# Evolution of plasmon excitations across the phase diagram of the cuprate
superconductor La2-xSrxCuO4

M. Hepting<EMAIL_ADDRESS>Max-Planck-Institute for Solid State Research,
Heisenbergstraße 1, 70569 Stuttgart, Germany  T. D. Boyko  Canadian Light
Source, Saskatoon, Saskatchewan S7N 2V3, Canada  V. Zimmermann  Max-Planck-
Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart,
Germany  M. Bejas  Facultad de Ciencias Exactas, Ingeniería y Agrimensura and
Instituto de Física de Rosario (UNR-CONICET), Avenida Pellegrini 250, 2000
Rosario, Argentina  Y. E. Suyolcu  Max-Planck-Institute for Solid State
Research, Heisenbergstraße 1, 70569 Stuttgart, Germany  P. Puphal  Max-Planck-
Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart,
Germany  R. J. Green  Department of Physics & Engineering Physics, University
of Saskatchewan, Saskatoon, Saskatchewan, Canada  Stewart Blusson Quantum
Matter Institute, University of British Columbia, Vancouver, British Columbia
V6T 1Z1, Canada  L. Zinni  Facultad de Ciencias Exactas, Ingeniería y
Agrimensura (UNR), Avenida Pellegrini 250, 2000 Rosario, Argentina  J. Kim
Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439,
USA  D. Casa  Advanced Photon Source, Argonne National Laboratory, Argonne,
Illinois 60439, USA  M. H. Upton  Advanced Photon Source, Argonne National
Laboratory, Argonne, Illinois 60439, USA  D. Wong  Helmholtz-Zentrum Berlin
für Materialien und Energie, Hahn-Meitner-Platz 1, D-14109 Berlin, Germany  C.
Schulz  Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-
Platz 1, D-14109 Berlin, Germany  M. Bartkowiak  Helmholtz-Zentrum Berlin für
Materialien und Energie, Hahn-Meitner-Platz 1, D-14109 Berlin, Germany  K.
Habicht  Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-
Platz 1, D-14109 Berlin, Germany  Institute of Physics and Astronomy,
University of Potsdam, Karl-Liebknecht-Straße 24/25, D-14476 Potsdam, Germany
E. Pomjakushina  Laboratory for Multiscale Materials Experiments (LMX), Paul
Scherrer Institute (PSI), CH-5232 Villigen, Switzerland  G. Cristiani  Max-
Planck-Institute for Solid State Research, Heisenbergstraße 1, 70569
Stuttgart, Germany  G. Logvenov  Max-Planck-Institute for Solid State
Research, Heisenbergstraße 1, 70569 Stuttgart, Germany  M. Minola  Max-Planck-
Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart,
Germany  H. Yamase  International Center of Materials Nanoarchitectonics,
National Institute for Materials Science, Tsukuba 305-0047, Japan  A. Greco
<EMAIL_ADDRESS>Facultad de Ciencias Exactas, Ingeniería y Agrimensura
and Instituto de Física de Rosario (UNR-CONICET), Avenida Pellegrini 250, 2000
Rosario, Argentina  B. Keimer  Max-Planck-Institute for Solid State Research,
Heisenbergstraße 1, 70569 Stuttgart, Germany

###### Abstract

We use resonant inelastic x-ray scattering (RIXS) at the O $K$\- and Cu
$K$-edges to investigate the doping- and temperature dependence of low-energy
plasmon excitations in La2-xSrxCuO4. We observe a monotonic increase of the
energy scale of the plasmons with increasing doping $x$ in the underdoped
regime, whereas a saturation occurs above optimal doping $x\gtrsim 0.16$ and
persists at least up to $x=0.4$. Furthermore, we find that the plasmon
excitations show only a marginal temperature dependence, and possible effects
due to the superconducting transition and the onset of strange metal behavior
are either absent or below the detection limit of our experiment. Taking into
account the strongly correlated character of the cuprates, we show that
layered $t$-$J$-$V$ model calculations accurately capture the increase of the
plasmon energy in the underdoped regime. However, the computed plasmon energy
continues to increase even for doping levels above $x\gtrsim 0.16$, which is
distinct from the experimentally observed saturation, and reaches a broad
maximum around $x=0.55$. We discuss whether possible lattice disorder in
overdoped samples, a renormalization of the electronic correlation strength at
high dopings, or an increasing relevance of non-planar Cu and O orbitals could
be responsible for the discrepancy between experiment and theory for doping
levels above $x=0.16$.

##  I Introduction

In spite of intense experimental and theoretical research efforts,
superconducting cuprates have retained an enigmatic character for more than
three decades. On the one hand, it is well known that their high-temperature
superconductivity emerges when charge carriers are doped into the CuO2 planes,
suppressing the long-range antiferromagnetic order that is prevalent in the
Mott insulating parent compounds [1, 2, 3, 4]. For instance, in the
prototypical cuprate La2-xSrxCuO4, the doping of $p\approx 0.07$ holes per
CuO2 unit is sufficient to stabilize superconductivity [5], and the highest
transition temperature $T_{c}$ is realized for $p\approx 0.16$. On the other
hand, a consensus on the microscopic mechanism mediating the superconductivity
in cuprates is still lacking. This elusive nature of superconductivity in
cuprates is mostly rooted in the fact that—in contrast to conventional
superconductors—cuprates are strongly correlated systems and their Cooper
pairs do not condense from a uniform and well-understood metallic state.
Instead, the superconducting dome as a function of charge carrier doping is
embedded in a variety of other enigmatic phases that also exhibit a strongly
correlated character, and either promote or compete with superconductivity in
a nontrivial fashion. Most prominently, these phases comprise the pseudogap,
spin and charge order, as well as the strange metal regime [1].

Insights into the electronic structure and the dynamics of doped charge
carriers in these diverse phases can be gained by various spectroscopic
techniques. For instance, x-ray absorption spectroscopy (XAS) and electron-
energy loss spectroscopy (EELS) at the O $K$-edge or Cu $L$-edge have revealed
how doped holes distribute within the CuO2 planes, that is, they reside within
the hybridized Cu $3d_{x^{2}-y^{2}}$ and O 2$p_{x,y}$ orbitals, with a
dominant influence of the latter orbitals [6, 7, 8, 9]. This behavior of the
doped holes is commonly translated into a low-energy effective model where the
quasiparticles are Zhang-Rice singlets (ZRS) [10, 11], which correspond to
plaquettes of holes and oxygen ions around a Cu ion, taking on the same role
as fully occupied or empty sites in an effective single-band Hubbard model
[12]. The single-band approach is also employed in the $t-J$ model [13, 14,
15, 16], which additionally discards all doubly occupied states. Nevertheless,
an accurate description of specific properties of cuprates, such as $pd$
charge-transfer related effects, requires the explicit consideration of all
three planar orbitals (Cu $3d_{x^{2}-y^{2}}$ and O 2$p_{x,y}$) [17, 18, 19,
20, 21, 22, 23, 24, 25, 26], and in some cases also non-planar orbitals [27,
28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41].

A different technique for investigating the charge dynamics of cuprates, both
in the normal and superconducting state, is infrared/optical spectroscopy [42,
43, 44]. Yet, in contrast to EELS, optical spectroscopy is a probe that is
essentially limited to the center of the Brillouin zone (BZ), due to the small
momentum of optical photons.

Recently, resonant inelastic x-ray scattering (RIXS) [45] has emerged as a
versatile tool to study electronic and magnetic excitations in cuprates,
providing both momentum resolution and a relatively high energy resolution.
Along these lines, soft x-ray RIXS at the Cu $L_{3}$ and O $K$-edge was
employed to investigate collective spin excitations [46, 47, 48, 49], and hard
x-ray RIXS at the Cu $K$-edge was used to probe electronic inter- and
intraband transitions [50, 51, 52, 53, 54, 55]. Moreover, collective charge
excitations were detected in Cu $L_{3}$ and O $K$-edge RIXS experiments, which
were attributed to acoustic plasmons [56, 57, 58, 59]. More recently, a high-
resolution Cu $L_{3}$-edge RIXS study revealed that these low-energy plasmon
branches are not strictly acoustic, but they exhibit an energy gap at the two-
dimensional BZ center [60].

Plasmons are a fundamental collective excitation of the charge carrier density
in metallic systems, mediated by long-range Coulomb interactions. An isotropic
three-dimensional metal usually exhibits only one optical plasmon branch,
while systems with stacked conducting planes that interact via poorly-screened
interlayer Coulomb interactions exhibit a set of acoustic plasmon branches
that disperse almost linearly towards zero energy at the BZ center [61, 62,
63, 64, 65]. However, in the presence of single-electron tunneling between the
planes, the latter branches are not strictly acoustic, but exhibit a gap at
the BZ center, which is proportional to the interlayer hopping integral
$t_{z}$ [60, 66]. In doped cuprates, which are composed of stacked conducting
CuO2 planes separated by dielectric spacer layers, early transmission EELS and
optical spectroscopy experiments already detected the optical plasmon, which
manifests itself as a plasma edge in reflectance spectra and a peak in the
loss function around 1 eV [67, 68]. Yet, direct evidence of the acoustic-like
plasmon branches was only provided later by high-resolution RIXS with
polarization analysis [56] and the capacity to independently vary the in- and
out-of-plane momentum transfer to probe the distinct plasmon dispersion along
different directions in the BZ [56, 58]. The plasmon dispersion is accurately
captured by a large-$N$ theory of the layered $t$-$J$-$V$ model [66, 69, 70,
58, 60], which intrinsically accounts for the strongly correlated character of
cuprates. Other methods to describe the plasmon dispersion in cuprates include
the random phase approximation (RPA) [71, 72], a combination of determinant
quantum Monte Carlo (DQMC) and RPA in a layered Hubbard model [56], and an
extended variational wave function approach [73].

In a broader context, low-energy plasmons in cuprates attracted considerable
attention already before their detection with RIXS [67, 68], as they were
suggested to play a role in the superconducting pairing [72], and their
evolution across the superconducting transition could reflect possible savings
of kinetic energy of the charge carriers [74]. Moreover, low-energy plasmon-
phonon modes [75, 76] were proposed to mediate superconductivity, or
contribute constructively to the high $T_{c}$ of cuprates [77]. In addition,
the evolution of plasmon excitations across the phase diagram of cuprates
might encode critical information about the charge carrier dynamics in the
normal state. Specifically, a debate is ongoing about the nature of the charge
carriers in the strange metal state [78, 79], which is characterized by a
linear-in-temperature resistivity [80]. Several theories suggest that the
standard Landau quasiparticle description for the charge carriers breaks down
in this regime [81, 82, 83], inhibiting the emergence of collective charge
excitations. As a consequence, strange metals could hamper the propagation of
plasmons, while they rapidly decay into a quantum critical continuum [84].
Indeed, the lack of well-defined plasmon peaks in EELS data acquired in a
reflection geometry was interpreted as a sign of such an anomalous plasmon
damping in overdoped cuprates, [85, 86], which is still discussed
controversially [87, 88].

In this work, we use soft x-ray RIXS at the O $K$-edge and hard x-ray RIXS at
the Cu $K$-edge to comprehensively map the evolution of low-energy plasmon
excitations across the phase diagram of La2-xSrxCuO4. Whereas the doping
dependence of plasmon excitations in electron-doped cuprates was addressed in
earlier studies [56, 57], the evolution of the plasmon as a function of hole-
doping has not been systematically investigated by RIXS, to the best of our
knowledge. We observe plasmon excitations for several Sr substitution levels
from $x=0.05$ to 0.4 and at various temperatures, including the strange metal
regime around $x=0.2$, where previous works proposed the decay of the plasmon
into an energy- and momentum-independent continuum [85, 86, 84]. We model the
doping dependence of the plasmon energy in the framework of layered
$t$-$J$-$V$ model calculations, which capture the experimentally observed
trends but show a deviation for dopings above $x\gtrsim 0.16$ (optimal
doping). We discuss the presence of disorder, a renormalization of the
electronic correlation strength in the overdoped regime, and an increasing
relevance of non-planar Cu and O orbitals as possible origins for this
discrepancy.

##  II Results

###  II.1 Electronic transport

Figure 1: In-plane resistivity $\rho_{ab}$ of La2-xSrxCuO4 films with various
hole-doping levels $x$. The $x$ = 0.05 curve is scaled by a factor 0.25. The
gray symbols in the inset correspond to the superconducting transition
temperatures $T_{c}$ of the films. The orange symbol indicates the $T_{c}$ of
the La1.8Sr0.2CuO4 single-crystal.

Thin films of La2-xSrxCuO4 were synthesized by ozone-assisted molecular beam
epitaxy (MBE) (see Appendix A for details). Our film with the lowest doping
($x$ = 0.05) is non-superconducting and becomes insulating at low temperatures
[Fig. 1], whereas the subsequent doping level $x$ = 0.1 shows a clear
superconducting transition. This is consistent with previous La2-xSrxCuO4 thin
film studies [89, 90, 91] and bulk measurements, where the onset of the
superconducting dome is situated around $x\sim 0.07$ [5]. Note that the long-
range antiferromagnetic (AFM) order, which is prevalent in the parent compound
La2CuO4, vanishes already around $x\sim 0.016$ [92]. For higher dopings, spin
and charge stripe ordered states emerge and are most pronounced around $x\sim
0.125$. In the bulk, a low-temperature-tetragonal (LTT) and low-temperature
less orthorhombic (LTLO) structural phase transition also occur in this doping
regime [93]. In thin epitaxial La2-xSrxCuO4 thin films, however, these
structural transitions are suppressed [90].

As expected, we detect the highest superconducting transition of our film
series for the optimal doping $x$ = 0.16, that is, $T_{c}$ = 41 K. The $T_{c}$
decreases slightly in the subsequent film with $x$ = 0.2, where the electrical
resistivity varies almost perfectly linear as a function of temperature in the
normal state at least up to 300 K [Fig. 1]. Such $T$-linear behavior is
commonly regarded as a key manifestation of the strange metal phenomenology
[81, 79, 82]. Previous transport studies on La2-xSrxCuO4 found that the
$T$-linear behavior is restored at low temperatures when the superconducting
transition is suppressed in strong magnetic fields, and that the strange metal
phase emanates in a fan-like shape from a putative quantum critical point at
$p^{*}\sim 0.19$ [94]. In addition to a series of thin films, we have
synthesized a single-crystal with a doping concentration close to $x$ = 0.2.
The temperature dependence of the transport and the $T_{c}$ of the single-
crystal are closely similar to the corresponding film, and will be discussed
in more detail below in the context of the Cu $K$-edge RIXS measurements.

For the overdoped $x$ = 0.3 film, we observe a decrease of $T_{c}$ to 26 K.
The highly overdoped $x$ = 0.4 film exhibits an onset of a superconducting
transition around 5 K. This behavior in the latter two samples differs from
reports on bulk La2-xSrxCuO4, where the superconducting dome typically
terminates around $x\sim 0.25$ [92], and from prior thin film studies that
reported only a slightly extended dome [89]. In the case of our films, we
attribute the substantially extended superconducting dome to the highly
oxidizing growth atmosphere [95] and the stabilizing effect of the epitaxial
strain of the substrate, which can possibly enhance the solubility limit of Sr
[96] and alleviate the formation of oxygen vacancies [97, 98]. A similar trend
was recently observed in ozone-assisted MBE grown La2-xCaxCuO4 films, where
the superconducting transition persisted for Ca-substitutions as high as $x$ =
0.5 [99].

Figure 2: (a)-(f) O $K$-edge XAS of La2-xSrxCuO4 films with various hole-
doping levels $x$. The spectra were collected at $T$ =300 K with
$\sigma$-polarized photons at an incident angle $\theta=35^{\circ}$. The peak
associated with the doped holes and the upper Hubbard band (UHB) peak are
indicated. (g)-(l) O $K$-edge RIXS of the La2-xSrxCuO4 films, for various
momenta along the $H$ direction, while $K=0$ and $L^{*}\approx 0.37$. Orange
symbols correspond to $H$ = 0.04, light red symbols to $H$ = 0.08, and red
symbols to $H$ = 0.12. The spectra were taken with $\sigma$-polarized photons
at energies tuned to the hole-peak in the corresponding XAS. A high-energy
background was subtracted from the spectra (see Appendix B). The solid gray
lines are fits, which include the dispersive plasmon (blue shaded peak) and
the non-dispersive bimagnon (dashed black peak). The other contributions to
the fit are omitted for clarity (for details of the fitting procedure see
Appendix B). Curves for different momenta are offset in the vertical direction
for clarity.

###  II.2 X-ray absorption at the O $K$-edge

The XAS signal across the near-edge fine structure of the O $K$-edge contains
hallmark fingerprints of the electronic structure of the cuprates [7, 31, 8,
21, 11, 33]. Figures 2a-f display the O $K$-edge XAS of the La2-xSrxCuO4 films
with dopings between $x$ = 0.05 and 0.4. The peak near 529.5 eV is
particularly pronounced for low doping levels and corresponds to the upper
Hubbard band (UHB), which arises as the on-site part of the Coulomb repulsion
$U$ splits the Cu 3$d$ band into a fully occupied lower Hubbard band (LHB) and
the empty UHB. The Cu 3$d$ band-derived features are visible in the O $K$-edge
XAS due to the strong 2$p$-3$d$ hybridization between O and Cu. The peak in
the XAS at lower energies ($\sim$ 528 eV) typically emerges in hole-doped
cuprates [100], and is associated with the ZRS states [11]. With increasing
hole density, spectral weight is transferred from the UHB peak to the hole-
peak. Notably, we observe that with increasing Sr-substitution the intensity
of the hole-peak increases continuously and the peak energy shifts to lower
energies, even up to $x$ = 0.4. This trend is consistent with the notion of a
continuous creation of new doped hole states and that the ZRS stays intact up
to highest doping levels [21, 11].

###  II.3 RIXS at the O $K$-edge

Figure 3: Doping-dependence of the plasmon energy. The circles are the plasmon
energies extracted from fits of the O $K$-edge RIXS spectra of the
La2-xSrxCuO4 films in Figs. 2g-l. The energies are grouped according to the
different in-plane momentum transfer $H$ (light to dark red). The gray
triangles correspond the plasmon energy at the three-dimensional Brillouin
zone center $H=K=L=0$ (optical plasmon), reported for La2-xSrxCuO4 single-
crystals in Ref. 44. The dashed lines are computed plasmon energies obtained
from layered $t$-$J$-$V$ model calculations. The labels next to the dashed
arrows indicate the momenta for which the calculations were performed.

In order to gain deeper insights into the dynamics of the doped holes and the
ZRS state for each doping level, we measured RIXS with the incident photon
energy tuned to the maximum of the hole-peak in the corresponding XAS from
Figs. 2a-f. In the following, we denote the momentum transfer by $(H,K,L)$ in
reciprocal lattice units $(2\pi/a,2\pi/b,2\pi/c)$, where $a$, $b$, and $c$ are
the lattice constants of La2-xSrxCuO4. However, the crystallographic unit cell
of La2-xSrxCuO4 contains two CuO2 planes, whereas only the distance $d=c/2$
between adjacent CuO2 planes is relevant for the periodicity of the plasmon
dispersion [56]. Hence, we utilize the index $L^{*}=L/2$ instead.

The obtained RIXS spectra for in-plane momenta $H$ = 0.04, 0.08, and 0.12 are
displayed in Figs. 2g-l. The out-of-plane momentum transfer of the spectra is
on average $L^{*}\sim$ 0.37 (see Appendix A for details). A broad background
contribution in the RIXS signal from fluorescence and $dd$-excitations [48] is
already subtracted from the spectra in Figs. 2g-l. The raw spectra without
background subtraction and details about the fitting procedure can be found in
Appendix B.

In Figs. 2g-l, the spectra are decomposed into several components, including
the elastic line centered at zero-energy loss, a non-dispersive feature around
510 meV, and a dispersive peak that evolves between 200 and 500 meV. Along the
lines of previous O $K$-edge RIXS studies, we assign the $\sim$510 meV peak
(dashed black line in Figs. 2g-l) to a non-dispersive bimagnon excitation [48,
49, 58, 59]. Note that long-range AFM order is absent for the investigated
doping levels, hence, this excitation corresponds strictly speaking to a bi-
paramagnon [46, 47] and not a bimagnon. The essentially unchanged energy scale
of our observed non-dispersive feature is consistent with the evolution of
paramagnon excitations in Cu $L$-edge RIXS experiments on La2-xSrxCuO4 films
with doping levels between $x$=0.1 and 0.4 [47]. In the following, we will
focus on the dispersive peak (solid blue line in Figs. 2g-l), which we
attribute to a low-energy plasmon excitation. Notably, the bimagnon and the
plasmon peak overlap strongly for momenta between $H$ = 0.04 and 0.12 [Figs.
2g-l], which precluded the identification of two distinct peaks in early RIXS
experiments on La2-xSrxCuO4 [49, 101]. Nevertheless, recent O $K$-edge RIXS
studies with improved energy resolution [58, 59] clearly discerned the two
excitations and determined the plasmon energy as a function of momentum
transfer $H$. Similarly, we extract the energy of the dispersive plasmon peak
from our fits, while keeping the energy of the bimagnon peak fixed. For
La1.84Sr0.16CuO4, we obtain closely similar plasmon energies [Fig. 2i and Fig.
3] as reported in Ref. 58 for the same doping (see Appendix C). Furthermore,
our extracted plasmon energies for La1.9Sr0.1CuO4 [Fig. 2h and Fig. 3] are not
far from those reported for La1.88Sr0.12CuO4 in Ref. 59 (see also Appendix C).
Yet, to the best of our knowledge, the plasmon energy for other doping levels
of La2-xSrxCuO4 has not yet been reported. Thus, we closely inspect Figs. 2g-l
in the following to extract a possible systematic trend in the doping-
dependence of the plasmon excitation.

Figure 2g shows the RIXS spectra for the lightly doped case $x=0.05$, where
our fits reveal that the broad shoulder next to the elastic line contains the
bimagnon and the plasmon excitation, similar to the $x=0.1$ and 0.16 cases
[Figs. 2h,i]. However, the energy scale and the dispersion of the plasmon for
$x=0.05$ are substantially smaller than for $x=0.1$ and 0.16. This difference
is also visible in Fig. 3, which plots the doping dependence of the plasmon
energy for each momentum $H$, suggesting a monotonic increase of the energy as
a function of $x$. An increase of the plasmon energy with increasing doping
concentration was also observed in RIXS experiments on the electron-doped
cuprate La2-xCexCuO4 [56, 57]. Note that this trend does not depend on the
details of our fitting procedure, such as the exact treatment of the bimagnon
peak. For instance, a fit that captures both the plasmon and the bimagnon
feature by a single broad peak function still yields an increase of the peak
energy with increasing doping concentration (see Appendix B).

For dopings beyond $x=0.16$, we find that the plasmon energy does not continue
to grow, but instead tends to saturate for $x=0.2$, 0.3, and 0.4 [Figs. 2j-l
and Fig. 3]. The onset of a flattening of the plasmon energy increase in the
overdoped regime was already noticed for electron-doped La2-xCexCuO4 and
attributed to a variation in the band dispersion and Fermi surface
reconstruction [56], or strong correlations [57]. However, the investigated
range for La2-xCexCuO4 was between $x=0.11$ and 0.18, whereas our present
study on La2-xSrxCuO4 extends over a much wider range from $x=0.05$ to 0.4,
which can allow for more robust conclusions.

Besides the systematic evolution of the plasmon energy [Fig. 3], we find that
the integrated intensity of the plasmon peak changes only marginally,
especially in the overdoped regime [Figs. 2j-l]. On the other hand, a small
decrease in the linewidth of the plasmon for high dopings might be discernible
in Figs. 2j-l. Yet, because of the strong overlap with the bimagnon, the
statistical error of the integrated intensities and linewidths extracted from
the fits are large and we refrain from further discussions of these two
quantities.

Nonetheless, the RIXS data in Figs. 2g-l clearly demonstrate that plasmon
excitations pervade the cuprate phase diagram at least from $x=0.05$ to 0.4
and for momenta as large as $H$ = 0.12. In particular, the unaffected
emergence of plasmon quasiparticles in the overdoped regime raises the
question, whether such an observation is compatible with previous proposals
based on reflection EELS experiments [85, 86, 88] that suggested the vanishing
of coherent plasmon excitations specifically in the strange metal regime of
cuprates. Hence, in the following we focus on the doping level $x=0.2$, which
is situated within the strange metal region in the La2-xSrxCuO4 phase diagram
[94, 90].

###  II.4 RIXS at the Cu $K$-edge

Figure 4: Cu $K$-edge RIXS of the La1.8Sr0.2CuO4 single-crystal. (a)
Electrical transport of the crystal. (b) XAS across the Cu $K$-edge measured
with $\pi$\- (solid line) and $\sigma$-polarized (dashed line) incident
photons. (c) Cu $K$-edge RIXS spectra for various in-plane momenta along the
diagonal direction ($H=K$), while the out-of-plane momentum is fixed to
$L^{*}=6.5$. The fit (gray line) to the experimental data (filled symbols)
includes the plasmon peak (blue shaded peak profile), an orbital excitation
(dashed black line), and other contributions (not shown here) that are
described in Appendix B. Curves for different momenta are offset in the
vertical direction for clarity. (d) RIXS spectra for momenta along the $L^{*}$
direction, while $H=0.09$ and $K=0$ are fixed. (e,f) Plasmon energy (red
squares) extracted from the fits (e) along the diagonal direction and (f)
along the $L^{*}$ direction. The solid black lines are the corresponding
plasmon dispersions computed in the $t$-$J$-$V$ model.

To corroborate the existence of plasmon excitations in the strange metal
regime and to obtain complementary information, we use RIXS at a different
absorption edge, that is, the Cu $K$-edge. An advantage of the high energy of
the hard x-ray photons tuned to the Cu $K$-edge is the wide range of
reciprocal space that becomes accessible for RIXS. However, due to the
increased penetration depth of the photons, we carry out the RIXS measurements
on a bulk single-crystal instead of the La1.8Sr0.2CuO4 thin film that was used
for the O $K$-edge experiment above. Our single-crystal exhibits a $T_{c}$ of
approximately 38 K and a similar $T$-linear transport behavior [Fig. 4a] as
the La1.8Sr0.2CuO4 thin film [Fig. 1], indicating that the sample is
excellently suitable for complementary RIXS measurements.

Figure 4b shows the XAS spectra of the La1.8Sr0.2CuO4 single-crystal across
the Cu $K$-edge measured with $\pi$\- and $\sigma$-polarized incident photons,
respectively. The main absorption line for the former polarization is centered
at 8993.7 eV (peak A), while the latter polarization yields pronounced peaks
around 8998.1 eV (peak B) and 9004.9 eV (peak C). The shapes of our XAS
spectra are in close agreement with those of overdoped La1.7Sr0.3CuO4 in Ref.
55. A relative shift of the absolute energy scale of approximately 1 eV in
Ref. 55 is likely due to a different energy calibration.

In order to identify the resonance energy where the emergence of the plasmon
excitation is expected in RIXS, we briefly recall previous assignments of the
Cu $K$-edge XAS features of cuprates. Note that low-energy plasmons were not
observed in previous Cu $K$-edge RIXS experiments on hole-doped cuprates,
whereas recent Cu $K$-edge RIXS studies on electron-doped cuprates revealed a
low-energy excitation that is compatible with the typical plasmon dispersion
[102]. Yet, the XAS of electron-doped cuprates includes several distinctions
from the hole-doped counterparts [51]. In hole-doped La2-xSrxCuO4, the XAS
peak A was attributed to well-screened intermediate states, and previous RIXS
studies detected charge-transfer (CT) excitations around 4 eV energy loss in
resonance to this energy [53, 55, 50]. For resonant energies in the region of
peak B and C, which are associated with poorly screened states, molecular-
orbital excitations around 7.5 eV energy loss were observed with RIXS [53, 55,
50]. However, both types of high-energy excitations are clearly not associated
with low-energy plasmon modes. Instead, the plasmon excitation must be located
in the CT gap, where indeed spectral weight below 3 eV energy loss was
reported, which showed a dependence on the in-plane momentum transfer [53, 54,
55]. This broad spectral weight, whose intensity resonates at the energy of
peak A, was initially attributed to an interband excitation from a low-lying
band below the Fermi energy to the ZRS band [54]. In contrast, an analysis of
the spectral intensity specifically around 1 eV in Ref. 55 suggested an
intraband excitation as the origin, although this assignment is incompatible
with theoretical calculations [103]. Nonetheless, Ref. 55 pointed out that for
incident energies around peak A, the RIXS intensity inside the CT gap contains
an appreciable contribution from another type of excitation, possibly
unresolved due to an insufficient energy resolution of $\Delta E\sim$ 350 meV
in the RIXS experiment [55].

Hence, we focus on incident energies in the region around peak A in our RIXS
experiment, and use a much improved energy resolution of $\Delta E\sim$ 80
meV. To reduce the strong elastic scattering at the Cu $K$-edge, we employ
$\pi$-polarized photons and keep the scattering angle close to $90^{\circ}$
for all measurements. This condition is realized for out-of-plane momenta
revolving around $L^{*}=6.5$ and small in-plane momenta. Further details about
the scattering geometry are given in Appendix A.

Figure 4c shows our Cu $K$-edge RIXS spectra for different in-plane momenta
along the diagonal direction with $H=K$, while the out-of-plane momentum is
fixed to $L^{*}=6.5$. The RIXS spectra were taken with an incident photon
energy of 8993 eV, which is slightly below the maximum of the XAS peak A [Fig.
4b]. Below 2 eV energy loss in the spectra, we resolve two prominent inelastic
features (for details about the fitting procedure, see Appendix B). The
feature centered around 1.5 eV energy loss (dashed black line in Fig. 4c)
shows only little or no dispersion as a function of momentum transfer, whereas
the feature at lower energies (solid blue line in Fig. 4c) disperses strongly
for varied momentum transfer. Both features resonate within an incident energy
interval of more than 1 eV above and below the maximum of XAS peak A. Upon
increasing the incident energy, we find that the inelastic feature that
appears at 1.5 eV for an incident energy of 8993 eV moves to higher energy
losses. In contrast, we observed that the low-energy feature exhibits a Raman-
like character and remains essentially at the same energy loss for different
incident energies (not shown here).

Furthermore, the latter feature shows a distinct out-of-plane dispersion [Fig.
4d], while the 1.5 eV feature remains at a constant energy loss within the
experimental error. These behaviors indicate that the 1.5 eV feature likely
corresponds to an inter- or intraorbital excitation, as proposed for the
features detected in Refs. 53, 54, 55. In contrast, the behaviors of our
dispersive feature at lower energies are compatible with a plasmon excitation,
in analogy to the characteristic in- and out-of-plane dispersion of the
plasmon mode recently identified in an O $K$-edge RIXS study on optimally
doped La1.84Sr0.16CuO4 [58], as well as that of electron-doped La2-xCexCuO4 in
an Cu $L$-edge RIXS study [56]. Moreover, the energy range of the dispersion
of the low-energy feature in the Cu $K$-edge RIXS spectra is comparable to the
range of the plasmon energy of the $x$ = 0.2 film in the O $K$-edge RIXS
spectra in Fig. 2j and Fig. 3. However, we note that the O $K$-edge RIXS
spectra are acquired along the axial $H$ direction for $L^{*}=0.37$, whereas
the Cu $K$-edge spectra in Fig. 4c are along the diagonal direction ($H=K$)
for $L^{*}=6.5$, which prevents a direct comparison. We rule out an assignment
of the low-energy mode to a bimagnon excitation, as magnetic excitations in
layered cuprates exhibit a two-dimensional character [56], which would be at
odds with the out-of-plane momentum dependence of our low-energy mode in Fig.
4d. Note that previous Cu $K$-edge RIXS studies detected a bimagnon excitation
in (underdoped) La2-xSrxCuO4, with a maximum in the intensity at the in-plane
momentum ($\pi$, 0) and vanishing intensity at (0, 0) [104]. On the other
hand, it was suggested that the bimagnon intensity in O $K$-edge RIXS is
maximal around (0, 0), as the two types of RIXS measurements are sensitive to
different parts of the bimagnon continuum [48]. The observation of a
bi-(para)magnon in our O $K$-edge RIXS data [Figs. 2g-l] and its absence (or
very low intensity) in our Cu $K$-edge spectra [Figs. 4c,d] is therefore
consistent with this notion.

For a definitive assignment of the dispersive mode in Figs. 4c,d, we turn to
layered $t$-$J$-$V$ model calculations (for details see Appendix D).
Specifically, we determine the plasmon energy as the maximum of the computed
imaginary part of the charge susceptibility
$\chi^{\prime\prime}(\textbf{q},\omega)$ [66, 69, 70, 58, 60] to model the
dispersion observed in Fig. 4c along the ($H$,$H$) direction for fixed
$L^{*}=6.5$, as well as the dispersion in Fig. 4d along the out-of-plane
direction for fixed $H$ = 0.09 and $K$ = 0. For an unbiased comparison between
the experimental and modeled dispersion [Figs. 4e,f], we compute
$\chi^{\prime\prime}(\textbf{q},\omega)$ of La1.8Sr0.2CuO4 using the same
parameters as in Ref. 60 for La1.84Sr0.16CuO4, and only adjust the different
hole-doping level $x$ = 0.2 in the calculation. Remarkably, Figs. 4e,f reveal
that both the in- and out-of-plane dispersion extracted from our experimental
data essentially match with the computed dispersion, although the model
appears to overestimate the plasmon energy. This small but systematic
discrepancy will be discussed in more detail below. Nonetheless, the
semiquantitative agreement between the experimental and the computed
dispersion corroborates our identification of the plasmon mode in the Cu
$K$-edge RIXS spectra, which in turn substantiates the emergence of plasmon
quasiparticles in La2-xSrxCuO4 in the $T$-linear resistivity regime.

###  II.5 Temperature dependence of the plasmon excitation

Figure 5: Temperature dependence of the plasmon excitation. (a) Cu $K$-edge
RIXS spectra of the La1.8Sr0.2CuO4 crystal at $T$ = 17, 35, and 100 K,
respectively. The momentum transfer is $H=K$ = 0.13 and $L^{*}=6.5$. (b,c) O
$K$-edge RIXS spectra of the (b) La1.84Sr0.16CuO4 and (c) La1.7Sr0.3CuO4 film
at $T$ = 15 and 100 K, respectively. The momentum transfer is $H$ = 0.1, $K$ =
0, and $L^{*}=0.5$.

In addition to the dependence of the plasmon excitation on the doping level
$x$ at a fixed temperature [Fig. 2], we investigate the temperature dependence
of the plasmon for specific doping levels in the following. Figure 5a displays
the Cu $K$-edge RIXS spectra of the La1.8Sr0.2CuO4 crystal at $T$ = 17, 35,
and 100 K. Within the experimental error, the plasmon peak remains unchanged
between 17 and 35 K, corresponding to measurements below and above the
superconducting transition ($T_{c}\sim 32$ K), respectively. Even an increase
of the temperature to 100 K affects the plasmon excitation only marginally,
that is, a subtle increase in the linewidth of the peak, likely due to thermal
broadening of the collective excitation. To confirm the minor temperature
dependence of the plasmon, we also perform O $K$-edge RIXS measurements on
La2-xSrxCuO4 films with $x$ = 0.16 and 0.3 [Figs. 5b,c], which exhibit a
$T_{c}$ of 41 and 26 K, respectively. Similarly to Fig. 5a, the temperature
dependence of the plasmon peak in the latter two samples is small between 15
and 100 K and almost indiscernible from the noise in the data, although a
general trend of a linewidth broadening at higher temperatures appears
evident.

The observed absence of strong renormalization effects of the plasmon energy
and linewidth across $T_{c}$ is consistent with detailed ellipsometry
experiments of the optical plasmon branch in Bi-based cuprates in Ref. 74.
These experiments indicated that the temperature evolution of the optical
plasmon branch with an energy of approximately 1 eV exhibits at most subtle
anomalies at the superconducting transition. The detection of such subtle
anomalies in the acoustic-like branches in Fig. 5 would require RIXS
measurements with much higher statistics and denser temperature sampling
intervals around $T_{c}$.

Moreover, to detect significant renormalization effects of the acoustic-like
plasmons across $T_{c}$, it may be necessary to conduct the RIXS measurements
in immediate vicinity to the in-plane BZ center ($H=K\approx 0$).
Specifically, at the in-plane BZ center the energy of acoustic-like plasmons
becomes minimal, and can be comparable to the energy scale of the
superconducting gap $2\Delta$, given that the interlayer-hopping $t_{z}$ of
the material [60] is relatively small. Such proximity or a crossover of the
two energy scales could affect the plasmon properties substantially. While
measurements in the present study were performed at momenta as small as $H$ =
0.04, the intense elastic line in the RIXS spectra progressively obscures the
plasmon peak when approaching the in-plane BZ center. Thus, our present RIXS
measurements do not provide sufficient information to ascertain or exclude a
relationship between plasmons and superconductivity. Nevertheless, future RIXS
investigations leveraging an improved energy resolution could yield new
insights into this subject.

###  II.6 Theoretical modeling of the doping dependence

After our confirmation of the emergence of the plasmon excitation for $x$ =
0.2 by Cu $K$-edge RIXS [Fig. 4] and our finding of a marginal temperature
dependence [Fig. 5], we return to the O $K$-edge RIXS data from Fig. 3 for a
detailed analysis of the doping-dependence in the framework of the $t$-$J$-$V$
model [66, 69, 70, 58, 60]. Specifically, Fig. 3 compares the experimentally
extracted plasmon energies (symbols) with the computed energies (dashed
lines), which correspond to the maximum of the imaginary part of the charge
susceptibility $\chi^{\prime\prime}(\textbf{q},\omega)$ calculated as a
function of the doping $x$. The data are grouped according to the momenta $H$
= 0.04, 0.08, and 0.12, respectively, while $K=0$ and $L^{*}=0.37$. Similarly
to our calculation in Figs. 4e,f, we employ the model parameters of
La1.84Sr0.16CuO4 derived in Ref. 60 (for details see Appendix D), and only
adjust the doping level. For completeness, we also present the computed
optical plasmon branch, and superimpose the optical plasmon energy of
La2-xSrxCuO4 crystals (gray triangles) reported in optical reflectivity
experiments [44]. Hence, these data points correspond to the plasmon energy at
the three-dimensional BZ center ($H=K=L=0$), and were derived from the
position of the peak in the loss function [44].

Notably, the computed plasmon energies in Fig. 3 agree well with the
experimentally observed energies for low dopings until approximately $x\approx
0.16$. This indicates that $t$-$J$-$V$ model calculations not only capture the
behavior of plasmons in cuprates at a fixed doping level [66, 69, 70, 58, 60],
but also describe their doping dependence without additional tuning of the
parameters, at least for small $H$ and small dopings. The observed increase of
the plasmon energy with increasing charge carrier concentration $x$ is
intuitively expected for a system with some correspondence to a simple free
electron model, where the plasmon energy scales with the charge carrier
density. Yet, according to the basic plasmon theory [105], the plasmon energy
should decrease with increasing hole doping, as it is proportional to
$\sqrt{n/m^{*}}$, where $n$ is the electron density and $m^{*}$ the effective
electron mass. Moreover, in RPA calculations (see Appendix E for details) the
plasmon energy follows the band filling effect, i.e., the plasmon energy
decreases with increasing hole-doping [Fig. 6]. Specifically, the plasmon
energy computed in RPA is maximal for $x=0$, while it decreases with
increasing $x$, and eventually vanishes for $x=1$, where the band becomes
completely empty. Obviously, our observed increase of the plasmon energy in
hole-doped La2-xSrxCuO4 until $x\approx 0.16$ is incompatible with the notion
of the basic plasmon theory [105] and the RPA results. A different theoretical
calculation for La2-xSrxCuO4 [76], which includes Hartree and exchange-
correlation contributions, considered a mixed plasmon-phonon mode at very
small in-plane momenta ($H\lesssim 0.005$). Whereas the plasmon-like branch of
the mixed mode is essentially doping-independent between $x=0.156$ and
$0.297$, the phonon-like branch increases in energy. However, the present RIXS
data were acquired at much larger momenta which implies that such a coupling
to phonons is not relevant for our plasmon dispersion in Fig. 3. In addition,
Ref. 76 considered the case of an uncoupled plasmon, which shows an increase
in energy for dopings between $x=0.156$ and $x=0.297$ for momenta $H\lesssim
0.005$, whereas it is almost doping-independent for slightly larger momenta.
For a direct comparison to the present RIXS data, it will be interesting to
carry out future calculations in analogy to Ref. 76, but at larger momenta
which coincide with the RIXS experiment and also for dopings below $x=0.156$.

In the present study, we employ the $t$-$J$-$V$ model to explore the entire
doping range of La2-xSrxCuO4 at momenta that directly correspond to the RIXS
experiment. The $t$-$J$-$V$ model takes into account strong correlations which
are known to be responsible for the insulating state at half-filling ($x=0$)
[4]. Accordingly, Figs. 3 and 6 illustrate that the plasmon energy computed in
the $t$-$J$-$V$ model tends towards zero for $x=0$, which is consistent with
the notion that insulators cannot exhibit plasmon excitations composed of
conduction electrons (or holes). Note, however, that the plasmon energy at
$x=0$ is expected to be non-zero also in the present calculations, because a
finite exchange interaction $J$ induces a finite bandwidth even at $x=0$, if
antiferromagnetic order is not considered [66]. Nonetheless, the present
theory is applicable at least down to $x=0.05$ [Figs. 3 and 6], which is the
lowest hole-doping for which plasmons were detected in the experiment [Fig.
2].

Interestingly, above optimal doping and especially in the highly overdoped
regime, the plasmon energies determined by RIXS tend to saturate around a
doping that is lower than the prediction of the $t$-$J$-$V$ calculation,
except for the momentum $H=0.04$ [Fig. 3]. The optical plasmon energy
determined from optical reflectivity experiments [44] even decreases between
$x$ = 0.2 and 0.34, although we note that the decrease instead of a saturation
might be due to possible sample quality issues for the highest doping in Ref.
44. In general, the calculations suggest that the saturation trend is due to a
broad peak structure of the plasmon energy as a function of doping [Fig. 6],
which emerges naturally when considering that the plasmon energy tends towards
zero for $x=0$ and 1. In particular, in the limit $x=1$, both the $t$-$J$-$V$
model calculations and the RPA results exhibit a strong decrease of the
plasmon energy. This suggests that the $t$-$J$-$V$ model recovers the band
filling effect when approaching the empty-band limit at $x=1$. However, Fig. 3
indicates that the $t$-$J$-$V$ model calculation predicts a saturation of the
plasmon energy at a doping level higher than that observed experimentally.

Figure 6: Doping-dependence of the plasmon energy computed in the layered
$t$-$J$-$V$ model (dashed lines) and the RPA (dotted lines). The parameters
used in the RPA calculations are the same as those used for the $t$-$J$-$V$
model calculation, except for $V_{c}/t=14$, which was adjusted to match the
experimental values of the optical plasma frequency ($\omega_{pl}^{opt}\sim
0.85$) at $x=0.16$ [43, 44].

##  III Discussion and Conclusion

Our RIXS measurements at the O $K$-edge revealed that from very low dopings up
to $x\approx 0.2$ the plasmon energy increases with increasing hole-doping
[Fig. 3]. The increase in this doping regime is accurately captured by
$t$-$J$-$V$ model calculations. The RPA calculations yield an opposite trend
[Fig. 6], namely, a decrease of the plasmon energy with increasing hole-
doping. This discrepancy points to the critical importance of an explicit
inclusion of strong electron correlations into the modeling of plasmon
excitations in cuprates, which is naturally implemented in the framework of
the $t$-$J$-$V$ calculations. However, our study reveals that the $t$-$J$-$V$
model predicts a monotonic increase of the plasmon energy at least up to
$x\approx 0.55$, whereas the experimental data indicates that the plasmon
energy increase tends to saturate beyond $x\approx 0.2$ [Fig. 3]. In the
following, we discuss three possible origins of this quantitative difference.

First, we address potential experimental shortcomings and note that the doping
dependence of the plasmon energy probed by RIXS is reminiscent of that of the
optical plasmon energy observed by optics experiments [44, 99]. This
similarity rules out that the saturation is due to errors or inaccuracies of
our RIXS measurements or data analysis. However, we cannot fully rule out that
structural defects or chemical disorder of the dopant ions in our samples play
a role in the saturation. In fact, the details of the distribution of the
dopant ions [106, 107] as well as the emergence of oxygen vacancies [98] and
the effects of out-of-plane substitutional disorder [108, 109, 110, 111] are
currently under debate for cuprates. Along these lines, we note that the room-
temperature resistivity of the $x$ = 0.4 film lies above that of the $x$ = 0.3
film [Fig. 1], whereas the resistivities decrease sequentially from $x$ = 0.05
to 0.3. This suggests that the electronic properties are affected by
structural defects and/or disorder at least in our $x$ = 0.4 film. Yet, we
stress that a Raman spectroscopy study reported that the formation of oxygen
vacancies can be greatly alleviated in our ozone-assisted MBE grown
La2-xSrxCuO4 films for dopings below $x$ = 0.35 [97]. Moreover, the onset of
the plasmon energy saturation is observed already between $x=0.16$ and 0.2,
whereas the effects of disorder typically emerge only in the overdoped and
highly overdoped regime [108, 109, 110, 111, 98]. These considerations suggest
that while disorder might have some effect on the propagation of the
collective plasmon excitation for our overdoped samples, it cannot fully
account for the observed plasmon energy saturation.

Second, there might be an overestimation of correlations in the $t$-$J$-$V$
model, which pushes the crossover from a strongly correlated to a moderately
or weakly correlated regime to higher dopings. In more detail, the broad
maximum of the plasmon energy between $0.4\lesssim x\lesssim 0.6$ in the
$t$-$J$-$V$ calculation in Fig. 6 can be regarded as a crossover between
strong and weak correlation physics. At high dopings, electrons are more
dilute and the local constraint in the $t$-$J$-$V$ model that prohibits double
occupancy of electrons as well as the nearest-neighbor exchange interactions
are less effective. This is the reason why the $t$-$J$-$V$ and RPA
calculations show the same trend in the limit of $x=1$ in Fig. 6. If the
present $t$-$J$-$V$ calculations overestimate the correlation effect for high
dopings in cuprates, a weaker correlation is expected to yield the broad peak
at lower doping, which would be in better agreement with the experimental data
in Fig. 3. Whether the experimental plasmon energy also exhibits a crossover
behavior as predicted by the theory and hence starts to decrease for some
doping level above $x=0.4$ is an interesting topic for future experiments if
La2-xSrxCuO4 films with such high dopings become available.

Third, the activation of additional orbital degrees of freedom at large doping
might play a significant role in the doping dependence. Such an activation is
not captured by the one-band $t$-$J$-$V$ model, which is an effective model
for the three-band Hubbard model, where the Cu $3d_{x^{2}-y^{2}}$ orbitals are
hybridized with the 2$p_{x}$ and 2$p_{y}$ orbitals of the in-plane oxygen
ions, forming the ZRS band [12]. Yet, there are theoretical indications that
other planar [17, 18, 19, 20, 22, 23, 24, 25, 26] and non-planar orbitals [27,
28, 30, 31, 33, 32, 36, 35, 38, 29, 41] should be taken into account in
comprehensive models of cuprates. In the overdoped regime, specifically the Cu
$3d_{3z^{2}-r^{2}}$, interstitial $4s$, and the $2p_{z}$ orbital of the apical
oxygen were proposed to become relevant [30, 34, 37, 39, 40]. For instance, a
six-band Hubbard model, which includes the Cu $3d_{3z^{2}-r^{2}}$ and O
$2p_{z}$ and $2p_{-z}$ orbitals, was employed to compute the doping dependence
of the optical conductivity of La2-xSrxCuO4 [34]. The obtained doping
dependence is consistent with the experimentally observed continuous increase
in optical spectral weight below 1.5 eV with increasing hole-doping [99, 44].
Yet, the detailed dependence of the optical plasmon energy in the overdoped
regime [Fig. 3] is difficult to discern in Ref. 44 and might be affected by
disorder in the overdoped single-crystalline samples. On the other hand, a
recent ellipsometry study on highly oxidized La2-xCaxCuO4 films [99] carefully
decomposed the optical spectral weight below 1.5 eV into a Drude peak and
contributions from intra-ZRS transitions, revealing that the former component
as well as the (bare) plasma frequency saturate above $x\approx 0.2$. By
contrast, the second component associated with the intra-ZRS transitions and
the Cu $3d_{3z^{2}-r^{2}}$ and O $2p_{z}$ orbitals increases continuously up
to the highest measured doping. The sum of both components is again compatible
with the spectral weight evolution calculated in Ref. 34. These distinct
behaviors of the components in the optical spectra are strikingly reminiscent
of our RIXS and XAS data. In particular, it seems likely that the peak in our
RIXS spectra predominantly reflects the dynamics of the Drude charge carriers
in the planar orbitals, whereas holes that are added to the system beyond
$x\approx 0.2$ might be mostly associated with non-planar orbitals and do not
contribute to the collective plasmon excitation probed by RIXS. On the other
hand, the continuous increase of the XAS spectral weight of the hole-peak up
to $x$ = 0.4 [Fig. 2a-f] might reflect the total number of doped holes, both
in the planar and non-planar orbitals. In this context, note that our XAS
includes contributions from both planar and non-planar orbitals due to the
employed measurement geometry (see Appendix A for details). Hence, future XAS
and RIXS experiments in a measurement geometry [31] that is exclusively
sensitive to charge carriers in the non-planar orbitals are highly desirable
for confirming the above scenario and gaining further insights into the
mobility of holes in the highly overdoped regime of cuprates.

Besides the observed saturation of the plasmon energy, our results provide
fresh insights into the debate on whether plasmons can exist at all doping
levels in the cuprate phase diagram, or if specific phases preclude their
emergence. In particular, previous reflection EELS experiments on
Bi2.1Sr1.9CaCu2O8+x [86, 85] concluded that well-defined plasmon excitations
are replaced by a featureless, temperature- and momentum-independent continuum
in the strange metal regime, although the authors specified in a recent
comment [88] that they identified the spectral feature as a plasmon excitation
for momenta $H\leq 0.12$. For larger momenta, it was suggested that the decay
of the plasmon into a continuum is unusually strong [86, 85, 88], and that the
putative quantum critical nature of the continuum can be captured by
holographic theories [84]. Yet, our combined O $K$\- and Cu $K$-edge RIXS
results on La2-xSrxCuO4 clearly indicate that dispersive plasmon excitations
exist for dopings between $x$ = 0.05 and 0.4, including the strange metal
regime at $x\sim 0.2$ for momenta at least up to $H$ = 0.15. Moreover, between
$x$ = 0.05 and 0.2, the excitation energies are theoretically well described
by the $t$-$J$-$V$ model without any fine-tuning of the parameters, which is a
long-established model for cuprates. However, the linewidths of the plasmon
excitations in Figs. 4c,d are comparable to the respective plasmon energies
for all measured momenta, indicating that the mode is appreciably damped. In
general, several mechanisms can be responsible for a damping of plasmons in
the long-wavelength limit around the BZ center, including scattering with
phonons and Umklapp scattering [112], which can provide the momentum necessary
to couple the plasmon to the electron-hole continuum. Furthermore, the damping
around the BZ center depends on the detailed balance between the correlation
strength and the size of the plasmon gap due to the interlayer hopping $t_{z}$
[113]. At large momenta, a pronounced plasmon decay is expected even in a
conventional Lindhard model picture due to Landau damping when the plasmon
traverses the energy scale of intra- and interband transitions [87]. A
definite distinction between the presence of these mechanisms and a quantum
critical decay [84] appears unfeasible based on the present RIXS data, whereas
a comprehensive study of the plasmon linewidth evolution for both small and
large momenta might give critical insights. In particular, while previous
reflection EELS studies were focused on relatively large momenta [86, 85],
future Cu $K$-edge RIXS experiments are anticipated to provide detailed
information about the entire BZ.

In a broader context, we point out that a cautious use of the terminology for
the plasmon phenomenology in cuprates is important. For instance, Ref. 65
refers to the ensemble of branches between the optical plasmon and the lowest-
energy (quasi-)acoustic branch in terms of a plasmon continuum. Such a
continuum is not directly related to the decay of a nominally sharp plasmon
excitation into a continuum due to various classical or quantum critical
damping mechanisms [112, 113, 87, 84]. In fact, the degree of the continuum-
like character of the plasmon branch ensemble is determined by the total
number $n$ of CuO2 planes of a system, as the number of (quasi-)acoustic
branches is $n-1$ [67]. Notably, for a given in-plane momentum, the signal
probed by the reflection EELS technique is integrated over all (or at least a
broad range of) out-of-plane momenta [65]. In contrast, the momentum
resolution of RIXS is relatively high in all directions of reciprocal space
[114], which implies that especially for thin-film samples with a small number
of CuO2 planes the out-of-plane integration can be close to the limit of
covering only a single branch. Whether this intrinsic difference between the
two experimental techniques is responsible for the distinct spectral features
observed below 1 eV energy loss remains to be tested in future experiments. In
particular, future Cu $K$-edge RIXS studies on Bi2.1Sr1.9CaCu2O8+x and other
hole-doped cuprates are highly desirable for establishing the universality of
plasmon excitations in the strange metal regime of the cuprates and clarifying
the nature of the plasmon decay.

## Acknowledgments

We thank A.V. Boris, G. Kim, and C. Falter for fruitful discussions. A.G.
acknowledges the Max-Planck-Institute for Solid State Research in Stuttgart
for hospitality and financial support. H.Y. was supported by JSPS KAKENHI
Grant No. JP20H01856, Japan. Part of the research described in this paper was
performed at the Canadian Light Source, a national research facility of the
University of Saskatchewan, which is supported by the Canada Foundation for
Innovation (CFI), the Natural Sciences and Engineering Research Council
(NSERC), the National Research Council (NRC), the Canadian Institutes of
Health Research (CIHR), the Government of Saskatchewan, and the University of
Saskatchewan. 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.

##  Appendix A Experimental details

A series of La2-xSrxCuO4 films with a thickness of 15 unit cells was grown by
ozone-assisted molecular beam epitaxy (MBE) on (001) oriented LaSrAlO4 (LSAO)
substrates. The nominal Sr substitution levels were $x$ = 0.05, 0.1, 0.16,
0.2, 0.3, and 0.4. The hole-doping concentration in La2-xSrxCuO4 is directly
proportional to the Sr-substitution level ($p=x$). The Van-der-Pauw method was
used to determine the in-plane electrical resistivity. The $T_{c}$ values for
$x$ = 0.1, 0.16, 0.2, 0.3, and 0.4 films are 35 K, 41 K, 32 K, 26 K, and below
2 K, respectively. The $x$ = 0.05 film did not show an onset of a
superconducting transition. In addition to the films, a single-crystal with a
stoichiometry close to La1.8Sr0.2CuO4 was grown by the optical floating zone
technique. A standard Crystal Systems CSC FZ 1000 equipped with 300 W lamps
was used and the crystal was grown at a rate of 1 mm/h in mixed Ar with 20% O2
atmosphere at 3 bar. The refinement of single-crystal x-ray diffraction (XRD)
data on two pieces broken off from the crystal indicated a Sr content of
$x=0.196\pm 0.03$ and $0.17\pm 0.03$, respectively. Energy dispersive x-ray
spectroscopy (EDS) on the crystal surface yielded $x=0.21$ when normalized to
a nominally ideal Cu stoichiometry. The $T_{c}$ of the crystal was about 38 K.

The doping dependence of the plasmon excitation in the La2-xSrxCuO4 films was
measured using O $K$-edge RIXS at the REIXS beamline of the Canadian Light
Source (CLS). The RIXS spectra were collected at 300 K using a Rowland circle
spectrometer with a combined energy resolution $\Delta E\sim$ 190 meV and
linearly polarized photons ($\sigma$-polarization). The films were mounted
such that the $c$\- and the $a/b$-axes were lying in the scattering plane. The
crystal structure of La2-xSrxCuO4 is orthorhombic, but as the difference
between the $a$ and $b$ axes is small, and the films are twinned in the
$ab$-plane, we do not distinguish between $a$ and $b$ in the following. The
scattering angle $2\theta$ was kept fixed at $90^{\circ}$, while the angle
$\theta$ between the sample surface and the incident x-rays was varied to
probe different momentum transfers. The XAS data were taken with
$\sigma$-polarized photons at an incident angle $\theta=35^{\circ}$ in partial
fluorescence yield using a silicon drift detector, collecting only the O
$K_{\alpha}$ emission line.

We denote the momentum transfer by $(H,K,L)$ in reciprocal lattice units
$(2\pi/a,2\pi/b,2\pi/c)$, where $a$, $b$, and $c$ are the lattice constants of
La2-xSrxCuO4. Note that the crystallographic unit cell of La2-xSrxCuO4
contains two CuO2 planes, whereas only the distance $d=c/2$ between adjacent
CuO2 planes is relevant for the periodicity of the plasmon dispersion [56].
Hence, the index $L^{*}=L/2$ is used in the following. Since the scattering
angle $2\theta$ was fixed for the measurements at the REIXS beamline, each
variation of the in-plane momentum transfer also leads to a (minor) change of
the out-of-plane momentum. Specifically, a change of $H$ from 0.04 to 0.12 in
the O $K$-edge RIXS measurements involves a concomitant change of $L^{*}$ from
0.393 to 0.346. However, since the plasmon dispersion as a function of $L^{*}$
is relatively shallow between $\sim$0.3 and 0.4 [66], we neglect the change of
$L^{*}$ in the following and employ the averaged value of $L^{*}=0.37$ for the
analysis of the O $K$-edge RIXS data.

Additional O $K$-edge RIXS measurements on selected La2-xSrxCuO4 films were
conducted at the U41-PEAXIS beamline [115, 116] of BESSY II at the Helmholtz-
Zentrum Berlin (HZB). The spectra were collected at various temperatures
between 15 and 100 K, with a combined energy resolution $\Delta E\sim$ 90 meV.
Typically, $\sigma$-polarized photons are used for soft x-ray measurements of
plasmon excitations in cuprates [56, 58, 57, 60], as the charge signal is
enhanced over the magnetic signal. However, since only $\pi$-polarization was
available for our RIXS experiment at U41-PEAXIS, the temperature-series was
taken at momenta that correspond to incident angles $\theta$ close to normal
incidence. Accordingly, the projection of the polarization vector of the
electric field of the $\pi$-polarized photons onto the CuO2 planes was
maximized, enhancing the charge signal as much as possible. The scattering
angle $2\theta$ at U41-PEAXIS can be varied across a broad range. For our
measurement, $2\theta$ was approximately $137^{\circ}$. The corresponding in-
plane momentum transfer was $H$ = 0.1 and the out-of-plane momentum transfer
$L^{*}=0.5$.

The Cu $K$-edge RIXS measurements on the La1.8Sr0.2CuO4 single-crystal were
carried out at the MERIX spectrometer at sector 27 at the Advanced Photon
Source (APS). RIXS spectra were collected between 17 and 100 K. The incident
photons were monochromatized by a four-bounce monochromator with
asymmetrically cut Si (4, 0, 0) crystals. A spherical diced Ge (3, 3, 7)
analyzer was used and the flight path between the analyzer and sample/detector
was approximately 1 meter. The combined energy resolution was $\Delta E\sim$
80 meV. The incident angle $\theta$ and the scattering angle $2\theta$ could
be varied independently from each other. To minimize the strong elastic
scattering signal in the RIXS spectra, $\pi$-polarized photons and a
scattering angle $2\theta$ close to $90^{\circ}$ was chosen. Specifically, for
all measured $H$, $K$, and $L$ values of momentum-dependence of the plasmon
excitation, the angle $2\theta$ remained close to $90^{\circ}$. The scattering
geometry of the RIXS experiment was such that the $c$-axis and the diagonal
between the $a$\- and $b$-axis of La1.8Sr0.2CuO4 were lying in the scattering
plane. The XAS data were collected in the fluorescence yield mode with $\pi$\-
and $\sigma$-polarized photons, respectively, at a shallow incident angle
$\theta$ (close to grazing incidence). Hence, the polarization vector of the
electric field of the incoming photons lies within the $a/b$ plane of the
crystallographic unit cell for $\sigma$-linearly polarized photons, whereas it
is mostly parallel to the $c$-axis for $\pi$-polarized photons, but with a
small projection on the $a/b$ plane due to a small $\theta$.

##  Appendix B RIXS raw data and fits

Figure 7a shows all components of a representative fit of the O $K$-edge RIXS
spectra of the La2-xSrxCuO4 films of Fig. 2 of the main text. The displayed
RIXS spectrum was taken on the $x$ = 0.16 film at momentum $H$ = 0.08, $K$ =
0, and $L^{*}$ = 0.37. The elastic peak was modeled by a Gaussian and the
other contributions in the spectra by anti-symmetrized Lorentzians [56, 58],
convoluted with the energy resolution of $\Delta E$ = 190 meV via Gaussian
convolution. The anti-symmetrized Lorentzian profiles ensure zero intensity at
zero energy loss (prior to convolution) for the inelastic features. The
inelastic features are assigned to a phonon, a plasmon, and a non-dispersive
bimagnon. In the fits, the energy of the bimagnon peaks was kept fixed, while
all other fit parameters were varied. Furthermore, the spectra contain a broad
and intense background (bkg) feature, which peaks beyond 2 eV energy loss.
This feature consists of fluorescence and $dd$-excitations [48]. In the fits,
we included the tail of the feature below 2.1 eV energy loss. For clarity, the
signal originating from this feature was subtracted in the RIXS spectra shown
in Fig. 2 of the main text. A collection of the raw data of the O $K$-edge
RIXS spectra of the La2-xSrxCuO4 films is shown in Fig. 8.

Figure 7: Representative fits of the RIXS spectra. (a) Fit of the O $K$-edge
RIXS spectrum of the $x$ = 0.16 film taken at momentum $H$ = 0.08, $K$ = 0,
and $L^{*}$ = 0.37. The individual contributions of the fit are described in
the text. (b) Fit of the Cu $K$-edge RIXS spectrum of the $x$ = 0.2 single-
crystal taken at momentum momentum $H$ = 0.09, $K$ = 0.09, and $L^{*}$ = 6.5.
Figure 8: (a)-(f) Raw RIXS spectra (filled symbols) of the La2-xSrxCuO4 films
with various hole-doping levels $x$ taken at the O $K$-edge, together with
fits (gray lines). Curves for different momenta are offset in the vertical
direction for clarity.

The observed trend of an increase of the peak energy with increasing doping
concentration up to $x=0.16$ does not depend on the details of our fitting
procedure, such as the exact treatment of the bimagnon peak. For instance, a
fit that captures both the plasmon and the bimagnon feature by a single broad
peak function yields similar results to Fig. 3 of the main text. This is
illustrated in Fig. 9, which shows the energy of the plasmon peak (blue
squares, two-peak fit) extracted from a fit that includes separate plasmon and
bimagnon peaks (see also Fig. 3 of the main text), together with the energy of
a single peak (light blue circles, single-peak fit) that comprises both the
plasmon and the bimagnon feature.

Figure 9: Results for alternative fits of the RIXS spectra of the La2-xSrxCuO4
films. (a) Fit results for the $x=0.05$ film. The blue squares (two-peak fit)
indicate the plasmon energy extracted from a fit that includes separate
plasmon and bimagnon peaks, in analogy to Fig. 7a. The light blue circles
(single-peak fit) indicate the energy of a single peak that comprises both the
plasmon and the bimagnon feature (fit not shown here). (b) Fit results for the
$x=0.16$ film.

The components of a representative fit of the Cu $K$-edge RIXS spectra of the
La1.8Sr0.2CuO4 single-crystal of Fig. 4 of the main text is given in Fig. 7b.
The displayed RIXS spectrum was taken at momentum $H$ = $K$ = 0.09, and
$L^{*}$ = 6.5. The elastic peak was modeled by a Gaussian and the other
contributions in the spectra by anti-symmetrized Lorentzians, convoluted with
the energy resolution of $\Delta E$ = 80 meV via Gaussian convolution. The
inelastic features are assigned to a phonon, a plasmon, a non-dispersive
orbital excitation around 1.5 eV energy loss (orbital exc. 1), and an
additional orbital excitation at higher energy losses (orbital exc. 2) [54]. A
collection of the raw data of the Cu $K$-edge RIXS spectra is shown in Fig.
10.

Figure 10: Raw RIXS spectra (filled symbols) of the La1.8Sr0.2CuO4 single-
crystal taken at the Cu $K$-edge, together with fits (gray lines). Curves for
different momenta are offset in the vertical direction for clarity. (a) RIXS
spectra for various in-plane momenta along the ($H,H$) direction, while the
out-of-plane momentum is fixed to $L^{*}=6.5$. (b) RIXS spectra for momenta
along the $L^{*}$ direction, while $H=0.09$ and $K=0$ are fixed.

##  Appendix C Comparison to literature

Figure 11 shows the plasmon (bimagnon) energies of two representative
La2-xSrxCuO4 film ($x=0.1$ and 0.16) as a function of the momentum transfer
$H$. In addition, the plasmon (bimagnon) energies taken at similar momenta and
dopings in Refs. [58, 59] are superimposed. The plasmon (bimagnon) energies
determined in the present work are similar to those from Refs. [58, 59].

Figure 11: Comparison of the fit results of the RIXS spectra with literature.
(a) Blue squares (dark gray triangles) are the plasmon (bimagnon) energies
extracted from the O $K$-edge RIXS spectra of the La2-xSrxCuO4 film with
$x=0.16$. Blue circles (light gray triangles) are the plasmon (bimagnon)
energies of a La2-xSrxCuO4 single-crystal with $x=0.16$ from Ref. 58. (b) Fit
results of the La2-xSrxCuO4 film with $x=0.1$ and the results of Ref. 59 on a
La2-xSrxCuO4 single-crystal with $x=0.12$.

##  Appendix D Theoretical details

For the theoretical description of plasmon excitations in cuprates, we employ
the layered $t$-$J$-$V$ model [66, 69], which incorporates the three-
dimensional character of the plasmons [56] via the long-range (interlayer)
Coulomb interaction $V$:

$H=-\sum_{i,j,\sigma}t_{ij}\tilde{c}^{\dagger}_{i\sigma}\tilde{c}_{j\sigma}+\sum_{\langle
i,j\rangle}J_{ij}\left(\vec{S}_{i}\cdot\vec{S}_{j}-\frac{1}{4}n_{i}n_{j}\right)\\\
+\frac{1}{2}\sum_{i,j}V_{ij}n_{i}n_{j}\,$ (1)

where the sites $i$ and $j$ run over a three-dimensional lattice. The hopping
$t_{ij}$ takes a value $t$ $(t^{\prime})$ between the first (second) nearest-
neighbors on the square lattice. The hopping integral between layers is scaled
by $t_{z}$. $\langle i,j\rangle$ denotes a nearest-neighbor pair of sites. The
exchange interaction $J_{ij}=J$ is considered only inside the plane; the
exchange term between the planes ($J_{\perp}$) is much smaller than $J$ [117].
$\tilde{c}^{\dagger}_{i\sigma}$ ($\tilde{c}_{i\sigma}$) is the creation
(annihilation) operators of electrons with spin $\sigma(=\uparrow,\downarrow)$
in the Fock space without any double occupancy.
$n_{i}=\sum_{\sigma}\tilde{c}^{\dagger}_{i\sigma}\tilde{c}_{i\sigma}$ is the
electron density operator and $\vec{S}_{i}$ is the spin operator. $V_{ij}$ is
the long-range Coulomb interaction on the lattice and is given in momentum
space by [118]:

$V({\bf q})=\frac{V_{c}}{A(q_{x},q_{y})-\cos q_{z}}\,,$ (2)

where $V_{c}=e^{2}d(2\epsilon_{\perp}a^{2})^{-1}$ and

$A(q_{x},q_{y})=\alpha(2-\cos q_{x}-\cos q_{y})+1\,.$ (3)

Here $\alpha=\frac{\tilde{\epsilon}}{(a/d)^{2}}$, where
$\tilde{\epsilon}=\epsilon_{\parallel}/\epsilon_{\perp}$, and
$\epsilon_{\parallel}$ and $\epsilon_{\perp}$ are the dielectric constants
parallel and perpendicular to the planes, respectively; $e$ is the electric
charge of electrons; $a$ is the lattice spacing in the planes and the in-plane
momentum $q_{\parallel}=(q_{x},q_{y})$ is given in units of $1/a$; similarly
$d$ is the distance between the planes and the out-of-plane momentum $q_{z}$
is given in units of $1/d$.

To compute the plasmon excitations we use the large-$N$ formalism given in
Ref. 66. The charge-charge correlation function can be calculated as

$\chi({\bf q},\mathrm{i}\omega_{n})=N\left(\frac{x}{2}\right)^{2}D_{11}({\bf
q},\mathrm{i}\omega_{n})\,.$ (4)

Here $D_{11}$ is the element $(1,1)$ of the $6\times 6$ bosonic propagator

$\displaystyle[D_{ab}({\bf q},\mathrm{i}\omega_{n})]^{-1}=[D^{(0)}_{ab}({\bf
q},\mathrm{i}\omega_{n})]^{-1}-\Pi_{ab}({\bf q},\mathrm{i}\omega_{n})\,,$ (5)

where $\omega_{n}$ are the Matsubara frequencies and $a$,$b$ run from $1$ to
$6$. The factor $N$ in Eq. (4) comes from the sum over the $N$ fermionic
channels after the extension of the spin index from $2$ to $N$. The
corresponding expressions for the bare bosonic propagator $D^{(0)}_{ab}({\bf
q},\mathrm{i}\omega_{n})$ and the bosonic self-energy $\Pi_{ab}({\bf
q},\mathrm{i}\omega_{n})$ are given in Ref. 66.

The large-$N$ formalism successfully captured the experimentally observed low-
energy plasmon dispersions in various cuprates for specific doping levels in
prior works [69, 70, 58, 60]. As also used in Ref. 60 to fit and reproduce the
experimental plasmon dispersion in La1.84Sr0.16CuO4, we use the parameters
$t/2=0.35$ eV, $t^{\prime}/t=-0.2$, $t_{z}/t=0.01$, $J/t=0.3$, $V_{c}/t=31$,
and $\alpha=3.5$ for La2-xSrxCuO4 in the present work. The number of planes is
$N_{z}=30$ and temperature $T=0$.

##  Appendix E Summary of the RPA formalism

In the RPA the charge correlation function is given by [105]

$\displaystyle\chi_{\rm RPA}({\bf
q},\mathrm{i}\omega_{n})=\frac{\chi^{(0)}({\bf
q},\mathrm{i}\omega_{n})}{1-V({\bf q})\chi^{(0)}({\bf
q},\mathrm{i}\omega_{n})},$ (6)

where $\chi^{(0)}({\bf q},\mathrm{i}\omega_{n})$ is the Lindhard function
which accounts for the particle-hole continuum. The electron dispersion
$E_{\bf{k}}$ is

$\displaystyle E_{\bf{k}}=E_{\bf{k}}^{\parallel}+E_{\bf{k}}^{\perp},$ (7)

where

$\displaystyle E_{\bf{k}}^{\parallel}=-2t(\cos k_{x}+\cos
k_{y})-4t^{\prime}\cos k_{x}\cos k_{y}-\mu\,,$ (8) $\displaystyle
E_{\bf{k}}^{\perp}=-2t_{z}(\cos k_{x}-\cos k_{y})^{2}\cos k_{z}\,.$ (9)

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