The nature of plasmon excitations in hole-doped cuprate superconductors. Physical

High T c superconductors show a rich variety of phases associated with their charge degrees of freedom. Valence charges can give rise to charge ordering or acoustic plasmons in these layered cuprate superconductors. While charge ordering has been observed for both hole- and electron-doped cuprates, acoustic plasmons have only been found in electron-doped materials. Here, we use resonant inelastic x-ray scattering to observe the presence of acoustic plasmons in two families of hole-doped cuprate superconductors (La 1 . 84 Sr 0 . 16 CuO 4 and Bi 2 Sr 1 . 6 La 0 . 4 CuO 6 þ δ ), crucially completing the picture. Interest-ingly, in contrast to the quasistatic charge ordering which manifests at both Cu and O sites, the observed acoustic plasmons are predominantly associated with the O sites, revealing a unique dichotomy in the behavior of valence charges in hole-doped cuprates.

(Received 1 August 2020; revised 18 October 2020; accepted 13 November 2020; published 14 December 2020) High T c superconductors show a rich variety of phases associated with their charge degrees of freedom. Valence charges can give rise to charge ordering or acoustic plasmons in these layered cuprate superconductors. While charge ordering has been observed for both hole-and electron-doped cuprates, acoustic plasmons have only been found in electron-doped materials. Here, we use resonant inelastic x-ray scattering to observe the presence of acoustic plasmons in two families of hole-doped cuprate superconductors (La 1.84 Sr 0.16 CuO 4 and Bi 2 Sr 1.6 La 0.4 CuO 6þδ ), crucially completing the picture. Interestingly, in contrast to the quasistatic charge ordering which manifests at both Cu and O sites, the observed acoustic plasmons are predominantly associated with the O sites, revealing a unique dichotomy in the behavior of valence charges in hole-doped cuprates. DOI: 10.1103/PhysRevLett.125.257002 The electronic structure of high temperature superconducting layered cuprates [1] may be understood in terms of a hybridization between the Cu 3d x 2 −y 2 and O 2p σ orbitals, and a strong on-site Coulomb repulsion between electrons on the Cu sites [2][3][4][5]. When holes are introduced [see Fig. 1(a)], they reside preferentially in the so-called "charge-transfer band" (CTB) which is composed primarily of O orbitals [6]. In contrast, doped electrons enter the upper Hubbard band (UHB) and primarily reside on the Cu sites [5]. Despite this asymmetry in the electronic structure, charge order, a complex phase of periodically modulated charge-carrier density, has been observed ubiquitously on both the electron-and hole-doped sides of the phase diagram [7].
Surprisingly, a more widely observed mode of collective charge-density oscillation, the acoustic plasmon [8], has been rather elusive for the cuprates. In contrast to threedimensional (3D) metals, where long-range Coulomb interactions give rise to isotropic long-wavelength optical-like gapped plasmons, out-of-phase oscillations of charges in neighboring planes of two-dimensional (2D) layered electron gases, form acoustic plasmons, whose energy tends to zero for small in-plane wave vectors [see Fig. 1(b)] [9,10]. Because of confinement of the doped charges to CuO 2 planes and poor screening of out-of-plane Coulomb interactions by intervening dielectric blocks [see Fig. 1(c)], acoustic plasmons are also expected in the layered cuprates [11][12][13].
A remarkable discovery has been the recent observation of acoustic plasmons in electron-doped La 2−x Ce x CuO 4 (LCCO) and Nd 2−x Ce x CuO 4 [14,15] using Cu L 3 resonant inelastic x-ray scattering (RIXS). The excitations were found to have the strong out-of-plane dispersion expected for plasmons in layered systems. The situation in holedoped cuprates, however, has remained rather controversial. While Cu L 3 RIXS did not detect plasmons in several families [16][17][18], O K RIXS did detect excitations in La 2−x ðBr; SrÞ x CuO 4 that were interpreted as incoherent intraband transitions [19]. In zero-momentum optical investigation of Bi 2 Sr 2 CaCu 2 O 8 , apart from observation of optical plasmons at 1.1 eV, a low-energy non-Drude behavior was contemplated to be due to a band of acoustic plasmons [20]. However, electron energy-loss spectroscopy, a traditional probe for studying plasmons, observed only a high energy (∼1 eV) overdamped optical plasmon for small in-plane wave vectors in Bi 2.1 Sr 1.9 CaCu 2 O 8þδ [21]. Acoustic plasmons in layered systems originate from the presence of conduction electrons and the long-range nature of the Coulomb interaction. Their absence in holedoped cuprates would conflict with our general understanding of the collective behavior of the doped charges.
In this Letter, we report that acoustic plasmons are indeed present in hole-doped cuprates from an extensive O K RIXS study of La 1.84 Sr 0.16 CuO 4 (LSCO) and Bi 2 Sr 1.6 La 0.4 CuO 6þδ (Bi2201) over a wide range of inand out-of-plane momenta. The discovery of acoustic plasmons in the hole-doped systems remarkably illustrates the universal existence of low-energy collective excitations besides phonons and spin fluctuations across the cuprate phase diagram. Surprisingly, the observed acoustic plasmons are predominantly associated with the O sites in these hole-doped cuprates. Our results will therefore stimulate more studies of doped-hole charge dynamics, taking into account the three band model in the cuprates [3,4].
Spectroscopically, Cu L 3 RIXS and O K RIXS directly probe the charge and magnetic excitations associated with the Cu 3d and O 2p orbitals, respectively, at the corresponding absorption peaks [ Fig. 1(a)]. In order to compare the excitations associated with the two orbitals, highresolution RIXS spectra were collected at Cu L 3 (ΔE ≃ 0.045 eV) and O K (ΔE ≃ 0.043 eV) edges, at I21-RIXS beam line, Diamond Light Source, United Kingdom [22,23], in the scattering geometry shown in Fig. 1(c). All data presented here were obtained with incident σ polarization (perpendicular to the scattering plane) to enhance the charge excitations [14]. Single crystals of LSCO and Bi2201 were cooled to their respective T c s of 38 and 34.5 K, consistent with optimal hole doping of p ¼ 0. 16, and x-ray absorptions (XAS) were collected in total electron yield mode [see Fig. 1(d)] [23]. A survey was first made near the in-plane zone center with a fixed scattering angle on Bi2201 [Figs. 1(e) and 1(f)]. The lowenergy inelastic spectra at Cu L 3 resonance are dominated by paramagnons without any noticeable signs of plasmons, similar to reports on other hole-doped systems [16][17][18]. At the O K edge hole-peak [6] however, a mode is found below 1 eV, dispersing toward the zero energy.
We next collected RIXS spectra by varying the incident energy (E i ) across the hole peak in the O K XAS of LSCO at (h ¼ 0.03, l ¼ 1.00) [ Fig. 1(d)] [6], as shown in Fig. 1(g). We denote momentum transfers along h, k, and l directions We find a broad feature at ∼0.5 eV shifting toward higher energies with increasing E i . With doping, the probability of scattering from doped charges in the intermediate state of RIXS increases [42]. Moreover, the energy shift of the magnetic excitations associated with incoherent charge excitations is enhanced in σ-polarized RIXS [43]. Thus, this feature can be ascribed to bimagnon excitations with an itinerant character [42]. We find an additional sharp mode, at ∼0.13 eV, whose energy remains constant with E i . This is a signature of its coherent nature [15] and is in contrast with previous O K RIXS results [19]. This feature cannot be due to two-particle electron-hole-like excitations, which are incoherent in nature. Neither can it be due to single magnons or paramagnons since ΔS ¼ 1 spin-flip processes are forbidden at the O K edge [42]. To ascertain its origin, we explored further its dispersion in energymomentum space.
The broad feature seen in Fig. 1(g) is almost nondispersive in the h direction, further confirming its assignment as bimagnons [42]. This can be seen from ðh; EÞ maps collected at constant l values for LSCO and Bi2201 (Fig. 2). In contrast, the sharp mode disperses toward zero energy near the in-plane zone center in both systems. The RIXS spectra were fitted with a sum of Gaussians for the elastic line, phonons, and damped harmonic oscillator functions for the sharp mode and the broad feature [Figs. 2(e) and 2(f)] [23]. The dispersion and reduction of amplitude and width toward the in-plane zone center of this mode, is reminiscent of the acoustic plasmon behavior observed in electron-doped LCCO [14]. The spectral weight of this mode moves to lower energy as the doping is reduced (see Supplemental Material, Fig. S15 [23]). This is expected and consistent with the behavior of plasmons in LCCO [14,15]. However, the mode is strongly damped in comparison to LCCO (see Supplemental Material, Fig. S12 [23]), reflecting the stronger correlations (e.g., pseudogap) near optimal doping in hole-doped cuprates [5,15].
The most stringent test for identifying the mode as plasmons in these systems is their l dispersion. In the outof-plane direction, plasmons in layered electron systems have a periodicity of 2π=d (where d is the interlayer spacing), which corresponds to l ¼ 2 in these systems, with a minimum in energy at l ¼ 1; 3; 5; … [ Fig. 1(c)]. LSCO and Bi2201 have interlayer spacings differing by a factor of ∼2, allowing us to probe separate portions of this period. The sharp mode observed in Fig. 2, is found to disperse to a minimum energy value at l ¼ 1 for both systems. This can be seen in the (l; E) maps collected at fixed h values shown in Figs. 3(a)-3(d). This behavior fundamentally proves the presence of acoustic plasmons in hole-doped cuprates. We can exclude the previous interpretation of these excitations as incoherent intraband charge or electron-hole excitations which are 2D [19], without significant l dependence [12]. We note that a limited out-of-plane dispersion study has also been done recently on underdoped LSCO [44].
The cuprates are strongly correlated electron systems [1]. As such, it is interesting to compare our experimental results with the recently developed calculations of plasmons in the framework of a t-J-V model [12], although generic plasmon behavior can also be described by random-phase-approximation calculations [11,13]. For discussing the nature and origin of the 3D charge excitations in LSCO and Bi2201 we employed the minimal layered t-J-V model [12,23,45]: Here, t ij represents the hopping parameter and J ij the exchange parameter. The 3D form of long-range Coulomb interaction V ij used in Eq. (1) in momentum space is [46] VðQÞ where V c ¼e 2 dð2ϵ ⊥ a 2 Þ −1 and Aðq x ;q y Þ ¼ αð2 − cosq x − cosq y Þ þ 1 with α ¼ ϵ k =ϵ ⊥ =ða=dÞ 2 , e the elementary charge, and high frequency in-(ϵ k ) and out-of-plane (ϵ ⊥ ) dielectric constants. The imaginary part of charge susceptibility χ 00 c ðQÞ, ω, obtained from the model resembles well the spectral shape of the plasmons for both systems, as shown in Fig. 3(f) at different (h, l) values. This demonstrates that the charge excitations in RIXS, although influenced by resonance and polarization effects, can fundamentally be related to the charge-density response function. At l values close to 2, the much larger suppression of plasmons compared to theory could be due to their decay through the incoherent charge channels associated with bimagnons or through umklapp scattering [14]. Severe suppression of the charge excitations in this region forbids us from detecting the optical plasmon branch in Bi2201 (see Supplemental Material, Fig. S6) [23]. In Figs. 4(a) and 4(b) we show that both the h and l direction plasmon dispersions extracted by fitting the RIXS spectra for LSCO and Bi2201 are also represented well by the t-J-V model optimized independently for each material. The higher acoustic plasmon velocities in Bi2201 than LSCO seen in Figs. 4(a) and 4(b), arise mainly due to the larger interlayer spacing, considering the two systems have similar carrier densities and Fermi velocities (see Supplemental Material, Sec. V) [9,14,23]. Nevertheless, the remarkable similarity of the plasmon dispersions in the two different families suggests their ubiquitous existence in hole-doped cuprates.
In order to shed light on the possible existence of plasmons at Cu sites, we use the model described above to calculate the expected plasmon energies for both systems along the (h, l) paths corresponding to the data for fixed scattering geometry. Neither in Bi2201 [ Fig. 1(e)] nor in LSCO (Supplemental Material, Fig. S2 [23]), there is any spectral weight evident in Cu L 3 RIXS spectra that can be assigned to plasmons [16,17]. This contrasts with the strong plasmon signals observed at the O sites. The  amplitude of plasmon is found to be highest close to l ¼ 1 for the O K RIXS [ Fig. 3 and Supplemental Material, Fig. S12(c) [23] ]. In the Cu L 3 RIXS experiments close to the in-plane zone center, we probe near l ¼ 3 for Bi2201 and near l ¼ 1.8 for LSCO. Plasmons were, however, clearly observed in Cu L 3 RIXS of electron-doped LCCO at similar l values as in LSCO [14]. The nonobservance of plasmons at Cu sites in the present study of LSCO and Bi2201 could be therefore due to a combined effect of an l dependence of plasmon spectral weight and a strong O 2p character of the doped charges [6,45,47]. Further studies are required to clarify such site-dependent behavior, since in the Zhang-Rice singlet state scenario for hole-doped cuprates, a strong coupling is expected between the doped holes in the O 2p σ orbitals and the intrinsic holes of the Cu 3d x 2 −y 2 orbitals [45].
It is interesting to discuss our findings in the context of the charge order type of density modulation observed in hole-doped cuprates. When charge order is present, it has been observed using both Cu L 3 -edge and O K-edge resonant x-ray scattering [48][49][50], both for short (∼15 Å) [50] and long (∼200 Å) [48] correlation lengths. In common with other charge-density waves, the order has a valence charge modulation and associated atomic displacements [51], making it possible to be observed by nonresonant x-ray scattering techniques [52]. Thus, it is likely that, in these systems, the charge ordering signal observed at the Cu L 3 absorption peak is primarily due to atomic displacements caused by electron-phonon coupling, while the dominant signal at the O K hole peak reflects directly the valence charge modulation [49,53]. Because of the much higher frequencies of the dynamic plasmons, it may be that they couple weakly to the phonons, further reducing the possibility to observe any signature directly from the Cu 3d orbitals.
The general existence of acoustic plasmons besides phonons and spin fluctuations in layered cuprates will lead to more investigations of charge dynamics in connection with the pseudogap phase, non-Fermi liquid behavior, and perhaps the superconductivity in cuprates [1]. Our results also suggest that the charge dynamics in hole-doped cuprates are mostly associated with the O sites, highlighting the importance of the three band model in the cuprates [3,4]. Going beyond, it would be interesting to utilize the site sensitivity of the RIXS technique to characterize plasmon behavior in other layered superconductors, like iron pnictides [54], having strong out-of-plane band dispersions, or the newly found nickelates in which 2D Ni 3d states strongly hybridize with 3D rare-earth 5d states [55].