laboratory, Oak
Ridge, Tennessee 37830, USA  E. S. Bozin  Department of Condensed Matter
Physics and Materials Science, Brookhaven National Laboratory, Upton, New York
11973, USA  I. K. Robinson  Department of Condensed Matter Physics and
Materials Science, Brookhaven National Laboratory, Upton, New York 11973, USA
London Centre for Nanotechnology, University College, Gower St., London WC1E
6BT, UK  G. D. Gu  Department of Condensed Matter Physics and Materials
Science, Brookhaven National Laboratory, Upton, New York 11973, USA  R. Osborn
S. Rosenkranz  Materials Science Division, Argonne National Laboratory,
Lemont, IL 60439, USA  J. M. Tranquada  M. P. M. Dean<EMAIL_ADDRESS>Department of Condensed Matter Physics and Materials Science, Brookhaven
National Laboratory, Upton, New York 11973, USA

###### Abstract

Although charge-density wave (CDW) correlations exist in several families of
cuprate superconductors, they exhibit substantial variation in CDW wavevector
and correlation length, indicating a key role for CDW-lattice interactions. We
investigated this interaction in La1.875Ba0.125CuO4 using single crystal x-ray
diffraction to collect a large number of CDW peak intensities, and determined
the Cu and La/Ba atomic distortions induced by the formation of CDW order.
Within the CuO2 planes, the distortions involve a periodic modulation of the
Cu-Cu spacing along the direction of the ordering wave vector. The charge
ordering within the copper-oxygen layer induces an out-of-plane breathing
modulation of the surrounding lanthanum layers, which leads to a related
distortion on the adjacent copper-oxygen layer. Our result implies that the
CDW-related structural distortions do not remain confined to a single layer
but rather propagate an appreciable distance through the crystal. This leads
to overlapping structural modulations, in which CuO2 planes exhibit
distortions arising from the orthogonal CDWs in adjacent layers as well as
distortions from the CDW within the layer itself. We attribute this striking
effect to the weak $c$-axis charge screening in cuprates and suggest this
effect could help couple the CDW between adjacent planes in the crystal.

##  I Introduction

Although the nature of superconductivity in cuprates remains controversial,
there is increasing evidence that charge and spin correlations might play an
important role [1, 2, 3, 4]. Experimental studies have shown that CDW
correlations are ubiquitous in underdoped cuprates [5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15], while modern numerical calculations suggest that CDWs are an
intrinsic feature of minimal model Hamiltonians relevant to these materials
[16, 17, 18, 19]. Cuprates such as La2-xBaxCuO4 are composed of two-
dimensional CuO2 layers which support CDW order, separated by electronically
inert buffer layers. While the CDW is primarily an electronic phenomenon, the
rearrangements of charge associated with the CDW necessarily involves atomic
displacements, which can play a substantial role in stabilizing the order. For
example, measurements of phonons show a strong coupling between CDWs and the
in-plane Cu-O bond stretching phonon mode [20, 21, 22, 23, 24, 25, 26].
Furthermore in La2-xBaxCuO4, the material of interest for this study, static
CDW order only develops below the low-temperature tetragonal (LTT) structural
phase transition [27, 5, 28, 29, 30, 31]. Within this phase, the O atoms along
one of the Cu-O bond directions are displaced out of the plane (in alternating
directions), while in the orthogonal direction the O atoms remain within the
CuO2 planes. This broken rotational symmetry has long been believed to pin CDW
order [32], leading to a short-range CDW-structure interaction that rotates
$90^{\circ}{}$ from plane-to-plane due to the screw axis symmetry.
La2-xSrxCuO4, which lacks the LTT structural phase, supports a CDW order that
is far weaker than that of La2-xBaxCuO4 [11, 33]. Further evidence of CDW-
structure coupling is seen in the temperature-dependence of CDW-domain
formation, which demonstrates that the CDW domains are pinned by the
structural domain walls [34]. In this work, we use x-ray scattering to
determine the atomic displacements associated with the CDW.

Advances in synchrotron x-ray instrumentation have made it practical to
measure superlattice peaks over a large volume of reciprocal space [35]. In
the present paper, we exploit these developments to collect an extensive set
of diffraction data from the CDW Bragg peaks in La1.875Ba0.125CuO4. From a
fitting analysis, we are able to determine the CDW-induced displacements of
the Cu and La atoms. Despite the participation of O atoms in the CDW, we do
not comment on O displacements, due to the low sensitivity of x-rays to O
atoms. We find that charge order induces a longitudinal modulation of Cu-Cu
spacings within the CuO2 plane. We also find that the La atoms show
substantial displacement along the $c$ axis, contributing to screening of the
charge order. Remarkably, the modulation of the La layers leads to a similar
modulation in the nearest-neighbor CuO2 layer (transverse to the direction of
its CDW wave vector). We attribute these long-range CDW effects to the weak
$c$-axis charge screening in cuprates. The propagation of the distortions
through the crystal helps to communicate the stripe orientation between the
next-nearest-neighbor layers. Lastly, we find evidence that the CDW modulation
is aligned along the direction of the tilted in-plane Cu-O bonds.

##  II Data Collection

La1.875Ba0.125CuO4 single crystal samples were grown at Brookhaven National
Laboratory using the floating zone method. These crystals are from the same
source that was used previously [5] and exhibited a consistent CDW transition
temperature. The samples were cut with a razor blade and a piece approximately
200 µm in all directions was selected for the measurement. X-ray diffraction
data were collected at the 6-ID-D beamline at the Advanced Photon Source at
the Argonne National Laboratory. The beam size was $500\times 500$ µm2 and the
x-ray energy was set to 87 keV to collect a large volume of reciprocal space
and to minimize signal distortions due to absorption. For La1.875Ba0.125CuO4
the attenuation length for 87 keV x-rays is $\sim 600$ µm. The samples were
chosen to be smaller than the beam size and isotropic in shape in order to
mitigate variation in intensity due to absorption effects or the amount of
sample in the beam. We selected a sample with a mosaic spread of less than the
$0.1^{\circ}$ experimental resolution for the data collection.

The sample was cooled using a helium cryocooler, and data were collected by
continuously rotating the sample while reading out the Pilatus 2M CdTe
detector every $0.1^{\circ}$. The images were processed by mapping each pixel
into reciprocal space and then binning the pixels into a 3-dimensional grid of
voxels. Bright Bragg peaks were masked on the raw images prior to binning to
improve the visibility of weak features such as the CDW peaks. Two further
different rotations were made at different sample tilt angles, in order to
provide comprehensive mapping of reciprocal space and to fill in unpopulated
voxels. The data shown correspond to a total measurement time of 4 hours.

Figure 1: La1.875Ba0.125CuO4 CDW measurements over a large reciprocal space
range at 28 K. (a) Extended reciprocal space data ranging from $L=10$ to
$L=20$, showing diffuse scattering from the high-temperature tetragonal
(HTT)structure (at integer positions) and low-temperature tetragonal
(LTT)peaks at ($\frac{1}{2}$, $\frac{1}{2}$, $L$) positions. (b) $(H0L)$
reciprocal space map showing Bragg peaks and CDW signal appearing as diffuse
rods located at $\pm 0.24$ to either side of the Bragg peaks. (c) The region
indicated by the white rectangle in panel (b) summed over the $K$ and $L$
directions. The background was fit with a quadratic function over a small
region surrounding the known peak position. Four voxels covering 0.08
reciprocal lattice units (RLU)were summed over to encompass the CDW peak
width, as illustrated by the vertical red dashed lines. The points within this
region were summed to give the CDW peak intensity. Error bars shown are the
square root of the total photon counts. The faint streaks that form along
random reciprocal space directions are detector artifacts as described in the
text.

Figure 1(a) illustrates the large extent of reciprocal space measured, and
shows approximately one-sixth of the full dataset. The data were collected at
28 K, well below the simultaneous LTT structural phase transition and CDW
ordering at 54 K. Throughout this work we will refer to reciprocal space
positions using reciprocal lattice units (RLU), in terms of the unit cell of
the high-temperature tetragonal (HTT) phase with $a=b=3.78$ Å and $c=13.19$ Å.
We use this notation to maintain consistency with literature, even though
La1.875Ba0.125CuO4 goes through two structural transitions with decreasing
temperature, entering the low-temperature orthorhombic (LTO) phase at 240 K
and the low-temperature tetragonal (LTT)phase at 54 K. The unit cell of the
LTO and LTT phases is larger and rotated by $\sim 45^{\circ}$, leading to
superlattice peaks at $(\frac{1}{2},\frac{1}{2},L)$ positions. These
superlattice peaks are visible in the horizontal cuts of the reciprocal space
shown in Fig. 1(a).

Figure 2:  Comparison between measured CDW diffraction intensity and
structural models with modulation of either one or both formula unit layers of
the unit cell. (a) Plots of the $K=0$ diffraction planes. The left panel shows
the experimental data. Blank areas in the diffraction planes denote areas
where a CDW intensity could not be definitely extracted. The middle and right
panels shows the calculated peak intensities for the $Bbmb$ space group of
irreducible representation (irrep)S1 using modulations of either one or both
planes in the crystal structure (one-layer and two-layer models respectively).
In all panels, the radius is proportional to the intensity. (b) and (c)
Measured vs. calculated intensity for the best fit values of the model
parameters. The dashed red line traces the ideal case of the measured and
calculated intensity being equal. The better fit in panel (b) proves that the
CDW induces distortions in both planes of the La1.875Ba0.125CuO4 unit cell.
For readability, the errorbars are shown for only 1% of the data points.

The data also show faint but clearly visible CDW signal at wavevectors of
$\vec{q}=\pm(0.24,0,0.5)$ and $\vec{q}=\pm(0,0.24,0.5)$ surrounding the
structural Bragg peaks, as shown in Fig. 1(b). The CDW peaks are extended and
rod-like in the $L$ direction, and are centered at half-integer $L$ positions.
These are well-known features of the CDW in La1.875Ba0.125CuO4 — all
detectable CDW peaks are of the type $\vec{q}=(\pm 0.24,0,\pm 0.5)$ or
$\vec{q}=(0,\pm 0.24,\pm 0.5)$ [36, 28, 5]. The rod-like structure is due to
the short correlation length (a few unit cells) in the out-of-plane direction,
while the half-integer $L$ peak position is a result of the out-of-phase
stacking of the charge pattern between adjacent unit cells. While the low
resolution of our measurement precludes a precise estimate of the CDW
correlation lengths, our observations are consistent with the previously
reported anisotropic correlation lengths of approximately 200 Å in-plane and a
few unit cells out of plane.

The intensities of the CDW peaks were extracted from this data set by summing
over each four-voxel-wide region surrounding the peaks and subtracting the
background as shown in Fig. 1(c). Despite masking of bright Bragg peaks, some
streaks of intensity remain in the data due to ‘blooming’ from saturation of
the detector. The reciprocal space map was manually inspected in the region of
each peak to check for bright streaks at the CDW peak position, and
contaminated peaks were excluded from the data set. The error in the peak
intensity was estimated as the square root of the intensity. Due to twinning,
and lattice symmetry, the CDW produces peaks along both $H$ and $K$, so for
simplicity we can choose to focus on either $H$ and $K$ without losing any
information. For the analysis, we chose to analyze peaks with the CDW along
$H$ i.e. peaks with $H\approx n\pm 1/4$ where $n$ is an integer. The data were
also symmetrized by reflection in the $K=0$ mirror plane.

##  III Structural Model and Fitting

While there are CDW modulations in two orthogonal in-plane directions, there
is no evidence for significant coupling between these modulations. This is
consistent with the CDW orientation being tied to the in-plane structural
anisotropy of the LTT structure. Hence, we will analyze the modulation for
just one direction, assuming that equivalent results apply for the second. We
emphasize that the full crystal structure will necessarily include distortions
due to both CDW components. Since no higher harmonic CDW peaks have ever been
observed, the structural models were constructed assuming simple sinusoidal
modulations of the atomic positions.

The software Isodistort [37, 38] was used in our analysis to identify CDW
component structures compatible with the different types of symmetry breaking
that may occur in the LTT crystal structure. In the analysis, we simulated our
observed set of CDW Bragg peaks assuming a commensurate wave vector
$\vec{q}=(0.25,0,0.5)$ from the CDW domains that lie along $H$, and identified
four possible irreducible representations (irreps) of the parent $P4_{2}/ncm$
space group labeled S1, S2, S3, and S4. These irreps denote the different ways
that the distortion can break the crystal symmetry, grouping together
distortion modes that maintain a given symmetry operation. S1 and S3 both
maintain a vertical mirror plane symmetry which is broken in S2 and S4. We
will show later that the data support the preservation of the vertical mirror
plane. S1 and S3 differ in the relative orientation of the CDW modulation with
respect to the bent in-plane Cu-O bonds within the LTT phase. Structural
refinements, which we will also outline later, support the S1 CDW component
structure, which has the CDW modulation parallel to the bent in-plane Cu-O
bonds. Isodistort files describing the atomic displacements involved in the
different symmetry modes are provided in the Supplemental Material [39].

The individual distortion modes are sinusoidal modulations of atomic positions
along the direction of the ordering vector. Phase transitions often involve
only a single irreducible representation, so each of these possible CDW
component structures was refined individually against the experimental data.
Each of these models includes only a subset of the free parameters required to
describe the atomic positions within a suitable supercell, greatly simplifying
the fitting procedure. The distortion mode model of Isodistort was converted
to the crystal structure, which was used to compute the CDW peak intensities
for comparison with the experimental data. The atomic form factors provided by
Waasmaier _et al_. [40] were used. The overall scale factor for the data was
determined by modeling the intensity of 240 weak Bragg peaks (using the known
LTT structure) and comparing with the observed intensity. A commensurate
ordering vector was used for the symmetry analysis, despite the fact that the
actual wave vector of the scattering is slightly incommensurate. It is thought
that this shifted wave vector arises from a mixture of locally commensurate
charge stripe structures [41, 31], and we will show later that the measured
diffraction patterns are minimally sensitive to the phase of the CDW with
respect to the underlying lattice.

We also note that since the CDW exists within the LTT structure, symmetry
dictates that additional CDW peaks of the form $\vec{q}=(0.5\pm 0.24,0.5,0.5)$
should exist. These are indeed present in our model, but with an intensity
several orders of magnitude weaker than the $\vec{q}=(0.24,0,0.5)$ peaks,
consistent with the absence of observable peaks of this type in the
experimental data.

The CDW component structural models of irreps S2 and S4 were found to predict
very little intensity in the $K=0$ plane. Our experimental data show that the
brightest peaks lie in this plane, and so these irreps could be immediately
discarded. S1 and S3 each allow a number of possible distorted structures, and
the two highest symmetry structures for each irreps were refined against the
experimental data for a total of four candidate structures. For both S1 and
S3, the two highest symmetry distortions had space groups $Bbmb$ and $Bmmb$.
Although these space groups are both orthorhombic, we note that this is a
consequence of including only one of the two CDW domains, and does not reflect
the symmetry of the full crystal structure (which will include both CDW
ordering vectors). We do not observe any difference between the $a$ and $b$
lattice constants within the experimental resolution, however, we note that
our measurement is not sensitive to small changes in lattice constants.

Figure 3:  The two refined CDW component structural distortions found to best
fit the data corresponding to the $Bbmb$ and $Bmmb$ space groups of the S1
irreps. These structures show the modulation of the atomic positions away from
the flat planes of the HTT structure due to the CDW propagating horizontally
in Layer 1, as denoted by $\leftrightarrow$. Layer 2 also hosts a CDW
propagating into the page. For clarity this is not shown, but it is denoted by
$\otimes$. The modulations result in a $4\times 2\times 2$ supercell compared
with the HTT structure, as shown by the black outlines. The CDW order causes
in-plane modulation of the copper positions on the central plane (blue solid
line), and a breathing type distortion is transmitted to the surrounding
layers of atoms (modulation shown by dashed lines). Remarkably, the distortion
involves not only Layer 1, which hosts the CDW, but also the adjacent Layer 2.
Based on the observed distortion, the probable pattern of hole density
modulation is illustrated by the orange shading. The two refined distortions
are very similar, differing only in the phase of the modulation with respect
to the crystal structure. The size of the distortion has been increased by a
factor of 100 for visibility, and the structures were plotted using Vesta
[42].

Initial modelling using each of the candidate CDW component structures was
done allowing all of the La and Cu $x$ and $z$ modulation amplitudes to vary.
Along the reciprocal space $K$ direction the peak intensities decrease
rapidly, suggesting that the distortions in the $y$ direction perpendicular to
the CDW are small compared to displacements along $x$ and $z$. Attempts to
refine the $y$ displacements did not lead to statistically significant
improvements to the fit so these modulations were therefore fixed to zero.
Several of the refined modulations were found to be small in comparison to
their computed errors. These values were also fixed to zero, resulting in
minimal models with only 6 free parameters for each of the candidate
structures. Adding further parameters to these models did not substantially
improve the goodness of fit. In particular, we performed fits in which the
La/Ba atoms were fixed and the Cu and O were free, but these failed to
reproduce the data. Fits in which all atoms (La/Ba, Cu, and O) were free
resulted in only minor improvements in the fit considering the large number of
additional parameters added. This is not surprising considering that the ratio
of atomic numbers of La to O is 57:8, so La contributes $(57/8)^{2}\approx 51$
times more strongly to the x-ray scattering for a similar displacement. The
oxygen distortions were therefore not included in the final refinement.

##  IV Results and Discussion

The unit cell of La2-xBaxCuO4 contains two La1.875Ba0.125CuO4 formula units,
related at low temperature by a screw axis parallel to the crystal $c$ axis.
Each formula unit layer consists of a CuO2 plane and its surrounding lanthanum
atoms. Throughout this discussion, we will refer to layers with the CDW
ordering wave vector along the direction of the bent Cu-O-Cu bonds as ‘Layer
1’. The remaining formula unit layers, with the wave vector along the straight
Cu-O-Cu bonds, will be referred to as ‘Layer 2’. Fig. 3 shows how the crystal
structure is constructed out of alternating layers 1 and 2. These two layer
types are thought to account for the two CDW domains, with the ordering vector
in adjacent layers rotated by 90∘, consistent with the screw axis symmetry.
The data chosen consists of peaks from only one of the two domains, and so we
initially considered structural models in which the CDW induced distortion was
confined to a single layer type. Regardless of the structural model used,
allowing only a single layer to distort produced only very poor fits to the
data, and failed to reproduce the qualitative features seen in the diffraction
pattern as shown in Fig. 2.

Good fits to the data could only be achieved by allowing both layers to
distort. When this was done, the qualitative features of the diffraction
pattern could be reproduced with all four of the remaining CDW component
structures: space groups $Bbmb$ and $Bmmb$ for each of the S1 and S3 irreps.
In all cases the $\chi^{2}$ values were $\sim 1$, however the fits for the S3
structures were slightly worse ($\chi^{2}=1.06$ for both $Bbmb$ and $Bmmb$) in
comparison with S1 ($\chi^{2}=1.02$ for both $Bbmb$ and $Bmmb$). We note that
the S1 and S3 CDW component structures all have in common a vertical mirror
plane including the ordering vector and the $c$ axis ($ac$ plane in the HTT
notation). This mirror plane is also compatible with a truly incommensurate
structure, as it leaves the ordering wave vector fixed. In contrast, a
symmetry operation (such as a $bc$ mirror plane) which transforms the ordering
vector $q$ to $-q$ cannot be simply translated into the incommensurate case.

The refined structures for S1 and S3 are very similar, the only difference
being which copper-oxygen plane hosts the charge order. In the S1 structures
Layer 1 hosts charge order, with the CDW ordering vector along the direction
of the bent copper-oxygen bonds. In the S3 structures Layer 2 hosts the charge
order, and the ordering vector is along the straight copper-oxygen bonds. The
reason for the similarity in the fits is that the two layers have only small
structural differences. Nevertheless, the slight difference in the fit quality
suggests that the charge ordering vector is in the direction of the bent
copper-oxygen-copper bonds. The refined structures were very similar, and from
this point forward we will focus on the S1 structures.

The refined distortions are on the order of $10^{-3}$ Å, and are given in
Table 1. These values are based on the assumption that the CDW occupies 100%
of the crystal. If the phase fraction is smaller, then the distortions would
be rescaled accordingly. Measurements suggest that static charge order
occupies the majority of the sample volume, albeit with significant disorder
[43, 44, 45, 46]. Another source of uncertainty is in the overall scaling of
the CDW distortions. Due to the small CDW peak intensities, they all scale as
the square of the displacements, so this scale factor cannot be independently
fit with just the CDW peaks. We determined this scaling by comparing with
structural Bragg peaks, which introduces possible uncertainty from self-
absorption and multiple-scattering effects. Our use of a sample that is
significantly smaller than the x-ray attenuation length means that this
uncertainty is smaller than the uncertainty coming from the domain population.
The relative sizes of the different distortion modes are not affected by these
sources of uncertainty.

Figure 2(a) compares the measured diffraction pattern with the intensities
predicted by the higher symmetry (space group $Bbmb$) CDW component structure
of the S1 irrep. Plots of the measured intensity against the calculated
intensity are provided in Fig. 2(b), such that a hypothetical perfect fit
would have points that fall on the red dotted line. We see in Fig. 2(b) that
the fit quality of the model with distortions in both copper-oxygen layers
(left panel) is significantly better than case with distortions in just one
layer (right panel). The two-layer model is thus able to fit 3500 peak
intensities using only 6 free parameters within a scatter comparable to the
experimental errorbar. The very similar structure of the S1 irrep with $Bmmb$,
rather than $Bbmb$, space group symmetry gives a similar fit to the data.
Examining the refined structures (shown in Fig. 3) suggests the probable
reason for this similarity. The $Bbmb$ and $Bmmb$ symmetry structures of the
S1 irrep differ only in the phase of the distortion with respect to the
crystal structure. Based on the equal goodness of fit, we conclude that this
phase cannot be determined based on our data set.

Table 1: Maximum CDW-related atomic displacements, with refined values for the $Bbmb$ and $Bmmb$ CDW component structures of irrep S1. The $4\times 2\times 2$ HTT supercell lattice parameters are $a=15.12$ Å, $b=7.56$ Å, $c=26.38$ Å. The estimated error values in parenthesis are the square root of the diagonal elements of the estimated covariance matrix, as derived from the output of the Scipy leastsq optimization function. Isodistort output files describing the distortion modes of these structures are provided as Supplemental Material [39].  Distortion Mode | $Bbmb$ | $Bmmb$
---|---|---
| parameters | parameters
| (Å$\times 10^{-3}$) | (Å$\times 10^{-3}$)
A’_1(a) (Layer 2, La x) | 0.19(1) | 0.13(1)
A’_2(a) (Layer 2, La x) | - | -
A’_3(a) (Layer 1, La y) | - | -
A’_4(a) (Layer 2, La z) | - | -
A’_5(a) (Layer 2, La z) | 0.68(1) | 0.494(8)
A’_6(a) (Layer 1, La z) | 1.356(6) | 1.843(8)
A”(a) (Layer 1, La x) | 0.33(1) | 0.49(2)
Bu_1(a) (Layer 1, Cu x) | 2.5(1) | 1.74(8)
Bu_2(a) (Layer 2, Cu y) | - | -
Bu_3(a) (Layer 1, Cu z) | - | -
Bu_4(a) (Layer 2, Cu z) | 1.42(4) | 1.95(6)

Our modeling shows that the CDW in La1.875Ba0.125CuO4 induces a longitudinal
modulation of the copper atoms of one of the copper-oxygen planes (Layer 1),
likely due to modulation of the hole population within this layer. The charge
modulation also leads to a breathing-type distortion in the layers of
lanthanum atoms immediately above and below this plane. The breathing
distortion additionally propagates further into the crystal and affects
adjacent layers. This finding is remarkable, since the adjacent copper-oxygen
layers are expected to support their own charge density wave modulations,
rotated by $90^{\circ}$ about the $c$ axis. The propagation of the distortion
into these adjacent layers means that in regions of the crystal where the two
CDW domains coexist, the structure is described by two wave vectors rather
than the simple 1D modulation usually proposed. If the modulation along the
charge-stripe direction couples to the charge, then, given a $4a$
commensurability in two directions at $x=1/8$, it might underly the doping
dependence of charge order and superconductivity in La2-xBaxCuO4 [47, 48] and
La${}_{2-x-y}R_{y}$SrxCuO4 with $R=$ Nd [49, 50] and Eu [6, 51] in which the
wavevector is essentially independent of doping for dopings above 1/8.

Our results can be compared with previous x-ray diffraction measurements [36],
which used a structural model of distortions in a single layer to fit a far
smaller data set of 70 CDW peak intensities. The authors found that the most
important distortions were the longitudinal modulation of copper atoms due to
the charge modulation in the copper-oxygen plane, and the modulation of
lanthanum atoms along the $c$ axis in response. Our results show that these
modulations are indeed significant, but that the breathing modulation also
propagates into the adjacent layers.

It is also instructive to compare the distortion in La1.875Ba0.125CuO4 to the
previously solved CDW structural distortion in YBa2Cu3O6+x [52]. The first
notable difference is that the copper-oxygen chain layers of YBa2Cu3O6+x are
completely undistorted; in contrast La1.875Ba0.125CuO4 hosts modulations that
propagate through the crystal structure leading to modulation in all layers.
The distortion of the copper-oxygen square planes also differs between the two
materials. Rather than the longitudinal distortions found in the copper-oxygen
planes of La1.875Ba0.125CuO4, the distortions within the YBa2Cu3O6+x square
layers are mostly along the out-of plane $c$ direction. Despite these
differences, the overall magnitude of the distortions are comparable in both
materials. These results highlight the importance of the structural
environment and its interaction with electronic order in the copper-oxygen
planes. Of course, the zero-field CDW order in YBa2Cu3O6+x has a very short
in-plane coherence length and is symmetric about a Cu-O chain layer. It would
be interesting to compare with the high-field 3D CDW order in YBa2Cu3O6+x [53,
54], but this is more challenging to measure.

Since the CDW in cuprates is driven by interactions within the CuO2 planes,
our detection of CDW-induced $c$ axis displacements of Cu and La might be
considered counter-intuitive. However, the cuprates are known to have very
weak electronic screening of Coulomb interactions along the $c$-axis. This is
seen in the giant $c$-axis lattice constant change upon photoexcitation [55,
56] and the large oscillator strength of several $c$-axis-polarized phonon
modes in optical conductivity and Raman measurements [57, 58, 59, 60],
including some that involve Cu and La motions [61]. Inelastic x-ray scattering
measurements of La1.875Ba0.125CuO4 also observe a substantial CDW-induced
softening of $c$-axis longitudinal acoustic phonons [30]. Prior neutron pair
distribution function measurements also associated the CDW with $c$-axis
displacements [62]. It is further worth noting that each La3+ ion sits just
above (or below) a square of four in-plane O2- ions, so displacing La atoms
involves a large modification in the $c$-axis Coulomb interactions. These
facts make our observed long-range displacements quite natural.

##  V Conclusions

Based on a comprehensive measurement of several thousand CDW peaks, we have
determined the atomic displacements associated with CDW order in
La1.875Ba0.125CuO4. The structural distortion involves a periodic modulation
of the copper atoms along the direction of the CDW and an out-of-plane
breathing modulation of the surrounding lanthanum layers. Despite the
widespread belief that CDW-related distortions should rotate by $90^{\circ}$
from layer-to-layer, this breathing modulation, in fact, propagates further
through the crystal and induces a similar distortion on the adjacent copper-
oxygen layers. We conclude that the effects of charge ordering cannot be
considered to be confined to a single plane, but manifest as a complex longer-
range distortion that should be described by two wave vectors rather than the
simple 1D modulation usually proposed. We attribute this surprising effect to
the weak $c$-axis charge screening in cuprates.

The supporting data for the plots in this article are openly available from
the Zenodo database [63].

###### Acknowledgements.

We thank C. Mazzoli for discussions and D. Robinson for assistance with x-ray
scattering experiments. Work at Brookhaven is supported by the Office of Basic
Energy Sciences, Materials Sciences and Engineering Division, U.S. Department
of Energy (DOE) under Contract No. DE-SC0012704. Work at Argonne National
Laboratory (MJK, RO, SR, single crystal diffuse scattering measurements and
data reduction) was supported by the U.S. Department of Energy, Office of
Science, Basic Energy Sciences, Materials Sciences and Engineering Division.
This research used resources of the Advanced Photon Source, a U.S. Department
of Energy (DOE) Office of Science User Facility operated for the DOE Office of
Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357.

## References

  * Kivelson  _et al._ [2003] S. A. Kivelson, I. P. Bindloss, E. Fradkin, V. Oganesyan, J. M. Tranquada, A. Kapitulnik, and C. Howald, How to detect fluctuating stripes in the high-temperature superconductors, Rev. Mod. Phys. 75, 1201 (2003).
  * Fujita  _et al._ [2012] M. Fujita, H. Hiraka, M. Matsuda, M. Matsuura, J. M. Tranquada, S. Wakimoto, G. Xu, and K. Yamada, Progress in Neutron Scattering Studies of Spin Excitations in High-$T_{c}$ Cuprates, J. Phys. Soc. Jpn. 81, 011007 (2012).
  * Fradkin  _et al._ [2015] E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Colloquium : Theory of intertwined orders in high temperature superconductors, Rev. Mod. Phys. 87, 457 (2015).
  * Keimer  _et al._ [2015] B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, From quantum matter to high-temperature superconductivity in copper oxides, Nature 518, 179 (2015).
  * Hücker  _et al._ [2011a] M. Hücker, M. v. Zimmermann, G. D. Gu, Z. J. Xu, J. S. Wen, G. Xu, H. J. Kang, A. Zheludev, and J. M. Tranquada, Stripe order in superconducting La2-xBaxCuO4 ($0.095\leq{}x\leq{}0.155$), Phys. Rev. B 83, 104506 (2011a).
  * Fink  _et al._ [2011] J. Fink, V. Soltwisch, J. Geck, E. Schierle, E. Weschke, and B. Büchner, Phase diagram of charge order in La1.8-xEu0.2SrxCuO4 from resonant soft x-ray diffraction, Phys. Rev. B 83, 092503 (2011).
  * Gupta  _et al._ [2021] N. K. Gupta, C. McMahon, R. Sutarto, T. Shi, R. Gong, H. I. Wei, K. M. Shen, F. He, Q. Ma, M. Dragomir, B. D. Gaulin, and D. G. Hawthorn, Vanishing nematic order beyond the pseudogap phase in overdoped cuprate superconductors, Proc. Natl. Acad. Sci. USA 118, 10.1073/pnas.2106881118 (2021).
  * Ghiringhelli  _et al._ [2012] G. Ghiringhelli, M. Le Tacon, M. Minola, S. Blanco-Canosa, C. Mazzoli, N. B. Brookes, G. M. De Luca, A. Frano, D. G. Hawthorn, F. He, T. Loew, M. M. Sala, D. C. Peets, M. Salluzzo, E. Schierle, R. Sutarto, G. A. Sawatzky, E. Weschke, B. Keimer, and L. Braicovich, Long-Range Incommensurate Charge Fluctuations in (Y,Nd)Ba2Cu3O6+x, Science 337, 821 (2012).
  * Hücker  _et al._ [2014] M. Hücker, N. B. Christensen, A. T. Holmes, E. Blackburn, E. M. Forgan, R. Liang, D. A. Bonn, W. N. Hardy, O. Gutowski, M. v. Zimmermann, S. M. Hayden, and J. Chang, Competing charge, spin, and superconducting orders in underdoped ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{y}$, Phys. Rev. B 90, 054514 (2014).
  * da Silva Neto  _et al._ [2014] E. H. da Silva Neto, P. Aynajian, A. Frano, R. Comin, E. Schierle, E. Weschke, A. Gyenis, J. Wen, J. Schneeloch, Z. Xu, S. Ono, G. Gu, M. Le Tacon, and A. Yazdani, Ubiquitous Interplay Between Charge Ordering and High-Temperature Superconductivity in Cuprates, Science 343, 393 (2014).
  * Thampy  _et al._ [2014] V. Thampy, M. P. M. Dean, N. B. Christensen, L. Steinke, Z. Islam, M. Oda, M. Ido, N. Momono, S. B. Wilkins, and J. P. Hill, Rotated stripe order and its competition with superconductivity in ${\mathrm{La}}_{1.88}{\mathrm{Sr}}_{0.12}{\mathrm{CuO}}_{4}$, Phys. Rev. B 90, 100510 (2014).
  * Comin  _et al._ [2014] R. Comin, A. Frano, M. M. Yee, Y. Yoshida, H. Eisaki, E. Schierle, E. Weschke, R. Sutarto, F. He, A. Soumyanarayanan, Y. He, M. L. Tacon, I. S. Elfimov, J. E. Hoffman, G. A. Sawatzky, B. Keimer, and A. Damascelli, Charge Order Driven by Fermi-Arc Instability in Bi2Sr2-xLaxCuO6+δ, Science 343, 390 (2014).
  * Tabis  _et al._ [2017] W. Tabis, B. Yu, I. Bialo, M. Bluschke, T. Kolodziej, A. Kozlowski, E. Blackburn, K. Sen, E. M. Forgan, M. v. Zimmermann, Y. Tang, E. Weschke, B. Vignolle, M. Hepting, H. Gretarsson, R. Sutarto, F. He, M. Le Tacon, N. Barišić, G. Yu, and M. Greven, Synchrotron x-ray scattering study of charge-density-wave order in ${\mathrm{HgBa}}_{2}{\mathrm{CuO}}_{4+\delta}$, Phys. Rev. B 96, 134510 (2017).
  * Kohsaka  _et al._ [2007] Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, An Intrinsic Bond-Centered Electronic Glass with Unidirectional Domains in Underdoped Cuprates, Science 315, 1380 (2007).
  * Webb  _et al._ [2019] T. A. Webb, M. C. Boyer, Y. Yin, D. Chowdhury, Y. He, T. Kondo, T. Takeuchi, H. Ikuta, E. W. Hudson, J. E. Hoffman, and M. H. Hamidian, Density Wave Probes Cuprate Quantum Phase Transition, Phys. Rev. X 9, 021021 (2019).
  * Zheng  _et al._ [2017] B.-X. Zheng, C.-M. Chung, P. Corboz, G. Ehlers, M.-P. Qin, R. M. Noack, H. Shi, S. R. White, S. Zhang, and G. K.-L. Chan, Stripe order in the underdoped region of the two-dimensional hubbard model, Science 358, 1155 (2017).
  * Huang  _et al._ [2017] E. W. Huang, C. B. Mendl, S. Liu, S. Johnston, H.-C. Jiang, B. Moritz, and T. P. Devereaux, Numerical evidence of fluctuating stripes in the normal state of high-$T_{c}$ cuprate superconductors, Science 358, 1161 (2017).
  * Huang  _et al._ [2018] E. W. Huang, C. B. Mendl, H.-C. Jiang, B. Moritz, and T. P. Devereaux, Stripe order from the perspective of the Hubbard model, npj Quantum Mater. 3, 22 (2018).
  * Jiang  _et al._ [2021] S. Jiang, D. J. Scalapino, and S. R. White, Ground-state phase diagram of the $t$-$t^{\prime}$-$J$ model, Proc. Natl. Acad. Sci. U.S.A. 118, e2109978118 (2021).
  * Pintschovius  _et al._ [2004] L. Pintschovius, D. Reznik, W. Reichardt, Y. Endoh, H. Hiraka, J. M. Tranquada, H. Uchiyama, T. Masui, and S. Tajima, Oxygen phonon branches in YBa2Cu3O7, Phys. Rev. B 69, 214506 (2004).
  * Reznik  _et al._ [2006] D. Reznik, L. Pintschovius, M. Ito, S. Iikubo, M. Sato, H. Goka, M. Fujita, K. Yamada, G. D. Gu, and J. M. Tranquada, Electron–phonon coupling reflecting dynamic charge inhomogeneity in copper-oxide superconductors, Nature 440, 1170 (2006).
  * Tejsner  _et al._ [2020] T. Tejsner, A. Piovano, A. Ţuţueanu, A. T. Rømer, B. O. Wells, J.-C. Grivel, M. Boehm, and L. Udby, Anomalous dispersion of longitudinal optical phonons in oxygen-doped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4+\delta}$, Phys. Rev. B 101, 100504 (2020).
  * Peng  _et al._ [2020] Y. Y. Peng, A. A. Husain, M. Mitrano, S. X.-L. Sun, T. A. Johnson, A. V. Zakrzewski, G. J. MacDougall, A. Barbour, I. Jarrige, V. Bisogni, and P. Abbamonte, Enhanced Electron-Phonon Coupling for Charge-Density-Wave Formation in ${\mathrm{La}}_{1.8-x}{\mathrm{Eu}}_{0.2}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4+\delta}$, Phys. Rev. Lett. 125, 097002 (2020).
  * Lin  _et al._ [2020] J. Q. Lin, H. Miao, D. G. Mazzone, G. D. Gu, A. Nag, A. C. Walters, M. García-Fernández, A. Barbour, J. Pelliciari, I. Jarrige, M. Oda, K. Kurosawa, N. Momono, K.-J. Zhou, V. Bisogni, X. Liu, and M. P. M. Dean, Strongly correlated charge density wave in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$ evidenced by doping-dependent phonon anomaly, Phys. Rev. Lett. 124, 207005 (2020).
  * Wang  _et al._ [2021] Q. Wang, K. von Arx, M. Horio, D. J. Mukkattukavil, J. Küspert, Y. Sassa, T. Schmitt, A. Nag, S. Pyon, T. Takayama, H. Takagi, M. Garcia-Fernandez, K.-J. Zhou, and J. Chang, Charge order lock-in by electron-phonon coupling in La1.675Eu0.2Sr0.125CuO4, Sci. Adv. 7, eabg7394 (2021).
  * Lee  _et al._ [2021] W. S. Lee, K.-J. Zhou, M. Hepting, J. Li, A. Nag, A. C. Walters, M. Garcia-Fernandez, H. C. Robarts, M. Hashimoto, H. Lu, B. Nosarzewski, D. Song, H. Eisaki, Z. X. Shen, B. Moritz, J. Zaanen, and T. P. Devereaux, Spectroscopic fingerprint of charge order melting driven by quantum fluctuations in a cuprate, Nat. Phys. 17, 53 (2021).
  * Fujita  _et al._ [2004] M. Fujita, H. Goka, K. Yamada, J. M. Tranquada, and L. P. Regnault, Stripe order, depinning, and fluctuations in ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_{4}$ and ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.075}{\mathrm{Sr}}_{0.050}{\mathrm{CuO}}_{4}$, Phys. Rev. B 70, 104517 (2004).
  * Wilkins  _et al._ [2011] S. B. Wilkins, M. P. M. Dean, J. Fink, M. Hücker, J. Geck, V. Soltwisch, E. Schierle, E. Weschke, G. Gu, S. Uchida, N. Ichikawa, J. M. Tranquada, and J. P. Hill, Comparison of stripe modulations in La1.875Ba0.125CuO4 and La1.48Nd0.4Sr0.12CuO4, Phys. Rev. B 84, 195101 (2011).
  * Miao  _et al._ [2017] H. Miao, J. Lorenzana, G. Seibold, Y. Y. Peng, A. Amorese, F. Yakhou-Harris, K. Kummer, N. B. Brookes, R. M. Konik, V. Thampy, G. D. Gu, G. Ghiringhelli, L. Braicovich, and M. P. M. Dean, High-temperature charge density wave correlations in La1.875Ba0.125CuO4 charge locking, Proceedings of the National Academy of Sciences 114, 12430 (2017).
  * Miao  _et al._ [2018] H. Miao, D. Ishikawa, R. Heid, M. Le Tacon, G. Fabbris, D. Meyers, G. D. Gu, A. Q. R. Baron, and M. P. M. Dean, Incommensurate phonon anomaly and the nature of charge density waves in cuprates, Phys. Rev. X 8, 011008 (2018).
  * Miao  _et al._ [2019] H. Miao, R. Fumagalli, M. Rossi, J. Lorenzana, G. Seibold, F. Yakhou-Harris, K. Kummer, N. B. Brookes, G. D. Gu, L. Braicovich, G. Ghiringhelli, and M. P. M. Dean, Formation of incommensurate charge density waves in cuprates, Phys. Rev. X 9, 031042 (2019).
  * Tranquada  _et al._ [1995] J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Evidence for stripe correlations of spins and holes in copper oxide superconductors, Nature 375, 561 (1995).
  * Croft  _et al._ [2014] T. P. Croft, C. Lester, M. S. Senn, A. Bombardi, and S. M. Hayden, Charge density wave fluctuations in ${\text{La}}_{2-x}$${\text{Sr}}_{x}$${\text{CuO}}_{4}$ and their competition with superconductivity, Phys. Rev. B 89, 224513 (2014).
  * Chen  _et al._ [2019] X. M. Chen, C. Mazzoli, Y. Cao, V. Thampy, A. M. Barbour, W. Hu, M. Lu, T. A. Assefa, H. Miao, G. Fabbris, G. D. Gu, J. M. Tranquada, M. P. M. Dean, S. B. Wilkins, and I. K. Robinson, Charge density wave memory in a cuprate superconductor, Nature Communications 10, 1435 (2019).
  * Krogstad  _et al._ [2020] M. J. Krogstad, S. Rosenkranz, J. M. Wozniak, G. Jennings, J. P. C. Ruff, J. T. Vaughey, and R. Osborn, Reciprocal space imaging of ionic correlations in intercalation compounds, Nat. Mater. 19, 63 (2020).
  * Kim  _et al._ [2008] Y.-J. Kim, G. D. Gu, T. Gog, and D. Casa, X-ray scattering study of charge density waves in ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_{x}{\mathrm{CuO}}_{4}$, Phys. Rev. B 77, 064520 (2008).
  * [37] H. T. Stokes, D. M. Hatch, and B. J. Campbell, Isodistort, isotropy software suite.
  * Campbell  _et al._ [2006] B. J. Campbell, H. T. Stokes, D. E. Tanner, and D. M. Hatch, ISODISPLACE: a web-based tool for exploring structural distortions, Journal of Applied Crystallography 39, 607 (2006).
  * [39] See supplemental material at [url will be inserted by publisher].
  * Waasmaier and Kirfel [1995] D. Waasmaier and A. Kirfel, New analytical scattering-factor functions for free atoms and ions, Acta Crystallographica Section A 51, 416 (1995).
  * Tranquada  _et al._ [1999] J. M. Tranquada, N. Ichikawa, and S. Uchida, Glassy nature of stripe ordering in ${\mathrm{La}}_{1.6-x}{\mathrm{Nd}}_{0.4}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$, Phys. Rev. B 59, 14712 (1999).
  * Momma and Izumi [2011] K. Momma and F. Izumi, VESTA3 for three-dimensional visualization of crystal, volumetric and morphology data, Journal of Applied Crystallography 44, 1272 (2011).
  * Savici  _et al._ [2005] A. T. Savici, A. Fukaya, I. M. Gat-Malureanu, T. Ito, P. L. Russo, Y. J. Uemura, C. R. Wiebe, P. P. Kyriakou, G. J. MacDougall, M. T. Rovers, G. M. Luke, K. M. Kojima, M. Goto, S. Uchida, R. Kadono, K. Yamada, S. Tajima, T. Masui, H. Eisaki, N. Kaneko, M. Greven, and G. D. Gu, Muon spin relaxation studies of magnetic-field-induced effects in high-t[sub c] superconductors, Phys. Rev. Lett. 95, 157001 (2005).
  * Hunt  _et al._ [2001] A. W. Hunt, P. M. Singer, A. F. Cederström, and T. Imai, Glassy slowing of stripe modulation in ${(\mathrm{L}\mathrm{a},\mathrm{E}\mathrm{u},\mathrm{N}\mathrm{d})}_{2-x}({\mathrm{S}\mathrm{r},\mathrm{B}\mathrm{a})}_{x}{\mathrm{CuO}}_{4}:\mathrm{A}{}^{63}\mathrm{Cu}$ and ${}^{139}\mathrm{La}$ NQR study down to 350 mK, Phys. Rev. B 64, 134525 (2001).
  * Chen  _et al._ [2016] X. M. Chen, V. Thampy, C. Mazzoli, A. M. Barbour, H. Miao, G. D. Gu, Y. Cao, J. M. Tranquada, M. P. M. Dean, and S. B. Wilkins, Remarkable Stability of Charge Density Wave Order in ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}{\mathrm{CuO}}_{4}$, Phys. Rev. Lett. 117, 167001 (2016).
  * Thampy  _et al._ [2017] V. Thampy, X. M. Chen, Y. Cao, C. Mazzoli, A. M. Barbour, W. Hu, H. Miao, G. Fabbris, R. D. Zhong, G. D. Gu, J. M. Tranquada, I. K. Robinson, S. B. Wilkins, and M. P. M. Dean, Static charge-density-wave order in the superconducting state of ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_{x}{\mathrm{CuO}}_{4}$, Phys. Rev. B 95, 241111 (2017).
  * Hücker  _et al._ [2011b] M. Hücker, M. v. Zimmermann, G. D. Gu, Z. J. Xu, J. S. Wen, G. Xu, H. J. Kang, A. Zheludev, and J. M. Tranquada, Stripe order in superconducting La2-xBaxCuO4 ($0.095\leq{}x\leq{}0.155$), Phys. Rev. B 83, 104506 (2011b).
  * Moodenbaugh  _et al._ [1988] A. R. Moodenbaugh, Y. Xu, M. Suenaga, T. J. Folkerts, and R. N. Shelton, Superconducting properties of ${\mathrm{La}}_{2\mathrm{-}\mathrm{x}}$${\mathrm{Ba}}_{\mathrm{x}}$${\mathrm{CuO}}_{4}$, Phys. Rev. B 38, 4596 (1988).
  * Ichikawa  _et al._ [2000] N. Ichikawa, S. Uchida, J. M. Tranquada, T. Niemöller, P. M. Gehring, S.-H. Lee, and J. R. Schneider, Local Magnetic Order vs Superconductivity in a Layered Cuprate, Phys. Rev. Lett. 85, 1738 (2000).
  * Crawford  _et al._ [1991] M. K. Crawford, R. L. Harlow, E. M. McCarron, W. E. Farneth, J. D. Axe, H. Chou, and Q. Huang, Phys. Rev. B 44, 7749 (1991).
  * Suryadijaya  _et al._ [2005] Suryadijaya, T. Sasagawa, and H. Takagi, Oxygen isotope effect on charge/spin stripes in La1.8-xEu0.2SrxCuO4, Physica C 426–431, 402 (2005).
  * Forgan  _et al._ [2015] E. M. Forgan, E. Blackburn, A. T. Holmes, A. K. R. Briffa, J. Chang, L. Bouchenoire, S. D. Brown, R. Liang, D. Bonn, W. N. Hardy, N. B. Christensen, M. V. Zimmermann, M. Hücker, and S. M. Hayden, The microscopic structure of charge density waves in underdoped YBa2Cu3O6.54 revealed by x-ray diffraction, Nature Communications 6, 10064 (2015).
  * Gerber  _et al._ [2015] S. Gerber, H. Jang, H. Nojiri, S. Matsuzawa, H. Yasumura, D. A. Bonn, R. Liang, W. N. Hardy, Z. Islam, A. Mehta, S. Song, M. Sikorski, D. Stefanescu, Y. Feng, S. A. Kivelson, T. P. Devereaux, Z.-X. Shen, C.-C. Kao, W.-S. Lee, D. Zhu, and J.-S. Lee, Three-dimensional charge density wave order in YBa2Cu3O6.67 at high magnetic fields, Science 350, 949 (2015).
  * Chang  _et al._ [2016] J. Chang, E. Blackburn, O. Ivashko, A. T. Holmes, N. B. Christensen, M. Hücker, R. Liang, D. A. Bonn, W. N. Hardy, U. Rütt, M. v. Zimmermann, E. M. Forgan, and S. M. Hayden, Magnetic field controlled charge density wave coupling in underdoped YBa2Cu3O6+x, Nat. Commun. 7, 11494 (2016).
  * Gedik  _et al._ [2007] N. Gedik, D.-S. Yang, G. Logvenov, I. Bozovic, and A. H. Zewail, Nonequilibrium Phase Transitions in Cuprates Observed by Ultrafast Electron Crystallography, Science 316, 425 (2007).
  * Radović  _et al._ [2008] Z. Radović, N. Božović, and I. Božović, Photoinduced expansion of cuprate superconductors: Evidence of strong electron-lattice coupling, Phys. Rev. B 77, 092508 (2008).
  * Uchida  _et al._ [1996] S. Uchida, K. Tamasaku, and S. Tajima, $c$-axis optical spectra and charge dynamics in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$, Phys. Rev. B 53, 14558 (1996).
  * Homes  _et al._ [2012] C. C. Homes, M. Hücker, Q. Li, Z. J. Xu, J. S. Wen, G. D. Gu, and J. M. Tranquada, Determination of the optical properties of La2-xBaxCuO4 for several dopings, including the anomalous $x=\frac{1}{8}$ phase, Phys. Rev. B 85, 134510 (2012).
  * Sugai [1989] S. Sugai, Phonon Raman scattering in (${\mathrm{La}}_{1\mathrm{-}\mathrm{x}}$${\mathrm{Sr}}_{\mathrm{x}}$${)}_{2}$${\mathrm{CuO}}_{4}$ single crystals, Phys. Rev. B 39, 4306 (1989).
  * Gozar  _et al._ [2003] A. Gozar, B. S. Dennis, T. Siegrist, Y. Horibe, G. Blumberg, S. Komiya, and Y. Ando, Inhomogeneous ${\mathrm{CuO}}_{6}$ tilt distribution and charge-spin correlations in ${\mathrm{La}}_{2-x-y}{\mathrm{Nd}}_{y}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$ around the commensurate hole concentration, Phys. Rev. B 68, 052511 (2003).
  * Henn  _et al._ [1997] R. Henn, A. Wittlin, M. Cardona, and S. Uchida, Dynamics of the $c$-polarized infrared-active modes in La${}_{2-x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$, Phys. Rev. B 56, 6295 (1997).
  * Bozin  _et al._ [2015] E. S. Bozin, R. Zhong, K. R. Knox, G. Gu, J. P. Hill, J. M. Tranquada, and S. J. L. Billinge, Reconciliation of local and long-range tilt correlations in underdoped La2-xBaxCuO4, Phys. Rev. B 91, 054521 (2015).
  * Sears  _et al._ [2023] J. Sears, Y. Shen, M. J. Krogstad, H. Miao, E. S. Bozin, I. K. Robinson, G. D. Gu, R. Osborn, S. Rosenkranz, J. M. Tranquada, and M. P. M. Dean, Data repository for: Structure of charge density waves in La1.875Ba0.125CuO4 (2023), https://doi.org/10.5281/zenodo.7653187.