It is well known that the microstructures of the transition-metal oxides1, 2, 3, including the high-transition-temperature (high-Tc) copper oxide superconductors4, 5, 6, 7, are complex. This is particularly so when there are oxygen interstitials or vacancies8, which influence the bulk properties. For example, the oxygen interstitials in the spacer layers separating the superconducting CuO2 planes undergo ordering phenomena in Sr2O1+yCuO2 (ref. 9), YBa2Cu3O6+y (ref. 10) and La2CuO4+y (refs 11–15) that induce enhancements in the transition temperatures with no changes in hole concentrations. It is also known that complex systems often have a scale-invariant structural organization16, but hitherto none had been found in high-Tc materials. Here we report that the ordering of oxygen interstitials in the La2O2+y spacer layers of La2CuO4+y high-Tc superconductors is characterized by a fractal distribution up to a maximum limiting size of 400 μm. Intriguingly, these fractal distributions of dopants seem to enhance superconductivity at high temperature.
Figures at a glance
Figure 1: Mixed real- and reciprocal-space images of dopant ordering. 
a, The X-ray microdiffraction apparatus is located at the European Synchrotron Radiation Facility (ESRF) and features an electron undulator providing 12–13-keV X-rays to crystal optics followed by a tapered glass capillary, which produces a 1-μm2 beam spot at the sample. A charge-coupled area detector (CCD; right hand side) records the X-rays scattered by the sample. The intensity, I(Q2), of the superstructure satellites due to the Q2 ordering of oxygen interstitials in the La2CuO4.1 crystal is integrated over square subareas of the images recorded by the CCD detector in reciprocal-lattice units (r.l.u.) and then normalized to the intensity (I0) of the tail of the main crystalline reflections at each point (x, y) of the sample reached by the translator. b, Incommensurate order is highly inhomogeneous, even for an optimal (Tc = 40 K) superconducting sample of La2CuO4.1. The intensities of the superstructure satellites are presented on a logarithmic scale as a false-colour image. The scale bar corresponds to 100 μm. The intense red–yellow peaks in the two-dimensional colour map represent locations in the sample with high strength of the three-dimensional i-O ordering, and dark blue indicates spots of disordered i-O domains. The scanning images show few regions with intense satellite μXRD reflections and many regions with weak satellite μXRD reflections. c, Real-space view of the ordered domains that give rise to the Q2 superstructure imaged on the CCD detector. It highlights the i-O ions (blue dots) in the c–b plane of the Fmmm crystal structure of La2CuO4. The i-O located at the (1/4, 1/4, 1/4) site in the La2O2+y spacer layers pair to form linear stripes in the orthorhombic a direction with a period of nearly four lattice units along the b axis in the a–b plane. The stripes alternate in different layers with a c-axis periodicity of two lattice units. The red octahedra indicate the CuO6 octahedral coordination units in the CuO2 plane.
Figure 2: Scale-free fractal distribution and power-law statistical analysis of ordered i-O domains. 
a, b, The position dependence of the Q2 superstructure intensity I(Q2)/I0 for two typical samples obtained by following different annealing–quenching protocols, resulting in Tc = 40 K (a) and Tc = 16+32 K (b) phases. Visual inspection of a and b shows that the spikes corresponding to ordered microdomains are more isolated for the more disordered sample with lower Tc than for the high-Tc sample, indicating that the nucleation and growth of Q2 regions proceeds to smaller length scales for shorter annealing times. c, The probability distribution, P(x), of the Q2 XRD intensity x = I(Q2)/I0 scales at sufficiently high intensity as a power-law distribution with exponential cut-off x0. The data are fitted by the function described in the text. The fitted power-law exponent is given by α = 2.6 ± 0.2 independently of the sample critical temperature, and the cut-off increases from 7 < x0 < 9 for the Tc = 16+32 K samples to 28 < x0 < 33 for the Tc = 40 K samples. In the plot, we show that the P(x) distributions, when properly rescaled, collapse on the same universal curve (solid line). d, Spatial correlation function, G(r), where r = |Ri − Rj|, calculated (Supplementary Information) for the intensities at the spots Rk mapped in a and b. The spatial correlation function does not have the standard exponential behaviour but instead obeys a power law, G(r) ∝ r−ηexp(−r/ξ), with cut-off ξ, as expected, near a critical point (see text for details). The correlation length, ξ, increases with increasing
I
, and in the illustrated cases for Tc = 16+32 K and 40 K has respective values 180 ± 30 μm and 400 ± 30 μm.
Figure 3: Nucleation and growth of i-O superstructures.
