Letter | Published:

# Evolution of charge order topology across a magnetic phase transition in cuprate superconductors

## Abstract

Charge order is now accepted as an integral constituent of cuprate high-temperature superconductors, one that is intimately related to other electronic instabilities including antiferromagnetism and superconductivity1,2,3,4,5,6,7,8,9,10,11. Unlike conventional Peierls density waves, the charge correlations in cuprates have been predicted to display a rich momentum space topology depending on the underlying fermiology12,13,14,15,16,17,18. However, charge order has only been observed along the high-symmetry Cu–O bond directions. Here, using resonant soft X-ray scattering, we investigate the evolution of the full momentum space topology of charge correlations in T′-(Nd,Pr)2CuO4 as a function of electron doping. We report that, when the parent Mott insulator is doped, charge correlations first emerge with full rotational symmetry in momentum space, indicating glassy charge density modulation in real space possibly seeded by local defects. At higher doping levels, the orientation of charge correlations is locked to the Cu–O bond directions, restoring a more conventional long-ranged bidirectional charge order. Through charge susceptibility calculations, we reproduce the evolution in topology of charge correlations across the antiferromagnetic phase boundary and propose a revised phase diagram of T′-Ln2CuO4 with a superconducting region extending toward the Mott limit.

## Main

In cuprates, an unconventional momentum–space electronic structure rapidly surfaces after doping carriers into the parent Mott insulator. The unconventional fermiology of lightly doped cuprates is embodied by the ‘pseudogap’ regime, where coherent quasiparticle excitations are only found in a small part of the Brillouin zone19,20. In this emergent state, electronic carriers in the CuO2 planes concomitantly organize into periodically modulated patterns in real space1,2,3,4,5,6,7,8,9,10,11. Visualization and elucidation of the microscopic link between these density waves and the many-body fermiology is a stepping stone towards understanding the nature of charge order and its relationship to Mott physics, the pseudogap and high-temperature superconductivity. In hole-doped cuprates, charge order has been mapped out to a great extent3, yet its origin and driving mechanism remain unclear. In Bi-based compounds, scanning tunnelling microscopy (STM)2,4,7,8 and resonant X-ray scattering (RXS)7,8 experiments have proposed that an instability of the pseudogapped Fermi surface might underlie the development of charge ordering. This proposal captures the temperature and doping dependence of charge order3,7,8, and has been supported by charge susceptibility calculations in various implementations16,17,18. On the other hand, recent STM and resonant inelastic X-ray scattering studies detect charge order outside the pseudogap phase (that is, in the very underdoped or overdoped limits), leaving the exact relationship between the charge order and fermiology unsettled21,22.

Electron-doped cuprates are an alternative platform with which to gain new perspectives on this problem, given their analogies to hole-doped cuprates in the phenomenology of charge order10,11. In contrast to hole-doped systems, here antiferromagnetic (AFM) correlations persist over a wide doping range (Fig. 1a) and deeply influence the many-body fermiology of electron-doped cuprates, as reflected in the appearance of an AFM pseudogap and the evolution of the Fermi surface topology with doping20,23. Furthermore, in these systems charge order populates a doping region characterized by strong AFM correlations10,11, setting an ideal stage to study the interplay between the spin and charge degrees of freedom in cuprates.

The intimate relationship between AFM and charge order has been explored in numerous theoretical studies12,13,14,15 that have focused on density wave instabilities localized along the Cu–O bond directions, in accordance with experimental reports1,2,3,4,5,6,7,8,9,10,11. At the same time, theory suggests that strong AFM correlations might induce charge instabilites in broader regions of momentum space (Q-space)12,13,15. Charting the topology of charge correlations in electron-doped cuprates is thus a new opportunity to probe the connection between charge order, antiferromagnetism and many-body fermiology in cuprates.

Here, we investigate the evolution of the momentum-space topology of charge correlations in T′-Ln2CuO4 (Ln = Nd, Pr) thin films as a function of electron doping. In contrast to the widely studied Ce-doped superconductors (T′-Ln2–xCexCuO4), thin films of T′-Ln2CuO4 can host superconductivity without cation substitution when subjected to post-growth reduction annealing procedures24. The possibility of superconductivity in the ‘undoped’ limit questions the very Mott nature of the parent T′-cuprates, and alternative scenarios involving a Slater mechanism have been proposed24,25. On the other hand, recent angle-resolved photoemission spectroscopy (ARPES), X-ray photoemission spectroscopy and X-ray absorption spectroscopy studies concluded that superconducting T′-Ln2CuO4 is actually electron-doped, presumably due to oxygen vacancies26,27,28. However, the exact doping–temperature phase diagram remains unclear due to difficulties in estimating the intrinsic electron concentration in these samples.

Figure 1b illustrates the schematics of our experimental set-up. In RXS experiments, the in-plane component of momentum transfer (Q||) is scanned by rotating the sample about the axis perpendicular to the scattering plane (θ). To access the full Q-space structure of charge correlations, we acquired successive θ scans for different orientations of the CuO2 planes, obtained by varying the azimuthal angle ϕ. The range of Q-space canvassed by this method is schematically shown in the inset of Fig. 1b.

We first demonstrate the presence of enhanced charge correlations in T′-Nd2CuO4 (referred to as T′-NCO#1) along the Cu–O bond directions (ϕ = 0°). Figure 1c,d presents a series of RXS scans as a function of photon energy and temperature. When the photon energy is tuned to the maximum of the Cu–L3 absorption edge (E ≈ 932 eV), the scattered intensity displays clear peaks in momentum space centred at |Q||| ≈ 0.2 reciprocal lattice units (r.l.u.) (Fig. 1c). The scattering peak rapidly vanishes as the photon energy is tuned off resonance, confirming that it originates from electrons in the CuO2 planes. As displayed in Fig. 1d, the intensity of this peak smoothly diminishes with increasing temperature, but survives up to room temperature, as in other electron-doped cuprates10,11. After subtracting the room-temperature background from the 12 K data (inset of Fig. 1d), we extract a correlation length of ξ ≈ 5–6 unit cells or ~20–25 Å. Such short-ranged charge correlations have been seen in other cuprates, (Nd,La)2–xCexCuO4 (NCCO, LCCO), Bi2Sr2–xLaxCuO6+δ (BSLCO) and HgBa2CuO4+δ (HgBCO)8,9,10,11.

The presence of periodic charge modulations in the CuO2 planes indirectly reflects the intrinsic carrier doping in the chemically undoped T′-Ln2CuO4 (refs. 26,27,28). The observed wavevector of charge modulations (Qc) can be used to estimate the carrier concentration from experimental measurements of Qc versus doping in both hole- and electron-doped cuprates3,5,8,10,11. Previous RXS studies reported that Qc in electron-doped cuprates (NCCO, LCCO) increases with higher electron content, similarly to hole-doped cuprates (BSLCO, HgBCO and YBCO). From previous estimates10,11, the observed wavevector of T′-NCO#1 (Qc ≈ 0.2 r.l.u.) maps onto an electron density of n ≈ 0.07 ± 0.02 per Cu, placing our sample at the low-doping limit of the phase diagram. This assignment of doping level is consistent with temperature-dependent Hall measurements performed on the same T′-Ln2CuO4 films (see Supplementary Information).

In this low-doping limit, we successively performed RXS scans at various azimuthal angles ϕ to map the complete Q-space topology of charge correlations. Surprisingly, as shown in Fig. 2a, broad but clear diffraction peaks are detected along all investigated momentum directions (ϕ = 0, 15, 30, 45, 65 and 90°; see Supplementary Figs. 1 and 2 for the full data sets). To investigate this feature in more detail, we subtracted a slowly varying fluorescence background from each scan and plotted the residual resonant peaks (Fig. 2b). The peaks display almost identical intensity, wavevector (Qc ≈ 0.2 ± 0.02 r.l.u.) and linewidth (half-width at half-maximum of ~0.07 ± 0.02 r.l.u.), regardless of the probed momentum direction. Furthermore, the detailed temperature and photon-energy dependence of the scattering peaks at representative azimuthal angles (0, 15 and 45°, Fig. 2g,h) fall into a single curve within experimental errors, indicating these structures arise from the same state. The two-dimensional plot in Fig. 2c summarizes this observation, highlighting the fact that, at low doping levels, the charge correlations in the CuO2 plane possess full (Cinf) rotational symmetry in Q-space. This result is in marked contrast to all previous reports of bond-oriented charge order in cuprates (Fig. 2d)3,5,6,7,8,9,10,11, and demonstrates that, on electron doping to the parent Mott/Slater insulator, charge correlations initially develop a complex Q-space structure that was entirely unanticipated. To rule out an extrinsic origin to the observed structure factor, the single crystallinity (or C4 symmetry) of the sample was confirmed before and after the RXS experiments (Supplementary Fig. 3). Therefore, the charge correlations in T′-NCO#1 manifest a higher rotational symmetry than the underlying lattice.

To gain additional insights into the observed topology of structure factor S(Q), we simulated a possible charge density map δρ(r). In doing so, we adopted randomly assigned reciprocal space complex phases, and thus δρ(r) represents only one of the possible realizations corresponding to S(Q) (see Methods). The simulated map of δρ(r) in Fig. 2e depicts a ‘glassy’ state with an apparent tendency to periodic ordering at wavevector Qc (with a period of five unit cells), but without any preference in orientation. This is again in sharp contrast to the case of bond-oriented charge order, whose simulated δρ(r) is shown in Fig. 2f (see also Supplementary Fig. 4).

To elucidate this new phenomenology, we recall that, within Lindhard theory29, the linearized charge density perturbation ρ(Q) induced by an external potential ϕ(Q) is obtained as ρ(Q) = χ(Qϕ(Q), where the generalized charge susceptibility χ(Q) is directly linked to the underlying electronic band structure. In this framework, a local potential can induce Friedel oscillations of charge density (akin to quasiparticle interference modulations observed in STM), to which RXS is sensitive, as revealed by recent theoretical studies17,30,31. Given that the experimental observable, structure factor S(Q), is proportional to the squared amplitude of the charge density (so that $$S\left( {\mathbf{Q}} \right) \propto \left| {\rho \left( {\mathbf{Q}} \right)} \right|^2 = \left| {\chi \left( {\mathbf{Q}} \right) \cdot \phi \left( {\mathbf{Q}} \right)} \right|^2$$), this formulation of the scattering process allows us to analyse the RXS information not only in terms of genuine long-range ordering with spontaneous symmetry breaking, but also as a direct reflection of generic many-body instabilities in the Fermi surface, encoded in the interacting charge susceptibility.

To articulate this scenario, in Fig. 3 we compare the Fermi surface topology, the Q-space maps of the static charge susceptibility and the Q-space topology of observed RXS peaks. Figure 3a-e and 3f-j, respectively, show the calculated Fermi surfaces and static charge susceptibilities of T′-NCO at various representative doping levels. The calculation is based on the momentum-resolved density fluctuation model, which has been used to capture certain experimental characteristics of charge order in hole-doped cuprates from instabilites in the Fermi surface (see Methods for details)18. In electron-doped systems, the longitudinal spin and charge susceptibilities become mixed in the presence of commensurate AFM correlation32, which results in the spin–charge interaction being distinct from the incommensurate stripe physics of hole-doped cuprates (see Supplementary Information for details). In the low-doping limit (n = 0.05, Fig. 3a) with strong AFM correlations, the electrons first fill the upper Hubbard states at the antinodes, developing electron pockets around (±π, 0)/(0, ±π) that are very circular in this limit due to the presence of strong (±π, ±π) AFM scattering. Correspondingly, the interacting charge susceptibility is enhanced uniformly along all azimuthal directions at |Q||| = 0.2 r.l.u. (Fig. 3f,l), in remarkable accordance with the observed Q-space structure of the scattering peaks in T′-NCO#1 (Fig. 3k,m). This analysis demonstrates that the RXS structures in Fig. 2 reflect the rotationally symmetric particle–hole scattering channels connecting low-energy states across the Fermi surface, which materialize in real space possibly through the defect- or impurity-induced Friedel oscillations in charge density17,30,31. We note that oxygen vacancies, a likely source of doping in T′-Ln2CuO4, might provide the pinning potential ϕ(Q) in this framework.

The dramatic change in Fermi surface topology occurs near the AFM critical point, where the termination of Néel order accompanies the collapse of the Mott gap, resulting in the formation of hole pockets at (±π/2, ±π/2) (Fig. 3c,d). With the weakening of (π,π) AFM scattering, the charge susceptibility becomes anisotropic, and is enhanced around the high-symmetry Cu–O bond directions (Fig. 3h,i), recovering the C4 symmetry of the underlying lattice. This trend continues up to the paramagnetic limit (n = 0.17, Fig. 3e), where the electron and hole pockets merge to form a single large Fermi surface centred at (±π, ±π). We note that the doping evolution of the Fermi surface presented here is in close agreement with experimental Fermi surfaces obtained from ARPES and quantum oscillation experiments on electron-doped cuprates20,23. These calculations thus suggest that AFM order in electron-doped cuprates promotes charge correlations in all momentum directions in the low-doping limit, as detected by the present RXS experiments, while a more conventional bidirectional charge order emerges as AFM order is removed at larger doping.

To confirm the crossover from Cinf to C4 topology of the charge correlation, we performed RXS measurements on the T′-Pr2CuO4 (T′-PCO#1) sample with higher intrinsic doping. As displayed in Fig. 3o, we observed a sharp RXS peak at |Q||| = 0.32 ± 0.02 r.l.u. along the Cu–O bond direction (see Supplementary Fig. 5 for details of RXS experiments on T′-PCO#1). From the empirical doping dependence of Qc—or equivalently by comparing Qc to the maximum wavevector of the calculated charge susceptibility (Fig. 3n)—the value Qc = 0.32 ± 0.02 r.l.u. maps to an electron density n ≈ 0.17 ± 0.02 (ref. 11; see also Supplementary Information for the corresponding Hall measurements). This observation confirms that low-doping charge correlations with Cinf rotational symmetry morph into a disconnected topology with charge order peaks pinned to the Cu–O bond directions in the high-doping limit, forming a more conventional type of bidirectional charge order as schematically displayed in Fig. 3p.

It is worth noting that the charge order in the high-doping limit (T′-PCO#1) is rather long-range, with a correlation length of ξ ≈ 20 unit cells or ~80 Å, which is about four times larger than the low-doping limit (T′-NCO#1). This pronounced enhancement in spatial coherence is consistent with proposals that the charge order correlation length is affected by the degree of directionality in the underlying electronic susceptibility17. We note that the long correlation length of T′-PCO#1 is comparable to that in YBa2Cu3O6+δ (YBCO), where charge order directly competes with superconductivity5. On the other hand, the short correlation length of T′-NCO#1 is akin to that of BSLCO and HgBCO (refs. 8,9), where the direct competition between charge order and superconductivity is not fully established. This fact hints at a possible crossover in the nature of charge order as a function of electron doping: from an incipient ordering phenomenon in the low-doping regime, to a long-range-ordered ground state at higher electron doping as the electronic susceptibility acquires strong directionality. At the same time, the analogy between T′-NCO#1 and BSLCO and HgBCO raises an important question on the nature of charge order in these hole-doped systems: whether it is a bona fide symmetry-broken phase or rather an incipient state possibly initiated by quasiparticle scattering across the Fermi surface as observed here. Altogether, our results, combined with previous reports on hole-doped cuprates3,5,6,7,8,9, hint at a common link between the low-energy fermiology and the spatial organization of the electronic state in hole- and electron-doped cuprates.

The established connection paves the way to use Qc as a measure of intrinsic electron density. This inference is especially valuable for all T′-structured cuprates, whose unknown oxygen stoichiometry usually hinders the exact determination of their true electron density24,26,27,28,33. In Fig. 4b we plot the calculated doping dependence of Qc as extracted from the maximum of the charge susceptibility (Fig. 4a). For comparison, we also plot the Ce-doping dependence of Qc from previous studies of NCCO and LCCO11. A common trend is apparent; nonetheless, the NCCO data appear to be slightly offset to higher doping levels. To some degree, this deviation is natural because the Ce concentration cannot represent the sole factor determining carrier density in electron-doped cuprates, as emphasized in recent photoemission studies26,27,28,33. This consideration further underscores the importance of the explicit determination of intrinsic electron concentrations in T′-cuprates.

In addition to T′-NCO#1 (Qc ≈ 0.2 r.l.u.) and T′-PCO#1 (Qc ≈ 0.32 r.l.u.), as presented above, we measured two additional samples with RXS peaks at Qc ≈ 0.22 r.l.u. (T′-PCO#2) and Qc ≈ 0.25 r.l.u. (T′-NCO#2) (Supplementary Fig. 6). Based on the relationship between Qc and n, we infer that our samples cover a wide doping range from n ≈ 0.05 to 0.17, as shown in Fig. 4b. Interestingly, all our samples were subjected to the post-growth annealing and show superconductivity with Tc = 24.4, 23.1, 24.6 and 24.8 K (for T′-NCO#1, T′-PCO#2, T′-NCO#2 and T′-PCO#1, respectively), as shown in Fig. 4c. This finding reflects a persistence of the superconducting phase into the low-doping region, in contrast to the conventional phase diagram of electron-doped cuprates (Fig. 1a). This conclusion lends support to the very recent ARPES studies on bulk T′-Pr2–xLaCexCuO4–δ, which also report a revised phase diagram with an extended superconducting dome based on the intrinsic electron density estimated from Luttinger’s theorem33. Our experimental observations challenge the conventional understanding of charge order and the phase diagram of electron-doped cuprates, and provide insights into a unified description of charge order phenomenology.

## Methods

### Sample growth and characterizations

The thin films of T′-NCO and T′-PCO used in this work were grown by molecular beam epitaxy under ultrahigh vacuum using Nd, Pr and Cu metal sources and atomic oxygen generated in situ from a radiofrequency oxygen source35. Reflection high energy electron diffraction and electron impact emission spectroscopy were used to monitor and control the growth of the NCO and PCO films on (001) SrTiO3 substrates in real time. High-resolution reciprocal space mapping data show that films with a thickness of 100 nm are grown fully relaxed. The films were subjected to a two-step annealing process36. The superconducting transition was measured by electrical transport and magnetometry.

### RXS experiments

RXS experiments were performed at the UE46_PGM-1 beamline of BESSY II (T′-NCO#1, T′-NCO#2), the REIXS beamline of the Canadian Light Source (T′-PCO#1), and Beamline 4.0.2 of the Advanced Light Source (T′-PCO#2). Experiments were conducted at 12 K (unless specified) and under high vacuum (better than 10−9 torr). Samples were oriented in situ using Bragg reflections. All measurements were conducted at the Cu–L3 absorption edge with out-of-scattering-plane (σ) incoming polarization to maximize the sensitivity to charge scattering. Momentum–space scans were obtained by rocking the sample angle at a fixed detector position. The temperature dependence series was acquired by both heating and cooling the samples, yielding consistent results.

### Simulations of the real space charge fluctuations

The structure factor S(Q) for T′-NCO#1 is modelled by $$S\left( {\mathbf{Q}} \right) = \exp \left( {\frac{{4\ln 2\left( {\left| {\mathbf{Q}} \right| - Q_{\mathrm{c}}} \right)^2}}{{w^2}}} \right)$$, where the Qc is the wavevector centroid of charge correlations, and w is the full-width at half-maximum. S(Q) for BSLCO is similarly modelled by a 2D Gaussian function centred at (±Qc, 0), (0, ±Qc). The real space charge–charge correlation function C(r) is obtained by a discrete 2D Fourier transform of S(Q). To simulate the real space charge density fluctuation δρ(r), we first introduce a matrix of random reciprocal-space phases ϕ(Qm) and subsequently simulate the charge fluctuation as $$\delta {\it{\rho }}\left( {\mathbf{r}} \right) = \mathop {\sum }\limits_m \sqrt {S({\mathbf{Q}}_{\mathbf{m}})} \cos ({\mathbf{Q}}_{\mathbf{m}} \cdot {\mathbf{r}} + ({\mathbf{Q}}_{\mathbf{m}}))$$, where m labels the discretized Fourier components37.

### Momentum-resolved density fluctuation calculations

We take an effective Hamiltonian near the AFM saddle point

$$H = \mathop {\sum }\limits_{k,\sigma } \left[ {\xi _kc_{k,\sigma }^\dagger c_{k,\sigma } + Um\sigma c_{k + Q,\sigma }^\dagger c_{k,\sigma }} \right] + {\mathrm{h.c.}}$$

where the non-interacting dispersion ξk is taken from tight-binding calculation of previous works, U is the on-site Hubbard interaction, and m represents the staggered magnetization32. In the commensurate AFM state with Q = (π, π), the longitudinal spin and charge susceptibilities become mixed in the Umklapp scattering channels, while the transverse spin susceptibilities remain decoupled. Based on the above Hamiltonian, we compute the correlation functions with many-body corrections implemented within the random-phase approximations and self-energy corrections. In the framework of the momentum-resolved density fluctuation model, both the single-particle Green’s function and the two-body correlation functions are calculated self-consistently with the self-energy corrections. We included Bethe–Salpeter-type vertex corrections obeying Ward’s identity. The AFM gap and chemical potential are also computed self-consistently for each doping with a doping dependent U adopted from previous studies32. See Supplementary Information for a detailed description of the calculation method.

## Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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## Author information

### Affiliations

1. #### Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA

• Mingu Kang
• , Jonathan Pelliciari
•  & Riccardo Comin
2. #### Department of Physics, University of California, Berkeley, CA, USA

• Alex Frano
• , Nicholas Breznay
• , Mo Chen
• , Keto Zhang
• , Alejandro Ruiz
• , Zeyu Hao
• , Sylvia Lewin
•  & James Analytis

• Alex Frano
4. #### Helmholtz-Zentrum Berlin für Materialien und Energie, Berlin, Germany

• Enrico Schierle
•  & Eugen Weschke

• Ronny Sutarto
•  & Feizhou He
6. #### Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

•  & Elke Arenholz
7. #### NTT Basic Research Laboratories, NTT corporation, Atsugi, Japan

• Yoshiharu Krockenberger
•  & Hideki Yamamoto

• Tanmoy Das

### Contributions

M.K., J.P., E.S., A.F. and N.B. conducted the RXS experiments and analysed the data with help from M.C., K.Z., A.R., Z.H., S.L. and J.A. E.W., R.S., F.H., P.S. and E.A. maintained the X-ray beamlines and supported RXS experiments. T.D. performed the calculations. Y.K. and H.Y. grew the thin films, performed the transport measurements and analysed the data. R.C. conceived the experiment and directed the project. M.K. and R.C. wrote the manuscript with input from all other co-authors.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Riccardo Comin.

## Supplementary information

1. ### Supplementary Information

Supplementary Table 1, Supplementary Figures 1–7 and Supplementary References 1–14