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Superstructure formation and the structural phase diagram of YBa2Cu3O6+x
Physica C 317–318 Ž1999. 259–269
Superstructure formation and the structural phase diagram of YBa 2 Cu 3 O6qx N.H. Andersen a,) , M. von Zimmermann b, T. Frello a , M. Kall ¨ a, D. Mønster c , b P.-A. Lindgard , H.F. Poulsen a , O. Schmidt a , ˚ a, J. Madsen a, T. Niemoller ¨ J.R. Schneider b, Th. Wolf d , P. Dosanjh e, R. Liang e, W.N. Hardy e a
Risø National Laboratory, DK-4000 Roskilde, Denmark HASYLAB at DESY, Notkestrasse 85, D-22603 Hamburg, Germany c ˚ UNI-C, Oluf Palmes Alle´ 38, DK-8200 Arhus N, Denmark d Forschungszentrum Karlsruhe, ITP, D-76021 Karlsruhe, Germany e UniÕersity of British Columbia, VancouÕer, Canada V6T 2E7
b
Abstract The structural ordering properties of oxygen in YBa 2 Cu 3 O6qx have been studied by neutron and high-energy synchrotron X-ray diffraction and by computer simulations based on an extension of the Asymmetric Next Nearest Neighbour Interaction ŽASYNNNI. model. The observed structural phases are the tetragonal disordered and five orthorhombic ordered phases that result from Cu–O chains formation along the b-axis and ordering with different periodicity na along the a-axis: ortho-I Ž a., ortho-II Ž2 a., ortho-III Ž3a., ortho-V Ž5a. and ortho-VIII Ž8 a.. Only the tetragonal and the ortho-I structure have long range order. The structural phase diagram of the superstructure ordering has been established from the experimental data, and it is concluded that the short-range superstructure ordering results from the formation of finite size domains that freeze before long range order is established. By an extension of the 2D ASYNNNI lattice gas model to include Coulomb interactions between oxygen atoms on chains that are 2 a apart, we account for the observed structural phases, and confirm that the superstructures freeze into finite size domains at low temperatures. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Cu–O chains; Oxygen ordering; Diffraction; Model simulations
1. Introduction It is well established that superconductivity in YBa 2 Cu 3 O6qx ŽYBCO. depends on the oxygen stoichiometry, x, as well as on details of the oxygen ordering. From the very large number of experimen-
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Corresponding author. E-mail: [email protected]
tal w1–12x and theoretical w13–28x studies that have been performed over the last 10 years it has become clear that the structural ordering results from the formation and alignment of Cu–O chains in the CuO x basal plane of the structure. However, despite the significant efforts there is still no satisfactory understanding of the oxygen ordering properties, in particular the superstructure ordering close to room temperature and below, and how it influences the electronic structure, the charge transfer and the ap-
0921-4534r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 9 . 0 0 0 6 6 - 0
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pearance of superconductivity. A likely reason is that local structural details are important and they can only be obtained from diffraction data in combination with theoretical models that reproduce the experimental results. However, since the oxygen diffusion kinetics becomes very slow at room temperature it becomes hard or impossible to obtain thermodynamic equilibrium and unique structural data. Therefore, reliable model predictions may only be obtained from simulations that reproduce the major non-equilibrium properties observed experimentally. The structural ordering originates from Cu–O chains that start to form at relatively high temperatures in the tetragonal disordered phase. The chains align along one of the crystallographic axis, usually taken as the b-axis, at the transition into the orthorhombic phase. At lower temperatures, the chains tend to grow in length and align in superstructures with different sequences of full Cu–O and empty Cu-vacancy chains and periodicity along the a-axis. These superstructures are significantly perturbed by structural defects, and they are governed by rather weak Coulomb interactions that are comparable to or smaller than the thermal energy, whereas the interactions giving rise to the chain formation are significantly larger. Accordingly, the effective oxygen chain diffusion is very slow and tends to freeze out by itself or by pinning to impurities before equilibrium is established. Sample preparation under controlled conditions of high purity single crystals is, therefore, essential for experimental studies and for comparison between results from different structural techniques and with theories. We have performed structural studies of the oxygen ordering in YBa 2 Cu 3 O6qx as function of temperature and oxygen stoichiometry, 0 - x - 1, on carefully prepared and annealed high purity single crystals by use of neutron scattering and X-ray diffraction, using high-energy Ž100 keV. photons from a synchrotron source. In the cases where crystals are studied by both diffraction techniques, the results are in agreement. The observed structural phases are the tetragonal oxygen disordered, and five orthorhombic oxygen ordered structures. Of the observed phases only the tetragonal and the fundamental ortho-I structure establish long range order. The ortho-I phase is shown in Fig. 1 for the ideal case with x s 1.0 where all the
Fig. 1. Idealised structural phases in YBa 2 Cu 3 O6qx observed experimentally and by Monte Carlo simulation studies of the extended ASYNNNI model. T is the tetragonal disordered, and OI–OVIII refer to the orthorhombic ortho-I– ortho-VIII phases discussed in the text. The optimal oxygen stoichiometries expected for the structural phases are given in parenthesis.
chains are fully occupied, but it appears for all x below the tetragonal to orthorhombic phase transition as alignment of finite length chain fragments Žsee Fig. 5.. The transition temperatures, T T – I Ž x ., separating the tetragonal and ortho-I phase have been determined from neutron powder diffraction studies w29x. The four different superstructure sequences of essentially full Cu–O and empty Cu chains have periodicities along the a-axis: 2 a Ž ortho-II., 3a Ž ortho-III., 5a Ž ortho-V. and 8 a Ž orthoVIII., respectively. These are shown schematically in Fig. 1 for the ideal stoichiometries and ordered structures. In reduced lattice units, the corresponding superstructure peaks appear at modulation vectors Q s Ž nh m , 0, 0., where n is an integer and h m s 1rm for m s 2, 3, 5 and 8, respectively. However, the dominant peaks are at n s 2 for ortho-V and n s 3 for ortho-VIII. These superstructures have all been observed by electron microscopy w1–4x at the same oxygen stoichiometries w4x. The ortho-II w5,6,8,10,12x and ortho-III w7,9,11x have previously been observed as bulk structural phases by neutron w6,9x and X-ray w5,7–12x diffraction techniques. In the present studies on properly annealed single crystals, we find evidence for bulk phase ortho-V and ortho-VIII superstructure ordering from X-ray diffraction. The origin of the finite size superstructure ordering is analysed
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by studies of the thermal hysteresis in the ordering process and the ordering kinetics following quenches from above the ordering temperature. The experimental data are compared with model calculations obtained by computer simulations with Monte Carlo technique of the ordering properties using an extension of the 2D anisotropic lattice gas model, called the Asymmetric Next Nearest Neighbour Interaction ŽASYNNNI. model w13x. With the extension, the model is proven to account for most of the so far unexplained structural features. In the present paper, we give a brief account of the experimental w8–12x and model w22,24,28x studies of the superstructure ordering process and discuss the results obtained. For more detailed information we refer to papers in the reference list. The paper is organised in the following way: in Section 2, we give a brief account for the experimental background of sample preparation and scattering techniques. In Section 3, we present experimental data showing evidence of the ortho-II, ortho-III, ortho-V and ortho-VIII superstructure formation and present the structural phase diagram. Section 4 contains results of the ortho-II superstructure ordering kinetics. In Section 5, we present the extensions of the ASYNNNI lattice gas model and give results, which show that it may be used to account for essential features of the structural ordering process. Finally in Section 6, we make a summary of the results and conclude on our understanding of the superstructure ordering properties.
2. Experimental background High quality single crystals of weights between 10 and 100 mg have been prepared from high purity powders Žtypically 99.99% or 99.999% purity. by the flux growth method as described in Refs. w30,31x. The crystals were prepared with the appropriate oxygen stoichiometry in a gasvolumetric equipment w29x using ceramic YBCO powder as buffer material, resulting in an accuracy of the oxygen stoichiometry better than D x s 0.02. Neutron scattering studies were performed at the TAS1 triple axis spectrometer at the DR3 reactor at Risø National Laboratory using pyrolytic graphite as monochromator and analyser crystals and a neutron
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energy of 5 meV. Two cooled Be-filters were used to remove the higher order reflections from the monochromator. The transverse and longitudinal resolu˚ y1 and 0.019 A˚ y1 , respectively. tions were 0.011 A X-ray diffraction studies were performed at the triple axis diffractometer at the high-energy synchrotron beam line BW5 at HASYLAB in Hamburg, using an incident radiation of 100 keV, which has penetration depths of ; 1 mm in YBCO. Monochromator and analyser crystals were either Ž200. SrTiO 3 or Ž111. SirTaSi 2 , both having a mosaic spread of 50Y , ˚ y1 at the giving a longitudinal resolution of 0.0075 A Ž200. reflection of YBCO. The transverse resolution ˚ y1 at the Ž200., limited by the was 0.0015 to 0.003 A crystal mosaicity of 0.05 to 0.18. The vertical resolution was defined by slits before the sample and the detector. They were usually set to integrate one quarter to one half of a reciprocal lattice unit in the perpendicular direction. The crystals were wrapped in Al-foil and mounted in a small furnace designed for an Eulerian cradle. The furnace, which was operated in the temperature range between 258C and 2508C with an accuracy of 18C, was filled with Ar gas at a pressure of 0.3 bar to prevent oxygen uptake.
3. Superstructure ordering In previous studies of the ortho-II superstructure ordering it was observed that only short-range order develops. The finite size ordering was suggested to result from impurity based random fields that give a logarithmic time dependence of the domain growth, as observed experimentally from quench studies. We have performed new studies of the ortho-II superstructure ordering on other single crystals prepared with x s 0.50, and annealed by different heat treatments. In all cases only short-range order was observed. Here, we shall report on the results from a heating and subsequent cooling cycle of a high purity single crystal, which was initially annealed for an extensive period of 70 days at 808C and slowly cooled to room temperature by 18Crh. 808C was chosen because this is the optimal temperature for domain growth, as will be shown in the next paragraph. This treatment resulted in the longest in-plane correlation lengths observed so far Žsee below.. The
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heating and cooling cycle was performed within about 1 h. The ortho-II superstructure reflections corrected for the experimental resolution are well described by anisotropic Lorentzian scattering functions to the power y: 2
S Ž q . s Ar Ž 1 q Ž qhrG h . q Ž qkrG k . q Ž qlrG l .
2 y
.
2
Ž 1.
where j m s 1rGm is the correlation length along m and q is the deviation from the ordering modulation vector Q. For critical fluctuations y s 1 is expected, whereas the scattering from finite size domains with sharp boundaries is expected to follow Porod’s law: y s Ž d q 1.r2 for a d-dimensional Ising Žor lattice gas. system. Room temperature scans along aU Ž h., bU Ž k . and cU Ž l . were performed at the Ž2.5, 0, 5. reciprocal point and used to derive the correlation ˚ j b s 158 A, ˚ j c s 25 A˚ .. The lengths Ž j a s 52 A, experimental data and a fit to Eq. Ž1. for the scan along h are shown in Fig. 2. The Lorentzian power is yX f 3r2, as expected for 3D domains with sharp
Fig. 2. Synchrotron X-ray diffraction scans in the h-direction between Ž2, 0, 0. and Ž3, 0, 0. for three oxygen compositions x. The three panels show superstructures of: ortho-II at hs 2.5 for x s 0.50 Župper., ortho-III for x s 0.77 at hs 2.33 and 2.67 Žmiddle., and combined ortho-V and ortho-II for x s 0.62 at hs 2.4 and 2.6, and hs 2.5 Žlower..
Fig. 3. Temperature variation of ortho-II superstructure ordering properties of a high purity single crystal with x s 0.50, measured by v-scans at Ž2.5, 0, 5. using synchrotron X-ray diffraction. Prior to the studies the crystal was annealed for 70 days at 808C to develop large ortho-II domains Žsee the text.. Open symbols refer to data from the initial heating, and closed symbols from the subsequent cooling within approximately 1 h. Upper panel shows the peak intensity, middle panel the HWHM and lower panel the X X exponent y obtained from fits to Lorentzians of power y . The solid line in the middle panel is the result of a fit to the HWHM data in the critical region with a fixed exponent n s 0.63 as for a 3D Ising system.
boundaries and with our experimental conditions, where the scattering function is integrated along the vertical direction Žintegration in one direction of a Lorentzian to the power y gives a power yX s y y 1r2.. The temperature variation during heating and subsequent cooling of the peak intensity, the HWHM Žhalf-width at half maximum. of an v-scan Žrocking curve. and the power parameter yX are shown in Fig. 3. The ortho-II superstructure intensity disappears when the temperature exceeds 1508C. During heating the peak intensity, representing the scattering from the crystal when it is annealed for 70 days is significantly higher than during subsequent cooling below 1008C. However, the decrease in intensity is compensated by an increase of the HWHM, such that the integrated intensity is essentially independent of thermal history. This indicates that the total volume and the order parameter of the ortho-II domains depend only on temperature. Above 1008C the scat-
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tering functions for heating and cooling are essentially equal and are given by a Lorentzian of power y s 1. During heating and cooling the HWHM is constant below 1008C, but above this temperature it increases consistent with critical behaviour and an exponent of n s 0.63 as for a 3D Ising system Žsee the heating data in Fig. 3.. That 1008C is a suitable choice of transition temperature is supported further by the fact that the maximum slope of the peak intensity is at 1008C, and that the y exponent starts to increase below this temperature due to slow growth of well ordered domains. Ortho-II superstructure ordering with 3D character has also been observed at x s 0.35, 0.36 w10x and 0.42 but with smaller correlation lengths and lower transition temperatures Žsee Fig. 5 below.. Ortho-III superstructure with finite size domains and 2D character has previously been observed by X-ray and neutron diffraction in a detwinned crystal with x s 0.77 w9x. Similar ortho-III structures have been observed by X-ray diffraction in twinned crystals with x s 0.72 and 0.82 Žcf. Fig. 5.. The result of the h-scan is reproduced in Fig. 2. The characteristic in-plane correlation length is in all cases about four times smaller than for ortho-II. The 2D character of these superstructures is evident from the lack of superstructure correlation along the c-axis. For the crystal with x s 0.72 this has been corroborated by examination of the HWHM of the superstructure reflection above Tc . A critical exponent n s 0.92Ž8., which is in good agreement with the expectation for a 2D Ising system: n s 1, was found. Further, the superstructure peak shapes are properly accounted for using a simple Lorentzian, which is of the expected line shape in the present case for 2D domains with sharp boundaries when the scattering intensity is integrated along one direction. In a crystal prepared with x s 0.62 we observe a superstructure phase with a mixture of ortho-V and ortho-II correlations Žsee Fig. 2.. The peak overlap makes it difficult to analyse the characteristics of the peak shapes. However, the ortho-II reflections show superstructure modulations along the c-axis indicating 3D ordering, whereas correlations along the caxis could not be verified in the ortho-V peaks. By heating, the peak intensity of the ortho-V peaks decreases above Tc s 408C, whereas the ortho-II peaks gain intensity up to 758C and then decreases.
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On subsequent cooling within 1 h the superstructure freezes in with only ortho-II correlations. Superstructure peaks with ordering vector Q s Ž3r8, 0, 0. corresponding to ortho-VIII ordering were observed in a crystal with x s 0.67. The scattering profiles obtained at room temperature for hscans between Ž2, 0, 3. and Ž3, 0, 3. are nicely fitted with two Lorentzians centred close to the expected values h s 2.375 and 2.625 for ortho-VIII. Attempts to fit with a combination of ortho-V and ortho-III superstructure intensities gave significantly poorer agreement. Therefore, fits to two Lorentzians with variable position and widths were performed during subsequent heating and cooling. In the critical region above Tc s 508C, the peak position shifts gradually first from h s 2.375 Ž ortho-VIII. to h s 2.40 Ž ortho-V. and then above 908C to h s 2.33 Ž ortho-III. as shown in Fig. 4. On subsequent cooling within 1 h
Fig. 4. Temperature variation of the superstructure ordering observed by synchrotron X-ray diffraction from h-scans between Ž2, 0, 3. and Ž3, 0, 3. in a crystal prepared with x s 0.67. Open symbols are from the initial heating and closed symbols from subsequent cooling within approximately 1 h. Upper panel shows the peak intensity, middle panel the HWHM and lower panel the peak position obtained by fitting two Lorentzians to the scattering profiles. Notice that the peak positions hs 2.375, 2.4 and 2.33 correspond to ortho-VIII, ortho-V and ortho-III, respectively. The solid curve in the middle panel is a fit to HWHM data in the critical region giving an exponent of n s 0.79Ž3., intermediate between the expectation for 3D Ž n s 0.63. and 2D Ž n s1. Ising systems.
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Fig. 5. Structural phase diagram of the superstructure ordering in YBa 2 Cu 3 O6qx . The superstructure phases: ortho-II ŽOII., orthoIII ŽOIII., ortho-V ŽOV. and ortho-VIII ŽOVIII. develop only short-range order. 3D ordering is observed for ortho-II superstructures. Ortho-III, ortho-V and ortho-VIII are predominantly 2D. The upper data point at x s 0.50 is from Ref. w8x, and the one at x s 0.77 is from Ref. w9x.
Section 3 under pseudo-equilibrium conditions during heating and cooling for x s 0.50. Using X-ray diffraction and the same high-purity crystal as in Section 3 we have verified directly that dynamical scaling applies. Direct measurements of the peak profiles along h, k and l of the Ž2.5, 0, 5. superstructure reflection, as function of time is too slow to allow for the desired time resolution. However, based on a sequence of identical quenches from 1708C to 758C, we have measured the time dependence of the intensity at 21 different points along h and l and thereby established the time dependence of these peak profiles. In these studies the vertical integration Žalong k . is not complete at the early time after a quench. With the assumption that the correlation length j b scales with j a , we have included the finite vertical resolution in the analysis of the peak profiles. The results of the temporal variation of the peak intensity, the correlation lengths j a and j c , and the integrated intensity are shown in Fig. 6. Our
the superstructure is frozen into ortho-V, and the ortho-VIII correlations are not recovered within this short time scale. Scans along l did not indicate 3D ordering, but analysis of the HWHM peak width of the h-scans gives a critical exponent n s 0.79Ž3., which is in between the values for 2D and 3D Ising type ordering. Crystals prepared with composition x s 0.87 and higher have the ortho-I structure with no sign of superstructure formation at room temperature. Based on those facts and our experimental results above, we suggest the superstructure phase diagram shown in Fig. 5.
4. Ortho-II superstructure ordering kinetics In a previous dynamical study of the ortho-II superstructure formation following a quench from ortho-I, a time dependent growth mechanism was observed with a change from algebraic to logarithmic growth at a length scale comparable to the distance between impurities w8x. In this study, the characteristic length was determined from the peak intensity by the assumption that dynamical scaling applies. This implies that the integrated intensity is constant, and thereby that the total volume of the ordered domains is constant. This was established in
Fig. 6. Ortho-II superstructure oxygen ordering kinetics observed by X-ray synchrotron diffraction at Ž2.5, 0, 5. in a high purity crystal with x s 0.5. The results are obtained by fitting a Lorentzian squared function convoluted with the resolution function to the data, as explained in the text. The upper panel shows the peak intensity Žsolid points, left ordinate. and the temperature profile Ždashes line, right ordinate., the middle panel shows the correlation lengths j a Žalong the a-axis. and j c Žalong the c-axis., and the lower panel shows the integrated intensity.
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Fig. 7. Ortho-II superstructure oxygen ordering kinetics measured by neutron diffraction on a very long time scale using the same high-purity crystal with x s 0.50 as for the X-ray diffraction studies presented in Fig. 6. Measurements of the Ž0.5, 0, 0. superstructure peak intensity were performed after a quench from 1708C to 758C and after subsequent rapid changes to other temperatures as marked on the figure.
results show clearly that the peak intensity and the correlation lengths increase after a quench from high temperatures into the ortho-II phase, but the integrated intensity is constant shortly after the quench. Notice that the peak intensity and the integrated intensity decrease immediately after a rapid increase of the temperature to 908C, and that the correlation lengths start to grow faster. These results clearly indicate that the ordering process is governed by growth of domains that have internal thermodynamic equilibrium. Dynamical scaling requires that the domain pattern evolves in a self-similar way with a characteristic length scale that increases as function of time. For an isotropic system, a constant integrated intensity assures that the characteristic length scale determining the domain growth may be derived from the peak intensity of the structure factor. However, for an anisotropic system the pattern would only look selfsimilar if also the growth rates along the main axes, i.e., the correlation lengths, scale with time. Analysis of the correlation lengths presented in Fig. 6 shows that this is indeed the case for j a and j c . From other studies w12x, which will be published elsewhere, we find evidence that also j b scales with j a and j c , as was assumed in our data analysis.
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The ordering kinetics following a temperature quench from 1708C to 758C and subsequent rapid changes to a sequence of different temperatures has been studied by measuring the peak intensity of the Ž1r2, 0, 0. superstructure peak by neutron diffraction. The results are shown in Fig. 7. They reveal the same properties and support the conclusions already established from the X-ray data, but this time on a much longer time scale Žnotice that one point of the neutron data in Fig. 7 covers the whole time-span of the X-ray data in Fig. 6.. From the data it is observed that the fastest domain growth at this late time of the ordering process occurs at temperatures between 758C and 858C, and not as at earlier times, where the X-ray data indicates 908C to be better. Accordingly, we use 808C for annealing our crystals to obtain large ortho-II domains. It should be mentioned, that more extensive measurements show that the domain growth is consistent with a logarithmic variation at late times, and that long range ortho-II ordering is not obtained within a foreseeable period of time.
5. Model studies of oxygen ordering Many model studies of the oxygen ordering in YBCO are based on the 2D ASYNNNI lattice gas Žor Ising. model, which has the three effective pair interactions between the oxygen atoms in the CuO x basal plane: V1 , V2 and V3 visualised in Fig. 8. From Monte Carlo simulations with V1rk B s 5430 K and V3 s 0.12 V1 repulsive, and V2 s y0.36 V1 attractive, it was found that the ASYNNNI model accounts successfully for the concentration dependence
Fig. 8. The effective oxygen pair interactions in the 2D ASYNNNI model used to study the oxygen ordering in the CuO x basal plane of the YBa 2 C 3 O6qx structure. V1 , V2 , and V3 are the effective interactions in the original ASYNNNI model. V5 is the additional pair interaction introduced into the extended model. Filled dots are Cu atoms, open circles are possible oxygen sites.
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of the transition temperature, T T – I , between the tetragonal and the orthorhombic ortho-I phase w15,22,29x. It also predicts the ortho-II phase, but the ordering temperature between ortho-I and ortho-II, TI – II , is significantly higher than found experimentally Žcf. Refs. w15,22x and Fig. 5.. A 3D version of the ASYNNNI model with a weak attractive nearest neighbour interaction, V4 s y0.02 V1 along the caxis, has also been studied w24x. By rescaling V1rk B s 4490 K, it establishes a similar good agreement with the TT – I data, and it opens a significant gap between the TT – I and T I – II . Although finite size ortho-II domains have been observed below T I – II in the simulations, the general trend is that both the 2D and 3D ASYNNNI models predict long-range orthoII order. It is not surprising that none of these models predicts the ortho-III, ortho-V and ortho-VIII structures, which may only be stabilised by more longrange interactions. We have extended the 2D ASYNNNI model by adding a repulsive Coulomb type interaction, V5 , between oxygen pairs that are separated by 2 a without intervening Cu atoms Žsee Fig. 8. w28x. We have estimated the V5 parameter using a screened Coulomb potential w20x and found that 0 F V5 F 0.04 V1 is a realistic range, and studied its consequences for the ordering properties. Using a parallelized code we equilibrate and average a system of 32 independent
128 = 128 ensembles during stepwise quenching from high temperatures. After each temperature step, the system is allowed to equilibrate for 5000 MCS ŽMonte Carlo Steps per site., and the structure factor SŽ q ., Cu–O chain lengths and oxygen co-ordination numbers for the Cu atoms were calculated by averaging typically 10 successive states separated by 10 MCS. Fig. 9 shows the structure factor calculation along the h-direction and corresponding snapshots for three different oxygen concentrations x s 0.50, 0.60 and 0.66. The results are obtained with V5 s 0.04 V1 and a reduced temperature T U ' k B TrV1 s 0.07, which is well inside the ordered phases. The extended ASYNNNI model stabilises the ortho-III superstructure by construction Ž x s 0.66 in Fig. 9. but surprisingly it also shows superstructures with clear signs of ortho-V and ortho-VIII correlations Žcf. Fig. 1 and x s 0.60 in Fig. 9., although these phases may only be stabilised by even longer-range interactions. Further, it is clear that the V5 parameter obstructs the formation of long range ortho-II order by formation of finite size anti-phase and transverse twin domains. As expected, this tendency is less significant for smaller values of V5 . However, we also observe the formation of finite size ortho-II domains for V5 s 0.02 V1. This is visualised in Fig. 10 where the power y, obtained by fitting Eq. Ž1. to the simulated structure factor data for x s 0.50 and
Fig. 9. Results of Monte Carlo simulation studies of the oxygen ordering in YBa 2 Cu 3 O6qx by use of the extended 2D ASYNNNI model with V5 s 0.04 V1. Leftmost figure shows the structure factors calculated along the h-direction for three different oxygen compositions, x s 0.50, 0.60 and 0.66. The reduced temperature of the studies is T U ' k B TrV1 s 0.07, which is well inside the ordered phases of the observed superstructures. All three structure factors: ortho-II at h s 0.5 Ž x s 0.50., ortho-III at h s 0.33 and 0.67 Ž x s 0.66., and ortho-V at h s 0.40 and 0.60 Ž x s 0.60. have finite widths and thereby finite size domains. The three corresponding snapshots Žonly one quarter of the 128 = 128 ensemble is shown., corroborate that finite size domains with anti-phase boundaries are formed for all the superstructures. Local domains of different types of superstructures are found in the snapshots, in particular for the x s 0.60 system, where also ortho-VIII correlations may be found.
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Monte Carlo technique based on the ASYNNNI model with realistic model parameters.
6. Summary and conclusions
Fig. 10. Results of Monte Carlo simulation studies of the extended ASYNNNI model for x s 0.50. T U s k B T r V1 is the reduced temperature. Ža. shows the influence of the V5 parameter on the structural phase transition temperatures separating the tetragonal T phase, the orthorhombic OI Ž ortho-I., and the OII Ž ortho-II. phases. Žb. shows the result for V5 s 0.02 V1 of the power y obtained by fitting a 2D Lorentzian of power y Žcf. Eq. Ž1.. to the simulated structure factor data. A gradual increase from y s1.0 to y s1.5 is observed at the onset of ordering, as expected for 2D finite size domains with sharp boundaries.
V5 s 0.02 V1 , is shown as function of temperature. We find that the onset of ordering results in a gradual change from y s 1 to y f 1.5, which is characteristic of scattering from finite size 2D domains with sharp boundaries. Calculations of the Cu–O chain length also show that it is finite for V5 s 0.02 V1 even for T U ™ 0, whereas it diverges towards infinite for V5 s 0 w28x. Fig. 10 also shows how the simulated ordering temperatures change as function of V5 . A significant decrease is observed, and the gap between T T – I and TI – II increases. If the gap for V5 s 0.02 V1 is combined with the one established from simulations with the 3D ASYNNNI model w24x, we obtain a value of ; 170 K which is approaching the experimental one of ; 250 K. It remains to be studied whether the extended ASYNNNI model may establish quantitative agreement with the experimental diffraction data by a suitable choice of interaction parameters. The use of interaction parameters that are independent of temperature and oxygen composition have been questioned, but the results of our combined experimental w11x and model studies give no evidence that throws doubt on this fixed parameter approach. Clearly, the effect of strain should be studied. This has been done in a mean-field approach w19x, but so far not by
In this paper, we have presented an overview of some of our previous and also very recent experimental results and model studies of the oxygen ordering in YBCO for x ) 0.3. From high energy X-ray and neutron diffraction studies on high quality crystals, that are carefully prepared and annealed, we have verified that the superstructure correlations with ortho-II, ortho-III, ortho-V and ortho-VIII character, earlier observed by electron diffraction w4x, represent bulk structural properties of YBCO. From the diffraction studies, we have established the structural ‘phase diagram’ for oxygen ordering in YBCO as function of oxygen stoichiometry, x, and temperature, T. It is not a real thermodynamic phase diagram since long range order is only observed in the form of regular Bragg peaks in the tetragonal oxygen disordered, and the basic orthorhombic ortho-I phase, which always appears on cooling before the shortrange ordered superstructures. Analyses of the c-axis correlations and the critical exponent n for the peak width above the critical temperatures show that only the ortho-II superstructure develops clear 3D ordering. Hysteresis in the ortho-II superstructure ordering properties during heating and cooling through TI – II shows evidence for quasi-equilibrium ordering, leading to short-range ordering due to formation of finite size domains with anti-phase boundaries, while having internal thermodynamic equilibrium. This is corroborated by the experimental observation of the structure factor line-shape in the ordered phase, which is a Lorentzian raised to the power yX f 3r2. It can also be concluded from the ordering kinetics following a quench into the ordered phase where dynamical scaling has been shown to apply. In our survey determining the phase diagram, we found no evidence of other than the mentioned superstructures. We cannot preclude that other chain ordering structures may develop in carefully prepared crystals and long time annealing at sufficiently low temperatures. It is, however, expected that the slow oxygen kinetics in the ordering process pre-
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vents the development of more complex phases with the ordering sequences predicted theoretically using the infinite chain approximation. The assumption behind this ordering scheme may also be questioned by the recent value of the average Cu–O chain length deducted from NMR studies w32x, which indicate that it is only a few lattice units long at the TI – II ordering temperature for x s 0.5. Below TI – II the ordering kinetics is rapidly slowing down and the domain sizes freeze in. These observations are corroborated by our model studies with the 2D extended ASYNNNI model, which reproduce the average chain length at TI – II and suggest a non-critical slow increase at lower temperature w28x. The extension of the ASYNNNI model to include a simple additional interaction parameter V5 increases the predictive power of this model significantly w28x. Many of the properties that were unexplained in the original version have been accounted for on a semi-quantitative level by this extension. These include the observation of superstructures with ortho-III, ortho-V and ortho-VIII type correlations and the formation of only finite size domains of all the superstructures. Further improvement of model predictions is obtained if the effects of the previously studied 3D coupling are included in the ASYNNNI model w24x. In this way, the observed strong suppression of the TI – II relative to TT – I is almost accounted for. It may be argued that more long-range interactions of the Coulomb type as well as strain should be included. This is certainly needed to explain mesoscopic phenomena as the observed formation of twin-domains. However, for the local structure the parameters in the ASYNNNI model are sufficient, being effective oxygen–oxygen interactions that rapidly decrease in strength with distance. Since the dominant pair interactions, V1 and V2 , have formed Cu–O chains when the inter-chain interactions, V3 and V5 , become effective, there is only a need for effective interactions between nearest neighbour oxygen atoms on different chains. A further extension of the 2D ASYNNNI model should therefore account for the effective Coulomb interaction between oxygen pairs on chains that are 3a or more apart. These interactions are even smaller than V5 ; 100 K P k B and they will thermodynamically only play a role at such low temperatures that the ordering is frozen in due to slow oxygen diffusion.
The detailed information obtained in the present study from numerous experimental scattering investigations and the semi-quantitative agreement with the results of the simple extended ASYNNNI model study have clarified many unsolved problems about the oxygen ordering in YBCO. This information may be used as a basis for new theoretical studies of the electronic structure in the Cu–O chains and the charge transfer leading to superconductivity.
Acknowledgements The Danish Technical Science Research Council supports TF, and The Danish Natural Science Research Council supports synchrotron X-ray diffraction studies through DanSynch. DM acknowledges the hospitality of Edinburgh Parallel Computing Centre as a visitor sponsored by the EU-TMR program. The Danish Natural Science Research Council has provided support for the computing activities at UNI-C. Collaboration with J.V. Andersen, J. Berlin, H. Casalta, T. Fiig, R. Hadfield, O.G. Mouritsen and P. Schleger on initial studies preceding this work is gratefully acknowledged.
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Superstructure formation and the structural phase diagram of YBa 2 Cu 3 O6qx N.H. Andersen a,) , M. von Zimmermann b, T. Frello a , M. Kall ¨ a, D. Mønster c , b P.-A. Lindgard , H.F. Poulsen a , O. Schmidt a , ˚ a, J. Madsen a, T. Niemoller ¨ J.R. Schneider b, Th. Wolf d , P. Dosanjh e, R. Liang e, W.N. Hardy e a
Risø National Laboratory, DK-4000 Roskilde, Denmark HASYLAB at DESY, Notkestrasse 85, D-22603 Hamburg, Germany c ˚ UNI-C, Oluf Palmes Alle´ 38, DK-8200 Arhus N, Denmark d Forschungszentrum Karlsruhe, ITP, D-76021 Karlsruhe, Germany e UniÕersity of British Columbia, VancouÕer, Canada V6T 2E7
b
Abstract The structural ordering properties of oxygen in YBa 2 Cu 3 O6qx have been studied by neutron and high-energy synchrotron X-ray diffraction and by computer simulations based on an extension of the Asymmetric Next Nearest Neighbour Interaction ŽASYNNNI. model. The observed structural phases are the tetragonal disordered and five orthorhombic ordered phases that result from Cu–O chains formation along the b-axis and ordering with different periodicity na along the a-axis: ortho-I Ž a., ortho-II Ž2 a., ortho-III Ž3a., ortho-V Ž5a. and ortho-VIII Ž8 a.. Only the tetragonal and the ortho-I structure have long range order. The structural phase diagram of the superstructure ordering has been established from the experimental data, and it is concluded that the short-range superstructure ordering results from the formation of finite size domains that freeze before long range order is established. By an extension of the 2D ASYNNNI lattice gas model to include Coulomb interactions between oxygen atoms on chains that are 2 a apart, we account for the observed structural phases, and confirm that the superstructures freeze into finite size domains at low temperatures. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Cu–O chains; Oxygen ordering; Diffraction; Model simulations
1. Introduction It is well established that superconductivity in YBa 2 Cu 3 O6qx ŽYBCO. depends on the oxygen stoichiometry, x, as well as on details of the oxygen ordering. From the very large number of experimen-
)
Corresponding author. E-mail: [email protected]
tal w1–12x and theoretical w13–28x studies that have been performed over the last 10 years it has become clear that the structural ordering results from the formation and alignment of Cu–O chains in the CuO x basal plane of the structure. However, despite the significant efforts there is still no satisfactory understanding of the oxygen ordering properties, in particular the superstructure ordering close to room temperature and below, and how it influences the electronic structure, the charge transfer and the ap-
0921-4534r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 9 . 0 0 0 6 6 - 0
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pearance of superconductivity. A likely reason is that local structural details are important and they can only be obtained from diffraction data in combination with theoretical models that reproduce the experimental results. However, since the oxygen diffusion kinetics becomes very slow at room temperature it becomes hard or impossible to obtain thermodynamic equilibrium and unique structural data. Therefore, reliable model predictions may only be obtained from simulations that reproduce the major non-equilibrium properties observed experimentally. The structural ordering originates from Cu–O chains that start to form at relatively high temperatures in the tetragonal disordered phase. The chains align along one of the crystallographic axis, usually taken as the b-axis, at the transition into the orthorhombic phase. At lower temperatures, the chains tend to grow in length and align in superstructures with different sequences of full Cu–O and empty Cu-vacancy chains and periodicity along the a-axis. These superstructures are significantly perturbed by structural defects, and they are governed by rather weak Coulomb interactions that are comparable to or smaller than the thermal energy, whereas the interactions giving rise to the chain formation are significantly larger. Accordingly, the effective oxygen chain diffusion is very slow and tends to freeze out by itself or by pinning to impurities before equilibrium is established. Sample preparation under controlled conditions of high purity single crystals is, therefore, essential for experimental studies and for comparison between results from different structural techniques and with theories. We have performed structural studies of the oxygen ordering in YBa 2 Cu 3 O6qx as function of temperature and oxygen stoichiometry, 0 - x - 1, on carefully prepared and annealed high purity single crystals by use of neutron scattering and X-ray diffraction, using high-energy Ž100 keV. photons from a synchrotron source. In the cases where crystals are studied by both diffraction techniques, the results are in agreement. The observed structural phases are the tetragonal oxygen disordered, and five orthorhombic oxygen ordered structures. Of the observed phases only the tetragonal and the fundamental ortho-I structure establish long range order. The ortho-I phase is shown in Fig. 1 for the ideal case with x s 1.0 where all the
Fig. 1. Idealised structural phases in YBa 2 Cu 3 O6qx observed experimentally and by Monte Carlo simulation studies of the extended ASYNNNI model. T is the tetragonal disordered, and OI–OVIII refer to the orthorhombic ortho-I– ortho-VIII phases discussed in the text. The optimal oxygen stoichiometries expected for the structural phases are given in parenthesis.
chains are fully occupied, but it appears for all x below the tetragonal to orthorhombic phase transition as alignment of finite length chain fragments Žsee Fig. 5.. The transition temperatures, T T – I Ž x ., separating the tetragonal and ortho-I phase have been determined from neutron powder diffraction studies w29x. The four different superstructure sequences of essentially full Cu–O and empty Cu chains have periodicities along the a-axis: 2 a Ž ortho-II., 3a Ž ortho-III., 5a Ž ortho-V. and 8 a Ž orthoVIII., respectively. These are shown schematically in Fig. 1 for the ideal stoichiometries and ordered structures. In reduced lattice units, the corresponding superstructure peaks appear at modulation vectors Q s Ž nh m , 0, 0., where n is an integer and h m s 1rm for m s 2, 3, 5 and 8, respectively. However, the dominant peaks are at n s 2 for ortho-V and n s 3 for ortho-VIII. These superstructures have all been observed by electron microscopy w1–4x at the same oxygen stoichiometries w4x. The ortho-II w5,6,8,10,12x and ortho-III w7,9,11x have previously been observed as bulk structural phases by neutron w6,9x and X-ray w5,7–12x diffraction techniques. In the present studies on properly annealed single crystals, we find evidence for bulk phase ortho-V and ortho-VIII superstructure ordering from X-ray diffraction. The origin of the finite size superstructure ordering is analysed
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by studies of the thermal hysteresis in the ordering process and the ordering kinetics following quenches from above the ordering temperature. The experimental data are compared with model calculations obtained by computer simulations with Monte Carlo technique of the ordering properties using an extension of the 2D anisotropic lattice gas model, called the Asymmetric Next Nearest Neighbour Interaction ŽASYNNNI. model w13x. With the extension, the model is proven to account for most of the so far unexplained structural features. In the present paper, we give a brief account of the experimental w8–12x and model w22,24,28x studies of the superstructure ordering process and discuss the results obtained. For more detailed information we refer to papers in the reference list. The paper is organised in the following way: in Section 2, we give a brief account for the experimental background of sample preparation and scattering techniques. In Section 3, we present experimental data showing evidence of the ortho-II, ortho-III, ortho-V and ortho-VIII superstructure formation and present the structural phase diagram. Section 4 contains results of the ortho-II superstructure ordering kinetics. In Section 5, we present the extensions of the ASYNNNI lattice gas model and give results, which show that it may be used to account for essential features of the structural ordering process. Finally in Section 6, we make a summary of the results and conclude on our understanding of the superstructure ordering properties.
2. Experimental background High quality single crystals of weights between 10 and 100 mg have been prepared from high purity powders Žtypically 99.99% or 99.999% purity. by the flux growth method as described in Refs. w30,31x. The crystals were prepared with the appropriate oxygen stoichiometry in a gasvolumetric equipment w29x using ceramic YBCO powder as buffer material, resulting in an accuracy of the oxygen stoichiometry better than D x s 0.02. Neutron scattering studies were performed at the TAS1 triple axis spectrometer at the DR3 reactor at Risø National Laboratory using pyrolytic graphite as monochromator and analyser crystals and a neutron
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energy of 5 meV. Two cooled Be-filters were used to remove the higher order reflections from the monochromator. The transverse and longitudinal resolu˚ y1 and 0.019 A˚ y1 , respectively. tions were 0.011 A X-ray diffraction studies were performed at the triple axis diffractometer at the high-energy synchrotron beam line BW5 at HASYLAB in Hamburg, using an incident radiation of 100 keV, which has penetration depths of ; 1 mm in YBCO. Monochromator and analyser crystals were either Ž200. SrTiO 3 or Ž111. SirTaSi 2 , both having a mosaic spread of 50Y , ˚ y1 at the giving a longitudinal resolution of 0.0075 A Ž200. reflection of YBCO. The transverse resolution ˚ y1 at the Ž200., limited by the was 0.0015 to 0.003 A crystal mosaicity of 0.05 to 0.18. The vertical resolution was defined by slits before the sample and the detector. They were usually set to integrate one quarter to one half of a reciprocal lattice unit in the perpendicular direction. The crystals were wrapped in Al-foil and mounted in a small furnace designed for an Eulerian cradle. The furnace, which was operated in the temperature range between 258C and 2508C with an accuracy of 18C, was filled with Ar gas at a pressure of 0.3 bar to prevent oxygen uptake.
3. Superstructure ordering In previous studies of the ortho-II superstructure ordering it was observed that only short-range order develops. The finite size ordering was suggested to result from impurity based random fields that give a logarithmic time dependence of the domain growth, as observed experimentally from quench studies. We have performed new studies of the ortho-II superstructure ordering on other single crystals prepared with x s 0.50, and annealed by different heat treatments. In all cases only short-range order was observed. Here, we shall report on the results from a heating and subsequent cooling cycle of a high purity single crystal, which was initially annealed for an extensive period of 70 days at 808C and slowly cooled to room temperature by 18Crh. 808C was chosen because this is the optimal temperature for domain growth, as will be shown in the next paragraph. This treatment resulted in the longest in-plane correlation lengths observed so far Žsee below.. The
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heating and cooling cycle was performed within about 1 h. The ortho-II superstructure reflections corrected for the experimental resolution are well described by anisotropic Lorentzian scattering functions to the power y: 2
S Ž q . s Ar Ž 1 q Ž qhrG h . q Ž qkrG k . q Ž qlrG l .
2 y
.
2
Ž 1.
where j m s 1rGm is the correlation length along m and q is the deviation from the ordering modulation vector Q. For critical fluctuations y s 1 is expected, whereas the scattering from finite size domains with sharp boundaries is expected to follow Porod’s law: y s Ž d q 1.r2 for a d-dimensional Ising Žor lattice gas. system. Room temperature scans along aU Ž h., bU Ž k . and cU Ž l . were performed at the Ž2.5, 0, 5. reciprocal point and used to derive the correlation ˚ j b s 158 A, ˚ j c s 25 A˚ .. The lengths Ž j a s 52 A, experimental data and a fit to Eq. Ž1. for the scan along h are shown in Fig. 2. The Lorentzian power is yX f 3r2, as expected for 3D domains with sharp
Fig. 2. Synchrotron X-ray diffraction scans in the h-direction between Ž2, 0, 0. and Ž3, 0, 0. for three oxygen compositions x. The three panels show superstructures of: ortho-II at hs 2.5 for x s 0.50 Župper., ortho-III for x s 0.77 at hs 2.33 and 2.67 Žmiddle., and combined ortho-V and ortho-II for x s 0.62 at hs 2.4 and 2.6, and hs 2.5 Žlower..
Fig. 3. Temperature variation of ortho-II superstructure ordering properties of a high purity single crystal with x s 0.50, measured by v-scans at Ž2.5, 0, 5. using synchrotron X-ray diffraction. Prior to the studies the crystal was annealed for 70 days at 808C to develop large ortho-II domains Žsee the text.. Open symbols refer to data from the initial heating, and closed symbols from the subsequent cooling within approximately 1 h. Upper panel shows the peak intensity, middle panel the HWHM and lower panel the X X exponent y obtained from fits to Lorentzians of power y . The solid line in the middle panel is the result of a fit to the HWHM data in the critical region with a fixed exponent n s 0.63 as for a 3D Ising system.
boundaries and with our experimental conditions, where the scattering function is integrated along the vertical direction Žintegration in one direction of a Lorentzian to the power y gives a power yX s y y 1r2.. The temperature variation during heating and subsequent cooling of the peak intensity, the HWHM Žhalf-width at half maximum. of an v-scan Žrocking curve. and the power parameter yX are shown in Fig. 3. The ortho-II superstructure intensity disappears when the temperature exceeds 1508C. During heating the peak intensity, representing the scattering from the crystal when it is annealed for 70 days is significantly higher than during subsequent cooling below 1008C. However, the decrease in intensity is compensated by an increase of the HWHM, such that the integrated intensity is essentially independent of thermal history. This indicates that the total volume and the order parameter of the ortho-II domains depend only on temperature. Above 1008C the scat-
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tering functions for heating and cooling are essentially equal and are given by a Lorentzian of power y s 1. During heating and cooling the HWHM is constant below 1008C, but above this temperature it increases consistent with critical behaviour and an exponent of n s 0.63 as for a 3D Ising system Žsee the heating data in Fig. 3.. That 1008C is a suitable choice of transition temperature is supported further by the fact that the maximum slope of the peak intensity is at 1008C, and that the y exponent starts to increase below this temperature due to slow growth of well ordered domains. Ortho-II superstructure ordering with 3D character has also been observed at x s 0.35, 0.36 w10x and 0.42 but with smaller correlation lengths and lower transition temperatures Žsee Fig. 5 below.. Ortho-III superstructure with finite size domains and 2D character has previously been observed by X-ray and neutron diffraction in a detwinned crystal with x s 0.77 w9x. Similar ortho-III structures have been observed by X-ray diffraction in twinned crystals with x s 0.72 and 0.82 Žcf. Fig. 5.. The result of the h-scan is reproduced in Fig. 2. The characteristic in-plane correlation length is in all cases about four times smaller than for ortho-II. The 2D character of these superstructures is evident from the lack of superstructure correlation along the c-axis. For the crystal with x s 0.72 this has been corroborated by examination of the HWHM of the superstructure reflection above Tc . A critical exponent n s 0.92Ž8., which is in good agreement with the expectation for a 2D Ising system: n s 1, was found. Further, the superstructure peak shapes are properly accounted for using a simple Lorentzian, which is of the expected line shape in the present case for 2D domains with sharp boundaries when the scattering intensity is integrated along one direction. In a crystal prepared with x s 0.62 we observe a superstructure phase with a mixture of ortho-V and ortho-II correlations Žsee Fig. 2.. The peak overlap makes it difficult to analyse the characteristics of the peak shapes. However, the ortho-II reflections show superstructure modulations along the c-axis indicating 3D ordering, whereas correlations along the caxis could not be verified in the ortho-V peaks. By heating, the peak intensity of the ortho-V peaks decreases above Tc s 408C, whereas the ortho-II peaks gain intensity up to 758C and then decreases.
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On subsequent cooling within 1 h the superstructure freezes in with only ortho-II correlations. Superstructure peaks with ordering vector Q s Ž3r8, 0, 0. corresponding to ortho-VIII ordering were observed in a crystal with x s 0.67. The scattering profiles obtained at room temperature for hscans between Ž2, 0, 3. and Ž3, 0, 3. are nicely fitted with two Lorentzians centred close to the expected values h s 2.375 and 2.625 for ortho-VIII. Attempts to fit with a combination of ortho-V and ortho-III superstructure intensities gave significantly poorer agreement. Therefore, fits to two Lorentzians with variable position and widths were performed during subsequent heating and cooling. In the critical region above Tc s 508C, the peak position shifts gradually first from h s 2.375 Ž ortho-VIII. to h s 2.40 Ž ortho-V. and then above 908C to h s 2.33 Ž ortho-III. as shown in Fig. 4. On subsequent cooling within 1 h
Fig. 4. Temperature variation of the superstructure ordering observed by synchrotron X-ray diffraction from h-scans between Ž2, 0, 3. and Ž3, 0, 3. in a crystal prepared with x s 0.67. Open symbols are from the initial heating and closed symbols from subsequent cooling within approximately 1 h. Upper panel shows the peak intensity, middle panel the HWHM and lower panel the peak position obtained by fitting two Lorentzians to the scattering profiles. Notice that the peak positions hs 2.375, 2.4 and 2.33 correspond to ortho-VIII, ortho-V and ortho-III, respectively. The solid curve in the middle panel is a fit to HWHM data in the critical region giving an exponent of n s 0.79Ž3., intermediate between the expectation for 3D Ž n s 0.63. and 2D Ž n s1. Ising systems.
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Fig. 5. Structural phase diagram of the superstructure ordering in YBa 2 Cu 3 O6qx . The superstructure phases: ortho-II ŽOII., orthoIII ŽOIII., ortho-V ŽOV. and ortho-VIII ŽOVIII. develop only short-range order. 3D ordering is observed for ortho-II superstructures. Ortho-III, ortho-V and ortho-VIII are predominantly 2D. The upper data point at x s 0.50 is from Ref. w8x, and the one at x s 0.77 is from Ref. w9x.
Section 3 under pseudo-equilibrium conditions during heating and cooling for x s 0.50. Using X-ray diffraction and the same high-purity crystal as in Section 3 we have verified directly that dynamical scaling applies. Direct measurements of the peak profiles along h, k and l of the Ž2.5, 0, 5. superstructure reflection, as function of time is too slow to allow for the desired time resolution. However, based on a sequence of identical quenches from 1708C to 758C, we have measured the time dependence of the intensity at 21 different points along h and l and thereby established the time dependence of these peak profiles. In these studies the vertical integration Žalong k . is not complete at the early time after a quench. With the assumption that the correlation length j b scales with j a , we have included the finite vertical resolution in the analysis of the peak profiles. The results of the temporal variation of the peak intensity, the correlation lengths j a and j c , and the integrated intensity are shown in Fig. 6. Our
the superstructure is frozen into ortho-V, and the ortho-VIII correlations are not recovered within this short time scale. Scans along l did not indicate 3D ordering, but analysis of the HWHM peak width of the h-scans gives a critical exponent n s 0.79Ž3., which is in between the values for 2D and 3D Ising type ordering. Crystals prepared with composition x s 0.87 and higher have the ortho-I structure with no sign of superstructure formation at room temperature. Based on those facts and our experimental results above, we suggest the superstructure phase diagram shown in Fig. 5.
4. Ortho-II superstructure ordering kinetics In a previous dynamical study of the ortho-II superstructure formation following a quench from ortho-I, a time dependent growth mechanism was observed with a change from algebraic to logarithmic growth at a length scale comparable to the distance between impurities w8x. In this study, the characteristic length was determined from the peak intensity by the assumption that dynamical scaling applies. This implies that the integrated intensity is constant, and thereby that the total volume of the ordered domains is constant. This was established in
Fig. 6. Ortho-II superstructure oxygen ordering kinetics observed by X-ray synchrotron diffraction at Ž2.5, 0, 5. in a high purity crystal with x s 0.5. The results are obtained by fitting a Lorentzian squared function convoluted with the resolution function to the data, as explained in the text. The upper panel shows the peak intensity Žsolid points, left ordinate. and the temperature profile Ždashes line, right ordinate., the middle panel shows the correlation lengths j a Žalong the a-axis. and j c Žalong the c-axis., and the lower panel shows the integrated intensity.
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Fig. 7. Ortho-II superstructure oxygen ordering kinetics measured by neutron diffraction on a very long time scale using the same high-purity crystal with x s 0.50 as for the X-ray diffraction studies presented in Fig. 6. Measurements of the Ž0.5, 0, 0. superstructure peak intensity were performed after a quench from 1708C to 758C and after subsequent rapid changes to other temperatures as marked on the figure.
results show clearly that the peak intensity and the correlation lengths increase after a quench from high temperatures into the ortho-II phase, but the integrated intensity is constant shortly after the quench. Notice that the peak intensity and the integrated intensity decrease immediately after a rapid increase of the temperature to 908C, and that the correlation lengths start to grow faster. These results clearly indicate that the ordering process is governed by growth of domains that have internal thermodynamic equilibrium. Dynamical scaling requires that the domain pattern evolves in a self-similar way with a characteristic length scale that increases as function of time. For an isotropic system, a constant integrated intensity assures that the characteristic length scale determining the domain growth may be derived from the peak intensity of the structure factor. However, for an anisotropic system the pattern would only look selfsimilar if also the growth rates along the main axes, i.e., the correlation lengths, scale with time. Analysis of the correlation lengths presented in Fig. 6 shows that this is indeed the case for j a and j c . From other studies w12x, which will be published elsewhere, we find evidence that also j b scales with j a and j c , as was assumed in our data analysis.
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The ordering kinetics following a temperature quench from 1708C to 758C and subsequent rapid changes to a sequence of different temperatures has been studied by measuring the peak intensity of the Ž1r2, 0, 0. superstructure peak by neutron diffraction. The results are shown in Fig. 7. They reveal the same properties and support the conclusions already established from the X-ray data, but this time on a much longer time scale Žnotice that one point of the neutron data in Fig. 7 covers the whole time-span of the X-ray data in Fig. 6.. From the data it is observed that the fastest domain growth at this late time of the ordering process occurs at temperatures between 758C and 858C, and not as at earlier times, where the X-ray data indicates 908C to be better. Accordingly, we use 808C for annealing our crystals to obtain large ortho-II domains. It should be mentioned, that more extensive measurements show that the domain growth is consistent with a logarithmic variation at late times, and that long range ortho-II ordering is not obtained within a foreseeable period of time.
5. Model studies of oxygen ordering Many model studies of the oxygen ordering in YBCO are based on the 2D ASYNNNI lattice gas Žor Ising. model, which has the three effective pair interactions between the oxygen atoms in the CuO x basal plane: V1 , V2 and V3 visualised in Fig. 8. From Monte Carlo simulations with V1rk B s 5430 K and V3 s 0.12 V1 repulsive, and V2 s y0.36 V1 attractive, it was found that the ASYNNNI model accounts successfully for the concentration dependence
Fig. 8. The effective oxygen pair interactions in the 2D ASYNNNI model used to study the oxygen ordering in the CuO x basal plane of the YBa 2 C 3 O6qx structure. V1 , V2 , and V3 are the effective interactions in the original ASYNNNI model. V5 is the additional pair interaction introduced into the extended model. Filled dots are Cu atoms, open circles are possible oxygen sites.
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of the transition temperature, T T – I , between the tetragonal and the orthorhombic ortho-I phase w15,22,29x. It also predicts the ortho-II phase, but the ordering temperature between ortho-I and ortho-II, TI – II , is significantly higher than found experimentally Žcf. Refs. w15,22x and Fig. 5.. A 3D version of the ASYNNNI model with a weak attractive nearest neighbour interaction, V4 s y0.02 V1 along the caxis, has also been studied w24x. By rescaling V1rk B s 4490 K, it establishes a similar good agreement with the TT – I data, and it opens a significant gap between the TT – I and T I – II . Although finite size ortho-II domains have been observed below T I – II in the simulations, the general trend is that both the 2D and 3D ASYNNNI models predict long-range orthoII order. It is not surprising that none of these models predicts the ortho-III, ortho-V and ortho-VIII structures, which may only be stabilised by more longrange interactions. We have extended the 2D ASYNNNI model by adding a repulsive Coulomb type interaction, V5 , between oxygen pairs that are separated by 2 a without intervening Cu atoms Žsee Fig. 8. w28x. We have estimated the V5 parameter using a screened Coulomb potential w20x and found that 0 F V5 F 0.04 V1 is a realistic range, and studied its consequences for the ordering properties. Using a parallelized code we equilibrate and average a system of 32 independent
128 = 128 ensembles during stepwise quenching from high temperatures. After each temperature step, the system is allowed to equilibrate for 5000 MCS ŽMonte Carlo Steps per site., and the structure factor SŽ q ., Cu–O chain lengths and oxygen co-ordination numbers for the Cu atoms were calculated by averaging typically 10 successive states separated by 10 MCS. Fig. 9 shows the structure factor calculation along the h-direction and corresponding snapshots for three different oxygen concentrations x s 0.50, 0.60 and 0.66. The results are obtained with V5 s 0.04 V1 and a reduced temperature T U ' k B TrV1 s 0.07, which is well inside the ordered phases. The extended ASYNNNI model stabilises the ortho-III superstructure by construction Ž x s 0.66 in Fig. 9. but surprisingly it also shows superstructures with clear signs of ortho-V and ortho-VIII correlations Žcf. Fig. 1 and x s 0.60 in Fig. 9., although these phases may only be stabilised by even longer-range interactions. Further, it is clear that the V5 parameter obstructs the formation of long range ortho-II order by formation of finite size anti-phase and transverse twin domains. As expected, this tendency is less significant for smaller values of V5 . However, we also observe the formation of finite size ortho-II domains for V5 s 0.02 V1. This is visualised in Fig. 10 where the power y, obtained by fitting Eq. Ž1. to the simulated structure factor data for x s 0.50 and
Fig. 9. Results of Monte Carlo simulation studies of the oxygen ordering in YBa 2 Cu 3 O6qx by use of the extended 2D ASYNNNI model with V5 s 0.04 V1. Leftmost figure shows the structure factors calculated along the h-direction for three different oxygen compositions, x s 0.50, 0.60 and 0.66. The reduced temperature of the studies is T U ' k B TrV1 s 0.07, which is well inside the ordered phases of the observed superstructures. All three structure factors: ortho-II at h s 0.5 Ž x s 0.50., ortho-III at h s 0.33 and 0.67 Ž x s 0.66., and ortho-V at h s 0.40 and 0.60 Ž x s 0.60. have finite widths and thereby finite size domains. The three corresponding snapshots Žonly one quarter of the 128 = 128 ensemble is shown., corroborate that finite size domains with anti-phase boundaries are formed for all the superstructures. Local domains of different types of superstructures are found in the snapshots, in particular for the x s 0.60 system, where also ortho-VIII correlations may be found.
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Monte Carlo technique based on the ASYNNNI model with realistic model parameters.
6. Summary and conclusions
Fig. 10. Results of Monte Carlo simulation studies of the extended ASYNNNI model for x s 0.50. T U s k B T r V1 is the reduced temperature. Ža. shows the influence of the V5 parameter on the structural phase transition temperatures separating the tetragonal T phase, the orthorhombic OI Ž ortho-I., and the OII Ž ortho-II. phases. Žb. shows the result for V5 s 0.02 V1 of the power y obtained by fitting a 2D Lorentzian of power y Žcf. Eq. Ž1.. to the simulated structure factor data. A gradual increase from y s1.0 to y s1.5 is observed at the onset of ordering, as expected for 2D finite size domains with sharp boundaries.
V5 s 0.02 V1 , is shown as function of temperature. We find that the onset of ordering results in a gradual change from y s 1 to y f 1.5, which is characteristic of scattering from finite size 2D domains with sharp boundaries. Calculations of the Cu–O chain length also show that it is finite for V5 s 0.02 V1 even for T U ™ 0, whereas it diverges towards infinite for V5 s 0 w28x. Fig. 10 also shows how the simulated ordering temperatures change as function of V5 . A significant decrease is observed, and the gap between T T – I and TI – II increases. If the gap for V5 s 0.02 V1 is combined with the one established from simulations with the 3D ASYNNNI model w24x, we obtain a value of ; 170 K which is approaching the experimental one of ; 250 K. It remains to be studied whether the extended ASYNNNI model may establish quantitative agreement with the experimental diffraction data by a suitable choice of interaction parameters. The use of interaction parameters that are independent of temperature and oxygen composition have been questioned, but the results of our combined experimental w11x and model studies give no evidence that throws doubt on this fixed parameter approach. Clearly, the effect of strain should be studied. This has been done in a mean-field approach w19x, but so far not by
In this paper, we have presented an overview of some of our previous and also very recent experimental results and model studies of the oxygen ordering in YBCO for x ) 0.3. From high energy X-ray and neutron diffraction studies on high quality crystals, that are carefully prepared and annealed, we have verified that the superstructure correlations with ortho-II, ortho-III, ortho-V and ortho-VIII character, earlier observed by electron diffraction w4x, represent bulk structural properties of YBCO. From the diffraction studies, we have established the structural ‘phase diagram’ for oxygen ordering in YBCO as function of oxygen stoichiometry, x, and temperature, T. It is not a real thermodynamic phase diagram since long range order is only observed in the form of regular Bragg peaks in the tetragonal oxygen disordered, and the basic orthorhombic ortho-I phase, which always appears on cooling before the shortrange ordered superstructures. Analyses of the c-axis correlations and the critical exponent n for the peak width above the critical temperatures show that only the ortho-II superstructure develops clear 3D ordering. Hysteresis in the ortho-II superstructure ordering properties during heating and cooling through TI – II shows evidence for quasi-equilibrium ordering, leading to short-range ordering due to formation of finite size domains with anti-phase boundaries, while having internal thermodynamic equilibrium. This is corroborated by the experimental observation of the structure factor line-shape in the ordered phase, which is a Lorentzian raised to the power yX f 3r2. It can also be concluded from the ordering kinetics following a quench into the ordered phase where dynamical scaling has been shown to apply. In our survey determining the phase diagram, we found no evidence of other than the mentioned superstructures. We cannot preclude that other chain ordering structures may develop in carefully prepared crystals and long time annealing at sufficiently low temperatures. It is, however, expected that the slow oxygen kinetics in the ordering process pre-
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vents the development of more complex phases with the ordering sequences predicted theoretically using the infinite chain approximation. The assumption behind this ordering scheme may also be questioned by the recent value of the average Cu–O chain length deducted from NMR studies w32x, which indicate that it is only a few lattice units long at the TI – II ordering temperature for x s 0.5. Below TI – II the ordering kinetics is rapidly slowing down and the domain sizes freeze in. These observations are corroborated by our model studies with the 2D extended ASYNNNI model, which reproduce the average chain length at TI – II and suggest a non-critical slow increase at lower temperature w28x. The extension of the ASYNNNI model to include a simple additional interaction parameter V5 increases the predictive power of this model significantly w28x. Many of the properties that were unexplained in the original version have been accounted for on a semi-quantitative level by this extension. These include the observation of superstructures with ortho-III, ortho-V and ortho-VIII type correlations and the formation of only finite size domains of all the superstructures. Further improvement of model predictions is obtained if the effects of the previously studied 3D coupling are included in the ASYNNNI model w24x. In this way, the observed strong suppression of the TI – II relative to TT – I is almost accounted for. It may be argued that more long-range interactions of the Coulomb type as well as strain should be included. This is certainly needed to explain mesoscopic phenomena as the observed formation of twin-domains. However, for the local structure the parameters in the ASYNNNI model are sufficient, being effective oxygen–oxygen interactions that rapidly decrease in strength with distance. Since the dominant pair interactions, V1 and V2 , have formed Cu–O chains when the inter-chain interactions, V3 and V5 , become effective, there is only a need for effective interactions between nearest neighbour oxygen atoms on different chains. A further extension of the 2D ASYNNNI model should therefore account for the effective Coulomb interaction between oxygen pairs on chains that are 3a or more apart. These interactions are even smaller than V5 ; 100 K P k B and they will thermodynamically only play a role at such low temperatures that the ordering is frozen in due to slow oxygen diffusion.
The detailed information obtained in the present study from numerous experimental scattering investigations and the semi-quantitative agreement with the results of the simple extended ASYNNNI model study have clarified many unsolved problems about the oxygen ordering in YBCO. This information may be used as a basis for new theoretical studies of the electronic structure in the Cu–O chains and the charge transfer leading to superconductivity.
Acknowledgements The Danish Technical Science Research Council supports TF, and The Danish Natural Science Research Council supports synchrotron X-ray diffraction studies through DanSynch. DM acknowledges the hospitality of Edinburgh Parallel Computing Centre as a visitor sponsored by the EU-TMR program. The Danish Natural Science Research Council has provided support for the computing activities at UNI-C. Collaboration with J.V. Andersen, J. Berlin, H. Casalta, T. Fiig, R. Hadfield, O.G. Mouritsen and P. Schleger on initial studies preceding this work is gratefully acknowledged.
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