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Thermodynamics of oxygen ordering in YBa₂Cu₃O₆₊ Schleger, Paul R. 1992

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THERMODYNAMICS OF OXYGEN ORDERING IN YBa2Cu306+.ByPaul Richard SchlegerB.Sc., The University of British Columbia, 1987A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESPHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIA1992© Paul Richard Schleger, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of  aft,c,The University of British ColumbiaVancouver, CanadaDate 1)€„,,44bei- /9) (Signature)DE-6 (2/88)AbstractAn apparatus has been built to study and manipulate the oxygen in high temperaturesuperconductors. It uses the principle of cryogenically assisted volumetric titration toprecisely set changes in the oxygen content of high-T, samples. The apparatus hasbeen used to study the thermodynamics of oxygen in YBa2Cu306+x in order to helpdetermine the correct model for oxygen thermodynamics as well as to provide standardcurves for materials preparation by other methods. In particular, extensive measurementshave been made on the oxygen pressure isotherms as a function of x for temperaturesbetween 450°C and 650°C. The measurement technique also allows one to extract thethermodynamic response function, (ax/O,a)T, Ca is the chemical potential), which issensitive to the oxygen configuration and which can be calculated by any candidatetheory of the oxygen thermodynamics. Several existing theoretical models for the oxygenordering thermodynamics are presented and compared to the experimental results. Themodels considered are classed into two basic approaches: lattice gas models and defectchemical models. It is found that the lattice gas models which assume static effectivepair interactions between oxygen atoms, do not fit the experimental data very well,especially in the orthorhombic phase. The defect chemical models, which incorporateadditional degrees of freedom (spin and charge) due to the creation of electronic defects,fit significantly better, but make crude assumptions for the configurational entropy ofoxygen atoms.Using a commonly accepted picture for the creation of mobile electron holes andunpaired spins on the copper sites, it is possible to relate these quantities in terms ofshort range cluster probabilities defined in mean field approximations to the 2D lattice11gas models. Based upon this connection, a thermodynamical model is developed, whichtakes into account interactions between oxygen atoms and the additional spin and chargedegrees of freedom, assuming a narrow band, high temperature limit for the motion ofthe charge carriers. The model, containing the nearest-neighbour oxygen interaction(0.241eV) and the single site oxygen binding energy (-0.82eV - D/2; D is the dissocia-tion energy of an oxygen molecule) as the only adjustable parameters, is compared toexperimental results for the chemical potential, kT(Ox/a,a)T, fractional site occupancies,structural phase diagram, the number of monovalent coppers, and the total number ofmobile electron holes. Qualitative agreement is found for all compared quantities, andquantitative agreement is found for the chemical potential, fractional site occupanciesand kT(ax/a,u)T in the orthorhombic phase. Improvements to the model are outlinedwhich should result in a quantitative fit to all results, in particular the valence and holecount vs. x. In addition to illuminating what is lacking in the commonly used twodimensional lattice gas models, the theory may form the basis for accurately predictingthe electron hole count of the Cu02 plane of YBa2Cu306+s as a function of the samplepreparation conditions.PrefaceThe incredible attention given to high temperature superconductors since their discoveryin 1986 is partly due to the promise of "inexpensive" superconductivity at high tem-peratures, perhaps even room temperature. But also, these compounds have attractedmuch interest because of their unusual physical properties. Of particular interest in thecuprate superconductors is the existence of Cu02 planes with strong two dimensionalcharacter, whose properties are controlled by electron hole doping. At low doping theseplanes give rise to semiconductive behaviour and antiferromagnetic order. At higherdoping levels, they become metallic and superconducting (at low temperature). The re-maining elements of the structure apparently serve only to structurally maintain the twodimensional copper-oxide lattice and to dope the planes with holes.There are a variety of ways to manipulate the external structure and create electronholes in the planes. One of the simplest cuprate superconductors from an overall struc-tural point of view is YBa2Cu306+„ where the hole doping is controlled by the additionof oxygen to sites external to the Cu02 planes. This leads to an indirect charge transferof electrons from the planes to the layer where the additional oxygen resides. However,the degree of oxidation is not the only parameter which controls the planar hole concen-tration; the additional oxygen displays various types of long range atomic ordering, whichis seen to modify the number of holes doped. The basic structural and electronic prop-erties of YBa2Cu306+x are presented in chapter 1. In order to understand the electronicproperties of YBa2Cu306+x, and in particular the mechanism of superconductivity, it isof interest to obtain precise information about the connection between oxygen orderingand hole doping.ivSince the added oxygen becomes fairly mobile at temperatures significantly lower thanthe melting point of these materials (300K < T < 1100K < melting point) much of therequired information can come from studies of the temperature and x (oxygen concentra-tion) dependence of the oxygen ordering thermodynamics. The oxygen chemical potentialit is a quantity which gives information concerning the thermodynamics. It can be ob-tained by measurements of the equilibrium oxygen partial pressure of the surroundingenvironment (it --, In P). The fine details of the chemical potential are revealed throughthe derivative, (Ox/ap)T, which acts as a thermodynamic response function. The exper-imental aim of this thesis was to map (ax/a,u)T over a significant range of temperatureand oxygen concentration. The experimental details are presented in chapter 3.One can distinguish two main approaches to describe the thermodynamics of this sys-tem (described in detail in chapter 2). One of them is based upon a pure configurationallattice gas model with interactions between oxygens. The other consists of describingthe oxygen interactions in a very primitive fashion, but to take into account additionaldegrees of freedom caused by the creation of electronic defects (defect chemistry models).Neither of these approaches are able to explain the complete set of thermodynamic datafor this system, as will be seen in chapter 4, where our experimental results are presentedand discussed. In chapter 5, the two approaches are merged to provide a more completedescription, capable of fitting virtually all thermodynamic results for temperatures above450K with just two free parameters. Since the behaviour of the electronic defects playsa major role in the thermodynamic model, this analysis provides a route to determiningthe planar hole count from the sample preparation conditions. This model also showswhat is missing from the very popular lattice gas models as well as casting some doubtconcerning the origin of the order-disorder transition.vTable of ContentsAbstract^ iiPreface^ ivList of Tables^ xList of Figures^ xiAcknowledgements^ xvNotation^ xvi1 Brief summary of the oxygen ordering problem in YBa2Cu306+.- 11.1 The structure of YBa2Cu306+s and the oxygen ordering problem 21.1.1^Basic structural overview ^ 21.1.2^Structural stability and long range atomic order ^ 41.1.3^The structural phase diagram ^ 81.2 Oxygen dependence of the basic electronic properties ^ 91.2.1^The insulating region ^ 91.2.2^The superconducting region ^ 101.2.3^What determines Tc? 121.3 Measurement of the oxygen chemical potential as a probe of the oxygenordering and charge transfer ^ 142 Introduction to lattice gas models in YBa2Cu306+. 17vi2.1 Definition of relevant thermodynamic quantities  ^182.1.1 The reaction function and the chemical potential  ^182.1.2 The thermodynamic response function (Ox/DOT^ 192.2 The 2D ASYNNNI lattice gas model for YBa2Cu306+s  222.2.1 Criticisms and modifications of the 2D ASYNNNI model ^ 272.3 The cluster variation method ^  292.3.1 CVM square approximation  312.3.2 Predictions of the CVM square, pair and point approximations^352.4 Defect chemical models ^  412.4.1 General introduction  412.4.2 Reaction model of Verweij and Feiner ^  442.5 Summary ^  503 Experiment^ 523.1 Experimental setup ^  523.1.1 Design concept  ^523.1.2 Design requirements ^  543.1.3 Deoxygenation apparatus  553.2 Measurement ^  633.2.1 Deoxygenation procedure^  653.2.2 Measurement of the oxygen pressure isotherms and (ax/a,u)T^673.3 Systematic errors and their corrections  ^723.3.1 Temperature drift of the volumes ^  723.3.2 Thermomolecular pressure gradient  743.3.3 Impurity gas correction  ^763.3.4 Dead volume correction ^  78vii4 Results and Existing Theories 814.1 Experimental results ^ 814.1.1^Oxygen pressure isotherms ^ 814.1.2^The thermodynamic response function (Oxfait)T ^ 854.2 Comparison to other work ^ 874.2.1^Oxygen pressure isotherms 884.2.2^The thermodynamic response function (ax/att)T ^ 894.2.3^Orthorhombic to tetragonal transition ^ 934.2.4^Discussion of the comparisons made 954.3 Fit to existing theories ^ 974.3.1^Defect chemical models 974.3.2^Lattice gas models ^ 1035 The Extended CVM model for YBa2Cu306+x 1075.1 Connection between electronic defects and cluster configurations ^ 1085.1.1^Counting the holes ^ 1125.1.2^Counting the spins 1145.1.3^CVM free energy with hole and spin degrees of freedom ^ 1175.2 Results and comparisons to experiment ^ 1205.2.1^Comparison to kT(Ox/a,a)T data 1205.2.2^Comparison to the oxygen chemical potential: determination of thesite energy ^ 1265.2.3^Comparison to the fractional site occupancies measured by neutrondiffraction ^ 1305.2.4^Phase diagram 136viii5.2.5 Predictions for the copper valence and hole count and comparisonto XAS measurements ^  1385.3 Commentary on the approximations made ^  1425.3.1 Possible limitations and complications  1445.3.2 Question about the nature of the O-T transition ^ 1465.3.3 Suggestions for enhancements and future work  1485.4 Summary ^  1536 Concluding remarks^ 155Appendices^ 158A Calculation of the chemical potential for a diatomic gas...^158B The natural iteration method...^ 161B.1 CVM square approximation  161B.2 Adding the spin and hole entropy ^  165C Sample preparation and characterization^ 168C.1 Sample preparation ^  168C.1.1 Final oxygen content ^  169C.1.2 Impurity levels in the YBa2Cu306+s ^  170C.2 Sample characterization ^  170Bibliography^ 175ixList of Tables2.1 Calculation for the Kikuchi-Barker coefficients for the CVM square ap-proximation ^ 322.2 Predictions for the order-disorder transition for various CVM approxima-tions^ 362.3 Configuration function for the oxygen 2p holes for the defect chemicalmodel of Verweij in the orthorhombic phase^ 493.1 Calibration values for the volumes of the deoxygenation apparatus . .^. 623.2 Values of A*, B* and C* given by Furuyama for oxygen. ^ 754.1 Cubic spline interpolation of the oxygen pressure plotted in figure 4.1^. . 844.2 Relative uncertainties of oxygen pressures at various representative pres-sure ranges for data listed in table 4.1 ^ 855.1 Number of sites per unit cell available for hole distribution for the differentpossible assumptions of the Verweij model^ 1135.2 Spin degeneracy factor for the various cases of the Verweij model. ^. .^. 117C.1 Some physical parameters of the YBa2Cu306+x master batch ^ 172List of FiguresChapter 11.1 Unit cell structure of YBa2Cu306 and YBa2Cu307  ^31.2 Schematic representation of the structural phase diagram  ^61.3 Phase diagram for the electron system ^  111.4 Plot of T, vs. x for various works  ^13Chapter 22.1 Schematic Diagram of the CuOx basal plane^  262.2 Phase diagram as predicted by the CVM square, pair and point approxi-mations and the 2D ASYNNNI TMFSS calculation of Aukrust et al. . . . 372.3 Predictions for kT(Ox/ay)T and the long range order parameter for theCVM square, pair and point approximations at kT/V --= 0.4. ^ 392.4 Comparison of kT(ax/(907, for the CVM point, pair and square approxi-mations to the Monte Carlo results of Rikvold et al^ 402.5 Schematic diagram of the change in Cu(1) valence as a function of itsnearest neighbour oxygen occupation ^  46Chapter 33.1 Schematic diagram of the experimental setup to measure the oxygen pres-sure isotherms of YBa2Cu3061, as a function of x ^ 563.2 Schematic of the gas handling system showing the labeling of the valves ^ 583.3 Plot of the furnace dead volume as a function of temperature with a 33gsample of YBa2Cu306 loaded in the sample space^  61xi3.4 Block diagram of the instrumentation used in the isotherm measurements. 643.5 Basic flow chart for data acquisition program to measure oxygen pressureisotherms ^  693.6 Block diagram for the communication between the MKS Baratron pressuretransducers and the personal computer. ^  713.7 Plot of the estimated thermomolecular pressure gradient vs. pressure forvarious temperatures. ^  773.8 Comparison of the vapour pressure at 450°C vs. x with and without theimpurity gas corrections ^  79Chapter 44.1 Plot of the measured oxygen pressure isotherms^  824.2 Plot of kT(Ox/O,u)T vs. x in YBa2Cu306+x  864.3 Comparison of the oxygen pressure isotherms between this work and thedata of McKinnon et al. and Meuffels et al. ^  904.4 Comparison of kT(Oxia,a)T between McKinnon at al. and this work . . ^ 914.5 Plot of the structural phase diagram of YBa2Cu3064„: theory and exper-iment^  944.6 Comparison of kT(ax/a,u)T between experiment and the model of Voroninet al ^994.7 Comparison of kT (0 x 10 ,a)T between the model of Verweij and Feiner. andexperiment at 550°C in the orthorhombic phase ^  1024.8 Comparison of kT(Ox/a,u)T between the predictions of the pure lattice gasmodels and experiment at 550°C ^  104Chapter 55.1 Valence of Cu(1) for various nn oxygen configurations ^ 113xii5.2 Comparison of kT(Oxla,a)7, for the extended CVM models with chain andplane hole distribution and the data at 550°C^  1215.3 Comparison of kT(Ox1.0,07, for the extended CVM models with a restrict-ed hole distribution and the data at 550°C. ^  1225.4 Plot of kT(ax/a,u)T for the best fit cases for the electron hole distribution. 1245.5 Plot of € + D/2 vs. T determined by comparing the f(II)F(II) model toexperiment. ^  1275.6 Plot of the experimentally determined chemical potential isotherms andthe predictions of the f(II)F(II) model. ^  1295.7 Plot of kT(ax/a,u)T vs. x at the temperatures of the experiment using thef(II)F(II) model ^  1315.8 Comparison of the fractional site occupancy vs. ,u,IkTo between the neu-tron diffraction data and the f(II)F(II) model. ^  1345.9 Comparison of the fractional site occupancy vs. T between the neutrondiffraction data and the f(II)F(II) model^  1355.10 Phase diagram as predicted by the f(II)F(II), CVM square and the 2DASYNNNI model (TMFSS)^  1375.11 Comparison of the phase diagram between the f(II)F(II) model and exper-iment^  1395.12 Prediction of the f(II)F(II) model for the number of Cui+ and comparisonto the XAS data of Tolentino et al. ^  1405.13 Plot of the amount of oxygen 2p holes vs. x from the f(II)F(II) model andcomparison to the schematic behaviour deduced by Tolentino et al. . . . 1435.14 CVM 4+5 point cluster for the basal plane oxygen. ^ 1505.15 CVM 3x3 point cluster ^  1525.16 Minimal size clusters defining an ortho-II and ortho-I region^ 152Appendix CC.1 Room temperature conductivity of YBa2Cu306+s vs. x for x < 0.08 .^171C.2 Resistivity vs. T for YBa2Cu306.987 ^  173C.3 Magnetization vs. T for YBa2Cu306.987 in a 10 gauss field^ 174xivAcknowledgementsI wish to thank first and foremost Walter Hardy, my research supervisor, for his continualguidance throughout the years. His kindness, open-mindedness and experience madework in the lab an enjoyable experience.To Ian Affieck, Jess Brewer and Jim Carolan, I extend the warmest thanks for ac-cepting to be on the Ph.D. committee and for always displaying quite a bit of interestmy work.I would like to thank Bingxin Yang for his early work in building the the main partsof the deoxygenation apparatus and for his preliminary vapour pressure measurements.Throughout the years, many individuals have come and gone who were at some pointdirectly involved in the sample preparation and characterization. I thank Reinhold Krah-n, David Brawner, Mark Norman, Alex O'Reilly, and Ruixing Liang. But especially, Iextend my warmest thanks to Pinder Dosanjh, who has been a constant companion inthe lab since 1986.The breakthrough in the theoretical understanding of the thermodynamics in thissystem came only this summer during my trip to Paris. Without this visit, chapter 5would have probably never been written. I thank Walter Hardy for letting me leave forsuch a long time, Mike Hayden for organizing things at l'Ecole Normale Superieure, andthe ENS as well as the Laboratoire de Physique des Solides d'Orsay for the use of theirlibraries during my stay. I thank especially Helene Casalta for her help in the analysis ofthe data and the write up of the thesis.Finally, I would like to say a special thanks to Peter Palffy-Muhoray for getting mehere in the first place.XVNotationSome symbols in the thesisy chemical potentialP pressureT temperatureH enthalpyS entropyE internal energySH partial enthalpySS partial entropyF Helmholtz free energyO Gibbs free energyG grand potentialfi reaction functionk Boltzmann's constantx susceptibilityN number of particles (eg. oxygen atoms)N, number of (oxygen) sitesM number of unit cellsV. nearest-neighbour effective pair interactionVci, next-nearest-neighbour effective pair interaction mediated by a copper atomVv next-nearest-neighbour effective pair interaction (non-mediated)€ site energyD dissociation energy of an oxygen moleculexviW band widtha oxygen sublattice denoted by a circle in figures/3 oxygen sublattice denoted by a square in figuresx oxygen content in YBa2Cu306-Exxor oxygen concentration at the O-T transitionx oxygen concentration on a sublatticez° oxygen concentration on sublattices long range order parameter (= (x" — x13)1 x)y23 pair cluster probabilityzi3ki square cluster probability[hole] number of electron holes per unit cell[Cul] number of monovalent copper per unit cell[Cu] number of divalent copper in the basal plane per unit celln number of sites availible for hole occupation per unit cellg spin degeneracy factorAcronymsAF AntiferromagneticASYNNNI Asymmetric next-nearest-neighbour interactionBCD Binary coded decimalCVM Cluster variation methodEPI Effective pair interactionLMTO-ASA Linearized muffin-tin orbital, atomic sphere approximationNI Natural iterationNMR Nuclear magnetic resonanceNR Newton-RaphsonO-T Orthorhombic to tetragonalSC SuperconductingTEP Thermoelectric PowerTGA Thermogravimetric analysisTMFSS Transfer matrix finite size scalingXAS X-ray absorption spectroscopyxviiiChapter 1Brief summary of the oxygen ordering problem in YBa2Cu306+In the past five years there have been a number of materials discovered which can becalled high temperature superconductors. Of these, the cuprate superconductors werediscovered first and are by far the most studied. Initially, Bednorz and Milner discoveredthe (La,Ba)2Cu04 system to be superconducting with a maximum critical temperature7', of 35K[1]. Such a critical temperature is higher than the maximum Te expected instandard theoretical models of the time (i.e. electron-phonon coupling BCS theory).For this, they received the Nobel prize in physics a very short time later in 19871. Thediscovery of superconducting (La,Ba)2Cu04 triggered an enormous research effort and thediscovery of several other copper-oxide superconductors with higher critical temperatures.The first system which displayed superconductivity above the boiling point of nitrogenwas YBa2Cu307, discovered by Wu et al. in February 1987{4It is partly for this reason that YBa2Cu306+, is the most studied material of thehigh 7', superconductors. The existence of bulk superconductivity above the boilingpoint of liquid nitrogen promises a substantial reduction of operating costs of systemsutilizing superconductive elements. But later, it also turned out that the YBa2Cu306-Fssystem can be made very pure, and its properties are easily controlled by the variationof the oxygen content. Thus, because of its early discovery and promising physicalcharacteristics, YBa2Cu306+s is perhaps the best understood material of the cupratesuperconductors. Much of the attention is still focused on this system, even though'See reference [2] for their Nobel prize lecture, which gives an insight into the research and thoughtprocesses which lead to this discovery.1Chapter 1. Brief summary of the oxygen ordering problem in YBa2 Cu3 06-Ex^2it is not the simplest system, both from a structural and electronic point of view. InYBa2Cu306+s, the electronic properties depend in a non-trivial way upon the atomicconfiguration of the oxygen. This is on the one hand convenient, since one can vary theelectronic properties by manipulating the oxygen; but on the other hand, the interplaybetween the oxygen configurations and the behaviour of the electrons is quite complexand is only now beginning to be understood in detail.First, we will present the basic structure and structural phenomena which are observedin YBa2Cu306+s, and then illustrate what effect these have on the electronic propertiesof this system.1.1 The structure of YBa2Cu306+s and the oxygen ordering problem1.1.1 Basic structural overviewThe structure of YBa2Cu306+, is depicted in figure 1.1. It is a layered material consistingof 2 Cu02 planes separated by an yttrium atom, two BaO layers sandwiching the Cu02bi-layer and a CuOs layer. The Cu(1) and 0(1), 0(5) are called the chain-site coppersand oxygens respectively. This CuOs layer is also called the basal plane. The 0(4) site isoften referred to as the apical oxygen or interstitial site. For a comprehensive review ofthe early structural studies of YBa2Cu306+s we refer the reader to the review of Beyersand Shaw[4].The stoichiometry can be varied from x = 0 to 1, by the addition of oxygen to theCuOs layer. This is accomplished by annealing a sample at an appropriate temperatureand partial pressure of oxygen gas. The binding energy of the chain site oxygen is lowenough that at elevated temperatures, the system develops a finite oxygen vapour pres-sure. The equilibrium oxygen pressure depends on the basal plane oxygen concentrationand on the temperature (cf. figure 4.1 for a plot of the equilibrium oxygen pressurea^ -1.-CityYB a 2C U 3070(5)chainsCu(2)o(2)• copper0 oxygene barium• yttriumChapter I. Brief summary of the oxygen ordering problem in YBa2Cu3 06-Fs^3YBa2Cu306Figure 1.1: Unit cell structure of YBa2Cu306 and YBa2Cu307. The variousnon-equivalent sites for the copper and oxygen are labelled according to the standardconvention for this system. The open squares indicate unoccupied oxygen sites. Atx = 0, the 0(1) and 0(5) sites are unoccupied. At x = 1, the 0(1) site is fully occupiedand the 0(5) site is empty. At intermediate concentrations, there is a finite probabilityfor 0(5) site occupation, but simultaneous occupation of the 0(1) and 0(5) site with-in a unit cell (nn oxygen occupation) is very small, if not zero. Note that the Cu(1) inYBa2Cu306 is two-fold coordinated and four-fold coordinated in YBa2Cu307. The Cu(2)are in a five-fold pyramidal coordination. The thick lines emphasize this coordination ofthe copper.Chapter 1. Brief summary of the oxygen ordering problem in YBa2 Cu3 06+.^4vs. x and T for YBa2Cu306+x). By setting an appropriate temperature and pressureenvironment, one can control the oxygen stoichiometry.The occupation of oxygen at low x is random with an equal probability to occupy the0(1) as well as the 0(5) site, although simultaneous occupation of the 0(1) and 0(5)site is suppressed due to a strong, direct, nearest neighbour coulomb repulsion. If theoccupation of the 0(1) and 0(5) sites is equal, the system is tetragonal with equal aand b lattice constants. As the oxygen content is increased, the system undergoes anorder-disorder transition, where the occupation of the 0(5) site becomes depleted andthe occupation of the 0(1) site is enhanced. This gives rise to the formation of onedimensional Cu-0 chains along the b-axis, so that the CuOs layer is also called copperoxide chains. As a result of the formation of chains, the b-axis length expands and thea-axis contracts, so that the system has orthorhombic symmetry. The 0(2) and 0(3)sites in the Cu02 planes are always occupied. There is some evidence that the 0(4) sitecan be depleted upon deoxygenation, but the amount is quite small, smaller than theerror bars determining the occupation[5].1.1.2 Structural stability and long range atomic orderIt is commonly said that YBa2Cu306+s should phase separate at low temperatures intoregions of different, discrete values of oxygen content (eg. YBa2Cu306 and YBa2Cu307).However, it is clear, experimentally, that YBa2Cu306+x is kinetically stable. At lowtemperature, the mobility of oxygen in the basal plane is low enough so that phaseseparation is never observed. The exact point at which phase separation should occur isnot known and estimates are necessarily model dependent. Khachaturyan et al.[7, 8, 9]have suggested that phase separation into YBa2Cu306 and YBa2Cu307 should occureven above room temperature, whereas de Fontaine et al.[10] propose that there are manyother stable ordered structures with intermediate oxygen concentrations, such that phaseChapter 1. Brief summary of the oxygen ordering problem in YBa2Cu3 06+x^5separation is very unlikely. Certainly, the experimental claims for the observation of amiscibility gap' are few (cf. for example, two publications of Tetenbaum et al.[11, 12]).There are other indications why phase separation is an unlikely event in YBa2Cu306+x,at least on a macroscopic scale: the superconducting critical temperature 71, depends onx and the transition widths are observed to be narrow for all values of x[13] (cf. alsofigure 1.4). If phase separation were to occur, then one would expect either a two steptransition or at least significant broadening of the transition in certain regions of x.In order to discuss the appearance of various types of long range atomic order, it isuseful to show a two dimensional representation of the basal plane, where the ordering ofoxygen takes place. The bottom three images in figure 1.2 shows the three well establishedtypes of atomic ordering observed in YBa2Cu306-Fx •In the tetragonal phase, the (average) occupancy of the 0(1) and 0(5) site is equal.In the ortho-I phase, one observes a sublattice splitting, where the occupancy of the0(1) and 0(5) sites are distinct and the tendency is to form chains along the b-axis.The ideal structure for this phase appears at x = 1, when one has fully established aone dimensional chain structure with a fully occupied 0(1) and empty 0(5) sites. In theortho-II phase, the 0(1) sublattice splits, giving rise to alternating full and empty chainsalong b. At x 0.5, this structure is ideal. Above x = 0.5, oxygens take up positions inthe empty chains and below x = 0.5, oxygen is removed from the fully developed chains.The tetragonal and ortho-I phases are very well established and easily detected, sincethe ordering of oxygen gives rise to a macroscopic orthorhombic distortion (i.e. 3D reg-istry of the 2D distortion). However, the ortho-II phase is more subtle and was detectedinitially only by surface sensitive probes such as transmission electron microscopy (cf.2A miscibility gap is a region in (x ,T) space, where the free energy is lower if one phase separatesinto two regions of distinct concentration. As the temperature is lowered, the difference in concentrationincreases. At zero temperature, one would have only the end members of the system present (i.e.YBa2Cu306 and YBa2Cu307).Chapter 1. Brief summary of the oxygen ordering problem in YBa2 C113 06-Fs^69002'2 700ia)ciE 500F-3000.0 0.2 0.4 0.6 0.8 1.0x in YBa2Cu306+x••0•0•0••--•-••0^•-•-••0••••••••O 0 0 0 0 0 0 0 0 0 0 00•••0•••13•0••^13•0•0•0•0•0•0O 0 0 0 • 0 0 0 0 0 0 0••0•0•0•0•0•0^••••••••0• .-_..0^• 0 0 • 0 0 0 0 0 0o • o /0•0•0•0•0^0•13•0••-•--••0•DO 0 0^0 0 0 0 0 0 00•0•0•0•0/0••^.--.—■ • 0 • .4,1.----.—•—•--.O 0 0 0^0 0 0 0 0 0 00 • o • •--.--• •0•0•0^0•0O 0 0 0 0 • 0••0•0•0•••080•-•-•-•--• • D • •-•-•-•--•O 0 0 0 0 011-0-11--•-•-•-•-•-•-•-•-•-11O 0 0 0 0 0••0•0•••• - Ill . .O 0 0 0 0 019-0-•-•-111-•-•-•-•-•-•-•--•O 0 0 0 0 0•-■-• . O• •^•0 0 0 0 0 0•--•--• • 0• ••• • •••O 0 0 0 0 0.--,--.--.--11--,--111--4,--1*--1,--.--,--.•• •0 •0•0 •00 0 0 0 0•-•-•-•-•-•-•-•-•-•-•-e-/1Tetragonal Ortho—II Ortho—IFigure 1.2: Schematic representation of the structural phase diagram. The top graphshows the phase boundaries vs. x and T according to the calculation of Hilton et al[14].The bottom three figures show a typical image of the basal plane in the three phases. Thesquares and circles correspond to the 0(1) and 0(5) sites respectively. The small filledcircles correspond to the Cu(1). Other filled elements correspond to occupied oxygensites. In the tetragonal phase, short chains are formed, but are randomly oriented. Inthe ortho-II phase, the tendency is to form chains along the b-axis (horizontally), butwith every other horizontal chain empty. In the ortho-I phase, chains are formed alongb in every unit cell.Chapter 1. Brief summary of the oxygen ordering problem in YBa2Cu306+.^7for example ref. [15]). The surface probes showed the existence of the ortho-II phasebetween 0.28 < x < 0.65. However, the disadvantage of these measurements is that onlythe surface of the sample is probed. There have been few attempts using bulk techniquessuch as X-ray[16] or neutron diffraction[17, 18]. Mostly, these investigations suffered theproblem of having only few superstructure reflections to analyse. These experiments es-tablished the stability of the ortho-II phase, but the observed correlation lengths wereshort (less than 10 lattice spacings along the a direction, perpendicular to the chains).Only very recently have Zeiske et al.[19] established through single crystal X-ray diffrac-tion for a x = 0.51(5) sample the existence of a longer range ortho-II structure (18 latticespacings along a, 135 along b and 6 along c at room temperature).Electron diffraction has also detected other types of long range order. In particu-lar a 2•\/a x 2-Va phase, consisting of alternating half full and quarter full chains atx 0.35[20, 21]. However, it was disputed to not be the result of oxygen orderingbut rather surface desorption of copper and barium[22, 23]. In the meantime, super-structure reflections corresponding to such a symmetry have been observed by neutrondiffraction[24], but that only 15% of the of the basal plane oxygen contribute to the su-perstructure. It was argued by de Fontaine et al.[25] that the 2.\/2-a x 2V2-a phase consistsof many three-fold coordinated coppers, which are believed to be energetically unfavor-able compared to two-fold or four-fold coordination[26]. On the other hand, Aligia etal.[27, 28, 29] claim that the the 2N/a x 2/a phase is indeed stable and that the oxygenordering model of de Fontaine cannot be correct (cf. chapter 2 for a brief description ofthese models). These are very recent publications, showing that the issue of the correctoxygen ordering model and the interpretation of experimental results lending support toone or the other model is still under intense investigation. It should be noted that bothmodels predict the stability of the ortho-II phase.Chapter I. Brief summary of the oxygen ordering problem in YBa2Cu306 -1-.^81.1.3 The structural phase diagramThe boundary between the disordered tetragonal phase and the orthorhombic phase (0-Ttransition) depends on temperature[4, and references therein]. It has been measured bymany groups using quite different techniques (cf. Voronin et al.[30] and chapter 4 for alist of some of the O-T transition results). A representation of the phase diagram is givenin figure 1.2 (see also figure 4.5). This is the result of a calculation by Hilton et al.[14] forthe 2D lattice gas model of de Fontaine[10] for the oxygen ordering thermodynamics (cf.chapter 2). This calculation fits the experimental T-(ortho-I) transition well. However,the (ortho-I)-(ortho-II) phase boundary has not been determined experimentally, so thatthis boundary curve is perhaps only qualitatively correct. The phase diagram only con-siders the existence of the tetragonal, ortho-I and ortho-II phases. Theoretical models,such as the lattice gas model of de Fontaine or Aligia will also predict the existence of ahirarchy of higher order superstructures at low temperatures; but except for the disputed2 /a x 2'N/2ct phase, none other have been seen by bulk probes, presumably due to theextremely slow kinetics at low temperature.We see that the tetragonal phase exists for low x and high T, the ortho-I phase forhigh x and the ortho-II phase for low T and intermediate values of x close to 0.5. Theextension of the ortho-I phase down to low temperature, close to the tetragonal phaseboundary is difficult to verify, since the kinetics at these temperatures are very slow.Other phase diagram calculations[31, 32, 33, for example] do not have such an extension.Rather, they predict the T-(ortho-I) phase boundary to terminate at a tricritical point atthe top of the ortho-II phase. The justification for the use of these models is still underdebate, so that the fine details of theoretical phase diagram have yet to be finalized'.3In fact, chapter 5 of this thesis will show that severe modifications of the standard lattice gas modelsare necessary in order to be consistent with certain experiments, but that the modifications do notstrongly alter the phase diagram.Chapter 1. Brief summary of the oxygen ordering problem in YBa2 CU3 06+x^9Experimentally, only the O-T phase boundary, i.e. the boundary between tetragonal andorthorhombic (I or II) symmetry, has been measured.1.2 Oxygen dependence of the basic electronic propertiesThe variation of the structural arrangements of the chain site oxygen, by changing thetotal oxygen content from x = 0 to 1 and also by quenching in varying types of longrange order (for a given oxygen concentration), has a dramatic effect upon the electronicproperties of YBa2Cu306-1-x•1.2.1 The insulating regionFigure 1.3 shows a schematic representation of the phase diagram. At x = 0, YBa2Cu306-1,is an antiferromagnetic insulator of the charge transfer type[34]. The coppers in theCu02 plane are divalent (Cu2+) with the unpaired electron spin having 3D long rangeantiferromagnetic order below 415K[35]. The chain site coppers are monovalent andnon-magnetic. As oxygen is added to the chains, nearest neighbour Cu(1) to the insertedoxygen are oxidized and become divalent4. However, at some point (x 0.2) longerCu-0 chains are formed for which it is energetically favorable to transfer electrons fromthe Cu02 planes to the chains: one has doped holes in the planes creating a p-type semi-conductor. This hole doping quickly destroys the 3D antiferromagnetic order. This isdue to the fact that the hole is not completely localized, has a spin which interacts withthe Cu2+ spin, thus creating disorder which supresses the in-plane, 2D antiferromagneticcorrelation length[37]. It is the coupling between ordered 2D domains in the third dimen-sion which triggers the 3D antiferromagnetic order[38]. A reduction of the 2D correlation4The term divalent must be understood to be the formal valence state and just means that the holehas a high probability to be on the copper site. This is a formal valence since there exists a significant0(2p)-Cu(3d) hybridization[36].Chapter 1. Brief summary of the oxygen ordering problem in YBa2 C113 06+x^10length by hole doping destroys the 3D order, but the 2D antiferromagnetism still existsover an appreciable length scale.At the metal-insulator transition, no more 3D long range antiferromagnetic orderis observed and superconductivity is seen immediately; there is no intermediate zone,as observed in (for example) (La2_,Sry)Cu04. The curves plotted are taken from theinelastic neutron scattering measurements of Rossat-Mignod et al. [35]. The exact detailsof the curves depend on the sample preparation (cf. Brewer et al.[39]) so that the dataplotted in figure 1.3 is only a qualitative representation of the system.1.2.2 The superconducting regionThe order-disorder transition for the oxygen configurations in the basal plane, giving riseto the formation of long chains, results in a large transfer of holes to the planes. The sys-tem undergoes a metal insulator transition which coincides with the O-T transition[32].The charge carriers in the Cu02 planes are electron holes transferred from the chains. Inthe metallic state, YBa2Cu306+x becomes superconducting with a 71, that depends on xand the degree of oxygen order.Referring to figure 1.3, we see that 7', developes a plateau at x '..-_- 0.6, which coincideswith the stability region of the ortho-II phase (see also figure 1.4). This plateau does notappear in quenched samples which do not allow for the formation of the ortho-II phase.It is quite clearly associated with the existence of the ortho-II phase. A second plateauis observed close to x = 1, where a maximum is seen at x = 0.93. Further additionof oxygen in fact reduces I', slightly (much recent attention has been focussed on thisregion, where significant changes in the electronic and magnetic structure is observed (cf.refs. [40, 41])).600500400gz 300200100^ 0o0.0 0.2 0.4 0.6 0.8 1.0x in YBa2Cu306+),50200250150H100SemiconductingTetragonalAFMetallicOrthorhombicScChapter 1. Brief summary of the oxygen ordering problem in YBa2Cu306+x^11Figure 1.3: Phase diagram for the electron system as measured by Rossat-Mignod etal.[35]. The scale on the left indicates the Neel temperature TN for the transition to3D long range antiferromagnetic order (AF). The scale on the right indicates 7', forthe transition to the superconducting state (SC). In the tetragonal phase, the systemis semiconducting and there is a transition to 3D long range antiferromagnetic orderfor the Cu(2) spins. At the metal-insulator transition, which coincides with the O-Ttransition, the system no longer exhibits 3D long range AF order, but instead becomes asuperconductor. Note that although the 3D long range order dissapears at the onset ofthe SC phase, there are indications that short range magnetic correlations coexist withSC[45]. There are two superconducting plateaus in T. The first plateau at T o 60Kcoincides with the region of stability of the ortho-II phase and is due to a stagnation ofthe hole doping of the planes. The origin of the second plateau is less clear. It mightbe due to a saturation effect of 7', on the number of holes (see text). See also figure 1.4which plots 71, vs. x from several experiments.Chapter I. Brief summary of the oxygen ordering problem in YBa2Cu3 064-x^121.2.3 What determines Tc?The existence of two plateaus, at 60K and 90K, has sometimes been suggested to bedue to the existence of two different superconducting phases. However, there is ampleevidence that the plateau at 60K is due to a stagnation of the number of holes dopedto the planes, as x increases. This is clearly supported by the valence bond sum ofthe in-plane Cu(2) bonds to the oxygen, which correlates directly with 71, vs. x[42]5.Additionally, Zubkus et al. [32], have clearly shown that no plateau exists when T, isplotted against the number mobile holes given by Hall effect measurements. A plotof T, vs. the number of mobile holes per unit cell, nH, gives an inverted parabolacentered about nH 1, which is strikingly similar to the shape of the T, vs. y curve inLa2_y SryCu04 and in agreement with the valence bond sum analysis of Whangbo andTorardi[43], who predict that T, should be an inverted parabolic function of nH•Much of the early confusion and debate over the origin of the T, vs. x curve inYBa2Cu306+x was due to the complex nature of hole transfer from the chains to theplanes and its intrinsic connection to oxygen ordering. Very strong evidence that oxygenordering modifies the charge transfer came from a series of papers[48, 13, 49, 46] whichfound that room temperature oxygen ordering occurs, increasing T, without modifyingthe total oxygen content. For example, in figure 1.4 we plot T, vs. x data of variousgroups. In particular, the solid diamond is a data point of Jorgensen et al.[46] which wasquenched from high temperature and inititally displayed no superconductivity. However,the sample quickly began to develop a finite T, which saturated at 20K in six days. Itbecame very clear that not only the total oxygen content, but also the degree of oxygenorder is important in determining the number of holes doped in the planes. There havebeen many models presented in the literature which calculate the hole doping from "first'The valence bond sum of the Cu(2), when corrected for the "steric" effect of the Cu(2)-0(3) bondstretching due to oxygen addition, gives a measure of the number of holes doped to the planes[42, 43].Chapter 1. Brief summary of the oxygen ordering problem in YBa2Cu306+x^1310080602'1- 40201^10Jorgensen 500°C- Ill Cava 415°CVCava 440°_ •Poulsen Monte CarloOBrewer 500°C•Jorgensen quench0 „^0.0 0.2 0.4 0.6 0.8 1.0x in YBa2Cu306+xFigure 1.4: Plot of 71 vs. x for various works. The temperature indicated after thename in the legend indicates the quenching temperature. Jorgensen 500°C is from ref.[5],Cava 415°C and Cava 440°C is from ref.{441, Brewer 500°C is from ref.[45]. Jorgensenquench is the saturated 7', for a sample quenched from high temperature and annealed atroom temperature for one week. Initially, this sample showed no superconductivity[46].Poulsen Monte Carlo refers to the 7', vs. x calculation of Poulsen et al.[47].Chapter 1. Brief summary of the oxygen ordering problem in YB a2 Cu3 06-Ex^14principles" (cf. for example [50, 51, 52, 53, 54]). However, these calculations separatethe problem of oxygen ordering and hole doping by assuming, a priori, the existenceof oxygen ordered structures. There are also phenomenological models[55, 47, 56, 57]which assume a simple relationship between oxygen order and charge transfer and thencalculate the hole doping by examining the configurational thermodynamics of oxygenordering. A successful fit to the 7', vs. x data was first acheived by Poulsen et al.[47],by a phenomenological minimal model connecting the hole doping to the distribution ofordered domains and utilizing the 2D lattice gas model of de Fontaine et al.[10] for thecounting of ordered domains. A plot of some T, vs. x data is shown in figure 1.4 togetherwith the prediction of Poulsen. However, the model assumes a linear 7', dependence upondoping, whereas some experiments suggest that the 90K plateau is due to a saturationeffect of Tc[32, 58], so that the agreement at high x might be fortuitous. A more recentcalculation by Lapinskas et al.[57], using the same 2D lattice gas model, but solving forthe oxygen ordering using a mean-field type approximation and assuming that 7', variesquadratically with the hole count also gives a very good fit of the 7', vs. x data (we donot plot the curve of Lapinskas since it is very similar to the curve of Poulsen).1.3 Measurement of the oxygen chemical potential as a probe of the oxygenordering and charge transferWe have briefly described the structural aspects of YBa2Cu306+„ with an emphasison the types of long range atomic ordering observed for the chain site oxygen. It wasillustrated that the basic electronic properties are essentially determined by the planarelectron hole count and that this hole count is set by the detailed configurations of thechain site oxygen. One of the goals of the experimental work in this thesis, and thetheoretical analysis of the data, is to improve upon the understanding of the connectionChapter 1. Brief summary of the oxygen ordering problem in YBa2Cu3 06+x^15between the oxygen ordering and hole doping.Experimentally, the object is to measure the chain site oxygen chemical potentialat temperatures between 450°C and 650°C as a function of x. This is obtained bymeasurements of the equilibrium oxygen partial pressure of the surrounding environment(cf. appendix A). A thermodynamic model that can fit the chemical potential willgive the oxygen interactions, which are set by the electronic structure of the material.Thus, a measurement of the chemical potential indirectly provides information about theelectronic structure.As one would expect, the oxygen chemical potential has been measured by innu-merable groups, using a variety of methods'. What has not been commonly done is tomeasure the derivative of the chemical potential with respect to x, specifically the quan-tity (ax/a,a)T. The only measurements of (ax/a,a)T, aside from the results presentedhere, were made by W.R. McKinnon et al.[59) at one temperature, 650°C. There areseveral ways to view the meaning of (ax/(9,a)T; but the most pragmatic one is to saythat by taking the derivative, one is strongly magnifying the fine details of the chemicalpotential.We will see that the standard lattice gas models, which are quite suscessful in pre-dicting the correct ground states and phase diagram, as well as modelling 7', vs. x,are not capable of fitting (ax/a,a)T. By using arguments based upon well establishedresults concerning the electronic structure and its connection with charge transfer andhole doping, it will be shown that the creation of charge carriers directly influences theoxygen thermodynamics in a very unusual manner, and which a standard lattice gasmodel cannot generate. In essence, the creation of charge carriers giving rise to themetal-insulator transition has a strong influence upon the oxygen thermodynamics. Al-though this somewhat complicates the theory, one must directly include the hole creation6We will not go into the details of the different methods used.Chapter 1. Brief summary of the oxygen ordering problem in YBa2 CU3 06+x^16mechanism into the model, so that it becomes possible to self-consistently solve for boththe oxygen ordering thermodynamics and the charge transfer. A full theory for this isnot developed, since a solution to the microscopic model for the electronic system is verycomplex. Instead, an approximate model will presented which results in a fairly goodquantitative fit to the data and acts as a guide to a more complete theory.Chapter 2Introduction to lattice gas models in YBa2Cu306+sLattice gases are systems where the particles forming the gas take on discrete positionson a lattice, and where the kinetic energy of the gas is negligible. The particles are, inprinciple, interacting with each other. These interactions can be long range, extendingover many lattice sites, or short range going to nearest or next-nearest neighbours. Thereis a direct one-to-one correspondence between a lattice gas and the Ising model in afield[60], which is important, since it allows one to carry over many fundamental ideas ofthe Ising model. The theory of lattice gases is essentially concerned with the computationof the configurational entropy of the particles which are placed on the lattice. Thereare many books which discuss the theory of lattice gases[61, 60], sometimes under thename of, for example, configurational thermodynamics[62] or order-disorder in alloys[63].This chapter will be concerned with presenting and defining the relevant thermodynamicquantities, with a strong emphasis on (ax/att)T and outlining the various approachesone can use to solve the system. Also, since defect chemical models are in some formquite closely connected to the concept of the lattice gas, the general idea of the defectchemical model will be discussed.17Chapter 2. Introduction to lattice gas models in YBa2Cu306+x^ 182.1 Definition of relevant thermodynamic quantities2.1.1 The reaction function and the chemical potentialBefore outlining the solution of the lattice gas model, some relevant thermodynamicquantities will be discussed in order to introduce some of the physical connections betweenexperimental observations and their theoretical implications. We will follow the notationused by Verweij et al. [64, 65]. In any thermodynamic model, one can start by writingdown the Legendre transform defining the Gibb's free energy (I) in terms of the enthalpyH and the entropy S:0 = H — TS^ (2.1)In a system with N particles distributed on N, sites, the chemical potential is defined as1 (a0)II — No .9c ) T,P(2.2)where c=N/No. In the case of YBa2Cu306+x, in fact x = 2c (cf. chapter 1) so that wewrite2 (01.No ax ) T,pUsing equation 2.1 and defining the partial enthalpy and partial entropy as2 (OHt5H -= —No ax) 7,,pSS  -- —2 [ OS)No aX ) Tpwe can also write the chemical potential as(2.3)(2.4)y .--- SH —TSS^ (2.5)In the range of oxygen pressures physically realized, the change in volume due to thechange in pressure is negligible. So we drop the specification ... , P and can write it =Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fx^19it(x, T). Essentially, we are dealing with an incompressible system where the conjugatevariables (P, V) do not come into play. The pressure of oxygen is just determining thechemical potential it. In this sense, the Helmholtz free energy F, and the Gibbs freeenergy 43. are the same; the enthalpy is just the internal energy.The chemical potential, the partial enthalpy and the partial entropy are directlymeasurable quantities which have been determined in a variety of ways.Suppose one has a system in which the entropy can be decomposed into a sum ofseparate entropic contributions S = S°+S1+S2+...+Si+..., where S° is the concentrationindependent part of the entropy, then it is convenient to define a reaction function as(-6Si(x))fi(x) = exp ^k (2.6)with the understanding that f(x) and 8S(x) are in general also functions of other vari-ables too, such as T and various order parameters, if they exist. In a pure lattice gasmodel, the reaction function is just a configuration function since the only entropy de-fined is the configurational entropy of the lattice gas and hence the emphasis on writingthe reaction function as a function of x. Writing the thermodynamic variables in termsof the reaction function is convenient when one is examining the system within sim-ple approximations, where the the number of defined order parameters is small, as forexample, in the Bragg-Williams (or CVM point) approximation. The Bragg-Williamsapproximation forms, in fact, the basis for the defect chemical models, where one hasnot only the configurational entropy for the oxygen but also entropic contributions dueto valence changes in the solid as a function of oxygen content.2.1.2 The thermodynamic response function (0x/a,u)TThe thermodynamic response function measured in this experiment is essentially theinverse of the derivative of ,tt vs. x. By expressing it in terms of the reaction function,Chapter 2. Introduction to lattice gas models in YBa2 C113 06-1-sone obtains that -1_x^Ex akT^7 1 + ^ )TkT° itt Twhere(^)ft(x)\ ]-1_ ^(x) TE  as 7-7^[ (2.8)—fi This form might not seem at first sight very informative. However, what is made clear isthe relationship between the partial enthalpy and kT(Ox/a,u)T. If the partial enthalpyis independent of the oxygen concentration, then kT(Oxiait)T just depends on the re-action function. Moreover, if the temperature is high enough then kT(3x/Oit)T is againonly dependent upon the reaction function. Looking at this equation from another view-point, in defect chemical models and lattice gas models, the reaction function is usuallyindependent of temperaturel Hence, if kT(Ox/O,u)T is measured to be independent oftemperature, it means that the partial enthalpy is quite irrelevant to the determinationof kT(Ox/a,u)T. This point will be important when an extension to current theoriesis needed to explain the discrepancy between the measurements of (Ox/OOT and thepredictions of the theory.Fluctuation viewpoint of (axl0p)7,Since it can be shown that the lattice gas system is analogous to the magnetic system ofthe Ising model[60, 66, 63, 67], in principle all of the general ideas developed for magneticsystems follow through to the lattice gas model. Here, just a few results will be discussed.We refer the reader to the books of Ducastelle[63] and Thompson[67] for a completediscussion of the the thermodynamics of the Ising model and its correspondence to thelattice gas. From this correspondence, one realizes that (0x10,02, is in fact analogous'The reaction function will be independent of temperature if the interactions are either much loweror much higher than kT. In the case where the interactions are very strong, then this just leads to anexclusion principle for the configurational entropy.20(2.7)Chapter 2. Introduction to lattice gas models in YBa2 C1-13 06-Fs^ 21to the magnetic susceptibility. One can introduce the isothermal ordering susceptibilityas the derivative of the order parameter m with respect to the corresponding symmetrybreaking field 1468]:(am)x = ah T^ (2.9)In an Ising model, m is the magnetization and h is the applied magnetic field. TheHamiltonian of a lattice gas in the grand canonical ensemble is analogous to the Isingmodel in the canonical ensemble. The chemical potential plays the role of the field(h ,a/2) and the site occupation number plays the role of the Ising spin (m = x — 1).In this way, one sees that (.9x/a,u)T=x. In the lattice gas models considered here,the site occupation probability x is always non-zero. Thus, it would be wrong to callx an order parameter of the system. Instead, these systems undergo phase transitionswhere sublattices become inequivalent. Differences in the occupation probabilities ofthe sublattices become non-zero at the phase transition and these are identified as trueorder parameters. Thus, to distinguish (ax/19)T from fluctuations of the true orderparameters of the system, (Oxia,u)T is called the non-ordering susceptibility[31, 69].Analogous to the specific heat and compressibility, one can make the identification,using the definition of the thermal average in the grand canonical ensemble, that':kT^T = N0((x2) — (x)2)^ (2.10)OX This equation implies that (ax/a,a)T is truly a thermodynamic response function. Also,equation 2.10 has practical implications for the calculation of thermodynamic responsefunctions in Monte Carlo simulations: (ax/a,u)T can be calculated from the straight for-ward evaluation of the difference in the thermal average (x2) and (x)2. Furthermore, onecan show that the non-ordering susceptibility (ax/a/47, is related to the pair correlation2This is in fact a form of the fluctuation-dissipation theorem for 2 point correlation functions[66, 61].Chapter 2. Introduction to lattice gas models in YBa2 013 06+x^ 22function. By definition, the pair correlation function isG (k , l) = (ckc i ) — (ck )(q )^ (2.1 1 )where ck is the occupation number at site k, Therefore one can write[67]ki, ()ax = 1 v,G(r)1,t)7,^2where r is the vector connecting the site k to site 1. This illustrates the sensitivity of(0x/0/L)T to phase transitions: at the critical point, the correlation lengths diverge.To summarize, it has been shown that (0x10,07, is a susceptibility analogous to themagnetic susceptibility in a magnetic system. In certain cases, kT(Ox/OF)T is dominatedby the entropy, which will typically result in a temperature independence of kT(Ox/0,a)T.It is a true thermodynamic response function, which is proportional to the thermody-namic fluctuations and correlations, and should exhibit a significant structure at criticalpoints.2.2 The 2D ASYNNNI lattice gas model for YBa2Cu3064-.The essential structural detail, inherent in any lattice gas system, is the underlying latticeonto which particles may be placed. In YBa2Cu306+x, the active element of the system,which is behaving like a lattice gas, is the oxygen in the copper-oxide basal plane (cf.figure 1.1). A schematic representation of this 2D structure is shown in figure 2.1. Thereare 2 possible oxygen sites and one occupied copper site per unit cell. The oxygen inthe basal plane is weakly bound to the lattice so that at elevated temperatures it formsa finite oxygen vapour pressure. It is essentially a 2D square lattice (the orthorhombicdistortion is ignored), with 4 nearest neighbours (nn) and 4 next-nearest neighbours(nnn). The interaction between nnn oxygen must be distinguished between two types.Two nun sites are separated by a copper, whereas the other two are not. In the disordered,(2.12)Chapter 2. Introduction to lattice gas models in YBa2Cu306-Es^ 23tetragonal phase, all oxygen sites are equivalent. In the ortho-I phase, the lattice splitsinto two inequivalent sublattices, denoted a and /3, where an oxygen site on sublattice ais surrounded by four nn oxygen sites of sublattice /3. This phase transition, as well asother transitions, such as the occurrence of the cell-doubled ortho-II phase are a result ofcompeting interactions between nn and nnn oxygens. In the 2D ASYNNNI3 model, oneassumes that the oxygens interact with each other via effective pair interactions (EPI's)to within second nearest neighbours. These pair interactions are defined in figure 2.1 andsome comments about their nature will be made in the next section.Effective pair interactionsThe effective pair interactions which are used to model the oxygen ordering process inYBa2Cu306+x are an estimate of the effect of the structural modifications to the elec-tronic orbitals and band structure which binds the material together to a solid. Thefollowing is a synopsis of the idea behind the effective pair interactions, as discussed indetail in the book of Ducastelle on the order and phase stability in alloys[63].The formal-ism developed, which allows one to approximate the effect of the atomic configurationson the band structure, comes essentially from the theory of order and phase stability inmetallic alloys. By defining certain ordered atomic structures one can, using band theo-ry, determine the effect of varying the atomic configurations on the band structure. Inalloys, the interplay between band structure and atomic configurations can be solved us-ing, for example, the coherent potential approximation[63] for the electronic sub-systemand a mean field theory for the atomic configurations. From this analysis emerges therealization that the effect of the band structure can be modeled by introducing effectiveinteractions between atoms. However, it is only a first approximation to assume thatthese effective interactions between atoms are constant in x and T. In fact, Ducastelle3ASYmmetric Next Nearest Neighbour InteractionChapter 2. Introduction to lattice gas models in YBa2Cu306 -fs^ 24shows that the sign of the interactions may change depending on the number of electronspresent. The approximation to assume that one can solve the configurational problemby assuming effective interactions between atoms given by the details of the band struc-ture and to assume that these interactions are constants has only really been clearlyjustified in well behaved, wide-band metals. YBa2Cu306+s is not such an entity. N-evertheless, constant effective pair interactions have been used extensively to study thethermodynamics of oxygen ordering in YBa2Cu306+. Some of the justification for suchan approximation is the relative insensitivity of the phase diagram to modifications ofthe EPI's[70, 71].One of the more popular models is the 2D ASYNNNI model mentioned above. Onecan make some basic chemical arguments to predict the sign and order of magnitudeof the interactions[25]. The nearest neighbour interaction Vin is expected to be largeand positive due to a strong direct coulomb repulsion between nearest neighbour sites.Similarly, the next-nearest neighbour interaction without an intervening copper, Vv, isexpected to be positive for the same reason, but its magnitude should be smaller sincethe distance is larger. The next-nearest neighbour interaction via the copper, Vc,„ isexpected to be attractive due to the 0-Cu bonding of the Cu-3d orbital with the 0-2porbital.Although we have defined the interactions only in two dimensions, the EPIs takeinto account the full three dimensional electronic structure of the system. In metals,the strength of the EPIs rapidly decrease for more distant nearest neighbours [25, andreferences therein]. They can be calculated from first principles using band structurecalculations of a partially ordered system[63], although full advantage has not been tak-en of the formalism and techniques available for this problem. Indeed, it is question-able whether the standard alloy approach for calculating the effective pair interactionsChapter 2. Introduction to lattice gas models in YBa2 C113 06+x^ 25is appropriate for YBa2Cu306+472, 73, 74, 29, and chapter 5]. Nevertheless, very ap-proximate values for the energy of such configurations have been calculated throughLMTO-ASA4 total energy calculations[75]. This estimate for the nn interaction V„„, andanisotropic nnn interactions Vc„ and Vv results in the correct stable ground states asseen experiment ally[76].Also, tight-binding calculations show that a 3-fold coordinated Cu(1) is energeticallyunfavorable compared to a 2 fold or square planar (4-fold) coordinated Cu(1)[26]5. Such asituation, favouring 2- or 4-fold coordination over 3-fold coordination would be equivalentto introducing an attractive interaction between nnn oxygens separated by a copper.There has been extensive work done on this 2D ASYNNNI model. We refer to thePhD thesis of Henning Fries Poulsen[77] for a good review and introduction to variousaspects of the ASYNNNI model in relation to YBa2Cu3064.x, especially concerning theground state stability, low temperature properties and Monte Carlo simulations. Veryrecently, a least-squares fit to the structural phase diagram has been made, using transfermatrix finite size scaling (TMFSS) calculations of the ASYNNNI model, to obtain thebest fit values for the EPI parameters[14]:VflnVc„Vv==2800K—2380K270K (2.13)Thus, the physical picture is of oxygen atoms which take up positions on a squarelattice and who interact with each other up to next-nearest neighbours with anisotropicinteraction parameters. The strong repulsive nn interaction results in the maximumphysically realizable oxygen content of x=1 corresponding to half filling of the available4Linearized-Muffin-Tin-Orbital Atomic-Sphere-Approximation'The apical oxygens are assumed to be always present and so the Cu(1) is always 2 fold or highercoordinated.Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fx^ 26Figure 2.1: Schematic Diagram of the CuOs basal plane. The lattice is split into twointerpenetrating sublattices. Sublattice a is denoted as circles and sublattice 0 is denotedas squares. The coppers are depicted as small solid circles. The effective pair interactions,V71727 VCU and Vv, for the 2D ASYNNNI model are defined as shown. The dotted linescorrespond to the square clusters used for the CVM square approximation (discussed insection 2.3.1). Note that no distinction is made in the CVM square between a squarewith and without a central copper.Chapter 2. Introduction to lattice gas models in YBa2Cu3 06+x^ 27oxygen sites in the basal plane. The strong and attractive nnn interaction mediated viathe copper results, together with the non copper mediated, weaker and repulsive nnninteraction, in the occurrence of chain formation and the Ortho-II phase. A necessarycondition for correctly predicting the observed well established ground states is thatV. < 0 < Vv < V7in. Naturally, even in the case of Vct, and Vv being zero, one stillsees a transition to an ordered phase containing chains due to the strong nn repulsion,but the Ortho-II phase is not seen and the chain lengths are significantly shorter in thetetragonal phase.The resultant phase diagram of the 2D ASYNNNI model with these interactions wasmentioned in chapter 1. However, as was discussed in the chapter 1, YBa2Cu306+xundergoes a metal-semiconductor transition. This is expected to cause the effectivepair interactions to change, since this transition entails a modification of the electronicstructure. This point is mentioned in almost all papers on 2D lattice gas models inYBa2Cu306+s, but is usually then ignored.2.2.1 Criticisms and modifications of the 2D ASYNNNI modelIn a series of papers by A.A. Aligia[27, 78, 79, 28, 70, 80, 29], it is argued that:1. All effective pair interactions should be repulsive, as one would expect in to see inan insulator, except possibly Vct, at low x.2. Non-zero hopping of the charge carriers gives rise to screening effects which will re-duce the strength of the interactions exponentially with distance, with some screen-ing length.3. Charge transfer to the planes will reduce the next-nearest neighbour Vc24. Thisreduction is calculated from first principles from the extended Hubbard model.Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fx^ 28However, in ref. [70], (Ox/O,a)T is specifically calculated and the result seems on-ly vaguely qualitatively correct, although it is quantitatively better than the pure 2DASYNNNI model. The model is quite complex, utilizing interactions of up to 6th near-est neighbours and an x dependent Vcii due to charge transfer (the exact reduction ofVc„ can only be estimated due to the uncertainties in the parameters of the extendedHubbard model). It predicts the stability, as well as the correct relative magnitudes ofthe superlattice peaks, for the three well defined phases, tetragonal, ortho-I and ortho-II,and the 2 '/a x 2\/a phase. On the other hand, the pure 2D ASYNNNI model cannotpredict the stability of the 2 V-2-ct x 2-‘7a phase, since interactions higher than nnn arerequired. The specific picture of Aligia has very recently been heavily attacked by deFontaine et al.[25] and a rebuttal has been submitted by Aligia and Garces[29].Recently, V.E. Zubkus et al. [71] also considered the effect of the metal-semiconductortransition upon the effective pair interaction and the resultant phase diagrams, but themodification was purely empirical6. Also, there are claims that the EPI's are not stronglydependent on the oxygen concentration[26, 81]. We see that the precise details of theoxygen effective pair interactions are still being sorted out.The discussion as to the nature and value of the effective pair interactions is veryimportant for the determination of the ground states and stability of YBa2Cu306-1-x•Fortunately, at high temperatures, where only the tetragonal and ortho-I phase remain,the system becomes very insensitive to longer range interactions and is dominated by Yinonly, which all models agree should be large and repulsive. Calculations in this thesis willbe thus restricted to nearest-neighbour interaction models. This should be understoodas an approximation to the full description using longer range interactions, justified bythe fact that in the temperature range of the the experiment, the nn model agrees well'It should be noted that the purpose of the paper of Zubkus is to study the behaviour of the structuralphase diagram as a function of the three effective pair interactions, and not to derive the concentrationdependence of these interactions.Chapter 2. Introduction to lattice gas models in YBa2Cu306-1-x^29with the higher order models. Before illustrating this insensitivity in section 2.3.2, wewill present a commonly used mean-field theory for the solution of lattice gas models.2.3 The cluster variation methodThe most difficult problem one encounters when trying to solve the lattice gas problem(without reverting to Monte Carlo methods) is the calculation of the configurationalentropy. The reason for the difficulty lies in the complexity of counting the possibleconfigurations of a lattice. As an illustration of this point, let us consider writing downthe free energy for a particular cluster of N, sites. Given a particular configuration ofoxygens in the cluster, one needs to calculate the internal energy and the entropy. ie .F= (E) — kT ln SINo(2.14)where C/ is the number of possible configurations with the same internal energy (E). Thecalculation of the internal energy is easy, since it is a straight-forward counting of thenumber of bonds of a particular type. For example, in the 2D ASYNNNI model, wherethere are 3 effective pair interaction parameters, one has[72],(E). E nrvrc•^ (2.15)r=1,2,3where nr is the number of bonds of type r per lattice site, V, are the effective pair in-teractions, defined in equation 2.13, and is the pair correlation function for a bond oftype r. The difficulty now arises in the determination of the configurational entropy forthe cluster of N, sites having an internal energy, (E). It is quite clear that, in general,it is impossible to enumerate the number of configurations with a given energy whenthe number of sites is large. One needs to utilize approximate methods to determinethe configurational entropy. One technique useful for the calculation of the configura-tional entropy is the cluster variation method (CVM) first introduced by Kikuchi[82]. AChapter 2. Introduction to lattice gas models in YBa2 013 06+x 30comprehensive review of the configurational thermodynamics of solid solutions has beengiven by D. de Fontaine[62]. Later, de Fontaine presented a tutorial introduction to thecluster variation method with emphasis on YBa2Cu306+,[33]. The specific presentationof the CVM equations for the 2D ASYNNNI model in YBa2Cu306d„ was given by L.T.Wille[72]. Essentially, the CVM method is a mean field theory which calculates the de-generacy of a small cluster exactly, but larger clusters are treated in the superpositionapproximation. The specific type of CVM approximation is denoted by the maximalcluster considered. In the point approximation, the cluster is just a single lattice siteand is identical to the Bragg-Williams approximation. The CVM pair approximationtakes a bond as the maximal cluster and it is identical to the quasi-chemical or Betheapproximation. For a square lattice, there exists the square approximation which is suit-able, as in the CVM pair, for models with nn interactions only, and uses the square asits maximal cluster. For next-nearest neighbour interactions, one needs to go to higherclusters in order to correctly take into account the higher order interactions. For the 2DASYNNNI model in YBa2Cu306, the 4+5 point approximation is very often used (cf.for example [33, 83, 72, 84, 85]), although even higher order clusters have been used[70].We refer the reader to the original paper by Kikuchi[82], the reviews by de Fontaine[62,33] and the book by Ducastelle[63] for details on the derivation of the CVM equations.Here, the final result will be quoted. Using the notation of Wille[72], the free energy iswritten as= E 71,14-C — kT Exi(J) In xj(J) (2.16)v0 r=1,2,... j Jwhere the first term was defined above in equation 2.15, x3(J) is the cluster probabilityfor a cluster of type j with a particular configuration of occupied and unoccupied sitesdenoted by J. The sum over j runs over all subclusters" up to and including the maximal7A subcluster is a cluster which is contained within the maximal cluster. For example, a nn bond isa subcluster of the square.Chapter 2. Introduction to lattice gas models in YBa2 CU3 06+s^ 31one. The sum over J runs over all possible configurations that a particular cluster maytake on. The -yi is the Kikuchi-Barker coefficient for the cluster j and can easily becalculated recursively by[86, 72, 33]-n1L=^_ E mja=i+1(2.17)where mL is the number of clusters per unit cell of the maximal cluster, mi is the numberof clusters of type i, nil? is the number of subclusters of type j contained within thecluster of type i. Next follows a brief derivation for the CVM square approximation. Thesolution for the 4+5 point approximation is derived in the review of de Fontaine[33] andthe paper of Wille[72].2.3.1 CVM square approximationThe CVM square approximation is similar to the CVM pair, in that it is useful forcalculations involving nn interactions. Obviously, it is useful only for square lattices,whereas the CVM pair can be used for any type of lattice. However, since the maximalcluster is the square as opposed to the bond, it is expected to be more accurate and givea better approximation to the configurational entropy.This approximation is capable of giving rise to an order-disorder transition which inYBa2Cu306i„ is identified with the O-T transition. In the disordered phase, there is nodistinction between the two sublattices and the total number of oxygen atoms situatedon either sublattice is equal. The first task in deriving the entropy expression is to writedown the maximal cluster and all its subclusters, then to calculate the -yi's according toequation 2.17. Table 2.1 shows how one arrives at these -yi's for the tetragonal phase.Chapter 2. Introduction to lattice gas models in YBa2Cu3061-.^ 32type i mi j=1 2m;?3 4 -yipointbondanglesquare1234oo-og_ole:12211 213214421-120-1Table 2.1: Calculation for the Kikuchi-Barker coefficients for the CVM square approxi-mation. i is the cluster type index, mi is the number of clusters of type i per lattice siteand nyli is the number of subclusters of type i contained within the cluster of type j.Thus, the number of configurations, DT, in the tetragonal phase is given byfo-o}2f2Tf°}Where we have used the CVM notation for the cluster product:(2.18){i} = F1.1(N0x3(J))!-In order to take into account the existence of the order-disorder transition, one needs tosplit up the lattice into two sublattices, a and as shown in figure 2.1. In this situation,the point cluster probability, xi(J), and the angle cluster probability, x3(J) each split upinto two distinct cluster probabilities, xcl(J), x(J) and x(J), 4J) respectively. In thiscase, it is easy to show that the number of configurations in the orthorhombic phase, fiois given by{0-0}2C20 =  ^ (2.19){^1{012 fol2Finally, using Stirling's approximation for the factorials, one arrives at an equation ofthe form 2.16. Since we only have 4 cluster probabilities in this problem, we rewrite forthe sake of clarityChapter 2. Introduction to lattice gas models in YBa2CU3 06+x^ 33point 4(J) = x^x(J) =bond^x2(J) =square x4(J) = zijkiwith i, j, k,1 E {0,1}, where the indices i, j,k,1 refer to the individual atoms on a clusterasand where the value of the index indicates an occupied (1) or empty (0) site. Usingthis notation, the configurational entropy in the CVM square approximation becomesSsquarekNo= 2 E yij in yij 1zkllnzk1 - -2 E in +i,j,k,1^ ixl? ln x}^(2.20)Using the same notation, we can write down the internal energy of the systemE EijklZijkl^ (2.21)N°^i,j,k,1where cijkl^17.011/2 =^k)(j /)/4 is the energy of the square cluster config-uration, with Vnn, being the effective nn pair interaction energy. This formulation, infact, allows one to take into account three or four body interactions, since they are justrepresented as additional terms in Eijki. Here, we have just assumed nn pair interactions.Also note that the right hand part of the expression for Eiji,/ is only true if one uses theconvention i, j, k, I E {0,1}.The cluster probabilities, x, 43, yij and zijki are not all independent. They areinterrelated by[87]Xi^E Zijkl^ (2.22)Chapter 2. Introduction to lattice gas models in YBa2Cu3 06+s^ 34and the normalization condition0x =-- E Zijkli,k,1Yij = E Zijklk,1(2.23)(2.24)E zi3ki =1^ (2.25)i,3,k,/so that for the CVM square one has 15 independent cluster configuration probabilities (orcluster distributions) to solve for, some of which are degenerate due to the symmetries ofthe square, further reducing the number of independent variables. Nevertheless, one stillhas 6 independent variables remaining. The typical method would be to differentiate thefree energy with respect to the six independent variables and use the Newton-Raphsoniteration method to minimize the free energy. Unfortunately, for a system of 5 or morevariables, it is very difficult to get the NR method to converge. Also, one needs to takeanalytic derivatives and insert them into the computer code, giving ample possibility formistakes if the equations are complicated enough. A much simpler method applicable tosome types of clusters was introduced by Kikuchi [87] to solve the free energy minimizationproblem. It is called the Natural Iteration method and takes advantage of the specificnature of free energy minimization problems'. In Appendix B the NI method is described.Here, it suffices to say that one minimizes the grand potential at fixed ft and T, G =E — TS — No,ux, i.e.9GsquareNo E EijklZijkl [1-FkT E ZijkihlZijki — 2 E yii In yii + — Excy ln e +2 . "i,j,k,1^ i,j^ x1_ it E _(i+ i + k + 1)zijkii,j,k,1 4x'q ln 2}1(2.26)8The NI method works only in specific cases where the maximal cluster is of the single type (i.e. notin the CVM point) [63].9Note: pN = pNoc = Nopx.Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fs^ 35where we have written the last term in its most symmetric form.With the grand potential minimized, one can calculate experimentally measurablequantities, such as the p, (040)T, the O-T transition, xoT, the long range order pa-rameter, s (xia — 4)/x, and the short range order parameter, q = Yii (= number ofnn sites occupied).For reference sake, since the next section will compare the predictions of the variousapproximations, we also give the form for the grand potential to be minimized in theCVM pair and CVM point approximations. In the CVM pair we have:Gpair N,-FkT [2 E yia ln^—1^.^.—P E^-V-1*3x,'' ln x7 + E 4 ln x'3}i^I(2.27)And, in the CVM point we have:Gpoint E Eiixia^13N,1-kkT {E ln +^xi? In^}11 ,. + (2.28)2.3.2 Predictions of the CVM square, pair and point approximationsIn this section, the predictions of these CVM approximations will be examined andcompared to "exact" results of Monte Carlo simulations and TMFSS calculations in orderto determine which approximation is appropriate for the calculation of kT(ax/Op)T. Wewill compare the predictions for the phase diagram, the long range order parameter andkT(ax/(9,a)T. The comparison for the phase diagram for the CVM point, CVM pairChapter 2. Introduction to lattice gas models in YBa2 013 06+x^ 36Approximation kToT/VnnCVM point 1.0CVM pair 0.7212CVM square 0.6057Best known 0.567Table 2.2: Predictions for the order-disorder transition at it = 0 for various CVM ap-proximations. Values taken from the review of de Fontaine[62]and nn interaction Monte Carlo calculations has been done by McKinnon et al. in theoriginal paper on measurements of kT(.940,tt)T in YBa2Cu306.4,[59]. Figure 2.2 showsthe predictions for the structural phase diagram, for the same CVM approximationsin addition to a more recent 2D ASYNNNI model calculation[31]. We see that as themaximal cluster size is increased, one gradually approaches the prediction of the TMFSSand Monte Carlo calculations of Aukrust et al[31]1°. A measure for the accuracy of theseapproximations is given by examining the it = 0 transition temperature (at it = 0, x = 1).Table 2.2 presents a comparison of the predicted order-disorder (i.e. O-T) transition TOTat ft = 0 for the various approximations[62], which shows that at high temperatures, theCVM square is within about 10% of the exact result for the transition temperature.At lower temperatures, however, in the vicinity of the tricritical point and the occur-rence of the ortho-II phase, these nn CVM approximations rapidly begin to fail. Thisis expected, since they do not account for the existence of the ortho-II phase. The va-lidity of the CVM square approximation is thus dependent upon the temperature scalephysically realized in YBa2Cu306+s. If the maximum temperature of the ortho-II phaseis small enough, the CVM square approximation should be usable. In the latest TMF-SS calculation of Hilton et al.[14], the maximum temperature of the ortho-II phase is'Although Aukrust uses 14IkT=1, Vc/kT--,--0.5 and VvIkT=0.5, which are not the currentlyaccepted values for the EPIs, the resultant phase diagram at high T is not very different from morecurrent calculations.Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fx 371.00.80.61---' 0.40.2_i---- CVM Point^../ ••••^ CVM Pair^ CVM Square x, ••••TMFSS i,,,,,,,///////////// / /^/ // /^/-_. ....................^—-_--/,0.0 0.2 0.4 0.6x in YBa2Cu306+x0.0^' 1 1 i^. 0.8^1.0Figure 2.2: Phase diagram as predicted by the CVM square, pair and point approx-imations and the 2D ASYNNNI TMFSS calculation of Aukrust et al[31]. The linescorrespond to the boundary between the tetragonal, disordered phase at low x and theorthorhombic, ordered phase at high x. See figure 1.2 for a definition of the differentregions.Chapter 2. Introduction to lattice gas models in YBa2Cu306+,^ 38at ,550K. The measurements of kT(Ox/O,u)Tpresented in this work are between 723Kand 923K, so that one could imagine that the effects of higher correlations due to thenext-nearest neighbour interactions are not dominant. This will be more obvious in thefollowing paragraphs, when (Ox/(3,07, is compared for the different approximations.From a practical point of view, it is important to see the predictions of these approxi-mations for the non-ordering susceptibility. First, figure 2.3 shows a plot of kT(Ox/O,u)Tand the long range order parameter s for the CVM point, pair and square approximationsat kT /V,,=0.4 (the value 0.4 is a reasonable but quite arbitrary choice). The long rangeorder parameter is defined asa 0x1 — xis =--   (2.29)xT€ +We see that there is a very prominent feature in kT(Ox/aft)T at the order-disorder phasetransition, which is as expected (cf. section 2.1.2). Below the peak, one is in the tetrago-nal phase and the long range order parameter is zero as seen in the bottom graph. Abovethe transition, one is in the ortho-I phase and the order parameter rises to one. At thetransition, the order parameter rises very rapidly, but continuously, indicating a secondorder phase transition. In the tetragonal phase for very low x, kT(Ox/(9,a)T has thesame slope as a non-interacting lattice gas for particles distributed randomly over all 2xbasal plane oxygen sites', but the fluctuations are quickly suppressed due to the nearestneighbour interaction. In the orthorhombic phase, the different CVM approximationshave qualitatively different shapes, except near x=1. For x '--1, kT(Ox/O,a)T has theshape of a non-interacting lattice gas for particles distributed over x sites, with a smallcorrection due to some finite nn occupancies. In the intermediate regime, however, thecurves are quite different. The curvature changes sign from the CVM point to the CVMsquare.random11I11 the random case, kT(Oxlap)2,^= x(1— x12), with a maximum of 0.5 at x=1 and a slopeof 1 at x=0.Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fx 390.50.4—;"-: 0.3—0-}-13_,->< 0.20.1CVM square with kT/V„=0.4^ CVM pair- - CVM point••■ ••■■...........................•■•••0.8^ '0.0 0.2 0.4 0.6x in YBa2Cu306.fxFigure 2.3: Predictions for kT(Ox/aA)T and the long range order parameter for the CVMsquare, pair and point approximations at kT/V;in = 0.4.02D ASYNNI Monte CarloCVM squareCVM pair-- CVM point0.600.500.400.300.200.100.00Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fs^ 400.0 0.2 0.4 0.6 0.8 1.0x in YBa2Cu306+xFigure 2.4: Comparison of kT(Ox/O,u)T for the CVM point, pair and square approx-imations to the Monte Carlo results of Rikvold et al[69]. The temperatures used are:kT/V=0.862 for CVM point, kT/177,---0.495 for CVM pair and kT/V=0.1 for CVMsquare.Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fx^ 41In order to decide as far as (ax/a,u)T is concerned which maximal cluster of theCVM approximation is sufficient to qualitatively describe the 211 ASYNNNI model, weplot in figure 2.4 the CVM predictions and the Monte Carlo results for the 2D ASYNNNImodel[69]. It is clear that the CVM square approximation is already quite sufficient indescribing (Oxia,a)T, especially in the orthorhombic phase. Later, when these predic-tions are compared to experiment, it will become clear that the discrepancies betweenthe 2D ASYNNNI model and the CVM square are very small compared to the discrep-ancy between theory and experiment, so that the CVM square should form an adequateplatform from which to propose modifications in order to improve the fit to experiment.This is said in light of the fact that the 211 ASYNNNI model is very successful in correct-ly predicting many structural phenomena (eg. phase diagram and stable ground states)but not the chemical potential, which contains this structural information in addition toother effects.2.4 Defect chemical models2.4.1 General introductionAs mentioned in section 2.2, the effective pair interactions could, in principle, be con-centration dependent. There are many effects which could give rise to a concentrationdependence (cf., for example, ref. [63]), such as band structure changes, elastic forces,magnetic effects, etc. The detailed calculation for the effect of the metal-insulator tran-sition upon the effective hamiltonian of the lattice gas problem has not been carried out,and the discussion of this would be beyond the scope of this thesis.However, one approach which has been taken in the analysis of the chemical potentialmeasurements of YBa2Cu306+x is the so called defect chemical approach. Althoughdefect chemical models are not strictly lattice gas models, there are similarities in thatChapter 2. Introduction to lattice gas models in YBa2 Cu3 06+x^ 42defect chemical models usually employ the Bragg-Williams approximation to describethe entropy of the various configurational variables of the system. In this sense theyare lattice gas models; however they contain variables other than the configurationalvariable of the particles placed on the lattice. We note that for oxide superconductorsand YBa2Cu306+x in particular, the addition of oxygen into the basal plane cannot bemade without the introduction of electronic defects as well. Since Cu is a transitionmetal, the electronic defect can be accommodated by a change in valence of the Cu.Therefore, the introduction of non-stoichiometry involves the introduction of holes whichmay be loosely bound to a particular atom. In other words, the placement of oxygenonto the basal plane creates holes which may be free to hop from site to site and giverise to p-type semiconductive behaviour. Also, there is the effect of charge transferfrom the basal plane to the Cu02 bi-layer. These effects, giving rise to, amongst otherthings, the metal-insulator transition, could in principle be important to the oxygenthermodynamics One way to see this is to realize that the stoichiometry is controlledby the oxygen partial pressure. Thus, changes in the oxygen pressure will result in amodification of the electronic structure. The response of the system to changes in thechemical potential will depend on how the electronic structure behaves'.Essentially, in the defect chemical approach, a localized picture of the electronic de-fects is used where the holes are considered to be particles which can be placed on certainatoms. Typically, a random configurational entropy for the hole placement on the latticeis used. The expression for the hole entropy depends on the specific picture proposed.There have been many such defect chemical models proposed for YBa2Cu306+„ Specif-ically, there are basal plane reaction models, where it is assumed that the complete12And of course, one measure of the response of the system to changes in II is the susceptibility(OslOp)T.^Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fs^ 43reaction takes place in the basal plane, including the charge distribution (cf. for exam-ple refs. [88, 89, 90, 91, 30, 92]). Also, there is the alternative model of Verweij andFeiner[64, 65] where the effect of charge transfer to the Cu02 plane is crudely taken intoaccount. In all models, with the exception of the model proposed by Voronin et al.[30],no prediction is made for the O-T transition. This is because a regular solution model isassumed, and the O-T transition is introduced artificially by proposing different modelsfor the two phases. For example, it is assumed that above the order-disorder transition(i.e. x > xoT), the long range order parameter s is immediately equal to 1 and the 0sublattice is empty.Since a random entropy is assumed in these models, it is very simple to write down thereaction function defined in equation 2.6. In particular, the reaction function is composedof three parts: a configurational term for the oxygen fc(x), a configurational term for theholes (or valence options of the Cu) fh(x), and a spin term of the holes fs(x). Assumingsuch a reaction function, the chemical potential is given by (cf. equation 2.5)= SH — TSS° kTln (fc(x)fh(x)f.(x)) (2.30)where SH contains all the formation enthalpies for the defined particles and SS° is theconcentration independent part of the entropy. The non-ordering susceptibility is givenby (cf. equation 2.7)kT°P)a( :T) T)-1Ex 1 +kTfc(x)fh(x)fs(x) (2.31)Mx)fh(x)fs(x)-F fc(x)fi,(x)fs(x)+ fc(x)fh(x)Mx)where the prime indicates the partial derivative with respect to x at constant T.The determination of the reaction function is dependent upon the exact model chosen.In particular, one needs to determine the configurational entropy for the holes (or Cu7-7Chapter 2. Introduction to lattice gas models in YBa2Cu306+r^ 44valence). Early defect chemical models did not have much (and some times incorrect!)information about the valence situation in these materials. Consequently, we will notreproduce here all of the proposed oxidation thermodynamic models which exist in theliterature, since many are now out of date. Instead, we will present the model of Verweijand Feiner[64, 65], since it gives the "flavour" of the defect chemical models, seems tobe consistent with the current valence picture in YBa2Cu306+s and forms part of theextended CVM model of chapter Reaction model of Verweij and FeinerAt x=0, all the Cu(1) is two-fold coordinated in the Cul+ valence state. This means thatthe outer shell is 3d19 and there are no unpaired electrons: the Cul+ is non-magnetic.Adding one oxygen to the basal plane, the two neighbouring coppers "donate" an electronto the oxygen and change their valence from Cul+ to Cu2+ (cf. center graph in figure 2.5).The Cu2+ atom is now three fold coordinated and contains one unpaired electron (3d9):the Cu2+ is magnetic. What happens when a second oxygen is added is a more difficultquestion. If the oxygen is isolated, then it is again the same. However, if an oxygenis added to the next nearest neighbour site, then the central copper is now four-foldcoordinated (cf. bottom graph in figure 2.5). It may share its two outer electrons withthe two neighbouring oxygens, making it effectively a Cu3+(3d8). It is also possible toassume that the copper stays Cu2+ and an oxygen 2p hole is created. More likely, thehole is shared between the oxygen and copper (cf. chapter 5). One sees that there areseveral options to choose from. Can the Cu2+ valence "hop" from site to site? Is thereCu3+? Should one take into account charge transfer to the Cu02 plane? Is the nature ofthe chain holes different from the planar holes? Must one take into account holes in the2p, orbital of the 0(4) site? It should be noted that in this description we are speakingabout the formal valence, corresponding to the oxidation state of the copper. In fact,Chapter 2. Introduction to lattice gas models in YBa2C11306-Fx^ 45these compounds are too covalent to really speak about charges and their location.The model of Verweij[64] takes into account some of the above mentioned valenceoptions but is restricted to the orthorhombic phase. It is assumed that the occupancyof oxygen is random and restricted to one sublattice (i.e. s = 1). The valence of thecopper is determined by the surrounding oxygen configuration, with certain restrictions.It is assumed that the change from Cul+ to Cu2+ is completely localized; the Cul+ andCu2+ valence may not "hop" from site to site. This means that configurations suchas 0-Cul+-0, Vo-Cu2+-Vo are forbidden (Vo is an oxygen vacancy, 0 is an oxygen).Charge neutrality is given then by the creation of holes, which may hop and also go tothe Cu02 plane. In other words, it is assumed that at x = 0, all Cu(1) is Cul+. Isolatedoxygens create two Cu2+ as shown in the center graph of figure 2.5, but that there is noconfigurational entropy associated with the existence of these Cu2+: the valence the Cu2+is bound to that site. Only the addition of a second oxygen next to the Cu2+ creates ahole that may hop and give rise to a configurational entropy. The rule is thus, that thecopper is Cul+ when neighboured by two oxygen vacancies, Cu2+ otherwise. Additionalcharge compensation is afforded through the creation of holes (either on the Cu or onthe oxygen). These holes give rise to a configurational entropy.NotationWe will present briefly the notation of Verweij in order to discuss this model further.A more general and complete derivation of the relevant equations will be delayed untilchapter 5. Denoting [Cu} as the number of Cu2+ in the basal plane per unit cell and[hole] as the number of holes per unit cell, charge neutrality dictates that (Note: if wehave M unit cells, N, = 2M.)[Cu] + [hole] = 2x^ (2.32)Chapter 2. Introduction to lattice gas models in YBa2 Cu3 06+.^ 46+1 +1+2^+2^+ 1+1^+2^+2^+2^+1^•+ holeFigure 2.5: Schematic diagram of the change in Cu(1) valence as a function of its nearestneighbour oxygen occupation. Vacant oxygen sites are drawn as large open circles. Oc-cupied oxygen sites are large filled circles. The coppers are drawn as small solid circlesand their valence is shown above each copper. In the top graph, there are no oxygens sothe valence is 1+. In the center graph, the isolated oxygen changes the valence of thetwo nearest neighbour coppers to 2+. In the bottom graph, there is an oxygen "chain"of length 2. The center copper stays 2+ and a mobile hole is created.Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fs^ 47In a random solution model for the oxygen configuration limited to one sublattice (i.e.orthorhombic phase), the number of Cul+ is given by [Cu'] = (1 — x)2. (1 — x) is theprobability to find a vacancy and (1—x)2 is the probability to find 2 consecutive vacancies.Since the Cu in the basal plane is either Cul+ or Cu2+, [Cub2p+] --= 1 — [Cul] ---= x(2 — x),so that the number of holes is given by [holes] = x2. One now must decide where theseholes reside (either in the chains, in the planes, on the copper, on the oxygen or a mixtureof everything). Also, there is the question of the spin of any free Cu2+ and the spin ofthe holes. One can propose that the spin is free or that it forms singlet, doublet, triplet,etc. states with neighbouring spins.Verweij introduces the following notation to specify the various options. This notationwill be useful in chapter 4, when the predictions of Verweij will be compared to experimentand in chapter 5 where this model will be combined with the cluster variation methodin order to improve the fit to the data.1. Location of the holes:(a) All holes are in the Cu02 planes and are "free", i.e. holes are on the oxygenand distributed over all 2N0 planar oxygen sites. This is denoted as F.(b) All holes are in the planes and are "bound" to the copper, i.e. distributedover the N, planar copper sites. This is denoted as B.(c) All holes are in the CuO, chains and are "free", i.e. holes are on the oxygenand distributed over all lix/V, chain oxygen sites. This is denoted as f.(d) All holes are in the CuOx chains and are "bound" to the Cu2+ sites havingtwo oxygen nearest neighbours, i.e. distributed over the [Cub2p1N0 coppersites which are in the interior portion of the chain fragments. This is denotedas b.xf(x) =(1 — x)(2.33)Chapter 2. Introduction to lattice gas models in YBa2Cu306+.^ 48(e) The holes are divided between the chains and planes in the proportion to thenumber of sites available in each subsytem, i.e. F-f, F-b, B-f, or B-b.2. Behaviour of the spin:(a) If the hole is free, then the spin may be free (D, or may form a doublet (II)or quartet (IV) with the neighbouring Cu2+ spins.(b) If the hole is bound, then the spin may be free (D, or form a singlet (I) ortriplet (III) with the Cu2+ to which it is bound.The following reaction functions are thus defined:andf(x) = 1 (2) 2xs ^ (2.34)where g is the spin degeneracy factor. g = 2 for , 1/2 for I or II, 3/2 for III and 1 forIV.Validity of the modelNaturally, some of these special cases seem unphysical, especially the cases where theholes are constrained to be either on the chains or the planes. If the holes were restrictedto be in the chains, then the planes would not be doped and no superconductivity wouldoccur! If the holes were restricted to the planes, then there would be 1 hole in theplanes per unit cell at x = 1, which is higher than calculated (cf. for example refs.[50, 53, 51, 93, 56]) or measured (cf. for example refs. [94, 95]). It could be possiblethat at lower temperatures, a modification of the charge transfer takes place, but thisis unlikely since, as Verweij points out, there is no evidence either in conductivity, HallChapter 2. Introduction to lattice gas models in YBa2Cu306-Fx^49Hole location fh(x)2^2xF or B:[2nx— x2]2F-f or B-f:2n-Fx—x212x[2n +xx — x2] [^2n + x.F-b or B-b.x2 2Y{ 2n+x21r^x^-12xf:b:(1 — x)1[1 —xTable 2.3: Configuration functions for the oxygen 2p holes for the defect chemical modelof Verweij in the orthorhombic phase. See text for notation. For F models n = 4 and forB models n = 2.effect or specific heat measurements that a significant change in the hole distributiontakes place.Essentially, this is a localized description of charge carriers, where the configurationindex is assumed to be a good quantum number. This might seem physically unrea-sonable, since the charge carriers in the metallic state are typically described by a bandpicture with delocalized states. However, if the band width W is much less than kT, thenall single particle states in the band have an equal probability to be occupied, and theband picture in this limit is indistinguishable from the lattice gas picture of the holes [64}.One can extend the model of Verweij to the tetragonal phase, and ask if the predictionsfor the copper valence and hole count are consistent with experimental results from, forexample, XAS13 measurements. In the tetragonal phase, a random solution model forthe oxygen will give [Cul] = (1 — 1-x)4, implying that [Cu] = 1 — (1 — -12-x)4 and hence'3X-Ray Absorption SpectroscopyChapter 2. Introduction to lattice gas models in YBa2 013 06+x^ 50[hole] = 2x + (1 — -x)4 —1. These equations for the valence and hole count are preciselywhat is deduced from recent XAS measurements of Tolentino et al.[94] in the limit oflow x. Also, the corresponding equations for the orthorhombic phase presented abovein section 2.4.2 are consistent with the findings of Tolentino, in the x —* 1 limit. Inthe intermediate regime, Tolentino finds that the behaviour of the hole count is morecomplex than a random solution model could predict, requiring a refinement due to theoxygen ordering effect. This is not surprising, since a random solution model for thetetragonal phase grossly underestimates the effect of the nn oxygen interactions'''. Inthe orthorhombic phase, once a significant sublattice splitting has developed, the nninteractions are no longer important and it is more reasonable to postulate a randomsolution model.Since the number of possibilities for this model are quite numerous, the presentationof (ax/.9,01, for this model will be delayed until chapter 4, when the experimental resultsare shown.2.5 SummaryWe have given in this chapter a resume of the two basic approaches used to study thethermodynamics of oxygen in YBa2Cu306+r. One the one hand, there are the pure oxy-gen ordering lattice gas models which take into account the oxygen-oxygen interactions.These models are very complex and predict a rich structural behaviour. On the otherhand, one has the defect chemical models, which utilize a very basic picture for the oxy-gen interactions, but also include other configurational variables, such as hole and spindegrees of freedom. Although Verweij's basic idea concerning the hole count is consistentwith both theory and experiment, it is quite obvious that the random solution approach"This is clearly the reason why Verweij and Feiner did not extend their theory to the tetragonalphase.Chapter 2. Introduction to lattice gas models in YBa2CU3 06+x^ 51for the oxygen configuration needs to be refined in order to coincide with the structuralpredictions of the lattice gas models. As far as the pure lattice gas model is concerned,however, it is clear that ignoring the additional degrees of freedom is potentially a seriousproblem. The predictions of these two approaches will be compared to the experimentalresults of (Oxiatt)7, in chapter 4. It will become clear that the two models need to bemerged so that a consistent picture results. But first, the experimental setup for themeasurement of the non-ordering susceptibility will be presented.Chapter 3ExperimentThis chapter is composed of three parts whose purpose is to explain in detail the designconcept of the experimental apparatus, the measurement procedure and the processingand corrections applied to the raw data to ultimately end up with the oxygen pressureisotherms and (Oxlay)T. As was outlined in Chapter 1, the goal of the experimentis to obtain accurate curves of (0x10,a)T in order to test the theoretical models forthe oxygen ordering in YBa2Cu306+s. Although the apparatus built specifically hadthis experiment in mind, it should be noted at the outset that this setup can be usedfor other purposes such as, for example: preparation of samples with a well controlledoxygen stoichiometry and oxygen isotope substitution. Indeed, this apparatus has alsobeen used to prepare deoxygenated samples of YBa2Cu306+x for muon spin rotationstudies[39, 96, 97, 45, 98], and 170 substituted samples to help locate the site of thepositive muon in YBa2Cu306+s[99]. Accordingly, this chapter not only describes theexperimental method used for the present work, but also provides enough information toact as a reference for the general use of the apparatus.3.1 Experimental setup3.1.1 Design conceptThe apparatus was designed for the study and manipulation of high 'I', oxides which haveproperties dependent on their oxygen stoichiometry and whose oxygen content can be52Chapter 3. Experiment^ 53altered and studied at temperatures up to about 800°C. Typically, these high I', samples,when heated, develop a finite oxygen pressure due to the existence of certain loosely boundoxygen sites within the structure. There are several experimental techniques one canuse to study the relationship between oxygen pressure, temperature and stoichiometry(Voronin et al.[30] have a comprehensive list of oxygen chemical potential experiments).The method chosen here to study YBa2Cu306+, was to control the oxygen stoichiometryand to measure the equilibrium oxygen pressure at constant temperature.One way to do this is to have an appropriately designed gas handling system whichcan be used to extract oxygen from the sample. Unfortunately, as oxygen is removedfrom the sample, the equilibrium pressure decreases rapidly (P-- exp x). This means thatone can run into serious trouble if the desired target oxygen concentration of the materialis low enough. The key idea to get around this problem is to include in the gas handlingsystem an extra volume which can be cooled to liquid helium temperatures. A volumeat liquid helium temperature will act as pump and remove oxygen from the sample. Onecan keep track of the amount of oxygen removed by warming the the volume to roomtemperature and measuring the subsequent oxygen pressure. This "liquid helium wand"enables one to set an accurate decrease in the oxygen content.For the measurements of (ax10,a)T, in YBa2Cu306+, the simplest and potentiallymost accurate method' is to start with a sample with a low oxygen content, xa-_0, andthen use the gas handling system to set a small increase in the oxygen content, Ax, bytitration and measuring the rise in the equilibrium oxygen pressure, P. i.e:, ax)^2Ax (— ':--2 (3.1)ay T A (log P)•To obtain an accurate measurement of (ax/Oft)T, one has to be able to set small andprecise changes in x and to measure the equilibrium pressure over several orders of'There is one disadvantage, however, and it arises when the oxygen used to titrate contains impurities.cf. Section 3.3.Chapter 3. Experiment^ 54magnitude (between x f_-_-0.05 and 0.95, the pressure changes by six orders of magnitudein YBa2Cu3061-x)•3.1.2 Design requirementsThe main requirement for the apparatus was to allow for a precise enough titration andpressure measurement to enable one to take the derivative of the experimental curve withan acceptable amount of scatter. Aside from the obvious desire for accurate measurementsof the pressure, there are several other aspects of the design that had to be kept in mind:1. That part of the gas handling system involved in measuring the equilibrium oxygenpressure should be made as small as possible to minimize the "dead volume", sinceoxygen in the dead volume has to be accounted for. The smaller the correction,the more accurate the results will be. On the other hand, connection tubes shouldnot be too small, otherwise at low enough pressures, thermal transpiration canintroduce significant errors (cf. section 3.3).2. One should have good control and measurement of the sample temperature duringan experimental run, since the equilibrium pressures depend sensitively on temper-ature. For example, for YBa2Cu306+s, log P ,s, T.3. It is important to control the temperature of all parts of the the gas handlingsystem, since the oxygen content is deduced from knowledge of the pressure ofoxygen in reference volumes of the gas handling system, plus corrections for oxygenin dead volumes.4. If equilibration times are long, the lowest measurable pressure can be limited by thebackground outgassing of the walls. Thus, the design should not contain plasticsor other materials that may act as absorbants and subsequent sources of impurityChapter 3. Experiment^ 55gases. In addition, that part of the sample holder which will be at high temperatureshould be highly inert. Stainless steel, for example, will oxidize at high enoughtemperature and pressure, acting as a sink for the oxygen gas. Quartz is a goodchoice.5. Finally, several key valves of the gas handling system and all of the instrumentationshould be under computer control, so that the data acquisition can be automated.This is necessitated by the fact that long equilibration times are required for eachmeasurement of pressure and (Ox/ay)T.3.1.3 Deoxygenation apparatusOverviewFigure 3.1 shows a schematic of the experimental setup used for the oxygen pressure and(ax/(9,a)T measurements. One sees on the left the gas handling system and on the rightthe quartz sample volume extending into a single zone tube furnace (Lindberg model59544 1200°C). Starting from the sample space on the right and moving to the left, theindividual components are: The sample, consisting of roughly 60g of material, in theform of 3/4 inch diameter, 3g pellets, wrapped in Pt foil, and placed into the quartztube; an evacuated quartz bulb, containing two stainless steel radiation shields, insertedinto the sample space to minimize the dead volume, to cut down on the radiation strikingthe stainless steel connector to the gas handling system and to minimize errors in thetemperature of the sample. The connection to the gas handling system is made usinga quartz/Pyrex graded joint followed by a Housekeeper seal. The Housekeeper seal iswelded to a Varian Conflat flange.Whenever a new sample is mounted, a liquid nitrogen cold trap is inserted as shown.This freezes out any water and CO2 that the sample might have absorbed. For a 55g41''Chapter 3. Experiment^ 56Figure 3.1: Schematic diagram of the experimental setup to measure the oxygen pressureisotherms of YBa2C11306+x as a function of x. The dead volume of the main quartz tubeholding the sample is minimized by the use of a sealed inner quartz tube. This innertube incorporates radiation baffles to reduce temperature gradients. The Pt vs. Pt/13%Rh thermocouple is situated just outside the outer quartz tube close to the sample. Analumina and fire brick encasement surrounds the thermocouple and quartz tube.Chapter 3. Experiment^ 57sample of YBa2Cu306+,, the amount of water and CO2 caught by the cold trap can beas much as 10 Torr in Vs+Vp+Vd (cf. Figure 3.2) which, if not removed, would resultin a very large error in the oxygen pressure measurements. The liquid nitrogen cold trapis not used during the (ax/0,a)T measurement runs, as will be explained in section 3.2.The more detailed view of the gas handling system is shown in figure 3.2. Here,various important volumes and valves have been labeled. All tubes are stainless steeland connections either welded, where appropriate, or made using Cajon VCO connectors,which use silver plated stainless steel washers in a compression seal. The valves are NuproBN series bellows valves, two of which are air operated and under computer control. Theentire system thus contains only two small pieces of plastic material, namely from the airoperated valves, which have a Kel-F plastic plunger tips to reduce the sealing force. Thebackground outgassing from the stainless steel walls and the quartz assembly is roughly1mTorr in 24 hours, once it is well pumped out.The liquid helium wand is a 3/8 inch thin wall stainless steel tube welded to a 1/2inch diameter, 1 metre long flexible vacuum line connected via a Cajon VCO to valve 2.Inserting the wand into a liquid helium storage dewar extracts oxygen from the sample.All stainless steel volumes are temperature controlled to ,--0.1°C by being in directcontact with copper plates and/or tubes connected to a Haake refrigerated circulatingwater bath.The sample temperature is measured with a Pt-Pt13%Rh thermocouple (Aesar, sec-ondary standard grade, +1°C absolute calibration) which is mounted outside of the quartztube next to the sample. The entire thermocouple and sample region is fitted into analumina and fire brick assembly to ensure a homogeneous and well defined temperature2.2It was found that without this assembly temperature readings varied significantly, depending onthermocouple placement.Chapter 3. Experiment^ 58Figure 3.2: Schematic of the gas handling system showing the labeling of the valves (smallv's) and of the volumes (large V's). Valves 5 and 6, labeled with an A, are pneumaticallyoperated under computer control.Chapter 3. Experiment^ 59Volume calibrationInitially, the volume of the reference container, Vr, is calibrated by measuring the weightchange of the container after backfilling with degassed, distilled water. Then, the re-maining volumes in the system are all calibrated by stepwise titration of oxygen into theappropriate volume and subsequent removal of the gas either by freezing into the heliumwand or by evacuation through the pump.a) Reference volume calibrationThe reference volume is removed from the system and the pressure transducer detached.The VCO connection to the air actuated valve, V5, on the reference volume is sealed and aHoke packed valve is fitted onto the opening for the pressure transducer at the other end.After evacuation, the weight of the container is measured using a Sartorius 1264 MP scaleaccurate to 0.01g and capable of measuring weights of up to 3000.00g. Then, through thevalve, degassed, distilled water is backfilled into the volume and the assembly is weighedagain. Before sealing the volume containing the water, its temperature is measured.This procedure was repeated several times and gave consistent results. Finally, thevalve is removed and its volume is measured with calipers and also by adding waterto its volume. This volume (L..s'_ 0.5cm3) is subtracted from the calculated volume andthe volume of the pressure transducer (=2.80cm3 from manufacturer's specification) isadded. The uncertainty quoted is a result from the reproducibility measurements as wellas taking into consideration uncertainties in temperature and density of water due topossible absorption of gases[100]. Finally, V, is reconnected to the system.b) Subsidiary volume calibrationsThe reference volume, Vr, is filled with roughly 3 atm of 02 gas. To calibrate Vs, forexample, one evacuates Vs., then repeatedly (100 times for Vs) removes oxygen from V,by titrating into an evacuated Vs, measuring the new pressure in V, and evacuating V.Chapter 3. Experiment^ 60through v1. One can show that by this procedure, a plot of logft-, vs. i (Pi and Ti arethe pressure and temperature in V, at the ith titration) will give a straight line whoseslope, m, is given by(exp(—m) — 1) 17„. (3.2)In this manner all volumes, or combination of volumes were calibrated at room temper-ature.For the sample volume (also called dead volume), Vd, a new calibration was requiredwhenever the sample was changed, since the calibration was made with the sample inplace. In addition, since the sample space is at an elevated temperature, one needs tocalibrate Vd as a function of temperature, i.e. Vd is an effective volume due to theexistence of the temperature gradient. From the ideal gas law, the effective dead volumeshould go asVd(T) =14, + ^ (3.3)(T T0)where Voo, A and T, are parameters to be fitted. For the dead volume calibration, thesample is deoxygenated to a low value in order that the partial pressure of oxygen due tothe sample is negligible, and N2 gas is used as the buffer gas in order not to reoxygenatethe sample. Figure 3.3 shows a plot and the best fit line to equation 3.3 for a 33g sampleof YBa2Cu306. It is assumed that surface adsorption of N2 gas onto the sample is notsignificant. There is no indication of the incorporation of nitrogen into the structure ofYBa2Cu306+,. Finally, it should be noted that the pressure dependence of the effectivedead volume was checked and found to not change significantly within the resolution ofthe instrumentation. Such a pressure dependence could, in principle, arise due to varyingthermal transport conditions along the quartz tube as a function of pressure, which wouldchange the temperature profile along the quartz tube.Table 3.1 shows the values for the various volumes obtained by the above calibrationAChapter 3. Experiment^ 6180cr),, 78E076a)E=Tj>74pcria)cna) 72.>-.6a)w7068200 300 400 500 600 700Temperature (°C)Figure 3.3: Plot of the furnace dead volume as a function of temperature with a 33gsample of YBa2Cu306 loaded in the sample space.Chapter 3. Experiment^ 62Volume SizeV, 998.13 +^0.08cm3Vs 14.756 +^0.001cm3V, 13.199 +^0.01cm3V, 127.49 ±^0.05cm3V d ''' 4 7 c m, 3 and T dep.Table 3.1: Calibration values for the volumes of the deoxygenation apparatusprocedure. It should be noted that the most crucial volume combinations were calibratedwith the most care, taking into account the valve positions (ie. whether they were openedor closed).InstrumentationNow follows a brief description of the instrumentation used for the measurements (cf.Figure 3.4). During the course of the experiments, the oxygen pressure in the samplespace is measured using two MKS Baratron type 310 pressure transducers (1 Torr fullscale and 1000 Torr full scale) giving a measurement range between 1 mTorr and 1000Torr with an accuracy of about 0.1% of reading. The analog output of the MKS Baratrontype 170M controller is fed into an HP 3478a 5 digit voltmeter to improve the precisionof the pressure reading (The Baratron controller is configured with a 4- digit readoutunit, which gives digitization noise during signal averaging). The pressure in the referencevolume is monitored by a separate pressure transducer (Sensotec, 50 PSIA rating, +0.1%f.s.) mounted on the top of V. The sample temperature is measured with respect to anice bath using a Pt-Pt 13%Rh thermocouple connected to an HP 3478a voltmeter. Thesample temperature and pressure are both monitored on a Philips PM 8252A two penchart recorder. A home built, computer controlled solenoid switch is used to control thetwo air operated valves. Finally, all the instrumentation is connected via the IEEE-488Chapter 3. Experiment^ 63interface bus to a personal computer (IBM compatible 286) enabling automated dataacquisition.3.2 MeasurementIn this section, the method used to deoxygenate samples of YBa2Cu306.fs and the exper-imental procedure for the measurement of the oxygen pressure isotherms and (ax/att)Twill be described. As mentioned in Chapter 1, it is generally accepted that YBa2Cu306-Fscan have oxygen concentrations roughly ranging from x = 0 --+ 1. It is thought that thereexists a strong nearest neighbour repulsion between oxygen atoms in the chains and thisresults in the oxygen vapour pressure increasing dramatically, perhaps even diverging,at x=13. This means that the oxygen content is not very strongly dependent on theannealing pressure when x is close to 1. As a result, the preparation of samples underappropriate conditions resulting in an oxygen content close to one is a good referencestate to use for all subsequent processing of samples. In other words, if one prepares asample with oxygen content close to one, then the uncertainty of this oxygen content willbe small and an accurate removal of oxygen from this reference state will maintain thesmall uncertainty. One could, of course, prepare samples with an oxygen content close tox=0, but the temperature and pressure treatment required is very severe (T>800°C andvery low oxygen partial pressures) and can result in sample decomposition. Therefore,it was decided to prepare samples in the x 1 reference state and then deoxygenateto the desired oxygen content (cf. Appendix C for a detailed description of the samplepreparation and characterization). It turns out that under the conditions used for thereference state preparation, the oxygen content is x=0.987.3At x=1, the 2D square lattice for the chain site-oxygen is half full.Chapter 3. Experiment^ 64Figure 3.4: Block diagram of the instrumentation used in the isotherm measurements.Chapter 3. Experiment^ 653.2.1 Deoxygenation procedureAfter a sample is prepared in its x=0.987 reference state, it is then deoxygenated toa (known) low value for the isotherm measurement. For a typical measurement run,about 55g of YBa2Cu306.987 is wrapped in Pt foil and placed into the quartz tube.The tube and the liquid nitrogen cold trap are connected to the gas handling system,and the furnace set to 100°C. For roughly one hour the sample is pumped on with adiffusion pump to remove most of the water that the sample had absorbed. At 100°C,YBa2Cu306.987 does not develop a significant oxygen partial pressure and thus no errorin x is introduced through this pumping procedure. At this point the furnace is set to300°C and the vacuum pump is isolated from the system. Liquid nitrogen is added to thecold trap as a check for leaks in the system. If the pressure in the system does not dropto zero, then it means that nitrogen has leaked into the sample space and, most likely,that the Confiat gasket seal is not leak tight. A tightening of the bolts on the connectortypically solves the problem. Once it is verified that no large leak exists, the furnace isset to 650°C, with the cold trap maintained at liquid nitrogen temperature.After reaching operating temperature, the helium wand is inserted into a liquidhelium storage dewar and the deoxygenation procedure begins. From the integral of thepressure vs. time plotted on the chart recorder, one can estimate the amount of oxygenremoved from the sample. Typically, 12 hours are required to take the sample to x=0.15at 650°C. After an appropriate pumping time, the sample space is isolated by closingv6 and v4. The helium wand is removed from the liquid helium and warmed up toroom temperature. Once temperature equilibrium is reached, the pressure is measuredin (Vw+Vp+Vx-FV,) and the amount, Ax, of oxygen removed is calculated according to:(AP)Vx^ (3.4)-ymTWhere -y = 46.7737 Torrcm3for YBa2Cu306+s assuming a starting composition of x=0.987gKChapter 3. Experiment^ 66(i.e. molar mass = 666.015—b). And, where AP is the pressure measured in volume Vat temperature T from a sample with an initial mass m at x=0.987.Once the amount of deoxygenation is determined, the extracted oxygen is pumpedout and the sample is annealed at 650°C for several hours and then slow cooled to roomtemperature overnight. In the case where one desires just a simple deoxygenated sam-ple for other experiments, the sample is annealed for two days and then slow cooledor quenched, depending on the requirements. In this manner, one can set very accu-rate changes in oxygen content to within +0.15%. The dominant errors are due to theuncertainty in the volume calibration (+0.1%) and the pressure measurement (+0.1%).Without a temperature control of the volumes through the use of a water bath, thisuncertainty would be roughly doubled. It should be noted that care must be taken tomeasure the sample weight carefully and in a consistent manner. The potential problemin measuring the weight is not the uncertainty of the measurement itself' but that thesample weight changes if exposed to air due to the absorption of (probably) water[101].However, if a consistent measurement approach is used, and samples are not left exposedto the air for more than several minutes, the uncertainty in the mass of the sample isnegligible'.After cooling to room temperature, samples are weighed again and the oxygencontent checked from the change in mass using:Ax (1^m(x)  (M(YBCO) m(0.987))^M(0) )^(3.5)Where m(0.987) and m(x) is the mass before and after deoxygenation, respectively,M(YBCO) is the molar mass of YBa2Cu306.987 and M(0) is the molar mass of atomicoxygen. This measurement of Ax is not as accurate as the determination using pressures4Measurements were made with a Mettler AE 163 scale. Range: 0-30g ±0.01mg or 0-100g ±0.1g.5Sample mass increases by about 0.006% to a saturating value in roughly 20 minutes when exposedto the air.Chapter 3. Experiment^ 67and serves just as a check to verify that no errors were made during deoxygenation.At this point, one can either remove some material for other experiments or conductthe (ax /N)T measurement as will be described in the next section. For the vapourpressure measurements, the YBa2Cu306+x sample is deoxygenated to about x'-'0.15. Alower value was not chosen in order to avoid possible long term contamination of thesample due to reaction with the platinum foil or to decomposition.3.2.2 Measurement of the oxygen pressure isotherms and (ax/N)TIn this section, the procedure for the isotherm and (ax/ait)T measurements will bedescribed. It will be mostly concerned with the details about the automated data acqui-sition, since the basic idea is very straight forward. After deoxygenation, the sample isreturned to the apparatus, but this time the liquid nitrogen cold trap is not included.In figure 3.4 one sees the basic setup for this measurement. For clarity, those volumesnot required for the isotherm measurements are not depicted although physically thesystem remains unchanged. Since the sample has at this point a low oxygen content, itis possible to to heat the sample to 350°C and still not have a measurable oxygen vapourpressure. Thus, it is not necessary to include the liquid nitrogen cold trap since any gasesevolving from the sample are not oxygen and can be safely pumped out. At this point,a calibration of the dead volume is made as a function of temperature using the methoddescribed above in section 3.1.3.Once the dead volume is calibrated, the furnace temperature is set to the desiredtemperature for the vapour pressure run. The oxygen reservoir is pressurized with about1400 Torr of ultra high purity oxygen (Linde UHP, rated at <21 ppm of impurities).After the sample reaches equilibrium, the computer data acquisition program is started.The equilibration time was defined to be roughly twice the time it took to see no changesin pressure greater than 0.1%. Under computer control, oxygen is titrated into theChapter 3. Experiment^ 68sample space from the oxygen reservoir via the air actuated valves. The vapour pressureis monitored until equilibrium is reached. This procedure is repeated until the vapourpressure reached a value close to 1000 Torr. Once an isotherm is measured, the sample isagain deoxygenated, and the whole process was repeated at a different temperature. Theisotherm measurements ran 24 hours a day under complete computer control. For lowvapour pressures, the time required to see no change in pressure within the resolution ofthe apparatus is around 12-14 hours and decreases to 2-4 hours for pressures above 100Torr.Computer data acquisitionSince the measurement of the vapour pressure isotherms using small increments in x toextract (Ox/OOT requires equilibration times of up to 14 hours per measurement point,it was necessary to automate the data acquisition in order to complete the experiment ina reasonable period of time. Such long equilibration times greatly complicated the taskof taking data, since the computer program had to not only measure several parameters,but also control two valves. The complexity of the computer program coupled with verylong waiting periods made the development period for the software surprisingly long,since incipient bugs in the program often became apparent only after taking many datapoints.Some of the major hurdles to overcome in the software were, for example: a) con-trolling the valves and verifying whether a requested action took place; b) keeping trackof the elapsed time over a period of several days (this has to do with the details of how aPC keeps track of time, how the system clock is read and how to detect when a day hasgone by); c) reading and writing to the MKS Baratron controller to measure pressure,change the head ranges, check if the data was valid etc. The basic algorithm of the dataacquisition program is depicted in figure 3.5. The issues mentioned above as well asIWhile x < xma.IRead reservoirpressureTitrate IWhile t < tmaxRead P,T,tYesI Store P,T,tin file vs. timeII^Pause'Store all important]data in file vs. xChapter 3. Experiment^ 69( ST.;RT )I Enter operating,parametersFigure 3.5: Basic flow chart for data acquisition program to measure oxygen pressureisotherms. P,T,t are, respectively, the pressure, sample temperature and elapsed time.xma, is the maximum desired oxygen content, tnias is the desired equilibration time, APand At correspond to the change in pressure and time since the last file save, AP, andAt, are the maximum desired changes in pressure and time before the data is written toa file.Chapter 3. Experiment^ 70details concerning graphics are omitted. The program is essentially a double loop, wherethe outer loop titrates and increments the oxygen content and the inner loop measuresthe subsequent pressure as a function of time. An attempt was made to develop a moresophisticated algorithm to decide when equilibrium was reached, but this task turned outto be very complicated when the condition of robustness was added. Instead, the pro-gram simply uses the elapsed time as a criterion for equilibration. This is not a problemsince the resolution of the apparatus decreases as a function of oxygen pressure.The measurement of the oxygen vapour pressure through the MKS Baratrons ismade difficult due to the non-standard method MKS uses to allow computer interfacing.Instead of the IEEE-488 standard, the MKS Baratron controller communicates to theoutside world via binary coded decimal (BCD) information. It is necessary not onlyto read the pressure but also to control the pressure head range and ensure that thepressure reading is valid. In other words, it is necessary to measure and control theBaratrons, which thus requires some sort of hand-shaking and communications protocol.All this needs to be done via the non-standard BCD input and output ports of the MKScontroller.Figure 3.6 shows how the physical connections are made. All communicationsbetween the IEEE-488 port of the computer need to be translated from and to BCDinformation. This is done using an ICS Model 4880 Instrument Coupler, consisting of oneIEEE-488 port, a 50 pin BCD output port and a 50 pin BCD input port. The BCD outputof the instrument coupler is connected to the MKS Baratron Range Selector/ElectronicsUnit to control the range selection of the pressure heads. The BCD output of the MKSBaratron Digital Readout Unit is connected to the BCD input of the instrument couplerto obtain information regarding the current range selection, error status and pressurereading. Each pressure reading needs to be triggered and a special handshaking protocolis followed according to the manual for the MKS Baratron controller. The pin assignmentsChapter 3. Experiment^ 71Figure 3.6: Block diagram for the communication between the MKS Baratron pressuretransducers and the personal computer.Chapter 3. Experiment^ 72for each BCD port are different and a unique cables are required.Together with the data acquisition software, this setup enables one to complete themeasurement of an oxygen pressure isotherm vs. x with minimal human intervention.After deoxygenation, the program is started and one just has to maintain the ice bath forthe Pt thermocouple, occasionally check that waiting times for equilibrium are reasonableand once for every isotherm, one also has to physically change from the 1 Torr pressurehead to the 1000 Torr pressure head using the manual switch of the multiplexer; it is notpossible to control the head selection from the computer. Once the data is taken for anisotherm, several corrections to the data are made to account for some of the systematicerrors.3.3 Systematic errors and their correctionsAside from calibration errors for the volumes and the pressure transducers, there areseveral systematic errors identified, some of which are corrected for in the final analysis.The largest correction at low pressures is due to the existence of impurities (probably N2gas) in the source oxygen used for titration. At high pressures, the corrections for theoxygen in the dead volume becomes significant. In addition, the temperature drift of thereference volumes and furnace are also accounted for. One other error which is consideredbut not accounted for is the effect of thermal transpiration. Each of the mentioned errorswill be discussed in order of importance.3.3.1 Temperature drift of the volumesAlthough all volumes are temperature controlled to 0.1°C, it is possible to measure de-viations of less than that by measuring the pressure in the reference volume during thecourse of the experiment. Since the reference volume, Vr, is isolated from the sampleChapter 3. Experiment^ 73space, one can use the change in pressure as a thermometer. In this way, the titratedamount of oxygen is corrected for the instantaneous temperature of the reference volumeat the moment of the titration, using the following formula:p prTR,  t t —1 Dr (3.6)Where T,RT and P,r is the temperature and pressure of Vr at the ith titration.In addition, any temperature drift of the furnace is also accounted for through themeasurement of the furnace temperature using the Pt thermocouple. The effective deadvolume is corrected for the drift in furnace temperature using equation 3.3. Since one istrying to measure (ax/a 11)T at constant T, it is in principle necessary to correct (ax/(9,,4Tfor the drift of the furnace temperature. Fortunately, this was not necessary, since theno measurable fluctuations of the oxygen pressure were attributable to variations of thesample temperature.It should be noted that the temperature controller itself has to be temperaturecontrolled in order to minimize furnace drift and make the correction to (0.4.9,a)T in-significant. By flowing water from a second circulating water bath, through copper tubessoldered to copper plates, which encased the furnace temperature controller, one is able tominimize the temperature drift of the furnace. This setup is necessary since the furnacetemperature controller uses a platinell II thermocouple referenced to room temperature.Although the sensor installed in the micro-chip controller' has a compensation algorithmto account for error signals due to room temperature drift, one still experiences a greaterchange in the furnace temperature than the room temperature fluctuations.6Lindberg programmable temperature controller model 818P.Chapter 3. Experiment^ 743.3.2 Thermomolecular pressure gradientA potentially serious systematic error arises from the thermomolecular effect (also knownas thermal transpiration). This effect is a well known in low temperature physics, partic-ularly in 'Ile vapour pressure thermometry [102]. It arises when the mean free path of theatoms or molecules becomes comparable to the separation of the enclosing walls. In sucha situation, the atoms collide more often with the containment walls than with themselvesand their behaviour will be dependent on the atom-wall interaction. If one has a thincapillary tube which is held at different temperatures at either end and the mean freepath is sufficiently long, there will exist an equilibrium pressure gradient between the twoends. A rough estimate shows' that, at 450°C, the mean free path becomes comparableto the size of the quartz tube below about 5 mTorr. The relationship correlating thepressure gradient to the temperature and average pressure is not very straight-forward.Only in the limit of A/D --+ 00 (A is the mean free path and D is the capillary diameter)does the correction become simple. In this case one has that P2/P1 = VT2/711[104]. Inthis extreme case, the correction to the pressure measured at room temperature couldbe as high as a factor of 2 for high temperatures.In the not so extreme limit, one needs to become specific and the behaviour dependson the type of gas, the capillary geometry and the capillary material. A closed formexpression for such a case does not exist and one needs to look at empirical forms toapproximate experimental findings. A relatively successful equation, which is essentiallyempirical in nature, was proposed by Takaishi et al.[105] relating the pressures at the hotand cold ends of a capillary tube:P2 _ AX2 + BX + CV7Y + A-P1 — AX2 + BX + C-I-X— + 1'Using a mean free path, A = kTIVird2P, where d is the collision diameter of the gas[103], thevalue, d=3.75)1. for N2 and a quartz tube diameter of 5mm.Chapter 3. Experiment^ 75A* = 8.6x105B*^1.7x103C* =^10K ^2Torr.mmTorr.mm( KTorr.mm)Table 3.2: Values of A*, B* and C* given by Furuyama for oxygen.T2 < T1A^A*(T*)-2B = B*(T*)C^C*(T*)-11T* = -2 (Ti + T2)X = P2d (3.7)where Pi, Ti and P2 T2 are the pressures and temperatures and the hot and cold end-s, respectively and A*, B* and C* are temperature independent constants for the gas.Although this equation was proposed for gases at temperatures less than those experi-enced in this experiment, there is ample evidence that they should work even at highertemperatures[106, 107]. Unfortunately, the geometry in this experiment is not one of acapillary with a circular cross section. Due to the insertion of the tight fitting quartzbulb, the appropriate cross section is anular: i.e. a ring with a 2cm diameter and 2-4mmthickness. In addition the thickness is not uniform due to irregularities in the quartz bulband tube. Nevertheless, it is important to know when such a correction is important andhow quickly it saturates. Using the values for oxygen given in ref.[108] (cf. table 3.2),and a rough guess for the effective capillary diameter, D=5mm, figure 3.7 shows an esti-mate of the correction for this effect. One sees that above 1 Torr there is no significantcorrection but that at 10 mTorr the correction is as much as 40%. This systematic errorChapter 3. Experiment^ 76is difficult to account for quantitatively, and, as will be shown in the next section, theimpurity gas correction will dominate this error significantly so that it does not becomeimportant to try to correct for thermal transpiration.3.3.3 Impurity gas correctionThe most significant systematic error at low pressures is the accumulation of impuritygases in the sample space as one progressively titrates oxygen from the reference volumeinto the sample space. To illustrate what happens, consider the following: at low oxygencontent, YBa2Cu306+x has a very low oxygen vapour pressure, say 10 mTorr. Duringtitration, oxygen at 3 atm pressure is transferred to the "airlock" volume, V. Thenthis oxygen, corresponding to a small amount of x in YBa2Cu306+x, is titrated into thesample space. At the moment of titration, the pressure jumps to about 200 Torr. In thisgas there is not only oxygen but also a small amount of impurities. Since the amounttitrated corresponds only to a small change in x, once equilibrium is reached, the newoxygen partial pressure will be only slightly elevated above the previous one. This meansthat almost all of initial 200 Torr of oxygen will be incorporated into the sample. Whatremains is the oxygen left to create the new equilibrium oxygen partial pressure plus anyimpurities, which are now much more concentrated. One has gone from 200 Torr to 20mTorr of oxygen, but all the impurities remain.Using the ideal gas law and the method of titration, one can account for the impuritygas effect as follows:,T T4. + )3i -1P,:upir Vr piv VTaiP: Vs +^1VdA =  ^(3.8)Piv-Vdwhere i denotes the ith titration, a is the impurity concentration in the reference volume,)3 is the impurity concentration in the sample space, Pr is the total pressure in Vr and=Chapter 3. Experiment^ 771.6From 450°C to 650°Cin 50°C incrementsassuming D.4mm -1.00.001 0.01^0.1^1^10Pressure at cold end P2 (Torr)Figure 3.7: Plot of the estimated thermomolecular pressure gradient vs. pressure forvarious temperatures using equation 3.7.Chapter 3. Experiment^ 78Pv is the total pressure in V d (i.e. vapour pressure measured).This impurity gas effect can be negligible if the source oxygen used is pure enough.The oxygen used in this experiment is rated at <21ppm of impurities. At this level,using the values for the various volumes, this effect should not be significant. However,figure 3.8 shows a comparison between two isotherm measurements made for differentstarting compositions. The top curve shows a plot without the impurity gas correction,the bottom has the correction applied. One sees that the measured pressure is stronglyaffected by even a very small impurity concentration of 121 ppm. This impurity level,determined by requiring the difference between the two curves to be a minimum, is about6 times greater than that stated by the manufacturer. One possibility is that improperhandling of the oxygen regulator caused this increase in the impurity concentration. Theuncertainty in this correction dominates the error for pressures less than 1 Torr.3.3.4 Dead volume correctionOne final correction which needs to be discussed is the effect of the dead volume. Asthe oxygen content of the sample is progressively increased by repeated titration, theamount of oxygen remaining in the dead volume also increases as a result of the increasedequilibrium oxygen partial pressure. At some point, the amount of oxygen in the deadvolume becomes significant compared to the amount of oxygen in the sample. Thecorrection for this dead volume effect is easily accounted for. For YBa2Cu306+s in thisapparatus, it becomes significant above about 10 Torr.Combining equations 3.4, 3.3 and 3.8 one obtains an equation to calculate the newvalue for the oxygen content after titration:=11 1^1( [(1 —^)Pir_ - (1 - )P2v_^, [  (1 - M)Piv^(1 - /32^ 1 .^(3.9)t 7m Tri^v d^TiRTTzfiT1In general, one can say that the impurity gas effect dominates the error bars in thelow pressure region and that the uncertainty in the dead volume correction dominates in11 111 1Not corrected for impurity gas effect11111^-1^-I^-I^-I^I^A^I^10OP•subift9g),e000^Corrected for 0.0121%impurity gases0.001 1^1^I^I^1^I^I^I^I^I^I^I^I^I^I^I^I^I^I^1^1^I1^I 10.1Chapter 3. Experiment^ 790.2 0.3 0.4 0.5 0.6 0.7x in YBa2Cu306_,),Figure 3.8: Comparison of the vapour pressure at 450°C vs. x with and without theimpurity gas corrections. The top graph shows the raw vapour pressure for two separateruns without any impurity gas corrections. It is possible to merge the two curves if oneapplies the impurity gas correction with a source gas impurity level of 121 ppm. Theimpurity gas corrections become negligible for pressures above 1 Torr, so for the majorityof the data points measured, this correction is not significant.Chapter 3. Experiment^ 80the high pressure region. In the next chapter, the experimental results for the isothermand (Ox/0,a)T measurements will be presented.Chapter 4Results and Existing TheoriesThis chapter first presents the oxygen isotherm and (0x10,a)T measurements and dis-cusses some of the basic observations and conclusions one can draw from the data. Then,predictions from some of the existing theories on the oxygen ordering thermodynamicswill be compared to the experimental curves. In particular, the defect chemical model ofVerweij and Feiner, and the 2D lattice gas model will be discussed. It should be stressed,however, that to date no complete description of the thermodynamics of oxygen orderingin YBa2Cu306+, exists, and that no one theory is capable of explaining all of the results.4.1 Experimental results4.1.1 Oxygen pressure isothermsThe experimental results of this investigation can be summarized in two graphs, namelyfigures 4.1 and 4.2. Isotherms were measured at six different temperatures: 450°C, 475°C,500°C, 550°C, 600°C and 650°C. Two reproducibility measurements were conducted at450°C and 600°C and give an indication of the absolute uncertainty of the measurement.The two 600°C curves are obtained from 2 different batches of YBa2Cu306+x manu-factured 6 months apart. In addition, after these measurements were completed, somepellets were reground and 9g of powdered YBa2Cu306+, were placed into the apparatusand a measurement run at 550°C was repeated. Since the amount of powder that couldsafely be placed into the apparatus was limited to about 10g, one could not go to high81+4-++00°^oo°0° +1E cfP00^14- o° +41- CPo AAAA0 # 4_^1,1,••••A<> xLA.XX• •A•^X•AA^X•A X-7.10000.0011001010.10.01Chapter 4. Results and Existing Theories^ 820.0 0.2 0.4 0.6 0.8 1.0x in YBa2Cu306+xFigure 4.1: Plot of the measured oxygen pressure isotherms. The data was taken at: 0650°C, + 600°C, a 550°C, A 500°C, 0 475°C, and x 450°C. The curves at 600°C and450°C each contain two separate data runs and the close agreement gives an indicationof the uncertainties in this measurement. The curve at 550°C includes the experimentwith a powdered sample.Chapter 4. Results and Existing Theories^ 83pressures without quickly running into the dead volume effect (cf. section 3.3.4). Thus,for the powder measurement, the experimental run was halted at 4 Torr. It is obviousthat the two curves overlap sufficiently to prove that there is no qualitative difference inthe oxygen ordering properties between powdered and ceramic samples, making it veryprobable that strain effects due to anisotropic thermal expansion in the ceramics are notimportant to the bulk oxygen ordering properties of these substances.The scatter from the reproducibility measurements give an indication of the abso-lute uncertainty in these measurements. For low pressures the errors are dominated byuncertainties in the impurity gas corrections. At high pressures the errors are dominatedby uncertainties in the dead volume corrections. Table 4.2 shows the estimated uncer-tainties in the oxygen pressure at various representative pressures. These uncertaintieswere calculated by examining the reproducibility measurements conducted at 600°C and450°C. The absolute uncertainty in x is estimated at ,0.005 and was determined by look-ing at the shift required to minimize the difference between the two isotherms measuredat 600°C.In order to present the data in a form useful for both internal and external users, acubic spline interpolation of the isotherms was made; the results are given in Table 4.1.The data in brackets are extrapolated pressures for x values slightly beyond the range ofthe measurements actually made. This table has been proven especially useful for mate-rials preparation, where it has allowed the determination of proper annealing conditionsfor preparing samples, including single crystals, with a well defined oxygen content.The focus of the discussion now turns to the physics that one might learn fromthese measurements. Relevant information lies in the thermodynamic response function(49x/ap)T, which is a form of susceptibility, and measures the response of the system tochanges in the chemical potential.Chapter 4. Results and Existing Theories^ 84x Oxygen Pressure in YBa2Cu306.4, (Torr)±0.005 450°C 475°C 500°C 550°C 600°C 650°C0.12 (0.109)0.14 (0.032) 0.2090.16 0.070 0.3640.18 (0.003) (0.007) 0.129 0.5850.20 0.005 0.012 (0.045) 0.203 0.8820.22 0.007 0.018 0.068 0.289 1.2700.24 0.010 0.025 0.099 0.411 1.7800.26 0.014 0.033 0.140 0.573 2.4470.28 0.019 0.044 0.194 0.786 3.3140.30 0.025 0.057 0.266 1.056 4.4270.32 0.033 0.075 0.360 1.403 5.8520.34 0.043 0.097 0.482 1.861 7.6960.36 0.057 0.126 0.639 2.451 10.080.38 (0.021) 0.076 0.164 0.831 3.211 13.140.40 0.028 0.100 0.214 1.075 4.186 17.130.42 0.036 0.132 0.279 1.405 5.446 22.340.44 0.047 0.175 0.367 1.837 7.105 29.150.46 0.062 0.232 0.486 2.410 9.287 38.140.48 0.083 0.307 0.618 3.174 12.13 50.060.50 0.108 0.401 0.795 4.178 15.85 65.820.52 0.139 0.511 1.038 5.500 20.86 86.870.54 0.176 0.630 1.315 7.239 27.57 114.90.56 0.214 0.790 1.639 9.362 36.36 152.80.58 0.272 0.993 2.058 11.74 47.77 203.60.60 0.355 1.268 2.634 14.62 60.64 267.30.62 0.455 1.623 3.259 18.32 75.92 337.70.64 0.571 2.073 4.155 23.12 95.53 426.40.66 0.733 2.666 5.322 29.38 121.4 542.00.68 0.971 3.476 6.887 37.68 155.3 693.40.70 1.285 4.600 9.025 48.98 200.90.72 1.701 6.144 11.89 64.21 262.80.74 2.265 8.305 15.94 85.51 348.80.76 3.073 11.44 21.70 115.8 (466.3)0.78 4.251 16.07 29.80 159.80.80 5.988 23.05 42.41 225.70.82 8.622 34.15 61.85 328.20.84 12.53 52.81 93.23 490.60.86 19.42 86.42 (146.9)0.88 31.82 151.6 -0.90 55.00 297.40.92 106.1 (672.5)0.94 242.3Table 4.1: Cubic spline interpolation of the oxygen pressure plotted in figure 4.1Chapter 4. Results and Existing Theories^ 85Pressure(Torr)EstimatedUncertainty100 0.3%10 0.8%1 2%0.5 5%0.05 10%0.005 50%Table 4.2: Relative uncertainties of oxygen pressures at various representative pressureranges for data listed in table The thermodynamic response function (ax/s9,07,As stated in section 3.1.1, the non-ordering susceptibility (Oxfatt)T is obtained using therelation kT(Oxiatt)T=2Ax/A(ln P), where Ax is the externally controlled change in xand A(ln P) is the measured change in ln P. The result of this operation on the oxygenpressure isotherms is shown in figure 4.2. Several important features are immediatelyapparent. The most obvious feature is the jump close to x = 0.6. This jump, whoseposition varies with temperature, is identified with the orthorhombic to tetragonal (0-T) transition, and will be discussed in more detail later. The other striking feature isthat, outside the region of the jump, kT(5x/a,u)7, is independent of T to within theresolution of the experiment. A direct implication of this is that entropic contributionsmust dominate the chemical potential (cf. section 2.1.2). This is an important point forlater considerations, when it is seen that existing lattice gas models, which also predict aT independent kT(ax/att)T, don't fit this data and one is therefore restricted to proposingpurely entropic extensions to the theoryl.A final basic observation, which has practical consequences and theoretical ones1-One should not be inclined to discard the lattice gas models, since they do correctly predict manyother experimental results.x T=450°C• T=475°CA T=500°Co T=550°C+T=600°Co T=650°CChapter 4. Results and Existing Theories^ 860. 0.2 0.4 0.6 0.8 1.0x in YBa2Cu306_,xFigure 4.2: Plot of kT(Ox/MT vs. x in YBa2Cu306-FsChapter 4. Results and Existing Theories^ 87also, is that the width of the O-T transition is less than the size of the Ax incrementsused (which is about 0.025). This is not obvious from figure 4.2 but can be seen infigure 4.7. Looking closely, one sees that there are only one, or at most two, data pointsin the jump. Since one is taking the numerical derivative, one always expects one pointwhich is situated within the jump, even for a discontinuity in (ax/(9,t). Therefore, thefact that so few points are seen within the jump, implies that the transition width inthese samples is small, smaller than the Ax of the experiment. It has been suggested[69]that the lack of agreement between the data of McKinnon et al.[59] and the MonteCarlo calculations of the 2D ASYNNNI model of Rikvold et al.[69] is a result of sampleinhomogeneity and perhaps impurity phases. As will be shown in the next section, theagreement between our data and McKinnon's data is quite good and hence the suggestionthat inhomogeneities are the cause for the disagreement between theory and experimentis very weak. If sample homogeneity were the cause, then one would not expect thatsamples with transition widths of of less than 0.03 in x would give rise to quantitativelydifferent answers for roughly the entire range of x than those predicted by the bare 2DASYNNNI model (cf. section 4.3.2).In summary then, the kT(Ox/O,u)T curves are essentially temperature independent,with a sharp jump at the O-T transition. The fact that the transition width is small,coupled with the observation of temperature independence makes a strong argument thatthese curves are indeed a representation of the intrinsic properties of this material.4.2 Comparison to other workThere have been many measurements of the chemical potential of oxygen through vari-ous means, ranging from standard thermogravimetric analysis (TGA) to electrochemicalcells, to volumetric titration (cf. [30] and references therein). The very large numberChapter 4. Results and Existing Theories^ 88of experimental investigations into the oxygen thermodynamics makes the compilationand comparison of the various works and techniques a daunting task. Fortunately, thishas been done by Voronin et al.[30] in order to generate a self-consistent, but empirical,set of equations describing the thermodynamics of the YBCO system (The comparisonof Voronin's equations to the experimental results of this work will be presented later).Instead of repeating this compilation of results, our data will just be compared to thework of McKinnon et al.[59] and Meuffels et al.[109].The only other existing data on (ax/(9,a)T is by McKinnon et al., made at 650°Cand is an obvious work to compare to. For comparison of the vapour pressure isotherms,the data of Meuffels was chosen, since the measurements were made with a high degree ofaccuracy and with many data points. In addition, the data was acquired as a function ofT at (roughly) constant x, so that a comparison of Meuffels' work to our data representsthe intersection of two virtually perpendicular paths in (x, T) space, making it a verysensitive test of the global reproducibility of oxygen pressure data in YBa2Cu306+x.4.2.1 Oxygen pressure isothermsFigure 4.3 shows a plot of oxygen pressure data for the three different investigations. Thesolid lines are straight line interpolations of our oxygen pressure isotherms, that is, nosmoothing of the data has been made. Plotted on top of these are the interpolated oxygenpressures of Meuffels et al. corresponding to the temperatures used in this work, whichMeuffels was kind enough to provide for this comparison. Finally, at 650°C, the data ofMcKinnon et al. is also plotted. It is immediately apparent that the agreement betweenthese experiments is very good, except for possible small shifts in x for a particularisotherm and perhaps some small discrepancy in the slopes between the different works.This agreement is much better than the data plotted for isotherm measurements from1988 to 1989 which was compiled and presented by Voronin et al.[30], and is perhaps aChapter 4. Results and Existing Theories^ 89representation of the fact that measurement procedures have improved since then.4.2.2 The thermodynamic response function (ax/apt)TThere is no other data on (Oxfatt)T which exists, except that of McKinnon et al., whoin fact were the first ones to measure this. This measurement was conducted at onetemperature, 650°C. Figure 4.4 shows a comparison of McKinnon's data to ours. Wesee that now that the derivative has been taken in order to generate (ax/ait)T and tofocus in on the fine details of the chemical potential, the agreement does not appear asgood. There is quantitative agreement only close to the transition. Above and belowthe O-T transition, in the region where the limits of experimental resolution are beingapproached, the curves depart, McKinnon's data being higher in the tetragonal phaseand lower in the orthorhombic. A critical examination of both experimental setups wouldbe needed to decide which data was a better representation of the intrinsic behaviour.There is agreement, however, on the value of x at the O-T transition, and themagnitude of the jump in (ax/ait)T at the transition. Aside from attempting to analyzethe systematic errors in either experiment, there are two points which one could make:the first point is fairly circumstantial. In our investigation, it is found that the (ax/a,u)Tcurves are essentially T independent. At different temperatures, the value of the pressure,which corresponds to the same value of x can be orders of magnitude different, so thateach curve of (ax/ay)T is made at quite different pressures. The magnitude and typeof correction applied to the data is significantly different in the various pressure regimes.Therefore the fact that we end up with T independent curves is highly unlikely to be dueto improper corrections.The second point is perhaps more solid. This comes directly from the paper ofMcKinnon in which a relationship is proposed between measurements of (axja,u)T andmeasurements of (ax/an. In fact, this proposal is made more solid by the observationChapter 4. Results and Existing Theories^ 901000010001001010.10.010.0 0.2 0.4 0.6 0.8 1.0x in YBa2Cu3064xFigure 4.3: Comparison of the oxygen pressure isotherms between this work (solid lines)and the data of McKinnon et al.[59] and Meuffels et al.[109] The open circles are thedata of Meuffels et al. and the diamonds are the results of McKinnon et al. The solidlines are a straight line interpolation of our isotherms. The temperatures are the sameas in figure 4.1. Except for possible small constant shifts in x, the agreement betweenthe various experiments is very good.0.200.15o This work T=5500COThis work T=6500C• McKinnon et al. T=6500CMcKinnon isobar w. 5H=-0.80e0.000.05Chapter 4. Results and Existing Theories^ 910.0 0.2 0.4 0.6 0.8 1.0x in YBa2Cu306,xFigure 4.4: Comparison of kT(Ox/O,u)T between McKinnon at al.[59] and this work.Chapter 4. Results and Existing Theories^ 92here that kT(Ox/Oit)T is independent of temperature. This T independence implies, asequation 2.7 shows, that the partial enthalpy is independent of the oxygen concentration.Consequently, it is possible to writeSH kT1nF(x) (4.1)where the only x dependence is in F. Using this equation, together with equation A.9which connects the chemical potential to the oxygen pressure, one can show that[59](ö iaX) 1 T 611 — — -TikTT (aa Tx) p (4.2)where hc.,) = 0.0980eV is the vibrational ground state energy of the oxygen molecule,relative to the potential energy minimum given by the dissociation energy. Therefore forthe case where the partial enthalpy is x independent, one has a simple relationship be-tween (Oxfatt)T and (ax/aT)p. McKinnon uses this relationship to generate a (ax/(9,)2,curve from measurements of x vs. T at constant pressure (i.e. isobars). The solid line infigure 4.4 is a cubic spline interpolation of the kT(ax/8it)7, data generated by McKin-non from his isobar measurements. However, McKinnon uses in his paper SH = —0.9eV,which is very close to the commonly accepted value for SH —0.92eV[65, 89, 109, andequation 5.14 in section 5.2.2]. In order to match both data sets, we have plotted theMcKinnon's isobar curve in figure 4.4 using SH = —0.80eV, which is close to, but not infull agreement with, the accepted value for SH. This curve generated from the tempera-ture scan at constant P is strikingly similar our data. The shape of the curve both aboveand below the transition is virtually identical. Thus, as far as theoretical fits to the datais concerned, it will be assumed that our curves represents the intrinsic behaviour of thesystem, although it should be kept in mind that the only other measurement of (ax/ait)Tis slightly different.Chapter 4. Results and Existing Theories^ 934.2.3 Orthorhombic to tetragonal transitionFinally, as can be seen from the (ax/O,u)7, curves, there exists a jump which is associatedwith the O-T transition. Figure 4.5 shows a plot of the O-T transition temperature vs.the oxygen concentration at the transition, xoT, as measured by various works and aspredicted by the 2D ASYNNNI model. The solid circles are the values of xoT obtainedfrom the (ax/ait)2, curves. The lower error bar is the position at the onset of the jumpin (axiatt)2, and the upper error bar is the position of the peak. The short dashedline is the best fit line to the data of Meuffels et al. [109], also obtained from vapourpressure measurements. This line, as mentioned by Meuffels, is in good agreement withthe data of Specht et al. [110] obtained by X-ray diffraction. The long dashed line isfrom Gerdanian et al. [111]. This line, which they plot as a guide to the eye from theirexperimental results, comes from in-situ measurements of the electrical resistivity vs.oxygen pressure and T. It is apparent that all these investigations are quite consistentin their measurement of xoT. There are many more such measurements, (cf. Gerdanianet al. in ref. [111]) and they all lie within the same band shown in figure 4.52• Alsoincluded in the figure, for later discussion, is a solid line representing the most recenttransfer matrix finite size scaling (TMFSS) calculation of the phase diagram of the 2DASYNNNI model. Again, the good agreement of our data to various other works andthe predictions of theory is a convincing argument that the (ax/ap)T curves are reliableenough to merit an attempt to pinpoint which theoretical model can best describe thedata.2With the exception of an early experiment of Jorgensen et al.[112], which finds ZOT values whichare lower by about 0.1. However, this experiment, which was perhaps the first, was conducted when notmuch was known about this system.1000400800600•This work— — Gerdanian (Expt)^ Meuffels (Expt)^ Hilton (TMFSS)/ 1///// _----,1Chapter 4. Results and Existing Theories^ 940.2 0.3 0.4 0.5 0.6 0.7 0.8x in YBa2Cu306_,„Figure 4.5: Plot of the structural phase diagram of YBa2Cu306+1. The solid circles arefrom the position of the jump in the (ax/a/L)T measurements, with the lower error barrepresenting the onset of the jump and the upper bar the peak of (ax/O,u)T. The shortdashed line is the best fit straight line to the data of Meuffels et al. in ref. [109]. Thelong dashed line is the guide to the eye plotted by Gerdanian et al. of ref. [111]. Thesolid line is the phase diagram predicted by the TMFSS calculation of the 2D ASYNNNImodel.Chapter 4. Results and Existing Theories^ 954.2.4 Discussion of the comparisons madeIt is evident that one has a very good control over the absolute oxygen concentration inYBa2Cu306+x. Looking at the differences in x between the data plotted in figure 4.3, onecan say that the absolute uncertainty in x for any carefully prepared sample is certainlybetter than 0.01. For a series of samples made with relative differences in oxygen content,this relative uncertainty is much less. This makes YBa2Cu306+x a very good system touse in the study of high Te superconductivity, even considering the fact that the chargetransfer is influenced by the oxygen content as well as the oxygen ordering (cf. nextparagraph).What is clear from experimental observations, for example the dependence of T, onoxygen ordering[48, 13, 49, 46], and model calculations of the same phenomenon[47, 56], isthat one needs to not only control the total oxygen content, but also the degree of oxygenorder. This problem is emphasized by the example in NMR, where two recent 63CuNMR experiments on YBCO samples with x'-.0.6 gave quantitatively different resultswhich is most likely explained by a difference in hole doping[113, 114]. Such a problemis most severe when an experiment in YBa2Cu306+x is made for only one value of x,and much less if a series of experiments for different x is made. But even here, insidiousproblems can occur. For example, a series of samples prepared with varying x in a TGAapparatus may not all be annealed at the same temperature due to the limitation in thecontrol of the oxygen partial pressure. If samples were to be prepared with low oxygenconcentrations, say 0.05, 0.1, 0.15 etc., the required oxygen partial pressure changes byorders of magnitude very quickly for a given fixed temperature and to be able to controlthe oxygen content with such an apparatus one needs to also play with the annealingtemperature. But, since the degree of oxygen order depends not only on x but also on Tand time, such a technique introduces an unknown and uncontrolled variation in oxygenChapter 4. Results and Existing Theories^ 96order and hence hole doping.It is clear from the experiments showing a time dependence of T, for quenchedsamples stored at room temperature[48, 13, 49, 46], that the degree of oxygen ordervaries even at room temperature, since it is shown that x is not changing. This is aclear indication that the degree of oxygen order influences the charge transfer. It doesnot mean that the quenching temperature is irrelevant for the final state of order, sincethe ordering process at room temperature may only be accomplished by local oxygenrearrangements using short jumps. Rapid quenches from elevated temperatures can lockthe system in a metastable state of lower order, so that allowing the sample to "anneal"at room temperature will not fully remove the degree of disorder caused by quenching.Thus, samples with a different time-temperature profiles but identical oxygen contentcan have varying degrees of hole doping. These complications are only overcome byeither having a complete description of the relationship between oxygen order and holeconcentration or by experimental techniques such as post-annealing the sample sealedin a small quartz tube, so that samples with different oxygen contents retain the sametime-temperature profile. Unfortunately, such post-annealing techniques do not seem tobe common practice and sometimes are not practical or possible.It is the hope that these (Ox/(9,a)T measurements will be able to contribute to theunderstanding of the relation between hole doping, oxygen content, oxygen ordering andtemperature. Although there exist studies, for example of in-situ X-ray absorption edgestudies vs. x[94], which give a relationship between hole count and oxygen content at lowT, and phenomenological models which seem to able to predict 71, vs. x and ordering[47],no complete microscopic theory exists which can, as a whole, predict the thermodynamicsof oxygen and its ordering as well as the resultant charge transfer. In the next section,the existing theories which make predictions about (ax/19,a)T will be be compared to theexperimental results. It will be shown that the 2D ASYNNNI model, which predicts veryChapter 4. Results and Existing Theories^ 97well the structural properties vs. x and T, seems to fail in the prediction of (Ox/Op)T,whereas certain defect chemical models, which make very simplified assumptions aboutthe oxygen ordering, are capable of quantitatively fitting (ax/a,u)T in certain cases.4.3 Fit to existing theoriesIn the analysis of the oxygen thermodynamics in YBa2Cu306+x, there seem to be twocamps. In one, the problem is approached from an interacting 2D lattice gas model forthe oxygen[33] in which the effective interactions represent the effect of the 3D electronicstructure of the system[62]. These effective pair interactions are assumed to be x and Tindependent, as an approximation to the real situation. The other approach is to use thedefect chemical formalism in order to take account of the fact that as one puts oxygeninto the system, additional degrees of freedom exist which contribute to the entropy. Inparticular, formal valences of the neighbouring Cu atoms change and additional holes arecreated (See chapter 2 for a description of these models).4.3.1 Defect chemical modelsThere have been a number of investigations analyzing oxygen pressure data from a defectchemical point of view (see, for example refs [89, 90, 88, 30, 92]). In these models (withthe exception of Voronin [30]), it is assumed that the interaction between oxygen atomsis negligible, but that the change of the Cu valence with oxygen content is taken intoaccount which gives rise to extra terms in the entropy. Naturally, since the effect ofoxygen interactions is neglected, these models do not predict an O-T transition. Theyare thus forced to propose specific models which are different for each phase.Chapter 4. Results and Existing Theories^ 98The model of VoroninThe study of Voronin takes into account the oxygen ordering using the Bragg-Williamsor CVM point approximation. Figure 4.6 shows a plot of kT(5xj0,07, using the modelof Voronin, along with our data.This fit is quite good, considering that the scatter in the source data used to obtainthe parameters is considerably larger than the scatter of the (ax/MT data. In detail,however, the fit is only qualitatively correct. The curvature in the orthorhombic phase isfor the most part incorrect and there is a strong temperature dependence in the tetragonalphase away from the O-T transition. It is also clear that the temperature dependenceof the position of the O-T transition is stronger in the model of Voronin. Althoughthis fit is, to date, the best one can find, it contains several deficiencies. The model isessentially the CVM point approximation in that it splits up the system into varioussublattices and assumes a random entropy of mixing for species on a given sublattice.Interactions between species, written as an excess free energy term, are defined within thisregular solution model and no higher order correlation functions are introduced. Thus,the definition of the equations to an arbitrary range of interactions results in essentiallya power series expansion in the point occupation probability x of the chemical potential,with arbitrary temperature dependent coefficients. For the calculation of (ax/(9,)7, itmeans that one has a fairly arbitrary additional term to the random solution model forthe oxygen occupancy and copper valence, that can in principle result in a good fit. Inaddition, the model does not take into account the existence of Cu2+ and its possibleeffect on the entropy of mixing. Instead, this effect is taken care of by the power seriesexpansion. Although the model has a sound basis, one ends up with a 9 parameter fitto the data and the formalism does not lend itself well to the determination of the holecount in the Cu02 plane.0.200.15x T=450°C0 T=550°C0 T=650°C— Voronin et al.0.00Chapter 4. Results and Existing Theories^ 990.0 0.2 0.4 0.6 0.8 1.0x in YBa2Cu306_,„Figure 4.6: Comparison of kT(Ox/ött)T between experiment and the model of Voroninet al[30]. Note the strong T dependence of kT(ax/a,u)T in the tetragonal phase and thedifferent curvature in the orthorhombic phase for the model of Voronin.Chapter 4. Results and Existing Theories^ 100The basic approach of this model does have some very positive elements. Oneway to look at how the equations are generated is as follows: one has a descriptionfor the oxygen configuration and their interactions, which leads to expressions for theentropy and internal energy of the oxygen system. Then, from the variables defined inthe configurational problem for the oxygen (in this case just the concentration, x, and thefractional site occupancy) one writes down the connection between these variables andthe other degrees of freedom of the system (in this case the copper valence). Then, withthe other degrees of freedom defined, one writes down the entropy and internal energy forthese other degrees of freedom. This approach will be used in the next chapter to proposea model with just two free parameters and a deeper insight into the interplay betweenoxygen configuration and copper valence (and hence the hole count). But first, theelements leading to the proposition will be described. This requires describing the defectchemical model of Verweij and Feiner, which forms the basis for making the connectionbetween oxygen configuration and hole count and showing the results of the 2D latticegas model calculations for (ax/a,u)T, which will be used to generate the description ofthe oxygen ordering.The model of Verweij and FeinerFrom measurements of the partial enthalpy 811 and partial entropy SS, by a measurementof the oxygen pressure vs. T at constant x, Verweij and Feiner have proposed several de-fect chemical models [64, 65, 88]. The measurements were made using a feedback controlsystem to maintain a constant x within the sample, which is placed in a system with afinite dead volume. We will concentrate here on the most recent proposition of Verweijand Feiner. Although the experimental setup did not allow for very accurate measure-ments, the finding was that, to within experimental accuracy, in the orthorhombic phaseChapter 4. Results and Existing Theories^ 101811 is independent of x and T. The conclusion one can draw from this is that the inter-actions between the particles involved in the reaction mechanism are either much higheror much lower than kT. One the one hand, small interactions can be ignored, whereaslarge interactions just give rise to exclusion principles, which will result in a constant811 3 . In addition, if concurrent microscopic reactions take place, then they most likelyinvolve the same enthalpy change, implying that the reactions occur with equal a prioriprobability. Using data on x vs. T at constant pressure in a TGA apparatus, and theimplications mentioned above, Verweij and Feiner proposed several models to fit the data(cf. section 2.4.2). Due to the limited accuracy of the data, Verweij and Feiner had eightremaining candidates able to fit the oxygen pressure curves. Using our kT(Ox/Ott)T dataat 550°C, we were able to refine this choice. Each model was examined and the four bestfits are plotted in figure 4.7.Note that since these models are for the orthorhombic phase only. The notation forthe various fits is defined in chapter 2. As a reminder, however, b(I)F(II) implies thatthe holes are distributed to the planes and chains. The holes in the chains are bound (b)to the copper in the interior of the chain fragments in a singlet state and the holes inthe planes are distributed on planar oxygen sites in a doublet state. f(II)F(II) is similar,except that the holes in the chains are free (0, i.e. distributed over all oxygen in thebasal plane. F(IV) is the case where all the holes are in the planes, distributed over theoxygens with the spin in a quartet state with the two neighbouring Cu2+. Finally, b(I)is the case where the holes are only in the chains, bound to the copper in a singlet state.These fits are qualitatively and quantitatively much better than the curves ofVoronin. They have the correct curvature and fall, with the exception of b(I), with-in the scatter of the experimental data. Unfortunately, three of the best candidates are'This in fact somewhat justifies the use of the CVM point approximation in the model of Voronin inthe orthorhombic phase.0.207--1- 0 15^°•^00 —-ot. 0.10 -0 Data at 550°C^ b(I)-F(II)^-^ f(II)-F(II)F(IV)b(I)0.05 -Chapter 4. Results and Existing Theories^ 1020.250.00 ^0.5 0.6 0.7 0.8 0.9 1.0x in YBa2Cu306_,xFigure 4.7: Comparison of kT(ax/ait)Tbetween the model of Verweij and Feiner andexperiment at 550°C in the orthorhombic phase. Depicted are the four cases of themodel of Verweij which seem to fit the data best. The nomenclature for the variouscurves is defined in section 2.4.2.Chapter 4. Results and Existing Theories^ 103very different in their implications. Some could, in principle, be eliminated through otherinvestigations which could decide, definitively, where the hole resides and how the spincorrelates with neighbouring spins. Before any of these curves are eliminated, however, aserious deficiency should be pointed out in this model. The model of Verweij and Feinerassumes that, in the orthorhombic phase, one sublattice is completely empty (i.e. s = 1).This is definitely not the case over the entire orthorhombic phase, as clearly evidenced byneutron diffraction[112, 5, 115]. The assumption that s --,---- 1 for the entire orthorhombicrange should result in significant errors, particularly as one approaches the transition,since in this region s is changing rapidly. This could qualitatively change the behaviourof the curve. Thus, one should not draw too many conclusions before attempting to takeinto account the effect of the variation of the long range order parameter, which is thesubject of the next chapter.4.3.2 Lattice gas modelsDetails regarding the 2D lattice gas models were presented in chapter 2. Figure 4.8 showsa comparison of kT(Oxia,u)T for the experimental run at 550°C, CVM calculations andthe Monte Carlo results of Rikvold et al.[69]. Clearly, in the orthorhombic phase, theagreement is not even qualitatively correct. In the tetragonal phase, the agreement isbetter, but not better than the model of Voronin. It is obvious that the pure lattice gasmodels, even the very accurate 2D ASYNNNI Monte Carlo results, have a key elementmissing. However, as was discussed in chapter 1, the 2D ASYNNNI model is very suc-cessful in predicting many other experimental results. One might be inclined to say thatthere is something basically wrong with the (ax/a,u)T measurements, but this is unlikelyin light of the reproducibility of the data and the agreement of the chemical potentialdata between many experimental groups.Chapter 4. Results and Existing Theories^ 104 0.600.500.400.300.200.100.00I-_-o-o1='-_0.0 0.2 0.4 0.6 0.8 1.0x in YBa2Cu306_,„Figure 4.8: Comparison of kT(Ox/Op)T between the predictions of the pure lattice gasmodels and experiment at 550°C. The following symbols are used: solid line is the CVMsquare approximation, dotted line is the CVM pair approximation, diamonds are theMonte Carlo results of Rikvold et al. [69] and the circles are the experimental data at550° C.Chapter 4. Results and Existing Theories^ 105So, what is wrong with these lattice gas models? The clue lies in the weak tem-perature dependence of the kT(ax/a,a)T curves, in a given phase. Since these curvesare essentially T independent, except for the position of the O-T transition, one realizesthat the thermodynamics is dominated by the entropy, and not the internal energy (cf.chapter 2). So, proposing modifications of the 2D lattice gas internal energy, say byincluding anisotropic elastic interactions, would not have much hope in improving thefit: large modifications to the internal energy would be required and kT(Ox/O,a)T wouldtend to become T dependent. Instead, one needs to examine the entropy. It cannotbe a problem with the CVM configurational entropy already defined in the lattice gasproblem, since the "exact" results from Monte Carlo are very similar to the CVM square.Instead, one should look at the possibility that there are other degrees of freedom in theproblem which give rise to added terms in the entropy, but do not contain much "internalenergy".Naturally, the obvious degree of freedom left out of the 2D lattice gas model is theresult of hole creation. The success of the model of Verweij in predicting the kT(ax/a/L)Tdata is because it makes a fairly reasonable approximation for what the oxygens are doingand includes a model for the configurational entropy of the holes which is consistent withother experimental observations (cf. section 2.4.2 and section 4.3.1). Thus a possiblesolution is to use the basic approach of Verweij and Feiner, but to use an interactinglattice gas model for the oxygen and define the hole and spin entropy in terms of theoxygen configuration. These additional terms in the free energy could, in principle,alter the shape of the phase diagram and the behaviour of the order parameters, sothat there is a danger of losing the agreement between the lattice gas model and thestructural data (i.e. the phase diagram and stable ground states). But, if the expressionfor the number of holes is not too strongly dependent on the order parameters, then thestructural modifications should not be too severe. Since the number of holes depends onChapter 4. Results and Existing Theories^ 106short range oxygen configurations only (cf section 2.4.2, or refs. [47, 56]), it is natural touse the CVM approximation for this problem. In the description of the valence pictureaccording to Verweij et al.[64] and Tolentino et al.[94], described in section 2.4, the totalmobile hole count is determined by the nearest neighbour configuration of oxygen. Thesolution of the CVM equations gives the required short range order parameters neededto determine the hole count. In the next chapter, this extended CVM model will bepresented.Chapter 5The Extended CVM model for YBa2Cu306+xIn the previous chapter, it was shown that the 2D ASYNNNI model is not capable offitting the dependence of the chemical potential on x and T. However, a simple modelfor the orthorhombic phase making a reasonable assumption for the oxygen order andincluding additional entropic terms due to the hole creation was very successful. It wasconcluded that one cannot ignore the fact that electronic defects are created upon oxygenaddition. On the other hand, the success of the 2D lattice gas models in predicting manyof the structural phenomena indicates that one should not simply discard them. Instead,one should try to merge the two descriptions. The crucial piece of evidence from the(ax10,)T data which motivates our particular attempt is the result that kT(Ox/Oit)Tis essentially independent of temperature, except in the O-T transition region. This iswhy Verweij's model works. Thus, the system appears to be dominated by the entropyand the internal energy plays a minor role at these temperatures. However, the modelof Verweij and Feiner makes a crude assumption for the configurational entropy of theoxygens, in that they are assumed to be randomly placed on the a sublattice. In thelimit of high x, this is not so bad. But, closer to the O-T transition and in the tetragonalphase, a random solution model cannot work.One is therefore forced to retain a more complete description of the oxygen sub-system. In this chapter, the configurational entropy and spin entropy of the holes willbe added to the lattice gas description for the free energy of oxygen. This is not totallytrivial, since the number of holes depends upon the oxygen configuration. One needs107Chapter 5. The Extended CVM model for YBa2Cu3 06+x^ 108to postulate a connection between the oxygen configuration and the hole count which isconsistent with experimental observations and current theoretical ideas.In principle, it is necessary to minimize the total free energy of the two systems. Forexample, one minimizes the grand potential given by the sum of the oxygen configura-tional grand potential and the grand potential of the electronic system. The two systemsare connected, however, and it is necessary to solve the equations self-consistently. Inthe following, the basic approach is to write down the grand potential for the oxygenthermodynamics and propose a relationship between the oxygen configurations and thetotal electron hole count. This then restricts one to minimize the electronic free energyin the canonical ensemble given the total hole count found from the oxygen sub-system.Proposing an electronic free energy can be complex, since the material has many atomsper unit cell with charge transfer and a metal-insulator transition. But, in principle,given the total hole count, one can minimize the electronic free energy. We will use avery simple approximation, assuming that the charge carriers move in a narrow band andseparating the charge and spin degrees of freedom. This allows one to directly insert thesolution for the electronic free energy into the grand potential.5.1 Connection between electronic defects and cluster configurationsBefore presenting the extension to the lattice gas model, it is useful to describe qualita-tively the mechanism of electron hole creation. Some elements discussed here were brieflymentioned in section 2.4.2. Here we go into some detail concerning the justification ofthe approach used.1. At x = 0, the basal plane copper Cu(1) are two fold coordinated, bonding withthe 2 apical oxygen sites above and below. The formal valence is Cul+ and isdesignated as being monovalent.Chapter 5. The Extended CVM model for YBa2Cu306+.^ 1092. Adding one isolated oxygen to the basal plane will place it between two Cu(1) andit will bond to these two. The Cu(1) is oxidized and acts as a charge reservoirgiving one 3d electron to the oxygen'. The formal valence becomes Cu2+ for thetwo neighbouring coppers of the isolated oxygen. In essence, one is creating twoelectron holes, one on each copper; however, these are completely localized, so thatthey do not play a role in the configurational problem.3. Filling up the lattice this way works until the probability to put two oxygens oneither end of a copper becomes appreciable; one has formed a chain of length 2.The central copper is now four-fold coordinated (cf. figures 2.5 and 5.1 to get apicture of this). At this point there are in principle several options, which will beelaborated upon in the next section. We will see that the only case which seemsconsistent with experiment is where the extra hole resides on the oxygen.Whatever the case may be, it is reasonable to assume that the nature of the hole cre-ated by adding a second oxygen next to a copper is different than the first. We willdesignate this as a real hole. It is a relevant hole since it is believed that it is the activeelement of the electronic subsystem, giving rise to the metal-insulator transition and tosuperconductivity.JustificationOne can make some very clear arguments in support of such a picture, and also determinethe nature of the extra hole. These are based upon the estimates for the Hubbard modelparameters in YBa2Cu306+x[36]. In fact, calculations made for the Cu02 planes use smallCu-0 clusters which are geometrically identical to a small chain site Cu-0 cluster. Such'This is a very ionic viewpoint for the bonding. In fact, the bonding is covalent and the electronspends some time on each of the atom species. It is sometimes said that the electron sits somewherein the bonds. Nevertheless, the ionic description is commonly used (see the next paragraph on thejustification of this picture). A. Sleight gives a good expose for this issue in the cuprates in ref [116].Chapter 5. The Extended CVM model for YBa2Cu306+T^ 110cluster calculations are believed to be a good approximation to the Hubbard parametersfor the entire Cu02 plane. Thus, they should be also applicable to small clusters in thechains. At x 0, the Cu02 plane band is half filled. The energy to put these holes onthe copper is substantially smaller than to put them on the oxygen. One has placed onehole on each copper oxidizing it to a Cu2+. These holes are localized due to the largecopper on-site coulomb repulsion Ud, giving an antiferromagnetic insulator. Similarly,one can say that adding an isolated oxygen to the chains will also place two holes on thecopper, creating 2 Cu2+ and that these two holes are completely localized.In the Cu02 plane, adding more holes beyond half filling will "place" the extra holeson the oxygen, since the on-site coulomb repulsion is larger than the difference betweenthe oxygen and copper site energy2, Ep — Ed. These extra holes have a finite hoppingprobability, giving rise to conduction. Similarly, by placing two oxygens in the chainswith a copper in between, one will create four holes. The first three may be placed onthe coppers to create three Cu2+. The fourth hole will have a large probability to be onthe oxygen, and can hop.An important approximation that will be made, following the model of Verweij andFeiner, is that conduction takes place in a narrow band allowing one to use a purelyconfigurational entropy for the electron holes. This approximation should be justifiedsomewhat before continuing. The hopping integral t, gives the band-width W for thecharge carriers, but is model dependent and influenced by the degree of hybridization.In the extreme case of the t — J model, where no double occupancy of the copper site isallowed, the hopping probability is given by the oxygen-oxygen hopping term tpp. This2Eventhough the hole is largely on the oxygen, there is a finite probability to find it on the copper,so that we write "place" in quotation marks. The hole has largely oxygen 2p character, but there issignificant hybridization.Chapter 5. The Extended CVM model for YBa2 013 06-Fs^ 111is estimated to be between 0.4eV[117] and 0.65eV[36]. This does not quite put one inthe narrow-band limit at the temperatures of the experiment, since T is of the order of0.09eV. But, the temperature needs only to be within a factor of four of the band widthbefore a configurational entropy is valid (i.e. T> W/4)[118].An experimental test to check if one is in the narrow band limit would be to seea temperature independent thermoelectric power[119, 118]. Early thermoelectric power(TEP) experiments saw a temperature independent TEP above room temperature[120,121, 122, 123]. But a very recent TEP measurement by Cohn et al.[118], on an untwinnedsingle crystal of YBa2Cu306+x, distinguished between chain and plane TEP. They foundthat the TEP along the chains was clearly in the narrow band limit, but not the planarTEP. The measurements were made up to 325K, so that it is possible to still see satu-ration at high temperature for the planar TEP. Thus, the assumption of a narrow bandlimit for charge carrier conduction (i.e. the use of a pure particle picture for the holeentropy) is possibly an extreme point of view. Nevertheless, it is perhaps a reasonablefirst step, since one is probably not very far from that limit'To summarize, holes are first placed on the copper, but Cu(3d) holes are localized,creating Cu2+. Extra holes have a large probability to be on the oxygen and may hop.However, a pure ionic picture for the extra holes might not be appropriate so that in thefurther analysis, we will allow for all possibilities for the behaviour of the extra holes, as inthe model of Verweij and Feiner. The results for opposing cases compared to experimentmay give insight into possible improvements.'Also, one should realize that the interpretation of thermoelectric power data is often quite compli-cated due to numerous effects (see the discussion in ref. [118] or [123]), and that the t — J model is reallyjust an approximation to the real situation. These considerations act as a guide but are not definitive.Chapter 5. The Extended CVM model for YBa2Cu306+x^ 1125.1.1 Counting the holesUsing the above picture, one can write down a connection between the (relevant) holecount and the oxygen configuration. We will use the framework of the CVM approxi-mation, which provides the appropriate short range cluster probabilities. In particular,the CVM square cluster will be used, since it is the four nearest neighbour oxygen sitessurrounding the copper which determine its valence and therefore the hole count4. Re-ferring to section 2.3.1, and to the notation for the basic square cluster, the number ofCul+ per unit cell is simply given by the probability to have the four nn oxygens to thecentral copper empty, i.e. {i, j, k, 1} = {0,0 , 0, 0} (cf. figure 5.1):[Cul+] = z0000 (5.1)According to the description for the Cu valence, the formal valence of the basal planecopper is either Cul+ or Cu2+. It should be emphasized that the number of Cu2+ in thebasal plane simply reflects the number of electron holes completely localized to the Cu-0basal plane bond. Thus, with this understanding, we have[Cu} -- 1 — [Cu l ] = 1 — z0000 (5.2)Using the condition of charge neutrality (equation 2.32) [Cu] + [hole] = 2x, we obtainan expression for the number of extra holes per unit cell (i.e. those not localized to thecopper).[hole] = 2x + z0000 — 1 (5.3)A significant approximation is made in equations 5.1-5.3. When writing the proba-bility for a square, the central copper is in fact ignored. In figure 2.1, we have drawn twosquares. One contains a copper and the other does not. In the formulation of the CVM4Note that, so far, nothing has been said about charge transfer to the Cu02 planes. This is a morecomplicated problem.Chapter 5. The Extended CVM model for YBa2 C113 06-Ex^ 113+ holeFigure 5.1: Valence of Cu(1) for various nn oxygen configurations. The cluster shownis the one used for the CVM square approximation. The first cluster corresponds toi + j + k + 1 = 0, i.e. z0000 and gives the number of Cul+. The middle shows howi + j + k + 1 = 1 gives one Cu2+. The cluster on the right shows the creation of a holeby the addition of a second oxygen to the cluster. Note: there are other configurationswhich can give rise to a hole (for example i = j = 1 and k = 1 = 0), but these arestrongly supressed since they involve nn oxygen occupancies.Holedistributionn(available sitesper unit cell)b [hole]B 2f xF 4b-B [hole] + 2b-F [hole] + 4f-B x + 2f-F x + 4Table 5.1: Number of sites per unit cell available for distribution of electron holes, n, forthe different possible assumptions of the Verweij model. [hole] is the number of holes perunit cell, x is the oxygen content of YBa2Cu3 06+x. The notation of Verweij is used tospecify the different cases (cf. section 2.4.2).Chapter 5. The Extended CVM model for YBa2Cu306-1-x^ 114square, this distinction is ignored and the cluster probabilities for the square with andwithout a copper are the same. The CVM square is a nearest-neighbour interaction mod-el. In a 2D ASYNNNI model, there would be a distinction between these two squares,since the one with the copper contains two Vcii bonds and the one without contains twoVv bonds. However, in order to distinguish these two clusters, one needs to go to theCVM 4+5 point approximation, resulting in 25 independent cluster probabilities for theortho-I phase. Since (Ox/a,a)T for the CVM square is not much different than for the2D ASYNNNI, it is hoped that one is at high enough temperature that ignoring thisdistinction is not too serious.In general, one should distinguish chain site holes from holes on the Cu02 planes.Depending on the specifics of the charge transfer, it might be necessary to distinguishthe entropy of the chain site holes from the planar hole entropy. However as a firstapproximation as it was discussed at the start of this section, it seems reasonable touse the idea of Verweij and Feiner, which is to randomly distribute the holes over allavailable hole sites. Denoting M as the number of unit cells, if there are n sites availableper unit cell for the placement of the holes, then after applying Stirling's approximation(ln N! N1nN — N when N oo) one has for the hole entropy:[hole]Shol„ = knM lnn — [hole]) + k[hole]M ln ^)n[hole] )(5.4)Table 5.1 lists n, the number of sites available for the placement of holes per unit cell forthe various options of the Verweij and Feiner model.5.1.2 Counting the spinsEach free electron spin in the system contributes 2 states. Thus, if the number of freespins in the system changes, it should be taken into account in the expression for thefree energy; there is an extra entropic contribution due to the number of spin states.Chapter 5. The Extended CVM model for YBa2Cu306-kx^ 115We will now describe how one can count the number of spin states as a function ofthe square cluster probability. At x = 0, the Cu(1) is non-magnetic with a filled 3d."shell. The Cu(2) (planar coppers), however, are Cu2+, with an unpaired electron spin.At TN = 415K[35], the spins order anti-ferromagnetically. Above TN the spins arestill correlated, but the correlation length decreases with increasing temperature. Theisotropic Cu-Cu super-exchange interaction J is estimated to be Lsz 1500K[35, 124]. Thus,it is expected that at the temperatures of the vapour pressure experiment (723 — 923K)there is enough thermal energy to consider the planar Cu2+ spins to be uncorrelated.We have therefore two free copper spins per unit cell at x = 0. When isolated oxygenis added, one converts two monovalent coppers to a divalent state in the chains (Cul+becomes Cu2+). It is probably reasonable to assume that the copper-copper exchangeinteraction in the chains is of the same order as the planar Cu2+, so that these spins canalso be considered uncorrelated and free.What happens when a hole is created? In section 2.4.2, the various possibilitieswere presented. The options depend on whether the hole can be considered to be on thecopper or on the oxygen. First, let us examine a hole on oxygen.Hole on oxygenIf the hole is on the oxygen, one has three different possibilities for the spin behaviour.To see this, consider that the oxygen is neighboured by 2 Cu2+ (this is true for the chainsand planes). Since the spin of the Cu2+ is free, these two coppers have a total of 4 states:{-r•T, T•1, , I.11, where the "." is an oxygen. Adding a hole to the oxygen puts a spinbetween the two Cu2+. The following is then possible:Chapter 5. The Extended CVM model for YBa2C11306-Fx^ 116Hole spin is free: 4 states -4 8 states,forms a doublet: 4 states -> 2 states,forms a quartet: 4 states --> 4 states.where doublet and quartet specifiy the possible ground states for the three spins. Thespecification that the spins form a doublet or a quartet is saying that the excited statesare thermally inaccessible, and the number of states is given by the degeneracy of theground state."Hole on copper"If the hole is "on the copper", again one has three possibilities. Before adding a hole, theCu2+ is free and has two states: ft ,11. Adding a hole, one has the following options:^Hole spin is free: 2 states^4 states,forms a singlet: 2 states^1 state,forms a triplet: 2 states ---> 3 states.Tithe hole corresponds to the physical removal of an electron from the Cu(3d)orbital, then the singlet state would be the only physically reasonable case. However, itis also possible to view a hole with mainly oxygen 2p character to be bound to a coppersite. For example, one way to view the Zhang-Rice singlet [125] is to say that the relevantentity which hops from site to site is the singlet (quasi-particle). So that, although thehole is physically an oxygen 2p hole, the relevant number of sites available for hopping isdetermined by the copper. For the counting of the number of spin states, it is possible toimagine that elevated temperatures will make spin configurations other than the singletstate accessible.The degeneracy factor g (modification of the number of spin states per copper) issummarized (using the notation of Verweij) in table 5.2. The spin entropy is thus givenpossible spin pairingdegeneracy factor gapplicable toI^II^III IV1^1^3^12^2^2b,B f,F b,B f,F122allChapter 5. The Extended CVM model for YBa2 CU3 06+x^ 117Table 5.2: Spin degeneracy factor for the various cases of the Verweij model. meansthat the hole spin is free and I, II, III, IV means that the hole spin forms a singlet,doublet, triplet and quartet, respectively, with nearest copper spins. The bottom rowindicates in which specific case the pairing is possible.by (M is the number of unit cells):Sspin = k (2 + — [hole])M ln 2 + k[hole]M ln(2g) (5.5)where the first term corresponds to the number of unmodified and uncorrelated divalentcopper spins and the second to the number of spin states of a copper-hole pair. Usingthe charge balance condition, [Cu] + [hole] 2x, gives,Sspin, = k(1 x)M ln 4 — k[hole]M ln 2(97) (5.6)We see that the second term vanishes for g = 2, which is the case for free hole spins (i.e.(D). For the other cases, the sign is negative, but the importance of the second termdepends on the type of pairing. It is very important to note that this equation is validonly when the type of pairing in the chains is the same as the planes. This is probably areasonable assumption since NMR does not detect different spin susceptibilities for thedifferent sites[126]: YBa2Cu306+x is considered to contain a single spin fluid.5.1.3 CVM free energy with hole and spin degrees of freedomAs mentioned at the start of the chapter, one needs to simultaneously minimize the twocoupled free energies for the oxygens and the electrons. In the above it was assumed thatthe solution to the electronic problem is given by a random entropy of mixing with aChapter 5. The Extended CVM model for YBa2 C113 061-x^ 118negligible internal energy compared to the temperature. Thus, the electronic free energyproblem is automatically solved and can be added directly to the configurational freeenergy of the oxygen'. In this approximation, the effect of the holes and spins can beviewed as additional x and T dependent effective interactions for the oxygen system.This is made clear when the hole and spin entropy are written explicitly as functions ofthe square cluster configuration probabilities zijki.We see that according to equation 5.3, the entropy of the holes and spins can bewritten as a function of the oxygen cluster configurations. Formally, one can rewriteequation 5.3 using the notation of section 2.3.1:[hole] =^(2aijk/^Cijkl — 1)Zijk1 = E hijklZijkl^ (5.7)i,j,k, i,j,k,1whereCijkl =aijki =1 1 if(i+j+k+/), 00 otherwise1—2 (i+j+k4-1)(5.8)(5.9)We see that cijk/ tells us if we have a Cu l+ and aiiki how many oxygens there are in thecluster. With the equation for the hole written in such a form, one can immediately seethe effect of Sspin on the CVM free energy. The expression 5.6 for Sspin becomes:Ss inP^—0111 (2)E ci3kiziJki + kin g E a/z„ki + ki ln (2)^(5.10)No 2^gadditive constant:E [Eipa + kT-1 in (2g)ciikil zijkiT S square^T Shole 2N0kT(21ng)] E aijklZijkl(5.11)2i,j,k,1^ i,j,k,1 ^gAdding the the entropy of the spins and holes to equation 2.26 gives, to within an'In a less naive picture, one would have to re-solve the electronic free energy for every configurationof oxygen proposed.Chapter 5. The Extended CVM model for YBa2Cu3 06-Es^ 119where Ssqua„ and S hole are given by equations 2.20 and 5.4. The spin entropy gives anadded temperature dependent term for the energy Ei3k1 of a cluster. In fact, it makesempty clusters energetically unfavorable compared to clusters containing oxygen atoms.It also adds a constant to the chemical potential term, which is irrelevant for the freeenergy minimization. The effect of the addition of the hole entropy is not so easily seen,but it also essentially adds concentration and T dependent interactions, whose sign maychange under certain circumstances. Using this formalism, one can again use the NaturalIteration method to minimize the free energy (cf. Appendix B). The solution of 5.11using the NI method gives the solution of zj3k/ vs. it and T. From this, one can obtainthe oxygen concentration x = x(it,T), the long range order parameter s (it, T) andshort range order parameters, such as the nn occupation probability. In the following weuse for the cluster energy:eijkl == 141n (i k)(j +1)/4 (5.12)where Yin, is the nn repulsive interaction defined in section 2.2 (In particular, cf. fig-ure 2.1). In the free energy minimization, one factors out kT and thus the problem isdefined in terms of a rescaled temperature kT/Vnn and chemical potential it/kT. Notethat we have not included a single particle site energy. The site energy can be absorbedinto the chemical potential since it will have the form € aijkizi3kr. In calculatingkT(Oxfatt)T, a site energy term and the second spin term will drop out. Thus, as faras a comparison of kT(Ox/(9,02, to experiment is concerned, a detailed knowledge of thesite energy is not needed. For comparisons to other data, such as it vs. T, the site energywill have to be included, and can, in general, depend on temperature'. Thus, the 'it' inequation 5.11 is not directly the chemical potential measured in experiment, but containsadded terms which need to be subtracted if a direct comparison to experimental data is6For example, c may change due to thermal expansion.Chapter 5. The Extended CVM model for YBa2Cu306+s^ 120to be done (with the exception of (Ox/ap)T). This will become more clear later.5.2 Results and comparisons to experiment5.2.1 Comparison to kT(Ox/ap)T dataFor each possibility of the spin and hole behaviour described in section 5.1, kT(Ox/ait)Twas calculated and plotted against the experimental result at 550°C. The curve at 550°Cwas chosen since it has the lowest noise and spans a reasonable range of x. A commontemperature of kT = 0.3Vni, was used for all of the calculations, since 14,7, is estimatedto be 2800K[14], which would put T roughly at 550°C. kT(ax/Oit)T was calculated bytaking the numerical derivative of the resulting x vs. it curve. Note that, as mentionedin the previous section, any T dependent site energy terms do not affect the kT(ax/ap)Tcurves. The plots have been separated according to the particular hole distribution caseslisted in table 5.1. In figure 5.2, the various cases for holes distributed both in the chainsand planes are shown.It is clear that the only curves which can fit the data reasonably well are thosewith a spin degeneracy factor of These cases correspond to the physical picture wherethe spin of the hole has a strong anti-ferromagnetic coupling to the copper (either thesingle copper for hole on copper or the two neighbouring coppers for hole on oxygen). Ifthese models are physically reasonable, then it implies that the spin of the hole wouldlike to form a singlet state with the spin on the copper site. This is in agreement withmost current theoretical and experimental viewpoints concerning the spin behaviour,which put the hole mainly on oxygen with a strong anti-ferromagnetic coupling to theneighbouring divalent copper spins (Zhang-Rice singlet[125]), although in this situationthe temperature is much higher than normally considered.Figure 5.3 shows the results for the cases where the holes are restricted to lie0. 5. The Extended CVM model for YBa2 C113 064-s^ 121••• - "• .I/^\-^/-^/ \II' - . • . ' • ••‘/^......!^•%.^•-^/ /• %^•."-'■^_-^ \.... /^..^ ....^.% -% • %/,, ..•^_I/^#f^'..• /I^le^"AIt\ % '..-^1;^otetter^\ • -^#6tertue^, %A^-r^e^s,^...),.^0414k^' •;f(II)F(II)^';'.,•i 6) 411 ^f(IV)F(IV)b(I)B(I)^- ^b((II)B(III)^1--^f(1/2)F(1/2) --^b(112)B(112)if1^I^f^i I^I^Ii^1^1^ij-^i^1^i^I....^..,_^. .r .r^\r ■_^. ■. ■.^■-— -.-- i^.^-.^.. .// 4.^% . \.4^\/cereal,^\ - /^teltge^.r \ ■^-ek.%//^I / 64 \% i^•• , \-^11. ii — /^• Al \^-b(I)F(II)^%- - - - f(II)B(I)--^b(1/2)F(1/2) --^f(1/2)B(1/2).1,^.1_,^.1.0.2 0.5 0.8^0.2 0.5 0.8x in YBa2Cu306+xFigure 5.2: Comparison of kT(ax/aft)T for the extended CVM models with chain andplane hole distribution and the data at 550°C. The top left graph is for holes on oxygen,top right is for holes on copper, bottom left for holes on copper in the chains and holeson oxygen in the planes, and bottom right for holes on oxygen in the chains and holes oncopper in the planes. Note that the only cases which can fit the data reasonably well arethose with a spin degeneracy factor of (i.e. f(II)F(II), b(I)B(I), b(I)F(II) and f(II)B(I),which all assume spin singlet for hole on copper and spin doublet for hole on oxygen)..............^et-re•olossigeB(1)B(III)- F(II)- F(IV)^ B(1/2)^ F(1/2)0.30.2b(I)^ b(III)- - 411)- f(IV)^ b(1/2)^ f(1/2)0.0 ^0.0^0.2^0.4^0.6^0.8^1.0x in YBa2Cu306+x0.20.1Chapter 5. The Extended CVM model for YBa2Cu306+s^ 122Figure 5.3: Comparison of kT(Ox/a,u)T for the extended CVM models with a restrictedhole distribution and the data at 550°C. The top graph corresponds to cases where thehole is only in the planes, whereas the bottom assumes only chain holes to be present.Holes bound to the copper in the chains, in particular b(III) and b(1/2), work quitewell in the tetragonal phase, but have the same problems as the standard CVM squareapproximation in the orthorhombic phase. Some plane-only cases seem to work well inthe orthorhombic phase. Note that b(1/2) is the same as the standard CVM square.Chapter 5. The Extended CVM model for YBa2 CU3 06-Es^ 123either in the planes or the chains. Chain-only cases (bottom graph) have the samequalitative shape as the standard CVM square approximation (In fact, b(1/2)=-CVMsquare). Although one might imagine a hybrid model where one uses a chain-only casefor the tetragonal phase and a combined one for the orthorhombic, it is difficult toformulate such a scenario which allows a determination of the O-T transition in a self-consistent manner. However, a rough approach is outlined in section 5.3.1, which wouldessentially result in a chain-only model for the tetragonal phase. For now, we will notconsider these chain-only models further.Some of the plane-only models (top graph) seem to fit fairly well in the orthorhombicphase and seem just as good as the combined chain-and-plane cases in the tetragonalphase. Overall, B(1/2) seems the best among the plane-only cases. Note that all of theplane-only models have quite a small jump at the O-T transition; smaller than all of thecombined models. We will retain B(1/2) as a representative example of the plane-onlymodels for the next figure.Figure 5.4 shows a plot of the best fit cases. For comparison, the standard CVMsquare model is also plotted. One should recall that the CVM square is very close to thepredictions of the full 2D ASYNNNI model, and so, should be regarded as representingthe 2D ASYNNNI as well. It is clear that all of these cases are clear improvements tothe original CVM model in the orthorhombic phase. Although quantitatively, the CVMsquare seems better in the tetragonal phase, one could argue that the extended modelsare qualitatively correct in that they have a maximum in the tetragonal phase. Overall,certainly the extended models seem to do a much better job than the standard latticegas model.In fact, while all of the possible cases for the hole dynamics were presented forcompleteness sake, most are physically unreasonable. Should one particular case haveB(1/2)f(II)F(II)b(I)B(I)b(I)F(II)f(II)B(I)Standard CVM000000.30.0Chapter 5. The Extended CVM model for YBa2Cu3 06+x^ 1240.0 0.2 0.4 0.6 0.8 1.0x in YBa2Cu306,xFigure 5.4: Plot of kT(axfatt)T for the best fit cases for the electron hole distribution.Included is also the standard CVM square approximation for comparison. The tempera-tures for each case has been chosen in order to have the O-T transition coinciding with theexperiment. The open circles is the data at 550°C. For the CVM square, the minimumxoT is roughly 0.6, so that a fit of the O-T transition for this case is not possible.Chapter 5. The Extended CVM model for YBa2 013 06-Fx^ 125resulted in a spectacular fit, then one could have tried to re-examine its possible validity.But, as it turned out, none of these fits are really outstanding. Therefore, when comparingto other experimental results, it is useful to decide which specific case should be used inthe further analysis. Of the cases plotted in figure 5.4, none Scan really be judged muchbetter than others simply by looking at the graph. One needs to resort to other physicalmeasurements to be able to decide.Two very recent X-ray absorption spectroscopy (XAS) measurements clearly showthat there are holes in the chains as well as the planes[95, 94], so that the remainingplane-only case should be discarded. In particular, angle-resolved XAS on an untwinnedYBa2Cu307 crystal clearly shows twice the chain hole occupancy that a random holedistribution model would predict[95]. Thus, the B(1/2) case is clearly unreasonable,since raising the temperature could decrease the chain site hole occupation, but notremove it. This XAS experiment also shows that the random occupation model used forall cases is really only a first approximation'. From the in-situ and ex-situ XAS studyof Tolentino et al.[94], it is clear that at low values of x the holes are localized to thechains, so that B(1/2) becomes even more unreasonable.There is ample evidence that the holes have oxygen 2p character (cf. section 5.1).But it is possible that for the particle picture of the holes, the oxygen 2p hole spendsa significant amount of time close to the copper, so that it can be considered to be a"hole on copper". It is not clear what choice for the hole (b or f) dynamics, withinthe approximation of a random solution model for these holes, is a better representationof the real situation. The tendency of the author would be to say that the "hole onoxygen" viewpoint (i.e. f models) seems more correct, since the hole is seen to havemainly oxygen 2p character. Thus, we choose the f(II)F(II) case for further comparisons7Both these recent XAS experiments by Krol et al.[95] and Tolentino et al.[94] also see a significanthole occupation of the interstitial 0(4) oxygen site at x = 1. Tolentino, however, sees this occupationvanish at x = 0.8, so that the 0(4) site was not included as a candidate hole site in this work.Chapter 5. The Extended CVM model for YBa2 CU3 06+x^ 126to other experimental investigations, but with the understanding that, in principle, amore refined model for the hole dynamics should be developed before making a finalchoice for the most accurate representation of the interplay between oxygen ordering andhole creation.5.2.2 Comparison to the oxygen chemical potential: determination of thesite energyA plot of the f(II)F(II) model prediction for the oxygen pressure requires a knowledgefor the single site binding energy for an oxygen atom in the basal plane. There are manyeffects which can make the site energy temperature dependent. At this stage it is perhapsnot useful to make an estimation for the site energy, but to just assume a temperaturedependence and fit the model to the experiment. The effective chemical potential ,ti ofequation 5.11 can be written as1= —2 (Po2 D) c (5.13)where E is the site energy and D = 5.08eV[127] is the dissociation energy of an oxygenmolecule. ,a02 is the chemical potential of molecular oxygen given by equation A.9.In order to obtain a true temperature scale from the theory, it is necessary to use adefinite value for It is possible to conduct a fit of the theory to experiment using V„„and € as fitting parameters, but the CVM square approximation, which forms the basisfor this model, does poorly in the prediction of the O-T transition. Instead, the value ofHilton et al.[14] will be used: Yin = 2800K, which is what a best fit of the 2D ASYNNNIto O-T transition data gives. Since the model fits best in the orthorhombic phase, thetheoretical and experimental chemical potentials vs. x in the orthorhombic phase aresubtracted to give c + D/2 as a function of each temperature measured. Figure 5.5 showsa plot of E-F D/2 at the various temperatures, deduced from the difference between theoryChapter 5. The Extended CVM model for YBa2Cu3 06-Ex^ 127-0.88700-0.76-0.785-'^-0.80a)a^-0.82+(.0-0.84-0.86750 800 850 900 950Temperature (K)Figure 5.5: Plot of E + D/2 vs. T determined by comparing the f(II)F(II) model toexperiment. The open circles are determined from the average difference (in the or-thorhombic phase) between experimental isotherms and the f(II)F(II) model assuming a14,,, = 2800K.Chapter 5. The Extended CVM model for YBa2Cu306+.^ 128and experiment.1c + —2D = —(0.820 + 0.004)eV (5.14)We see that c + D/2, deduced from the fit of the f(II)F(II) model to experiment, isindependent of temperature to within the resolution of the experiment. This agreesfairly well with the constant term deduced by Shaked et al.[128] in a fit to in-situ neutrondiffraction data using the CVM pair approximation with an empirical x and T dependentsite energy term (Shaked finds the constant term to be -0.818eV). In the f(II)F(II) model,however, there is no need to introduce a T and x dependent site energy to fit the data.In a sense, we have explained the origin of the postulated x and T dependence of the siteenergy which has been introduced in many works proposing models for the the oxygenthermodynamics [129, 130, 131, 132, 128, 133]. Also note that since we have chosenV. = 2800K, and the temperatures experienced in the experiment are T = 723 ... 923,the number of nn sites occupied is very small (< 1.5%). Therefore, the internal energyis essentially given by € + D/2, since the contribution due to nn populations is negligible(including the nn occupancies in the internal energy calculation is roughly a 1% correctionat 650° Cr . Thus, the heat of solution is essentially given by SH = E+ D/2, which shouldagree with the heat of solution determined by other methods (eg. calorimetrically).We should stress that the fit to the chemical potential was made in the orthorhombicphase, so that equation 5.14 is really only valid in the orthorhombic phase. In the opinionof the author, the model does not fit well enough in the tetragonal phase to merit anattempt to extract the heat of solution for the tetragonal phase. Examining the literature,one finds varying values for the heat of solution. Taking a rough average[65, 89, 134] gives8 H'-.-' —0.94eV. We note in particular Verweij et al.[65], who find SH independent of8If Vn,-, = 2800K, then even at the highest temperature of the experiment (=923K) the effect ofthe nn interaction essentially gives one a nn exclusion principle, and the specific value of V„,, is fairlyirrelevant.-13-14-15-16-17-18-19-200.0 0.2 0.4 0.6 0.8 1.0--------__-Chapter 5. The Extended CVM model for YBa2Cu306-1-s^ 129x in YBa2Cu306.fxFigure 5.6: Plot of the experimentally determined chemical potential isotherms and thepredictions of the f(II)F(II) model. Since the site energy was determined by looking atthe orthorhombic phase, the fit seems much better in that phase. Temperatures usedare: 0 650°C, + 600°C, o 550°C, A 500°C and x 450°C.Chapter 5. The Extended CVM model for YBa2 C113 06-1-r^ 130x and T in the orthorhombic phase. The average value found in the literature does notagree completely with 5.14, but the disagreement is not spectacular. Finally, one remarksthat the value given in 5.14 is very close to the value of 811 used to fit McKinnon's isobarmeasurement to the (ax/a,a)T data (cf. figure 4.4).Now, essentially, the model is complete. One has specified the nn interactionthe site energy given by equation 5.14 and the specific model f(II)F(II), which is solvedby minimizing equation 5.11 as a function of 1a02 and T. Figure 5.6 plots p,02/kT vs. xand compares to the experiment. We see that the fit is very good in the orthorhombicphase, but less so in the tetragonal phase, which is as expected. At 450°C, where amaximum of orthorhombic phase is seen, the fit is excellent over the entire orthorhombicrange. For reference sake we make a final plot in figure 5.7 of kT(Oxia,u)T for the choiceof parameters used and the corresponding temperatures of the experiment.5.2.3 Comparison to the fractional site occupancies measured by neutrondiffractionOne of the predictions which can be made by an oxygen ordering model is the behaviour ofthe fractional site occupancy of the two sublattices in the basal plane, x„ and x09. In-situ,high temperature powder neutron diffraction can measure the occupation probability ofthe atoms in the crystal lattice. Measuring the occupancy of the 0(1) and 0(5) site givesdirectly x, and xo (cf. figure 1.1). To the best of the knowledge of the author, a precisein-situ high temperature measurement of the fractional site occupancy has been doneonly by Jorgensen et al. at Argonne National Laboratory[112, 5]. In the first paper[112],the fractional site occupancy was measured at constant oxygen pressure as a function oftemperature. In the second paper[5], a measurement was conducted at a constant 490°Cvs. oxygen pressure.'Recall that the long range order parameter is .s x, — x#. 5. The Extended CVM model for YBa2 013 06-1-s^ 1310.0 0.2 0.4 0.6 0.8 1.0x in YBa2Cu306+xFigure 5.7: Plot of kT(Ox/(9,07, vs. x at the temperatures of the experiment usingthe f(II)F(II) model. The symbols are the same as in figure 5.6 and correspond toexperiment. The solid line is the f(II)F(II) model with 177,, = 2800K at the correspondingtemperatures of the experiment.Chapter 5. The Extended CVM model for YBa2Cu306+5^ 132Before comparing the predictions of f(II)F(II) model to Jorgensen's data, a fewwords should be said concerning the absolute value of the oxygen concentration. Thereis a profusion of estimates for the maximum obtainable oxygen content in YBa2Cu306+5at 1 atm pressure. In Jorgensen's work, it is estimated that the maximum oxygen contentis x = 0.91(3), and this is reflected in the plots of the fractional site occupancy. In thiswork, and in many others, it is estimated that the maximum obtainable oxygen content isvery close to x 1. In particular, we assume that the absolute oxygen content at 400°Cin 1 atm of oxygen is x = 0.987. (cf. appendix C). Since the model fits the chemicalpotential data very well, especially at low temperature, any calculation will result in anoxygen content close to Xmas = 1 at low temperature, instead of Jorgensen's value ofXmas 0.91. Thus, one would not expect a good fit to the neutron data without somefurther analysis.We would like to argue that the estimate of Jorgensen for a maximum oxygencontent of Xmas 0.91 is too low and that thus his data should be rescaled to correspondto an oxygen content quite close to x = 1 at low temperatures and high oxygen pressures.Aside from the fact that current estimates for the maximum value are between Xmas =0.98... 1.0, the kT(Ox/aft)T shown in figure 4.2 and the chemical potential curves infigure 5.6 also provide evidence that Jorgensen's estimate is too low. In particular, thekT(Oxfait)T data clearly tends to zero at x =1 and not x = 0.9. This suggests that ourestimate for Xmas fs2 0.987 is very reasonable, if one believes any of the oxygen orderingmodels proposed so far. Also, examining figure 5.6, it is clear that the model fits very wellin the orthorhombic phase. It is not possible to modify the curvature of the theoreticalcurve by an adjustment of either 17,, or c. Adjusting simply shifts the curve up or downand Vrari essentially just sets the position of the O-T transition. The shape of the curveis determined by the hole entropy, which contains no adjustable parameters. A constantshift in x for the chemical potential data with a corresponding shift in e for the theoryChapter 5. The Extended CVM model for YBa2 C113 06+x^ 133curve would just worsen the fit.In light of the above considerations, we believe it is reasonable to scale neutrondiffraction results so that they predict x = 0.987 at T = 400°C and P = 760Torr.Figure 5.8 plots the fractional site occupancy (i.e. xa and xo) vs. chemical potential at aconstant temperature of To = 490°C. The open circles are the rescaled neutron diffractionresults of Jorgensen et al.[5] and the small points are the unscaled data. The fit to thescaled data is excellent. Also included, for comparison, is the prediction of the standardCVM square approximation, which quite clearly predicts a much too rapid increase inthe long range order parameter. The effect of the site energy is to just produce a shiftalong the x-axis.Figure 5.9 plots the fractional site occupancy vs. temperature at a constant oxygenpressure pressure of latm. Again, the fit to the rescaled data is excellent. Note that forthis plot, the site energy is important and can modify the shape of the graph.It is important to realize that the model presented in this chapter really containsjust the value of f as the only adjustable parameter. In addition, f drops out in thedetermination of many quantities. There has been a presentation of many differentpossibilities for the behaviour of the holes for completeness sake, but there are reallyonly a few which are physically reasonable. The choice of Vnn was made to optimize theagreement with the 2D ASYNNNI model phase diagram, but for the quantities presentedhere, does not play a major role other than to set the position of the O-T transition.Since the dominant term in the free energy is the entropy, the shape of these curves doesnot depend strongly on temperature. The only strong effect of temperature is to modifythe point at which one switches from the tetragonal phase to the orthorhombic (cf. figure5.7).The choice for using the f(II)F(II) case for the hole behaviour is based largely upon.--^ CVM Square^.-..--0Jorgensen et al. ,,../I,,0=490°C,I.430.0-18^-16^-141.1 / k T 01-12-_--Chapter 5. The Extended CVM model for YBa2Cu3064,^ 134Figure 5.8: Comparison of the fractional site occupancy vs. plkTo between the neutrondiffraction data and the f(II)F(II) model at T, = 490°C. The open circles are the dataof Jorgensen et aL[5] rescaled so that x = 0.987 at T = 450°C and 760Torr. The smallpoints are the unscaled data. The solid line is the prediction of the f(II)F(II) modelwith 177,7, = 2800K and € given by equation 5.14. The dotted line is the prediction of thestandard CVM square approximation.Chapter 5. The Extended CVM model for YBa2 C1-13 06+x^ 1351.0o 0.8Cas0_r3 0.60a)._Cl)To 0.4c0-4=02 0.20.0350 450 550 650 750 850 950Temperature (°C)Figure 5.9: Comparison of the fractional site occupancy vs. T between the neutrondiffraction data and the f(II)F(II) model at P = latm. The open circles are the dataof Jorgensen et al.[112] resealed so that x = 0.987 at T = 450°C and 760Torr. Thepoints are the unscaled data. The solid line is the prediction of the f(II)F(II) model withV7in = 2800K and c given by equation 5.14.Chapter 5. The Extended CVM model for YBa2 C113 06-1-s^ 136the fact that it seems the most reasonable. One could have probably, right at the start,made a choice for hole distribution over all chain and plane oxygen sites with spin-singletpairing, since this is the most consistent picture with other investigations into the normalstate properties of the electron system. However, some of the other possibilities presentedseem to fit the (0.449/L)", data just as well, so that rigorously, one should not discardthe other possibilities, especially the "hole on copper" b(I)B(I) case. The entropy modelfor the holes is simply too crude to really make a judgement as to the validity of theindividual cases. A more serious theoretical analysis needs to be made in order to decideupon the best description for the hole behaviour and an improved fit to the (Ox/a,a)Tdata.5.2.4 Phase diagramNaturally, it is important to see the modification of the phase diagram upon the additionof the hole entropy. First, one should examine the modification of the standard CVMsquare with the new model to see the relative changes. Such a comparison will give anidea of the effect that the additions to the free energy will have on higher order models,such as the 2D ASYNNNI, which is already very successful in fitting the phase diagram.Figure 5.10 plots the phase diagram of the standard CVM square and the f(II)F(II)model. Included for reference is the phase diagram of the 2D ASYNNNI.We see that adding the entropic contribution due to the creation of holes to theCVM square approximation does not modify the phase diagram strongly. In fact, it causesa flattening of the O-T transition curve at intermediate temperatures, which is a desirableeffect, since experimentally, the O-T transition line is quite straight (cf. figure 5.11). Tocompare the phase diagram of the f(II)F(II) model to experiment is not actually veryuseful, since it is based upon the CVM square which over-estimates the transition byabout 10%. Nevertheless, it is perhaps instructive to show such a comparison, to checkChapter 5. The Extended CVM model for YBa2 CU3 06+s^ 1370.^0.4^0.6^0.8^1.0x in YBa2Cu306+),Figure 5.10: Phase diagram as predicted by the f(II)F(II), CVM square and the 2DASYNNNI model (TMFSS). The f(II)F(II) does not modify the phase diagram strongly.Chapter 5. The Extended CVM model for YBa2 CU3 06+x^ 138if the temperature scale, set by the choice of 14,7, is reasonable. To predict the phasediagram properly, one really needs to go to higher order approximations or resort toother method such at transfer matrix finite size scaling or Monte Carlo. Any agreementto such a lower order mean field approximation would be purely fortuitous unless a clearcase can be made to prove that the oxygen interactions are long range. We plot themodel in comparison to experiment in figure 5.11 with Vnn, = 2800K. We see that thecurve is in fair agreement, but one should be cautious, since the mean field theory chosenoverestimates the transition by about 10%.5.2.5 Predictions for the copper valence and hole count and comparison toXAS measurementsThe extended CVM model is based upon the establishment of a connection between theelectronic defects and the oxygen configurations. The success in fitting the structuraland thermodynamic quantities makes it an exciting prospect to be able to provide amicroscopic understanding of the hole creation and charge transfer mechanism. Sincethe electron hole and oxygen configurations are solved self-consistently, it is important toverify that the resultant predictions for the copper valence and hole count is consistentwith experimental observations and intuitive expectations. In other words, we have tocheck that electronic behaviour predicted by the oxygen configurations remains reason-able. The comparison to experiment will also provide clues for how the model may beimproved.We will use a very recent in-situ and ex-situ XAS study on YBa2Cu306+x con-ducted by Tolentino et al[94]. In this XAS measurement, in-situ measurements at hightemperature vs. oxygen pressure were conducted to directly measure the amount ofmono-valent copper in the system vs. x. Since the only site which may be mono-valent is•This work— — Gerdanian (Expt)^ Meuffels (Expt)^ Hilton (TMFSS)— — f(II)F(II)Chapter 5. The Extended CVM model for YBa201306-ps^ 13914001200.1 000a)E 800600400^.^,^I i^,^, 0.2 0.3 0.4 0.5 0.6 0.7 0.8x in YBa2Cu306_,xFigure 5.11: Comparison of the phase diagram between the f(H)F(H) model and exper-iment. See also figure 4.5 for further details on the curves plotted. The f(H)F(H) curveis plotted using = This is the same as for the curve of Hilton et al[14]. Theagreement is fair, suggesting that the choice of Yin for the f(II)F(II) is reasonable.1.0f(H)F(11)ATolentino ex-situ -0Tolentino in-situ IOTolentino in-situ H _0.80.20.0Chapter 5. The Extended CVM model for YBa2Cu3 06+s^ 1400.0 0.2 0.4 0.6 0.8 1.0x in YBa2Cu306,Figure 5.12: Prediction of the f(II)F(II) model for the number of Cu'+ and comparisonto the XAS data of Tolentino et al[94J. The solid line is the prediction of the model at490°C. The XAS ex-situ data corresponds to low temperature annealed samples. Thein-situ measurements were made at high temperature. Notice that the plateaus in theexperimental data correspond to the O-T transition. This plateau is absent in the model.Chapter 5. The Extended CVM model for YBa2Cu306-1-.^ 141the basal plane copper, we can use this data to check that the equilibrium oxygen config-urations obtained from the f(II)F(II) model predict the correct amount of Cu l+ throughequation 5.1. Also, Tolentino conducted ex-situ measurements vs. x at low temperature.In this ex-situ measurement, the samples are annealed at low temperature in a "zero"dead-volume arrangement, to provide a maximum of oxygen order and maintain a welldefined oxygen content.Figure 5.12 shows a plot of the number of mono-valent Cu per unit cell vs. x. Thetriangles correspond to the ex-situ, low temperature anneal data and the circles to thein-situ high temperature run. The solid line is the prediction for the f(II)F(II) modelat 490°C. (it was found that there is not much dependence of the theory line on T.)We see that the model is qualitatively consistent with the data. For x close to zero,[Cull-, 1 — 2x, clearly showing that each isolated oxygen destroys two Cul+. At x closeto 1, the slope is very small, since the main effect of removing oxygen from x = 1 isto reduce the hole count but not to create isolated coppers (cf. equation 5.1). What isclear, however, is that the data predicts a levelling off of [Cu'] at the order-disordertransition. The ex-situ data has a plateau at x 0.3, which corresponds to the positionof the O-T transition at low T. The high temperature data has a plateau at x 0.6 alsocorresponding to the 0-T transition'. The model does not generate such a plateau atthe O-T transition. A slight kink is visible, but not much else. In light of the very goodagreement between the model and the fractional site occupancy determined by neutrondiffraction, it is clear that a better agreement cannot be obtained by forcing a strongmodification of the behaviour of the oxygen configurations. Instead, it is more likely thatthe use of the CVM square formulation, which does not distinguish between two types ofsquare clusters in the basal plane, is at fault. It is important to realize that the amount of10A plateau at the order-disorder transition is a direct result of the fact that order will tend to clusteroxygens into chains giving rise to more coppers with no neighbouring oxygens[55].Chapter 5. The Extended CVM model for YBa2 C113 06+x^ 142Cul+ is not influenced by the behaviour of the holes; i.e. whether the holes are transferredto the planes etc. The Cu l+ count comes directly from the geometric determination ofoxygen clusters and so that the disagreement here at the O-T transition is most likely adirect reflection of choice of the CVM square approximation as the base model.Figure 5.13 plots the number of oxygen 2p holes predicted from the f(II)F(II) model,using equation 5.3. In comparison, we have also plotted the line proposed by Tolentino etal from their XAS measurements. This dashed line was generated by Tolentino from theCul+ data using the condition of charge neutrality ([Cu] + [hole] = 2x, and equation5.2). This dashed line is a low temperature estimate assuming the existence of the ortho-II phase. We see that at the order-disorder transition, (which for the dashed line occursat X 0.3) causes a steep rise in the hole count due to chain formation. A correspondingsteep rise should be present in the high temperature prediction of the f(II)F(II) model atx 0.6, but again, only a kink is seen. As for figure 5.12, this points to a deficiency inthe use of the CVM square cluster to calculate the hole count. At high x, the agreementis good. Starting from x = 1, the removal of one oxygen destroys 2 holes; i.e. breakingup a chain will destroy 2 electron holes.5.3 Commentary on the approximations madeAssuming a lattice gas picture for the holes distributed randomly over all available sitesamounts to a high temperature, narrow band approximation for the charge carriers.Actually, it is believed that there are two bands, the chain band and the plane band[135].Thermopower measurements show, that a narrow band picture might apply above roomtemperature (cf. section 5.1), implying that a pure configurational approach is perhapsa good approximation for the entropy of the charge carriers. Neglecting the existenceof a two band structure also assumes that the charge transfer gap is smaller than kT,f(II)F(II)— — — Tolentino schematic--_..-^/-Chapter 5. The Extended CVM model for YBa2Cu306+x^ 1431.0--a75 0.80c2 0.6a)ci)a)-5 0.4.c0._c.,° 0.2 0.4 0.6 0.8 1.0x in YBa2Cu306,xFigure 5.13: Plot of the amount of oxygen 2p holes from the f(II)F(II) model and com-parison to the schematic behaviour deduced by Tolentino et al[94]. The solid line iscalculated from the f(II)F(II) model. The dashed line is the schematic behaviour pro-posed by Tolentino which was deduced from the XAS measurement using the conditionof charge neutrality. The dotted lines have slopes of 1 and 2 and are included as a guideto the eye.Chapter 5. The Extended CVM model for YBa2 C113 06-Fx^ 144so that the energy difference between chain and plane populations can be ignored and arandom distribution is a valid assumption. Such a hypothesis is really the simplest onepossible, and results in not having to solve the free energy minimization problem for theelectronic subsystem. This is certainly not reasonable at low temperatures, when theentropy per carrier is seen to become T dependent. The hope is, however, that at thehigh temperatures of the vapour pressure experiment, this random solution model forthe charge carriers is a good first step.It should be pointed out that such a situation with two narrow bands is very unusualand not often mentioned in connection with defect structure calculations. There are quitesuccessful ab-initio methods for dealing with the order disorder phenomenon in alloys[63],where the band structure is fairly simple. In the high temperature superconductors, thereare many atoms per unit cell, there is the complicated process of the magnetism, chargetransfer and metal-insulator transition, so that the popular formalism for determiningthe order and phase stability is perhaps not even appropriate.5.3.1 Possible limitations and complicationsThe viewpoint that might be taken about this system is perhaps analogous to thediscovery that the electron has an added degree of freedom called spin. One has thethermodynamic problem of the placement of oxygen in the basal plane which gives aprediction for the chemical potential. The comparison to experiment, however, stronglyindicates the existence of added degrees of freedom. The above analysis is thus an attemptat giving an explanation of what these added degrees of freedom are: one cannot ignorethe creation of holes in this system. At x = 0 one has a completely filled plane and chainband. These do not contribute to the entropy of the system. But, as oxygen is loaded intothe basal plane, holes are created which are initially in the chains, but soon also transferto the planes. Also, the holes may hop from site to site. This gives rise to an addedChapter 5. The Extended CVM model for YBa2 013 06-Fs^ 145degeneracy of a particular oxygen configuration. In this "zeroth order" approximation,the oxygen configuration sets the number of holes and these holes give rise to a numberof degenerate states of the underlying electronic subsystem, which must be taken intoaccount in the oxygen thermodynamics.It is tempting to continue the analysis further and make perhaps more reasonableassumptions of the behaviour of the electronic subsystem, but there are several reasonswhy such an attempt might be futile:1. The relative positions of the bands may easily depend on oxygen content. Espe-cially, the charge transfer gap may vary. To take this into account self-consistentlywould require a very deep understanding of the interplay between the structuraland electronic degrees of freedom. A very recent publication of Uimin et al.[136],which uses the Kondo type approximation for the extended Hubbard model, anda configurational entropy for the chains to fit the XAS measurements of Tolentinocould form the basis for improving the Et of the (Ox/Op)T data.2. The effect of finite band widths might be important, such that casting away the pureparticle picture might be the next necessary step. This again, would be achievedby using the approach of Uimin.3. The basic cluster of the CVM square approximation, which is used to determine thecopper valence, is too simple. No allowance is made for the possible occurrence ofthe ortho-II phase. The CVM square approximation was used since it represents thelowest order approximation which is capable of simply connecting the Cu valenceand oxygen clusters. Higher order clusters with a more detailed mapping betweenconfigurations and hole counts might give the necessary modifications to improvethe fit. Indeed, this has been done by McCormack et al[56J, but the influence ofthis connection upon the oxygen ordering itself was completely ignored.Chapter 5. The Extended CVM model for YBa2 Otis 06-Es^ 1464. Extracting (ax/a/)T from oxygen pressure isotherms is an exacting business. Todate this has only been done here and by McKinnon et al.[59]. The two curvesdo not completely overlap, such that it might not be appropriate to quantitativelyfit these curves to such a high degree of accuracy, until the data has been furthercorroborated by other groups. Indeed, if one believes the curve of McKinnon (cf.figure 4.4), then the fit is fairly good in the tetragonal phase.Using the square as the basic cluster is not so bad as it might seem, since thesuperconducting plateau seen at x '--' 0.6 only occurs upon annealing at temperatureslower than those experienced in this work. This plateau has often been associated withthe occurence of the ortho-II phase and a stagnation of hole doping to the planes[47, 56].Such a plateau is not seen in quenched samples, and suggests that the temperatures arehigh enough to give a smooth evolution of the hole doping in the orthorhombic phase aspredicted in the present square cluster approximation. Whether it is appropriate to usean approach that gives smooth planar doping as a function of x across the O-T transitionis another question. In the model presented here, the high temperature limit for the chainand plane holes was used which gives a smooth evolution of the plane holes upon dopingeven across the O-T boundary. This is perhaps not very correct as will be discussed inthe next section.5.3.2 Question about the nature of the O-T transitionIt quite possible to imagine that the specifics of the chain-to-plane charge transfer mightbe the driving force behind the order-disorder O-T transition. In a more sophisticatedpicture, where one would take into account the mechanism of charge transfer, a situationcould arise where, as a function of x, there is a fundamental change in the behaviour of theadded holes. If this is the case, then the charge carrier entropy may take on completelyChapter 5. The Extended CVM model for YBa2 CU3 06+x^ 147different characteristics which initiates an order-disorder transition in the chain oxygens.Consider the following scenario: imagine that one could describe the electronic system bya narrow CuOs (i.e. basal plane) band and a Cu02 band, with the CuOs band lying lowerin energy than the Cu02 band. Then, add to this an effective strong nearest-neighbourrepulsion between electron holes. If T is comparable to or less than the band gap, butmuch lower than the nn repulsion, then all the holes will pile up in the basal plane untilthe nearest-neighbour exclusion forces holes to be transferred to the Cu02 planes. In sucha situation, below some critical concentration of holes, the entropy is determined by thecharacteristics of the CuOs basal plane. At higher hole concentrations, the CuOs bandis inert and the entropy is determined by the Cu02 planes. In such a case, the reductionin entropy associated with the ordering of the oxygen atoms is to some degree offsetby the increase in hole entropy caused by charge transfer'. Thus, the O-T transitionis in general affected by the mechanism of charge transfer. It is therefore unlikely thatthe O-T transition is determined solely by concentration independent oxygen effectivepair interactions. Such a point has also been stressed by L.A. Andreev, Y.S. Netchaevand co-workers[73, 74] one year ago. This conclusion also emerges from the very recent,detailed analysis of the XAS measurements of Tolentino[94] by Uimin et al.[137, 138, 136].In principle, one could incorporate nn electron hole interactions by utilizing a CVMapproximation with a maximal cluster containing two copper atoms (cf. next section).Also, one should consider the behaviour of the antiferromagnetic and supercon-ducting phase diagram in relation to the O-T transition. As discussed in the introduc-tory chapter, YBa2Cu306+s is an antiferromagnetic insulator (AF) in the tetragonalphase and a superconductor in the orthorhombic phase (cf. figure 1.2). In other per-ovskite phase superconductors, such as (La2,Srs)Cua4, there exists an intermediate"This is also one way to see that the electron hole entropy can effectively cause an attraction betweennnn oxygens via the copper.Chapter 5. The Extended CVM model for YBa2 C113 06+x^ 148region between the AF and superconducting phases. Such an intermediate region whereno long-range magnetic order is seen and where there is no superconductivity has n-ever been seen in YBa2Cu306+x. However, doping YBa2Cu306 by other means, suchas Ca substitution for Y, this intermediate region does exist. As for (La2_xSrx)Cu04,in (Yi_yCay)Ba2Cu306 the planes are doped through heterovalent substitution and thedoping is in principle a smooth function of y. One has direct control over the amoun-t of holes doped to the planes. The fact that the intermediate zone does not exist inYBa2Cu306-Fx implies that the metal-insulator transition, which coincides with the O-Ttransition, entails a large increase in hole doping at the transition. This is a fairly directexperimental proof that a large transfer of holes must take place at the onset of theorthorhombic phase. The oxygen vapour pressure experiments are carried out at highT whereas the AF and superconductivity experiments are at low T, so that it might bepossible to reconcile these findings by an argument involving the T dependence of theelectronic and structural behaviour. But it is clear from the (OxIOOT measurementsthat one cannot separate the two systems, and traditionally held beliefs concerning theset of approximations (in particular the static effective pair interaction viewpoint) usedfor the structural properties of YBa2Cu306+x should be critically re-examined.5.3.3 Suggestions for enhancements and future workThere are several obvious steps to take from here. Most were alluded to in sections 5.3.1and 5.3.2, but here we will be more specific. There are more refined approximations thatone can make both for the configurational problem of the oxygens and their connectionto the hole count as well as for the assumptions concerning the electronic structure. Wewill concentrate here on the subject of the oxygen configurations.It has been mentioned several times that the CVM square cluster ignores the dis-tinction between clusters with and without a central copper. In the next higher CVMChapter 5. The Extended CVM model for YBa2 CU3 06+x^ 149approximation for the 2D square lattice, the 4+5 point cluster allows a separation ofclusters containing different amounts of copper atoms. Figure 5.14 shows the maximalcluster of the 4+5 point approximation and the relevant square subcluster which onecould use for the determination of the Cu valence. One could use the same approach asin section 5.1, but the statistics for this square cluster might be different due to the morecomplex set of clusters used. Unfortunately, Kikuchi's natural iteration method cannotbe applied to the 4+5 point approximation making the free energy minimization quitedifficult.Perhaps a better step would be to also incorporate the distinction between chainand plane holes by using a nn exclusion principle for electron holes in the chain sites.This would require an even larger cluster for the CVM approximation, since one wouldneed to include statistics for clusters incorporating 2 or more central copper atoms. Sucha cluster would be the 3x3 point CVM cluster shown in figure 5.15. Writing down theconfigurational entropy for such a cluster is quite a complex problem and minimizing thefree energy would be very difficult once the hole entropy has been included, since the 3 x3point cluster is not of single type (there are 2 in-equivalent maximal clusters) so that theNI method would not work. It would probably be preferable then to use a Monte Carlomethod in the following manner:In a Monte Carlo simulation one only utilizes the expression for the internal energy.Since the particle picture for the electron holes neglects their internal energy, one wouldadd to the internal energy estimates for the configurational entropy of the chain and planeholes. The number of chain and plane holes would be given by the oxygen configuration.This would give a complicated temperature and oxygen configuration dependent "internalenergy". The chain and plane holes would be determined by counting the square clustersof oxygen surrounding a chain copper site which create a mobile hole just as in section 5.1.However, nn holes would require one hole to be transferred to the planes. One could set150Chapter 5. The Extended CVM model for YBa2Cu3 06+x^P I-^1^ asFigure 5.14: CVM 4+5 point cluster for the basal plane oxygen (Thick solid lines). Thindotted line corresponds to the appropriate sub-cluster to determine the copper valence.Note that there are two in-equivalent maximal clusters.Chapter 5. The Extended CVM model for YBa2Cu306-1,^ 151up a scheme where holes are distributed within their isolated oxygen chains until it isnecessary to transfer some to the planes, in order to not violate the nn exclusion principle.Poulsen et al.[47] conducted a Monte Carlo simulation of the pure 2D ASYNNNImodel at low T and counted the number of ortho-I and ortho-II regions as a function ofx. It was proposed that the minimal cluster which could define an ortho-I region wouldtransfer a number of holes to the planes such that spanning the entire basal plane withortho-I clusters would give the correct number of holes to give a 90K superconductor.Similarly, a minimal cluster which could define an ortho-II region donates enough holesso that a spanning region of ortho-II gives a 60K superconductor. The assumption isthat 7', is proportional to the number of planar holes, and that the number of planarholes depends on the proportionate amount of ortho-I and ortho-II regions. This modelis strikingly successful in predicting 71, vs. x. The hole creation scheme described abovewill predict a similar kind of behaviour. To see this, consider the minimal cluster ofan ortho-I phase shown in figure 5.16. It contains two nn coppers, both which havegenerated a hole. Since they are nearest neighbour coppers, one hole must be transferredto the plane. The planar hole count is 0.5 per unit cell. In a minimal cluster of ortho-IIphase, one has 8 coppers with 4 holes. In this case, there are 0.25 planar holes per unitcell. If the ortho-II phase becomes stabilized and pre-dominant at x L-: 0.66, then thisscheme will exhibit a plateau in the hole count: additional oxygens added to the ortho-IIphase will just create more localized Cu2+.This is a very simplified picture and would predict a 45K not a 60K superconductor(assuming that 7', is linear in hole doping), but the essential features would be present.Nevertheless, it is perhaps most fruitful to first improve the model along these lines beforerelaxing the narrow band configurational approach for the charge carriers. On a purelyspeculative vein, it is possible to imagine that taking into account the energetics of chargetransfer in more detail would be capable of generating a similar phase diagram with theChapter 5. The Extended CVM model for YBa2Cu3 06+x^ 152Figure 5.15: CVM 3x3 point cluster. Note that this cluster contains two central coppers.Figure 5.16: Minimal size clusters defining an ortho-II and ortho-I region. The figure onthe left shows a pure ortho-II phase and the right shows a pure ortho-I phase.Chapter 5. The Extended CVM model for YBa2 CU3 06-Fx^ 153existence of an ortho-II phase etc., assuming only a direct nn coulomb repulsion betweenoxygen atoms and letting the nature of hole creation and charge transfer dictate the rest.It remains to be seen if a complete self-consistent inclusion of the charge transfer willretain the essential features of the 2D ASYNNNI model.The essential feeling one gets from this analysis is that taking into account theelectronic structure in detail cannot leave one with x and T independent effective pairinteractions which would be able to correctly describe the oxygen thermodynamics. Ithas often been said that the reason for the poor fit of the 2D ASYNNNI model to theoxygen pressure data is due to long range elastic effects or shifts in the phonon spectrumresulting in the subsequent dismissal of oxygen chemical potential data as containing toomany effects to be a useful test for theoretical ideas. Thus, the continued use of a staticeffective pair interaction picture. The proper physical grounds for such claims have neverbeen properly addressed one should prehaps re-examine in detail the entire physical basisfor the pure 2D lattice gas model in YBa2Cu306+x•5.4 SummaryIt was shown that the inclusion of the behaviour of the electronic subsystem into thefree energy of the oxygen ordering problem drastically improves the agreement betweentheory and experiment. The proposed phenomenological extension to the standard CVMapproximation was based upon several experimental observations:1. The experimental kT(Oxiait)T curves were found to be essentially independent oftemperature except very close to the O-T transition. This implies that at thesetemperatures, the entropy dominates the chemical potential.2. The in-plane and bulk thermopower in YBa2Cu306+s is roughly independent ofT above room temperature. This suggests that a narrow-band picture might beChapter 5. The Extended CVM model for YBa2 CU3 06-Fx^ 154a reasonable approximation, where the entropy of the charge carriers is purelyconfigurational.3. From x-ray absorption spectroscopy, a functional ionic description for the valenceof the chain coppers emerged which allows one to connect the oxygen configurationto the creation of holes.4. The existence of a semiconductor-metal transition in the Cu02 planes when theoxygen concentration is increased as well as numerous other charge carrier depen-dent experiments clearly imply that electron holes created in the chains are at somepoint transferred to the planes.These findings together with the inability of the standard 2D ASYNNNI model to fitthe non-ordering susceptibility kT(ax/a,u)T lead us to propose a simple parameter freel2extension to the standard oxygen ordering models which takes into account the creationof holes. The extension was based upon a thermodynamically consistent hybrid modelbetween the standard 2D lattice gas model and the defect chemical model of Verweijand Feiner. The connection between these two models was made by identifying thesquare cluster configuration of oxygen with the creation of holes. Charge transfer wasintroduced by assuming a high temperature, random occupation of all candidate electronhole sites. The system was solved in a modified CVM square approximation which, inits standard form, agrees fairly well with the higher order 2D ASYNNNI model at thetemperatures experienced in the oxygen pressure experiment. It was seen that no strongmodification of the structural phase diagram is produced but that agreement is achievedwith virtually all experimental findings related to the oxygen thermodynamics and basicvalence behaviour.120ne might argue that proposing 22 different cases for the various hole options is not parameter free,but could just as easily have a priori eliminated the remaining 21 options on physical grounds. Thiswas not done for the sake of completeness of the theory and to make sure that no unphysical model issignificantly better than the one chosen.Chapter 6Concluding remarksMuch of what was said in the last sections of chapter 5 could have also been placed in thischapter. Instead, we will conclude by giving a historical perspective of the experimentaland theoretical work of the thesis. In recent years, there has been quite a shift inthe methods used to study and explain the oxygen thermodynamics in YBa2Cu306+,.The initial idea, presented early on by de Fontaine, was to propose short range oxygeninteractions in a lattice gas model. Its simplicity and initial successes probably accountedfor its popularity. However, also quite early on, there were indications that a pure latticegas model could not work. Already in early 1988, McKinnon could not fit his (ax/(9,)7-,data to a nearest-neighbour lattice gas model, which, as was shown in chapter 2, is almostidentical to the "full blown" 2D ASYNNNI model. Something was wrong, but there werealways plausible reasons for not obtaining a fit to the (ax/ait)T data: inhomogeneities inthe sample, too many extra effects, such as lattice dilation. For example, in 1991 Rikvoldet al. calculated kT(.9x/Oit)7, using Monte Carlo on the 2D ASYNNNI model (cf. figure4.8) claiming that good agreement was found and that the remaining discrepancies wereprobably due to sample inhomogeneities.At the same time, quite a few chemists were utilizing a completely different ap-proach to fit their thermodynamic data: the defect chemical model. They immediatelyrecognized the existence of additional degrees of freedom through the creation of elec-tronic defects, caused by the additional oxygen. These two camps, however, seemedcompletely disconnected. Defect chemical papers would not cite lattice gas papers and155Chapter 6. Concluding remarks^ 156vice versa. I had spoken to some chemists active in this field in the fall of 1991, and theyseemed quite unaware of the profusion of literature that exists concerning the lattice gasapproach for YBa2Cu306. The major deficiency of these models is the fact that theyignore the types of superstructures seen in YBa2Cu306+s, and mostly make no predictionfor the O-T transition.Having an affinity for statistical mechanics, and recognizing the obvious successof the lattice gas models to predict the ordered superstructures, I continued trying tomake the lattice gas models work. Furthermore, there are many lattice gas papers whichintroduced empirical on-site energies to successfully fit the thermodynamic data, mostnotably, Shaked et al. and Bakker et al. However, although some of the defect chemicalmodels did not have exactly the correct magnitude for kT (ax/a,u)T, there was the strikingfeature of predicting the correct curvature and a T independent kT(ax/ap)T, as was foundexperimentally. There had to be something correct about the defect chemical model.The breakthrough came upon reading the paper by Verweij and Feiner[64]. Here,more solid arguments were made for the behaviour of the electron holes, which agreedwith the more recent understanding and workable picture for the creation of holes andtheir connection to oxygen configurations. Unfortunately, the point of the paper was touse their own model and its agreement to the thermodynamic data as somewhat of aproof for the existence of charge carriers in a narrow band (less than 0.1eV); this paperis not often cited in the literature.After having made many calculations using the CVM mean field approximation forthe lattice gas model, it became apparent how one could merge the two "non-interacting"camps into a single, self-consistent model. The result was a model with very few (just 2)free parameters which seems to fit most of the available data.Chapter 6. Concluding remarks^ 157As for the future, it is obvious now that one must include the behaviour of theelectronic structure in detail, since it is the unusual electronic properties which are dom-inating the thermodynamics. Including the electrons in detail is now becoming feasible.It seems that one is getting a handle on the description of the electronic properties, inrelation to charge transfer and hole doping, at a first principles level. The recent paperby Uimin and Rossat-Mignod (Physica C, September 1992) seems very promising, al-though it is currently restricted to the orthorhombic phase (cf. section 5.3.1). Suddenly,the topic of oxygen ordering thermodynamics has moved from being mainly useful forsample preparation and the application of some interesting statistical mechanics models,to becoming directly involved in the discussion of the unusual electronic structure of thehigh T, materials. Papers that are published concerning this topic are now being writ-ten by individuals simultaneously involved in the search for a pairing mechanism. Thelanguage has evolved from "lattice gases" and "cluster variation method" to "Hubbardmodels" and "the large U limit".Appendix ACalculation of the chemical potential for a diatomic gas with molecules oflike atoms applied to 02 gasUsing the approach of Landau and Lifshitz[61] to describe the thermodynamic propertiesof a diatomic gas, this appendix is a resume of the derivation of the chemical potentialit of an ideal gas with molecules of like atoms. A specific calculation for 02 gas will bepresented at the end.The chemical potential can be determined from the expression for the free energyF. In the case of a diatomic gas, the expression for F is obtained by taking into accountthe translational, rotational and vibrational degrees of freedom of the molecule. Underthe Born-Oppenheimer approximation, the energy levels of a diatomic molecule can bewritten as the sum of three independent parts: the electron energy, the rotational energy,and the vibrational energy of the nuclei within the molecule. For a singlet' electronicstate, these levels may be written:1E19,K =-- ECI + nW(19 + ) + h2K(K + 1)/21 (A.1)where co is the electron energy, hco the vibrational quantum, V the vibrational quantumnumber, K the angular momentum of the molecule, I = m'rg the moment of inertia ofthe molecule (m' = mim2/(mi + m2) is the reduced mass of the two atoms and ro theequilibrium of the distance between the nuclei).Substituting the expression A.1 into the definition of the partition function, and1-Note that the electronic ground state of oxygen is in fact a very narrow triplet. This narrow widthjust gives rise to a degeneracy of the electronic state as shown later.158Appendix A. Calculation of the chemical potential for a diatomic gas...^159using the fact that F = —NkBT1n(Z), one obtains:3\F —NkBT1n (eV (  mT N 2h )^Feib + Fel + Neo^(A.2)where m = m1 + m2 is the mass of the molecule, N is the number of molecules and V isthe volume. The first term may be called the translational part, Ftr, since it arises fromthe degrees of freedom of the translational motion of the molecules. F„t and Fvib arerespectively the rotational and the vibrational free energies, and are defined next. Fel isthe electronic contribution, and for most cases is given by the degeneracy of the electronicground state, ge. Excited electronic states are typically very high in energy (eg. the firstexcited state of oxygen is at 11,300K), and do not contribute to the thermodynamicproperties. Thus,Fel = —NkbTlnge.^ (A.3)For the case of a molecule of like atoms, and if the temperature T satisfies T>> -C(which is typically in the range of temperature concerned [300K-1000K] for the case ofmolecular oxygen where 1.05K), the rotational free energy has the form:Frot —NkBT ln (kBT) — NkBT1n ( )^(A.4)the vibrational free energy is given by:Fyib NkBT1n (1 — exp (TcBhTw))^ (A.5)Replacing the expressions of F„t and Fb in the equation A.2 and assuming thatthe gas can be considered ideal (PV = NkBT), one obtains:[P(1 — exp GA`,)- ))F = NkBT1n^7B  I + NE0 (A.6)(kBT)with3eIge m=h2^27rh2(A.7)Appendix A. Calculation of the chemical potential for a diatomic gas...^160Using the fact that ft = (aaNF) Ty and taking as the origin of energy the ground statewhich corresponds to 9 = 0 and K = 0 in the equation A.1, one finally obtains thefollowing expression for IL:[P (1 — exp  ^hwu,, = kBT ln ^4-(kBT):1^2where P is the pressure and is given by the equation A.7.Application to molecular oxygen:For oxygen we have[139]:31.9988g/molehwr,ge==1580.361cm-11.2075A32273.875Kso that finally, the chemical potential per molecule of oxygen is:[P (1 — exp (&- ) )1 nu,with:= 1.1628 x 1018Torr/(eV)ihw2 = 97.9733n/eVPo, = kBT ln(kBT)i^2(A.8)(A.9)This expression agrees, to within 10-2eV, with the phenomenological expressionfor the chemical potential of molecular oxygen given in Barin and Knacke[140] in thetemperature range 300-1000K.Appendix BThe natural iteration method for the free energy minimization of CVMmodelsThis appendix describes the natural iteration method for the minimization of the clustervariational free energy. The NI method was first presented in 1973 by Ryoichi Kikuchifor the solution of the CVM equations[87]. It is an alternative to the standard Newton-Raphson approach to solving the CVM free energy. In his original paper, Kikuchi derivesthe NI method for the case of the CVM pair and tetrahedron approximation. We willderive the NI equations for the case of the CVM square approximation and then for theextended CVM model of chapter 5.B.1 CVM square approximationWe start by writing down the grand potential of the CVM square approximation givenby equation 2.26 of section 2.3.1.E EijklZijkli,j,k,1[i^'j? 2}]1-kkT E Zijki ln Z^ + X Inijki — 2 E yii In yij + —2 E x7 In x71aijklZijkl2^. .GN,(B.1)161(B.2)(B.3)(B.4)Xr3.3Y23^Zijklk,1Appendix B. The natural iteration method...^ 162where we refer the reader to section 2.3.1 for the definitions of the terms in the equation.Note that x7, xl!, and yii are functions of zijki throughso that the grand potential can be expressed as a function of only the square clusterprobabilities Zijki. Since the cluster probabilities are normalized, one has the conditionthat Eij,k,13ki 1, which one can add to equation B.1 by the introduction of a Lagrangemultiplier:E EijklZijkli,j,k,1[^ r(xn+1-PkT E r(ziiko - 2 Er(yii)+ —2 E^Er(Xji,j,k,1 i,a^ aE^A(1 — E ziik)2where we have used the notationr(x) = xlnx^ (B.6)We now have the unconstrained minimization problem for G with respect to the clusterprobabilities zijki. The strategy of the NI method is to take advantage of the symmetriesof zijki and the resulting symmetries of the point and pair cluster probabilities 4, 4and yi3. Looking at the definition of zi3k/ through the square cluster,N,}1(B.5)i,j,k,1Appendix B. The natural iteration method...^ 163we can identify symmetries in the cluster probabilities:Z.iki = Zkjil -= Zilkj = ZklijYij = Yil = Ykj = Ykl (B.7)which are obtained by looking at all the possible geometric operations which leave theimage of the cluster invariant (eg. rotation of the square by 1800). Naturally, in thedisordered phase, xa = x/3 and more symmetries arise, but we will not list them here.Using these symmetry relations, we can formally write41 (Ef(Yii)+Ef(y)+Ef(yki) +^(yid))i71^kj^k,11221 (E f(4) Er(X))3^1(E,c(4)+E.c(x00)(B.8)The existence of these symmetry relations is a necessary condition for the NI method towork. We rewrite equation B.5 using B.8 so that the expression for the grand potentialhas the symmetry of the maximal cluster. Then, setting the derivative of G with respectAppendix B. The natural iteration method...^ 164to zijki to zero and solving for Zijki gives:(YiiYakiYkat): exp —(ciiki — paijk//2 — A) }Zijki(414 XZX11)T^ kT(B.9)Now it has become apparent why one has written the pair and point cluster probabilitieswith the same symmetry of zijki. Without this step, it would not have been possible towrite equation B.9 in such a manner. Next, we use the normalization condition for zijkito obtain ):—A^(YiNilYkjYkl) 2^- (Eijki - /Lao,/ /2)exP IT r--- i,j,k,1 (Xf' X11 XZX113)1 exP^kT(B.10)At its minimum, the grand potential is given by ): G = NoA. Equations B.9 and B.10form the basis for the NI method. The calculation goes as follows:1. Specify a chemical potential it and a temperature T.2. Make an initial guess for x, x',? and yij. One can use the high temperature approx-a 0imation for y ziandom =3. Insert x7, xi? and yij into equation B.10 to calculate A.4. Insert A, 4, xi: and yij into equation B.9 to obtain a new estimate for Zijki.5. Use equations B.2-B.4 to obtain new values for x7, xi; and yij.6. Check for convergence (see next paragraph). If the equations have not converged,the go back to step 3.Kikuchi showed that this iteration scheme is virtually guaranteed to converge to theminimum of the free energy. One can monitor the evolution of the cluster probabilities todecide when the iteration has converged. The value of the free energy is less sensitive andconverges faster so that it is better to include the evolution of the cluster probabilities forthe convergence criterion. The number of iterations required for convergence of the CVMAppendix B. The natural iteration method...^ 165square approximation was roughly 100 steps away from the order disorder transition andup to 10000 steps very close to the transition. The maximum number of steps allowedfor the chemical potential calculations was 5000. For the phase diagram calculation, thelimit was set to 10000 iterations.B.2 Adding the spin and hole entropyThe addition of the configurational entropy of the holes and the spin entropy of the Cu2+valence for the oxygen ordering problem in YBa2Cu306+x. (cf. chapter 5) is easily takeninto account, since these additions are all expressed in terms of zip,/ (cf. section 5.1.3).Taking the derivative of the spin entropy (eq. 5.10) with respect to zi,k/ gives(.9.5spin)aZijkl )kNoi^kNin (22)ciikil2 I ln(g)aiki ^ 2(B.11)We recall that the number holes per unit cell is given by (cf. equation 5.7)[hole] = E (2ai + Cijkl 1)Zijk1^E hijklZijkl^(B.12)i,j,k,1^ i,j,k,1where ai jki^(i j k l)/2 gives the oxygen concentration of a square cluster andco,/ = 1 if the cluster produces a Cu'+ (cf. equations 5.9 and 5.9). Thus, taking thederivative of the hole entropy (eq. 5.4) gives(ashoie)aziJklkNo^n^)11'1 In — [hole]) hi3k1= 2 in — [hole])^[hole] )(B.13)where vijki is the derivative of the availible hole sites n wrt. Zijki. Inserting relationsB.11 and B.13 into the derivative of the grand potential gives finally the new NI iterationequations for zi3k/ and A:^Zijkl^1^1exp —(cijki — paiik//2 — kTcriiki/2 — A)ijkl " ijkl^ (B .14)kT_^I^i^exp (uA)^E X exp —(Eijki — paiik112 — kTo-iik112) (B.15)i,j,k,1 kTAppendix B. The natural iteration method...^ 166Xijki = x7x'q xc'xi3k I^ (B.16)Yijkl = giigilgkjgki (B.17)n^)vijkl ( n — [hole])3"Hijkl =(n — [hole])^[hole] )^(B.18)Note that it is crucial to express Hijki, hijki, and vi,k/ so that they contain the samesymmetries of zi30, as was done with the point and pair cluster probabilites. If this isnot done, then the right hand side of equation B.14 will not contain the same symmetriesas the left hand side'Specializing to the f(II)F(II) model, where one assumes that the hole is on theoxygen forming a spin doublet with the two neighbouring Cu2+ and distributed over alloxygens in the chains and planes, we getn = 4 + x = 4 + E aizijklvijkl =aijklaijkl12—2111(2) (aiiki^ciiki)(B.19)(B.20)This implies that the additional spin entropy, giving rise to criiki, effectively shift-s the chemical potential by an amount kT2 ln 2 and adds an effective cluster energykT ln(2)ciiki. The hole entropy adds a complicated term Hijki, which could be interpret-ed as another effective cluster energy —kT ln (140 /2. For the NI iteration scheme, oneneeds to add the evaluation of [hole] using equation B.12 to the appropriate steps, i.e.steps 2,3,4,5.In this fashion, the cluster probabilities were calculated at each temperature for200 values of the chemical potential. Using these values for zi3ki, x, =^— x, [hole],'Also note that if ijki has the form of the reaction functions of Verweij and Feiner given in table 2.3.By using equation B.18 together with tables 5.1 and 5.2, one can easily generate the reaction functionsof Verweij and Feiner, using their assumption for the number of oxygen 2p holes: [hole] = x2.Appendix B. The natural iteration method...^ 167[Cu1-1], etc. were calculated as a function of tt and T. k49.40,07, was determined bycalculating the change in oxygen concentration x for every change ,a. The phase diagramswere generated by making a detailed calculation of the long range order parameter s closeto the transition to detect when .s becomes non-zero. This is an inefficient algorithm interms of computer CPU time, but its simplicity minimized the chance for errors in thecalculation. Typical CPU times on an IBM RS/6000 560 were about 25 minutes for aphase diagram calculation and 2 minutes for the other calculations.Appendix CSample preparation and characterizationWe have put the information about the making of the master batch of YBa2Cu306+, andthe characterization of its physical properties into this appendix, because there really isno remaining mystery about the proper technique to prepare pure YBa2Cu306+x. Thiswas not true at the beginning of YBa2Cu3064.s research early in 1987. Some of thesamples prepared for the oxygen vapour pressure experiment were made at a time whenthe making of YBa2Cu306+s was still considered to be somewhat of an art. Thus,the description of the technique used to prepare the master batch might seem to somereaders fairly inefficient. After preparation, the batch of YBa2Cu306+x was extensivelycharacterized in order to confirm the expected high quality and to obtain estimates forthe upper limit of impurities. This is crucial since one important parameter needed todetermine the oxygen content during a vapour pressure run is the mass of the sample.Significant amounts of impurities would leave the real mass of YBa2Cu306+x uncertain.Also, "oxygen active" impurities would also produce an error in x by acting as a sourceand/or a sink for oxygen.C.1 Sample preparationThe master batch of YBa2Cu306+x was prepared using five 9's purity starting materialswhich had been preheated overnight to several hundred degrees C in order to removetrapped water. Y203, CuO and Ba2CO3 in stoichiometric proportions, giving a total ofabout 100g of starting materials, were mixed thoroughly by hand in a beaker and then168Appendix C. Sample preparation and characterization^ 169transferred in smaller batches to an automatic micro mill (Brinkman Instruments RetschMill). The material was mixed three times in the mill for approximately 10 minutes each.The powder was transferred to small, 1" high, 1/2" diameter UHP high density MgOcrucibles. Roughly 20 crucibles were needed to hold the entire batch. Small crucibleswere used to allow CO2 to escape and oxygen to enter easily. The batch was calcinedin a box furnace for 18 hours at the following temperatures: 845, 850, 870, 880, 890,897, 910 and 925°C. Inbetween firings, the powder was reground in the micro mill for 7minutes and sifted through a 70,am sieve. After each firing, a small sample was extractedto monitor the progress using X-ray diffraction. Finally, the powder was pressed, with apressure of approximately 10 tonnes per cm2, into 3g, 3/4" diameter pellets and sinteredin flowing oxygen in clean tube furnace (Lindberg single zone 1200C), dedicated solely tothe oxygenation of YBa2Cu306+s. The standard grade oxygen was first passed througha Pd catalyst at 500°C to catalyse CO to CO2, then fed through a liquid nitrogen coldtrap to freeze out any CO2 and water. The following oxygenation procedure was used:fast 10min 0 fast 10min^2hr s^0^2hr a o^Ohrs^0RT -->300°C^600°C fast 9250c, 120 C/hr 5300c, 6 Clhr 3700C100 Clhr RT (C. 1)The intermediate steps at 300°C and 600°C are to allow the carbon lubricant, which wasused to press the pellets, to burn off.C.1.1 Final oxygen contentThe final oxygen content of a sample prepared in the above fashion was estimated tobe x = 0.987 + 0.002. This value was determined by examining the room temperatureelectrical conductivity of a series of deoxygenated pellets which were prepared to havean oxygen content close to x = 0.00. By plotting the room temperature conductivity vs.x for such a series of samples and by assuming that at x = 0.00, YBa2Cu306+s is aninsulator (hole doping is zero), one can obtain a value for the fully oxygenated pelletsAppendix C. Sample preparation and characterization^ 170prepared in this manner. It was found (cf. figure C.1) that the conductivity a varieslinearly with x for x < 0.08 and an extrapolation to to a = 0 is easily made. For x < 0,the conductivity was found to increase with decreasing xl. Also, iodiometric titrationwas used to determine the oxygen content of a sample annealed at 450°C in 1 atm oxygenpressure. The oxygen concentration for such a sample was found to be x = 0.95 ± 0.01.Referring to figure 4.1 and to table 4.1, we see that this is in very good agreement withthe oxygen pressure isotherms.C.1.2 Impurity levels in the YBa2Cu306+sTwo tests were made to directly measure the impurities in the master batch. First,powder X-ray diffraction using a Rigaku 12kV rotating anode X-ray diffractometer with amonochromator stage showed no detectable levels of impurity phases. Second, inductivelycoupled plasma mass spectrometry (conducted by Elemental Research Inc.) showedsilicon as the major impurity at 160ppm. Such an impurity presumably comes fromthe grinding process using the micro mill. The next highest impurity detected was calciumat --, 24ppm.C.2 Sample characterizationThe master batches of YBa2Cu306+s created with the technique outlined abovewere extensively used in a number of experiments, most notably for muon spin rotation(OR) studies at TRIUMF, Vancouver, B.C. Canada[39, 96, 97, 45, 98, 99] and for a mea-surement of the Ginzburg-Landau parameter K2(T)[1411. Numerous other measurementshave been conducted with these samples; we will list only the two most pertinent for theverification of sample quality: Resistivity and magnetization. Resistivity measurements'One should add that we are not assuming that samples with x <0 are stable, or that they have notpartially decomposed in the process.8■,.....,1-LI ^-_---1 1 1Appendix C. Sample preparation and characterization^ 171-0.02 0.00 0.02 0.04 0.06 0.08x in YBa2Cu306_,xFigure C.1: Room temperature conductivity of YBa2Cu306+s vs. x for x < 0.08. Thesolid line is a visual fit to the three data points. The oxygen content was determined byassuming that the annealing conditions described in section C.1 gives x = 0.987. Noticethat the conductivity increased for the x '1.•_-' —0.018 data point.Appendix C. Sample preparation and characterization^ 172RT resistivity = 1.0rnf/cmV") = 93.7K= 93.4K^T(1°%)^92.8K(o) 92.7KAT(resistivity) = 0.9K^AT(magnetization)^3KMeissner fraction^25%Major Impurity: Si © 120 ppmTable C.1: Some physical parameters of the YBa2Cu306+s master batchwere conducted on a fully oxygenated sintered block of dimensions 0.77 x 1.52 x 4.43inm3,using an AC lock-in technique with a lrnA RMS driving current. Figure C.2 shows aplot of the resistivity vs. T for the fully oxygenated sample. Also included is an insertshowing the transition region. The transition width is less than 1K, and shows no signifi-cant "tail" indicating high sample quality and homogeneity (cf. table C.1 for a summaryof the pertinent characteristics).The magnetization was measured in a RF SQUID magnetometer (Quantum DesignMPMS SQUID) cooled in 10 gauss static field (cf. figure C.3). Applying a 10 gauss fieldto a zero-field cooled sample at low temperature gives a shielding signal of -0.178 emu/g.The estimated Meissner fraction is about 25% at 87K. Hysteresis curves at 100K showedno detectable ferromagnetic signal. No Curie term was detected at low temperatures.ConclusionThe YBa2Cu306+s material prepared for the oxygen vapour pressure experiment is ofthe highest quality. No impurities were found by X-ray diffraction either before or afterthe experiment. Measurements of the resistivity and magnetization indicate standardbehaviour of very pure, sintered YBa2Cu306-Fs•Appendix C. Sample preparation and characterization^ 1730. 130 180 230 280Temperature (K)Figure C.2: Resistivity vs. T for YBa2Cu306.987 (i.e. fully oxygenated). The insert showsthe transition region. The room temperature resistivity was measured to be 1.0m9cm.The midpoint of the transition is at 7(mid)--,-  93.4K and the 10%-90% transition widthis 0.9K.-0.02-0.03 ^1 ,, i n^i ,0.00oo-0.01oooo ooo --ooAppendix C. Sample preparation and characterization^ 17480^85^90^95^100Temperature (K)Figure C.3: Magnetization vs. T for YBa2Cu306.987 in a 10 gauss field.Bibliography[1] J.G. Bednorz and K.A. Muller. Possible high I', superconductivity in the Ba-La-Cu-0 system. Z. Phys. B, 64:189, 1986.[2] J.G. Bednorz and K.A. Miiller. Perovskite-type oxides-the new approach to high-T,superconductivity. Rev. Mod. Phys., 60(3):585, 1988.[3] M.K. Wu, J.R. Ashburn, C.J. Torng, P.H. Hor, R.L. Meng, L. Goa, Z.J. Huang,Y.Q. Wang, and C.W. Chu. High-pressure study of the new Y-Ba-Cu-O supercon-ducting compound system. Phys. Rev. Lett., 58:908, 1987.[4] R. Beyers and T.M. Shaw. The Structure of YBa2Cu307_8 and its derivatives,volume 42 of Solid State Physics, pages 135-212. 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