Doping evolution of the charge excitations and electron correlations in electron-doped superconducting La$_{2-x}$Ce$_{x}$CuO$_{4}$

TL;DR

This study uses resonant inelastic x-ray scattering to investigate how charge excitations and electron correlations evolve with doping in electron-doped cuprate superconductor La$_{2-x}$Ce$_{x}$CuO$_{4}$, revealing a transition from incoherent to coherent electronic states.

Contribution

It provides new insights into the doping-dependent evolution of charge excitations and electron correlations in La$_{2-x}$Ce$_{x}$CuO$_{4}$ using RIXS, highlighting the transition from incoherent to coherent quasiparticles.

Findings

Plasmons are identified as 3D charge excitations.

Plasmon energy increases monotonically with doping.

Evolution from broad to sharp plasmon peaks indicates increasing electron coherence.

Abstract

Electron correlations play a dominant role in the charge dynamics of the cuprates. We use resonant inelastic x-ray scattering (RIXS) to track the doping dependence of the collective charge excitations in electron doped La$_{2-x}$Ce$_{x}$CuO$_{4}$(LCCO). From the resonant energy dependence and the out-of-plane momentum dependence, the charge excitations are identified as three-dimensional (3D) plasmons, which reflect the nature of the electronic structure and Coulomb repulsion on both short and long length scales. With increasing electron doping, the plasmon excitations show monotonic hardening in energy, a consequence of the electron correlation effect on electron structure near the Fermi surface (FS). Importantly, the plasmon excitations evolve from a broad feature into a well defined peak with much increased life time, revealing the evolution of the electrons from incoherent states to coherent quasi-particles near the FS. Such evolution marks the reduction of the short-range electronic correlation, and thus the softening of the Mottness of the system with increasing electron doping.

Equations

$$ϵ(\vec{q},\omega )=1-v(\vec{q})\Pi (\vec{q},\omega ),$$

$$\displaystyle\omega_{p}(\vec{q})=\alpha Aq_{\parallel}\bigg{[}$$

$${\chi }_0(\vec{q},\omega )\sim \sum _{\vec{k}}{n}_{\vec{k}}(1-n_{\vec{k}+\vec{q}})\frac{E_{\vec{k}-\vec{q}}}{{\omega }^2-E_{\vec{k}-\vec{q}}^2},$$

$${\omega }_p=ax+b\to \Delta {\omega }_p=a(x-x_0)$$

$$\Delta {\omega }_p=a(\sqrt{x}-\sqrt{x_0})(\sqrt{x}+\sqrt{x_0})$$

Full text

I Introduction

The electronic behavior of metals is usually described using the standard Fermi liquid theory in terms of a single-particle spectral function of well-defined electron-like quasiparticles and a two-particle excitation spectrum dominated by long-lived collective charge excitations called plasmons Pines and Nozieres (1999); Mahan (2010). It is, however, generally agreed that Fermi liquid theory breaks down in the high-temperature superconducting cuprates and the search for a fully satisfactory replacement theory remains one of the most studied problems in condensed matter physics Keimer et al. (2015). Empirically, techniques such as angle-resolved photoemission spectroscopy (ARPES), scanning tunneling spectroscopy (STS) and quantum oscillations have given us a detailed picture of the cuprates’ electronic band structure Damascelli et al. (2003). Developing a comprehensive picture of the collective charge excitation in the cuprates has proved much more challenging. A key issue is limited $\vec{q}$-space access. Optical techniques are intrinsically limited to $\vec{q}\approx 0$ Basov and Timusk (2005); Devereaux and Hackl (2007). Although electron energy loss spectroscopy (EELS) has no such restriction in principle, accessing $c$-axis dispersion remains difficult Fink et al. (2001); Roth et al. (2014); Mitrano et al. (2018); Vig et al. (2017). RIXS has recently observed dispersive charge excitations in electron-doped cuprates opening new routes to characterize the charge dynamics of the high temperature superconductors Lee et al. (2014); Ishii et al. (2014); Dellea et al. (2017); Hepting et al. (2018). Importantly, RIXS also provides bulk sensitivity. Many properties of such excitations have, however, not been investigated in detail, especially compared to extensive RIXS studies of the magnetic response Braicovich et al. (2010); Le Tacon et al. (2011); Dean et al. (2012, 2013a); Dean (2015). This leads to contradictory explanations of the origin of the observed charge excitations in the cuprates either as intraband particle-hole excitations Ishii et al. (2014, 2017), new modes arising from symmetry breaking near quantum critical point Lee et al. (2014) or collective plasmon excitations Greco et al. (2016); Bejas et al. (2017); Greco et al. (2019); Eremin and Kochergin (2018); Hepting et al. (2018). It is vital to resolve the nature of these excitations, and in particular how the electronic state evolves upon doping, which can deepen our understanding of the electron correlation, including both the short- and long-range Coulomb interactions, in the cuprate unconventional superconductors.

Here we report RIXS measurements on electron doped LCCO Jin et al. (2011) at the Cu $L_{3}$-edge, focusing on charge excitations around the two dimensional BZ center with various $c$-direction momentum transfer. By using doping-concentration-gradient (combi) films Yu et al. (2017), we are able to provide much finer doping dependence than measured previously covering the optimal superconducting state to the non-superconducting metallic phase Lee et al. (2014); Ishii et al. (2014); Dellea et al. (2017); Hepting et al. (2018). We find that the charge excitations have mostly collective nature, and exhibit out-of-plane momentum transfer dependence. This strongly supports the plasmon nature of the observed peaks, as does the fact that the dispersion of charge excitations can be empirically fitted by a 3D plasmon model. As the electron doping increases from $x=0.11$ to $x=0.18$, the plasmon energy gently hardens and life time gradually increases. We explain these observations in connection with reduced effective mass and increased electronic quasiparticle coherence near the FS. Both are signs for a crossover from Mott physics to an itinerant picture, as doping drives the system from a high-temperature-superconducting to a metallic state.

II Charge excitation and its plasmon nature oscillation

Figure 1(b) plots Cu $L_{3}$-edge RIXS measurements of LCCO $x=0.17$. Distinct from measurements of hole-doped cuprates, where the low energy excitations are dominated by spin excitations Braicovich et al. (2010); Le Tacon et al. (2011); Dean et al. (2012, 2013a); Dean (2015), two low-energy dispersive modes are seen Lee et al. (2014); Ishii et al. (2014); Dellea et al. (2017); Hepting et al. (2018). The excitation at lower energy with stronger intensity can be assigned as spin fluctuations, following previous studies of hole-doped cuprates Braicovich et al. (2010); Le Tacon et al. (2011); Dean et al. (2012, 2013a); Dean (2015). The other excitation with higher energy and weaker intensity was absent in RIXS measurements of hole-doped systems, and appears to be unique to electron-doped cuprates Lee et al. (2014); Ishii et al. (2014); Dellea et al. (2017); Hepting et al. (2018). This feature disperses much more steeply, and merges into the high energy $dd$ excitations above $\sim$ 1.3 eV with in-plane wavevector $q_{\parallel}{}\geq 0.2$ reciprocal space units (r.l.u).

To characterize this excitation, we first examined its polarization dependence. In the RIXS process, the x-ray photon can exchange angular momentum $m_{J}$ with the sample, thus excitations of different characters can be enhanced or suppressed in different polarization channels Ament et al. (2009); Sala et al. (2011). Figure 2(a) and (b) compare the excitation intensity with $\sigma$- and $\pi$-polarized incident beams at different $q_{\parallel}$ points. The strongly dispersive feature in Figure 1(b) consistently shows stronger spectral intensity with $\sigma$-polarized incident beam. Such polarization dependence indicates that the excitation is related to $\Delta m_{J}=0$ or $2$ processes, namely pure charge excitations or bimagnon excitations. Bimagnon excitations, however, were suggested to disperse weakly near BZ center and appear as tails of the $\Delta m_{J}=1$ paramagnon excitations Onose et al. (2004); Chaix et al. (2018). Therefore, the higher energy feature is most likely to be charge excitations, consistent with previous interpretations Lee et al. (2014); Ishii et al. (2014); Dellea et al. (2017); Hepting et al. (2018).

Further characterization of this excitation was provided by examining its out-of-plane wavevector $q_{\perp}$ dependence, which probes the inter-CuO2 plane coupling. In Figure 2(c) and (d), we compare spectra taken at different $q_{\perp}$ with $q_{\parallel}$ fixed at different points. At $q_{\parallel}=(0.11,0)$, where both the spin excitation and the higher energy excitation can be clearly observed, the spin excitation energy stays constant for different $q_{\perp}$, consistent with the quasi-2D nature of this layered compound. On the other hand, the higher energy excitation behaves differently with strong $q_{\perp}$ dependence Peng et al. (2017). With a $q_{\perp}$ variation from 1.68 to 1.78, its energy at $q_{\parallel}=(0.06,0)$ shifts by $\sim$ 90 meV. Conversely, spin-related excitations are expected to show 2D character. Thus the strong $q_{\perp}$ dependence serves as a clear evidence supporting its charge fluctuation origin. Considering the strong dispersion of the charge excitations which extends $\sim$ 1 eV in the observable $q_{\parallel}$ range, the natural choice of the leading interaction to host such excitations would be the long-range Coulomb interaction Hepting et al. (2018). With a reasonable sized long-range Coulomb interaction to couple the electrons in different layers, plasmon excitations will become highly dispersive along the out-of-plane direction Das Sarma and Quinn (1982); Tselis and Quinn (1984); Jain and Allen (1985); Das Sarma and Hwang (2009).

To check the RIXS resonance behavior of the observed charge excitation, spectra at various incident x-ray energies across the Cu-$L_{3}$ edge were collected. Collective excitations are expected to exist at constant energy loss regardless of incident energy, whereas incoherent x-ray processes tend to occur at constant final energy Ament et al. (2011). As shown in Figure 3(b), both features from spin and charge excitations start to gain spectral weight at the rising edge of the Cu-$L_{3}$ edge white line, indicating they are stimulated by the same intermediate states, namely the ultra-fast transient double occupancy in the $2p^{6}3d^{9}\xrightarrow{}2p^{5}3d^{10}\xrightarrow{}2p^{6}3d^{9\ast}$ RIXS process Dean (2015). The spin excitation spectral weight peaks at the maximum of the absorption curve, consistent with observations in hole-doped cuprates Minola et al. (2015). Although both features share similar behaviors at the absorption rising edge, namely increasing in spectral weight and constant energy loss, they behave quite differently at the falling edge on the higher x-ray energy side. The spin excitation quickly diminishes, but the charge excitation seems to survive longer into the tail. More importantly, the charge excitation spectral weight reaches its maximum at $\sim$ 0.3 eV above the energy of the white line peak and then gradually reduces, accompanied by a noticeable shift in its energy. We emphasize that, although the shift is obvious, the size of the shift is much smaller than the increase of the incident x-ray energy, so the data rule out constant final energy processes as shown in Figure 3(c). We assign the charge excitation as primarily a collective plasmon, but a small contribution of incoherent electron-hole processes appear to be present above the white line Minola et al. (2015, 2017); Huang et al. (2016). It is worth noting that the observed $q_{\perp}$ dependence is only expected within a plasmon model; electron-hole excitation would be expected to be independent of $q_{\perp}$ Greco et al. (2016); Bejas et al. (2017); Greco et al. (2019).

III Plasmon doping dependence

With the charge excitation identified as a plasmon, we investigate its doping dependence. By using the LCCO combi films, we were able to obtain a fine doping dependence by simply translating the sample along its doping gradient direction while keeping the experimental conditions identical. Four $q_{\parallel}$ points were studied, namely $(0.06,0),(0.11,0),(0.04,0.04)$ and $(0.08,0.08)$, which all show similar doping dependent behaviors. In Figure 4, data at $q_{\parallel}=(0.06,0)$ and $(0.04,0.04)$ are shown as typical examples. More experimental results can be found in Supplemental Material (SM) SM .

Figure 4(a) shows a stacking plot of the RIXS spectra at $q_{\parallel}=(0.06,0)$. Two direct observations can be made before any quantitative analysis. From $x=0.11$ to $0.18$, the plasmon excitation evolves from a broad feature to a well-defined peak, and its energy gradually hardens, consistent with studies of single phase samples Hepting et al. (2018). Since the spectra were measured with identical experimental geometry, there is actually a small variation in the momentum transfer due to possible variation in lattice constants at different doping levels. As shown in Figure 2, the plasmon excitation energy strongly depends on $q_{\perp}$ values. With this concern, the variation in the momentum transfer was carefully considered. Figure 4(g) shows the converted $q_{\perp}$ values from the measured $c$-lattice constant variation of our combi film as a function of doping Yu et al. (2017). From $x=0.11$ to $0.18$, the change in $c$-lattice constant is about 0.4$\%$, leading to a variation in $q_{\perp}$ of 0.008. Importantly, Figure 2 shows that, in the vicinity of these $q_{\perp}$ values, the plasmon excitation hardens towards larger $q_{\perp}$. At higher doping level, towards which the measured RIXS signal hardens, the $q_{\perp}$ actually reduces. Thus it is safe to conclude that the hardening upon further doping is an intrinsic property of the plasmon excitation.

In Figure 4(b)-(c), RIXS spectra for two end doping levels, $x=0.11$ and $0.18$, are compared for clarity. The shaded peaks are the plasmon excitation components from our fitting SM . The extracted energy and intrinsic, deconvolved peak width are shown in Figure 4(e)-(f). At both $q_{\parallel}$ points, the plasmon energy hardens by $\sim 20\%$ and the width sharpens by $\sim 20\%$. A variation in $q_{\perp}$ of 0.1 at $q_{\parallel}=(0.06,0)$, as shown in Figure 2(c), leads to about 100 meV energy change in plasmon energy. A simple linear estimation suggests that the $c$-lattice constant reduction of 0.008 could lead to $\sim$ 8 meV softening in the measured doping range, which is far smaller than the energy variation in Figure 4(e) and can be neglected compared to the much larger doping induced plasmon energy hardening.

We emphasize that the two $q_{\parallel}$ points we compare were specifically chosen such that their in-plane and out-of-plane momentum transfers are the same, although $(0.06,0)$ is along the in-plane $(H,0)$ direction and $(0.04,0.04)$ is along the $(H,H)$ direction. The similarity of their RIXS response is further shown in Figure 4(d). At the same doping level, the charge excitation features for both $q_{\parallel}$ points almost overlap. Such $(H,0)-(H,Hf)$ symmetry is confirmed at another $\vec{q}$ pair, namely $(0.11,0,1.65)$ and $(0.08,0.08,1.65)$. The connection of such observations to the band structure of LCCO will be discussed in section V.

IV Microscopic consideration of the collective excitations

A plasmon is an emergent collective mode in many-body physics arising due to the materials’ electronic structure and Coulomb interactions including both the short-range interaction and the screened long-range interactions respectively Pines and Nozieres (1999); Mahan (2010). The dispersion of the plasmon is given by the zero of dielectric function

[TABLE]

where $v(\vec{q})$ is the Fourier transformation of Coulomb interaction, and $\Pi(\vec{q},\omega)$ is the polarizability function.

Exact calculations of the plasmon dispersion in strongly correlated metals is challenging, partially due to the lack of knowledge on the vertex contribution which is needed to derive the polarizability function Mahan (2010). Progress has been made studying the 2D extended Hubbard model van Loon et al. (2014) and the large-$N$ expansion of the layered $t-J-V$ model Greco et al. (2016). In particular, the plasmon excitation energy was suggested to be closely related to the single-particle density distribution. The spectral weight and the renormalization of its dispersion reflect the evolution of the on-site Coulomb repulsion $U$ and the long-range screened Coulomb interaction $V$. Thus RIXS could serve as a good bulk sensitive tool to probe the electronic structures of the cuprates, complimentary to other spectroscopic techniques.

Although the plasmon excitation in strongly correlated systems involves complicated processes, H. Hafermann et al. Hafermann et al. (2014) pointed out that the polarizability $\Pi(\vec{q},\omega)$ shares similar $\Pi(\vec{q},\omega)\sim\alpha^{2}\frac{\vec{q}^{2}}{\omega^{2}}$ behavior as the Lindhard function in random phase approximation (RPA) calculations, in the long-wavelength limit. Such a property of the polarization function in the long-wavelength limit, even for the correlated system, is a consequence of gauge invariance and local charge conservation, regardless of the interaction strength van Loon et al. (2014); Hafermann et al. (2014). Thus for small $q_{\parallel}$, to leading order, the electronic correlations only modify the scaling factor $\alpha$. This was numerically confirmed by the consistency of RPA and Dual-Boson calculations, where the vertex corrections from the dynamical mean field theory (DMFT) local contribution was taken into account only in the latter van Loon et al. (2014); Hafermann et al. (2014). Applying this approach, we investigate the dispersion of the observed charge excitation with $\Pi(q,\omega)\sim\alpha^{2}\frac{q^{2}}{\omega^{2}}$ in the low $q_{\parallel}\lessapprox 0.13$ r.l.u regime that we focus on here, and treating the Coulomb potential as that from charge on a lattice Greco et al. (2016); Becca et al. (1996). As a result the plasmon dispersion for a layered correlated system in the long wave-length limit can be approximated as

[TABLE]

This includes material parameters $a$($d$) representing in-plane (inter-plane) lattice constant, $\epsilon_{\parallel}$($\epsilon_{\perp}$) the dielectric constants parallel (perpendicular) to the plane, and $A=\sqrt{\frac{e^{2}}{2a^{2}d}}$. Equation 2 was used to fit the measured plasmon dispersion with free parameters $p_{1}=\alpha/\sqrt{\epsilon_{\parallel}}$ and $p_{2}=\epsilon_{\parallel}/\epsilon_{\perp}$. The fitting for $x=0.17$ is plotted as the light pink line in Figure 1(c), which agrees with the RIXS measurements quite well. With the fitted $p_{1}$ and $p_{2}$, the plasmon energy for $x=0.17$ at $\vec{q}=0$ is calculated to be 1.24 eV, which is consistent with infrared optical and EELS measurements Onose et al. (2004); Nücker et al. (1989); Mitrano et al. (2018).

Within Eq. 2, the doping dependence of the plasmon excitation is parameterized by $\alpha$, which encapsulates the electron correlation effects. It is interesting to compare this coefficient with the naive expectation that the plasmon energy scales like the square root of the carrier concentration, as predicted from the free electron model. In Figure 5, the extracted plasmon excitation energies are plotted as function of both $x$ and $\sqrt{x}$. At a lower $q_{\parallel}$ value of $(0.06,0)$, the plasmon excitation could be described to be linear with $x$ or $\sqrt{x}$, with similar degree of agreement. Such ambiguity lies in the fact that the measured doping range is not large enough to clearly resolve any power law behavior (see appendix D). At a higher $q_{\parallel}$ value of $(0.11,0)$, it is obvious that the plasmon excitation does not scale with either $x$ or $\sqrt{x}$. The disagreement with a simple free electron model comes as no surprise, and the degree of the disagreement serves as a measurement of the electron correlation effect in defining the electronic structures in the cuprates.

V Discussion

To help understand our observations, we examine the charge susceptibility via the Lindhard-Ehrenreich-Cohen expression for the charge dynamical susceptibility

[TABLE]

where $n_{\vec{k}}$ is the Fermi-Dirac distribution function, and $E_{\vec{k}-\vec{q}}$ is the excitation energy for a $|\vec{k}>\xrightarrow{}|\vec{k}+\vec{q}>$ transition. This equation can model the elements of our results that are consistent with an effective quasiparticle picture, such as how Eq. 2 can provide a reasonable description of the data. At low $\vec{q}$, bounded by $n_{\vec{k}}(1-n_{\vec{k}+\vec{q}})$, electron-hole pairs involving electrons deeply under the FS do not contribute to $\chi_{0}(\vec{q},\omega)$. Thus the low $q_{\parallel}$ plasmon we probe here is governed by the dynamics of the electrons near the FS, which are most relevant to the conducting behavior including superconductivity. In the low $q_{\parallel}$ limit, we can approximate $E_{\vec{k}-\vec{q}}\approx\vec{v}\cdot\vec{q}$ where $\vec{v}=\nabla_{\vec{k}}E_{\vec{k}-\vec{q}}$. The effective summation area in 2D reciprocal space, which contains electrons that contribute to the summation, is proportional to $\vec{q}\cdot\vec{k}_{F}$. Thus $\chi_{0}(\vec{q},\omega)$ is expected to be proportional to $\vec{v}\cdot\vec{k}_{F}$. It is worthwhile to clarify such FS sensitivity by considering the simple 2D electron gas case. Here the conducting band is defined as $E(\vec{k})=\frac{k^{2}}{2m}$, $\vec{v}=\vec{k}_{F}/m$ and $k_{F}^{2}$ is proportional to total carrier density $n$. Thus, the low $\vec{q}$ plasmon intensity scales with $\frac{n}{m}$ consistent with general expectations.

The above argument shows that the plasmon in LCCO measures the dot product of the Fermi velocity $\vec{v}$ with the Fermi wavevector $\vec{k}_{F}$, combined with the $\alpha$ factor (i.e. the prefactor in Eq. 2) from the correlation effects. To appreciate the applicability of the quasiparticle description to the plasmon excitation and the doping dependent evolution of $\vec{v}$, the one particle spectrum for LCCO was calculated with DMFT combined with density functional theory (DFT) Kotliar et al. (2006). The main results are shown in Figure 6, and more details can be found in Supplemental Material SM .

In the calculated spectrum, the electron correlation effects significantly reduce the quasiparticle lifetime away from the Fermi energy. While the states near the FS are of good coherent character. With increased electron itinerancy from $x=0.1$ to $0.2$, the near-FS region that hosts long lived quasiparticles extends further out, as shown in Figure 6. This explains the sharpening of the plasmon peak with increasing doping as observed in our RIXS data. As the plasmon excitation at low $q_{\parallel}$ originates from the dynamics of the electrons near the FS, its width measures the life time of the quasi-particles near the FS in the region accessible with $q_{\parallel}$. Upon increasing $q_{\parallel}$, the plasmon excitation starts to probe the incoherent states that suffer more from the electron correlation effect (reduced lifetime and increased vertex corrections). Thus the excitation quickly broadens and become hard to identify in the RIXS spectrum as observed.

Figure 7 gives a 2D view of the portion of the single particle spectrum where electrons can be excited into unoccupied hole states under momentum conservation constraints, as required in Eq. 3. Although the plasmon is a collective excitation involving a large number of electrons, plasmon excitation along different $\vec{q}$ directions selectively probes different portion of the electron states near the FS. The almost overlapping plasmon excitations at $q_{\parallel}=(0.06,0)$ and $q_{\parallel}=(0.04,0.04)$, as shown in Figure 4, may suggest that the FS is more rounded than that from our DMFT+DFT calculations, consistent with ARPES reports Armitage et al. (2002).

Following the above reasoning, we can speculate about the electron-hole asymmetry for the plasmon excitations which were, to the authors’ knowledge, never observed in Cu $L_{3}$-edge RIXS measurements of hole-doped cuprates, despite extensive studies of these materials Braicovich et al. (2010); Le Tacon et al. (2011); Dean et al. (2012, 2013a, 2013b); Dean (2015); Minola et al. (2015, 2017). The correlation effects could be still significant in the studied doping range, and coherent particles only live in a very small region in reciprocal space. Thus the plasmon excitation is expected to be weaker and broader, making it difficult to identify in the RIXS spectrum. Further, carriers in the hole-doped cuprates are known to have primarily oxygen rather than copper character, so O $K$-edge RIXS might be insightful Ishii et al. (2017). Another consideration is the weaker intensity for charge excitations in hole-doped cuprates than electron-doped cuprates caused by the RIXS process and the asymmetric electronic states Tohyama et al. (2015).

The hardening of the plasmon excitation upon increased doping is a combined effect from multiple factors. In the standard Ferimi liquid model, the plasmon excitation is expected to harden upon doping an electron pocket, as both $\vec{k}_{F}^{2}$ and $\vec{v}$ are expected to increase. On the other hand, as discusssed in section IV, our observed hardening does not follow simple $\sqrt{x}$ relation, emphasizing the contribution from the local correlation effect. It would be interesting to compare the RIXS measured plasmon spectrum to that from the 2D extended Hubbard model van Loon et al. (2014). From the doping dependent evolution of the plasmon excitation lifetime and energy, the two major observations we made here, a full picture of crossing over from a strongly correlated Mott system to itinerant electron liquid could be obtained.

VI Conclusions

Our RIXS measurements of combi films reveal the collective plasmon nature of the charge excitations in LCCO. The cuprate electronic structure and Coulomb interactions determine the plasmon behavior, and both effects are expressed in different ways in different $\vec{q}$ regions. In our measured small $\vec{q}$ region, the plasmon dispersion can be described by single coherent particle model and is mainly governed by the itinerant character of the electrons and the long-range Coulomb interaction, while the correlation effect serves as a renormalization to determine the plasmon energy scale. By fitting the dispersion with the 2D layered model, we extract effective parameters to describe the electron liquid that hosts high-temperature superconductivity. Our observed out-of-plane momentum dependence, demonstrates the 3D nature of the Coulomb interactions, bringing us to consider cuprates as more than a 2D CuO2 plane. The merits of appreciable momentum range and being bulk sensitive distinguish RIXS from other conventional techniques in probing the two-particle charge dynamics in the cuprates. With improved energy resolution to probe even smaller regions close to the FS, future RIXS measurements could provide unprecedented bulk sensitive information on the evolution of the electronic structure in the electron doped cuprates.

Acknowledgements.

We thank Zi-Yang Meng and Jiemin Li for helpful discussions and support related to this project. Work at ShanghaiTech University was supported by the ShanghaiTech University startup fund and partially supported by MOST of China under Grant No. 2016YFA0401000. JQL was also supported by the international partnership program of Chinese Academy of Sciences under Grant No. 112111KYSB20170059. MPMD acknowledges the support from the U.S. Department of Energy, Office of Basic Energy Sciences, Early Career Award Program under Award No. 1047478. Work at Brookhaven National Laboratory was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-SC0012704. Work at PSI is supported by the Swiss National Science Foundation through the Sinergia network Mott Physics Beyond the Heisenberg Model (MPBH). Xingye Lu acknowledges financial support from the European Community’s Seventh Framework Programme (FP7/20072013) under Grant agreement No. 290605 (Cofund; PSI-Fellow). MD was partially funded by the Swiss National Science Foundation within the D-A-CH programme (SNSF Research Grant 200021L 141325). KJ acknowledges the financial support from CAS interdisciplinary innovation team and National Key Basis Reaserch Program of China grant 2016YFA0300301, 2017YFA0303003, 2017YFA0302902. ZPY was supported by the NSFC (Grant No. 11674030), the Fundamental Research Funds for the Central Universities (Grant No. 310421113), the National Key Research and Development Program of China grant 2016YFA0302300. The calculations used high performance computing clusters at BNU in Zhuhai and the National Supercomputer Center in Guangzhou. We acknowledge Diamond Light Source for time on Beamline I21 and the Swiss Light Source at the Paul Scherrer Institut for beamtime at ADRESS under Proposal 20151449.

Appendix A Sample

A LCCO doping-concentration-gradient film was grown by pulsed laser deposition (PLD) at the Institute of Physics, Chinese Academy of Sciences. In our measured sample, $x$ varies from 0.10 to 0.19 along the chosen direction from edge to edge. Extensive characterization of this sample has been reported elsewhere Yu et al. (2017). The high quality of the sample was also evidenced by the smooth evolution of the measured XAS spectrum during our experiments (see SM Figure S1) SM . The choice of such gradient film allows a careful doping dependent survey in fine doping steps. More importantly, changing the doping concentration was done by simply translating the film along the doping gradient direction while keeping the experimental condition identical. Thus experimental error is largely minimized, and data for different dopings can be compared with great confidence. To avoid possible non-intrinsic edge effects from the growth, our experiments were focused on the doping range of 0.11 to 0.18, covering the optimal superconducting state to the non-superconducting metallic phase Jin et al. (2011); Saadaoui et al. (2015); Yu et al. (2017). The wave vectors used in our manuscript are indexed using the tetragonal ($I4/mmm$) space group with $a=b=4.01$ Å, $c$ continuously changes from 12.46 Å for $x=0.10$ to 12.40 Å for $x=0.19$.

Appendix B Experimental setup

Our experiments were carried out at the ADRESS beamline of the Swiss Light Source at the Paul Scherrer Institute Strocov et al. (2010); Ghiringhelli et al. (2006) and beamline I21 of the Diamond Light Source. The experimental geometry is sketched in Figure 1(a). Most of the data were taken with $\sigma$-polarized incident light, unless mentioned otherwise. At ADRESS, the scattered signal was collected at a fixed angle of 130 degrees, while this angle is of 146 degrees at beamline I21. This difference allows comparison of RIXS spectra at different out-of-plane momentum transfers ($q_{\perp}$) with the in-plane momentum transfer ($q_{\parallel}$) fixed at certain points. At each station, the momentum transfer was varied by rocking the sample. Most of the RIXS spectra were collected with the incident x-ray photon energies at the maximum of the absorption curve near the Cu $L_{3}$-edge SM , unless mentioned otherwise.

Along the gradient direction, the doping level $x$ varies from 0.10 to 0.19 in an 8 mm range, which is 0.011 per mm. The experiments were performed at the ADRESS beamline of the Swiss Light Source using the SAXES spectrometer and the I21 beamline of Diamond Light Source, with beamspots at the sample of $50(\text{H})\times 4(\text{V})$ $\mathrm{\mu m}^{2}$ and $30(\text{H})\times 10(\text{V})$ $\mathrm{\mu m}^{2}$, respectively. Thus, the variation of doping level $x$ of the sample within the beamspot is less than 0.0006, after considering beam footprint effects and can be regarded as homogeneous. The instrumental resolution is about 120 meV at the ADRESS beamline and 50 meV at I21 beamline at the Cu $L_{3}$ edge, estimated from the FWHM of the elastic scattering from carbon tape. All data presented were collected at a temperature of 20 K.

Appendix C plasmon dispersion fitting

The total RIXS spectrum is fitted with five components: A pseudo-Voigt function for the elastic line, an anti-symmetrized Lorentzian function multiplied by Bose-Factor and convoluted with resolution function for the paramagnon, a Gaussian function convoluted with the resolution function for the plasmon and a Gaussian function for the $dd$ excitations. Background scattering is treated with a polynominal function. Similar approaches have been used extensively in the RIXS literature Le Tacon et al. (2011); Dean et al. (2013a); Ishii et al. (2014); Hepting et al. (2018); Minola et al. (2015).

Appendix D linearity

Assuming $\omega_{p}$ follows a linear relation to $x$ as,

[TABLE]

Replace $x$ with ${\sqrt{x}}^{2}$, we have,

[TABLE]

The variation of $(\sqrt{x}+\sqrt{x_{0}})$ in the measured range measures the error if a $\omega_{p}\sim\sqrt{x}$ relation is enforced. For $x$ from 0.11 to 0.18, $\sqrt{x}+\sqrt{x_{0}}$ is from 0.7125 to 0.8051 with $\pm 6\%$ variation from the averaged value, which is not big enough to be well resolved from the experimental data.