SUPPLEMENTARY INFORMATION

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1 SI - N. Poccia et al. Evolution and Control of Oxygen Order in a Cuprate Superconductor SUPPLEMENTARY INFORMATION 1. X-ray diffraction experiments The diffraction data were collected on the x-ray diffraction beamline (XRD1) at Elettra storage ring as shown in Fig. S1. The X-ray beam, emitted by the wiggler source of 2 GeV electron storage ring, was monochromatized by a Si(111) double crystal monochromator, and focused on the sample. Crystals are cooled to liquid nitrogen temperatures (as low as 90 K) with a 700 series Oxford Cryosystems cryocooler. The temperature of the crystal has been monitored with an accuracy of ±1K. We have collected the data in the reflection geometry using a CCD detector assembly, with a photon energy of 12.4 KeV and a high flux X-ray beam available at third generation synchrotron radiation sources. The beam size can be tuned from approximately 400 " 400 to 100 "100 µm 2. The sample oscillation around the b axis is in the 0 < Φ < 20 range, where Φ is the angle between the direction of the photon beam and the a axis. We have investigated a portion reciprocal space recording the diffraction spots up to the maximum indexes 3, 3, 19 in the a*, b*, c* direction respectively. The experimental setup consists of a kappa diffractometer fully controllable from a remote computer. This experiment provides the diffraction intensity averaged over the sample surface area illuminated by the beam spot as shown in Fig.1a, showing the experimental setup in the paper. The supplementary figure 1 presents the x- ray beam line for XRD at Elettra storage ring. -1- NATURE MATERIALS 1

2 Figure S1 The X-ray diffraction XRD1 beam line at Elettra. The light source is a multipole wiggler with a useful range from 4 to 22 kev. The optics consist of a vertical collimating mirror, a doublecrystal Si(111) monochromator followed by a toroidal bendable focusing mirror. The experimental station is equipped with a 165 mm CCD detector from MarResearch. Crystals can be cooled to liquid nitrogen temperatures (routinely 100 K) with a 700 series Oxford Cryosystems cryo-cooler. Data collection is controlled through the Mar software running on a Linux PC workstation NATURE MATERIALS

3 SUPPLEMENTARY INFORMATION 2. Ordering of the oxygen interstitials. The La 2 CuO 4.1 single crystal, used in this experiment, was oxidized using electrochemical methods as discussed in ref. 1. The disordered oxygen interstitial phase shown in Fig. S2 is obtained by heating the sample above the order-to-disorder temperature and rapidly quenching the sample in the temperature at T<300K. Figure S2 The disordered phase of oxygen interstitials in i-o in R the rocksalt La 2 O 2 layers of La 2 CuO 4+y. The i-o ions occupy the O F (1 4,1 4,1 4) fluorite sites La 2 O R 2 O F y forming a random 2D glassy phase. A portion of the crystal made of 10a x 16b Fmmm unit cells with lattice units a= nm and b= nm is shown. We show that by illuminating selected portions of the La 2 CuO 4+y sample with disordered oxygen interstitials (shown in Figure S2) it is possible to form patches of ordered Q2 phase shown in Figure S3. The transition rate is controlled by the x-ray flux. Only the illuminated portions of the sample are converted into the fractal phase of ordered Q2 patches shown in Figure S NATURE MATERIALS 3

4 Figure S3 The Q2 ordered phase of oxygen interstitials in the rocksalt La 2 O R 2 layers of La 2 CuO 4+y. A portion of the crystal of 11a x 16b Fmmm unit cells is shown. The i-o ions occupy the O F (1 4,1 4,1 4) fluorite sites La 2 O R 2 O F y forming the ordered striped phase. The i-o stripes are shown to run along the crystalline a-axis (a= nm) with a period of 4 lattice units along the b axis (b= nm). 3. Surface resistivity Supplementary figure 4 shows the experimental set-up, for single-coil inductance measurements described in ref.2, made on the same crystal surface as probed by x-ray diffraction. The measurement system contains a simple electronic circuit consisting of a capacitor C, connected in parallel to a spiral coil with an inductance L and a resistance R that is placed on the sample surface. This circuit is mounted in a liquid Helium 3 cryostat (HELIOX 3 He). In this experimental system, the sample temperature is controlled and measured by an Oxford ITC-503 temperature controller, interfaced with a computer alongside other measuring devices. A change of impedance of the LC circuit is detected as a change of resonant frequency ν -4-4 NATURE MATERIALS

5 SUPPLEMENTARY INFORMATION and amplitude V of the oscillating signal. The LC parameters are chosen to set ν at 2 4 MHz, which is within the range of optimum operation of the oscillator. The very high sensitivity of the method arises from the strong mutual inductance between sample and coil in the single-coil geometry, in addition to the very high frequency stability of the oscillator. Figure S4 Surface resistivity experiment. Block diagram of the experimental setup for surface resistivity showing the coil sample arrangement, the LC circuit, and the marginal oscillator. The single-coil inductance technique, nondestructive and contact-less, enables us to measure the complex conductivity of the same crystal surface investigated by x-ray diffraction. A miniaturized pancake coil with a 0.5 mm diameter has been used. The temperature variation of the resonant 1 frequency "(T) = L(T)C - R(T)2 2 provides a measure of the temperature L(T) & M($) variation of the coil inductance L = L o " #µ o ' d$ where 1+ 2$%coth(d % L ) M (! ) is the mutual inductance between coil and sample at a given wave number!. Therefore the measured (! ) 2 0 /!, where! 0 is the reference -5- NATURE MATERIALS 5 0

6 frequency of the coil measured in the proximity of non-superconducting copper metal, is a measure of the surface resistivity at frequency!, proportional to the London penetration depth " L. The derivative of the surface resistivity of the La 2 CuO 4.1 samples quenched at low temperature at different stages of the i-o Q2 domain growth is shown in Figure S5. The curve (a) is for the quenched sample and (b), (c), (d), (e) after 0.5, 1.5, 3, and 15 hours of exposure time respectively. Figure S5 Derivative of the surface resistivity. Derivative of the response function of the surface resistivity experiment on La 2 CuO 4.1 samples quenched at low temperature at different stages a,b,c,d,e of the i-o Q2 domains growth. 4. Scaling parameters Fig. S6 shows the fitting parameters, t 0 and τ of the functional form % < I(t) > "B = C# ' 1" e & t"t0 $ ( *# t " t 0 ) ( ) y, the integrated intensity as a function of time in the domain growth regimes (3) and (4) where the ordering of i-o domains in -6-6 NATURE MATERIALS

7 SUPPLEMENTARY INFORMATION different LaO layers develops because of interactions between layers. The results of t 0 as a function of τ show that they scale as " = 300# ln(3# t 0 ). Figure S6 Scaling parameters. t 0 as a function of τ in the domain growth regime (3) and (4) characterized by narrowing of the diffraction line in the c-axis direction. 5. The Kolmogorov nucleation theory. The classical theory for the domain growth dynamics for nucleation and crystallisation is the Kolmogorov theory [3], according to which I(t) = (1" e (t /# )4 ) (named also Johnson-Mehl-Avrami-Kolmogorov, or JMAK equation for crystallisation). The red curve in figure S7 shows the result of fitting the data reported in the main figure 2a to this standard formula. The diffraction intensity increases from the background value I o to the maximum final value I max. The variation of the experimental data between the minimum and the maximum -7- NATURE MATERIALS 7

8 value is accounted for by a single parameter (τ ) fit, keeping the exponent α=4 fixed. The result of the fitting (red curve) clearly shows that the present domain growth process of oxygen interstitials strongly deviates from the standard 3D Kolmogorov behavior. Figure S7 Comparison with the standard Kolmogorov nucleation theory. Panel a) the fit of the domain growth dynamics of oxygen interstitial ordered domains measured by the time dependent variation of the intensity of the superstructure Q2 reflections <I> using the standard Kolmogorov nucleation theory I(t) = (1" e (t /# )4 ) with a single parameter τ (red curve). The result (red curve) is clearly a bad fit of the experimental data. Panel (b) the experimental value "ln(1" (1" < I > norm ) where < I > norm (t) = < I > (t)" < I > 0 is plotted in a loglog < I > max " < I > 0 scale NATURE MATERIALS

9 SUPPLEMENTARY INFORMATION Because of the highly anisotropic nature of La 2 CuO 4+δ, it is possible that the exponent α of the nucleation process could deviate from the Kolmogorov theoretical value of 4. We have accordingly searched for the experimental exponent by plotting the quantity "ln(1" (1" < I > norm ) where < I > norm (t) = < I > (t)" < I > 0 < I > max " < I > 0 in a log-log plot. This should yield a straight line with gradient α for the standard Kolmogorov formula as it is shown by the red line for α=4. The experimental data (block dots) clearly shows a continuous slowing down of the kinetics where the exponent α shows a continuous variation going from an initial value of 4 to 0.65 for at long times. The plot shows the slowing down of the crystallization dynamics and the presence of a more complex dynamics, indicating at least 3 different crystallisation regimes with quite different exponents α 1 =4, α 2 =2.3 and α 3 =0.65, in correspondence to the different exponents linking the anisotropic nucleation and growth processed described in the main text. Other Supporting Online Material for this manuscript includes the following: 1. Movie S1: Movie of time evolution of Q2 x-ray reflections in La 2 CuO 4+y shining the crystal with x-ray continuous flux. Movie of a portion of the reciprocal space of La 2 CuO 4+y displayed by the CCD camera measuring by x-ray diffraction reflection spots. The growing intensity of the q2 superstructure x-ray diffraction satellites as a function of time (in movie frames; Acquisition rate was 1050 sec/frame) records the photostimulated ordering of oxygen interstitial collected using a x-ray -9- NATURE MATERIALS 9

10 beam with a constant photon flux " P(0.1 nm ) = 5#10 14 N P(0.1 nm) # s $1 # cm $2 shinning the sample surface. The images have been mounted in the movie using fit2d and Matlab. Reference. 1. Chou, F. C. & Johnston, D. C. Phase separation and oxygen diffusion in electrochemically oxidized La 2 CuO 4+y : A static magnetic susceptibility study Physical Review B 54, (1996). URL 2. Di Castro, D. et al. Large fluctuations near the superconducting transition in the underdoped Bi2212 Physica C: Superconductivity 332, (2000). URL 3. A. K. Jena, M. C. Chaturvedi Phase Transformations in Materials. Prentice Hall. p. 24 (1992). 10 NATURE MATERIALS