Optimum inhomogeneity of local lattice distortions in La2CuO(4+y)

Abstract

Electronic functionalities in materials from silicon to transition metal oxides are, to a large extent, controlled by defects and their relative arrangement. Outstanding examples are the oxides of copper, where defect order is correlated with their high superconducting transition temperatures. The oxygen defect order can be highly inhomogeneous, even in optimal superconducting samples, which raises the question of the nature of the sample regions where the order does not exist but which nonetheless form the "glue" binding the ordered regions together. Here we use scanning X-ray microdiffraction (with a beam 300 nm in diameter) to show that for La(2)CuO(4+y), the glue regions contain incommensurate modulated local lattice distortions, whose spatial extent is most pronounced for the best superconducting samples. For an underdoped single crystal with mobile oxygen interstitials in the spacer La(2)O(2+y) layers intercalated between the CuO(2) layers, the incommensurate modulated local lattice distortions form droplets anticorrelated with the ordered oxygen interstitials, and whose spatial extent is most pronounced for the best superconducting samples. In this simplest of high temperature superconductors, there are therefore not one, but two networks of ordered defects which can be tuned to achieve optimal superconductivity. For a given stoichiometry, the highest transition temperature is obtained when both the ordered oxygen and lattice defects form fractal patterns, as opposed to appearing in isolated spots. We speculate that the relationship between material complexity and superconducting transition temperature T(c) is actually underpinned by a fundamental relation between T(c) and the distribution of ordered defect networks supported by the materials.

Fig. 1.

The maximum superconducting critical temperature, discovered so far by material research, increases by increasing the lattice complexity. The maximum critical temperature at ambient pressure (sold bars) and high pressure (dashed bars) in systems made by a single element (Ca under pressure) and two elements (magnesium diboride) and copper oxides with multiple elements in the chemical formula.

Fig. 2.

The coexistence of the LLD droplets and ordered Oi puddles in different spatial locations of La₂CuO_4+y for y ≈0.06 as seen by scanning X-ray microdiffraction. The pictorial view of the Q2-Oi puddles made of Oi ordered with Q2 superstructure (A) and of the Q3-LLD puddles made of ordered LLD with Q3 superstructure (B) in the bc crystal plane of the Fmmm structure of La₂CuO_4+y C. The three dimensional color plot imaging the position dependence of the Q3-LLD superstructure intensity I(Q3)/I₀ (values > 0) and of the Q2-Oi superstructure intensity I(Q2)/I₀ (values < 0). The scanning images show a few large disconnected Q2-Oi islands (negative blue-dark valleys) embedded in a matrix of the granular superconductor made of Q3-LLD (positive red-dark peaks). Data have been then normalized to the intensity (I₀) of the tail of the main crystalline reflections at each point (x, y). Visual inspection of both the mapping x–y position dependence of the integrated satellite peak intensity for Q2-Oi and Q3-LLD shows that from the scale of hundreds of nanometers to micrometers, the ordered Oi and the ordered LLD occupy distinct locations in space. The intensities of the superstructure satellites due to Q3-LLD and Q2-Oi ordering have been integrated over square sub-areas of the images recorded by the CCD detector in reciprocal-lattice units (r.l.u.). (D) The Q3-LLD superstructure intensity I(Q3)/I₀ and of the Q2-Oi superstructure intensity I(Q2)/I₀ are plotted as a function of each other. The resulting plot indicates a high degree of anti-correlation between the two type of domains characterized by different superstructures. (E) The schematic view of the spatial distribution of the LLD droplets (blue circles) and the ordered Oi puddles (red polygons). The grey backgrounds are regions of the sample where neither droplets or puddles are present.

Fig. 3.

(A) The CCD image of the Q3-LLD satellite in the b*–c* plane near the main Fmmm reflections of the underdoped La₂CuO_4+y single crystal, after the removal of the Q2-Oi satellite by rapid quenching after heating the sample above 350 K. Crystals are cooled to liquid nitrogen temperatures (as low as 85 K) with a 700 series Oxford Cryosystems cryocooler. (B) The position dependence of the Q3-LLD superstructure intensity I(Q3)/I₀ in the 2D color plots after the removal of the Q2-Oi superstructure intensity I(Q2)/I₀, by thermal annealing. (C) The intensity of the Q3-LLD XRD reflections is plotted as function of fluence ϕ or time for constant X-ray flux. The surface is illuminated by a X-ray flux ϕ_P(0.1 nm) = 5.10¹⁴ N_P·s^-1 cm^-2 keeping the temperature constant at 85 K. (D) The temperature evolution of the Q3-LLD satellite intensity in the range of 85 to 350 K collecting images every 2 K. The time evolution experiment has been carried out at the Elettra storage ring in Trieste. The X-ray beam, emitted by the wiggler source, was monochromatized at the 0.1 nm wavelength by a Si(111) double crystal monochromator and focused on the sample surface at the X-ray diffraction beamline (XRD1).

Fig. 4.

(A) The probability distributions, P(x), of the Q3-LLD XRD intensity x = I(Q3)/I₀ for single crystals of electrochemically doped La₂CuO_4+y from the underdoped state (y = 0.06) to the optimum doping range, 0.1 < y < 0.12. The curves follow a power law distribution P(x) ∝ x^-α exp(-x/x₀) with a variable exponential cut-off x₀. The curves of all samples as a function of x/x₀ collapse on the same curve. (B) The spatial correlation function, G(r), of the Q3-LLD XRD follows a power law distribution G(r) ∝ r^-η exp(-r/ξ). The correlation length ξ varies from 30 to 140 μm increasing with the doping range of the material investigated. The curves ξ^αG(r) of all samples as a function of r/ξ collapse onto the same curve.

Fig. 5.

(A, B, C) X-ray microdiffraction results for the position dependence of the Q3-LLD superstructure intensity I(Q3)/I₀ in the La₂CuO_4+y crystals with different critical temperature, T_c, 32, 34, and 37 K from A to C. The scanning XRD images show the better self organization of LLD droplets, proceeding to the higher T_c. D. The critical temperature T_c in the range 25 K < T_c < 37 K for five samples is plotted as a function of the cut-off parameter of the distribution of the LLD droplets density probed by the intensity distribution of the Q3-LLD superstructure satellites. Error bars in the critical temperature are of ± 1 K. The dashed line is the fit with a power law curve with exponent 0.4 ± 0.05, in agreement with theoretical predictions (56) for granular superconductivity on a scale invariant network.