Scale free distribution of oxygen interstitials wires in optimum doped HgBa$_2$CuO$_{4+y}$

Novel nanoscale probes are opening new venues for understanding unconventional electronic and magnetic functionalities driven by multiscale lattice complexity in doped high temperature superconducting perovskites. In this work we focus on the multiscale texture at supramolecular level of atomic oxygen interstitials (O-i) stripes in HgBa$_2$CuO$_{4+y}$ at optimal doping for the highest superconducting critical temperature $T_C$=94K. We report compelling evidence for the nematic phase of oxygen-interstitial O-i atomic wires with fractal-like spatial distribution over multiple scales by using scanning micro and nano X-ray-diffraction. The scale free distribution of O-i atomic wires at optimum doping extending from micron scale down to nanoscale has been associated with the intricate filamentary network of hole rich metallic wires in the CuO$_2$ plane. The observed critical opalescence provides evidence for the proximity to a critical point controlling the emergence of high temperature superconductivity at optimum doping


Introduction
The true nature and the key role of the complexity at nanoscale in hightemperature superconductors [1][2][3][4][5][6][7][8] and related systems [9] have sparked growing interest in the condensed matter research field, since the discovery of unconventional overdoped perovskites [10,11]. The development of novel synchrotron radiation sources and imaging techniques on the basis of focusing on the X-ray beam down to the nanoscale allow for the visualization of multiscale inhomogeneities of the supramolecular structure [12][13][14][15]. In this way, scale invariant textures have been observed in lattice, spin, and charge degrees of freedom in strongly correlated oxide perovskites [16][17][18][19][20][21] and at metal-insulator transitions [22]. These results have shown that scale invariance [23,24] in granular matter near a critical point has been favoring quantum coherence in high-temperature superconductivity [25][26][27][28][29]. Lattice complexity beyond the average crystalline structure effects in the mechanism of high Tc superconductivity at optimum doping is attracting high interest. In particular, the role of oxygen interstitials and vacancies spatial disposition has a strong impact on the electronic properties of superconducting perovskite materials [30][31][32]. The texture of oxygen interstitials (O-i) related with their selforganization due to their large mobility in the space layers above about 200 K was detected by advanced scanning micro-X-ray diffraction, in layered perovskites, such as nickelates La2NiO4+y [33,34] and in several families of cuprate high-temperature superconductors (HTS), such as doped La2CuO4+y (La214) [35][36][37][38][39][40][41][42], YBa2Cu3O6+y (Y123) [43][44][45], and Bi2Sr2CaCu2O8+y (Bi2212) [46][47][48][49]. In this work, we focus on HgBa2CuO4+y (Hg1201) at optimum doping [50][51][52][53][54][55][56][57][58][59][60], which is the simplest member of the mercury-based cuprate compounds that provides the highest superconducting transition temperature. HgBa2CuO4+y has a simple tetragonal average structure with average Cu-O bond length of 194 pm [54], showing self-organization of a nematic phase with co-existing O-i atomic stripes [56,57] running in the a-direction (100) and bdirection (010) of the ab plane. The presence of anisotropic structures at room temperature, and the understanding of their properties, are crucial to explain the still elusive room temperature nematicity observed in HTS [61,62]. However, while the nanoscale phase separation made of first puddles rich in O-i stripes anticorrelated with second puddles that show short-range charge density waves (CDW) has been visualized by scanning micro-X-ray diffraction [57], the imaging of the nanoscale phase separation on the mesoscale (100-1000 nm) in Hg1201 is still missing. Information on this intermediate scale appears to be quite important, since the oxygen interstitials form domains with a dimension ranging from few, up to tens of nanometers. Therefore, higher resolution in real space is required for the X-ray diffraction measurements to image oxygen interstitials ordering on the mesoscale. Thanks to the advances in X-ray, focusing on optics is nowadays possible to cover this length scale connecting the microscopic to the nano-and atomic-world using beams down to 100 nm in scanning mode.
In this work, we have used scanning micro-X-ray diffraction (SµXRD) with a focused beam of 1 × 1 µm 2 joint with scanning nano-X-ray diffraction (SnXRD) with a focused beam of 100 × 300 nm 2 for directly imaging the texture formed by the oxygen interstitials in Hg1201 at microscale and mesoscale. We exploited spatial statistic tools, such as the probability density function of O-i populations, spatial correlations, and connectivity to describe quantitatively the O-i textures from micron to mesoscale. The results confirm the power law behavior of the oxygen interstitials organized in stripes at micrometric scale in Hg1201 [57], which was observed first in La2CuO4+y at optimum doping [35,39]. In addition, at improved resolution, we have been able to visualize a percolation interface between the insulating CDW puddles and the oxygen wires [61,62].

Results
The HgBa2CuO4+y single crystals have been grown with a final oxygen treatment to establish the y concentration of oxygen interstitials of about y = 0.12 showing the superconducting optimum critical temperature (TC) of 94 K [51][52][53]. The crystal structure has been determined by standard X-ray diffraction. The structure, schematized in Figure 1a, has a tetragonal P4/mmm space group symmetry, and the unit cell parameters are: a = b = 3.874 Å, c = 9.504 Å at room temperature in agreement with reference [54].
Nematic phase of atomic oxygen wires has been investigated by highenergy X-ray diffraction measurements performed at the BW5 beamline of the DESY Synchrotron in Hamburg. The results provide compelling experimental evidence for the location of the oxygen interstitial (O-i) content, y, which occupies the interstitial site ½, ½, and 0 in the basal plane of HgBa2CuO4+y. The arrangement of O-i gives rise to intriguing diffraction features constituted by clear diffuse streaks crossing the Bragg peaks along all the three crystallographic directions [56][57][58]. In Figure 1b, we show a typical diffuse streak between the (3,3) and (3,4) reflections in the hk diffraction plane. Figure  1c shows the streak profile in red and the background in black. The normalized X-ray diffraction profile is shown in the lower panel, where the diffuse streak intensity has been quantified by the difference between the Bragg peaks profile and the background. In this way, we obtained the streak intensity, due to the O-i stripes. The connection of the observed streaks with the O-i arrangement in stripes has been proved by calculating the X-ray diffraction pattern of a HgBa2CuO4.12 lattice made by 50 × 50 × 20 unit cells. In this model structure, the O-i located with positive correlations along the a-and b-axes has been created by the Monte Carlo method [63] using the software package DISCUS [64]. Scale-free distribution of oxygen wires from microscale to mesoscale has been probed first by scanning micro XRD measurements (SµXRD) and thus scanning nano XRD measurements (SnXRD) to enhance the spatial resolution. Both SµXRD and SnXRD measurements have been carried out at the ID13 beamline of ESRF in Grenoble. The O-i stripes in Hg1201 run along the a(b) direction with no correlation along the c direction; therefore, they give streaks also in the hl plane. The streak intensity, Istreak, in the hl plane has been calculated for each frame collected in the reflection geometry during the scanning of the sample, similar to what has been carried out for the Istreak calculations in the hk plane (see Figure 1f,g).
In this way, we have built the maps of the streak intensity, Istreak, that visualize the O-i spatial distribution. The maps measured in SµXRD and SnXRD show rich stripe-ordered O-i (yellow regions) embedded in a matrix of disordered O-i (black regions), as shown in Figure 2a,b, respectively. To characterize this inhomogeneity, we used a spatial statistics approach. More specifically, we have computed the probability density function (PDF) spatial correlations, G(r), and percolation pathways of the streak intensity for both SµXRD and SnXRD. The PDF can be modelled by a power law behavior given by PDF(Istreak) = C(Istreak) -α , where C is a constant. We find the critical exponent α = 2.0 ± 0.1 at both micron and mesoscale; the Istreak of SnXRD has been scaled in a way that its PDF overlaps the PDF of Istreak in the SµXRD, as shown in Figure 2c. The intensity of streak values ranges from values smaller than 1 in the poor oxygen wire regions to values larger than 10 3 in the rich O-i wire zones. The results show the scale-free organization of O-i in the sample, as it was already found at microscale in other oxygen-doped cuprates. However, we observe that in the SµXRD maps, the size of oxygen wire domains does not exceed the single pixel size corresponding to an area of 1 × 1 µm 2 given by the scanning beam size. This limitation has been overcome by enhancing the real space resolution in XRD setup using SnXRD, where the beam size has been reduced to 0.1 × 0.3 µm 2 . Indeed, in this case, the submicron structural features of domains rich in oxygen atomic wires can be appreciated (see Figure 2b).
To quantify the submicron structural features, we have extracted the spatial correlation function G(r) shown in Figure 2d. We observe that the decay of G(r) corresponds to 2 µm in both SµXRD and SnXRD, indicating this distance as the typical size of domains rich in oxygen atomic wires. We have studied the connectivity between the domains rich in O-i wires using cluster analysis. More specifically, we have calculated all the forming clusters, selecting the cluster with the largest extent, as a function of the streak intensity, Istreak. When this extent is equal to the system size, there is a spanning cluster and the system percolates. Therefore, we have calculated the percolation threshold, P(Istreak), the spanning cluster size, and the number of forming clusters in both SµXRD and SnXRD maps in Figure 2a,b. The results are shown in Figure 2e,f. At micron resolution, we cannot find a percolation threshold due to the resolution limit. On the other hand, in SnXRD map, we can find small clusters forming up to the percolation threshold, P(Istreak) = 2.7. At this intensity value, a spanning cluster occurs as an intermediate space (red color in the maps of panels a and b) connecting zones rich in oxygen wires (yellow spots in the maps) and CDW puddles (black regions).
Band structure calculations of doped HgBa2CuO4+y at optimum doping with 0.12 < y < 0.16 have been carried out with the assumption of very large 1D and 2D superstructures of ordered O-i [60]. The band structure calculations agree on the strong inhomogeneity of hole doping in metallic CuO2 plane near and far from the O-i oxygen interstitial wire. Following the experimental atomic inhomogeneity of doped HgBa2CuO4+y crystal, its structure is formed by a nematic phase of O-i rich metallic wires made of HgBa2CuO (5) [61]. The charge density is strongly inhomogeneous and differs considerably between sites in the proximity and far away from the wire of oxygen interstitials. Figure 3 shows the local decomposition of the DOS at EF (in units of (atom eV) −1 ) for Hg12Ba24Cu12O48 and "striped" Hg12Ba24Cu12O48+n with n = 2 oxygen interstitials. Apical O2 and planar O1 are the apical Oap and planar Opl oxygen coordinated by Cu. A periodic boundary condition was considered as an artificial periodicity of 6-unit cells. The supercell Hg12Ba24Cu12O48+2 extends over 6-and 2-lattice constants along x and y, respectively, and two additional oxygen interstitials were inserted at O(3) site in the Hg-plane forming an atomic stripe running along y. At a distance of 3-unit cells from the oxygen interstitial wire, the local electronic structure becomes close to the undoped system. Panel 3a shows the total density of states (DOS) for superconducting HgBa2CuO4+y with the one-dimensional oxygen interstitials O-i in the [HgO-i] spacer layers and for the undoped cuprate. Figure 3b shows the partial DOS in the apical oxygen sites and Figure 3c the partial DOS in the atomic Hg plane.
The density of states at the Fermi energy, N(EF), is higher than the undoped system, about 3 eV −1 per elementary cell. The increase in DOS is limited to the Cu adjacent to the O-i wire; see Figure 3 and Table 1. Ba and Hg atoms are negligible N(EF) in the undoped case, but the striped O-rich system shows a large increase in DOS on the sites near oxygen interstitials [60]. Hybridization with the p-states on the oxygen interstitials creates a large s-and d-DOS on Hg, and increase in the p-DOS on apical and planar oxygen states. The increase in the local DOS can be understood as an effect due to the O-i atomic stripe, which forms electronic wires with high hole doping when EF is approaching the wire van Hove DOS peak. The local DOS is very small in more distant atoms in the other atomic layers (BaO, Hg) with values very similar to the undoped Hg1201. At two lattice units from the O-stripe, the DOS already seems to have restored its local character as an undoped Hg1201, except for the modest increase in Cud DOS. Moreover, this is corroborated by the electric charge density within the Cu atoms, as well as by an analysis of the folded FS [60]. The wire-wire transversal interaction between adjacent wires in the CuO2 plane is weak due to the chosen large distance (6a) along x. The wire-wire interaction along z is weak between the CuO2 planes in this layered cuprate perovskite. Correlation is not expected to be an issue for cuprates with doping well away from half-filling of the d-band in agreement with the angular correlation of positron annihilation radiation (ACAR) spectroscopy [59], which probes FSs and bands that evolve with doping in agreement with densityfunctional theory (DFT) calculations in the overdoped regime.

Discussion
We have used scanning nano-X-ray diffraction with a focused beam of 100 × 300 nm 2 for directly imaging the mesoscale texture formed by the oxygen interstitial stripes in HgBa2CuO4+y. The O-i stripes show a scale-free-like organization, as found at microscopic and atomic scales in scanning micro-Xray diffraction and previous scanning transmission microscopy measurements in different systems. These results demonstrate the self-similarity in the spatial texture of the oxygen interstitial stripes affecting the material properties. This specific ordering of defects changes dramatically the electronic structure with the rise in a sharp peak at the Fermi energy in the density of state. This indicates that the ordering of interstitial oxygen dopants in atomic stripes gives a metallic character to the CuO2 plane. Therefore, the positive effect on TC from the ordering of O-i might come from the hole doping that the excess O-i provides to the regions far from the dopants. This scenario is consistent with the recent discovery that the space far from the oxygen interstitial atomic stripes in Hg1201 hosts segregated CDW ordered puddles [57]. The compelling evidence of scale invariant distribution of metallic wires in optimum doping Hg1201 supports the proposal that the system showing the superconducting high-critical temperature is in the proximity of a critical point for nanoscale phase separation near a Lifshitz transition in multi-band correlated electronic materials [64][65][66][67] in diborides [68], cuprates [69], organics [70,71], pressurized hydrides [72,73] showing high-order van Hove singularity [74,75], which could be further amplified by spin orbit coupling [76] due to internal electric field gradients driven by local charge segregation in oxygen-doped Hg1201. The compelling evidence of the nematic phase with scale-free distribution of atomic wires in mercury-based superconducting perovskites at optimum doping indicate the presence of critical opalescence [77] of atomic wires at a critical point involving local charge, spin [78,79], and electron-phonon interactions in two-dimensional superconductors [11,80]. A quantum critical point connected with charge density waves has been predicted by Castellani and Arpaia [81,82].

Materials and Methods
High-energy X-ray diffraction. The O-i organization in Hg1201 has been investigated by high-energy X-ray diffraction (XRD) in transmission geometry at the BW5 beamline of the DESY Synchrotron in Hamburg, using 200 × 200 µm X-ray beam with 100 KeV from the source by a double-crystal Si(111) monochromator. The high-energy allowed us to probe the average bulk O-i organization. The beam line is equipped with a single-axis diffractometer with a motorized goniometric xyz stage head. The single crystal c-axis has been oriented parallel to the direction of the X-ray incoming beam. In this geometry, we can probe the lattice fluctuations on the ab plane. The diffraction patterns have been collected by an area detector at room temperature. We achieved evidence of diffuse streaks connecting Bragg peaks associated with oxygen interstitial (O-i) atomic stripes.
Scanning micro-X-ray diffraction (SµXRD) experiments were performed in reflection geometry using the ID13 beamline at ESRF, Grenoble, France. We applied an incident X-ray energy of 13 KeV. By moving the sample under a 1µm focused beam with an x-y translator, we scanned a sample area of 65 × 80 µm 2 , collecting 5200 different diffraction patterns at T = 100 K. For each scanned point of the sample, the intensity profile of the streak from (1,5) to (1,6) Bragg peaks in the hl plane was extracted.
In conclusion, the scanning nano-X-ray diffraction (SnXRD) experimental setup was equipped with a double-crystal monochromator and a Kirkpatrick-Baez mirror system supplied with a beam size of 100 × 300 nm 2 . The data have been recorded using a wavelength of 0.1 nm. A charge-coupled device FreLon area X-ray detector has been used for the experiment. The sample was mounted on a motorized goniometric xyz stage head. In this study, we achieved evidence of diffuse streaks connecting Bragg peaks associated with oxygen interstitial (O-i) atomic stripes. The diffraction patterns have been collected at room temperature and the intensity profile of the streak from (1,5) to (1,6) Bragg peaks was extracted. We scanned a sample area of 9 × 12 µm 2 moving the sample by means of a piezo yz stage with a step size of 100 and 300 nm along the vertical and horizontal directions, respectively.

Data Availability Statement:
The data that support the findings of this study are available from the corresponding author, G.C., upon reasonable request.

Data Availability Statement:
The data that support the findings of this study are available from the corresponding author, [G.C.], upon reasonable request.