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

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THERMODYNAMICS OF OXYGEN ORDERING IN YBa2Cu306+. By Paul Richard Schleger B.Sc., The University of British Columbia, 1987  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES PHYSICS  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA 1992 © Paul Richard Schleger, 1992  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of  aft,c,  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  1)€„,,44 bei- / 9)  Abstract  An apparatus has been built to study and manipulate the oxygen in high temperature superconductors. It uses the principle of cryogenically assisted volumetric titration to precisely set changes in the oxygen content of high-T, samples. The apparatus has been used to study the thermodynamics of oxygen in YBa2Cu306+x in order to help determine the correct model for oxygen thermodynamics as well as to provide standard curves for materials preparation by other methods. In particular, extensive measurements have been made on the oxygen pressure isotherms as a function of  x for temperatures  between 450°C and 650°C. The measurement technique also allows one to extract the thermodynamic response function,  (ax/O,a)T, Ca is the chemical potential), which is  sensitive to the oxygen configuration and which can be calculated by any candidate theory of the oxygen thermodynamics. Several existing theoretical models for the oxygen ordering thermodynamics are presented and compared to the experimental results. The models considered are classed into two basic approaches: lattice gas models and defect chemical models. It is found that the lattice gas models which assume static effective pair interactions between oxygen atoms, do not fit the experimental data very well, especially in the orthorhombic phase. The defect chemical models, which incorporate additional degrees of freedom (spin and charge) due to the creation of electronic defects, fit significantly better, but make crude assumptions for the configurational entropy of oxygen atoms. Using a commonly accepted picture for the creation of mobile electron holes and unpaired spins on the copper sites, it is possible to relate these quantities in terms of short range cluster probabilities defined in mean field approximations to the 2D lattice 11  gas models. Based upon this connection, a thermodynamical model is developed, which takes into account interactions between oxygen atoms and the additional spin and charge degrees of freedom, assuming a narrow band, high temperature limit for the motion of the 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 dissociation energy of an oxygen molecule) as the only adjustable parameters, is compared to experimental 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 of mobile electron holes. Qualitative agreement is found for all compared quantities, and quantitative agreement is found for the chemical potential, fractional site occupancies and kT(ax/a,u)T in the orthorhombic phase. Improvements to the model are outlined which should result in a quantitative fit to all results, in particular the valence and hole count vs. x. In addition to illuminating what is lacking in the commonly used two dimensional lattice gas models, the theory may form the basis for accurately predicting the electron hole count of the Cu02 plane of YBa2Cu306+s as a function of the sample preparation conditions.  Preface  The incredible attention given to high temperature superconductors since their discovery in 1986 is partly due to the promise of "inexpensive" superconductivity at high temperatures, perhaps even room temperature. But also, these compounds have attracted much interest because of their unusual physical properties. Of particular interest in the cuprate superconductors is the existence of Cu02 planes with strong two dimensional character, whose properties are controlled by electron hole doping. At low doping these planes give rise to semiconductive behaviour and antiferromagnetic order. At higher doping levels, they become metallic and superconducting (at low temperature). The remaining elements of the structure apparently serve only to structurally maintain the two dimensional copper-oxide lattice and to dope the planes with holes. There are a variety of ways to manipulate the external structure and create electron holes in the planes. One of the simplest cuprate superconductors from an overall structural point of view is YBa2Cu306+„ where the hole doping is controlled by the addition of oxygen to sites external to the Cu02 planes. This leads to an indirect charge transfer of 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 concentration; the additional oxygen displays various types of long range atomic ordering, which is seen to modify the number of holes doped. The basic structural and electronic properties of YBa2Cu306+x are presented in chapter 1. In order to understand the electronic properties of YBa2Cu306+x, and in particular the mechanism of superconductivity, it is of interest to obtain precise information about the connection between oxygen ordering and hole doping. iv  Since the added oxygen becomes fairly mobile at temperatures significantly lower than the melting point of these materials (300K < T < 1100K < melting point) much of the required information can come from studies of the temperature and x (oxygen concentration) dependence of the oxygen ordering thermodynamics. The oxygen chemical potential  it is a quantity which gives information concerning the thermodynamics. It can be obtained by measurements of the equilibrium oxygen partial pressure of the surrounding environment (it --, In P). The fine details of the chemical potential are revealed through the derivative, (Ox/ap)T, which acts as a thermodynamic response function. The experimental aim of this thesis was to map (ax/a,u)T over a significant range of temperature and oxygen concentration. The experimental details are presented in chapter 3. One can distinguish two main approaches to describe the thermodynamics of this system (described in detail in chapter 2). One of them is based upon a pure configurational lattice gas model with interactions between oxygens. The other consists of describing the oxygen interactions in a very primitive fashion, but to take into account additional degrees 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 data for this system, as will be seen in chapter 4, where our experimental results are presented and discussed. In chapter 5, the two approaches are merged to provide a more complete description, capable of fitting virtually all thermodynamic results for temperatures above 450K with just two free parameters. Since the behaviour of the electronic defects plays a major role in the thermodynamic model, this analysis provides a route to determining the planar hole count from the sample preparation conditions. This model also shows what is missing from the very popular lattice gas models as well as casting some doubt concerning the origin of the order-disorder transition.  v  Table of Contents  ii  Abstract^ Preface^  iv  List of Tables^  x  List of Figures^  xi xv  Acknowledgements^  xvi  Notation^ 1  Brief summary of the oxygen ordering problem in YBa2Cu306+.1.1  1.2  1.3  The structure of YBa2Cu306+s and the oxygen ordering problem  2  1.1.1^Basic structural overview ^  2  1.1.2^Structural stability and long range atomic order ^  4  1.1.3^The structural phase diagram ^  8  Oxygen dependence of the basic electronic properties ^  9  1.2.1^The insulating region ^  9  1.2.2^The superconducting region ^  10  1.2.3^What determines Tc?  12  Measurement of the oxygen chemical potential as a probe of the oxygen ordering and charge transfer ^  2  1  Introduction to lattice gas models in YBa2Cu306+. vi  14 17  2.1 Definition of relevant thermodynamic quantities  ^18  2.1.1 The reaction function and the chemical potential 2.1.2 The thermodynamic response function  ^18  (Ox/DOT ^  19  2.2 The 2D ASYNNNI lattice gas model for YBa2Cu306+s ^ 22 2.2.1 Criticisms and modifications of the 2D ASYNNNI model ^ 27 2.3 The cluster variation method ^ 2.3.1 CVM square approximation ^  29 31  2.3.2 Predictions of the CVM square, pair and point approximations^35 2.4 Defect chemical models ^  41  2.4.1 General introduction ^  41  2.4.2 Reaction model of Verweij and Feiner ^  44 50  2.5 Summary ^ 3 Experiment^  52  3.1 Experimental setup ^ 3.1.1 Design concept  52 ^52  3.1.2 Design requirements ^  54  3.1.3 Deoxygenation apparatus ^  55  3.2 Measurement ^ 3.2.1 Deoxygenation procedure ^  63 65  3.2.2 Measurement of the oxygen pressure isotherms and (ax/a,u)T^67 3.3 Systematic errors and their corrections  ^72  3.3.1 Temperature drift of the volumes ^  72  3.3.2 Thermomolecular pressure gradient ^  74  3.3.3 Impurity gas correction  ^76  3.3.4 Dead volume correction ^  vii  78  4  Results and Existing Theories  4.1  4.2  81  Experimental results ^  81  4.1.1^Oxygen pressure isotherms ^  81  4.1.2^The thermodynamic response function (Oxfait)T ^  85  Comparison to other work ^  87  4.2.1^Oxygen pressure isotherms ^  88  4.2.2^The thermodynamic response function (ax/att)T ^  89  4.2.3^Orthorhombic to tetragonal transition  4.3  ^  4.2.4^Discussion of the comparisons made ^  95  Fit to existing theories ^  97  4.3.1^Defect chemical models ^  97  4.3.2^Lattice gas models ^ 5  The Extended CVM model for YBa2Cu306+x  5.1  5.2  93  103 107  Connection between electronic defects and cluster configurations ^  108  5.1.1^Counting the holes ^  112  5.1.2^Counting the spins ^  114  5.1.3^CVM free energy with hole and spin degrees of freedom ^  117  Results and comparisons to experiment ^  120  5.2.1^Comparison to kT(Ox/a,a)T data ^  120  5.2.2^Comparison to the oxygen chemical potential: determination of the site energy ^  126  5.2.3^Comparison to the fractional site occupancies measured by neutron diffraction ^ 5.2.4^Phase diagram ^  viii  130 136  5.2.5 Predictions for the copper valence and hole count and comparison to XAS measurements ^  138  5.3 Commentary on the approximations made ^  142  5.3.1 Possible limitations and complications ^  144  5.3.2 Question about the nature of the O-T transition ^ 146 5.3.3 Suggestions for enhancements and future work ^ 148 153  5.4 Summary ^ 6 Concluding remarks^  155  Appendices^  158  A Calculation of the chemical potential for a diatomic gas... ^158 B The natural iteration method...^  161  B.1 CVM square approximation ^  161  B.2 Adding the spin and hole entropy ^  165  C Sample preparation and characterization^ C.1 Sample preparation ^  168 168  C.1.1 Final oxygen content ^  169  C.1.2 Impurity levels in the YBa2Cu306+s ^  170  C.2 Sample characterization ^ Bibliography^  170 175  ix  List of Tables  2.1  Calculation for the Kikuchi-Barker coefficients for the CVM square approximation ^  2.2  32  Predictions for the order-disorder transition for various CVM approximations ^  2.3  36  Configuration function for the oxygen 2p holes for the defect chemical model of Verweij in the orthorhombic phase ^  49  3.1  Calibration values for the volumes of the deoxygenation apparatus . . ^.  62  3.2  Values of A*, B* and C* given by Furuyama for oxygen.  ^  75  4.1  Cubic spline interpolation of the oxygen pressure plotted in figure 4.1 ^. .  84  4.2  Relative uncertainties of oxygen pressures at various representative pressure ranges for data listed in table 4.1 ^  5.1  85  Number of sites per unit cell available for hole distribution for the different possible assumptions of the Verweij model ^  113  5.2  Spin degeneracy factor for the various cases of the Verweij model. ^. .^.  117  C.1  Some physical parameters of the YBa2Cu306+x master batch ^  172  List of Figures  Chapter 1  1.1 Unit cell structure of YBa2Cu306 and YBa2Cu307  ^3  1.2 Schematic representation of the structural phase diagram  ^6  1.3 Phase diagram for the electron system ^ 1.4 Plot of T, vs. x for various works  11 ^13  Chapter 2  2.1 Schematic Diagram of the CuOx basal plane ^  26  2.2 Phase diagram as predicted by the CVM square, pair and point approximations and the 2D ASYNNNI TMFSS calculation of Aukrust et al. . . . 37 2.3 Predictions for kT(Ox/ay)T and the long range order parameter for the CVM square, pair and point approximations at kT/V --= 0.4. ^ 39 2.4 Comparison of kT(ax/(907, for the CVM point, pair and square approximations to the Monte Carlo results of Rikvold et al ^ 40 2.5 Schematic diagram of the change in Cu(1) valence as a function of its nearest neighbour oxygen occupation ^  46  Chapter 3  3.1 Schematic diagram of the experimental setup to measure the oxygen pressure isotherms of YBa2Cu3061, as a function of x ^  56  3.2 Schematic of the gas handling system showing the labeling of the valves ^ 58 3.3 Plot of the furnace dead volume as a function of temperature with a 33g sample of YBa2Cu306 loaded in the sample space ^ 61 xi  3.4 Block diagram of the instrumentation used in the isotherm measurements. 64 3.5 Basic flow chart for data acquisition program to measure oxygen pressure isotherms ^  69  3.6 Block diagram for the communication between the MKS Baratron pressure transducers and the personal computer. ^  71  3.7 Plot of the estimated thermomolecular pressure gradient vs. pressure for various temperatures. ^  77  3.8 Comparison of the vapour pressure at 450°C vs. x with and without the impurity gas corrections ^  79  Chapter 4 4.1 Plot of the measured oxygen pressure isotherms ^  82  4.2 Plot of kT(Ox/O,u)T vs. x in YBa2Cu306+x ^  86  4.3 Comparison of the oxygen pressure isotherms between this work and the data of McKinnon et al. and Meuffels et al. ^  90  4.4 Comparison of kT(Oxia,a)T between McKinnon at al. and this work . . ^ 91 4.5 Plot of the structural phase diagram of YBa2Cu3064„: theory and experiment ^  94  4.6 Comparison of kT(ax/a,u)T between experiment and the model of Voronin et al  ^99  4.7 Comparison of kT (0 x 10 ,a)T between the model of Verweij and Feiner. and experiment at 550°C in the orthorhombic phase ^  102  4.8 Comparison of kT(Ox/a,u)T between the predictions of the pure lattice gas models and experiment at 550°C ^  104  Chapter 5 5.1 Valence of Cu(1) for various nn oxygen configurations ^ 113 xii  5.2 Comparison of kT(Oxla,a)7, for the extended CVM models with chain and plane hole distribution and the data at 550°C ^  121  5.3 Comparison of kT(Ox1.0,07, for the extended CVM models with a restricted hole distribution and the data at 550°C. ^  122  5.4 Plot of kT(ax/a,u)T for the best fit cases for the electron hole distribution. 124 5.5 Plot of € + D/2 vs. T determined by comparing the f(II)F(II) model to experiment. ^  127  5.6 Plot of the experimentally determined chemical potential isotherms and the predictions of the f(II)F(II) model. ^  129  5.7 Plot of kT(ax/a,u)T vs. x at the temperatures of the experiment using the f(II)F(II) model ^  131  5.8 Comparison of the fractional site occupancy vs. ,u,IkTo between the neutron diffraction data and the f(II)F(II) model. ^  134  5.9 Comparison of the fractional site occupancy vs. T between the neutron diffraction data and the f(II)F(II) model ^  135  5.10 Phase diagram as predicted by the f(II)F(II), CVM square and the 2D ASYNNNI model (TMFSS) ^  137  5.11 Comparison of the phase diagram between the f(II)F(II) model and experiment ^  139  5.12 Prediction of the f(II)F(II) model for the number of Cui+ and comparison to the XAS data of Tolentino et al. ^  140  5.13 Plot of the amount of oxygen 2p holes vs. x from the f(II)F(II) model and comparison to the schematic behaviour deduced by Tolentino et al. . . . 143 5.14 CVM 4+5 point cluster for the basal plane oxygen. ^ 150 5.15 CVM 3x3 point cluster ^  152  5.16 Minimal size clusters defining an ortho-II and ortho-I region ^ 152  Appendix C C.1 Room temperature conductivity of YBa2Cu306+s vs. x for x < 0.08 .^171 C.2 Resistivity vs. T for YBa2Cu306.987 ^  173  C.3 Magnetization vs. T for YBa2Cu306.987 in a 10 gauss field ^ 174  xiv  Acknowledgements  I wish to thank first and foremost Walter Hardy, my research supervisor, for his continual guidance throughout the years. His kindness, open-mindedness and experience made work in the lab an enjoyable experience. To Ian Affieck, Jess Brewer and Jim Carolan, I extend the warmest thanks for accepting to be on the Ph.D. committee and for always displaying quite a bit of interest my work. I would like to thank Bingxin Yang for his early work in building the the main parts of the deoxygenation apparatus and for his preliminary vapour pressure measurements. Throughout the years, many individuals have come and gone who were at some point directly involved in the sample preparation and characterization. I thank Reinhold Krahn, David Brawner, Mark Norman, Alex O'Reilly, and Ruixing Liang. But especially, I extend my warmest thanks to Pinder Dosanjh, who has been a constant companion in the lab since 1986. The breakthrough in the theoretical understanding of the thermodynamics in this system came only this summer during my trip to Paris. Without this visit, chapter 5 would have probably never been written. I thank Walter Hardy for letting me leave for such a long time, Mike Hayden for organizing things at l'Ecole Normale Superieure, and the ENS as well as the Laboratoire de Physique des Solides d'Orsay for the use of their libraries during my stay. I thank especially Helene Casalta for her help in the analysis of the data and the write up of the thesis. Finally, I would like to say a special thanks to Peter Palffy-Muhoray for getting me here in the first place.  XV  Notation  Some symbols in the thesis y chemical potential  P pressure T temperature H enthalpy  S entropy E internal energy  SH partial enthalpy SS partial entropy F Helmholtz free energy O Gibbs free energy G grand potential fi reaction function  k Boltzmann's constant x susceptibility N number of particles (eg. oxygen atoms)  N, number of (oxygen) sites M number of unit cells V. nearest-neighbour effective pair interaction  Vci, next-nearest-neighbour effective pair interaction mediated by a copper atom Vv next-nearest-neighbour effective pair interaction (non-mediated) € site energy  D dissociation energy of an oxygen molecule xvi  W band width a oxygen sublattice denoted by a circle in figures /3 oxygen sublattice denoted by a square in figures x oxygen content in YBa2Cu306-Ex xor oxygen concentration at the O-T transition x oxygen concentration on a sublattice z° oxygen concentration on sublattice s long range order parameter (= (x" — x13)1 x) y23  pair cluster probability  zi3ki 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 cell n number of sites availible for hole occupation per unit cell g spin degeneracy factor  Acronyms AF Antiferromagnetic ASYNNNI Asymmetric next-nearest-neighbour interaction BCD Binary coded decimal CVM Cluster variation method  EPI Effective pair interaction LMTO-ASA Linearized muffin-tin orbital, atomic sphere approximation NI Natural iteration NMR Nuclear magnetic resonance NR Newton-Raphson O-T Orthorhombic to tetragonal SC Superconducting  TEP Thermoelectric Power TGA Thermogravimetric analysis TMFSS Transfer matrix finite size scaling XAS X-ray absorption spectroscopy  xviii  Chapter 1  Brief summary of the oxygen ordering problem in YBa2Cu306+  In the past five years there have been a number of materials discovered which can be called high temperature superconductors. Of these, the cuprate superconductors were discovered first and are by far the most studied. Initially, Bednorz and Milner discovered the (La,Ba)2Cu04 system to be superconducting with a maximum critical temperature 7', of 35K[1]. Such a critical temperature is higher than the maximum Te expected in standard 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. The discovery of superconducting (La,Ba)2Cu04 triggered an enormous research effort and the discovery of several other copper-oxide superconductors with higher critical temperatures. The first system which displayed superconductivity above the boiling point of nitrogen was YBa2Cu307, discovered by Wu et al. in February 1987{4 It is partly for this reason that YBa2Cu306+, is the most studied material of the high 7', superconductors. The existence of bulk superconductivity above the boiling point of liquid nitrogen promises a substantial reduction of operating costs of systems utilizing superconductive elements. But later, it also turned out that the YBa2Cu306-Fs system can be made very pure, and its properties are easily controlled by the variation of the oxygen content. Thus, because of its early discovery and promising physical characteristics, YBa2Cu306+s is perhaps the best understood material of the cuprate superconductors. 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 thought processes which lead to this discovery.  1  Chapter 1. Brief summary of the oxygen ordering problem in YBa2 Cu3 06-Ex^2  it is not the simplest system, both from a structural and electronic point of view. In YBa2Cu306+s, the electronic properties depend in a non-trivial way upon the atomic configuration of the oxygen. This is on the one hand convenient, since one can vary the electronic properties by manipulating the oxygen; but on the other hand, the interplay between the oxygen configurations and the behaviour of the electrons is quite complex and is only now beginning to be understood in detail. First, we will present the basic structure and structural phenomena which are observed in YBa2Cu306+s, and then illustrate what effect these have on the electronic properties of this system.  1.1 The structure of YBa2Cu306+s and the oxygen ordering problem 1.1.1 Basic structural overview The structure of YBa2Cu306+, is depicted in figure 1.1. It is a layered material consisting of 2 Cu02 planes separated by an yttrium atom, two BaO layers sandwiching the Cu02 bi-layer and a CuOs layer. The Cu(1) and 0(1), 0(5) are called the chain-site coppers and oxygens respectively. This CuOs layer is also called the basal plane. The 0(4) site is often referred to as the apical oxygen or interstitial site. For a comprehensive review of the early structural studies of YBa2Cu306+s we refer the reader to the review of Beyers and Shaw[4]. The stoichiometry can be varied from x = 0 to 1, by the addition of oxygen to the CuOs layer. This is accomplished by annealing a sample at an appropriate temperature and partial pressure of oxygen gas. The binding energy of the chain site oxygen is low enough that at elevated temperatures, the system develops a finite oxygen vapour pressure. The equilibrium oxygen pressure depends on the basal plane oxygen concentration and on the temperature (cf. figure 4.1 for a plot of the equilibrium oxygen pressure  Chapter I. Brief summary of the oxygen ordering problem in YBa2Cu3  YBa2Cu306  06-Fs  ^  3  YB a 2C U 307 0(5) chains  Cu(2)  o(2)  C  ity  a ^-1.-  • copper 0 oxygen barium yttrium •  e  Figure 1.1: Unit cell structure of YBa2Cu306 and YBa2Cu307. The various non-equivalent sites for the copper and oxygen are labelled according to the standard convention for this system. The open squares indicate unoccupied oxygen sites. At x = 0, the 0(1) and 0(5) sites are unoccupied. At x = 1, the 0(1) site is fully occupied and the 0(5) site is empty. At intermediate concentrations, there is a finite probability for 0(5) site occupation, but simultaneous occupation of the 0(1) and 0(5) site within a unit cell (nn oxygen occupation) is very small, if not zero. Note that the Cu(1) in YBa2Cu306 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 of the copper.  Chapter 1. Brief summary of the oxygen ordering problem in YBa2 Cu3 06+.  ^4  vs. x and T for YBa2Cu306+x). By setting an appropriate temperature and pressure  environment, one can control the oxygen stoichiometry. The occupation of oxygen at low x is random with an equal probability to occupy the 0(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 the occupation of the 0(1) and 0(5) sites is equal, the system is tetragonal with equal a and b lattice constants. As the oxygen content is increased, the system undergoes an order-disorder transition, where the occupation of the 0(5) site becomes depleted and the occupation of the 0(1) site is enhanced. This gives rise to the formation of one dimensional Cu-0 chains along the b-axis, so that the CuOs layer is also called copper oxide chains. As a result of the formation of chains, the b-axis length expands and the a-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) site can be depleted upon deoxygenation, but the amount is quite small, smaller than the error bars determining the occupation[5]. 1.1.2 Structural stability and long range atomic order It is commonly said that YBa2Cu306+s should phase separate at low temperatures into regions of different, discrete values of oxygen content (eg. YBa2Cu306 and YBa2Cu307). However, it is clear, experimentally, that YBa2Cu306+x is kinetically stable. At low temperature, the mobility of oxygen in the basal plane is low enough so that phase separation is never observed. The exact point at which phase separation should occur is not known and estimates are necessarily model dependent. Khachaturyan et al.[7, 8, 9] have suggested that phase separation into YBa2Cu306 and YBa2Cu307 should occur even above room temperature, whereas de Fontaine et al.[10] propose that there are many other stable ordered structures with intermediate oxygen concentrations, such that phase  Chapter 1. Brief summary of the oxygen ordering problem in YBa2Cu3 06+x^5  separation is very unlikely. Certainly, the experimental claims for the observation of a miscibility 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 on x and the transition widths are observed to be narrow for all values of x[13] (cf. also figure 1.4). If phase separation were to occur, then one would expect either a two step transition 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 is useful to show a two dimensional representation of the basal plane, where the ordering of oxygen takes place. The bottom three images in figure 1.2 shows the three well established types 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 the  0(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 a one dimensional chain structure with a fully occupied 0(1) and empty 0(5) sites. In the ortho-II phase, the 0(1) sublattice splits, giving rise to alternating full and empty chains along b. At x 0.5, this structure is ideal. Above x = 0.5, oxygens take up positions in the 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, since the ordering of oxygen gives rise to a macroscopic orthorhombic distortion (i.e. 3D registry of the 2D distortion). However, the ortho-II phase is more subtle and was detected initially 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 separates into two regions of distinct concentration. As the temperature is lowered, the difference in concentration increases. 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  ^  6  900 2' 2 700 ia) ci  E 500  F-  300  0.0 0.2 0.4 0.6 0.8 1.0 x in YBa2Cu306+x  ••0•0•0••--•-••0^•-•-••0•••••••• O  0 0 0 0 0^0 0 0 0 0 0  0•••0•••13•0••^13•0•0•0•0•0•0  .-_..  0 0 0 • 0^0 0 0 0 0 0 O ••0•0•0•0•0•0^••••••••0• 0^• 0 0 •^0 0 0 0 0 0 o • o /0•0•0•0•0^0•13•0••-•--••0•D 0 0^0^0 0 0 0 0 0 O ----.—•—•--. 0•0•0•0•0/0••^.--.—■ • 0 • .4,1.  O 0 0 0^0^0 0 0 0 0 0 0 • o • •--.--• •0•0•0^0•0 •• •0 •0•0 •0 0 0 0 0 •^0 0 0 0 0 0 O ••0•0•0•••080 •-•-•-•-•-•-•-•-•-•-•-e-/1  •-•-•-•--•  • D • •-•-•-•--•  0 0 0 0 0 O 11-0-11--•-•-•-•-•-•-•-•-•-11 O 0 0 0 0 0 ••0•0•••• - Ill . O  0 0 0 0 0  O  0  0 0 0 0  O  0  0  .  19-0-•-•-111-•-•-•-•-•-•-•--•  •-■-• . O• •^• 0 0 0 0 0 0 •--•--• • 0• ••• • ••• 0 0 0  .--,--.--.--11--,--111--4,--1*--1,--.--,--.  Tetragonal Ortho—II Ortho—I Figure 1.2: Schematic representation of the structural phase diagram. The top graph shows 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. The squares and circles correspond to the 0(1) and 0(5) sites respectively. The small filled circles correspond to the Cu(1). Other filled elements correspond to occupied oxygen sites. In the tetragonal phase, short chains are formed, but are randomly oriented. In the ortho-II phase, the tendency is to form chains along the b-axis (horizontally), but with every other horizontal chain empty. In the ortho-I phase, chains are formed along b in every unit cell.  Chapter 1. Brief summary of the oxygen ordering problem in YBa2Cu306+. ^7  for example ref. [15]). The surface probes showed the existence of the ortho-II phase between 0.28 < x < 0.65. However, the disadvantage of these measurements is that only the surface of the sample is probed. There have been few attempts using bulk techniques such as X-ray[16] or neutron diffraction[17, 18]. Mostly, these investigations suffered the problem of having only few superstructure reflections to analyse. These experiments established the stability of the ortho-II phase, but the observed correlation lengths were short (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 diffraction for a x = 0.51(5) sample the existence of a longer range ortho-II structure (18 lattice spacings along a, 135 along b and 6 along c at room temperature). Electron diffraction has also detected other types of long range order. In particular a 2•\/a x 2-Va phase, consisting of alternating half full and quarter full chains at x  0.35[20, 21]. However, it was disputed to not be the result of oxygen ordering  but rather surface desorption of copper and barium[22, 23]. In the meantime, superstructure reflections corresponding to such a symmetry have been observed by neutron diffraction[24], but that only 15% of the of the basal plane oxygen contribute to the superstructure. It was argued by de Fontaine et al.[25] that the 2.\/2-a x 2V2-a phase consists of many three-fold coordinated coppers, which are believed to be energetically unfavorable compared to two-fold or four-fold coordination[26]. On the other hand, Aligia et al.[27, 28, 29] claim that the the 2N/a x 2/a phase is indeed stable and that the oxygen ordering model of de Fontaine cannot be correct (cf. chapter 2 for a brief description of these models). These are very recent publications, showing that the issue of the correct oxygen ordering model and the interpretation of experimental results lending support to one or the other model is still under intense investigation. It should be noted that both models predict the stability of the ortho-II phase.  Chapter I. Brief summary of the oxygen ordering problem in YBa2Cu306 1 .^8 --  1.1.3 The structural phase diagram The boundary between the disordered tetragonal phase and the orthorhombic phase (0-T transition) depends on temperature[4, and references therein]. It has been measured by many groups using quite different techniques (cf. Voronin et al.[30] and chapter 4 for a list of some of the O-T transition results). A representation of the phase diagram is given in figure 1.2 (see also figure 4.5). This is the result of a calculation by Hilton et al.[14] for the 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 that this boundary curve is perhaps only qualitatively correct. The phase diagram only considers 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 a hirarchy of higher order superstructures at low temperatures; but except for the disputed 2 /a x 2'N/2ct phase, none other have been seen by bulk probes, presumably due to the extremely slow kinetics at low temperature. We see that the tetragonal phase exists for low x and high T, the ortho-I phase for high x and the ortho-II phase for low T and intermediate values of x close to 0.5. The extension of the ortho-I phase down to low temperature, close to the tetragonal phase boundary 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 at the top of the ortho-II phase. The justification for the use of these models is still under debate, 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 models are necessary in order to be consistent with certain experiments, but that the modifications do not strongly alter the phase diagram.  Chapter 1. Brief summary of the oxygen ordering problem in YBa2 CU3 06+x^9  Experimentally, only the O-T phase boundary, i.e. the boundary between tetragonal and orthorhombic (I or II) symmetry, has been measured.  1.2 Oxygen dependence of the basic electronic properties  The variation of the structural arrangements of the chain site oxygen, by changing the total oxygen content from x = 0 to 1 and also by quenching in varying types of long range order (for a given oxygen concentration), has a dramatic effect upon the electronic properties of YBa2Cu306-1-x•  1.2.1 The insulating region Figure 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 the Cu02 plane are divalent (Cu2+) with the unpaired electron spin having 3D long range antiferromagnetic order below 415K[35]. The chain site coppers are monovalent and non-magnetic. As oxygen is added to the chains, nearest neighbour Cu(1) to the inserted oxygen are oxidized and become divalent4. However, at some point (x 0.2) longer Cu-0 chains are formed for which it is energetically favorable to transfer electrons from the Cu02 planes to the chains: one has doped holes in the planes creating a p-type semiconductor. This hole doping quickly destroys the 3D antiferromagnetic order. This is due to the fact that the hole is not completely localized, has a spin which interacts with the Cu2+ spin, thus creating disorder which supresses the in-plane, 2D antiferromagnetic correlation length[37]. It is the coupling between ordered 2D domains in the third dimension which triggers the 3D antiferromagnetic order[38]. A reduction of the 2D correlation 4The term divalent must be understood to be the formal valence state and just means that the hole has a high probability to be on the copper site. This is a formal valence since there exists a significant 0(2p)-Cu(3d) hybridization[36].  Chapter 1. Brief summary of the oxygen ordering problem in YBa2  C113 06+x  ^  10  length by hole doping destroys the 3D order, but the 2D antiferromagnetism still exists over an appreciable length scale. At the metal-insulator transition, no more 3D long range antiferromagnetic order is 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 the inelastic neutron scattering measurements of Rossat-Mignod et al. [35]. The exact details of the curves depend on the sample preparation (cf. Brewer et al.[39]) so that the data plotted in figure 1.3 is only a qualitative representation of the system. 1.2.2 The superconducting region The order-disorder transition for the oxygen configurations in the basal plane, giving rise to the formation of long chains, results in a large transfer of holes to the planes. The system 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. In the metallic state, YBa2Cu306+x becomes superconducting with a 71, that depends on x and the degree of oxygen order. Referring to figure 1.3, we see that 7', developes a plateau at x '..-_- 0.6, which coincides with the stability region of the ortho-II phase (see also figure 1.4). This plateau does not appear 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 plateau is observed close to x = 1, where a maximum is seen at x = 0.93. Further addition of oxygen in fact reduces I', slightly (much recent attention has been focussed on this region, where significant changes in the electronic and magnetic structure is observed (cf. refs. [40, 41])).  Chapter 1. Brief summary of the oxygen ordering problem in YBa2Cu306+x  250  600 500  ^11  Semiconducting Tetragonal  Metallic  200  Orthorhombic  400 150H  gz 300 200  AF  100 Sc  100  o  50  ^ 0 0.0 0.2 0.4 0.6 0.8 1.0 x in YBa2Cu306+),  Figure 1.3: Phase diagram for the electron system as measured by Rossat-Mignod et al.[35]. The scale on the left indicates the Neel temperature TN for the transition to 3D long range antiferromagnetic order (AF). The scale on the right indicates 7', for the transition to the superconducting state (SC). In the tetragonal phase, the system is semiconducting and there is a transition to 3D long range antiferromagnetic order for the Cu(2) spins. At the metal-insulator transition, which coincides with the O-T transition, the system no longer exhibits 3D long range AF order, but instead becomes a superconductor. Note that although the 3D long range order dissapears at the onset of the SC phase, there are indications that short range magnetic correlations coexist with SC[45]. There are two superconducting plateaus in T. The first plateau at T o 60K coincides with the region of stability of the ortho-II phase and is due to a stagnation of the hole doping of the planes. The origin of the second plateau is less clear. It might be due to a saturation effect of 7', on the number of holes (see text). See also figure 1.4 which plots 71, vs. x from several experiments.  Chapter I. Brief summary of the oxygen ordering problem in YBa2Cu3  064-x  ^  12  1.2.3 What determines Tc?  The existence of two plateaus, at 60K and 90K, has sometimes been suggested to be due to the existence of two different superconducting phases. However, there is ample evidence that the plateau at 60K is due to a stagnation of the number of holes doped to the planes, as x increases. This is clearly supported by the valence bond sum of the 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, is plotted against the number mobile holes given by Hall effect measurements. A plot of T, vs. the number of mobile holes per unit cell, nH, gives an inverted parabola centered about nH 1, which is strikingly similar to the shape of the T, vs. y curve in La2_y SryCu04 and in agreement with the valence bond sum analysis of Whangbo and Torardi[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 in YBa2Cu306+x was due to the complex nature of hole transfer from the chains to the planes and its intrinsic connection to oxygen ordering. Very strong evidence that oxygen ordering modifies the charge transfer came from a series of papers[48, 13, 49, 46] which found that room temperature oxygen ordering occurs, increasing T, without modifying the total oxygen content. For example, in figure 1.4 we plot T, vs. x data of various groups. In particular, the solid diamond is a data point of Jorgensen et al.[46] which was quenched 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. It became very clear that not only the total oxygen content, but also the degree of oxygen order is important in determining the number of holes doped in the planes. There have been 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) bond stretching 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^13  100 80 2' 1-  1^1 0Jorgensen 500°C - Ill Cava 415°C VCava 440° _ •Poulsen Monte Carlo OBrewer 500°C •Jorgensen quench  60 40 20 0 ^„ 0.0 0.2 0.4 0.6 0.8 1.0 x in YBa2Cu306+x  Figure 1.4: Plot of 71 vs. x for various works. The temperature indicated after the name 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]. Jorgensen quench is the saturated 7', for a sample quenched from high temperature and annealed at room 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^14  principles" (cf. for example [50, 51, 52, 53, 54]). However, these calculations separate the problem of oxygen ordering and hole doping by assuming, a priori, the existence of oxygen ordered structures. There are also phenomenological models[55, 47, 56, 57] which assume a simple relationship between oxygen order and charge transfer and then calculate the hole doping by examining the configurational thermodynamics of oxygen ordering. 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 of ordered domains and utilizing the 2D lattice gas model of de Fontaine et al.[10] for the counting of ordered domains. A plot of some T, vs. x data is shown in figure 1.4 together with the prediction of Poulsen. However, the model assumes a linear 7', dependence upon doping, whereas some experiments suggest that the 90K plateau is due to a saturation effect of Tc[32, 58], so that the agreement at high x might be fortuitous. A more recent calculation by Lapinskas et al.[57], using the same 2D lattice gas model, but solving for the oxygen ordering using a mean-field type approximation and assuming that 7', varies quadratically with the hole count also gives a very good fit of the 7', vs. x data (we do not 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 oxygen ordering and charge transfer We have briefly described the structural aspects of YBa2Cu306+„ with an emphasis on the types of long range atomic ordering observed for the chain site oxygen. It was illustrated that the basic electronic properties are essentially determined by the planar electron hole count and that this hole count is set by the detailed configurations of the chain site oxygen. One of the goals of the experimental work in this thesis, and the theoretical analysis of the data, is to improve upon the understanding of the connection  Chapter 1. Brief summary of the oxygen ordering problem in YBa2Cu3 06+x^15  between the oxygen ordering and hole doping. Experimentally, the object is to measure the chain site oxygen chemical potential at temperatures between 450°C and 650°C as a function of x. This is obtained by measurements of the equilibrium oxygen partial pressure of the surrounding environment (cf. appendix A). A thermodynamic model that can fit the chemical potential will give the oxygen interactions, which are set by the electronic structure of the material. Thus, a measurement of the chemical potential indirectly provides information about the electronic structure. As one would expect, the oxygen chemical potential has been measured by innumerable groups, using a variety of methods'. What has not been commonly done is to measure the derivative of the chemical potential with respect to x, specifically the quantity (ax/a,a)T. The only measurements of (ax/a,a)T, aside from the results presented here, were made by W.R. McKinnon et al.[59) at one temperature, 650°C. There are several ways to view the meaning of (ax/(9,a)T; but the most pragmatic one is to say that by taking the derivative, one is strongly magnifying the fine details of the chemical potential. We will see that the standard lattice gas models, which are quite suscessful in predicting 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 established results concerning the electronic structure and its connection with charge transfer and hole doping, it will be shown that the creation of charge carriers directly influences the oxygen thermodynamics in a very unusual manner, and which a standard lattice gas model cannot generate. In essence, the creation of charge carriers giving rise to the metal-insulator transition has a strong influence upon the oxygen thermodynamics. Although this somewhat complicates the theory, one must directly include the hole creation 6We 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^16  mechanism into the model, so that it becomes possible to self-consistently solve for both the oxygen ordering thermodynamics and the charge transfer. A full theory for this is not developed, since a solution to the microscopic model for the electronic system is very complex. Instead, an approximate model will presented which results in a fairly good quantitative fit to the data and acts as a guide to a more complete theory.  Chapter 2  Introduction to lattice gas models in YBa2Cu306+s  Lattice gases are systems where the particles forming the gas take on discrete positions on a lattice, and where the kinetic energy of the gas is negligible. The particles are, in principle, interacting with each other. These interactions can be long range, extending over many lattice sites, or short range going to nearest or next-nearest neighbours. There is a direct one-to-one correspondence between a lattice gas and the Ising model in a field[60], which is important, since it allows one to carry over many fundamental ideas of the Ising model. The theory of lattice gases is essentially concerned with the computation of the configurational entropy of the particles which are placed on the lattice. There are many books which discuss the theory of lattice gases[61, 60], sometimes under the name of, for example, configurational thermodynamics[62] or order-disorder in alloys[63]. This chapter will be concerned with presenting and defining the relevant thermodynamic quantities, with a strong emphasis on (ax/att)T and outlining the various approaches one can use to solve the system. Also, since defect chemical models are in some form quite closely connected to the concept of the lattice gas, the general idea of the defect chemical model will be discussed.  17  Chapter 2. Introduction to lattice gas models in YBa2Cu306+x^ 18  2.1 Definition of relevant thermodynamic quantities 2.1.1 The reaction function and the chemical potential Before outlining the solution of the lattice gas model, some relevant thermodynamic quantities will be discussed in order to introduce some of the physical connections between experimental observations and their theoretical implications. We will follow the notation used by Verweij et al. [64, 65]. In any thermodynamic model, one can start by writing down the Legendre transform defining the Gibb's free energy (I) in terms of the enthalpy H and the entropy S: 0 = H — TS^  (2.1)  In a system with N particles distributed on N, sites, the chemical potential is defined as 1 (a0) II — No .9c )  (2.2) T,P  where c=N/No. In the case of YBa2Cu306+x, in fact x = 2c (cf. chapter 1) so that we write 2 (01. No ax )  T,p  (2.3)  Using equation 2.1 and defining the partial enthalpy and partial entropy as 2 (OH t5H -= — No ax) 7,,p 2 [ OS) --— SS No aX ) Tp  (2.4)  we can also write the chemical potential as y .--- SH —TSS^  (2.5)  In the range of oxygen pressures physically realized, the change in volume due to the change in pressure is negligible. So we drop the specification ... , P and can write it =  Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fx ^19  it(x, T). Essentially, we are dealing with an incompressible system where the conjugate variables (P, V) do not come into play. The pressure of oxygen is just determining the chemical potential it. In this sense, the Helmholtz free energy F, and the Gibbs free energy 43. are the same; the enthalpy is just the internal energy. The chemical potential, the partial enthalpy and the partial entropy are directly measurable 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 of separate entropic contributions S = S°+S1+S2+...+Si+..., where S° is the concentration independent 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 variables too, such as T and various order parameters, if they exist. In a pure lattice gas model, the reaction function is just a configuration function since the only entropy defined is the configurational entropy of the lattice gas and hence the emphasis on writing the reaction function as a function of x. Writing the thermodynamic variables in terms of the reaction function is convenient when one is examining the system within simple approximations, where the the number of defined order parameters is small, as for example, in the Bragg-Williams (or CVM point) approximation. The Bragg-Williams approximation forms, in fact, the basis for the defect chemical models, where one has not only the configurational entropy for the oxygen but also entropic contributions due to valence changes in the solid as a function of oxygen content.  2.1.2 The thermodynamic response function (0x/a,u)T The thermodynamic response function measured in this experiment is essentially the inverse 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-s  one obtains that  a _x^Ex ^x)T kT^7 1 kT ° itt T  +  where  E  20  -1  (2.7)  (^ ft(x)\ ]-1  as) T 7-7^[^ _ ^ —  fi (x)  (2.8)  This form might not seem at first sight very informative. However, what is made clear is the relationship between the partial enthalpy and kT(Ox/a,u)T. If the partial enthalpy is independent of the oxygen concentration, then kT(Oxiait)T just depends on the reaction function. Moreover, if the temperature is high enough then kT(3x/Oit)T is again only dependent upon the reaction function. Looking at this equation from another viewpoint, in defect chemical models and lattice gas models, the reaction function is usually independent of temperaturel Hence, if kT(Ox/O,u)T is measured to be independent of temperature, it means that the partial enthalpy is quite irrelevant to the determination of kT(Ox/a,u)T. This point will be important when an extension to current theories is needed to explain the discrepancy between the measurements of (Ox/OOT and the predictions of the theory. Fluctuation viewpoint of (axl0p)7, Since it can be shown that the lattice gas system is analogous to the magnetic system of the Ising model[60, 66, 63, 67], in principle all of the general ideas developed for magnetic systems 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 complete discussion of the the thermodynamics of the Ising model and its correspondence to the lattice 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 lower or much higher than kT. In the case where the interactions are very strong, then this just leads to an exclusion principle for the configurational entropy.  Chapter 2. Introduction to lattice gas models in YBa2 C1-13 06-Fs^  21  to the magnetic susceptibility. One can introduce the isothermal ordering susceptibility as the derivative of the order parameter m with respect to the corresponding symmetry breaking field 1468]:  (am)  x = ah  T^  (2.9)  In an Ising model, m is the magnetization and h is the applied magnetic field. The Hamiltonian of a lattice gas in the grand canonical ensemble is analogous to the Ising model 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 call  x an order parameter of the system. Instead, these systems undergo phase transitions where sublattices become inequivalent. Differences in the occupation probabilities of the sublattices become non-zero at the phase transition and these are identified as true order parameters. Thus, to distinguish (ax/19)T from fluctuations of the true order parameters 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^= N0((x2) — (x)2)^ (2.10) OXT  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 response functions in Monte Carlo simulations: (ax/a,u)T can be calculated from the straight forward evaluation of the difference in the thermal average (x2) and (x)2. Furthermore, one can show that the non-ordering susceptibility (ax/a/47, is related to the pair correlation 2This 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  ^  22  function. By definition, the pair correlation function is G ( k , l)  = (c kc i )  —  (c k )( 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,^2  (2.12)  where 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 the magnetic susceptibility in a magnetic system. In certain cases, kT(Ox/OF)T is dominated by 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 thermodynamic fluctuations and correlations, and should exhibit a significant structure at critical points.  2.2 The 2D ASYNNNI lattice gas model for YBa2Cu3064-. The essential structural detail, inherent in any lattice gas system, is the underlying lattice onto 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. There are 2 possible oxygen sites and one occupied copper site per unit cell. The oxygen in the basal plane is weakly bound to the lattice so that at elevated temperatures it forms a finite oxygen vapour pressure. It is essentially a 2D square lattice (the orthorhombic distortion 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,  Chapter 2. Introduction to lattice gas models in YBa2Cu306-Es ^  23  tetragonal phase, all oxygen sites are equivalent. In the ortho-I phase, the lattice splits into two inequivalent sublattices, denoted a and /3, where an oxygen site on sublattice a is surrounded by four nn oxygen sites of sublattice /3. This phase transition, as well as other transitions, such as the occurrence of the cell-doubled ortho-II phase are a result of competing interactions between nn and nnn oxygens. In the 2D ASYNNNI3 model, one assumes 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 and some comments about their nature will be made in the next section. Effective pair interactions The effective pair interactions which are used to model the oxygen ordering process in YBa2Cu306+x are an estimate of the effect of the structural modifications to the electronic orbitals and band structure which binds the material together to a solid. The following is a synopsis of the idea behind the effective pair interactions, as discussed in detail in the book of Ducastelle on the order and phase stability in alloys[63].The formalism developed, which allows one to approximate the effect of the atomic configurations on the band structure, comes essentially from the theory of order and phase stability in metallic alloys. By defining certain ordered atomic structures one can, using band theory, determine the effect of varying the atomic configurations on the band structure. In alloys, the interplay between band structure and atomic configurations can be solved using, for example, the coherent potential approximation[63] for the electronic sub-system and a mean field theory for the atomic configurations. From this analysis emerges the realization that the effect of the band structure can be modeled by introducing effective interactions between atoms. However, it is only a first approximation to assume that these effective interactions between atoms are constant in x and T. In fact, Ducastelle 3ASYmmetric Next Nearest Neighbour Interaction  Chapter 2. Introduction to lattice gas models in YBa2Cu306 fs^ 24 -  shows that the sign of the interactions may change depending on the number of electrons present. The approximation to assume that one can solve the configurational problem by assuming effective interactions between atoms given by the details of the band structure and to assume that these interactions are constants has only really been clearly justified in well behaved, wide-band metals. YBa2Cu306+s is not such an entity. Nevertheless, constant effective pair interactions have been used extensively to study the thermodynamics of oxygen ordering in YBa2Cu306+. Some of the justification for such an approximation is the relative insensitivity of the phase diagram to modifications of the EPI's[70, 71]. One of the more popular models is the 2D ASYNNNI model mentioned above. One can make some basic chemical arguments to predict the sign and order of magnitude of the interactions[25]. The nearest neighbour interaction Vin is expected to be large and positive due to a strong direct coulomb repulsion between nearest neighbour sites. Similarly, the next-nearest neighbour interaction without an intervening copper, Vv, is expected to be positive for the same reason, but its magnitude should be smaller since the distance is larger. The next-nearest neighbour interaction via the copper, Vc,„ is expected to be attractive due to the 0-Cu bonding of the Cu-3d orbital with the 0-2p orbital. Although we have defined the interactions only in two dimensions, the EPIs take into 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, and references therein]. They can be calculated from first principles using band structure calculations of a partially ordered system[63], although full advantage has not been taken of the formalism and techniques available for this problem. Indeed, it is questionable whether the standard alloy approach for calculating the effective pair interactions  Chapter 2. Introduction to lattice gas models in YBa2 C113 06+x^  25  is appropriate for YBa2Cu306+472, 73, 74, 29, and chapter 5]. Nevertheless, very approximate values for the energy of such configurations have been calculated through LMTO-ASA4 total energy calculations[75]. This estimate for the nn interaction V„„, and anisotropic nnn interactions Vc„ and Vv results in the correct stable ground states as seen experiment ally[76]. Also, tight-binding calculations show that a 3-fold coordinated Cu(1) is energetically unfavorable compared to a 2 fold or square planar (4-fold) coordinated Cu(1)[26]5. Such a situation, favouring 2- or 4-fold coordination over 3-fold coordination would be equivalent to 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 the PhD thesis of Henning Fries Poulsen[77] for a good review and introduction to various aspects of the ASYNNNI model in relation to YBa2Cu3064.x, especially concerning the ground state stability, low temperature properties and Monte Carlo simulations. Very recently, a least-squares fit to the structural phase diagram has been made, using transfer matrix finite size scaling (TMFSS) calculations of the ASYNNNI model, to obtain the best fit values for the EPI parameters[14]: Vfln  2800K  Vc„  =  —2380K  Vv  =  270K  (2.13)  Thus, the physical picture is of oxygen atoms which take up positions on a square lattice and who interact with each other up to next-nearest neighbours with anisotropic interaction parameters. The strong repulsive nn interaction results in the maximum physically realizable oxygen content of x=1 corresponding to half filling of the available 4Linearized-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 higher coordinated.  Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fx ^ 26  Figure 2.1: Schematic Diagram of the CuOs basal plane. The lattice is split into two interpenetrating sublattices. Sublattice a is denoted as circles and sublattice 0 is denoted as 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 lines correspond to the square clusters used for the CVM square approximation (discussed in section 2.3.1). Note that no distinction is made in the CVM square between a square with and without a central copper.  Chapter 2. Introduction to lattice gas models in YBa2Cu3 06+x^  27  oxygen sites in the basal plane. The strong and attractive nnn interaction mediated via the copper results, together with the non copper mediated, weaker and repulsive nnn interaction, in the occurrence of chain formation and the Ortho-II phase. A necessary condition for correctly predicting the observed well established ground states is that V. < 0 < Vv < V7in. Naturally, even in the case of Vct, and Vv being zero, one still sees 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 the tetragonal phase. The resultant phase diagram of the 2D ASYNNNI model with these interactions was mentioned in chapter 1. However, as was discussed in the chapter 1, YBa2Cu306+x undergoes a metal-semiconductor transition. This is expected to cause the effective pair interactions to change, since this transition entails a modification of the electronic structure. This point is mentioned in almost all papers on 2D lattice gas models in YBa2Cu306+s, but is usually then ignored.  2.2.1 Criticisms and modifications of the 2D ASYNNNI model In 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 in an insulator, except possibly Vct, at low x. 2. Non-zero hopping of the charge carriers gives rise to screening effects which will reduce the strength of the interactions exponentially with distance, with some screening length. 3. Charge transfer to the planes will reduce the next-nearest neighbour Vc24. This reduction is calculated from first principles from the extended Hubbard model.  Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fx ^  28  However, in ref. [70], (Ox/O,a)T is specifically calculated and the result seems only vaguely qualitatively correct, although it is quantitatively better than the pure 2D ASYNNNI model. The model is quite complex, utilizing interactions of up to 6th nearest neighbours and an x dependent Vcii due to charge transfer (the exact reduction of Vc„ can only be estimated due to the uncertainties in the parameters of the extended Hubbard model). It predicts the stability, as well as the correct relative magnitudes of the 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 cannot  predict the stability of the 2 V-2-ct x 2-‘7a phase, since interactions higher than nnn are required. The specific picture of Aligia has very recently been heavily attacked by de Fontaine 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-semiconductor transition upon the effective pair interaction and the resultant phase diagrams, but the modification was purely empirical6. Also, there are claims that the EPI's are not strongly dependent on the oxygen concentration[26, 81]. We see that the precise details of the oxygen effective pair interactions are still being sorted out. The discussion as to the nature and value of the effective pair interactions is very important 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 Yin only, which all models agree should be large and repulsive. Calculations in this thesis will be thus restricted to nearest-neighbour interaction models. This should be understood as an approximation to the full description using longer range interactions, justified by the 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 structural phase diagram as a function of the three effective pair interactions, and not to derive the concentration dependence of these interactions.  Chapter 2. Introduction to lattice gas models in YBa2Cu306-1-x ^29  with the higher order models. Before illustrating this insensitivity in section 2.3.2, we will present a commonly used mean-field theory for the solution of lattice gas models.  2.3 The cluster variation method The most difficult problem one encounters when trying to solve the lattice gas problem (without reverting to Monte Carlo methods) is the calculation of the configurational entropy. The reason for the difficulty lies in the complexity of counting the possible configurations of a lattice. As an illustration of this point, let us consider writing down the free energy for a particular cluster of N, sites. Given a particular configuration of oxygens in the cluster, one needs to calculate the internal energy and the entropy. ie .  F = (E) — kT ln SI No  (2.14)  where C/ is the number of possible configurations with the same internal energy (E). The calculation of the internal energy is easy, since it is a straight-forward counting of the number of bonds of a particular type. For example, in the 2D ASYNNNI model, where there are 3 effective pair interaction parameters, one has[72],  (E).  E nrvrc•^  (2.15)  r=1,2,3  where nr is the number of bonds of type r per lattice site, V, are the effective pair interactions, defined in equation 2.13, and is the pair correlation function for a bond of type r. The difficulty now arises in the determination of the configurational entropy for the 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 when the number of sites is large. One needs to utilize approximate methods to determine the configurational entropy. One technique useful for the calculation of the configurational entropy is the cluster variation method (CVM) first introduced by Kikuchi[82]. A  Chapter 2. Introduction to lattice gas models in YBa2  013  06+x  30  comprehensive review of the configurational thermodynamics of solid solutions has been given by D. de Fontaine[62]. Later, de Fontaine presented a tutorial introduction to the cluster variation method with emphasis on YBa2Cu306+,[33]. The specific presentation of 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 degeneracy of a small cluster exactly, but larger clusters are treated in the superposition approximation. The specific type of CVM approximation is denoted by the maximal cluster considered. In the point approximation, the cluster is just a single lattice site and is identical to the Bragg-Williams approximation. The CVM pair approximation takes a bond as the maximal cluster and it is identical to the quasi-chemical or Bethe approximation. For a square lattice, there exists the square approximation which is suitable, as in the CVM pair, for models with nn interactions only, and uses the square as its maximal cluster. For next-nearest neighbour interactions, one needs to go to higher clusters in order to correctly take into account the higher order interactions. For the 2D ASYNNNI 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 is written as = E 71,14-C — kT Exi(J) In xj(J) v0 r=1,2,... j J  (2.16)  where the first term was defined above in equation 2.15, x3(J) is the cluster probability for a cluster of type j with a particular configuration of occupied and unoccupied sites denoted by J. The sum over j runs over all subclusters" up to and including the maximal 7A subcluster is a cluster which is contained within the maximal cluster. For example, a nn bond is a subcluster of the square.  Chapter 2. Introduction to lattice gas models in YBa2 CU3 06+s^  31  one. The sum over J runs over all possible configurations that a particular cluster may take on. The -yi is the Kikuchi-Barker coefficient for the cluster j and can easily be calculated recursively by[86, 72, 33] -n1L  =^_ E a=i+1  mj  (2.17)  where mL is the number of clusters per unit cell of the maximal cluster, mi is the number of clusters of type i, nil? is the number of subclusters of type j contained within the cluster of type i. Next follows a brief derivation for the CVM square approximation. The solution for the 4+5 point approximation is derived in the review of de Fontaine[33] and the paper of Wille[72]. 2.3.1 CVM square approximation  The CVM square approximation is similar to the CVM pair, in that it is useful for calculations 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 maximal cluster is the square as opposed to the bond, it is expected to be more accurate and give a better approximation to the configurational entropy. This approximation is capable of giving rise to an order-disorder transition which in YBa2Cu306i„ is identified with the O-T transition. In the disordered phase, there is no distinction between the two sublattices and the total number of oxygen atoms situated on either sublattice is equal. The first task in deriving the entropy expression is to write down the maximal cluster and all its subclusters, then to calculate the -yi's according to equation 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-. ^ 32  mi j=1 1 1  m;? 2 3 2 3  4 4  -yi -1  1  2  4  2  1  2  0  1  -1  type point  i 1  o  bond  2  o-o  2  angle  3  g_o  2  square  4  le:  1  Table 2.1: Calculation for the Kikuchi-Barker coefficients for the CVM square approximation. i is the cluster type index, mi is the number of clusters of type i per lattice site and 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 by f2T  fo-o}2 f°}  (2.18)  Where we have used the CVM notation for the cluster product: {i} = F1.1(N0x3(J))!In order to take into account the existence of the order-disorder transition, one needs to split 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 up into two distinct cluster probabilities, xcl(J), x(J) and x(J), 4J) respectively. In this case, it is easy to show that the number of configurations in the orthorhombic phase, fio is given by {0-0}  2  (2.19) ^ {^1{012 fol2 Finally, using Stirling's approximation for the factorials, one arrives at an equation of C20 =  the form 2.16. Since we only have 4 cluster probabilities in this problem, we rewrite for the sake of clarity  Chapter 2. Introduction to lattice gas models in YBa2CU3 06+x^  33  point 4(J) = x^x(J) = bond^x2(J) = square x4(J) = zijki with i, j, k,1 E {0,1}, where the indices i, j,k,1 refer to the individual atoms on a cluster as  and where the value of the index indicates an occupied (1) or empty (0) site. Using this notation, the configurational entropy in the CVM square approximation becomes Ssquare  kNo  =2  E  yij in yij  1 zkllnzk1 - 2 i,j,k,1^  E in + i  xl? ln x}^(2.20)  Using the same notation, we can write down the internal energy of the system  E  N°^i,j,k,1  EijklZijkl^  (2.21)  where cijkl^17.011/2 =^k)(j /)/4 is the energy of the square cluster configuration, with Vnn, being the effective nn pair interaction energy. This formulation, in fact, allows one to take into account three or four body interactions, since they are just represented 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 the convention i, j, k, I E {0,1}. The cluster probabilities, x, 43, yij and zijki are not all independent. They are interrelated by[87] Xi^  E  Zijkl  ^  (2.22)  Chapter 2. Introduction to lattice gas models in YBa2Cu3 06+s^  0 =-x  E Zijkl  34  (2.23)  i,k,1  Yij  = E Zijkl  (2.24)  k,1  and the normalization condition  E zi3ki =1^  i,3,k,/  (2.25)  so that for the CVM square one has 15 independent cluster configuration probabilities (or cluster distributions) to solve for, some of which are degenerate due to the symmetries of the square, further reducing the number of independent variables. Nevertheless, one still has 6 independent variables remaining. The typical method would be to differentiate the free energy with respect to the six independent variables and use the Newton-Raphson iteration method to minimize the free energy. Unfortunately, for a system of 5 or more variables, it is very difficult to get the NR method to converge. Also, one needs to take analytic derivatives and insert them into the computer code, giving ample possibility for mistakes if the equations are complicated enough. A much simpler method applicable to some types of clusters was introduced by Kikuchi [87] to solve the free energy minimization problem. It is called the Natural Iteration method and takes advantage of the specific nature 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.9 Gsquare  No  E EijklZijkl E  E  -FkT[1 ZijkihlZijki — 2 yii In yii + — ln e + 2x . " i,j,k,1^ i,j^ _ it  1 E _(i+ i + k + 1)zijki  i,j,k,1 4  Excy  x'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. not in the CVM point) [63]. 9Note: pN = pNoc = Nopx.  Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fs ^ 35  where we have written the last term in its most symmetric form. With the grand potential minimized, one can calculate experimentally measurable quantities, such as the p, (040)T, the O-T transition, xoT, the long range order parameter, s (xia — 4)/x, and the short range order parameter, q = Yii (= number of nn sites occupied). For reference sake, since the next section will compare the predictions of the various approximations, we also give the form for the grand potential to be minimized in the CVM pair and CVM point approximations. In the CVM pair we have: Gpair  N, -FkT  [2 E yia ln^— 3  x,'' ln x7 +  E 4 ln x'3} I i^  1^.^. —P E^-V-1*  (2.27)  And, in the CVM point we have: Gpoint  N,  E Eiixia^13 -kkT  1  {E  1 ,.  ln  +^xi? In^}1  +  (2.28)  2.3.2 Predictions of the CVM square, pair and point approximations In this section, the predictions of these CVM approximations will be examined and compared to "exact" results of Monte Carlo simulations and TMFSS calculations in order to determine which approximation is appropriate for the calculation of kT(ax/Op)T. We will compare the predictions for the phase diagram, the long range order parameter and  kT(ax/(9,a)T. The comparison for the phase diagram for the CVM point, CVM pair  Chapter 2. Introduction to lattice gas models in YBa2  Approximation CVM point CVM pair CVM square Best known  013  06+x  ^  36  kToT/Vnn  1.0 0.7212 0.6057 0.567  Table 2.2: Predictions for the order-disorder transition at it = 0 for various CVM approximations. Values taken from the review of de Fontaine[62]  and nn interaction Monte Carlo calculations has been done by McKinnon et al. in the original paper on measurements of kT(.940,tt)T in YBa2Cu306.4,[59]. Figure 2.2 shows the predictions for the structural phase diagram, for the same CVM approximations in addition to a more recent 2D ASYNNNI model calculation[31]. We see that as the maximal cluster size is increased, one gradually approaches the prediction of the TMFSS and Monte Carlo calculations of Aukrust et al[31]1°. A measure for the accuracy of these approximations 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 TOT at ft = 0 for the various approximations[62], which shows that at high temperatures, the CVM 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 occurrence of the ortho-II phase, these nn CVM approximations rapidly begin to fail. This is expected, since they do not account for the existence of the ortho-II phase. The validity of the CVM square approximation is thus dependent upon the temperature scale physically realized in YBa2Cu306+s. If the maximum temperature of the ortho-II phase is small enough, the CVM square approximation should be usable. In the latest TMFSS 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 currently accepted values for the EPIs, the resultant phase diagram at high T is not very different from more current calculations.  37  Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fx  1.0  ---- CVM Point ^ ^ CVM Pair ^ CVM Square x, TMFSS i , ,, , , , , ••••  0.8 0.6  i ../  _  ••••  . .......  .............^—  _  /  /  1--' 0.4  / / / / / / / / /  0.2  -  / ///  _  ^  / /  / /^ / /  0.0^' 0.0 0.2 0.4 0.6 1  1  ,  i^.  0.8^1.0  x in YBa2Cu306+x Figure 2.2: Phase diagram as predicted by the CVM square, pair and point approximations and the 2D ASYNNNI TMFSS calculation of Aukrust et al[31]. The lines correspond to the boundary between the tetragonal, disordered phase at low x and the orthorhombic, ordered phase at high x. See figure 1.2 for a definition of the different regions.  Chapter 2. Introduction to lattice gas models in YBa2Cu306+,^ 38  at ,550K. The measurements of kT(Ox/O,u)Tpresented in this work are between 723K and 923K, so that one could imagine that the effects of higher correlations due to the next-nearest neighbour interactions are not dominant. This will be more obvious in the following 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 approximations for the non-ordering susceptibility. First, figure 2.3 shows a plot of kT(Ox/O,u)T and the long range order parameter s for the CVM point, pair and square approximations at kT /V,,=0.4 (the value 0.4 is a reasonable but quite arbitrary choice). The long range order parameter is defined as  x1a— 0 xi  s =-xT€ +  (2.29)  We see that there is a very prominent feature in kT(Ox/aft)T at the order-disorder phase transition, which is as expected (cf. section 2.1.2). Below the peak, one is in the tetragonal phase and the long range order parameter is zero as seen in the bottom graph. Above the transition, one is in the ortho-I phase and the order parameter rises to one. At the transition, the order parameter rises very rapidly, but continuously, indicating a second order phase transition. In the tetragonal phase for very low x, kT(Ox/(9,a)T has the same slope as a non-interacting lattice gas for particles distributed randomly over all 2x basal plane oxygen sites', but the fluctuations are quickly suppressed due to the nearest neighbour interaction. In the orthorhombic phase, the different CVM approximations have qualitatively different shapes, except near x=1. For x '--1, kT(Ox/O,a)T has the shape of a non-interacting lattice gas for particles distributed over x sites, with a small correction due to some finite nn occupancies. In the intermediate regime, however, the curves are quite different. The curvature changes sign from the CVM point to the CVM square. random 11I11 the random case, kT(Oxlap)2,^= x(1— x12), with a maximum of 0.5 at x=1 and a slope of 1 at x=0.  Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fx  0.5 0.4  CVM square with kT/V„=0.4 ^ CVM pair - - CVM point  —;" —0-: 0.3  ••■  ••■■  }-13>< -_,- 0.2 ...........................  0.1  •■•••  0.2 0.0 ' 0.0 0.2 0.4 0.6  0.8^1.0  x in YBa2Cu306.fx Figure 2.3: Predictions for kT(Ox/aA)T and the long range order parameter for the CVM square, pair and point approximations at kT/V;in = 0.4.  39  Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fs ^ 40  0.60 0.50  02D ASYNNI Monte Carlo CVM square CVM pair -- CVM point  0.40 0.30 0.20 0.10 0.00  0.0 0.2 0.4 0.6 0.8 1.0 x in YBa2Cu306+x  Figure 2.4: Comparison of kT(Ox/O,u)T for the CVM point, pair and square approximations 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 CVM square.  Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fx ^ 41  In order to decide as far as (ax/a,u)T is concerned which maximal cluster of the CVM approximation is sufficient to qualitatively describe the 211 ASYNNNI model, we plot in figure 2.4 the CVM predictions and the Monte Carlo results for the 2D ASYNNNI model[69]. It is clear that the CVM square approximation is already quite sufficient in describing (Oxia,a)T, especially in the orthorhombic phase. Later, when these predictions are compared to experiment, it will become clear that the discrepancies between the 2D ASYNNNI model and the CVM square are very small compared to the discrepancy between theory and experiment, so that the CVM square should form an adequate platform 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 correctly predicting many structural phenomena (eg. phase diagram and stable ground states) but not the chemical potential, which contains this structural information in addition to other effects.  2.4 Defect chemical models 2.4.1 General introduction As mentioned in section 2.2, the effective pair interactions could, in principle, be concentration dependent. There are many effects which could give rise to a concentration dependence (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 transition 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 potential measurements of YBa2Cu306+x is the so called defect chemical approach. Although defect chemical models are not strictly lattice gas models, there are similarities in that  Chapter 2. Introduction to lattice gas models in YBa2 Cu3 06+x^  42  defect chemical models usually employ the Bragg-Williams approximation to describe the entropy of the various configurational variables of the system. In this sense they are lattice gas models; however they contain variables other than the configurational variable of the particles placed on the lattice. We note that for oxide superconductors and YBa2Cu306+x in particular, the addition of oxygen into the basal plane cannot be made without the introduction of electronic defects as well. Since Cu is a transition metal, 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 which may be loosely bound to a particular atom. In other words, the placement of oxygen onto the basal plane creates holes which may be free to hop from site to site and give rise to p-type semiconductive behaviour. Also, there is the effect of charge transfer from the basal plane to the Cu02 bi-layer. These effects, giving rise to, amongst other things, the metal-insulator transition, could in principle be important to the oxygen thermodynamics One way to see this is to realize that the stoichiometry is controlled by the oxygen partial pressure. Thus, changes in the oxygen pressure will result in a modification of the electronic structure. The response of the system to changes in the chemical potential will depend on how the electronic structure behaves'. Essentially, in the defect chemical approach, a localized picture of the electronic defects is used where the holes are considered to be particles which can be placed on certain atoms. Typically, a random configurational entropy for the hole placement on the lattice is used. The expression for the hole entropy depends on the specific picture proposed. There have been many such defect chemical models proposed for YBa2Cu306+„ Specifically, there are basal plane reaction models, where it is assumed that the complete 12And 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 ^ 43  reaction takes place in the basal plane, including the charge distribution (cf. for example refs. [88, 89, 90, 91, 30, 92]). Also, there is the alternative model of Verweij and Feiner[64, 65] where the effect of charge transfer to the Cu02 plane is crudely taken into account. 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 is assumed, and the O-T transition is introduced artificially by proposing different models for 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 0 sublattice is empty. Since a random entropy is assumed in these models, it is very simple to write down the reaction function defined in equation 2.6. In particular, the reaction function is composed of three parts: a configurational term for the oxygen fc(x), a configurational term for the holes (or valence options of the Cu) fh(x), and a spin term of the holes fs(x). Assuming such 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 the concentration independent part of the entropy. The non-ordering susceptibility is given by (cf. equation 2.7) (  kT  °P)  7-7  Ex 1 +  :T) T)  -1  a  kT  fc(x)fh(x)fs(x)  Mx)fh(x)fs(x)-F fc(x)fi,(x)fs(x)+ fc(x)fh(x)Mx)  (2.31)  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 Cu  Chapter 2. Introduction to lattice gas models in YBa2Cu306+r ^ 44  valence). Early defect chemical models did not have much (and some times incorrect!) information about the valence situation in these materials. Consequently, we will not reproduce here all of the proposed oxidation thermodynamic models which exist in the literature, since many are now out of date. Instead, we will present the model of Verweij and Feiner[64, 65], since it gives the "flavour" of the defect chemical models, seems to be consistent with the current valence picture in YBa2Cu306+s and forms part of the extended CVM model of chapter 5. 2.4.2 Reaction model of Verweij and Feiner At x=0, all the Cu(1) is two-fold coordinated in the Cul+ valence state. This means that the 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 electron to 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 difficult question. If the oxygen is isolated, then it is again the same. However, if an oxygen is added to the next nearest neighbour site, then the central copper is now four-fold coordinated (cf. bottom graph in figure 2.5). It may share its two outer electrons with the two neighbouring oxygens, making it effectively a Cu3+(3d8). It is also possible to assume that the copper stays Cu2+ and an oxygen 2p hole is created. More likely, the hole is shared between the oxygen and copper (cf. chapter 5). One sees that there are several options to choose from. Can the Cu2+ valence "hop" from site to site? Is there Cu3+? Should one take into account charge transfer to the Cu02 plane? Is the nature of the chain holes different from the planar holes? Must one take into account holes in the 2p, orbital of the 0(4) site? It should be noted that in this description we are speaking about the formal valence, corresponding to the oxidation state of the copper. In fact,  Chapter 2. Introduction to lattice gas models in YBa2C11306-Fx ^ 45  these 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 valence options but is restricted to the orthorhombic phase. It is assumed that the occupancy of oxygen is random and restricted to one sublattice (i.e. s = 1). The valence of the copper 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+ and Cu2+ valence may not "hop" from site to site. This means that configurations such as 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 to the Cu02 plane. In other words, it is assumed that at x = 0, all Cu(1) is Cul+. Isolated oxygens create two Cu2+ as shown in the center graph of figure 2.5, but that there is no configurational 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 a hole that may hop and give rise to a configurational entropy. The rule is thus, that the copper is Cu l+ when neighboured by two oxygen vacancies, Cu2+ otherwise. Additional charge compensation is afforded through the creation of holes (either on the Cu or on the oxygen). These holes give rise to a configurational entropy.  Notation We 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 until chapter 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 we have M unit cells, N, = 2M.) [Cu] + [hole] = 2x^  (2.32)  Chapter 2. Introduction to lattice gas models in YBa2 Cu3 06+. ^ 46  +1  +2  +1  ^  +2  ^  +1  +1^+2^+2^+2^+1 ^•  + hole Figure 2.5: Schematic diagram of the change in Cu(1) valence as a function of its nearest neighbour oxygen occupation. Vacant oxygen sites are drawn as large open circles. Occupied oxygen sites are large filled circles. The coppers are drawn as small solid circles and their valence is shown above each copper. In the top graph, there are no oxygens so the valence is 1+. In the center graph, the isolated oxygen changes the valence of the two 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 ^ 47  In 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 the probability 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 these holes reside (either in the chains, in the planes, on the copper, on the oxygen or a mixture of everything). Also, there is the question of the spin of any free Cu2+ and the spin of the 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 notation will be useful in chapter 4, when the predictions of Verweij will be compared to experiment and in chapter 5 where this model will be combined with the cluster variation method in 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 oxygen and 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. distributed over 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 oxygen and 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 having two oxygen nearest neighbours, i.e. distributed over the [Cub2p1N0 copper sites which are in the interior portion of the chain fragments. This is denoted as b.  Chapter 2. Introduction to lattice gas models in YBa2Cu306+. ^ 48  (e) The holes are divided between the chains and planes in the proportion to the number 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) or triplet (III) with the Cu2+ to which it is bound. The following reaction functions are thus defined:  f(x) =  x (1 — x)  (2.33)  and f(x) = s  1 (2)  2x  ^  (2.34)  where g is the spin degeneracy factor. g = 2 for , 1/2 for I or II, 3/2 for III and 1 for IV. Validity of the model Naturally, some of these special cases seem unphysical, especially the cases where the holes are constrained to be either on the chains or the planes. If the holes were restricted to be in the chains, then the planes would not be doped and no superconductivity would occur! If the holes were restricted to the planes, then there would be 1 hole in the planes 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 possible that at lower temperatures, a modification of the charge transfer takes place, but this is unlikely since, as Verweij points out, there is no evidence either in conductivity, Hall  Chapter 2. Introduction to lattice gas models in YBa2Cu306-Fx ^49  Hole location F or B:  fh(x) 2^2x  [2nx— x2] 2  F-f or B-f: .  F b or B-b. -  [2n +xx — x2]  2n-Fx—x [^2n + x  2x  21  x2 2Y { 2n+x21 r^x^-12x  f: b:  (1 1  —  x) [1 —x  Table 2.3: Configuration functions for the oxygen 2p holes for the defect chemical model of Verweij in the orthorhombic phase. See text for notation. For F models n = 4 and for  B models n = 2. effect or specific heat measurements that a significant change in the hole distribution takes place. Essentially, this is a localized description of charge carriers, where the configuration index is assumed to be a good quantum number. This might seem physically unreasonable, since the charge carriers in the metallic state are typically described by a band picture with delocalized states. However, if the band width W is much less than kT, then all single particle states in the band have an equal probability to be occupied, and the band 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 predictions for the copper valence and hole count are consistent with experimental results from, for example, XAS13 measurements. In the tetragonal phase, a random solution model for the oxygen will give [Cul] = (1 — 1-x)4, implying that [Cu] = 1 — (1 — -12-x)4 and hence  '3X-Ray Absorption Spectroscopy  Chapter 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 precisely what is deduced from recent XAS measurements of Tolentino et al.[94] in the limit of low x. Also, the corresponding equations for the orthorhombic phase presented above in section 2.4.2 are consistent with the findings of Tolentino, in the x —* 1 limit. In the intermediate regime, Tolentino finds that the behaviour of the hole count is more complex than a random solution model could predict, requiring a refinement due to the oxygen ordering effect. This is not surprising, since a random solution model for the tetragonal phase grossly underestimates the effect of the nn oxygen interactions'''. In the orthorhombic phase, once a significant sublattice splitting has developed, the nn interactions are no longer important and it is more reasonable to postulate a random solution model. Since the number of possibilities for this model are quite numerous, the presentation of (ax/.9,01, for this model will be delayed until chapter 4, when the experimental results are shown.  2.5 Summary We have given in this chapter a resume of the two basic approaches used to study the thermodynamics of oxygen in YBa2Cu306+r. One the one hand, there are the pure oxygen 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 other hand, one has the defect chemical models, which utilize a very basic picture for the oxygen interactions, but also include other configurational variables, such as hole and spin degrees of freedom. Although Verweij's basic idea concerning the hole count is consistent with 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 tetragonal phase.  Chapter 2. Introduction to lattice gas models in YBa2CU3 06+x^  51  for the oxygen configuration needs to be refined in order to coincide with the structural predictions 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 serious problem. The predictions of these two approaches will be compared to the experimental results of (Oxiatt)7, in chapter 4. It will become clear that the two models need to be merged so that a consistent picture results. But first, the experimental setup for the measurement of the non-ordering susceptibility will be presented.  Chapter 3  Experiment  This chapter is composed of three parts whose purpose is to explain in detail the design concept of the experimental apparatus, the measurement procedure and the processing and corrections applied to the raw data to ultimately end up with the oxygen pressure isotherms and (Oxlay)T. As was outlined in Chapter 1, the goal of the experiment is to obtain accurate curves of (0x10,a)T in order to test the theoretical models for the oxygen ordering in YBa2Cu306+s. Although the apparatus built specifically had this experiment in mind, it should be noted at the outset that this setup can be used for other purposes such as, for example: preparation of samples with a well controlled oxygen stoichiometry and oxygen isotope substitution. Indeed, this apparatus has also been used to prepare deoxygenated samples of YBa2Cu306+x for muon spin rotation studies[39, 96, 97, 45, 98], and 170 substituted samples to help locate the site of the positive muon in YBa2Cu306+s[99]. Accordingly, this chapter not only describes the experimental method used for the present work, but also provides enough information to act as a reference for the general use of the apparatus.  3.1 Experimental setup 3.1.1 Design concept The apparatus was designed for the study and manipulation of high 'I', oxides which have properties dependent on their oxygen stoichiometry and whose oxygen content can be  52  53  Chapter 3. Experiment^  altered 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 bound oxygen sites within the structure. There are several experimental techniques one can use 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 stoichiometry and to measure the equilibrium oxygen pressure at constant temperature. One way to do this is to have an appropriately designed gas handling system which can be used to extract oxygen from the sample. Unfortunately, as oxygen is removed from the sample, the equilibrium pressure decreases rapidly (P-- exp  x). This means that  one can run into serious trouble if the desired target oxygen concentration of the material is low enough. The key idea to get around this problem is to include in the gas handling system an extra volume which can be cooled to liquid helium temperatures. A volume at liquid helium temperature will act as pump and remove oxygen from the sample. One can keep track of the amount of oxygen removed by warming the the volume to room temperature 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 potentially most accurate method' is to start with a sample with a low oxygen content, xa-_0, and then use the gas handling system to set a small increase in the oxygen content, Ax, by titration and measuring the rise in the equilibrium oxygen pressure, P. i.e: ,  ax  2Ax  (— )^ :--2^ ay ^' T A (log P)•  To obtain an accurate measurement of  (3.1)  (ax/Oft)T, one has to be able to set small and  precise 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^  54  magnitude (between x f_-_-0.05 and 0.95, the pressure changes by six orders of magnitude in YBa2Cu3061-x)• 3.1.2 Design requirements  The main requirement for the apparatus was to allow for a precise enough titration and pressure measurement to enable one to take the derivative of the experimental curve with an acceptable amount of scatter. Aside from the obvious desire for accurate measurements of 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 oxygen pressure should be made as small as possible to minimize the "dead volume", since oxygen 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 should not be too small, otherwise at low enough pressures, thermal transpiration can introduce significant errors (cf. section 3.3). 2. One should have good control and measurement of the sample temperature during an experimental run, since the equilibrium pressures depend sensitively on temperature. For example, for YBa2Cu306+s, log P ,s, T. 3. It is important to control the temperature of all parts of the the gas handling system, since the oxygen content is deduced from knowledge of the pressure of oxygen in reference volumes of the gas handling system, plus corrections for oxygen in dead volumes. 4. If equilibration times are long, the lowest measurable pressure can be limited by the background outgassing of the walls. Thus, the design should not contain plastics or other materials that may act as absorbants and subsequent sources of impurity  Chapter 3. Experiment^  55  gases. In addition, that part of the sample holder which will be at high temperature should be highly inert. Stainless steel, for example, will oxidize at high enough temperature and pressure, acting as a sink for the oxygen gas. Quartz is a good choice. 5. Finally, several key valves of the gas handling system and all of the instrumentation  should 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 each measurement of pressure and (Ox/ay)T. 3.1.3 Deoxygenation apparatus Overview  Figure 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 right the quartz sample volume extending into a single zone tube furnace (Lindberg model 59544 1200°C). Starting from the sample space on the right and moving to the left, the individual components are: The sample, consisting of roughly 60g of material, in the form of 3/4 inch diameter, 3g pellets, wrapped in Pt foil, and placed into the quartz tube; an evacuated quartz bulb, containing two stainless steel radiation shields, inserted into the sample space to minimize the dead volume, to cut down on the radiation striking the stainless steel connector to the gas handling system and to minimize errors in the temperature of the sample. The connection to the gas handling system is made using a quartz/Pyrex graded joint followed by a Housekeeper seal. The Housekeeper seal is welded 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 55g  41''  Chapter 3. Experiment^  56  Figure 3.1: Schematic diagram of the experimental setup to measure the oxygen pressure isotherms of YBa2C11306+x as a function of x. The dead volume of the main quartz tube holding the sample is minimized by the use of a sealed inner quartz tube. This inner tube 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. An alumina and fire brick encasement surrounds the thermocouple and quartz tube.  Chapter 3. Experiment^  57  sample of YBa2Cu306+,, the amount of water and CO2 caught by the cold trap can be as much as 10 Torr in Vs+Vp+Vd (cf. Figure 3.2) which, if not removed, would result in a very large error in the oxygen pressure measurements. The liquid nitrogen cold trap is 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 steel and 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 Nupro BN series bellows valves, two of which are air operated and under computer control. The entire system thus contains only two small pieces of plastic material, namely from the air operated valves, which have a Kel-F plastic plunger tips to reduce the sealing force. The background outgassing from the stainless steel walls and the quartz assembly is roughly 1mTorr 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 1/2 inch 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 direct contact with copper plates and/or tubes connected to a Haake refrigerated circulating water bath. The sample temperature is measured with a Pt-Pt13%Rh thermocouple (Aesar, secondary standard grade, +1°C absolute calibration) which is mounted outside of the quartz tube next to the sample. The entire thermocouple and sample region is fitted into an alumina and fire brick assembly to ensure a homogeneous and well defined temperature2. 2It was found that without this assembly temperature readings varied significantly, depending on thermocouple placement.  Chapter 3. Experiment^  58  Figure 3.2: Schematic of the gas handling system showing the labeling of the valves (small v's) and of the volumes (large V's). Valves 5 and 6, labeled with an A, are pneumatically operated under computer control.  Chapter 3. Experiment^  59  Volume calibration Initially, the volume of the reference container, Vr, is calibrated by measuring the weight change of the container after backfilling with degassed, distilled water. Then, the remaining volumes in the system are all calibrated by stepwise titration of oxygen into the appropriate volume and subsequent removal of the gas either by freezing into the helium wand or by evacuation through the pump.  a) Reference volume calibration The 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 a Hoke 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 scale accurate to 0.01g and capable of measuring weights of up to 3000.00g. Then, through the valve, degassed, distilled water is backfilled into the volume and the assembly is weighed again. Before sealing the volume containing the water, its temperature is measured. This procedure was repeated several times and gave consistent results. Finally, the valve is removed and its volume is measured with calipers and also by adding water to its volume. This volume (L..s'_ 0.5cm3) is subtracted from the calculated volume and the volume of the pressure transducer (=2.80cm3 from manufacturer's specification) is added. The uncertainty quoted is a result from the reproducibility measurements as well as taking into consideration uncertainties in temperature and density of water due to possible absorption of gases[100]. Finally, V, is reconnected to the system.  b) Subsidiary volume calibrations The reference volume, Vr, is filled with roughly 3 atm of 02 gas. To calibrate Vs, for example, 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^  60  through v1. One can show that by this procedure, a plot of logft-, vs. i (Pi and Ti are the pressure and temperature in V, at the ith titration) will give a straight line whose slope, m, is given by (exp(—m) — 1) 17„. (3.2) In this manner all volumes, or combination of volumes were calibrated at room temperature. For the sample volume (also called dead volume), Vd, a new calibration was required whenever the sample was changed, since the calibration was made with the sample in place. In addition, since the sample space is at an elevated temperature, one needs to calibrate Vd as a function of temperature, i.e. Vd is an effective volume due to the existence of the temperature gradient. From the ideal gas law, the effective dead volume should go as  Vd(T) =14, +  A  (T T0)  ^  (3.3)  where Voo, A and T, are parameters to be fitted. For the dead volume calibration, the sample is deoxygenated to a low value in order that the partial pressure of oxygen due to the sample is negligible, and N2 gas is used as the buffer gas in order not to reoxygenate the sample. Figure 3.3 shows a plot and the best fit line to equation 3.3 for a 33g sample of YBa2Cu306. It is assumed that surface adsorption of N2 gas onto the sample is not significant. There is no indication of the incorporation of nitrogen into the structure of YBa2Cu306+,. Finally, it should be noted that the pressure dependence of the effective dead volume was checked and found to not change significantly within the resolution of the instrumentation. Such a pressure dependence could, in principle, arise due to varying thermal transport conditions along the quartz tube as a function of pressure, which would change the temperature profile along the quartz tube. Table 3.1 shows the values for the various volumes obtained by the above calibration  Chapter 3. Experiment^  61  80 cr),, E 78 0 a) E = Tj  76  >74 p cri a) cn a) 72 .> -.6 a) w70 68 200 300 400 500 600 700 Temperature (°C) Figure 3.3: Plot of the furnace dead volume as a function of temperature with a 33g sample of YBa2Cu306 loaded in the sample space.  Chapter 3. Experiment^  Volume V, Vs V, V,  Vd  62  Size 998.13 +^0.08cm3 14.756 +^0.001cm3 13.199 +^0.01cm3 127.49 ±^0.05cm3 ''' 4 7 c m, 3 and T dep.  Table 3.1: Calibration values for the volumes of the deoxygenation apparatus procedure. It should be noted that the most crucial volume combinations were calibrated with the most care, taking into account the valve positions (ie. whether they were opened or closed).  Instrumentation Now 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 sample space is measured using two MKS Baratron type 310 pressure transducers (1 Torr full scale and 1000 Torr full scale) giving a measurement range between 1 mTorr and 1000 Torr with an accuracy of about 0.1% of reading. The analog output of the MKS Baratron type 170M controller is fed into an HP 3478a 5 digit voltmeter to improve the precision of the pressure reading (The Baratron controller is configured with a 4- digit readout unit, which gives digitization noise during signal averaging). The pressure in the reference volume 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 an ice bath using a Pt-Pt 13%Rh thermocouple connected to an HP 3478a voltmeter. The sample temperature and pressure are both monitored on a Philips PM 8252A two pen chart recorder. A home built, computer controlled solenoid switch is used to control the two air operated valves. Finally, all the instrumentation is connected via the IEEE-488  Chapter 3. Experiment^  63  interface bus to a personal computer (IBM compatible 286) enabling automated data acquisition.  3.2 Measurement In this section, the method used to deoxygenate samples of YBa2Cu306.fs and the experimental procedure for the measurement of the oxygen pressure isotherms and (ax/att)T will be described. As mentioned in Chapter 1, it is generally accepted that YBa2Cu306-Fs can have oxygen concentrations roughly ranging from x = 0 --+ 1. It is thought that there exists a strong nearest neighbour repulsion between oxygen atoms in the chains and this results 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 the annealing pressure when x is close to 1. As a result, the preparation of samples under appropriate conditions resulting in an oxygen content close to one is a good reference state to use for all subsequent processing of samples. In other words, if one prepares a sample with oxygen content close to one, then the uncertainty of this oxygen content will be small and an accurate removal of oxygen from this reference state will maintain the small uncertainty. One could, of course, prepare samples with an oxygen content close to x=0, but the temperature and pressure treatment required is very severe (T>800°C and very 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 deoxygenate to the desired oxygen content (cf. Appendix C for a detailed description of the sample preparation and characterization). It turns out that under the conditions used for the reference 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^  64  Figure 3.4: Block diagram of the instrumentation used in the isotherm measurements.  Chapter 3. Experiment^  65  3.2.1 Deoxygenation procedure After a sample is prepared in its x=0.987 reference state, it is then deoxygenated to a (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 a diffusion 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 error in x is introduced through this pumping procedure. At this point the furnace is set to 300°C and the vacuum pump is isolated from the system. Liquid nitrogen is added to the cold trap as a check for leaks in the system. If the pressure in the system does not drop to 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 connector typically solves the problem. Once it is verified that no large leak exists, the furnace is set to 650°C, with the cold trap maintained at liquid nitrogen temperature. After reaching operating temperature, the helium wand is inserted into a liquid helium storage dewar and the deoxygenation procedure begins. From the integral of the pressure vs. time plotted on the chart recorder, one can estimate the amount of oxygen removed from the sample. Typically, 12 hours are required to take the sample to x=0.15 at 650°C. After an appropriate pumping time, the sample space is isolated by closing v6 and v4. The helium wand is removed from the liquid helium and warmed up to room temperature. Once temperature equilibrium is reached, the pressure is measured in (Vw+Vp+Vx-FV,) and the amount, Ax, of oxygen removed is calculated according to:  (AP)V  x^ -ymT Where -y = 46.7737  Torrcm 3 gK  (3.4)  for YBa2Cu306+s assuming a starting composition of x=0.987  Chapter 3. Experiment^  66  (i.e. molar mass = 666.015—b). And, where AP is the pressure measured in volume V at 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 pumped out and the sample is annealed at 650°C for several hours and then slow cooled to room temperature overnight. In the case where one desires just a simple deoxygenated sample for other experiments, the sample is annealed for two days and then slow cooled or quenched, depending on the requirements. In this manner, one can set very accurate changes in oxygen content to within +0.15%. The dominant errors are due to the uncertainty 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, this uncertainty would be roughly doubled. It should be noted that care must be taken to measure the sample weight carefully and in a consistent manner. The potential problem in measuring the weight is not the uncertainty of the measurement itself' but that the sample 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 exposed to the air for more than several minutes, the uncertainty in the mass of the sample is negligible'. After cooling to room temperature, samples are weighed again and the oxygen content checked from the change in mass using:  m(x) (M(YBCO) Ax (1^ (3.5) m(0.987))^M(0) )^ 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 atomic oxygen. This measurement of Ax is not as accurate as the determination using pressures 4Measurements 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 exposed to the air.  Chapter 3. Experiment^  67  and 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 conduct the (ax /N)T measurement as will be described in the next section. For the vapour pressure measurements, the YBa2Cu306+x sample is deoxygenated to about x'-'0.15. A lower value was not chosen in order to avoid possible long term contamination of the sample due to reaction with the platinum foil or to decomposition. 3.2.2 Measurement of the oxygen pressure isotherms and (ax/N)T In this section, the procedure for the isotherm and (ax/ait)T measurements will be described. It will be mostly concerned with the details about the automated data acquisition, since the basic idea is very straight forward. After deoxygenation, the sample is returned 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 volumes not required for the isotherm measurements are not depicted although physically the system remains unchanged. Since the sample has at this point a low oxygen content, it is possible to to heat the sample to 350°C and still not have a measurable oxygen vapour pressure. Thus, it is not necessary to include the liquid nitrogen cold trap since any gases evolving 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 method described above in section 3.1.3. Once the dead volume is calibrated, the furnace temperature is set to the desired temperature for the vapour pressure run. The oxygen reservoir is pressurized with about 1400 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 changes in pressure greater than 0.1%. Under computer control, oxygen is titrated into the  Chapter 3. Experiment^  68  sample space from the oxygen reservoir via the air actuated valves. The vapour pressure is monitored until equilibrium is reached. This procedure is repeated until the vapour pressure reached a value close to 1000 Torr. Once an isotherm is measured, the sample is again deoxygenated, and the whole process was repeated at a different temperature. The isotherm measurements ran 24 hours a day under complete computer control. For low vapour pressures, the time required to see no change in pressure within the resolution of the apparatus is around 12-14 hours and decreases to 2-4 hours for pressures above 100 Torr. Computer data acquisition Since the measurement of the vapour pressure isotherms using small increments in x to extract (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 in a reasonable period of time. Such long equilibration times greatly complicated the task of 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 very long waiting periods made the development period for the software surprisingly long, since incipient bugs in the program often became apparent only after taking many data points. Some of the major hurdles to overcome in the software were, for example: a) controlling the valves and verifying whether a requested action took place; b) keeping track of the elapsed time over a period of several days (this has to do with the details of how a PC keeps track of time, how the system clock is read and how to detect when a day has gone 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 data acquisition program is depicted in figure 3.5. The issues mentioned above as well as  69  Chapter 3. Experiment^  (  ST.;RT  )  I Enter operating, parameters  I While x < xma.  Read reservoir I pressure  Titrate  I  While t < tmax  Read P,T,t  Yes  Iin file vs. time Store P,T,t  I I^  Pause  'Store all important] data in file vs. x  Figure 3.5: Basic flow chart for data acquisition program to measure oxygen pressure isotherms. 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, AP and At correspond to the change in pressure and time since the last file save, AP, and At, are the maximum desired changes in pressure and time before the data is written to a file.  Chapter 3. Experiment^  70  details concerning graphics are omitted. The program is essentially a double loop, where the outer loop titrates and increments the oxygen content and the inner loop measures the subsequent pressure as a function of time. An attempt was made to develop a more sophisticated algorithm to decide when equilibrium was reached, but this task turned out to be very complicated when the condition of robustness was added. Instead, the program simply uses the elapsed time as a criterion for equilibration. This is not a problem since the resolution of the apparatus decreases as a function of oxygen pressure. The measurement of the oxygen vapour pressure through the MKS Baratrons is made 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 the outside world via binary coded decimal (BCD) information. It is necessary not only to read the pressure but also to control the pressure head range and ensure that the pressure reading is valid. In other words, it is necessary to measure and control the Baratrons, 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 MKS controller. Figure 3.6 shows how the physical connections are made. All communications between the IEEE-488 port of the computer need to be translated from and to BCD information. This is done using an ICS Model 4880 Instrument Coupler, consisting of one IEEE-488 port, a 50 pin BCD output port and a 50 pin BCD input port. The BCD output of the instrument coupler is connected to the MKS Baratron Range Selector/Electronics Unit to control the range selection of the pressure heads. The BCD output of the MKS Baratron Digital Readout Unit is connected to the BCD input of the instrument coupler to obtain information regarding the current range selection, error status and pressure reading. Each pressure reading needs to be triggered and a special handshaking protocol is followed according to the manual for the MKS Baratron controller. The pin assignments  Chapter 3. Experiment^  71  Figure 3.6: Block diagram for the communication between the MKS Baratron pressure transducers and the personal computer.  Chapter 3. Experiment^  72  for each BCD port are different and a unique cables are required. Together with the data acquisition software, this setup enables one to complete the measurement 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 for the Pt thermocouple, occasionally check that waiting times for equilibrium are reasonable and once for every isotherm, one also has to physically change from the 1 Torr pressure head to the 1000 Torr pressure head using the manual switch of the multiplexer; it is not possible to control the head selection from the computer. Once the data is taken for an isotherm, several corrections to the data are made to account for some of the systematic errors.  3.3 Systematic errors and their corrections Aside from calibration errors for the volumes and the pressure transducers, there are several 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 N2 gas) in the source oxygen used for titration. At high pressures, the corrections for the oxygen in the dead volume becomes significant. In addition, the temperature drift of the reference volumes and furnace are also accounted for. One other error which is considered but not accounted for is the effect of thermal transpiration. Each of the mentioned errors will be discussed in order of importance.  3.3.1 Temperature drift of the volumes Although all volumes are temperature controlled to 0.1°C, it is possible to measure deviations of less than that by measuring the pressure in the reference volume during the course of the experiment. Since the reference volume, Vr, is isolated from the sample  Chapter 3. Experiment^  73  space, one can use the change in pressure as a thermometer. In this way, the titrated amount of oxygen is corrected for the instantaneous temperature of the reference volume at the moment of the titration, using the following formula:  TR,p t  pr 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 the measurement of the furnace temperature using the Pt thermocouple. The effective dead volume is corrected for the drift in furnace temperature using equation 3.3. Since one is trying to measure (ax/a 11)T at constant T, it is in principle necessary to correct (ax/(9,,4T for the drift of the furnace temperature. Fortunately, this was not necessary, since the no measurable fluctuations of the oxygen pressure were attributable to variations of the sample temperature. It should be noted that the temperature controller itself has to be temperature controlled in order to minimize furnace drift and make the correction to (0.4.9,a)T insignificant. By flowing water from a second circulating water bath, through copper tubes soldered to copper plates, which encased the furnace temperature controller, one is able to minimize the temperature drift of the furnace. This setup is necessary since the furnace temperature controller uses a platinell II thermocouple referenced to room temperature. Although the sensor installed in the micro-chip controller' has a compensation algorithm to account for error signals due to room temperature drift, one still experiences a greater change in the furnace temperature than the room temperature fluctuations. 6Lindberg programmable temperature controller model 818P.  Chapter 3. Experiment^  74  3.3.2 Thermomolecular pressure gradient A potentially serious systematic error arises from the thermomolecular effect (also known as thermal transpiration). This effect is a well known in low temperature physics, particularly in 'Ile vapour pressure thermometry [102]. It arises when the mean free path of the atoms or molecules becomes comparable to the separation of the enclosing walls. In such a situation, the atoms collide more often with the containment walls than with themselves and their behaviour will be dependent on the atom-wall interaction. If one has a thin capillary tube which is held at different temperatures at either end and the mean free path is sufficiently long, there will exist an equilibrium pressure gradient between the two ends. A rough estimate shows' that, at 450°C, the mean free path becomes comparable to the size of the quartz tube below about 5 mTorr. The relationship correlating the pressure 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]. In this extreme case, the correction to the pressure measured at room temperature could be as high as a factor of 2 for high temperatures. In the not so extreme limit, one needs to become specific and the behaviour depends on the type of gas, the capillary geometry and the capillary material. A closed form expression for such a case does not exist and one needs to look at empirical forms to approximate experimental findings. A relatively successful equation, which is essentially empirical in nature, was proposed by Takaishi et al.[105] relating the pressures at the hot and 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], the value, d=3.75)1. for N2 and a quartz tube diameter of 5mm.  75  Chapter 3. Experiment^  K  ^2  A* = 8.6x105  Torr.mm  B*^1.7x103  Torr.mm  C* =^10  (Torr.mm) K  Table 3.2: Values of A*, B* and C* given by Furuyama for oxygen. T2 < T1  A^A*(T*)-2 B = B*(T*) C^C*(T*)-1 1 T* = - (Ti + T2)  2  X = P2d  (3.7)  where Pi, Ti and P2 T2 are the pressures and temperatures and the hot and cold ends, 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 experienced in this experiment, there is ample evidence that they should work even at higher temperatures[106, 107]. Unfortunately, the geometry in this experiment is not one of a capillary with a circular cross section. Due to the insertion of the tight fitting quartz bulb, the appropriate cross section is anular: i.e. a ring with a 2cm diameter and 2-4mm thickness. In addition the thickness is not uniform due to irregularities in the quartz bulb and tube. Nevertheless, it is important to know when such a correction is important and how 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 estimate of the correction for this effect. One sees that above 1 Torr there is no significant correction but that at 10 mTorr the correction is as much as 40%. This systematic error  Chapter 3. Experiment^  76  is difficult to account for quantitatively, and, as will be shown in the next section, the impurity gas correction will dominate this error significantly so that it does not become important to try to correct for thermal transpiration. 3.3.3 Impurity gas correction The most significant systematic error at low pressures is the accumulation of impurity gases in the sample space as one progressively titrates oxygen from the reference volume into the sample space. To illustrate what happens, consider the following: at low oxygen content, YBa2Cu306+x has a very low oxygen vapour pressure, say 10 mTorr. During titration, oxygen at 3 atm pressure is transferred to the "airlock" volume, V. Then this oxygen, corresponding to a small amount of x in YBa2Cu306+x, is titrated into the sample space. At the moment of titration, the pressure jumps to about 200 Torr. In this gas there is not only oxygen but also a small amount of impurities. Since the amount titrated corresponds only to a small change in x, once equilibrium is reached, the new oxygen partial pressure will be only slightly elevated above the previous one. This means that almost all of initial 200 Torr of oxygen will be incorporated into the sample. What remains is the oxygen left to create the new equilibrium oxygen partial pressure plus any impurities, which are now much more concentrated. One has gone from 200 Torr to 20 mTorr of oxygen, but all the impurities remain. Using the ideal gas law and the method of titration, one can account for the impurity gas effect as follows:  =  A=  ,T T4. + )3i -1P,:u pir Vr piv VT  aiP: Vs +^1Vd ^( 3.8) Piv-Vd  where 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  77  Chapter 3. Experiment^  1.6  From 450°C to 650°C in 50°C increments assuming D.4mm -  1.0 ^ 0.001 0.01^0.1^1 10 Pressure at cold end P2 (Torr) Figure 3.7: Plot of the estimated thermomolecular pressure gradient vs. pressure for various temperatures using equation 3.7.  Chapter 3. Experiment^  Pv is the total pressure in  Vd  78  (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 different starting 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 strongly affected 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 about 6 times greater than that stated by the manufacturer. One possibility is that improper handling of the oxygen regulator caused this increase in the impurity concentration. The uncertainty in this correction dominates the error for pressures less than 1 Torr. 3.3.4 Dead volume correction  One final correction which needs to be discussed is the effect of the dead volume. As the oxygen content of the sample is progressively increased by repeated titration, the amount of oxygen remaining in the dead volume also increases as a result of the increased equilibrium oxygen partial pressure. At some point, the amount of oxygen in the dead volume becomes significant compared to the amount of oxygen in the sample. The correction for this dead volume effect is easily accounted for. For YBa2Cu306+s in this apparatus, it becomes significant above about 10 Torr. Combining equations 3.4, 3.3 and 3.8 one obtains an equation to calculate the new value for the oxygen content after titration: =11^11( [(1 —^)Pir_ - (1 - )P2v_ ^, [ (1 - M)Piv^(1 - /32^ t 7m^ Tri^  v d^TiRT^  1 .  ^(3.9)  TzfiT1  In general, one can say that the impurity gas effect dominates the error bars in the low pressure region and that the uncertainty in the dead volume correction dominates in  79  Chapter 3. Experiment^  1 0.1  Not corrected for impurity gas effect 1  111  1  1  I  11111^-1^-I^-I^-I^ ^A^I^  10  OP •  sub  9 e 0  ift  g),  00^Corrected for 0.0121%  impurity gases  0.001  1^I 1^1^I^I^1^I^I^I^I^I^I^I^I^I^I^I^I^I^I^1^1^I  0.2 0.3 0.4 0.5 0.6 0.7 x in YBa2Cu306_,),  Figure 3.8: Comparison of the vapour pressure at 450°C vs. x with and without the impurity gas corrections. The top graph shows the raw vapour pressure for two separate runs without any impurity gas corrections. It is possible to merge the two curves if one applies the impurity gas correction with a source gas impurity level of 121 ppm. The impurity gas corrections become negligible for pressures above 1 Torr, so for the majority of the data points measured, this correction is not significant.  Chapter 3. Experiment^  80  the high pressure region. In the next chapter, the experimental results for the isotherm and (Ox/0,a)T measurements will be presented.  Chapter 4  Results and Existing Theories  This 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 thermodynamics will be compared to the experimental curves. In particular, the defect chemical model of Verweij 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 ordering in YBa2Cu306+, exists, and that no one theory is capable of explaining all of the results.  4.1 Experimental results 4.1.1 Oxygen pressure isotherms The experimental results of this investigation can be summarized in two graphs, namely figures 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 at 450°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 manufactured 6 months apart. In addition, after these measurements were completed, some pellets were reground and 9g of powdered YBa2Cu306+, were placed into the apparatus and a measurement run at 550°C was repeated. Since the amount of powder that could safely be placed into the apparatus was limited to about 10g, one could not go to high  81  Chapter 4. Results and Existing Theories^  82  1000 100  10 1  +4-++ 0°^oo° 0 0 0° +1E 14- ocfP 0^ A 0° +414_^1,1, CPo AAA #  • •• • A<>  x  LA.  X  • • X A•^X • AA^X •A^X  -7.  0.1 0.01  0.001  0.0 0.2 0.4 0.6 0.8 1.0 x in YBa2Cu306+x  Figure 4.1: Plot of the measured oxygen pressure isotherms. The data was taken at: 0 650°C, + 600°C, a 550°C, A 500°C, 0 475°C, and x 450°C. The curves at 600°C and 450°C each contain two separate data runs and the close agreement gives an indication of the uncertainties in this measurement. The curve at 550°C includes the experiment with a powdered sample.  Chapter 4. Results and Existing Theories ^  83  pressures 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 obvious that the two curves overlap sufficiently to prove that there is no qualitative difference in the oxygen ordering properties between powdered and ceramic samples, making it very probable that strain effects due to anisotropic thermal expansion in the ceramics are not important to the bulk oxygen ordering properties of these substances. The scatter from the reproducibility measurements give an indication of the absolute uncertainty in these measurements. For low pressures the errors are dominated by uncertainties in the impurity gas corrections. At high pressures the errors are dominated by uncertainties in the dead volume corrections. Table 4.2 shows the estimated uncertainties in the oxygen pressure at various representative pressures. These uncertainties were calculated by examining the reproducibility measurements conducted at 600°C and 450°C. The absolute uncertainty in x is estimated at ,0.005 and was determined by looking at the shift required to minimize the difference between the two isotherms measured at 600°C. In order to present the data in a form useful for both internal and external users, a cubic 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 of the measurements actually made. This table has been proven especially useful for materials preparation, where it has allowed the determination of proper annealing conditions for 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 from these 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 to changes in the chemical potential.  Chapter 4. Results and Existing Theories ^  Oxygen Pressure in YBa2Cu306.4, (Torr) x 550°C 600°C 650°C 475°C 500°C ±0.005 450°C (0.109) 0.12 (0.032) 0.209 0.14 0.070 0.364 0.16 0.129 0.585 0.18 (0.003) (0.007) (0.045) 0.203 0.882 0.005 0.012 0.20 0.289 1.270 0.018 0.068 0.22 0.007 1.780 0.025 0.099 0.411 0.24 0.010 0.033 0.140 0.573 2.447 0.014 0.26 3.314 0.044 0.194 0.786 0.28 0.019 4.427 0.057 0.266 1.056 0.025 0.30 0.360 1.403 5.852 0.033 0.075 0.32 1.861 7.696 0.34 0.043 0.097 0.482 0.639 2.451 10.08 0.126 0.36 0.057 0.831 3.211 13.14 0.164 0.38 (0.021) 0.076 0.214 1.075 4.186 17.13 0.100 0.40 0.028 22.34 0.279 1.405 5.446 0.036 0.132 0.42 0.367 1.837 7.105 29.15 0.175 0.44 0.047 2.410 9.287 38.14 0.232 0.486 0.062 0.46 3.174 12.13 0.618 50.06 0.307 0.48 0.083 4.178 15.85 65.82 0.401 0.795 0.50 0.108 86.87 1.038 5.500 20.86 0.52 0.139 0.511 1.315 7.239 27.57 114.9 0.54 0.176 0.630 1.639 9.362 36.36 152.8 0.56 0.214 0.790 2.058 11.74 47.77 203.6 0.58 0.272 0.993 1.268 2.634 14.62 60.64 267.3 0.355 0.60 18.32 75.92 337.7 0.62 0.455 1.623 3.259 4.155 23.12 95.53 426.4 0.64 0.571 2.073 2.666 5.322 29.38 121.4 542.0 0.66 0.733 3.476 6.887 37.68 155.3 693.4 0.68 0.971 9.025 48.98 200.9 0.70 1.285 4.600 0.72 1.701 6.144 11.89 64.21 262.8 0.74 2.265 8.305 15.94 85.51 348.8 11.44 21.70 3.073 115.8 0.76 (466.3) 0.78 4.251 16.07 29.80 159.8 42.41 225.7 0.80 5.988 23.05 34.15 61.85 328.2 0.82 8.622 0.84 12.53 52.81 93.23 490.6 0.86 19.42 86.42 (146.9) 0.88 31.82 151.6 55.00 297.4 0.90 0.92 106.1 (672.5) 0.94 242.3  Table 4.1: Cubic spline interpolation of the oxygen pressure plotted in figure 4.1  84  Chapter 4. Results and Existing Theories^  Pressure (Torr) 100 10 1 0.5 0.05 0.005  85  Estimated Uncertainty 0.3% 0.8% 2% 5% 10% 50%  Table 4.2: Relative uncertainties of oxygen pressures at various representative pressure ranges for data listed in table 4.1 4.1.2 The thermodynamic response function (ax/s9,07, As stated in section 3.1.1, the non-ordering susceptibility (Oxfatt)T is obtained using the relation kT(Oxiatt)T=2Ax/A(ln P), where Ax is the externally controlled change in x and A(ln P) is the measured change in ln P. The result of this operation on the oxygen pressure isotherms is shown in figure 4.2. Several important features are immediately apparent. The most obvious feature is the jump close to x = 0.6. This jump, whose position varies with temperature, is identified with the orthorhombic to tetragonal (0T) transition, and will be discussed in more detail later. The other striking feature is that, outside the region of the jump, kT(5x/a,u)7, is independent of T to within the resolution of the experiment. A direct implication of this is that entropic contributions must dominate the chemical potential (cf. section 2.1.2). This is an important point for later considerations, when it is seen that existing lattice gas models, which also predict a  T independent kT(ax/att)T, don't fit this data and one is therefore restricted to proposing purely entropic extensions to the theoryl. A final basic observation, which has practical consequences and theoretical ones 1-One should not be inclined to discard the lattice gas models, since they do correctly predict many other experimental results.  Chapter 4. Results and Existing Theories ^  0.20 0.15 0.10 T=450°C • T=475°C A T=500°C o T=550°C +T=600°C o T=650°C x  0.05 0.00  0.0 0.2 0.4 0.6 0.8 1.0 x in YBa2Cu306_,x  Figure 4.2: Plot of kT(Ox/MT vs. x in YBa2Cu306-Fs  86  Chapter 4. Results and Existing Theories^  87  also, is that the width of the O-T transition is less than the size of the Ax increments used (which is about 0.025). This is not obvious from figure 4.2 but can be seen in figure 4.7. Looking closely, one sees that there are only one, or at most two, data points in the jump. Since one is taking the numerical derivative, one always expects one point which is situated within the jump, even for a discontinuity in (ax/(9,t). Therefore, the fact that so few points are seen within the jump, implies that the transition width in these 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 Monte Carlo calculations of the 2D ASYNNNI model of Rikvold et al.[69] is a result of sample inhomogeneity and perhaps impurity phases. As will be shown in the next section, the agreement between our data and McKinnon's data is quite good and hence the suggestion that inhomogeneities are the cause for the disagreement between theory and experiment is very weak. If sample homogeneity were the cause, then one would not expect that samples with transition widths of of less than 0.03 in x would give rise to quantitatively different answers for roughly the entire range of x than those predicted by the bare 2D ASYNNNI 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 that these curves are indeed a representation of the intrinsic properties of this material.  4.2 Comparison to other work There have been many measurements of the chemical potential of oxygen through various means, ranging from standard thermogravimetric analysis (TGA) to electrochemical cells, to volumetric titration (cf. [30] and references therein). The very large number  Chapter 4. Results and Existing Theories^  88  of experimental investigations into the oxygen thermodynamics makes the compilation and comparison of the various works and techniques a daunting task. Fortunately, this has 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 comparison of 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 the work 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°C and 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 of accuracy and with many data points. In addition, the data was acquired as a function of  T at (roughly) constant x, so that a comparison of Meuffels' work to our data represents the intersection of two virtually perpendicular paths in (x, T) space, making it a very sensitive test of the global reproducibility of oxygen pressure data in YBa2Cu306+x. 4.2.1 Oxygen pressure isotherms Figure 4.3 shows a plot of oxygen pressure data for the three different investigations. The solid lines are straight line interpolations of our oxygen pressure isotherms, that is, no smoothing of the data has been made. Plotted on top of these are the interpolated oxygen pressures of Meuffels et al. corresponding to the temperatures used in this work, which Meuffels was kind enough to provide for this comparison. Finally, at 650°C, the data of McKinnon et al. is also plotted. It is immediately apparent that the agreement between these experiments is very good, except for possible small shifts in x for a particular isotherm and perhaps some small discrepancy in the slopes between the different works. This agreement is much better than the data plotted for isotherm measurements from 1988 to 1989 which was compiled and presented by Voronin et al.[30], and is perhaps a  Chapter 4. Results and Existing Theories^  89  representation of the fact that measurement procedures have improved since then.  4.2.2 The thermodynamic response function (ax/apt)T There is no other data on (Oxfatt)T which exists, except that of McKinnon et al., who in fact were the first ones to measure this. This measurement was conducted at one temperature, 650°C. Figure 4.4 shows a comparison of McKinnon's data to ours. We see that now that the derivative has been taken in order to generate (ax/ait)T and to focus in on the fine details of the chemical potential, the agreement does not appear as good. There is quantitative agreement only close to the transition. Above and below the O-T transition, in the region where the limits of experimental resolution are being approached, the curves depart, McKinnon's data being higher in the tetragonal phase and lower in the orthorhombic. A critical examination of both experimental setups would be 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 the magnitude of the jump in (ax/ait)T at the transition. Aside from attempting to analyze the 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)T curves 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 that each curve of (ax/ay)T is made at quite different pressures. The magnitude and type of 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 due to improper corrections. The second point is perhaps more solid. This comes directly from the paper of McKinnon in which a relationship is proposed between measurements of (axja,u)T and measurements of (ax/an. In fact, this proposal is made more solid by the observation  Chapter 4. Results and Existing Theories^  90  10000 1000 100 10 1 0.1 0.01  0.0 0.2 0.4 0.6 0.8 1.0 x in YBa2Cu3064x  Figure 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 the data of Meuffels et al. and the diamonds are the results of McKinnon et al. The solid lines are a straight line interpolation of our isotherms. The temperatures are the same as in figure 4.1. Except for possible small constant shifts in x, the agreement between the various experiments is very good.  Chapter 4. Results and Existing Theories ^  0.20 0.15  0.05 0.00  o This work T=5500C OThis work T=6500C • McKinnon et al. T=6500C McKinnon isobar w. 5H=-0.80e  0.0 0.2 0.4 0.6 0.8 1.0 x in YBa2Cu306,x  Figure 4.4: Comparison of kT(Ox/O,u)T between McKinnon at al.[59] and this work.  91  Chapter 4. Results and Existing Theories^  92  here that kT(Ox/Oit)T is independent of temperature. This T independence implies, as equation 2.7 shows, that the partial enthalpy is independent of the oxygen concentration. Consequently, it is possible to write  SH kT1nF(x) (4.1) where the only x dependence is in F. Using this equation, together with equation A.9 which connects the chemical potential to the oxygen pressure, one can show that[59] (ö  1 X) ia 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 for the case where the partial enthalpy is x independent, one has a simple relationship between (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 in figure 4.4 is a cubic spline interpolation of the kT(ax/8it)7, data generated by McKinnon 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, and equation 5.14 in section 5.2.2]. In order to match both data sets, we have plotted the McKinnon's isobar curve in figure 4.4 using SH = —0.80eV, which is close to, but not in full agreement with, the accepted value for SH. This curve generated from the temperature scan at constant P is strikingly similar our data. The shape of the curve both above and below the transition is virtually identical. Thus, as far as theoretical fits to the data is concerned, it will be assumed that our curves represents the intrinsic behaviour of the system, although it should be kept in mind that the only other measurement of (ax/ait)T is slightly different.  Chapter 4. Results and Existing Theories^  93  4.2.3 Orthorhombic to tetragonal transition Finally, as can be seen from the (ax/O,u)7, curves, there exists a jump which is associated with 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 as predicted by the 2D ASYNNNI model. The solid circles are the values of xoT obtained from the (ax/ait)2, curves. The lower error bar is the position at the onset of the jump in (axiatt)2, and the upper error bar is the position of the peak. The short dashed line is the best fit line to the data of Meuffels et al. [109], also obtained from vapour pressure measurements. This line, as mentioned by Meuffels, is in good agreement with the data of Specht et al. [110] obtained by X-ray diffraction. The long dashed line is from Gerdanian et al. [111]. This line, which they plot as a guide to the eye from their experimental results, comes from in-situ measurements of the electrical resistivity vs. oxygen pressure and T. It is apparent that all these investigations are quite consistent in their measurement of xoT. There are many more such measurements, (cf. Gerdanian et al. in ref. [111]) and they all lie within the same band shown in figure 4.52• Also included in the figure, for later discussion, is a solid line representing the most recent transfer matrix finite size scaling (TMFSS) calculation of the phase diagram of the 2D ASYNNNI model. Again, the good agreement of our data to various other works and the predictions of theory is a convincing argument that the (ax/ap)T curves are reliable enough to merit an attempt to pinpoint which theoretical model can best describe the data. 2With the exception of an early experiment of Jorgensen et al.[112], which finds ZOT values which are lower by about 0.1. However, this experiment, which was perhaps the first, was conducted when not much was known about this system.  Chapter 4. Results and Existing Theories^  1000  94  /1  •This work — — Gerdanian (Expt) ^ Meuffels (Expt) ^ Hilton (TMFSS)  / / / /  _  /  800  -  -  600  400  -,  1  0.2 0.3 0.4 0.5 0.6 0.7 0.8 x in YBa2Cu306_,„  Figure 4.5: Plot of the structural phase diagram of YBa2Cu306+1. The solid circles are from the position of the jump in the (ax/a/L)T measurements, with the lower error bar representing the onset of the jump and the upper bar the peak of (ax/O,u)T. The short dashed line is the best fit straight line to the data of Meuffels et al. in ref. [109]. The long dashed line is the guide to the eye plotted by Gerdanian et al. of ref. [111]. The solid line is the phase diagram predicted by the TMFSS calculation of the 2D ASYNNNI model.  Chapter 4. Results and Existing Theories ^  95  4.2.4 Discussion of the comparisons made It is evident that one has a very good control over the absolute oxygen concentration in YBa2Cu306+x. Looking at the differences in x between the data plotted in figure 4.3, one can say that the absolute uncertainty in x for any carefully prepared sample is certainly better 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 to use in the study of high Te superconductivity, even considering the fact that the charge transfer is influenced by the oxygen content as well as the oxygen ordering (cf. next paragraph). What is clear from experimental observations, for example the dependence of T, on oxygen ordering[48, 13, 49, 46], and model calculations of the same phenomenon[47, 56], is that one needs to not only control the total oxygen content, but also the degree of oxygen order. This problem is emphasized by the example in NMR, where two recent 63Cu NMR experiments on YBCO samples with x'-.0.6 gave quantitatively different results which is most likely explained by a difference in hole doping[113, 114]. Such a problem is 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, insidious problems can occur. For example, a series of samples prepared with varying x in a TGA apparatus may not all be annealed at the same temperature due to the limitation in the control of the oxygen partial pressure. If samples were to be prepared with low oxygen concentrations, say 0.05, 0.1, 0.15 etc., the required oxygen partial pressure changes by orders of magnitude very quickly for a given fixed temperature and to be able to control the oxygen content with such an apparatus one needs to also play with the annealing temperature. But, since the degree of oxygen order depends not only on x but also on T and time, such a technique introduces an unknown and uncontrolled variation in oxygen  Chapter 4. Results and Existing Theories^  96  order and hence hole doping. It is clear from the experiments showing a time dependence of T, for quenched samples stored at room temperature[48, 13, 49, 46], that the degree of oxygen order varies even at room temperature, since it is shown that x is not changing. This is a clear indication that the degree of oxygen order influences the charge transfer. It does  not mean that the quenching temperature is irrelevant for the final state of order, since the ordering process at room temperature may only be accomplished by local oxygen rearrangements using short jumps. Rapid quenches from elevated temperatures can lock the 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 content can have varying degrees of hole doping. These complications are only overcome by either having a complete description of the relationship between oxygen order and hole concentration or by experimental techniques such as post-annealing the sample sealed in a small quartz tube, so that samples with different oxygen contents retain the same time-temperature profile. Unfortunately, such post-annealing techniques do not seem to be 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 the understanding of the relation between hole doping, oxygen content, oxygen ordering and temperature. Although there exist studies, for example of in-situ X-ray absorption edge studies vs. x[94], which give a relationship between hole count and oxygen content at low  T, 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 thermodynamics of 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 the experimental results. It will be shown that the 2D ASYNNNI model, which predicts very  Chapter 4. Results and Existing Theories^  97  well 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 about the oxygen ordering, are capable of quantitatively fitting (ax/a,u)T in certain cases.  4.3 Fit to existing theories In the analysis of the oxygen thermodynamics in YBa2Cu306+x, there seem to be two camps. In one, the problem is approached from an interacting 2D lattice gas model for the oxygen[33] in which the effective interactions represent the effect of the 3D electronic structure of the system[62]. These effective pair interactions are assumed to be x and T independent, as an approximation to the real situation. The other approach is to use the defect chemical formalism in order to take account of the fact that as one puts oxygen into the system, additional degrees of freedom exist which contribute to the entropy. In particular, formal valences of the neighbouring Cu atoms change and additional holes are created (See chapter 2 for a description of these models).  4.3.1 Defect chemical models There have been a number of investigations analyzing oxygen pressure data from a defect chemical point of view (see, for example refs [89, 90, 88, 30, 92]). In these models (with the exception of Voronin [30]), it is assumed that the interaction between oxygen atoms is negligible, but that the change of the Cu valence with oxygen content is taken into account which gives rise to extra terms in the entropy. Naturally, since the effect of oxygen interactions is neglected, these models do not predict an O-T transition. They are thus forced to propose specific models which are different for each phase.  Chapter 4. Results and Existing Theories^  98  The model of Voronin The study of Voronin takes into account the oxygen ordering using the Bragg-Williams or CVM point approximation. Figure 4.6 shows a plot of kT(5xj0,07, using the model of Voronin, along with our data. This fit is quite good, considering that the scatter in the source data used to obtain the 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 is for the most part incorrect and there is a strong temperature dependence in the tetragonal phase away from the O-T transition. It is also clear that the temperature dependence of the position of the O-T transition is stronger in the model of Voronin. Although this fit is, to date, the best one can find, it contains several deficiencies. The model is essentially the CVM point approximation in that it splits up the system into various sublattices 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 this regular solution model and no higher order correlation functions are introduced. Thus, the definition of the equations to an arbitrary range of interactions results in essentially a 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, it means that one has a fairly arbitrary additional term to the random solution model for the oxygen occupancy and copper valence, that can in principle result in a good fit. In addition, the model does not take into account the existence of Cu2+ and its possible effect on the entropy of mixing. Instead, this effect is taken care of by the power series expansion. Although the model has a sound basis, one ends up with a 9 parameter fit to the data and the formalism does not lend itself well to the determination of the hole count in the Cu02 plane.  Chapter 4. Results and Existing Theories ^  99  0.20 0.15  T=450°C 0 T=550°C 0 T=650°C — Voronin et al. x  0.00  0.0 0.2 0.4 0.6 0.8 1.0 x in YBa2Cu306_,„  Figure 4.6: Comparison of kT(Ox/ött)T between experiment and the model of Voronin et al[30]. Note the strong T dependence of kT(ax/a,u)T in the tetragonal phase and the different curvature in the orthorhombic phase for the model of Voronin.  Chapter 4. Results and Existing Theories^  100  The basic approach of this model does have some very positive elements. One way to look at how the equations are generated is as follows: one has a description for the oxygen configuration and their interactions, which leads to expressions for the entropy and internal energy of the oxygen system. Then, from the variables defined in the configurational problem for the oxygen (in this case just the concentration, x, and the fractional site occupancy) one writes down the connection between these variables and the other degrees of freedom of the system (in this case the copper valence). Then, with the other degrees of freedom defined, one writes down the entropy and internal energy for these other degrees of freedom. This approach will be used in the next chapter to propose a model with just two free parameters and a deeper insight into the interplay between oxygen configuration and copper valence (and hence the hole count). But first, the elements leading to the proposition will be described. This requires describing the defect chemical model of Verweij and Feiner, which forms the basis for making the connection between oxygen configuration and hole count and showing the results of the 2D lattice gas model calculations for (ax/a,u)T, which will be used to generate the description of the oxygen ordering. The model of Verweij and Feiner From measurements of the partial enthalpy 811 and partial entropy SS, by a measurement of the oxygen pressure vs. T at constant x, Verweij and Feiner have proposed several defect chemical models [64, 65, 88]. The measurements were made using a feedback control system to maintain a constant x within the sample, which is placed in a system with a finite dead volume. We will concentrate here on the most recent proposition of Verweij and Feiner. Although the experimental setup did not allow for very accurate measurements, the finding was that, to within experimental accuracy, in the orthorhombic phase  Chapter 4. Results and Existing Theories ^  101  811 is independent of x and T. The conclusion one can draw from this is that the interactions between the particles involved in the reaction mechanism are either much higher or much lower than kT. One the one hand, small interactions can be ignored, whereas large interactions just give rise to exclusion principles, which will result in a constant  811 3 . In addition, if concurrent microscopic reactions take place, then they most likely involve the same enthalpy change, implying that the reactions occur with equal a priori probability. Using data on  x vs. T at constant pressure in a TGA apparatus, and the  implications 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 eight remaining candidates able to fit the oxygen pressure curves. Using our  kT(Ox/Ott)T data  at 550°C, we were able to refine this choice. Each model was examined and the four best fits are plotted in figure 4.7. Note that since these models are for the orthorhombic phase only. The notation for the various fits is defined in chapter 2. As a reminder, however, b(I)F(II) implies that the 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 in the 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 the basal plane. F(IV) is the case where all the holes are in the planes, distributed over the oxygens 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 of Voronin. They have the correct curvature and fall, with the exception of b(I), within 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 in the orthorhombic phase.  Chapter 4. Results and Existing Theories^  102  0.25 0.20  --17 0 •^ 15^ ° 00 —  0 Data at 550°C ^ b(I)-F(II)^^ f(II)-F(II) F(IV) b(I)  -o  t. 0.10 0.05 0.00 ^ 0.5  0.6 0.7 0.8 0.9 1.0 x in YBa2Cu306_,x  Figure 4.7: Comparison of kT(ax/ait)Tbetween the model of Verweij and Feiner and experiment at 550°C in the orthorhombic phase. Depicted are the four cases of the model of Verweij which seem to fit the data best. The nomenclature for the various curves is defined in section 2.4.2.  Chapter 4. Results and Existing Theories ^  103  very different in their implications. Some could, in principle, be eliminated through other investigations which could decide, definitively, where the hole resides and how the spin correlates with neighbouring spins. Before any of these curves are eliminated, however, a serious deficiency should be pointed out in this model. The model of Verweij and Feiner assumes 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 by neutron diffraction[112, 5, 115]. The assumption that s --,---- 1 for the entire orthorhombic range should result in significant errors, particularly as one approaches the transition, since in this region s is changing rapidly. This could qualitatively change the behaviour of the curve. Thus, one should not draw too many conclusions before attempting to take into account the effect of the variation of the long range order parameter, which is the subject of the next chapter. 4.3.2 Lattice gas models Details regarding the 2D lattice gas models were presented in chapter 2. Figure 4.8 shows a comparison of kT(Oxia,u)T for the experimental run at 550°C, CVM calculations and the Monte Carlo results of Rikvold et al.[69]. Clearly, in the orthorhombic phase, the agreement is not even qualitatively correct. In the tetragonal phase, the agreement is better, but not better than the model of Voronin. It is obvious that the pure lattice gas models, even the very accurate 2D ASYNNNI Monte Carlo results, have a key element missing. However, as was discussed in chapter 1, the 2D ASYNNNI model is very successful in predicting many other experimental results. One might be inclined to say that there is something basically wrong with the (ax/a,u)T measurements, but this is unlikely in light of the reproducibility of the data and the agreement of the chemical potential data between many experimental groups.  Chapter 4. Results and Existing Theories ^  104  0.60 0.50  I- 0.40 -o_ -o 0.30 1=' _0.20  0.10 0.00  0.0 0.2 0.4 0.6 0.8 1.0 x in YBa2Cu306_,„  Figure 4.8: Comparison of kT(Ox/Op)T between the predictions of the pure lattice gas models and experiment at 550°C. The following symbols are used: solid line is the CVM square approximation, dotted line is the CVM pair approximation, diamonds are the Monte Carlo results of Rikvold et al. [69] and the circles are the experimental data at 550° C.  Chapter 4. Results and Existing Theories^  105  So, what is wrong with these lattice gas models? The clue lies in the weak temperature dependence of the kT(ax/a,a)T curves, in a given phase. Since these curves are essentially T independent, except for the position of the O-T transition, one realizes that 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 by including anisotropic elastic interactions, would not have much hope in improving the fit: large modifications to the internal energy would be required and kT(Ox/O,a)T would tend to become T dependent. Instead, one needs to examine the entropy. It cannot be a problem with the CVM configurational entropy already defined in the lattice gas problem, 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 the problem which give rise to added terms in the entropy, but do not contain much "internal energy". Naturally, the obvious degree of freedom left out of the 2D lattice gas model is the result of hole creation. The success of the model of Verweij in predicting the kT(ax/a/L)T data is because it makes a fairly reasonable approximation for what the oxygens are doing and includes a model for the configurational entropy of the holes which is consistent with other experimental observations (cf. section 2.4.2 and section 4.3.1). Thus a possible solution is to use the basic approach of Verweij and Feiner, but to use an interacting lattice gas model for the oxygen and define the hole and spin entropy in terms of the oxygen 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, so that there is a danger of losing the agreement between the lattice gas model and the structural data (i.e. the phase diagram and stable ground states). But, if the expression for the number of holes is not too strongly dependent on the order parameters, then the structural modifications should not be too severe. Since the number of holes depends on  Chapter 4. Results and Existing Theories^  106  short range oxygen configurations only (cf section 2.4.2, or refs. [47, 56]), it is natural to use the CVM approximation for this problem. In the description of the valence picture according to Verweij et al.[64] and Tolentino et al.[94], described in section 2.4, the total mobile hole count is determined by the nearest neighbour configuration of oxygen. The solution of the CVM equations gives the required short range order parameters needed to determine the hole count. In the next chapter, this extended CVM model will be presented.  Chapter 5  The Extended CVM model for YBa2Cu306+x  In the previous chapter, it was shown that the 2D ASYNNNI model is not capable of fitting the dependence of the chemical potential on x and T. However, a simple model for the orthorhombic phase making a reasonable assumption for the oxygen order and including additional entropic terms due to the hole creation was very successful. It was concluded that one cannot ignore the fact that electronic defects are created upon oxygen addition. On the other hand, the success of the 2D lattice gas models in predicting many of 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)T is essentially independent of temperature, except in the O-T transition region. This is why Verweij's model works. Thus, the system appears to be dominated by the entropy and the internal energy plays a minor role at these temperatures. However, the model of Verweij and Feiner makes a crude assumption for the configurational entropy of the oxygens, in that they are assumed to be randomly placed on the a sublattice. In the limit of high x, this is not so bad. But, closer to the O-T transition and in the tetragonal phase, a random solution model cannot work. One is therefore forced to retain a more complete description of the oxygen subsystem. In this chapter, the configurational entropy and spin entropy of the holes will be added to the lattice gas description for the free energy of oxygen. This is not totally trivial, since the number of holes depends upon the oxygen configuration. One needs 107  Chapter 5. The Extended CVM model for YBa2Cu3 06+x^  108  to postulate a connection between the oxygen configuration and the hole count which is consistent with experimental observations and current theoretical ideas. In principle, it is necessary to minimize the total free energy of the two systems. For example, one minimizes the grand potential given by the sum of the oxygen configurational grand potential and the grand potential of the electronic system. The two systems are connected, however, and it is necessary to solve the equations self-consistently. In the following, the basic approach is to write down the grand potential for the oxygen thermodynamics and propose a relationship between the oxygen configurations and the total electron hole count. This then restricts one to minimize the electronic free energy in 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 atoms per 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 a very simple approximation, assuming that the charge carriers move in a narrow band and separating the charge and spin degrees of freedom. This allows one to directly insert the solution for the electronic free energy into the grand potential.  5.1 Connection between electronic defects and cluster configurations  Before presenting the extension to the lattice gas model, it is useful to describe qualitatively the mechanism of electron hole creation. Some elements discussed here were briefly mentioned in section 2.4.2. Here we go into some detail concerning the justification of the approach used. 1. At x = 0, the basal plane copper Cu(1) are two fold coordinated, bonding with the 2 apical oxygen sites above and below. The formal valence is Cul+ and is designated as being monovalent.  Chapter 5. The Extended CVM model for YBa2Cu306+. ^  109  2. Adding one isolated oxygen to the basal plane will place it between two Cu(1) and it will bond to these two. The Cu(1) is oxidized and acts as a charge reservoir giving one 3d electron to the oxygen'. The formal valence becomes Cu2+ for the two neighbouring coppers of the isolated oxygen. In essence, one is creating two electron holes, one on each copper; however, these are completely localized, so that they do not play a role in the configurational problem. 3. Filling up the lattice this way works until the probability to put two oxygens on either 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 a picture of this). At this point there are in principle several options, which will be elaborated upon in the next section. We will see that the only case which seems consistent 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 created by adding a second oxygen next to a copper is different than the first. We will designate this as a real hole. It is a relevant hole since it is believed that it is the active element of the electronic subsystem, giving rise to the metal-insulator transition and to superconductivity. Justification One can make some very clear arguments in support of such a picture, and also determine the nature of the extra hole. These are based upon the estimates for the Hubbard model parameters in YBa2Cu306+x[36]. In fact, calculations made for the Cu02 planes use small Cu-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 electron spends some time on each of the atom species. It is sometimes said that the electron sits somewhere in the bonds. Nevertheless, the ionic description is commonly used (see the next paragraph on the justification 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 ^  110  cluster calculations are believed to be a good approximation to the Hubbard parameters for the entire Cu02 plane. Thus, they should be also applicable to small clusters in the chains. At x 0, the Cu02 plane band is half filled. The energy to put these holes on the copper is substantially smaller than to put them on the oxygen. One has placed one hole on each copper oxidizing it to a Cu2+. These holes are localized due to the large copper 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 the copper, 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 holes on the oxygen, since the on-site coulomb repulsion is larger than the difference between the oxygen and copper site energy2, Ep — Ed. These extra holes have a finite hopping probability, giving rise to conduction. Similarly, by placing two oxygens in the chains with a copper in between, one will create four holes. The first three may be placed on the coppers to create three Cu2+. The fourth hole will have a large probability to be on the oxygen, and can hop.  An important approximation that will be made, following the model of Verweij and Feiner, is that conduction takes place in a narrow band allowing one to use a purely configurational entropy for the electron holes. This approximation should be justified somewhat before continuing. The hopping integral t, gives the band-width W for the charge 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 is allowed, the hopping probability is given by the oxygen-oxygen hopping term tpp. This 2Eventhough 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 is significant hybridization.  Chapter 5. The Extended CVM model for YBa2  013 06-Fs  ^  111  is estimated to be between 0.4eV[117] and 0.65eV[36]. This does not quite put one in the narrow-band limit at the temperatures of the experiment, since T is of the order of 0.09eV. But, the temperature needs only to be within a factor of four of the band width before 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 see a 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 untwinned single crystal of YBa2Cu306+x, distinguished between chain and plane TEP. They found that the TEP along the chains was clearly in the narrow band limit, but not the planar TEP. The measurements were made up to 325K, so that it is possible to still see saturation at high temperature for the planar TEP. Thus, the assumption of a narrow band limit for charge carrier conduction (i.e. the use of a pure particle picture for the hole entropy) is possibly an extreme point of view. Nevertheless, it is perhaps a reasonable first 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 the further analysis, we will allow for all possibilities for the behaviour of the extra holes, as in the model of Verweij and Feiner. The results for opposing cases compared to experiment may give insight into possible improvements. 'Also, one should realize that the interpretation of thermoelectric power data is often quite complicated due to numerous effects (see the discussion in ref. [118] or [123]), and that the t J model is really just 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^  112  5.1.1 Counting the holes Using the above picture, one can write down a connection between the (relevant) hole count and the oxygen configuration. We will use the framework of the CVM approximation, 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 sites surrounding the copper which determine its valence and therefore the hole count4. Referring to section 2.3.1, and to the notation for the basic square cluster, the number of Cul+ per unit cell is simply given by the probability to have the four nn oxygens to the central 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 plane copper is either Cul+ or Cu2+. It should be emphasized that the number of Cu2+ in the basal plane simply reflects the number of electron holes completely localized to the Cu-0 basal plane bond. Thus, with this understanding, we have  [C u} -- 1 — [C u l ] = 1 — z0000 (5.2) Using the condition of charge neutrality (equation 2.32) [Cu] + [hole] = 2x, we obtain an expression for the number of extra holes per unit cell (i.e. those not localized to the copper). [hole] = 2x + z0000 — 1 (5.3) A significant approximation is made in equations 5.1-5.3. When writing the probability for a square, the central copper is in fact ignored. In figure 2.1, we have drawn two squares. One contains a copper and the other does not. In the formulation of the CVM 4Note that, so far, nothing has been said about charge transfer to the Cu02 planes. This is a more complicated problem.  Chapter 5. The Extended CVM model for YBa2  C113 06-Ex  ^  113  + hole Figure 5.1: Valence of Cu(1) for various nn oxygen configurations. The cluster shown is the one used for the CVM square approximation. The first cluster corresponds to i + j + k + 1 = 0, i.e. z0000 and gives the number of Cul+. The middle shows how i + j + k + 1 = 1 gives one Cu2+. The cluster on the right shows the creation of a hole by the addition of a second oxygen to the cluster. Note: there are other configurations which can give rise to a hole (for example i = j = 1 and k = 1 = 0), but these are strongly supressed since they involve nn oxygen occupancies.  Hole distribution b B f F b-B b-F f-B f-F  n (available sites per unit cell) [hole] 2 x 4 [hole] + 2 [hole] + 4 x+2 x+4  Table 5.1: Number of sites per unit cell available for distribution of electron holes, n, for the different possible assumptions of the Verweij model. [hole] is the number of holes per unit cell, x is the oxygen content of YBa2Cu3 06+x. The notation of Verweij is used to specify the different cases (cf. section 2.4.2).  Chapter 5. The Extended CVM model for YBa2Cu306-1-x ^  114  square, this distinction is ignored and the cluster probabilities for the square with and without a copper are the same. The CVM square is a nearest-neighbour interaction model. 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 two Vv bonds. However, in order to distinguish these two clusters, one needs to go to the CVM 4+5 point approximation, resulting in 25 independent cluster probabilities for the ortho-I phase. Since (Ox/a,a)T for the CVM square is not much different than for the 2D ASYNNNI, it is hoped that one is at high enough temperature that ignoring this distinction 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 distinguish the entropy of the chain site holes from the planar hole entropy. However as a first approximation as it was discussed at the start of this section, it seems reasonable to use the idea of Verweij and Feiner, which is to randomly distribute the holes over all available hole sites. Denoting M as the number of unit cells, if there are n sites available per 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: Shol„ = knM ln  [hole] ) ) + k[hole]M ln ^ n n — [hole] [hole] )  (5.4)  Table 5.1 lists n, the number of sites available for the placement of holes per unit cell for the various options of the Verweij and Feiner model.  5.1.2 Counting the spins Each free electron spin in the system contributes 2 states. Thus, if the number of free spins in the system changes, it should be taken into account in the expression for the free energy; there is an extra entropic contribution due to the number of spin states.  Chapter 5. The Extended CVM model for YBa2Cu306-kx ^  115  We will now describe how one can count the number of spin states as a function of the 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 are  still correlated, but the correlation length decreases with increasing temperature. The isotropic 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 oxygen is 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 exchange interaction in the chains is of the same order as the planar Cu2+, so that these spins can also be considered uncorrelated and free. What happens when a hole is created? In section 2.4.2, the various possibilities were presented. The options depend on whether the hole can be considered to be on the copper or on the oxygen. First, let us examine a hole on oxygen. Hole on oxygen  If 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 chains and 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 spin between the two Cu2+. The following is then possible:  ^  Chapter 5. The Extended CVM model for YBa2C11306-Fx ^  116  Hole 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. The specification that the spins form a doublet or a quartet is saying that the excited states are thermally inaccessible, and the number of states is given by the degeneracy of the ground state. "Hole on copper" If the hole is "on the copper", again one has three possibilities. Before adding a hole, the Cu2+ 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, it is also possible to view a hole with mainly oxygen 2p character to be bound to a copper site. For example, one way to view the Zhang-Rice singlet [125] is to say that the relevant entity which hops from site to site is the singlet (quasi-particle). So that, although the hole is physically an oxygen 2p hole, the relevant number of sites available for hopping is determined by the copper. For the counting of the number of spin states, it is possible to imagine that elevated temperatures will make spin configurations other than the singlet state accessible. The degeneracy factor g (modification of the number of spin states per copper) is summarized (using the notation of Verweij) in table 5.2. The spin entropy is thus given  Chapter 5. The Extended CVM model for YBa2  CU3 06+x^  possible spin pairing  1  I^II^III IV  degeneracy factor g  2  1^1^3^1  all  b,B f,F b,B f,F  applicable to  2  117  2^2^2  Table 5.2: Spin degeneracy factor for the various cases of the Verweij model. means that 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 row indicates 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 divalent copper spins and the second to the number of spin states of a copper-hole pair. Using the 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 term depends on the type of pairing. It is very important to note that this equation is valid only when the type of pairing in the chains is the same as the planes. This is probably a reasonable assumption since NMR does not detect different spin susceptibilities for the different sites[126]: YBa2Cu306+x is considered to contain a single spin fluid.  5.1.3 CVM free energy with hole and spin degrees of freedom As mentioned at the start of the chapter, one needs to simultaneously minimize the two coupled free energies for the oxygens and the electrons. In the above it was assumed that the solution to the electronic problem is given by a random entropy of mixing with a  Chapter 5. The Extended CVM model for YBa2 C113 061-x^  118  negligible internal energy compared to the temperature. Thus, the electronic free energy problem is automatically solved and can be added directly to the configurational free energy of the oxygen'. In this approximation, the effect of the holes and spins can be viewed 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 of the square cluster configuration probabilities zijki. We see that according to equation 5.3, the entropy of the holes and spins can be written as a function of the oxygen cluster configurations. Formally, one can rewrite equation 5.3 using the notation of section 2.3.1: [hole] =^(2aijk/^Cijkl — 1)Zijk1 =  E  hijklZijkl  ^  (5.7)  i,j,k,1  i,j,k,^  where 1 1 if(i+j+k+/), 0 Cijkl =  aijki =  (5.8)  0 otherwise 1 — (i+j+k4-1) 2  We see that cijk/ tells us if we have a Cu l+ and  aiiki  (5.9) how many oxygens there are in the  cluster. With the equation for the hole written in such a form, one can immediately see the effect of Sspin on the CVM free energy. The expression 5.6 for Sspin becomes: Ss in P^—0111 No^2^g  (2)E  ci3kiziJki  + kin g E  i,j,k,1^  a/z„ki  ln (2)^(5.10) + ki^g  i,j,k,1  2  Adding the the entropy of the spins and holes to equation 2.26 gives, to within an additive constant:  E  [Eipa  + kT-1 in (2g)ciikil zijki  T S square^T Shole  2N0  kT(21ng)]  E  aijklZijkl  (5.11)  'In a less naive picture, one would have to re-solve the electronic free energy for every configuration of oxygen proposed.  Chapter 5. The Extended CVM model for YBa2Cu3  where Ssqua„ and  S hole  06-Es  ^  119  are given by equations 2.20 and 5.4. The spin entropy gives an  added temperature dependent term for the energy  Ei3k1  of a cluster. In fact, it makes  empty clusters energetically unfavorable compared to clusters containing oxygen atoms. It also adds a constant to the chemical potential term, which is irrelevant for the free energy 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 may change under certain circumstances. Using this formalism, one can again use the Natural Iteration method to minimize the free energy (cf. Appendix B). The solution of 5.11 using the NI method gives the solution of zj3k/ vs.  it  and T. From this, one can obtain  the oxygen concentration x = x(it,T), the long range order parameter s (it, T) and short range order parameters, such as the nn occupation probability. In the following we use 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. figure 2.1). In the free energy minimization, one factors out kT and thus the problem is defined in terms of a rescaled temperature kT/Vnn and chemical potential it/kT. Note that we have not included a single particle site energy. The site energy can be absorbed into the chemical potential since it will have the form € aijkizi3kr. In calculating kT(Oxfatt)T, a site energy term and the second spin term will drop out. Thus, as far as a comparison of kT(Ox/(9,02, to experiment is concerned, a detailed knowledge of the site energy is not needed. For comparisons to other data, such as it vs. T, the site energy will have to be included, and can, in general, depend on temperature'. Thus, the 'it' in equation 5.11 is not directly the chemical potential measured in experiment, but contains added terms which need to be subtracted if a direct comparison to experimental data is 6For example,  c may change due to thermal expansion.  Chapter 5. The Extended CVM model for YBa2Cu306+s ^  120  to be done (with the exception of (Ox/ap)T). This will become more clear later.  5.2 Results and comparisons to experiment 5.2.1 Comparison to  kT(Ox/ap)T data  For each possibility of the spin and hole behaviour described in section 5.1, kT(Ox/ait)T was calculated and plotted against the experimental result at 550°C. The curve at 550°C was chosen since it has the lowest noise and spans a reasonable range of x. A common temperature of kT = 0.3Vni, was used for all of the calculations, since 14,7, is estimated to be 2800K[14], which would put T roughly at 550°C. kT(ax/Oit)T was calculated by taking the numerical derivative of the resulting x vs. it curve. Note that, as mentioned in the previous section, any T dependent site energy terms do not affect the kT(ax/ap)T curves. The plots have been separated according to the particular hole distribution cases listed in table 5.1. In figure 5.2, the various cases for holes distributed both in the chains and planes are shown. It is clear that the only curves which can fit the data reasonably well are those with a spin degeneracy factor of These cases correspond to the physical picture where the spin of the hole has a strong anti-ferromagnetic coupling to the copper (either the single copper for hole on copper or the two neighbouring coppers for hole on oxygen). If these models are physically reasonable, then it implies that the spin of the hole would like to form a singlet state with the spin on the copper site. This is in agreement with most current theoretical and experimental viewpoints concerning the spin behaviour, which put the hole mainly on oxygen with a strong anti-ferromagnetic coupling to the neighbouring divalent copper spins (Zhang-Rice singlet[125]), although in this situation the temperature is much higher than normally considered. Figure 5.3 shows the results for the cases where the holes are restricted to lie  Chapter 5. The Extended CVM model for YBa2  0.3  I  0.2  I' - . • . ' • ••‘  ......!^  •%.^•  -^/ /•^%^• I/^#f^'..• \% It^  -^1;^otetter^\ •  ; 0 .^  0.2  /  ....  %A^-^#6tertue ^,s,^...),  r^•e^ b(I)B(I)^ 411 i 6)^ ^b((II)B(III)^1 --^b(112)B(112)  f(II)F(II)^';'., ^f(IV)F(IV) --^f(1/2)F(1/2)  cereal,^\ k.%  Ie  % -^11.^ ii - - - - b(I)F(II)^%  --^b(1/2)F(1/2)  0.0  121  ."-'■^_ -^ /^ .^.. \ .. .^.^ % -% • _ % /,,^..•^ /I^le^"A '..  ^I^I 1^I^f^i^I if —^ -  /^4.^%  /^  0.1  414k^ ' •  i^1^1^ij-^i^1^i^I ....^.., .^. r^ _^ . r^\ r^■ _^.^■ .^■ .^■ /  ^  ••• - "• .  -^/ /^\ /^\ -^ I /^  C113 064-s  /  -  .- i^.^.^. . .^ .^\ 4^ . \  teltge^.r \ ■  -  ^ /^ /^64 \ ••^ ^ i^ ,\ • ^ Al \^— / -  f(II)B(I) --^f(1/2)B(1/2)  .1,^.1_,^.1.  0.2 0.5 0.8^0.2 0.5 0.8 x in YBa2Cu306+x  Figure 5.2: Comparison of kT(ax/aft)T for the extended CVM models with chain and plane 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 holes on oxygen in the planes, and bottom right for holes on oxygen in the chains and holes on copper in the planes. Note that the only cases which can fit the data reasonably well are those 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).  Chapter 5. The Extended CVM model for YBa2Cu306+s  0.3  0.2  .............^  •  et-re  olossige  ^  122  B(1) B(III) F(II) F(IV) ^ B(1/2) ^ F(1/2)  b(I) ^ b(III) - - 411) f(IV) ^ b(1/2) ^ f(1/2)  0.2  0.1  0.0 ^ 0.0^0.2^0.4^0.6^0.8^1.0  x in YBa2Cu306+x Figure 5.3: Comparison of kT(Ox/a,u)T for the extended CVM models with a restricted hole distribution and the data at 550°C. The top graph corresponds to cases where the hole 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 quite well in the tetragonal phase, but have the same problems as the standard CVM square approximation in the orthorhombic phase. Some plane-only cases seem to work well in the 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^  123  either in the planes or the chains. Chain-only cases (bottom graph) have the same qualitative shape as the standard CVM square approximation (In fact, b(1/2)=-CVM square). Although one might imagine a hybrid model where one uses a chain-only case for the tetragonal phase and a combined one for the orthorhombic, it is difficult to formulate such a scenario which allows a determination of the O-T transition in a selfconsistent manner. However, a rough approach is outlined in section 5.3.1, which would essentially result in a chain-only model for the tetragonal phase. For now, we will not consider these chain-only models further. Some of the plane-only models (top graph) seem to fit fairly well in the orthorhombic phase and seem just as good as the combined chain-and-plane cases in the tetragonal phase. Overall, B(1/2) seems the best among the plane-only cases. Note that all of the plane-only models have quite a small jump at the O-T transition; smaller than all of the combined models. We will retain B(1/2) as a representative example of the plane-only models for the next figure. Figure 5.4 shows a plot of the best fit cases. For comparison, the standard CVM square model is also plotted. One should recall that the CVM square is very close to the predictions of the full 2D ASYNNNI model, and so, should be regarded as representing the 2D ASYNNNI as well. It is clear that all of these cases are clear improvements to the original CVM model in the orthorhombic phase. Although quantitatively, the CVM square seems better in the tetragonal phase, one could argue that the extended models are 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 lattice gas model. In fact, while all of the possible cases for the hole dynamics were presented for completeness sake, most are physically unreasonable. Should one particular case have  Chapter 5. The Extended CVM model for YBa2Cu3 06+x^  0.3  124  B(1/2) f(II)F(II) b(I)B(I) b(I)F(II) f(II)B(I) Standard CVM  00 000  0.0  0.0 0.2 0.4 0.6 0.8 1.0 x in YBa2Cu306,x  Figure 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 temperatures for each case has been chosen in order to have the O-T transition coinciding with the experiment. The open circles is the data at 550°C. For the CVM square, the minimum xoT 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  ^  125  resulted 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 comparing to other experimental results, it is useful to decide which specific case should be used in the further analysis. Of the cases plotted in figure 5.4, none Scan really be judged much better than others simply by looking at the graph. One needs to resort to other physical measurements to be able to decide. Two very recent X-ray absorption spectroscopy (XAS) measurements clearly show that there are holes in the chains as well as the planes[95, 94], so that the remaining plane-only case should be discarded. In particular, angle-resolved XAS on an untwinned YBa2Cu307 crystal clearly shows twice the chain hole occupancy that a random hole distribution 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 not remove it. This XAS experiment also shows that the random occupation model used for all cases is really only a first approximation'. From the in-situ and ex-situ XAS study of Tolentino et al.[94], it is clear that at low values of x the holes are localized to the chains, 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 spends a 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, within the approximation of a random solution model for these holes, is a better representation of the real situation. The tendency of the author would be to say that the "hole on oxygen" viewpoint (i.e. f models) seems more correct, since the hole is seen to have mainly oxygen 2p character. Thus, we choose the f(II)F(II) case for further comparisons 7Both these recent XAS experiments by Krol et al.[95] and Tolentino et al.[94] also see a significant hole occupation of the interstitial 0(4) oxygen site at x = 1. Tolentino, however, sees this occupation vanish 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^  126  to other experimental investigations, but with the understanding that, in principle, a more refined model for the hole dynamics should be developed before making a final choice for the most accurate representation of the interplay between oxygen ordering and hole creation. 5.2.2 Comparison to the oxygen chemical potential: determination of the  site energy A plot of the f(II)F(II) model prediction for the oxygen pressure requires a knowledge for the single site binding energy for an oxygen atom in the basal plane. There are many effects which can make the site energy temperature dependent. At this stage it is perhaps not useful to make an estimation for the site energy, but to just assume a temperature dependence and fit the model to the experiment. The effective chemical potential ,ti of equation 5.11 can be written as 1 = — (Po2 D) c 2  (5.13)  where E is the site energy and D = 5.08eV[127] is the dissociation energy of an oxygen molecule. ,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 a definite 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 basis for this model, does poorly in the prediction of the O-T transition. Instead, the value of Hilton et al.[14] will be used: Yin = 2800K, which is what a best fit of the 2D ASYNNNI to O-T transition data gives. Since the model fits best in the orthorhombic phase, the theoretical and experimental chemical potentials vs. x in the orthorhombic phase are subtracted to give c + D/2 as a function of each temperature measured. Figure 5.5 shows a plot of E-F D/2 at the various temperatures, deduced from the difference between theory  Chapter 5. The Extended CVM model for YBa2Cu3 06-Ex^  127  -0.76 -0.78 5-'^ a) -0.80 a^-0.82 + (.0 -0.84 -0.86 -0.88 700  750 800 850 900 950 Temperature (K)  Figure 5.5: Plot of E + D/2 vs. T determined by comparing the f(II)F(II) model to experiment. The open circles are determined from the average difference (in the orthorhombic phase) between experimental isotherms and the f(II)F(II) model assuming a 14,,, = 2800K.  Chapter 5. The Extended CVM model for YBa2Cu306+.^  128  and experiment. 1 c + — D = —(0.820 + 0.004)eV 2  (5.14)  We see that c + D/2, deduced from the fit of the f(II)F(II) model to experiment, is independent of temperature to within the resolution of the experiment. This agrees fairly well with the constant term deduced by Shaked et al.[128] in a fit to in-situ neutron diffraction data using the CVM pair approximation with an empirical x and T dependent site 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 site energy which has been introduced in many works proposing models for the the oxygen thermodynamics [129, 130, 131, 132, 128, 133]. Also note that since we have chosen V. = 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 energy is 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% correction at 650° Cr . Thus, the heat of solution is essentially given by SH = E+ D/2, which should agree 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 orthorhombic phase, so that equation 5.14 is really only valid in the orthorhombic phase. In the opinion of the author, the model does not fit well enough in the tetragonal phase to merit an attempt 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] gives 8 H'-.-' —0.94eV. We note in particular Verweij et al.[65], who find SH independent of 8If Vn,-, = 2800K, then even at the highest temperature of the experiment (=923K) the effect of the nn interaction essentially gives one a nn exclusion principle, and the specific value of V„,, is fairly irrelevant.  Chapter 5. The Extended CVM model for YBa2Cu306-1-s ^  -13  -  -14  -  -15  -  -16  -  -17  -  -18  _  -19  -  -20  129  -  _  0.0  0.2 0.4 0.6 0.8 1.0 x in YBa2Cu306.fx  Figure 5.6: Plot of the experimentally determined chemical potential isotherms and the predictions of the f(II)F(II) model. Since the site energy was determined by looking at the orthorhombic phase, the fit seems much better in that phase. Temperatures used are: 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^  130  x and T in the orthorhombic phase. The average value found in the literature does not agree completely with 5.14, but the disagreement is not spectacular. Finally, one remarks that the value given in 5.14 is very close to the value of 811 used to fit McKinnon's isobar measurement to the (ax/a,a)T data (cf. figure 4.4). Now, essentially, the model is complete. One has specified the nn interaction the site energy given by equation 5.14 and the specific model f(II)F(II), which is solved by minimizing equation 5.11 as a function of 1a02 and T. Figure 5.6 plots p,02/kT vs. x and compares to the experiment. We see that the fit is very good in the orthorhombic phase, but less so in the tetragonal phase, which is as expected. At 450°C, where a maximum of orthorhombic phase is seen, the fit is excellent over the entire orthorhombic range. For reference sake we make a final plot in figure 5.7 of kT(Oxia,u)T for the choice of parameters used and the corresponding temperatures of the experiment. 5.2.3 Comparison to the fractional site occupancies measured by neutron  diffraction One of the predictions which can be made by an oxygen ordering model is the behaviour of the 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 of the atoms in the crystal lattice. Measuring the occupancy of the 0(1) and 0(5) site gives directly x, and xo (cf. figure 1.1). To the best of the knowledge of the author, a precise in-situ high temperature measurement of the fractional site occupancy has been done only 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 of temperature. In the second paper[5], a measurement was conducted at a constant 490°C vs. oxygen pressure. 'Recall that the long range order parameter is .s x, — x#.  Chapter 5. The Extended CVM model for YBa2  013 06-1-s  ^  131  0.20 0.15  0.00  0.0 0.2 0.4 0.6 0.8 1.0 x in YBa2Cu306+x  Figure 5.7: Plot of kT(Ox/(9,07, vs. x at the temperatures of the experiment using the f(II)F(II) model. The symbols are the same as in figure 5.6 and correspond to experiment. The solid line is the f(II)F(II) model with 177,, = 2800K at the corresponding temperatures of the experiment.  Chapter 5. The Extended CVM model for YBa2Cu306+5  ^  132  Before comparing the predictions of f(II)F(II) model to Jorgensen's data, a few words should be said concerning the absolute value of the oxygen concentration. There is a profusion of estimates for the maximum obtainable oxygen content in YBa2Cu306+5 at 1 atm pressure. In Jorgensen's work, it is estimated that the maximum oxygen content is x = 0.91(3), and this is reflected in the plots of the fractional site occupancy. In this work, and in many others, it is estimated that the maximum obtainable oxygen content is very close to x 1. In particular, we assume that the absolute oxygen content at 400°C in 1 atm of oxygen is x = 0.987. (cf. appendix C). Since the model fits the chemical potential data very well, especially at low temperature, any calculation will result in an oxygen content close to Xmas = 1 at low temperature, instead of Jorgensen's value of Xmas 0.91. Thus, one would not expect a good fit to the neutron data without some  further analysis. We would like to argue that the estimate of Jorgensen for a maximum oxygen content of Xmas 0.91 is too low and that thus his data should be rescaled to correspond to 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 in figure 5.6 also provide evidence that Jorgensen's estimate is too low. In particular, the kT(Oxfait)T data clearly tends to zero at x =1 and not x = 0.9. This suggests that our  estimate for Xmas fs2 0.987 is very reasonable, if one believes any of the oxygen ordering models proposed so far. Also, examining figure 5.6, it is clear that the model fits very well in the orthorhombic phase. It is not possible to modify the curvature of the theoretical curve by an adjustment of either 17,, or c. Adjusting simply shifts the curve up or down and Vrari essentially just sets the position of the O-T transition. The shape of the curve is determined by the hole entropy, which contains no adjustable parameters. A constant shift in x for the chemical potential data with a corresponding shift in e for the theory  Chapter 5. The Extended CVM model for YBa2  C113 06+x  ^  133  curve would just worsen the fit. In light of the above considerations, we believe it is reasonable to scale neutron diffraction 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 a constant temperature of To = 490°C. The open circles are the rescaled neutron diffraction results of Jorgensen et al.[5] and the small points are the unscaled data. The fit to the scaled data is excellent. Also included, for comparison, is the prediction of the standard CVM square approximation, which quite clearly predicts a much too rapid increase in the long range order parameter. The effect of the site energy is to just produce a shift along the x-axis. Figure 5.9 plots the fractional site occupancy vs. temperature at a constant oxygen pressure pressure of latm. Again, the fit to the rescaled data is excellent. Note that for this 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 contains just the value of f as the only adjustable parameter. In addition, f drops out in the determination of many quantities. There has been a presentation of many different possibilities for the behaviour of the holes for completeness sake, but there are really only a few which are physically reasonable. The choice of Vnn was made to optimize the agreement with the 2D ASYNNNI model phase diagram, but for the quantities presented here, 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 does not depend strongly on temperature. The only strong effect of temperature is to modify the point at which one switches from the tetragonal phase to the orthorhombic (cf. figure 5.7). The choice for using the f(II)F(II) case for the hole behaviour is based largely upon  Chapter 5. The Extended CVM model for YBa2Cu3064, ^  ^ CVM Square^.-..-0Jorgensen et al. ,,../  134  .--  3  ,,  -  I  I  0  =490 °C .490 _  -  -  0.0 -18^-16^-14 1  -12  1.1 / k T 0 Figure 5.8: Comparison of the fractional site occupancy vs. plkTo between the neutron diffraction data and the f(II)F(II) model at T, = 490°C. The open circles are the data of Jorgensen et aL[5] rescaled so that x = 0.987 at T = 450°C and 760Torr. The small points are the unscaled data. The solid line is the prediction of the f(II)F(II) model with 177,7, = 2800K and € given by equation 5.14. The dotted line is the prediction of the standard CVM square approximation.  Chapter 5. The Extended CVM model for YBa2  C1-13 06+x^  135  1.0 o  C as 0_  0.8  r3 0.6 0 a) ._ Cl)  To 0.4 c 0 -4= 0 2 0.2 0.0 350 450 550 650 750 850 950 Temperature (°C) Figure 5.9: Comparison of the fractional site occupancy vs. T between the neutron diffraction data and the f(II)F(II) model at P = latm. The open circles are the data of Jorgensen et al.[112] resealed so that x = 0.987 at T = 450°C and 760Torr. The points are the unscaled data. The solid line is the prediction of the f(II)F(II) model with V7in = 2800K and c given by equation 5.14.  Chapter 5. The Extended CVM model for YBa2 C113 06-1-s^  136  the 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-singlet pairing, since this is the most consistent picture with other investigations into the normal state properties of the electron system. However, some of the other possibilities presented seem to fit the (0.449/L)", data just as well, so that rigorously, one should not discard the other possibilities, especially the "hole on copper" b(I)B(I) case. The entropy model for the holes is simply too crude to really make a judgement as to the validity of the individual cases. A more serious theoretical analysis needs to be made in order to decide upon the best description for the hole behaviour and an improved fit to the (Ox/a,a)T data. 5.2.4 Phase diagram Naturally, it is important to see the modification of the phase diagram upon the addition of the hole entropy. First, one should examine the modification of the standard CVM square with the new model to see the relative changes. Such a comparison will give an idea 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 the CVM square approximation does not modify the phase diagram strongly. In fact, it causes a flattening of the O-T transition curve at intermediate temperatures, which is a desirable effect, since experimentally, the O-T transition line is quite straight (cf. figure 5.11). To compare the phase diagram of the f(II)F(II) model to experiment is not actually very useful, since it is based upon the CVM square which over-estimates the transition by about 10%. Nevertheless, it is perhaps instructive to show such a comparison, to check  Chapter 5. The Extended CVM model for YBa2  CU3 06+s  ^  137  0.6  0.2  0.0  0.2^0.4^0.6^0.8  ^  1.0  x in YBa2Cu306+), Figure 5.10: Phase diagram as predicted by the f(II)F(II), CVM square and the 2D ASYNNNI 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^  138  if the temperature scale, set by the choice of 14,7, is reasonable. To predict the phase diagram properly, one really needs to go to higher order approximations or resort to other method such at transfer matrix finite size scaling or Monte Carlo. Any agreement to such a lower order mean field approximation would be purely fortuitous unless a clear case can be made to prove that the oxygen interactions are long range. We plot the model in comparison to experiment in figure 5.11 with Vnn, = 2800K. We see that the curve is in fair agreement, but one should be cautious, since the mean field theory chosen overestimates the transition by about 10%. 5.2.5 Predictions for the copper valence and hole count and comparison to  XAS measurements The extended CVM model is based upon the establishment of a connection between the electronic defects and the oxygen configurations. The success in fitting the structural and thermodynamic quantities makes it an exciting prospect to be able to provide a microscopic understanding of the hole creation and charge transfer mechanism. Since the electron hole and oxygen configurations are solved self-consistently, it is important to verify that the resultant predictions for the copper valence and hole count is consistent with experimental observations and intuitive expectations. In other words, we have to check that electronic behaviour predicted by the oxygen configurations remains reasonable. The comparison to experiment will also provide clues for how the model may be improved. We will use a very recent in-situ and ex-situ XAS study on YBa2Cu306+x conducted by Tolentino et al[94]. In this XAS measurement, in-situ measurements at high temperature vs. oxygen pressure were conducted to directly measure the amount of mono-valent copper in the system vs. x. Since the only site which may be mono-valent is  Chapter 5. The Extended CVM model for YBa201306-ps^  1400 1200  139  •This work — — Gerdanian (Expt) ^ Meuffels (Expt) ^ Hilton (TMFSS) — — f(II)F(II)  .1 000 a) E 800 600 i^,^, 400^.^ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ,^I  x in YBa2Cu306_,x Figure 5.11: Comparison of the phase diagram between the f(H)F(H) model and experiment. See also figure 4.5 for further details on the curves plotted. The f(H)F(H) curve is plotted using = This is the same as for the curve of Hilton et al[14]. The agreement is fair, suggesting that the choice of Yin for the f(II)F(II) is reasonable.  Chapter 5. The Extended CVM model for YBa2Cu3 06+s^  1.0 0.8  140  f(H)F(11) ATolentino ex-situ 0Tolentino in-situ I OTolentino in-situ H _  0.2 0.0  0.0 0.2 0.4 0.6 0.8 1.0 x in YBa2Cu306,  Figure 5.12: Prediction of the f(II)F(II) model for the number of Cu'+ and comparison to the XAS data of Tolentino et al[94J. The solid line is the prediction of the model at 490°C. The XAS ex-situ data corresponds to low temperature annealed samples. The in-situ measurements were made at high temperature. Notice that the plateaus in the experimental data correspond to the O-T transition. This plateau is absent in the model.  Chapter 5. The Extended CVM model for YBa2Cu306-1-. ^  141  the basal plane copper, we can use this data to check that the equilibrium oxygen configurations obtained from the f(II)F(II) model predict the correct amount of Cu l+ through equation 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 well defined oxygen content. Figure 5.12 shows a plot of the number of mono-valent Cu per unit cell vs. x. The triangles correspond to the ex-situ, low temperature anneal data and the circles to the in-situ high temperature run. The solid line is the prediction for the f(II)F(II) model at 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 close to 1, the slope is very small, since the main effect of removing oxygen from x = 1 is to reduce the hole count but not to create isolated coppers (cf. equation 5.1). What is clear, however, is that the data predicts a levelling off of [Cu'] at the order-disorder transition. The ex-situ data has a plateau at x 0.3, which corresponds to the position of the O-T transition at low T. The high temperature data has a plateau at x 0.6 also corresponding to the 0-T transition'. The model does not generate such a plateau at the O-T transition. A slight kink is visible, but not much else. In light of the very good agreement between the model and the fractional site occupancy determined by neutron diffraction, it is clear that a better agreement cannot be obtained by forcing a strong modification of the behaviour of the oxygen configurations. Instead, it is more likely that the use of the CVM square formulation, which does not distinguish between two types of square clusters in the basal plane, is at fault. It is important to realize that the amount of 10A plateau at the order-disorder transition is a direct result of the fact that order will tend to cluster oxygens into chains giving rise to more coppers with no neighbouring oxygens[55].  Chapter 5. The Extended CVM model for YBa2  C113 06+x  ^  142  Cul+ is not influenced by the behaviour of the holes; i.e. whether the holes are transferred to the planes etc. The Cu l+ count comes directly from the geometric determination of oxygen clusters and so that the disagreement here at the O-T transition is most likely a direct 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 et al from their XAS measurements. This dashed line was generated by Tolentino from the Cul+ data using the condition of charge neutrality ([Cu] + [hole] = 2x, and equation 5.2). This dashed line is a low temperature estimate assuming the existence of the orthoII phase. We see that at the order-disorder transition, (which for the dashed line occurs at  X  0.3) causes a steep rise in the hole count due to chain formation. A corresponding  steep rise should be present in the high temperature prediction of the f(II)F(II) model at  x 0.6, but again, only a kink is seen. As for figure 5.12, this points to a deficiency in the use of the CVM square cluster to calculate the hole count. At high x, the agreement is good. Starting from x = 1, the removal of one oxygen destroys 2 holes; i.e. breaking up a chain will destroy 2 electron holes.  5.3 Commentary on the approximations made Assuming a lattice gas picture for the holes distributed randomly over all available sites amounts 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 room temperature (cf. section 5.1), implying that a pure configurational approach is perhaps a good approximation for the entropy of the charge carriers. Neglecting the existence of a two band structure also assumes that the charge transfer gap is smaller than kT,  Chapter 5. The Extended CVM model for YBa2Cu306+x^  1.0 --a75 0.8 0 c 2 0.6 a) ci) a) -5 0.4 .c 0._ c., °0.2 0.0  143  f(II)F(II) — — — Tolentino schematic  -  -  _ ..-^/  -  0.0 0.2 0.4 0.6 0.8 1.0 x in YBa2Cu306,x  Figure 5.13: Plot of the amount of oxygen 2p holes from the f(II)F(II) model and comparison to the schematic behaviour deduced by Tolentino et al[94]. The solid line is calculated from the f(II)F(II) model. The dashed line is the schematic behaviour proposed by Tolentino which was deduced from the XAS measurement using the condition of charge neutrality. The dotted lines have slopes of 1 and 2 and are included as a guide to the eye.  Chapter 5. The Extended CVM model for YBa2  C113 06-Fx  ^  144  so that the energy difference between chain and plane populations can be ignored and a random distribution is a valid assumption. Such a hypothesis is really the simplest one possible, and results in not having to solve the free energy minimization problem for the electronic subsystem. This is certainly not reasonable at low temperatures, when the entropy per carrier is seen to become T dependent. The hope is, however, that at the high temperatures of the vapour pressure experiment, this random solution model for the charge carriers is a good first step. It should be pointed out that such a situation with two narrow bands is very unusual and not often mentioned in connection with defect structure calculations. There are quite successful 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, there are many atoms per unit cell, there is the complicated process of the magnetism, charge transfer and metal-insulator transition, so that the popular formalism for determining the order and phase stability is perhaps not even appropriate. 5.3.1 Possible limitations and complications  The viewpoint that might be taken about this system is perhaps analogous to the discovery that the electron has an added degree of freedom called spin. One has the thermodynamic problem of the placement of oxygen in the basal plane which gives a prediction for the chemical potential. The comparison to experiment, however, strongly indicates the existence of added degrees of freedom. The above analysis is thus an attempt at giving an explanation of what these added degrees of freedom are: one cannot ignore the creation of holes in this system. At x = 0 one has a completely filled plane and chain band. These do not contribute to the entropy of the system. But, as oxygen is loaded into the basal plane, holes are created which are initially in the chains, but soon also transfer to the planes. Also, the holes may hop from site to site. This gives rise to an added  Chapter 5. The Extended CVM model for YBa2  013 06-Fs^  145  degeneracy 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 number of degenerate states of the underlying electronic subsystem, which must be taken into account in the oxygen thermodynamics. It is tempting to continue the analysis further and make perhaps more reasonable assumptions of the behaviour of the electronic subsystem, but there are several reasons why such an attempt might be futile: 1. The relative positions of the bands may easily depend on oxygen content. Especially, the charge transfer gap may vary. To take this into account self-consistently would require a very deep understanding of the interplay between the structural and electronic degrees of freedom. A very recent publication of Uimin et al.[136], which uses the Kondo type approximation for the extended Hubbard model, and a configurational entropy for the chains to fit the XAS measurements of Tolentino could 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 pure particle picture might be the next necessary step. This again, would be achieved by using the approach of Uimin. 3. The basic cluster of the CVM square approximation, which is used to determine the copper valence, is too simple. No allowance is made for the possible occurrence of the ortho-II phase. The CVM square approximation was used since it represents the lowest order approximation which is capable of simply connecting the Cu valence and oxygen clusters. Higher order clusters with a more detailed mapping between configurations and hole counts might give the necessary modifications to improve the fit. Indeed, this has been done by McCormack et al[56J, but the influence of this connection upon the oxygen ordering itself was completely ignored.  Chapter 5. The Extended CVM model for YBa2 Otis 06-Es^  146  4. Extracting (ax/a/)T from oxygen pressure isotherms is an exacting business. To date this has only been done here and by McKinnon et al.[59]. The two curves do not completely overlap, such that it might not be appropriate to quantitatively fit these curves to such a high degree of accuracy, until the data has been further corroborated 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 the superconducting plateau seen at x '--' 0.6 only occurs upon annealing at temperatures lower than those experienced in this work. This plateau has often been associated with the 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 are high enough to give a smooth evolution of the hole doping in the orthorhombic phase as predicted in the present square cluster approximation. Whether it is appropriate to use an approach that gives smooth planar doping as a function of x across the O-T transition is another question. In the model presented here, the high temperature limit for the chain and plane holes was used which gives a smooth evolution of the plane holes upon doping even across the O-T boundary. This is perhaps not very correct as will be discussed in the next section.  5.3.2 Question about the nature of the O-T transition It quite possible to imagine that the specifics of the chain-to-plane charge transfer might be the driving force behind the order-disorder O-T transition. In a more sophisticated picture, where one would take into account the mechanism of charge transfer, a situation could arise where, as a function of x, there is a fundamental change in the behaviour of the added holes. If this is the case, then the charge carrier entropy may take on completely  Chapter 5. The Extended CVM model for YBa2  CU3 06+x^  147  different characteristics which initiates an order-disorder transition in the chain oxygens. Consider the following scenario: imagine that one could describe the electronic system by a narrow CuOs (i.e. basal plane) band and a Cu02 band, with the CuOs band lying lower in energy than the Cu02 band. Then, add to this an effective strong nearest-neighbour repulsion between electron holes. If T is comparable to or less than the band gap, but much lower than the nn repulsion, then all the holes will pile up in the basal plane until the nearest-neighbour exclusion forces holes to be transferred to the Cu02 planes. In such a situation, below some critical concentration of holes, the entropy is determined by the characteristics of the CuOs basal plane. At higher hole concentrations, the CuOs band is inert and the entropy is determined by the Cu02 planes. In such a case, the reduction in entropy associated with the ordering of the oxygen atoms is to some degree offset by the increase in hole entropy caused by charge transfer'. Thus, the O-T transition is in general affected by the mechanism of charge transfer. It is therefore unlikely that the O-T transition is determined solely by concentration independent oxygen effective pair interactions. Such a point has also been stressed by L.A. Andreev, Y.S. Netchaev and 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 CVM approximation with a maximal cluster containing two copper atoms (cf. next section). Also, one should consider the behaviour of the antiferromagnetic and superconducting phase diagram in relation to the O-T transition. As discussed in the introductory chapter, YBa2Cu306+s is an antiferromagnetic insulator (AF) in the tetragonal phase and a superconductor in the orthorhombic phase (cf. figure 1.2). In other perovskite 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 between nnn oxygens via the copper.  Chapter 5. The Extended CVM model for YBa2  C113 06+x  ^  148  region between the AF and superconducting phases. Such an intermediate region where no long-range magnetic order is seen and where there is no superconductivity has never been seen in YBa2Cu306+x. However, doping YBa2Cu306 by other means, such as 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 the doping is in principle a smooth function of y. One has direct control over the amount of holes doped to the planes. The fact that the intermediate zone does not exist in YBa2Cu306-Fx implies that the metal-insulator transition, which coincides with the O-T transition, entails a large increase in hole doping at the transition. This is a fairly direct experimental proof that a large transfer of holes must take place at the onset of the orthorhombic phase. The oxygen vapour pressure experiments are carried out at high T whereas the AF and superconductivity experiments are at low T, so that it might be possible to reconcile these findings by an argument involving the T dependence of the electronic and structural behaviour. But it is clear from the (OxIOOT measurements that one cannot separate the two systems, and traditionally held beliefs concerning the set of approximations (in particular the static effective pair interaction viewpoint) used for the structural properties of YBa2Cu306+x should be critically re-examined. 5.3.3 Suggestions for enhancements and future work There are several obvious steps to take from here. Most were alluded to in sections 5.3.1 and 5.3.2, but here we will be more specific. There are more refined approximations that one can make both for the configurational problem of the oxygens and their connection to the hole count as well as for the assumptions concerning the electronic structure. We will concentrate here on the subject of the oxygen configurations. It has been mentioned several times that the CVM square cluster ignores the distinction between clusters with and without a central copper. In the next higher CVM  Chapter 5. The Extended CVM model for YBa2 CU3 06+x^  149  approximation for the 2D square lattice, the 4+5 point cluster allows a separation of clusters containing different amounts of copper atoms. Figure 5.14 shows the maximal cluster of the 4+5 point approximation and the relevant square subcluster which one could use for the determination of the Cu valence. One could use the same approach as in section 5.1, but the statistics for this square cluster might be different due to the more complex set of clusters used. Unfortunately, Kikuchi's natural iteration method cannot be applied to the 4+5 point approximation making the free energy minimization quite difficult. Perhaps a better step would be to also incorporate the distinction between chain and 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 would need to include statistics for clusters incorporating 2 or more central copper atoms. Such a cluster would be the 3x3 point CVM cluster shown in figure 5.15. Writing down the configurational entropy for such a cluster is quite a complex problem and minimizing the free energy would be very difficult once the hole entropy has been included, since the 3 x3 point cluster is not of single type (there are 2 in-equivalent maximal clusters) so that the NI method would not work. It would probably be preferable then to use a Monte Carlo method 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 would add to the internal energy estimates for the configurational entropy of the chain and plane holes. The number of chain and plane holes would be given by the oxygen configuration. This would give a complicated temperature and oxygen configuration dependent "internal energy". The chain and plane holes would be determined by counting the square clusters of 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 set  Chapter 5. The Extended CVM model for YBa2Cu3 06+x  ^P  150  I-  ^1 as  Figure 5.14: CVM 4+5 point cluster for the basal plane oxygen (Thick solid lines). Thin dotted 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,^  151  up a scheme where holes are distributed within their isolated oxygen chains until it is necessary 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 ASYNNNI model at low T and counted the number of ortho-I and ortho-II regions as a function of  x. It was proposed that the minimal cluster which could define an ortho-I region would transfer a number of holes to the planes such that spanning the entire basal plane with ortho-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 holes so that a spanning region of ortho-II gives a 60K superconductor. The assumption is that 7', is proportional to the number of planar holes, and that the number of planar holes depends on the proportionate amount of ortho-I and ortho-II regions. This model is strikingly successful in predicting 71, vs. x. The hole creation scheme described above will predict a similar kind of behaviour. To see this, consider the minimal cluster of an ortho-I phase shown in figure 5.16. It contains two nn coppers, both which have generated a hole. Since they are nearest neighbour coppers, one hole must be transferred to the plane. The planar hole count is 0.5 per unit cell. In a minimal cluster of ortho-II phase, one has 8 coppers with 4 holes. In this case, there are 0.25 planar holes per unit cell. If the ortho-II phase becomes stabilized and pre-dominant at x L-: 0.66, then this scheme will exhibit a plateau in the hole count: additional oxygens added to the ortho-II phase 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 before relaxing the narrow band configurational approach for the charge carriers. On a purely speculative vein, it is possible to imagine that taking into account the energetics of charge transfer in more detail would be capable of generating a similar phase diagram with the  Chapter 5. The Extended CVM model for YBa2Cu3 06+x^  152  Figure 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 on the 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  ^  153  existence of an ortho-II phase etc., assuming only a direct nn coulomb repulsion between oxygen 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 will retain the essential features of the 2D ASYNNNI model. The essential feeling one gets from this analysis is that taking into account the electronic structure in detail cannot leave one with x and T independent effective pair interactions which would be able to correctly describe the oxygen thermodynamics. It has often been said that the reason for the poor fit of the 2D ASYNNNI model to the oxygen pressure data is due to long range elastic effects or shifts in the phonon spectrum resulting in the subsequent dismissal of oxygen chemical potential data as containing too many effects to be a useful test for theoretical ideas. Thus, the continued use of a static effective pair interaction picture. The proper physical grounds for such claims have never been properly addressed one should prehaps re-examine in detail the entire physical basis for the pure 2D lattice gas model in YBa2Cu306+x•  5.4 Summary It was shown that the inclusion of the behaviour of the electronic subsystem into the free energy of the oxygen ordering problem drastically improves the agreement between theory and experiment. The proposed phenomenological extension to the standard CVM approximation was based upon several experimental observations: 1. The experimental kT(Oxiait)T curves were found to be essentially independent of temperature except very close to the O-T transition. This implies that at these temperatures, the entropy dominates the chemical potential. 2. The in-plane and bulk thermopower in YBa2Cu306+s is roughly independent of T above room temperature. This suggests that a narrow-band picture might be  Chapter 5. The Extended CVM model for YBa2  CU3 06-Fx  ^  154  a reasonable approximation, where the entropy of the charge carriers is purely configurational. 3. From x-ray absorption spectroscopy, a functional ionic description for the valence of the chain coppers emerged which allows one to connect the oxygen configuration to the creation of holes. 4. The existence of a semiconductor-metal transition in the Cu02 planes when the oxygen concentration is increased as well as numerous other charge carrier dependent experiments clearly imply that electron holes created in the chains are at some point transferred to the planes. These findings together with the inability of the standard 2D ASYNNNI model to fit the non-ordering susceptibility kT(ax/a,u)T lead us to propose a simple parameter freel2 extension to the standard oxygen ordering models which takes into account the creation of holes. The extension was based upon a thermodynamically consistent hybrid model between the standard 2D lattice gas model and the defect chemical model of Verweij and Feiner. The connection between these two models was made by identifying the square cluster configuration of oxygen with the creation of holes. Charge transfer was introduced by assuming a high temperature, random occupation of all candidate electron hole sites. The system was solved in a modified CVM square approximation which, in its standard form, agrees fairly well with the higher order 2D ASYNNNI model at the temperatures experienced in the oxygen pressure experiment. It was seen that no strong modification of the structural phase diagram is produced but that agreement is achieved with virtually all experimental findings related to the oxygen thermodynamics and basic valence 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. This was not done for the sake of completeness of the theory and to make sure that no unphysical model is significantly better than the one chosen.  Chapter 6  Concluding remarks  Much of what was said in the last sections of chapter 5 could have also been placed in this chapter. Instead, we will conclude by giving a historical perspective of the experimental and theoretical work of the thesis. In recent years, there has been quite a shift in the 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 oxygen interactions in a lattice gas model. Its simplicity and initial successes probably accounted for its popularity. However, also quite early on, there were indications that a pure lattice gas 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 almost identical to the "full blown" 2D ASYNNNI model. Something was wrong, but there were always plausible reasons for not obtaining a fit to the (ax/ait)T data: inhomogeneities in the sample, too many extra effects, such as lattice dilation. For example, in 1991 Rikvold et al. calculated kT(.9x/Oit)7, using Monte Carlo on the 2D ASYNNNI model (cf. figure 4.8) claiming that good agreement was found and that the remaining discrepancies were probably due to sample inhomogeneities. At the same time, quite a few chemists were utilizing a completely different approach to fit their thermodynamic data: the defect chemical model. They immediately recognized the existence of additional degrees of freedom through the creation of electronic defects, caused by the additional oxygen. These two camps, however, seemed completely disconnected. Defect chemical papers would not cite lattice gas papers and 155  Chapter 6. Concluding remarks^  156  vice versa. I had spoken to some chemists active in this field in the fall of 1991, and they seemed quite unaware of the profusion of literature that exists concerning the lattice gas approach for YBa2Cu306. The major deficiency of these models is the fact that they ignore the types of superstructures seen in YBa2Cu306+s, and mostly make no prediction for the O-T transition. Having an affinity for statistical mechanics, and recognizing the obvious success of the lattice gas models to predict the ordered superstructures, I continued trying to make the lattice gas models work. Furthermore, there are many lattice gas papers which introduced empirical on-site energies to successfully fit the thermodynamic data, most notably, Shaked et al. and Bakker et al. However, although some of the defect chemical models did not have exactly the correct magnitude for kT (ax/a,u)T, there was the striking feature of predicting the correct curvature and a T independent kT(ax/ap)T, as was found experimentally. 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 agreed with the more recent understanding and workable picture for the creation of holes and their connection to oxygen configurations. Unfortunately, the point of the paper was to use their own model and its agreement to the thermodynamic data as somewhat of a proof for the existence of charge carriers in a narrow band (less than 0.1eV); this paper is not often cited in the literature. After having made many calculations using the CVM mean field approximation for the 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^  157  As for the future, it is obvious now that one must include the behaviour of the electronic structure in detail, since it is the unusual electronic properties which are dominating 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, in relation to charge transfer and hole doping, at a first principles level. The recent paper by Uimin and Rossat-Mignod (Physica C, September 1992) seems very promising, although 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 for sample preparation and the application of some interesting statistical mechanics models, to becoming directly involved in the discussion of the unusual electronic structure of the high T, materials. Papers that are published concerning this topic are now being written by individuals simultaneously involved in the search for a pairing mechanism. The language has evolved from "lattice gases" and "cluster variation method" to "Hubbard models" and "the large U limit".  Appendix A  Calculation of the chemical potential for a diatomic gas with molecules of like atoms applied to 02 gas  Using the approach of Landau and Lifshitz[61] to describe the thermodynamic properties of a diatomic gas, this appendix is a resume of the derivation of the chemical potential it of an ideal gas with molecules of like atoms. A specific calculation for 02 gas will be presented at the end. The chemical potential can be determined from the expression for the free energy  F. In the case of a diatomic gas, the expression for F is obtained by taking into account the translational, rotational and vibrational degrees of freedom of the molecule. Under the Born-Oppenheimer approximation, the energy levels of a diatomic molecule can be written 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' electronic state, these levels may be written: E19,K =--- ECI + nW(19  1 + ) + h2K(K + 1)/21  (A.1)  where co is the electron energy, hco the vibrational quantum, V the vibrational quantum number, K the angular momentum of the molecule, I = m'rg the moment of inertia of the molecule (m' = mim2/(mi + m2) is the reduced mass of the two atoms and ro the equilibrium of the distance between the nuclei). Substituting the expression A.1 into the definition of the partition function, and 1-Note that the electronic ground state of oxygen is in fact a very narrow triplet. This narrow width just gives rise to a degeneracy of the electronic state as shown later.  158  Appendix A. Calculation of the chemical potential for a diatomic gas...^159  using 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 is the volume. The first term may be called the translational part, Ftr, since it arises from the degrees of freedom of the translational motion of the molecules. F„t and Fvib are respectively the rotational and the vibrational free energies, and are defined next. Fel is the electronic contribution, and for most cases is given by the degeneracy of the electronic ground state, ge. Excited electronic states are typically very high in energy (eg. the first excited state of oxygen is at 11,300K), and do not contribute to the thermodynamic properties. 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 of molecular 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 that the gas can be considered ideal (PV = NkBT), one obtains: - )) [P(1 — exp GA`,) F = NkBT1n^7B I + NE0 (kBT) with  eIge =  m  h2^27rh2  (A.6)  3  (A.7)  Appendix A. Calculation of the chemical potential for a diatomic gas... ^160  Using the fact that ft = (aaN F)  Ty  and taking as the origin of energy the ground state  which corresponds to 9 = 0 and K = 0 in the equation A.1, one finally obtains the following expression for IL: [P (1 — exp ^hw ,u, = kBT ln ^ 4-(kBT):1^2  (A.8)  where P is the pressure and is given by the equation A.7. Application to molecular oxygen: For oxygen we have[139]: 31.9988g/mole hw  =  1580.361cm-1  r,  =  1.2075A  ge  2273.875K  3  ) )1  so that finally, the chemical potential per molecule of oxygen is: Po, = kBT ln  [P (1 — exp -(&  nu,  (kBT)i^2  (A.9)  with: = 1.1628 x 1018Torr/(eV)i hw  2  = 97.9733n/eV  This expression agrees, to within 10-2eV, with the phenomenological expression for the chemical potential of molecular oxygen given in Barin and Knacke[140] in the temperature range 300-1000K.  Appendix B  The natural iteration method for the free energy minimization of CVM models  This appendix describes the natural iteration method for the minimization of the cluster variational free energy. The NI method was first presented in 1973 by Ryoichi Kikuchi for the solution of the CVM equations[87]. It is an alternative to the standard NewtonRaphson approach to solving the CVM free energy. In his original paper, Kikuchi derives the NI method for the case of the CVM pair and tetrahedron approximation. We will derive the NI equations for the case of the CVM square approximation and then for the extended CVM model of chapter 5.  B.1 CVM square approximation We start by writing down the grand potential of the CVM square approximation given by equation 2.26 of section 2.3.1.  G N,  E  EijklZijkl  i,j,k,1  -kkT[  E  Zijki  E  1 ln Z^ ijki — 2 yii In yij + —2  E x7 In x7  In + X 'j?  2}]  i^  1 2^. .  (B.1)  aijklZijkl  161  162  Appendix B. The natural iteration method... ^  where 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 through (B.2) (B.3)  Xr3. 3  (B.4)  Y23 ^Zijkl k,1  so that the grand potential can be expressed as a function of only the square cluster probabilities that  Zijki.  Eij,k,13ki  Since the cluster probabilities are normalized, one has the condition  1, which one can add to equation B.1 by the introduction of a Lagrange  multiplier:  N,  E  EijklZijkl  i,j,k,1  E  1 - Er(yii)+ —2 E^ r(xn+ Er(Xj  -PkT[^ r(ziiko 2  i,j,k,1^i,a^  2  a  E^A(1 — E ziik)  }1  (B.5)  i,j,k,1  where we have used the notation r(x) = xlnx^  (B.6)  We now have the unconstrained minimization problem for G with respect to the cluster probabilities zijki. The strategy of the NI method is to take advantage of the symmetries of zijki and the resulting symmetries of the point and pair cluster probabilities 4, 4 and  yi3.  Looking at the definition of zi3k/ through the square cluster,  Appendix B. The natural iteration method... ^  163  we can identify symmetries in the cluster probabilities: Z. i ki = Zkjil -= Zilkj = Zklij  Yij = Yil = Ykj = Ykl  (B.7) which are obtained by looking at all the possible geometric operations which leave the image of the cluster invariant (eg. rotation of the square by 1800). Naturally, in the disordered phase, xa = x/3 and more symmetries arise, but we will not list them here. Using these symmetry relations, we can formally write  41 (Ef(Yii)+Ef(y)+Ef(yki) +^(yid)) i71^kj^k,1  1  2  (E,c(4)+E.c(x00)  21 (E f(4)  Er(X))  3^1  (B.8)  The existence of these symmetry relations is a necessary condition for the NI method to work. We rewrite equation B.5 using B.8 so that the expression for the grand potential has the symmetry of the maximal cluster. Then, setting the derivative of G with respect  164  Appendix B. The natural iteration method...^  to zijki to zero and solving for Zijki  Zijki  gives:  (YiiYakiYk t) : exp —(ciiki — paijk//2 — A) } a (414 XZX11)T^  (B.9)  kT  Now it has become apparent why one has written the pair and point cluster probabilities with the same symmetry of zijki. Without this step, it would not have been possible to write equation B.9 in such a manner. Next, we use the normalization condition for zijki to 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.10 form 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 approximation for y ziandom  =  a0  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 the minimum of the free energy. One can monitor the evolution of the cluster probabilities to decide when the iteration has converged. The value of the free energy is less sensitive and converges faster so that it is better to include the evolution of the cluster probabilities for the convergence criterion. The number of iterations required for convergence of the CVM  ^  165  Appendix B. The natural iteration method... ^  square approximation was roughly 100 steps away from the order disorder transition and up to 10000 steps very close to the transition. The maximum number of steps allowed for the chemical potential calculations was 5000. For the phase diagram calculation, the limit was set to 10000 iterations.  B.2 Adding the spin and hole entropy The 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 taken into 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 kN kNo in (22)ciikil i^ 2 I ln(g)aiki ^ 2  (.9.5spin)  aZijkl  )  (B.11)  We recall that the number holes per unit cell is given by (cf. equation 5.7) [hole] = where  ai jki^(i  j  E (2ai +  E  Cijkl 1)Zijk1^ hijklZijkl i,j,k,1^ i,j,k,1  ^  (B.12)  k l)/2 gives the oxygen concentration of a square cluster and  co,/ = 1 if the cluster produces a Cu'+ (cf. equations 5.9 and 5.9). Thus, taking the derivative of the hole entropy (eq. 5.4) gives (ashoie) aziJkl  kNo^n^)11'1 In — [hole]) hi3k1 = 2 in — [hole])^[hole] )  where vijki is the derivative of the availible hole sites n wrt.  Zijki.  (B.13)  Inserting relations  B.11 and B.13 into the derivative of the grand potential gives finally the new NI iteration equations for zi3k/ and A: 1^1 ^Zijkl^  —(cijki — paiik//2 exp ijkl " ijkl^  — kTcriiki/2 — A) kT  _^I^i —(Eijki — paiik112 exp (uA)^E X^exp kT i,j,k,1^  —  kTo iik112) -  (B .14)  (B.15)  Appendix B. The natural iteration method...^  (B.16)  Xijki = x7x'q xc'xi3 k I^  (B.17)  Yijkl = giigilgkjgki^ Hijkl =  n^)vijkl ( n — [hole])3" (n —  166  [hole])^[hole] )^  (B.18)  Note that it is crucial to express Hijki, hijki, and vi,k/ so that they contain the same symmetries of zi30, as was done with the point and pair cluster probabilites. If this is not done, then the right hand side of equation B.14 will not contain the same symmetries as the left hand side' Specializing to the f(II)F(II) model, where one assumes that the hole is on the oxygen forming a spin doublet with the two neighbouring Cu2+ and distributed over all oxygens in the chains and planes, we get n=4+ x=4+ vijkl =  E  aizijkl  aijkl 1  2 aijkl  —2111(2) (aiiki^ciiki)  (B.19) (B.20)  This implies that the additional spin entropy, giving rise to criiki, effectively shifts the chemical potential by an amount kT2 ln 2 and adds an effective cluster energy kT 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, one needs 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 for 200 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 functions of 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 by calculating the change in oxygen concentration x for every change ,a. The phase diagrams were generated by making a detailed calculation of the long range order parameter s close to the transition to detect when .s becomes non-zero. This is an inefficient algorithm in terms of computer CPU time, but its simplicity minimized the chance for errors in the calculation. Typical CPU times on an IBM RS/6000 560 were about 25 minutes for a phase diagram calculation and 2 minutes for the other calculations.  Appendix C  Sample preparation and characterization  We have put the information about the making of the master batch of YBa2Cu306+, and the characterization of its physical properties into this appendix, because there really is no remaining mystery about the proper technique to prepare pure YBa2Cu306+x. This was not true at the beginning of YBa2Cu3064.s research early in 1987. Some of the samples prepared for the oxygen vapour pressure experiment were made at a time when the 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 some readers fairly inefficient. After preparation, the batch of YBa2Cu306+x was extensively characterized in order to confirm the expected high quality and to obtain estimates for the upper limit of impurities. This is crucial since one important parameter needed to determine 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 source and/or a sink for oxygen.  C.1 Sample preparation The master batch of YBa2Cu306+x was prepared using five 9's purity starting materials which had been preheated overnight to several hundred degrees C in order to remove trapped water. Y203, CuO and Ba2CO3 in stoichiometric proportions, giving a total of about 100g of starting materials, were mixed thoroughly by hand in a beaker and then 168  Appendix C. Sample preparation and characterization ^  169  transferred in smaller batches to an automatic micro mill (Brinkman Instruments Retsch Mill). 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 MgO crucibles. Roughly 20 crucibles were needed to hold the entire batch. Small crucibles were used to allow CO2 to escape and oxygen to enter easily. The batch was calcined in 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 7 minutes and sifted through a 70,am sieve. After each firing, a small sample was extracted to monitor the progress using X-ray diffraction. Finally, the powder was pressed, with a pressure of approximately 10 tonnes per cm2, into 3g, 3/4" diameter pellets and sintered in flowing oxygen in clean tube furnace (Lindberg single zone 1200C), dedicated solely to the oxygenation of YBa2Cu306+s. The standard grade oxygen was first passed through a Pd catalyst at 500°C to catalyse CO to CO2, then fed through a liquid nitrogen cold trap to freeze out any CO2 and water. The following oxygenation procedure was used: fast 10min 0  fast  9250c,  5300c,  3700C100  Ohrs^0 Clhr 10min^2hr s^ 2hr a 6 o^ 1200^ C/hr Clhr fast  RT -->300°C^600°C  RT  (C. 1)  The intermediate steps at 300°C and 600°C are to allow the carbon lubricant, which was used to press the pellets, to burn off. C.1.1 Final oxygen content The final oxygen content of a sample prepared in the above fashion was estimated to be x = 0.987 + 0.002. This value was determined by examining the room temperature electrical conductivity of a series of deoxygenated pellets which were prepared to have an 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 an  insulator (hole doping is zero), one can obtain a value for the fully oxygenated pellets  Appendix C. Sample preparation and characterization ^  170  prepared in this manner. It was found (cf. figure C.1) that the conductivity a varies linearly 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 titration was used to determine the oxygen content of a sample annealed at 450°C in 1 atm oxygen pressure. 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 with the oxygen pressure isotherms. C.1.2 Impurity levels in the YBa2Cu306+s Two 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 a monochromator stage showed no detectable levels of impurity phases. Second, inductively coupled plasma mass spectrometry (conducted by Elemental Research Inc.) showed silicon as the major impurity at 160ppm. Such an impurity presumably comes from the grinding process using the micro mill. The next highest impurity detected was calcium at --, 24ppm.  C.2 Sample characterization The master batches of YBa2Cu306+s created with the technique outlined above were 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 measurement of the Ginzburg-Landau parameter K2(T)[1411. Numerous other measurements have been conducted with these samples; we will list only the two most pertinent for the verification 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 not partially decomposed in the process.  171  Appendix C. Sample preparation and characterization^  LI^  8  1  -  _ -  ■,.....,  -  -  1  1  1  -0.02 0.00 0.02 0.04 0.06 0.08 x in YBa2Cu306_,x Figure C.1: Room temperature conductivity of YBa2Cu306+s vs. x for x < 0.08. The solid line is a visual fit to the three data points. The oxygen content was determined by assuming that the annealing conditions described in section C.1 gives x = 0.987. Notice that the conductivity increased for the x '1.•_-' —0.018 data point.  ^  Appendix C. Sample preparation and characterization ^  172  RT resistivity = 1.0rnf/cm V") = 93.7K = 93.4K ^T(1°%)^92.8K ^T(o)^92.7K AT(resistivity) = 0.9K AT(magnetization)^3K Meissner fraction^25% Major Impurity: Si © 120 ppm Table C.1: Some physical parameters of the YBa2Cu306+s master batch were 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 a plot of the resistivity vs. T for the fully oxygenated sample. Also included is an insert showing the transition region. The transition width is less than 1K, and shows no significant "tail" indicating high sample quality and homogeneity (cf. table C.1 for a summary of the pertinent characteristics). The magnetization was measured in a RF SQUID magnetometer (Quantum Design MPMS SQUID) cooled in 10 gauss static field (cf. figure C.3). Applying a 10 gauss field to 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 showed no detectable ferromagnetic signal. No Curie term was detected at low temperatures. Conclusion The YBa2Cu306+s material prepared for the oxygen vapour pressure experiment is of the highest quality. No impurities were found by X-ray diffraction either before or after the experiment. Measurements of the resistivity and magnetization indicate standard behaviour of very pure, sintered YBa2Cu306-Fs•  Appendix C. 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