# Charge Condensation and Lattice Coupling Drives Stripe Formation in
Nickelates

Y. Shen<EMAIL_ADDRESS>Condensed Matter Physics and Materials Science
Department, Brookhaven National Laboratory, Upton, New York 11973, USA  G.
Fabbris  Condensed Matter Physics and Materials Science Department, Brookhaven
National Laboratory, Upton, New York 11973, USA  Advanced Photon Source,
Argonne National Laboratory, Lemont, Illinois 60439, USA  H. Miao  Condensed
Matter Physics and Materials Science Department, Brookhaven National
Laboratory, Upton, New York 11973, USA  Material Science and Technology
Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA  Y.
Cao  Condensed Matter Physics and Materials Science Department, Brookhaven
National Laboratory, Upton, New York 11973, USA  Materials Science Division,
Argonne National Laboratory, Lemont, Illinois 60439, USA  D. Meyers  Condensed
Matter Physics and Materials Science Department, Brookhaven National
Laboratory, Upton, New York 11973, USA  Department of Physics, Oklahoma State
University, Stillwater, Oklahoma 74078, USA  D. G. Mazzone  Condensed Matter
Physics and Materials Science Department, Brookhaven National Laboratory,
Upton, New York 11973, USA  Laboratory for Neutron Scattering and Imaging,
Paul Scherrer Institut, CH-5232 Villigen, Switzerland  T. Assefa  Condensed
Matter Physics and Materials Science Department, Brookhaven National
Laboratory, Upton, New York 11973, USA  X. M. Chen  Condensed Matter Physics
and Materials Science Department, Brookhaven National Laboratory, Upton, New
York 11973, USA  K. Kisslinger  Center for Functional Nanomaterials,
Brookhaven National Laboratory, Upton, New York 11973, USA  D. Prabhakaran  A.
T. Boothroyd  Department of Physics, University of Oxford, Clarendon
Laboratory, Oxford, OX1 3PU, United Kingdom  J. M. Tranquada  Condensed Matter
Physics and Materials Science Department, Brookhaven National Laboratory,
Upton, New York 11973, USA  W. Hu  A. M. Barbour  S. B. Wilkins  C. Mazzoli
National Synchrotron Light Source II, Brookhaven National Laboratory, Upton,
New York 11973, USA  I. K. Robinson  M. P. M. Dean<EMAIL_ADDRESS>Condensed
Matter Physics and Materials Science Department, Brookhaven National
Laboratory, Upton, New York 11973, USA

###### Abstract

Revealing the predominant driving force behind symmetry breaking in correlated
materials is sometimes a formidable task due to the intertwined nature of
different degrees of freedom. This is the case for La2-xSrxNiO4+δ in which
coupled incommensurate charge and spin stripes form at low temperatures. Here,
we use resonant X-ray photon correlation spectroscopy to study the temporal
stability and domain memory of the charge and spin stripes in La2-xSrxNiO4+δ.
Although spin stripes are more spatially correlated, charge stripes maintain a
better temporal stability against temperature change. More intriguingly,
charge order shows robust domain memory with thermal cycling up to 250 K, far
above the ordering temperature. These results demonstrate the pinning of
charge stripes to the lattice and that charge condensation is the predominant
factor in the formation of stripe orders in nickelates.

Emergent phenomena in strongly correlated materials arise due to multifarious
interactions among charge, spin and lattice degrees of freedom. Such
complexity hampers the ability to understand their remarkable states and
realize new functionalities Chumak  _et al._ (2015). Identifying dominant
interaction is, however, challenging as different interactions act
simultaneously and can yield complex ground states with more than one form of
order Fradkin  _et al._ (2015). A representative phenomenon of this type is
the electronic stripes that appear in various strongly correlated materials
Mori  _et al._ (1998); Lee  _et al._ (2006); Ulbrich and Braden (2012); Comin
and Damascelli (2016). These effects have been considered extensively in
cuprate high-temperature superconductors, which host charge and sometimes spin
stripe order, typically with a simple factor-of-two relationship between the
charge and spin incommensurabilities Tranquada  _et al._ (1995); Mook  _et
al._ (2000); Hoffman  _et al._ (2002). Nickelates also host both
superconductivity and stripe order Tranquada  _et al._ (1994, 1996); Li  _et
al._ (2019), but no system has yet been shown to simultaneously host both
orders. The existence of stripe order in La4Ni3O8, which appears rather
similar to superconducting Nd1-xSrxNiO2 Zhang  _et al._ (2016, 2017, 2019);
Lin  _et al._ (2020), does, however, support the likely proximity of stripe
order and superconductivity. While static stripe order appears to suppress
bulk 3D superconductivity, some researchers have suggested that stripe
fluctuations may act to promote superconductivity Emery  _et al._ (1997);
Kivelson  _et al._ (1998); Agterberg  _et al._ (2020). Therefore,
understanding the driving forces behind charge and spin stripe formation and
dynamics in strongly correlated materials has attracted considerable attention
and may be crucial to understanding unconventional superconductivity. Stripe
formation has been studied in the past through detailed measurements of stripe
transition temperatures and correlation lengths Chen  _et al._ (1993); Cheong
_et al._ (1994); Lee and Cheong (1997); Yoshizawa  _et al._ (2000); Lee  _et
al._ (2001); Kajimoto  _et al._ (2001); Ghazi  _et al._ (2004); Freeman  _et
al._ (2004); Raczkowski  _et al._ (2006) and associated Landau model analysis
Wochner  _et al._ (1998); Zachar  _et al._ (1998). The problem has also been
addressed via model Hamiltonian analysis that suggested that lattice coupling
might be crucial to stabilize stripes Zaanen and Littlewood (1994); Hotta and
Dagotto (2004). The implementation of resonant X-ray photon correlation
spectroscopy (XPCS) at modern low-emittance synchrotron sources opens new
routes to directly probe stripe formation and dynamics Chen  _et al._ (2016);
Thampy  _et al._ (2017); Chen  _et al._ (2019); Ricci  _et al._ (2019).

Herein, we report the first resonant XPCS experiment to simultaneously probe
charge order (CO), spin order (SO) and lattice coupling in a stripe-ordered
material, focusing on the prototypical material La2-xSrxNiO4+δ (LSNO) with
$x=0.225$ $\delta=0.07$. Although SO is more correlated and stable at 70 K, CO
is more robust in temporal stability against temperature changes, which we
attribute to electron-phonon coupling (EPC). This is further supported by our
discovery that the CO domains are effectively pinned to the lattice and the
corresponding speckle patterns remain highly reproducible with thermal cycling
up to 250 K, well above the transition temperature $T_{\mathrm{CO}}$. SO,
however, is not directly coupled to the lattice and loses its domain memory
once the sample is warmed across the magnetic transition temperature
$T_{\mathrm{SO}}$. These results imply that charge condensation, and its
coupling to the lattice and disorder, is the driving force behind stripe
ordering.

Figure 1: Experimental configuration and charge order (CO)and spin order
(SO)superlattice peaks. (a) The instrumental setup for the measurements at
CSX. The X-ray beam is set to the Ni $L_{3}$-edge energy and tuned in order to
maximize the strength of the CO and SO intensity sup . It then propagates
through the pinhole and is scattered by the La2-xSrxNiO4+δ (LSNO)sample onto
the detector. For the domain memory study, a 0.5 $\upmu\text{m}$ thick Pt mask
was deposited on the sample sup . (b) An optical micrograph of the Pt mask on
the (110) surface of LSNO single crystal. (c) Temperature dependence of the
correlation lengths along [$H$, $H$, 0] and [0, 0, $L$] directions. The
correlation length is defined as $\xi=d/\textrm{HWHM}$ where HWHM stands for
half-width at half-maximum in reciprocal lattice units and $d$ is the unit
cell size in the appropriate direction inc . (d) Temperature dependence of the
peak heights evaluated from fitting of the CO and SO superlattice peaks, which
are normalized according to their values at 60 K. The signals were fitted with
a three-dimensional Lorentzian function. (e) Incommensurability defined by the
peak position of COQ vector as a function of temperature. The shaded areas
indicate the onset temperature range for CO and SO.

X-ray measurements were carried out at the Coherent Soft X-Ray (CSX) 23-ID-1
beamline at the National Syncrotron Light Source II with X-ray energy tuned to
the Ni $L_{3}$-edge (Fig. 1a). The LSNO single crystal was synthesized by the
floating-zone method with a Sr concentration of $x=0.225$ Prabhakaran  _et
al._ (2002). As shown later, the CO incommensurability is $\epsilon\approx
0.27$, larger than $x$, which is likely related to oxygen doping since
$\delta=0.07$ Freeman  _et al._ (2006). The sample’s surface normal was close
to the [$H$, $H$, 0] direction. Thus, we made ($H$, $H$, $L$) the scattering
plane and focused on peaks with ${\textbf{\em
Q}_{\mathrm{CO}}}=(\epsilon,\epsilon,1)$ and ${\textbf{\em
Q}_{\mathrm{SO}}}=(1/2-\epsilon/2,1/2-\epsilon/2,0)$ sup . The reciprocal
lattice units (r.l.u.) is defined in terms of Q = ($H$, $K$, $L$) =
$(2\pi/a,2\pi/b,2\pi/c)$ within the space group I4/mmm and $a=b=3.84$ Å,
$c=12.65$ Å. For the domain memory measurements, we used a 0.5 $\upmu\text{m}$
thick Pt mask, which had been deposited on the sample in order to reproducibly
illuminate the same sample volume independent of possible thermal drifts in
the sample position (Fig. 1b) sup .

We start by characterizing the superlattice peaks corresponding to CO and SO
at different temperatures using standard resonant X-ray diffraction. With
decreasing temperature, the peak heights first increase substantially through
the transition temperatures along with enhanced correlation lengths for both
CO and SO (Fig. 1c, d). Below $\sim$70 K, the peak heights drop and the
spatial correlations are relaxed, consistent with previous reports Hatton  _et
al._ (2002); Ghazi  _et al._ (2004); Schlappa  _et al._ (2009). The reason for
this is not uniquely determined, but it may be connected to a spin
reorientation at lower temperature Freeman  _et al._ (2004) or the influence
of spin exchange interactions Ghazi  _et al._ (2004). Throughout the
temperature range, the correlation lengths along [$H$, $H$, 0] direction are
much larger than those along [0, 0, $L$] and SO possesses a larger correlation
length than CO (Fig. 1c). Due to the critical fluctuations and short-range
correlations near the phase transitions, the onset temperatures,
$T_{\mathrm{CO}}$ and $T_{\mathrm{SO}}$, are not uniquely defined. We estimate
them both to occur between 96 and 114 K. Regarding the incommensurability, the
inter-site Coulomb repulsion tends to stabilize $\epsilon$ equal to the hole
concentration Sachan  _et al._ (1995), while the commensurability effect
optimizes stripe formation at $x=1/3$. The actual incommensurability is a
compromise of these two factors Yoshizawa  _et al._ (2000). With increasing
temperature, thermal fluctuations are expected to start to outcompete Coulomb
repulsion Hatton  _et al._ (2002); Ishizaka  _et al._ (2004); Miao  _et al._
(2019), driving the incommensurability closer to 1/3 at higher temperature
(Fig. 1e).

Figure 2: Speckle patterns of CO and SO. (a),(b) Representative detector
images around the CO and SO superlattice peaks measured with a 10
$\upmu\text{m}$ pinhole. The white pixels arise from the beamstop or detector
errors and are omitted from the data. (c),(d) Line cuts through the horizontal
red dashed lines in (a) and (b). The envelope of the peak is estimated by
smoothing and fitting processes that are shown as red and orange lines,
respectively. The black dashed lines are uniform fluorescent background
evaluated from fittings.

To elucidate the temporal stabilities of CO and SO, we employ XPCS to study
the domain distribution and its fluctuations. In XPCS, the coherent photons
scattered by different domains interfere with each other, leading to a complex
“speckle” pattern modulated by the usual diffraction lineshape Brauer  _et
al._ (1995); Shpyrko (2014); Chen  _et al._ (2016); Thampy  _et al._ (2017);
Ricci  _et al._ (2019); Lee  _et al._ . Figure 2a, b shows the representative
speckles of the CO and SO superlattice peaks at 70 K. The shape of the peak
envelope is determined by the spatial correlations and instrument geometry. In
particular, the horizontal width of the SO peak is mainly determined by the
correlations along the [-1, 1, 0] direction while the vertical width is
dominated by $c$ axis correlations, elongating the envelope vertically. For
the CO peak, the vertical width has less contribution from $c$ axis
correlations so that the envelope appears more isotropic. Meanwhile, the
distribution of the underlying stripe domains is encoded in the positions of
the speckles Chen  _et al._ (2019), and the shape of the speckles is
determined by the Fourier transform of the beam footprint projected onto the
detector. The non-zero $L$ component of the CO peak makes the footprint of the
beam more anisotropic. To show the speckle modulation more clearly, we present
in Fig. 2c, d the line cuts through the red dashed lines in Fig. 2a, b. The
peak envelope is estimated by two independent methods: smoothing with the
Savitzky-Golay filter and fitting with a squared Lorentzian function. The
sharp speckle modulation observed here indicates that the fluctuations for CO
and SO are slower than the time windows of the measurements, which is 1 s at
70 K sup . Otherwise the contrast of the interference patterns will be
significantly reduced Chen  _et al._ (2016).

Figure 3: Temporal stability of CO and SO. (a),(b) Time dependence of the
intermediate scattering functions at different temperatures. The solid lines
are guides to the eye. (c),(d) The scattering functions after certain time
delays. Figure 4: Domain memory in CO but not SO. (a),(b) Representative
speckle images before and after thermal cyclings which are indicated by the
curved arrows. The open circles stand for the cycling temperatures,
$T_{\mathrm{cycle}}$. For each measurement, we collected images at 70 K,
changed the temperature to $T_{\mathrm{cycle}}$ and waited for 10 minutes.
Then the sample was cooled back to 70 K and equilibrated for 30 minutes before
collecting another image. For both the heating and cooling processes, the
temperature ramping rate was fixed to 4 K/min. The white bar in the first
speckle image indicates $10^{-3}$ Å-1. (c) Temperature dependence of the
normalized speckle cross-correlation function, $\xi_{\mathrm{CC}}$. The solid
and dashed lines are guides to the eye. The shaded area indicates the range of
CO and SO transition temperatures.

In order to quantify the fluctuation timescale, we measure the time dependence
of the speckle patterns and calculate the normalized one-time correlation
function Chen  _et al._ (2016)

$\displaystyle g_{2}(\tau)=\frac{\langle I(t)I(t+\tau)\rangle}{\langle
I(t)\rangle^{2}}=1+\beta|F(\tau)|^{2},$ (1)

where $I$ represents the total intensity including background, $\tau$ is the
lag time and $\langle\ldots\rangle$ stands for the time and ensemble average.
The time-dependent evolution can be extracted from the intermediate scattering
function, $|F(\tau)|$, which describes the correlation of the speckle patterns
separated by a certain time delay. In a statically ordered system, $|F(\tau)|$
will remain unchanged while speckle dynamics causes it to drop as a function
of time delay. Distinct from La2-xBaxCuO4 (LBCO), in which the CO is static
over a timescale of at least two hours Chen  _et al._ (2016); Thampy  _et al._
(2017), $|F(\tau)|$ in LSNO decays after several minutes for both CO and SO,
indicating charge and spin dynamics (Fig. 3). Moreover, we find that CO and SO
are both most stable around 70 K when they have longest correlation lengths,
but SO is more stable than CO at 70 K. Although stripes involve a co-
modulation of both charge and spin Zachar  _et al._ (1998), we observe that
these have different thermal evolution. As temperature is driven away from 70
K, the temporal stability for SO decreases faster, indicating that SO is less
stable against temperature changes. A qualitatively, but not quantitatively
similar trend in SO was reported recently in Ref. Ricci  _et al._ (2019). The
longer timescales observed here may reflect sample discrimination in strontium
and oxygen compositions or improved coherent flux and stability at CSX
compared to the Advanced Light Source.

From simple energetic considerations, if an order is less temporally stable
and has shorter correlation lengths one would expect it to be more fragile to
thermal disturbance. The unexpected robustness of CO against temperature
changes indicates that CO is coupled to other degrees of freedom which
constrain the CO domains during/after the charge condensation (Fig. 3). Such
hypotheses can be examined more deeply in term of domain pinning memory
effects. Since the speckle positions are primarily determined by the positions
of the ordering domains, the comparison of speckle patterns collected at 70 K
before and after cycling the sample temperature to $T_{\mathrm{cycle}}$ can
evaluate whether the domain distributions are reproduced Chen  _et al._
(2019). The usage of Pt mask further ensures that the illuminated sample
volume is fixed throughout the thermal cycling (Fig. 1b). It turns out that
the speckle patterns of CO are rather similar with $T_{\mathrm{cycle}}$ up to
250 K, well above $T_{\mathrm{CO}}$ (Fig. 4a). The SO speckles, however,
change their positions once $T_{\mathrm{cycle}}$ crosses $T_{\mathrm{SO}}$
($\sim 100$ K) (Fig. 4b). This effect can be quantified by calculating the
normalized cross-correlation function $\xi_{\mathrm{CC}}$ which describes the
similarity between two speckle patterns Chen  _et al._ (2019); sup .
$\xi_{\mathrm{CC}}$ approaches zero when the two speckle images are different
while two identical images will give $\xi_{\mathrm{CC}}$ of one.
Correspondingly, we calculate $\xi_{\mathrm{CC}}$ for both CO and SO speckle
patterns with different $T_{\mathrm{cycle}}$ (Fig. 4c). The results again show
that CO domain distributions are essentially unchanged after thermal cycling
to a temperature far above $T_{\mathrm{CO}}$ while SO speckle pattern loses
reproducibility after the system is driven into the disordered state.

The domain memory effect of CO is caused by coupling to the host lattice.
Local potentials arising from structural disorder induced, for example, by Sr
doping, structural domain boundaries or octahedral tilts, provide nucleation
centers for the CO domains and effectively pin the domains during stripe
condensation. Since the average lattice structure of LSNO has translational
symmetry over a lengthscale smaller than CO wavelength it cannot, in self, pin
the CO domains into reproducible locations. In charge-ordered cuprate LBCO,
the speckle pattern of CO domains loses memory after the sample is heated
across the transition temperature from the low temperature orthorhombic (LTO)
phase into the high temperature tetragonal (HTT) phase Chen  _et al._ (2019).
Thus, it is expected that the pinning landscape for CO in LBCO is constrained
by twin boundaries created by the LTO structural distortion. In LSNO, the
lattice remains in the HTT phase and no long-range LTO distortion is observed
Hücker  _et al._ (2004). However, short-range stripe-related distortions have
been reported to persist up to high temperatures Abeykoon  _et al._ (2013). It
is possible that either these distortions, or local defects due to Sr-related
doping disorder, determine the pinning landscape of LSNO in a similar manner.

The pinning effect of CO to the structural disorder also evinces the relevance
of EPC in nickelates, which has been illustrated by the discovery of phonon
anomalies and nematic behaviors in LSNO Pashkevich  _et al._ (2000); Tranquada
_et al._ (2002); Merritt  _et al._ (2020); Zhong  _et al._ (2017). It has been
argued theoretically that without EPC CO will remain dynamic and not order
Hotta and Dagotto (2004). For structure-driven CO, phonons soften to zero
energy and drag the valence charge along with it to form spatial modulations.
Here, however, phonons are softened by a maximum of 20% Tranquada  _et al._
(2002), and charge stripes are formed to reduce Coulomb interactions. EPC
helps pin pre-formed charge stripes according to the lattice symmetry,
promoting the static CO. The presence of EPC further couples the CO domains to
structural disorder, which strengthens the CO against thermal fluctuations.
Consequently, when CO and SO lose correlations progressively upon heating or
cooling away from 70 K, the fluctuations of CO speckles increase more slowly
(Fig. 3).

SO behaves in a different way. During the formation of SO, the spins can align
either parallel or antiparallel to their quantization axis. This would disrupt
the reproducibility of SO speckles after thermal cycling across
$T_{\mathrm{SO}}$ even if the domain walls are in the same place (Fig. 4).
Moreover, the rotational degree of freedom provides an additional fluctuation
channel to the ordered spins, facilitating the loss of SO stability when
driven away from 70 K (Fig. 3). This is in line with the observation of spin
reorientation in LSNO at low temperatures Lee  _et al._ (2001); Freeman  _et
al._ (2004, 2006).

The robustness of CO stability and its pinning to the lattice demonstrate that
the stripe order in LSNO is charge driven. This directly verifies prior
theoretical predictions based on Landau theory of coupled charge and spin
order parameters Zachar  _et al._ (1998) and may reflect that stripe-order is
charge-driven in general. Our approach will be extendable to other materials
and even to other degrees of freedom such as orbital order, bringing a
powerful means to disentangle the formation mechanisms of intertwined ground
states.

###### Acknowledgements.

This material is based upon work supported by the U.S. Department of Energy
(DOE), Office of Basic Energy Sciences. Work at Brookhaven National Laboratory
was supported by the U.S. DOE, Office of Science, Office of Basic Energy
Sciences, under Contract No. DE-SC0012704. The work at Argonne National
Laboratory was supported by the U.S. Department of Energy, Office of Basic
Energy Sciences, under Contract No. DE-AC0206CH11357. D. G. M. acknowledges
funding from the Swiss National Science Foundation, Fellowship No.
P2EZP2_175092. This research used resources at the. 23-ID-1 beamline of the
National Synchrotron Light Source, a U.S. DOE Office of Science User Facility
operated for the DOE Office of Science by Brookhaven National Laboratory under
Contract No. DE-AC02-98CH10886.

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