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# Using strain to uncover the interplay between two- and three-dimensional
charge density waves in high-temperature superconducting YBa2Cu3Oy

I. Vinograd  These authors contributed equally to this work  Institute for
Quantum Materials and Technologies, Karlsruhe Institute of Technology,
Kaiserstr. 12, D-76131 Karlsruhe, Germany  4th Physical Institute – Solids and
Nanostructures, University of Göttingen, D-37077 Göttingen, Germany  S. M.
Souliou  These authors contributed equally to this work  Institute for Quantum
Materials and Technologies, Karlsruhe Institute of Technology, Kaiserstr. 12,
D-76131 Karlsruhe, Germany  A.-A. Haghighirad  Institute for Quantum Materials
and Technologies, Karlsruhe Institute of Technology, Kaiserstr. 12, D-76131
Karlsruhe, Germany  T. Lacmann  Institute for Quantum Materials and
Technologies, Karlsruhe Institute of Technology, Kaiserstr. 12, D-76131
Karlsruhe, Germany  Y. Caplan  Racah Institute of Physics, The Hebrew
University, Jerusalem 91904, Israel  M. Frachet  Institute for Quantum
Materials and Technologies, Karlsruhe Institute of Technology, Kaiserstr. 12,
D-76131 Karlsruhe, Germany  M. Merz  Institute for Quantum Materials and
Technologies, Karlsruhe Institute of Technology, Kaiserstr. 12, D-76131
Karlsruhe, Germany  Karlsruhe Nano Micro Facility (KNMFi), Karlsruhe Institute
of Technology, Kaiserstr. 12, D-76131 Karlsruhe, Germany  G. Garbarino  ESRF,
The European Synchrotron, 71, avenue des Martyrs, CS 40220 F-38043 Grenoble
Cedex 9  Y. Liu  Max Planck Institute for Solid State Research,
Heisenbergstraße 1, D-70569 Stuttgart, Germany  S. Nakata  Max Planck
Institute for Solid State Research, Heisenbergstraße 1, D-70569 Stuttgart,
Germany  K. Ishida  Current address: Institute for Materials Research, Tohoku
University, Sendai 980-8577, Japan  Max Planck Institute for Chemical Physics
of Solids, Nöthnitzer Str. 40, D-01187 Dresden, Germany  H. M. L. Noad  Max
Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, D-01187
Dresden, Germany  M. Minola  Max Planck Institute for Solid State Research,
Heisenbergstraße 1, D-70569 Stuttgart, Germany  B. Keimer  Max Planck
Institute for Solid State Research, Heisenbergstraße 1, D-70569 Stuttgart,
Germany  D. Orgad  Racah Institute of Physics, The Hebrew University,
Jerusalem 91904, Israel  C. W. Hicks  Max Planck Institute for Chemical
Physics of Solids, Nöthnitzer Str. 40, D-01187 Dresden, Germany  School of
Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, UK  M.
Le Tacon  Institute for Quantum Materials and Technologies, Karlsruhe
Institute of Technology, Kaiserstr. 12, D-76131 Karlsruhe, Germany

###### Abstract

Uniaxial pressure provides an efficient approach to control the competition
between charge density waves (CDWs) and superconductivity in underdoped
YBa2Cu3Oy. It can enhance the correlation volume of ubiquitous short-range 2D
CDW correlations, and induces a long-range 3D CDW otherwise only accessible at
large magnetic fields. Here, we use x-ray diffraction to study the strain and
doping evolution of these CDWs. No signatures of discommensurations nor pair
density waves are observed in the investigated strain-temperature parameter
space, but direct evidence for a form of competition between 2D and 3D CDWs is
uncovered. We show that the interplay between the 3D CDW, the 2D CDWs and
superconductivity is qualitatively well described by including strain effects
in simulations of a nonlinear sigma model of competing superconducting and CDW
orders. From a broader perspective, our results underscore the potential of
strain tuning as a powerful tool for probing and manipulating competing orders
in quantum materials.

##  I Introduction

A tendency towards charge ordering in the underdoped high-temperature
superconducting cuprates was predicted soon after their discovery [1, 2, 3].
However, it took decades of effort with a variety of experimental methods to
demonstrate this phenomenon and its ubiquity across the cuprate
superconductors [4, 5, 6, 7, 8, 9, 10, 11, 12]. This effort has raised many
new questions. One observation that lacks a clear explanation is the fact that
the competition between charge order and superconductivity is striking in some
cuprate families [9, 10, 13], but is less pronounced in others [11, 12, 14].
Another is the very substantial variation in correlation length, which varies
over two orders of magnitude amongst the cuprates families — from a few unit
cells to several tens of nm — without any obvious relation to the
superconducting critical temperature $T_{\text{c}}$. To address these issues,
it has proved useful to focus on materials with low disorder scattering, and
to probe them with external tuning parameters, such as magnetic field [10] or
pressure [15, 16, 17], that do not introduce disorder. While hydrostatic
pressure has been widely used to investigate unconventional superconductors
and correlated electron systems, recent studies have highlighted several
benefits of uniaxial stress. In the cuprates, uniaxial stress has been used to
probe charge stripes in La-based compounds [18, 19, 20], and to polarize them
[21, 22]. In YBa2Cu3O6.67, substantial suppression of $T_{\text{c}}$ has been
achieved with uniaxial stress [23], and in contrast to the introduction of
pair-breaking impurities [24] or magnetic fields [10] (which introduce
vortices), the suppression is homogeneous. Suppressing $T_{\text{c}}$ allows
alternative electronic orders to develop, and homogeneity allows the interplay
between these orders and the superconductivity to be studied with precision.

Figure 1:  Experimental setup and representative results of x-ray scattering
under uniaxial stress. Measurements were performed in transmission geometry
using a Razorbill CS200T strain cell. The YBCO6.67 sample was mounted onto an
exchangeable titanium cross. Sections of large ($0KL$) reciprocal space maps
at zero (left) and highest $a$ axis compression (right) show the appearance of
sharp 3D CDW peaks at integer $L$. Faint 2D CDW intensity in between is
present at all strains at half-integer $L$ on top of a diffuse phonon
background. Grey arrows mark sharp Bragg reflections that appear at $K=0.5$
due to higher harmonics with $\lambda/2$. The inset shows an enlarged part of
the $(0KL)$ map. To make the 2D CDW more apparent, the colour scale is linear
for small and logarithmic for large intensities.

YBa2Cu3O6.67 is centered within a doping range where an incommensurate, short-
range, 2D-correlated charge order (hereafter referred to as the 2D CDW) is
strongest [25]. Uniaxial compression along the $a$ axis (the shortest axis of
the orthorhombic structure) has been shown to suppress $T_{\text{c}}$ by up to
30% [23], and enhance the amplitude and correlation lengths of the 2D CDW [26,
27]. Beyond a threshold stress value, a long-range-ordered, 3D-correlated CDW
(hereafter referred to as the 3D CDW) emerges [26, 27]. 3D CDW order can also
be induced by a magnetic field, though to date, with lower amplitude and
shorter correlation lengths than the one induced by uniaxial stress [8, 28,
29, 30, 31, 32]. The 2D CDW is biaxial, with components that propagate along
both the $a$ and $b$ axes, while the 3D CDW is uniaxial, propagating along the
$b$ axis only. On the other hand, it the in-plane wavelengths of the 2D and 3D
CDWs are identical [26, 29] and the analysis of the in-plane correlation
lengths revealed that the 2D CDW consists of quasi-independent unidirectional
orders [33]. This has further been confirmed by the observation of
substantially different dependencies of amplitude and correlation lengths of
its $a$ and $b$ components on uniaxial stress [27]. Overall, this suggests
that the 2D and 3D CDWs might be intimately related.

A key question remains to be clarified: does the 3D CDW emerge from “patches”
of 2D CDWs that lock together, or is it a separate order, formed by different
charge carriers, whose periodicity locks to that of the background 2D CDWs?
Additionally, so far there is little information on where the 3D CDW exists in
strain-temperature space, which is crucial information for understanding its
interaction with the superconductivity. Here, we study underdoped YBa2Cu3Oy
under uniaxial stress with synchrotron hard x-ray diffraction. We show that
the 2D CDW amplitude stops growing when the 3D CDW onsets, and map the
boundaries of the 3D CDW phase in a strain-temperature phase diagram. We
further show that the strain-induced formation of the 3D CDW and its interplay
with the 2D CDW and superconductivity are qualitatively well described by
including the effects of strain in an extension of a nonlinear sigma model
[34] previously used to investigate the magnetic-field-induced formation of
the long-range 3D order in YBCO [35, 36]. In this picture, the long-range 3D
CDW emerges from the regions between disorder-induced 2D CDW domains (i.e.,
the regions where superconductivity forms in the absence of strain). The
growth of disordered 2D CDW domains is then prevented against increased
surface tension due to phase-mismatch at the boundary with phase-ordered 3D
CDW. Finally, we have also looked for signs of discommensurations, which would
indicate a tendency of the CDW to lock to the lattice, and for pair density
wave correlations on top of the 3D CDW, but have found no evidence for either.

Figure 2:  Strain dependence of 2D and 3D CDW intensity for YBCO6.67. a) 2D
$b$-CDW intensity from $K$-cuts at half-integer $L$ values for $a$ axis
compression. $a$ axis compression amplifies the $b$-CDW only up to the onset
of 3D ordering. b) $a$ axis and $b$ axis compression dependence of the
integrated 2D CDW and 3D CDW intensities from $K$-cuts at half-integer $L$=6.5
and $L$-cuts through $q_{\rm{CDW}}$, respectively. The vertical scales of the
integrated intensities (line cuts) at $T\sim T_{c}(\varepsilon_{xx})$
correspond to the same total integrated intensity, $I_{tot}$, based on
Supplementary Table II in Supplementary Note 2. The orange dashed line marks
the onset of long-range 3D CDWs at $\varepsilon_{xx}$=-0.55%. Figure 3:
$K$-cuts of the strain-induced 3D CDW. a) YBCO6.67 $K$-cuts through the 3D CDW
peak at integer $L$, maximal compression and $T=60.5$ K $\sim T_{\rm{c}}$ show
an intense peak at incommensurate $q_{\rm{CDW}}=$ 0.310(1) r.l.u., with
$\delta$ marking the incommensurability with respect to the closest
commensurate value, 1/3. Vertical grey lines mark where discommensurations
would lead to satellite peaks at $q_{\rm{CDW}}\pm 2\delta$. b) YBCO6.55
$K$-cut through the 3D CDW peak at integer $L$, maximal compression and $T=45$
K $<T_{\rm{c}}$ in the superconducting state, where the vertical grey line
marks where pair density waves are expected to produce a peak at
$q_{\rm{PDW}}=q_{\rm{CDW}}/2=$ 0.327(3) r.l.u./2.  Figure 4:  Temperature
dependence of 2D and 3D CDW intensity for YBCO6.67 and YBCO6.55. a)
Temperature dependence of 3D and 2D CDW intensities for YBCO6.67 at $a$ axis
compressions $\varepsilon_{xx}$=-0.7 % and -0.85 %. b) 3D and 2D CDW
intensities for YBCO6.55 at compressions of $\varepsilon_{xx}$=-0.57 % and
-0.82 %. 2D CDW (3D CDW) intensities are determined from $K$-cuts ($L$-cuts).
Lines are guides to the eye. For a comparison of 2D and 3D intensities, see
Table LABEL:tab:correlation_length. c) Strain-temperature phase diagram of the
3D CDW intensity for YBCO6.67 with the $T_{c}$ curve reproduced from ref.
[23]. Dashed orange lines mark the boundary of the 3D CDW phase. In the
corresponding diagram for YBCO6.55 in panel d) $T_{c}(\varepsilon_{xx})$ is
estimated from the linear extrapolation down to the onset of the 3D phase
[37]. Within the 3D CDW phase $T_{c}(\varepsilon_{xx})$ is based on the peak
in the 3D CDW intensity.

##  II Results

Experimental results. Hard x-ray diffraction was performed as a function of
temperature on uniaxially pressurized YBa2Cu3Oysingle crystals, as described
in the Methods section. We first discuss the effect of uniaxial compression
along the $a$ and $b$ directions on the 2D CDW, for samples with $\rm{y}=6.67$
($p=0.125$). As shown in the intensity maps in Fig. 1, the $b$ component of
the 2D CDW has an extremely broad profile along the $[0,0,L]$ direction,
centered at half-integer $L$, but is relatively sharp along the $[0,K,0]$
direction already in the absence of strain. The incommensurate wavevector of
the modulations are in perfect agreement with previous reports [9, 10, 25,
24]. To evaluate intensity along the $K$ direction, we integrate the intensity
over the range $0.4<L<0.6$, with results at a few strains shown in Fig. 2a.
For these measurements, the temperature is close to
$T_{\text{c}}(\varepsilon_{xx})$. For strains not exceeding the 3D CDW onset
strain, the 2D CDW is most intense at $T\approx
T_{\text{c}}(\varepsilon_{xx})$. It can be seen in Fig. 2a that the intensity
of the $b$ component of the 2D CDW approximately doubles between
$\varepsilon_{xx}=0$ and $-0.55$% (but the CDW ordering wave vector is only
weakly affected by strain). The integrated intensity at a denser set of
strains is shown in Fig. 2b, in which this doubling is again visible, along
with a reduction in intensity for $\varepsilon_{xx}<-0.55$%. Also shown in
Fig. 2b is the integrated intensity of the $b$ component under compression
along the $b$ axis. The effect of $b$ axis compression is opposite to that of
$a$ axis compression: the intensity shrinks. In Supplementary Fig. 5 of
Supplementary Note 2 we provide evidence confirming the growth of the $a$ axis
component of the 2D CDW under $b$ axis compression, in agreement with previous
findings [27]. Together, these data show that the $a$\- and $b$-CDWs are
quasi-independent order parameters, and emphasize the uniaxial nature of the
underlying order parameter.

The decrease in intensity of the 2D CDW for $\varepsilon_{xx}<-0.55$%
coincides with the onset of the 3D CDW. The 3D CDW manifests itself as a sharp
peak at integer $L$, signaling long-range, in-phase correlations along the $c$
axis [29, 30]; the 3D CDW intensity is also shown in Fig. 2b. It is worth
noting that this onset strain is considerably lower than that estimated in
previous x-ray scattering studies [26, 27], in which the lattice parameter
longitudinal to the direction of compression was not accessible, but is closer
to the location of an anomaly identified in the stress dependence of
$T_{\text{c}}$ in Ref. [23]. The 3D CDW peak is also seen for $y=6.55$
($p=0.108$) (see Fig. 3b), but not yet for $y=6.80$ ($p=0.140$), where,
admittedly, the largest strain achieved in the present work was lower: we
reached $\varepsilon_{xx}=-0.45$%. (See Supplementary Fig. 6 in Supplementary
Note 2.)

We turn next to the temperature dependence of the 2D and 3D CDW intensities.
Integrated intensities of the 2D and 3D CDWs versus temperature at selected,
fixed strains are shown in Fig. 4a for YBCO6.67, and Fig. 4b for YBCO6.55. At
both of the selected strains for both compositions, the 2D CDW intensity stops
increasing and (within experimental accuracy even seems to decrease) when the
3D CDW onsets, as was found above to occur for the strain scan on YBCO6.67. At
these selected strains, the onset temperature of the 3D CDW exceeds
$T_{\text{c}}(\varepsilon_{xx}=0)$ and so, presuming that $a$ axis compression
suppresses $T_{\text{c}}$ in YBCO6.55 as it does in YBCO6.67 (and as inferred
from thermal expansion in the zero stress limit [37]), it substantially
exceeds $T_{\text{c}}$ at these strains. Therefore, the end of the 2D CDW
growth at these strains is clearly tied to the onset of the 3D CDW, and not to
the onset of superconductivity as at $\varepsilon_{xx}\sim 0$. Intensity maps
of the 3D CDW in strain-temperature space, built up from temperature scans
over these and additional strains, are shown for YBCO6.67 and YBCO6.55 in Fig.
4c and d, respectively. For YBCO6.55 there is no direct measurement of
$T_{\text{c}}(\varepsilon_{xx})$, so within the 3D CDW phase we estimate this
quantity from the temperature where the 3D CDW signal intensity reaches a
maximum, before getting gradually suppressed. This indicates a substantial
range of overlap between the 3D CDW and superconductivity. It is on the one
hand consistent with measurements in high magnetic fields that systematically
find the 3D CDW onset field to lie below $H_{c2}$ [30] (at least of its lower
estimated bound), and on the other hand the 3D CDW signal being observed down
to at least 41 K, i.e. well below $T_{\text{c}}(\varepsilon_{xx})$ for
YBCO6.67 at the highest measured strain of Ref. [26] (the present data allow
us to re-estimate this value to be $\varepsilon_{xx}=-0.85$%). For YBCO6.55,
the 3D CDW onset lies at a slightly smaller strain, $\varepsilon_{xx}=-0.4$%,
lower than for YBCO6.67.

Comparing Fig. 3a and b, which show scattering peaks from the 3D CDW at
$\varepsilon_{xx}=-0.85$% in YBCO6.67 and YBCO6.55, respectively, it can be
seen that the peak is substantially narrower for YBCO6.55, indicating a longer
correlation length, even though the tendency to 3D CDW order is probably
stronger in YBCO6.67. This observation points to a substantial effect of
disorder: the $b$ axis correlation length of the ortho-II chains in YBCO6.55
is longer than that of the ortho-VIII chains in YBCO6.67.

We end this section with a description of structural changes induced by the 3D
CDW formation which can be quantified, despite experimental limitations on the
accessible part of the reciprocal space imposed by the strain cell. Within
experimental uncertainty, we observe the main structural changes in the
CuO2-planes as the samples are strained. From structural refinements of
YBCO6.67 we observe that the clearest response of the average structure
(single unit cell) to the 3D CDW onset with increasing strain is a growing
anisotropy of the buckling angle of the planar oxygen bonds as shown in
Supplementary Fig. 7 in Supplementary Note 3. Minor changes can be discerned
in the parameters of the planar Cu atoms, e. g. in the intra-bilayer distance,
but these are less evident than the change observed for the buckling angles of
the oxygen bonds. Nevertheless, smaller motions are to be expected for the
heavier Cu atoms. This also finds confirmation in the smaller isotropic
temperature factors, shown in Supplementary Fig. 8 in Supplementary Note 3.

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Figure 5: Theoretical CDW structure factors based on a non-linear sigma model.
(a) The CDW structure factor at the in-plane wave vector of the $b$-CDW peaks
as a function of the $c$ axis wave vector, $L$, for $T=0.383$ and various
levels of the change in the CDW-mass, $\Delta g_{s}$, that model the effect of
$a$ axis compression. The inset depicts the CDW structure factor at the
position of the $a$-peak under the same conditions. (b) The intensity of the
$L=1$ $b$-CDW peak in the $T$-$\Delta g_{s}$ plane. The broken line marks the
superconducting transition temperature. (c) The temperature dependence of the
CDW structure factor at the position of the $b$-CDW peaks for $L=1$ and
$L=1/2$ in a system with $\Delta g_{s}$=-0.2. (d) The $\Delta g_{s}$
dependence of the same quantities at $T=0.383$.

Model. In an effort to elucidate the experimental findings we consider a
nonlinear sigma model describing competition between superconductivity and CDW
orders [34, 35, 36]. It is formulated in terms of a three-dimensional complex
vector whose components correspond to a superconducting order parameter,
$\psi_{j\mu}({\bf r})$, and two complex CDW order parameters,
$\Phi_{j\mu}^{a,b}({\bf r})$. The latter describe density variations
$\delta\rho_{j\mu}({\bf r})=e^{i{\bf Q}_{a}\cdot{\bf r}}\Phi_{j\mu}^{a}({\bf
r})+e^{i{\bf Q}_{b}\cdot{\bf r}}\Phi_{j\mu}^{b}({\bf r})+{\rm c.c.}$, along
the $a$ and $b$ directions with incommensurate wave-vectors ${\bf Q}_{a,b}$.
The order parameters reside on bilayers that are indexed by $j$ and represent
the CuO2 bilayers of YBCO. The index $\mu=0,1$ denotes the bottom (top) layer.
In the following we will coarse grain each of the planes into a square lattice
whose lattice constant is the observed CDW wavelength, i.e., about 3 Cu-Cu
spacings. The model assumes the existence of some type of local order at every
lattice point and incorporates the competition between the different
components via the constraints

$|\psi_{j\mu}|^{2}+|{\bm{\Phi}}_{j\mu}|^{2}=1,$ (1)

where ${\bm{\Phi}}_{j\mu}=(\Phi_{j\mu}^{a},\Phi_{j\mu}^{b})^{T}$.

The vector of order parameters is governed by the Hamiltonian

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{j}\sum_{\mu=0,1}H_{j\mu}+\frac{\rho_{s}}{2}\sum_{j}\sum_{\bf
r}\Big{[}\tilde{U}{\bm{\Phi}}^{\dagger}_{j,0}{\bm{\Phi}}_{j,1}$
$\displaystyle+U{\bm{\Phi}}^{\dagger}_{j,1}{\bm{\Phi}}_{j+1,0}-\tilde{J}\psi_{j,0}^{*}\psi_{j,1}-J\psi_{j,1}^{*}\psi_{j+1,0}$
$\displaystyle+{\bm{V}}^{\dagger}_{j}\left(\gamma{\bm{\Phi}}_{j,0}+{\bm{\Phi}}_{j,1}+{\bm{\Phi}}_{j+1,0}+\gamma{\bm{\Phi}}_{j+1,1}\right)+{\rm
c.c.}\Big{]}.$

Henceforth, the superconducting stiffness, $\rho_{s}$, serves as the basic
energy scale. We model the Coulomb interaction between CDW fields within a
bilayer by a local coupling $\tilde{U}$, and denote the intra-bilayer
Josephson tunneling amplitude by $\tilde{J}$. The (weaker) Coulomb interaction
and Josephson coupling between nearest-neighbor planes belonging to
consecutive bilayers are denoted by $U$ and $J$, respectively. In the last
term, we include the Coulomb interaction between the disordered doped oxygens
on the chain layers and the CDW fields on the adjacent bilayers, assuming that
the coupling to the outer CuO2 planes is reduced by a factor $\gamma$ compared
to the coupling to the inner planes. We model the disorder as a collection of
randomly placed discs, each of radius $r_{d}$ and containing a constant
potential with a random phase that couples either to the $a$ or $b$ CDW
fields, ${\bm{V}}({\bf r})=V\sum_{l}f(|{\bf r}-{\bf
r}_{l}|)e^{i\theta_{l}}(p_{l},1-p_{l})^{T}$. Here, $p$ takes the values 0,1
with probability 1/2 and $f({\bf r})=1-\Theta(|{\bf r}|-r_{d})$, where
$\Theta(r)$ is the step function.

Within a layer the Hamiltonian reads

$\displaystyle\\!\\!\\!\\!\\!\\!\\!H_{j\mu}$ $\displaystyle=$
$\displaystyle\frac{\rho_{s}}{2}\sum_{\bf
r}\Big{[}\left|\bm{\nabla}\psi_{j\mu}\right|^{2}+\lambda|\bm{\nabla}{\bm{\Phi}}_{j\mu}|^{2}+g|{\bm{\Phi}}_{j\mu}|^{2}$
(3) $\displaystyle+\left(\Delta g-\Delta
g_{s}\right)|\Phi^{a}_{j\mu}|^{2}+\Delta g_{s}|\Phi^{b}_{j\mu}|^{2}\Big{]},$

where $\bm{\nabla}$ is the discrete gradient, $\lambda\rho_{s}$ is the CDW
stiffness and $g\rho_{s}$ is the effective CDW mass reflecting the energetic
penalty for CDW ordering. The presence of such a penalty ensures that
superconductivity prevails over the CDW order at $T=0$, at least in the
disorder-free regions. The mass anisotropy, encapsulated by the $\Delta g>0$
term, is included as a result of our assumption that the potential induced by
the chain layers favors ordering along the $b$ axis. Finally, and most
pertinently to the present study, we assume that the application of strain
causes an increase in the mass of the CDW component along the direction of the
strain, while reducing the mass of the transverse component. Specifically,
applying compressive strain in the $a$ direction corresponds to $\Delta
g_{s}<0$.

Theoretical results. Our primary interest lies in the $k$-space (measured from
${\bf Q}_{a,b}$) CDW structure factor

$\displaystyle S_{\alpha}({\bf q},L)$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{{\bf r}{\bf
r}^{\prime}}\sum_{jj^{\prime}}\sum_{\mu\mu^{\prime}}e^{-i\left[{\bf
q}\cdot\left({\bf r}-{\bf
r}^{\prime}\right)+2\pi\left(j-j^{\prime}+\frac{\mu-\mu^{\prime}}{3}\right)L\right]}$
(4) $\displaystyle\times\langle\Phi^{\alpha}_{j\mu}({\bf
r})\Phi^{*\alpha}_{j^{\prime}\mu^{\prime}}({\bf r}^{\prime})\rangle,$

where $N$ is the number of lattice points, and the averaging is over both
thermal fluctuations and disorder realizations. We have used the fact that in
YBCO the CuO2 planes within a bilayer are separated by approximately 1/3 of
the $c$ axis lattice constant.

We have calculated $S_{\alpha}({\bf q},L)$ by Monte Carlo simulations of Eqs.
(II,3) on a $32\times 32\times 32$ (16 bilayers) system, with $\lambda=1$,
$g=1.1$, $\Delta g=0.1$, $\tilde{J}=0.15$, $J=0.015$, $\tilde{U}=0.85$,
$U=0.12$, $V=1$, and $\gamma=0.15$. Each data point was averaged over 1000
disorder realizations with 8 disordered regions per bilayer and $r_{d}=3$
[38].

Our core result is presented in Fig. 5a, which depicts the structure factor at
a temperature near the superconducting $T_{c}$ and for various values of
$\Delta g_{s}$, emulating the effect of $a$ axis strain. For $\Delta g_{s}=0$
(absence of strain), both $S_{a}$ and $S_{b}$ show broad peaks centered around
$L=0.7$. This is a result of CDW domains that nucleate due to interaction with
the disorder [36]. Since a disordered region tends to induce the same CDW
pattern on both its flanking planes, an out-of-phase arrangement of the CDW
order tends to form in the $c$-direction. The observed skewness away from
$L=1/2$ is a result of the form factor in Eq. (4). Increasing $\Delta g_{s}$
towards negative values increases the energetic cost for nucleating $a$-CDWs
and leads to a decrease in the height of the corresponding peak (see inset).
An opposite trend is seen for the disorder-induced peak along the $b$
direction. However, a much more dramatic effect emerges in $S_{b}$ beyond a
characteristic value of $\Delta g_{s}$ in the form of a large sharp peak
centered at $L=1$. This signal originates from the regions between the
disorder-induced CDW domains. It reflects the tipping of the balance between
superconductivity and the $b$-CDW order in favor of the latter when the
$b$-CDW nucleation cost is sufficiently reduced. The Coulomb interaction
between the strain-induced CDW regions on neighboring bilayers favors an in-
phase CDW configuration along the $c$-direction, hence the peak at $L=1$. The
sharpness of the peak is a direct result of the fact that these regions are
not induced by disorder.

The interplay between the superconducting and CDW orders is summarized in Fig.
5b, which shows the superconducting $T_{c}$ and the height of the $L=1$ CDW
peak as a function of the temperature and $\Delta g_{s}$. Clearly, $T_{c}$
(calculated by finite size analysis of the superconducting correlation length
and stiffness) is a decreasing function of $|\Delta g_{s}|$ owing to the
enhancement of the competing CDW fluctuations. A large 3D CDW signal appears
for $\Delta g_{s}\lesssim-0.17$ at a temperature that increases with $\Delta
g_{s}$, peaks in the vicinity of $T_{c}$ of the strained system and then
rapidly diminishes at lower temperatures, as is also evident in Fig. 5c. This
behavior stands in contrast to behavior of $S_{b}(0,L=1/2)$ that similarly
increases when the temperature is lowered towards $T_{c}$ but then saturates.

Finally, Fig. 5d shows the $\Delta g_{s}$ dependence of the 2D and 3D CDW
signals near $T_{c}$. There is a clear and sharp onset of the 3D CDW peak at
$\Delta g_{s}\simeq-0.17$, whereas $S_{b}(0,L=1)$ essentially vanishes at
lower values of $a$ axis strain and for all values of $b$ axis strain
(positive $\Delta g_{s}$). The 2D signal, $S_{b}(0,L=1/2)$, on the other hand,
diminishes continuously with decreasing $a$ axis strain and increasing $b$
axis strain. This is because the accompanying increase in the effective mass
of the $b$-CDW successively reduces the magnitude of the CDW induced by the
disorder. Interestingly, and similarly to the experimental observations, we
find a saturation and then a slight downturn of the 2D CDW signal that
coincides with the onset of the 3D CDW order.

##  III Discussion

We start our discussion from the structure of the 3D CDW. Having access to
large reciprocal space maps and intense 3D CDW peaks, we can provide strong
constraints for deviations from a perfectly sinusoidal charge density
modulation in the long-range ordered phase. Previous studies on Bi-based
cuprates [39] have indicated that in this system the CDW is commensurate with
periodicity $\lambda=4a$ over the entire doping range where it appears. More
recently, high-field NMR data have been successfully analyzed in a picture
where the CDW is locally commensurate with $\lambda=3b$. In both cases it was
argued that sharp phase slips (discommensurations) at the CDW domain boundary
could shift the position at which the 3D CDW peak is observed in scattering
experiments to an effectively incommensurate value [39, 40]. In this case, one
would expect the presence of satellite peaks at $q_{\rm{CDW}}\pm 2\delta$
($\delta$ being the incommensurability to the commensurate 3D CDW wavevector,
$1/3$) [40]. At $T_{c}$, for YBCO6.67 we observe no indications for additional
satellite peaks along $b^{*}$ coming from such sharp discommensuration (we
cannot fully rule out the presence of spatially extended phase slips). We also
note that the $K$-cut shown in Fig. 3b for YBCO6.55 at $T=45$ K $<T_{c}$ does
not exhibit a peak at $q_{\rm{PDW}}=q_{\rm{CDW}}/2$. Such a peak is expected
if the observed CDW is due to a pair density wave (PDW), i.e., a modulated
superconducting condensate, at wavevector $q_{\rm{PDW}}$, and if the PDW
coexists with uniform superconductivity [41, 42]. Based on our measurements,
the PDW signal must be at least 50 times smaller than the signal of the long-
range 3D CDW. However, further calculations are needed to estimate the
associated periodic lattice displacement and the resulting x-ray intensity to
exclude the presence of PDWs.

Next, and before discussing the nature of the interplay between the 2D and 3D
CDWs, it is worth noting that from a structural point of view, the fact that
the formation of the 3D CDW mostly results in a displacement of planar oxygen
atoms is consistent with a larger response of the relative quadrupolar
splitting in 17O-NMR than in 63Cu-NMR when the 3D CDW is induced using high
magnetic fields [40]. Structural refinement of the x-ray diffraction data in
zero field shows that the 2D CDWs along $a$ and $b$ axes induce an anisotropic
modulation of the copper-oxygen-bonds with significant out-of-plane
displacements of the oxygen transverse to the CDW modulation [43]. In this way
the 2D CDW locally breaks rotational symmetry and has a nematic structural
character. In fact, various experimental probes associate nematicity in the
cuprates with the growth of spin and charge density wave correlations upon
cooling [44, 45, 46, 47]. Here, for $a$ axis compression, we observed an
increasing trend of the $a$ axis buckling angle close to the 3D CDW onset,
although the error bars are relatively large compared to this subtle effect.
In the unstrained orthorhombic unit cell the buckling angles along $a$ and $b$
axes differ only slightly. Hence, the increased strain-induced buckling
anisotropy can be regarded as a structural parameter possibly related to
nematicity. Whereas anisotropy in the buckling is expected to exist for 2D
CDWs, it is maximal in the 3D CDW phase. However, as the 3D CDW breaks
rotational and translational symmetry globally, the anisotropy of the 3D CDW
should not be considered as nematic [48]. Further, the buckling anisotropy
should not be regarded as an order parameter of the 3D CDW, as for applied
strain it is already finite at temperatures higher than that of the 3D CDW
formation. We conclude from this that a buckling anistropy might be a
necessary condition for 3D ordering in YBCO, but it is certainly not
sufficient. In any event, to better understand the constraints imposed by the
crystal structures on the formation of long-range charge order in the cuprates
(in particular in view of the recent reports on the overdoped side [49, 50]),
more systematic structural investigations are needed.

We now turn to the nature of the interplay between the CDW orders and
superconductivity. On the one hand, a large body of experiments has
unambiguously established that the 2D CDW and superconducting orders compete
in YBCO [9, 10, 25], as both the CDW peak amplitude and the CDW domain
correlation length decrease in the superconducting state. On the other hand,
the strongest 3D CDW signals are observed where the homogeneous
superconducting phase has been suppressed either by application of a large
magnetic field [32, 31, 30, 29] or by uniaxial pressurization [26, 27]. In
both cases, the maximal achievable strength of the applied perturbations was
not sufficient to completely suppress the superconducting state at the lowest
temperatures. However, while in the case of strain tuning the low-temperature
superconducting phase is homogeneous with no trace of the 3D CDW, the mixed
state of the type-II superconductor in a magnetic field contains halos of 3D
CDW around vortex cores.

In the experimental data presented above, we have shown that a form of
competition can also manifest itself between the 2D and 3D CDW orders, which
is best evidenced in the region of the strain-temperature space where
superconductivity has been suppressed by strain. Data in unstressed YBCO6.67
under applied magnetic field [32, 31, 30, 29] can also be interpreted in terms
of this 2D-3D CDW competition. Specifically, it was shown in Ref. [32] that as
YBCO6.67 is cooled under strong applied field there is a temperature range
where the 2D CDW intensity starts to decrease while the 3D CDW keeps
increasing. In the same study, a direct conversion of the 2D CDW to the 3D CDW
was hypothesized, but the data there do not allow clean disentanglement of the
mutual interaction among the 2D CDW, 3D CDW, and superconductivity. Our data
clearly indicate that the 2D CDW stops growing and even shrinks at the onset
of the 3D CDW, where the superconductivity would have set in, in the absence
of strain. In the superconducting state, the evolution of 2D CDWs is
complicated by their mutual competition with the 3D CDW and the simultaneous
competition with superconductivity.

Finally, we note that the different temperature- and strain-dependence of the
2D and 3D CDWs can naturally be explained in the framework of the
phenomenological model developed above and is rooted in the different
mechanisms that are responsible for their respective establishment. The 2D CDW
arises in domains that are dominated by their coupling to the disorder
potential. Hence, these domains continue to host the CDW even when
superconductivity sets in within the intervening disorder-free regions. In
contrast, the 3D CDW arises in the same intervening regions at large enough
strain. This qualitatively explains why the 2D CDW order is stronger in
YBCO6.67 than YBCO6.55 [25] (the former being more disordered than the latter)
and why on the contrary the 3D CDW order onsets at a smaller strain in
YBCO6.55 compared to YBCO6.67. The strain further reduces the energetic cost
of the CDW fluctuations, thus allowing them to appear at temperatures that can
even exceed the $T_{c}$ of the unstrained system. In turn, the growth of the
3D CDW correlations contribute, via their competition with superconductivity,
to the suppression of $T_{c}$. Below $T_{c}$ the 3D CDW eventually gives way
to the superconducting order, which constitutes the ground state of the clean
model. Within the same picture, the coincident saturation (or shrinkage) of
the 2D CDW with the establishment of strong 3D CDW can be traced to phase
mismatch between the two types of CDWs. The phase of the 2D CDW within the
disordered domains conforms to the local disorder arrangement, whereas the 3D
CDW seeks to establish a uniform phase throughout the system. The phase
mismatch along the boundaries of the disordered domains gives rise to an
effective surface tension, via the CDW elastic term in Eq. (3), that
consequently arrests the growth of the 2D CDW domains.
These results more generally highlight the inherent strengths of strain tuning
as a powerful approach to investigate and manipulate the intricate interplay
between competing orders in quantum materials, presenting promising prospects
for advancing our understanding of these systems.

## Methods

Samples. High-quality single crystals of YBa2Cu3Oy with oxygen concentrations
$6.55\leq\rm{y}\leq 6.80$ were grown, detwinned and characterized as
previously described in ref. [25]. The crystals were then cut and polished to
needles with dimensions of $\approx 200~{}\mu$m$\times 100~{}\mu$m$\times$2
mm, and were pressurized along their lengths using a Razorbill CS200T
piezoelectric-driven uniaxial stress cell mounted in a helium flow cryostat.
Information about the samples is summarized in Supplementary Table I in
Supplementary Note 1. To enable fast sample exchange, the mounting into the
stress cell was done via a flexible titanium support, as displayed in Fig. 1;
needles were mounted into these supports in advance, a slow, delicate process,
then the supports were mounted into the cell during the beamtime. To achieve
higher stresses, the selected samples were further thinned down laterally
using a Xenon plasma focused ion beam (PFIB), as described in Supplementary
Note 1. We checked that this procedure did not alter their superconducting
properties.

X-ray Diffraction. Hard x-ray diffraction was performed at the ID15B beamline
of the European Synchrotron Radiation Facility with a fixed photon energy of
30 keV ($\lambda$=0.413 Å), that enabled us to work in transmission geometry.
The scattered photons were collected using a state-of-the-art large area
Dectris Eiger2X CdTe 9M hybrid photon-counting detector. Additional
experimental details are given in the Supplementary Note 1 and the original
data are available under [51]. Rigaku’s CrysalisPro software [52] has been
used for data reduction and to obtain reciprocal space maps. Structural
refinements of the average unit cell (neglecting oxygen ordering in the chain
layer) have been performed using Jana2020 [53]. In spite of the large
dynamical range of the detector, the very high intensity difference between
the CDW features and the main Bragg reflections presents a measurement
challenge. To map both features, photon flux was adjusted by varying the
undulator gap [54]: high photon flux was used to study CDW superstructure
peaks, while low photon flux was used to record the lattice Bragg peaks
without saturating the detector. Additional details about the structure
refinements are given in the Supplementary Note 3.

Strain determination. Strains reported in this paper are derived from the
Bragg peak positions, eliminating error from uncertainty in strain
transmission to the sample. The determination of lattice parameters was based
on integration of about 400 Bragg reflections, a number that was limited by
the geometrical constraints imposed by the cryostat and the stress cell, but
which is nevertheless sufficient to allow accurate determination of the three
lattice constants $a$, $b$, and $c$. Therefore, we also obtain the Poisson’s
ratios, $\nu_{ij}\equiv-\varepsilon_{ii}/\varepsilon_{jj}$, shown in
Supplementary Fig. 2 in Supplementary Note 1. Knowledge of the Poisson’s
ratios is useful both as a reference for other measurements, where the
scattering geometry might not allow determination of the longitudinal strain
[26], and potentially as a thermodynamic probe of possible changes in the
electronic structure [55]. Note that throughout this paper we use the
engineering definition of strain, in which negative values denote compression.

## Acknowledgements

We thank M. Dušek, M. Hanfland, S. Kivelson, V. Petříček, J. Schmalian and R.
Willa for helpful discussions, A. K. Jaiswal for SQUID measurements, A. Ghiami
and M. Hesselschwerdt for technical support and T. Poreba and N. Maraytta for
support during the diffraction experiments at the ESRF and IQMT, respectively.
Self-flux growth was performed by the Scientific Facility ‘Crystal Growth’ at
Max Planck Institute for Solid State Research, Stuttgart, Germany. This work
was supported through the funding of the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation), projects 422213477 (TRR 288 projects B03 and A10)
and 449386310. S.M.S. acknowledges funding by the DFG – Projektnummer
441231589, M. F. acknowledges funding by the Alexander von Humboldt Foundation
and the Young Investigator Grant, H. M. L. N. acknowledges support from the
Alexander von Humboldt Foundation through a Research Fellowship for
Postdoctoral Researchers, K. I. acknowledges the Japan Society for the
Promotion of Science Overseas Research Fellowships and I. V. acknowledges the
Horizon Europe MSCA fellowship 101065694. We thank the European Synchrotron
Radiation Facility (ESRF) for provision of synchrotron radiation facilities
under proposals number HC-4226 and HC-4865.

Data availability The data reported in this study are available at [51]. The
data that support the findings of this study are available from the
corresponding author, M. L. T.(matthieu.letacon@kit.edu), upon reasonable
request.

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