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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

SUPERCONDUCTING MAGNETS AND SUPERCONDUCTIVITY: RESEARCH, TECHNOLOGY AND APPLICATIONS

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Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

SUPERCONDUCTING MAGNETS AND SUPERCONDUCTIVITY: RESEARCH, TECHNOLOGY AND APPLICATIONS

HENRY TOVAR Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

AND

JONATHON FORTIER EDITORS

Nova Science Publishers, Inc. New York

Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Superconducting magnets and superconductivity : research, technology and applications / [edited by] Henry Tovar and Jonathon Fortier. p. cm. Includes index. ISBN 978-1-61728-587-5 (E-Book) 1. Superconducting magnet. 2. Superconductivity. I. Tovar, Henry. II. Fortier, Jonathon. QC761.3.S87 2009 537.6'23--dc22 2009014688

Published by Nova Science Publishers, Inc.

New York

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CONTENTS Preface

vii

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Research and Review Studies

1

Chapter 1

Correlation Effects and Superconductivity in Cuprates: A Critical Account Sven Larsson

Chapter 2

Improved Flux Pinning Properties for RE123 Superconductors by Chemical Methods Yui Ishii, Jun-ichi Shimoyama and Hiraku Ogino

55

Chapter 3

Mechanical Characterization at Nanometric Scale of Ceramic Superconductor Composites J.J. Roa, X.G. Capdevila and M. Segarra

77

Chapter 4

Understanding the Roles of Heavy Ion and Gamma-Irradiations on the Magnetic and Transport Properties of Superconductors I.M. Obaidat, B.A. Albiss and M.K. Hasan (Qaseer)

169

Chapter 5

On the Melt Processing of Bi-2223 High-Tc Superconductor Challenges and Perspectives A. Polasek, E.T. Serra and F.C. Rizzo

197

Chapter 6

Vortex Theory of Inhomogeneous Superconductors B.J. Yuan

215

Chapter 7

Fabrication of Pyrochlore-based Buffer Layers for Coated Conductors via Chemical Solution Deposition Hechang Lei, Xuebin Zhu, Yuping Sun and Wenhai Song

257

Chapter 8

Radiation Shielding Schemes and Advanced Fabrication Techniques for Superconducting Magnets Laila A. El-Guebaly and Lester M. Waganer

275

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3

vi Chapter 9

Contents Thermal Stability Characteristics of High Temperature Superconducting Composites V.R. Romanovskii and K. Watanabe

Short Communication Thermo-Mechanical Pumps for a Large Superconducting Magnet in Space Operated by Use of Superfluid Helium G. Kaiser, A. Binneberg and J. Klier

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Index

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293

401 403

415

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PREFACE This book presents current areas of research in the field of superconductors and superconducting magnets. The ways in which these magnets produce stronger magnetic fields than ordinary iron-core electromagnets is explored. A review of the electronic structure of transition metal oxides and salts is also included in this book, specifically what concerns electron transfer, electron correlation, electron-nuclear coupling, and inter-metal interaction in cuprates. Combining a number of well-known theories of conventional superconductors, a general vortex theory for inhomogeneous superconductors is proposed. Ways to fabricate superconducting magnets in a faster, cheaper and more practical way is also presented. In Chapter 1, electronic structure of transition metal oxides and salts is reviewed, particularly what concerns electron transfer, electron correlation, electron-nuclear coupling, and inter-metal interaction in cuprates. A short resume of electronic correlation in molecular systems is first given. Electron pair transfer is treated in a many-electron real space approach using standard mixed-valence theories. The latter models have been successful in the past in describing the response of the electrons to the motion of the nuclei (electron – phonon interaction). The possibility for multiple oxidation states of the metal ion is typical for transition metal compounds. Mott-Hubbard-U is strictly defined, its dependence on breathing mode coordinates analysed, and the connection between U and the energy gap for superconductivity clarified. Delocalization is treated in terms of an extended Hush model and electron pair supercurrents derived at the Van Hove degeneracy. d-wave gap anisotropy is found to be consistent with the general atomic level model presented here. Softening of phonon half-breathing modes in inelastic neutron scattering (INS) can be connected to mixedvalence. The fundamental vibronic interaction between spin density wave (SDW) and charge density wave (CDW) states leads to a new phase with energy gap and electron pair carriers. Finally comments are made on why MV-2 systems delocalize as Fermi-Dirac systems (ordinary metals) while MV-3 systems delocalize and condense as Bose-Einstein systems (superconductors). As explained in Chapter 2, effects of dilute impurity doping to Y123 single crystals and Y123 melt-solidified bulks on their flux pinning properties were systematically studied. It has been already reported in our previous papers that the dilute Sr-doping to Ba site improved the critical current properties of Y123 crystals, and the dilute Lu-doping to Y site was also found to increase critical current density, Jc. In addition, the dilute impurity doping to Cu in the CuO-chain was found to dramatically enhance Jc under magnetic field without large decreases in Tc in contrast to the case of dilute impurity doping to the superconducting CuO2-

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Henry Tovar and Jonathon Fortier

plane by Zn or Ni. Furthermore, we have attempted to extend this method to Dy123 meltsolidifed bulks, and found that the dilute impurity doping to Cu in the CuO-chain certainly improved Jc, and the dilute Sr-doping is also effective particularly below 50 K when a small amount of Dy/Ba substitution occurred in the Dy123 phase. Additions of Tb-containing compounds were also attempted to improve Jc properties of Dy123 melt-solidified bulks. Although CeO2 addition has been well known to enhance Jc by reducing size of RE211 particles, it always accompanies slight decrease in Tc. Our newly developed method, introduction of fine BaTbO3 precipitates, was found to be more effective for enhancement of Jc without such decrease in Tc. In particular, Tb4O7 added Dy123 meltsolidified bulks exhibited remarkably improved Jc due to the generation of fine BaTbO3 precipitates with 0.1 ~ 0.2 μm in size in the Dy123 matrix. This result means Tb4O7 is a more effective additive than CeO2. The nanoindentation or indenter testing technique (ITT) is a functional and fast technique that can give us a lot of information about the mechanical properties of different materials at nanometric scale, from soft materials, such as copper, to brittle materials, such as ceramics. The principle of the technique is the evaluation of the response of a material to an applied load. In a composite material, if the size of the residual imprint resulting from a certain load is lower than the size of the studied phase, then is possible to determine its mechanical properties, and therefore its contribution to the global mechanical properties of the composite. Depending on the tipped indenter used, different equations should be applied to study the response of the material and calculate stress-strain curves and parameters such as hardness, Young’s modulus, toughness, yield strength and shear stress. These equations are related to the different deformation mechanisms (elastic, plastic or elastoplastic) that the material undergoes. In the case of most of the ceramic composites, when a spherical tipped nanoindenter is used, elastic deformation takes place, and Hertz equations can be used to calculate the yield stress, shear stress and the strain-stress curves. On the other hand, when a Berckovich indenter is used, plastic deformation takes place, then Oliver and Pahrr equations must be applied to evaluate the hardness, Young’s modulus and toughness. Nevertheless, in the hardness study, Indentation Size Effect (ISE) must be considered. In Chapter 3, the mechanical properties of a ceramic superconductor material have been studied. YBa2Cu3O7-δ (YBCO or Y-123) textured by Bridgman and Top Seeding Melt Growth (TSMG) techniques have been obtained and their mechanical properties studied by ITT. This material presents a phase transition from tetragonal to orthorhombic that promotes a change in its electrical properties, from insulating to superconductor, and that can be achieved by partially oxygenating the material. On the other hand, the structure of the textured material is heterogeneous, and two different phases are present: a Y-123 as a matrix and Y2BaCuO5 (Y-211) spherical inclusions. Moreover, the texture process induces an anisotropic structure, thus being the ab planes the ones that transport the superconductor properties while the c axis remains insulating. The purpose of this study is the characterization of the mechanical properties, in elastic and plastic range, of orthorhombic phases of YBCO samples textured by Bridgman and TSMG technique. With the ITT technique, the oxygenation process can be followed and its kinetics established.

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Preface

ix

The roles of point defects created by γ-irradiation and chemical doping along with the roles of extended defects such as grain boundary regions and columnar defects created by Pbion irradiation on the behavior of the critical current density, Jc have been investigated in several superconductors. The critical current density in YBa2Cu3O7-δ (YBCO) polycrystalline sample doped with Mn was found to be significantly enhanced. The enhancement of Jc was found to be more significant at the lower temperatures. Magnetization measurements were carried out on YBa2(Cu1-xBx)3O7-δ (x = 0, 0.05, 0.1) polycrystalline samples. Considerable increase in the hysterisis width of the magnetization M versus the applied magnetic field H with increasing boron concentration was observed in these samples suggesting an enhancement of the vortex pinning forces in these samples. A limitation of the role of point and columnar defects on enhancing Jc was found in all our samples. In Chapter 4 we report an explanation of the effect of γ-irradiation and Pb-ion irradiation doses on Jc in these materials. The explanation is based on combining several competing mechanisms of irradiation which we believe to take place mainly in the regions of the grain boundaries. The influences of these mechanisms were found to vary with the irradiation dose level. The effect of γ-irradiation on the normal state resistivity of MgB2 polycrystalline superconducting specimens was also investigated. An increase in the normal state resistivity and a broadening of the resistive transition to the superconducting state were observed with increasing γ-irradiation dose. Different temperature dependence of normal-state resistivity and different residual resistivity ratios, RRR were obtained for different doses. Although the production of high performance (Bi,Pb)2Sr2Ca2Cu3O10+x (Bi-2223) silver sheathed superconducting tapes has already reached industrial level, there is still room for further improvements. On the other hand, sintered Bi-2223 bulk parts frequently exhibit high porosity and poor electrical properties. Melt processing has been for long attempted as an alternative to sintering, in order to improve density, texture and microstructure of tapes and bulk parts based on this compound. However, recrystallizing Bi-2223 from the melt is challenging, due to the lack of knowledge of the Bi-2223/melt equilibrium, the phase narrow stability range and sluggish formation kinetics. In fact, Bi-2223 decomposes peritectically into liquid and solid phases, but most of the attempts to revert such decomposition reaction have failed, generating mainly Bi2Sr2CaCu2O8+x (Bi-2212) and Bi2Sr2CuO6+x (Bi-2201), together with secondary phases such as (Ca,Sr)2CuO3, CuO and Ca2PbO4. This may be partially attributed to kinetical reasons, since Bi-2212 and Bi-2201 form faster than Bi-2223. In addition, Ca and Cu segregation as well as Pb volatilization take place above the solidus line, hampering Bi-2223 recrystallization from the melt. Nonetheless, promising results have been reported in the literature, particularly in the case of partial peritectic decomposition. Other works indicated that Bi-2223 can be partially recovered after complete peritectic decomposition, as well as after full melting via the glass ceramic route. Therefore, the investigation of Bi-2223 melting and recrystallization may provide results of high scientifical and technological relevance. In Chapter 5, a general review of this theme is undertaken with the aim of shedding light on this complex issue. In Chapter 6, combining a number of well-known theories of conventional superconductors [1–6], which include Abrikosov vortex theory, Gorkov formalism, Extended Ginzberg- Landau theory, and de Gennes-Werthamer proximity coupling theory, a general vortex theory for inhomogeneous superconductors is proposed [7–9]. This general vortex theory is applied to three characteristic and simple geometries, a bulk superconductor with

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finite size, a thin film in parallel applied filed and a special two-components superlattice, respectively. In each of the three cases, spatial form of order parameter and magnetic field in the geometry are derived, which lead to an analytical expression of Gibbs free energy. For a finite bulk superconductor, the eigenvalue spectrum in central region of the bulk geometry is in similar to the case of an infinite geometry with lower symmetry. A vortex lattice in the central region prefers a hexagonal structure if only minimum free energy is concerned, however, when effects of repulsion between flux lines or arrayed defects are included, symmetry of a vortex lattice varies. For a thin film geometry in a parallel field, the solution shows that the inhomogeneity caused by limited thickness of a film enhances extended GL parameter, κ2, which proves that thin film superconductor tends to be type II superconductor. For a two-components (NS) superlattice in a parallel applied filed, we find that, in addition to a slowly spatial varying pair amplitude nucleating in N layers, a highly condensed pair amplitude confined in the other component layers (S) can also simultaneously satisfy both a minimum eigenenergy requirement and a minimum free energy requirement. The formation of a vortex lattice either in N or S layers is determined by the competition of the two mechanisms. The analysis to different geometries shows that there are two energy criteria in forming a vortex lattice. Both energy criteria are affected explicitly or implicitly by the inhomogeneities of a superconductor. Many efforts have been devoted to fabricating YBa2Cu3O7-δ (YBCO) coated conductors in the last decades, because it has the higher critical current density in the high magnetic field than Bi-based HTS tapes [1-3]. For high critical current density, highly biaxial textured YBCO films are essential. Now there are two methods to fabricate YBCO coated conductors: one is ion-beam assisted deposition (IBAD) [4] and other is rolling-assisted biaxially textured substrates (RABiTS) [5]. In Chapter 7, pyrochlore-based buffer layers will be prepared using chemical solution deposition (CSD) method on NiW (200) substrates. Firstly, the effects of annealing temperature and seed layers on LZO buffer layers will be discussed; secondly, the influences of films thickness and precursor solution concentration on the orientation of LZO buffer layers will be studied; thirdly, we will present the results of YBCO deposited on CeO2/LZO/LZO seed layer/NiW substrates; finally, a new pyrochlore-based Y2Ti2O7 (YTO) buffer layer will be reported and the effects of orientation of underlying LZO on YTO will also be discussed. As presented in Chapter 8, the tokamak has been perceived as the most promising magnetic confinement concept to achieve fusion energy. For economic reasons, superconducting rather than normal magnets will be used in commercial magnetic fusion plants to significantly reduce the circulating power fraction. A number of superconducting magnet designs have been developed over the past 50 years with both low and high temperature superconductors. For a practical fusion facility, these magnets must be well shielded from energetic fusion neutrons to achieve the predicted performance and service lifetime. With innovative manufacturing techniques, these magnets can be fabricated faster, cheaper, and more practically than conventional processes currently indicate. The thermal stability characteristics of the high-Tc superconducting composites like multifilamentary current-carrying elements of superconducting magnets are discussed in the framework of the macroscopic continuum approximation. The performed analysis was based on the zero- and one-dimensional static and transient thermo-electric models. Various types of the voltage-current characteristic of a superconductor are considered. The thermal runaway

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Preface

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.

conditions of the superconducting composite are investigated under the conditions cooled by cryocoolers with various operating temperatures, liquid helium or hydrogen coolants. The linear and nonlinear temperature dependences of the critical current were used. As a result, the evolution peculiarities of the stable and unstable thermal and electric modes as a function of sweep rate, volume fraction of the superconductor in a composite, and its cross section are formulated for the partially and fully penetrated states under the different cooling conditions. It was shown that permissible stable values of the current and electric field might be both lower and higher than those determined by use of the standard critical current criterion. The reasons leading to these regimes are discussed. Consequently, the unavoidable temperature rise of the superconducting composite before its transition to the normal state takes place. The latter depends on a broad shape of the voltage-current characteristic of the high-Tc superconductor and the current sharing between a superconducting core and a matrix. In the limiting case, a stable value of the temperature of a composite may equal the critical temperature of a superconductor. For these operating modes, the criterion of the complete thermal stability condition is written when the charging current will flow stably only in a matrix. It is also validated that there exists the thermal degradation mechanism of the currentcarrying capacity of a superconducting composite. According to this mechanism, the quench currents do not increase proportionally to the increase of the critical current of a composite. The performed analysis reveals also the connection between different criteria used to determine the thermal runaway conditions. In the framework of the nonlinear temperature dependence of the critical current, it is shown that the operating regimes may have manyvalued stable and unstable branches appearing in accordance with the nontrivial variation of the differential resistivity of a composite. These states exist, first of all, due to the temperature change of the quantity ∂Jc/∂T and are accompanied by the jump-like current-sharing mechanism. The thermal runaway parameters are numerically derived as a function of operating temperature accounting for the additional stable branches of the voltage-current characteristics. The formation peculiarities of these phenomena are discussed in Chapter 9.. In the Alpha Magnetic Spectrometer-02 experiment charged particles coming from cosmic radiation are separated in a large scale superconducting magnet. Two thermomechanical pumps (TMPs) are used to supply the current leads and the magnet coil with superfluid helium. These pumps, which use the Fountain-effect for operation, have been developed at the ILK Dresden. Due to the application the TMPs are required to pump a mass flow of 0.2 g/s. This mass flow rate is pumped from a superfluid helium reservoir at 1.8 K against a pressure head of 200 mbar. This equates to a temperature of 3.2 K in evaporation equilibrium. It is not possible to operate a TMP directly against an evaporator at a temperature higher than the lambda point (2.17 K). Therefore a special separator, operating by use of kinetic energy concurrence, was developed to interrupt the superfluid quantum state between the exits of the TMPs and the entrance of the evaporator. In the Short Communication, after introducing into basic principles essential for the function of the TMPs we will report about the development and tests of the TMPs in combination with the superfluid quantum state suppressor and the evaporator.

Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

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RESEARCH AND REVIEW STUDIES

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In: Superconducting Magnets and Superconductivity… ISBN: 978-1-60741-017-1 Editors: H. Tovar and J. Fortier, pp. 3-53 © 2009 Nova Science Publishers, Inc.

Chapter 1

CORRELATION EFFECTS AND SUPERCONDUCTIVITY IN CUPRATES: A CRITICAL ACCOUNT Sven Larsson Department of Chemistry, Chalmers University of Technology, S-412 96 Göteborg, Sweden

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Abstract Electronic structure of transition metal oxides and salts is reviewed, particularly what concerns electron transfer, electron correlation, electron-nuclear coupling, and inter-metal interaction in cuprates. A short resume of electronic correlation in molecular systems is first given. Electron pair transfer is treated in a many-electron real space approach using standard mixed-valence theories. The latter models have been successful in the past in describing the response of the electrons to the motion of the nuclei (electron – phonon interaction). The possibility for multiple oxidation states of the metal ion is typical for transition metal compounds. Mott-Hubbard-U is strictly defined, its dependence on breathing mode coordinates analysed, and the connection between U and the energy gap for superconductivity clarified. Delocalization is treated in terms of an extended Hush model and electron pair supercurrents derived at the Van Hove degeneracy. d-wave gap anisotropy is found to be consistent with the general atomic level model presented here. Softening of phonon half-breathing modes in inelastic neutron scattering (INS) can be connected to mixed-valence. The fundamental vibronic interaction between spin density wave (SDW) and charge density wave (CDW) states leads to a new phase with energy gap and electron pair carriers. Finally comments are made on why MV-2 systems delocalize as FermiDirac systems (ordinary metals) while MV-3 systems delocalize and condense as Bose-Einstein systems (superconductors).

1. Introduction There seems to be universal agreement that superconductivity in cuprates∗) is connected to “strong correlation effects”. This insight may originate from an influential paper by P.W. ∗

In the normal chemical usage the ending –ate, as in “cuprate(N)”, indicates a negative molecular ion where the central atom is in its Nth oxidation states while the oxygen ligands have the ordinary oxidation state -2. KCuO2 is thus potassiumcuprate(III).

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Anderson that appeared shortly after the so called high Tc superconductors had been discovered in 1986 [1], in the excellent textbook by P. Fulde [2], or date even further back [3]. The present paper is an overview of correlation effects in transition metal oxides and salts, particularly for cuprates and copper oxide, and its connection to phenomena such as conductivity, giant magnetic resistance (GMR), and superconductivity. The only accepted microscopic model for superconductivity so far, the general Bardeen, Cooper, Schrieffer (BCS) model [4], is a k-space theory. In the present work mixed valence and other real space theories will be reviewed and connected to superconductivity (references will be given later). Correlation effects are commonly divided into two main categories: (1) electronic correlation effects and (2) correlation effects between electrons and nuclei (electron-phonon interactions). In the present discussion, contrary to the case in many other treatments, electron correlation is not seen as a replacement for electron-phonon interactions. The latter interactions remain as a very essential part of the theory, though related to the motion of the electron pair rather than to the stability of the pair. To simplify the treatment we will first assume that we are dealing with transfer of electron pairs that are localized in real space rather than of delocalized electron pairs, expressible only in k-space. Nuclear coupling, then, means that the nuclei locally respond to the presence of the electron pair, momentarily destroying the translational symmetry of the crystal. The conditions for delocalization into translationally invariant wave functions will be discussed. It is well-known and easy to prove that correlation between electrons with different spin cannot be accounted for using a single Slater determinant [3]. Interestingly, currents also cannot exist if the electronic ground state is described by single Slater determinant, as is the case in the Bloch (band) model [5], at least if we are going to believe in a theorem by Bloch in a discussion by Bohm [6]. Hence the phenomenon of superconductivity cannot be correctly described using a single Slater determinant but requires a wave function that accounts for correlation. Long before the electron was discovered as a particle by Thomson in 1897, chemists had realized that metal ions may exist in different “oxidation states”. After that discovery, oxidation state for a period of about 30 years became associated with the number of electrons on the metal ion. Later, after quantum mechanical methods had been introduced, it was realized that the electrons are delocalized over both the transition metal ion and its neighbouring atoms (the ligands). It became clear that each electronic state is associated with characteristic equilibrium geometry. Geometry change when the number of electrons in the whole complex is changed is the most important observable [7-10], as will be reiterated below in sections 4 and 5. In the theory for Mixed Valence (MV) systems [10-12] change of oxidation state is thus connected to both change of the number of electrons and change of geometry. The theoretical treatment of the MV field [10-12] has been very successful, though ignored by most solid state physicists. Incidentally, high-Tc superconductors are very often related to MV systems [13-16] and this connection will be one of the themes of the present paper, particularly in sections 4 and 5. Models which do not consider geometry dependence on the number of electrons, such as the “resonating valence bond model” [1] might in fact be missing physics that is important for the explanation of superconductivity and other properties. Concepts such as electron-phonon coupling and negative-U compounds [17,18] are related to the mentioned geometry change, as will be shown in sections 5 and 8. The structural sensitivity as a function of the number of electrons is measured as reorganization energy (λ) (section 4).

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Correlation Effects and Superconductivity in Cuprates: A Critical Account

5

The oxidation state concept has been criticised, particularly by physicists [19,20] and this makes it even more important to repeat crystal field theory (CFT) and ligand field theory (LFT) here. With suitable parameters CFT predicts the correct ordering between the electronic states, but does not account for exchange of electron spin and charge density between metal and ligands. The latter type of delocalization is accounted for in the LFT model, which is a molecular orbital (MO) model that mixes metal and ligand atomic orbitals (covalency). For practical reasons the states are often still named according to the number (n) of d-electrons as dn, which unfortunately has caused confusion for cuprates where the covalency is sometimes large. Q Q Au+

Q Q

Q Q Q

Q

Au3+ Q

Q Q

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Figure 1. Au+ and Au3+ sites in Cs2Au2Cl6. Black spheres are Cl− ions. Arrow indicates motion along the displacement coordinate Q to reverse the site oxidation states [21].

As mentioned, there are many well-known cases where the presence of electron pairs on metal ions seems to be related to superconductivity or at least high conductivity. Good examples are BaBiO3 where Bi3+ sites alternate with Bi5+ sites [15,16,19,20]. Au+ is known to alternate with Au3+ in A2Au2X6 and A4Au2X8 (A is alkali; X is halogen), fig. 1 [21]. WO3 doped with Na turns into NaxWO3 where W is possible in oxidation states W4+, W5+, and W6+ [22] while Ti3+ may be involved in LiTi2O4 and disproportionate: 2Ti3+ ↔ Ti2+ + Ti4+ [23]. In superconducting cuprates the common oxidation state is 2+ [24-28]. No proven example of disproportionation 2Cu2+ ↔ Cu+ + Cu3+ in the same way as in the case of gold seems to be known, however, and is also unknown at which higher energy such a virtual state would be expected. The meaning and presence of a Cu3+ “state”, already the subject of many controversial discussions, will be clarified below. Antiferromagnetism, charge alternation, or high Tc superconductivity are common phenomena in systems with many possible oxidation states. Often this type of system has one “active” electron (or hole) per site on the average. The coupling (t = H12) between the active electrons on different sites (the theory and definitions will be given below) is large, although smaller than in the case of normal metals. If such compounds form crystals with square planar grids, there is a van Hove singularity [29] at the Fermi level. Antiferromagnetic systems

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Sven Larsson

appear because the electronic energy is lower if the electrons have alternant spin directions in the ground state. In a charge alternating system the electrons instead pair up on every second site. In both cases the gain in energy by electronic correlation is substantial. Both systems are insulators with a huge energy gap at the Fermi level. We will use the notation charge density wave (CDW) for the latter type of system and spin density wave (SDW) for the antiferromagnetic system. The (hypothetical) relative chemical potential, μ(CDW) − μ(SDW), between these two types of phase depends on chemical composition, doping level, or applied pressure. The corresponding energy difference at the atomic level is a crucial number in MV theory that may be related to Mott-Hubbard U [17,18,30,31]. A CDW phase is thus characterized by alternating valences, different in two units. The SDW phase has the valence state in between. There is a well developed and useful vibronic coupling model for this type of system which we will here refer to as MV-3 theory [1112,32]. There is a similar model if only two valence states are prominent, called MV-2, that may be considered as a vibronic form of Marcus theory [6]. Still confusing might be under what conditions the interaction between the sites in a solid leads to delocalization over the whole crystal. This type of delocalization depends on the ratio t/λ [8,33-35]. The present paper does not support the idea that t/U is the only important delocalization parameter. Superconductivity in cuprates with Tc > 30 K, discovered in 1986 [24], was apparently unexpected and unpredicted among the theoreticians. It was therefore assumed that one is dealing with a new type of superconductivity, with a different “pairing mechanism” [1]. Later direct experimental evidence for nuclear participation in high Tc superconductivity has been obtained from various experimental sources, however. Oxygen isotope effects in the CuO2 plane have been found on all copper oxide superconductors [36-39]. The angular resolved photoelectron scattering (ARPES) technique gives a rather direct illustration of the superconducting gap and shows, along with a number of other techniques, that cuprates belong to d-wave superconductors [38,39]. It has also been shown using inelastic neutron scattering (INS) that superconductors belong to a group of MV systems where phonon breathing modes are the main players [4044] and where the effects are particularly dramatic in superconducting cuprates [44]. Again this lends support to nuclear participation in the phenomenon of superconductivity. Fritz London was the first one to notice that the behaviour of a superconductor could be explained if the carriers are assumed to be electron pairs moving with a small effective mass [45,46]. The necessity of a “pairing mechanism” to off-set the large inter-electronic repulsion was never of any great concern to him, as it has never been to other quantum chemists. The emphasis on Mott-Hubbard U [1,30,47] in much of the physics literature is surprising, since this parameter is not related to effective mass. Ogg was the first one to claim observation of an electron pair [48], suggesting that a pair of electrons is located in cavities in liquid ammonia of the same sort as already confirmed for single electrons. Ogg’s claim turned out to be unrealistic. There followed accurate quantum chemical calculations where the two electrons were assumed to move in a flat cavity potential (with steep walls). It was shown that the electron pair has a positive energy compared to the background and therefore is unstable [49,50]. This does not prove, however, that electron pairs are unstable generally, as part of a closed shell and with a strongly attractive atomic nuclei present.

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Copper is known in stable oxidation states from 0 to +3. In the present treatment the emphasis will be on the mobility of the pair. The interaction between the CDW and the SDW phases is the key to the electron – nuclear correlation, and also the reason for the gap, as will be shown in sections 5 and 8. In section 7 a simplified treatment is used to demonstrate that the present theory is consistent with pair currents in the ground state. In section 8 the gap is derived in a heuristic way. In section 9 the result of inelastic neutron scattering (INS) on superconducting cuprates is discussed and in section 10 the results of ARPES experiments.

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2. Electronic Correlation in Molecules How important, then, is electronic correlation in different oxidation states of copper, in cuprates, and in the interaction between CDW and SDW phases. At least the latter type of correlation appears already in the simple hydrogen molecule as we will see below. The k-space theory used in solid state physics was originally based on the one-electron Hartree approximation [51]. The most obvious criticism of the independent electron model is that the corresponding many-electron wave function has to be a Slater determinant. Correlation between electrons with different spin is not accounted for [3]. The best possible Slater determinant in variational energy is the one where the orbitals are optimized in a selfconsistent field (SCF) treatment according to the (unrestricted) Hartree-Fock method [52]. The Slater determinant permits by its mathematical form correlation between electrons with the same spin. Slater has shown in a very convincing way [53] how this relates to an “exchange hole” seen by the electrons. Various suggestions have been made to add a “Coulomb hole” in the effective one-electron Hamiltonian [53-56]. This can be done and the best orbitals, the so called Brueckner orbitals, give the Slater determinant with the best possible overlap with the true many-electron wave function [53-56]. This improvement is of limited value, however, and this is also the case with the DFT method. The density functional theory (DFT) method [57-59] provides other orbitals than the Brueckner orbitals and a less perfect overlap with the true wave function. The total wave function is still a single Slater determinant. Nevertheless the DFT model is a practically useful method and a vast improvement over earlier methods. It is obviously the method of choice in modern calculations of important properties, including ferromagnetism [60]. The self-energy is automatically included and is less suitable for parametrization. Ab initio methods including electron correlation may be applied even to crystals, however, and this has been done for example in the research groups of Broer and de Graaf [61-63], Fulde [64-66], and Roos. The latter developed a multi-configurational SCF method called Complete Active Space SCF (CASSCF) [67-69] which will be used here. The idea behind all these methods has been to approach a correlated wave function in a general way. The first mentioned group usually approximates the system by a cluster of minimum size to represent the extended crystal, in order to allow an extensive treatment of electron correlation. Particularly important results have been obtained on systems where transition metal ions are involved. The interpretation of the results may require simplified models, of which some will be mentioned below. The most important correlation effect in the present context is configuration interaction (CI) between the degenerate orbitals at the van Hove singularity. Degeneracy generates many configurations in the wave function. One aim is to keep an analytical expression of the wave

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function. We will first illustrate why many Slater determinants are necessary in the wave function by studying the simple case of the hydrogen molecule. Hydrogen molecule. Let us call the two nuclei A and B and the corresponding atomic 1s-orbitals, a and b, respectively. The symmetry adapted, unnormalized molecular orbitals (MO’s) are a + b and a – b. a + b is the bonding orbital, doubly occupied in the ground state. The unnormalized two-electron wave function is an orbital product: G G G G G G Φ MO ( r1 , r2 ) = [a ( r1 ) + b ( r1 )][a ( r2 ) + b ( r2 )] = ( a + b )( a + b )

(1)

In the simplified expression (a + b)(a + b) it is assumed that the first factor is a function of the spatial coordinates of electron 1 and the second factor a function of the spatial coordinates of electron 2. The spin function in the ground state is the singlet spin function αβ − βα. If the spatial wave function eq.(1) is multiplied by αβ − βα, we obtain:

G G ΨMO (1,2) = (a + b)( r1 )(a + b)( r2 )[(α(ζ1 )β(ζ 2 ) − β(ζ1 )α(ζ 2 )] G G (a + b)( r1 )α (ζ 1 ) (a + b)( r1 )β(ζ 1 ) = G G (a + b)( r2 )α(ζ 2 ) (a + b)( r2 )β(ζ 2 )

(2)

The rightmost member is the (unnormalized) Slater determinant. Let us assume that electron 1 is fixed near nucleus A but far away from nucleus B. Since

G

a and b are exponential functions centered at A and B, respectively, a ( r1 ) is large while

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G b( r1 ) is small and negligible. Inserted in eq.(1) we thus obtain:

G G G G G G G Φ MO ( r1 , r2 ) = a ( r1 )[a ( r2 ) + b( r2 )] = C[a ( r2 ) + b( r2 )]

(3)

where C is a constant since electron 1 is fixed. We thus find that electron 2 is described by MO (a+b), equally localized on the two nuclei. There is no correlation in position with electron 1. This may be generalized to hold for any two electrons with different spins in a system with any number of electrons. The Slater determinant may be a good or bad approximation depending on the character of the physical system it describes. Chemical bonding is usually well described except at separation of the atoms when the bond distance tends to infinity. By Koopmans’ theorem the orbital energy is approximately the negative of the ionization energy. The valence bond (VB) wave function is a different two-electron wave function, directly formed from the atomic orbitals a and b:

G G G G G G Φ VB ( r1 , r2 ) = a ( r1 )b( r2 ) + b( r1 )a ( r2 ) = ab + ba

(4)

This wave function, first derived by Heitler and London [70], accounts for the chemical bonding between the two hydrogen nuclei in the H2 molecule in a more accurate way than the MO wave function. If we again assume that electron 1 is located near A, we obtain:

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G G G G G Φ VB ( r1 , r2 ) = a ( r1 )b( r2 ) = C ⋅ b( r2 )

9 (5)

meaning that electron 2 is described by a 1s-function centered on atom HB. In ΦVB electron 2 is thus near B if electron 1 is near A. In the same way we may show that if electron 2 is near A, electron 1 has to be near B. Thus the two electrons are correlated in their positions. The VB function has the same spin function (αβ − βα) as the MO wave function. The total wave function is thus:

G G G G ΨVB (1,2) = [a ( r1 )b( r2 ) + b( r1 )a ( r2 )](αβ − βα ) = (ab + ba )(αβ − βα )

(6)

This wave function cannot be written as a single Slater determinant. The electrons are space and spin correlated, as is easily checked in eq.(6): If electron 1 is at A with up-spin, electron 2 will be at B with down-spin. This is a typical correlated situation also in the ionic lattice. An adequate description of the antiferromagnetic ground state requires a correlated wave function of VB type. The general expression for a wave function that includes correlation effects may be written as: Ψ = C 0 Φ0 +

∑ C 1a Φ ai + ∑ C ai,,jb Φ ia,,jb + .... i ,a

(7)

i ,j a ,b

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where we have ordered the correlation contributions according to number of substitutions in the Slater determinant [71]. It is important that the VB wave function may be written as a CI expansion (of two Slater determinants in this case):

ΨVB (1,2) = (ab + ba ) = [(a + b)(a + b) − (a − b)(a − b)] / 2

(8)

The product (a+b)(a+b) corresponds to Φ0 and the product (a − b)(a − b) to Φ1234 [the counting in eq.(7) is over spin orbitals]. If the wave functions are properly normalized, the coefficient in front of the term ΦMO = (a+b)(a+b) in eq.(8) is much larger than the negative of the coefficient in front of the doubly substituted term (a−b)(a−b). Both ΨVB and ΨMO are approximations to the ground state wave function. In spite of the fact that it does not account for correlation, ΨMO turns out to give a rather good approximation at the equilibrium positions of the nuclei, provided that the orbials are optimized using Hartree-Fock or DFT. However, if the nuclei are separated, the relative error increases. The correct electronic states in the limit when the separation distance becomes very large, are obtained only in the VB approximation. An important reason to include correlation in the molecular case is in fact to obtain correct potential energy surfaces (PES’s) when the atoms are separated. In the singly excited state, approximately given by (a+b)(a−b), we lift one electron from (a + b) to (a − b). There is a singlet excited state and three triplet excited states:

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Sven Larsson S

Ψ1 (1,2 ) = [( a − b )( a + b ) + ( a + b )( a − b )]( αβ − βα ) = 2 ( aa − bb )( αβ − βα )

(9)

Ψ(1,2 ) = [( a − b )( a + b ) − ( a + b )( a − b )]( αβ + βα ) = ( ab − ba )( αβ + βα )

(10)

T

The other two triplet states have the spin functions αα or ββ. The triplet states dissociate to a H + H state like the valence bond state. In a crystal the triplet states correspond to the ferromagnetic state. The singlet state dissociates to an ionic state H+ + H− , corresponding to the wave function (aa + bb). The doubly excited state (a−b)(a−b) may be written as: S

Ψ2 (1,2 ) = ( a − b )( a − b )( αβ − βα ) = ( aa + bb − ab − ba )( αβ − βα )

(11)

This state is thus an ionic state orthogonal to the VB state. It is not trivial to derive the energy of this state, and also not necessary for the present ends. We refer to the treatment of ref. [72]. Energy

Ionic states Triplet state

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MO approximation

R

Ground state

Figure 2. PES for the states created from the 1s orbitals in the H2 molecule. R is the interatomic distance.

If the sum and difference of the so called zwitterionic states aa + bb and aa − bb is taken, we obtain functions aa and bb forming an electron pair on A or B, respectively. In a lattice the latter states correspond to the CDW state. Clearly electronic correlation is important to give a good description of a state where two additional electrons are on the same site. Still in the H− ion the correlation energy is only 10% and in the He atom only 4% of the self energy [54,55]. Even in the case of H− the Hartree-Fock method gives a quite accurate wave function. The energy conditions are completely different in a solid. A detailed treatment has been given by Prassides et al. [73,74]. Many examples exist of correlated states and some were mentioned in the introduction.

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The calculation of the MO ground state in a crystal is relatively simple using HartreeFock or DFT, or any other band model where the wave function is a single Slater determinant. Unfortunately in an oxide such as NiO the calculated Fermi level is located in a continuous band. Thus the model incorrectly predicts a metallic spectrum for NiO. The failure of the band model may be connected to the fact that the antiferromagnetic ground state cannot be written as a single Slater determinant. Correlation of the same Coulomb type as in the hydrogen molecule has to be accounted for in the wave function. The presence of a spin ordered state in the Mott insulators (MnO, FeO, CoO, NiO, CuO) may be due to weak interaction between the 3d electrons on neighbouring sites at the end of the transition series [30]. The monoxides in the beginning of the transition series (TiO and VO) have a larger overlap and tend to be metallic [31].

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Cyclobutadiene. A considerably larger system, cyclobutadiene (C4H4), may be regarded as a prototype molecule for “strong correlation” effects in square planar systems. The square and rectangular forms were treated in a great detail by Voter and Goddard [75]. The ground state has a rectangular shape where the short sides correspond to CC double bonds and the long sides to CC single bonds. Short and long CC bonds are thus alternating as in polyacetylene, due to the Peierls instability [76]. There are other PES, at a higher energy, which are characterized by triplet spin or by charge alternation. The work of Voter and Goddard [75] is particularly useful since it contains explicit expressions for wave functions that will be used later in section 7. There we will also use a simplified model of Hückel type [77]. Even this method reproduces the most important properties of cyclobutadiene and will be used here for square-planar systems in general. CASSCF. The CASSCF method is a many-configurational self-consistent field method [67-69]. The total wave function is a superposition of a number of configurations, as in eq.(7), although the ordering of the configurations is different. First the MO’s are divided up according to energies into (1) “frozen” orbitals which are doubly occupied and invariant, (2) “inactive” orbitals which are doubly occupied but optimized in an MCSCF procedure for each new CI configuration, (3) active orbitals that are included in the CI treatment, and (4) virtual orbitals which unoccupied and remain orthogonal to the MO’s with lower energy. The size of the full CI expansion may be several thousand configurations. As we may see above in the comparison with the hydrogen molecule it is possible to derive the CDW and SDW wave functions from the active orbitals in the CASSCF method. Later in sect. 7 it will be shown that CDW and SDW precursor states may be formed generally from the orbitals belonging to the van Hove singularity [18,29]. Koopmans’ theorem. The band model has been very successful in the description of solid state electronic structure, so successful that the energy bands have sometimes been considered as almost “observables” using photoelectron spectra, X-ray (XPS) or Ultraviolet (UPS). Koopmans’ theorem [78] states that the orbital energies (with reversed sign) for occupied spin orbitals in the Hartree-Fock method are approximations to XPS ionization energies: I i = E i , N − 1 − E N ≈ −ε i

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(12)

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while the orbital energies for the unoccupied orbitals with reversed sign are approximations to electron affinities: A a = E a , N + 1 − E N ≈ −ε a

(13)

Koopmans’ theorem is more accurate for ionization energies than for electron affinities. Energy differences between occupied and unoccupied orbitals should not be interpreted as approximations to excitation energies; a Fermi gap obtained that way is unreliable. Direct translation of Hartree-Fock (or DFT) orbital energies to measured one-electron excitation energies is usually unjustified.

3. Electron Structure of Embedded Transition Metal Ion

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The lowest electronic energy levels of a transition metal ion, say Ni2+ surrounded by oxygen ligands (H2O or O2−), are formed depending on the different possibilities for orientation of the orbitals of the open 3d shell relative to the ligands. Traditionally this is treated either in an atomic model, called crystal field theory (CFT), or in an MO model, called ligand field theory (LFT). CFT and LFT have a rather long history and therefore the exact definitions of the two models may vary between different textbooks. These two simple models form the background knowledge for all discussions on the electron structure of transition metal ions. They serve to interpret experimental spectra as well as the results of ab initio calculations. Crystal field theory. The CFT model was set up by Bethe, van Vleck and others [79-82]. The zeroth order approximation in the CFT model is the atomic ion in vacuum. A given number n of d-electrons form a multiplet spectrum of many-electron type. The electrostatic field from the ligands, reduced to point charges or point dipoles, is included as a perturbation on the energy levels. Since the field strength is parametrized anyway, nothing is gained by using more realistic charge distributions. The electronic structure involves only the 3d (4d, 5d) sub-shells. For example in the case of the ions Ni2+ and Cu3+, eight out of a total of ten electrons are present and used to occupy the 3d orbitals and form the atomic multiplet structure. The ground state is a spin triplet state for Ni2+, thus of the same spin type as for the free Ni2+ ion. In an octahedral complex there is a total of six ligands on the x, y, and z-axes. The energies of the d-orbitals pointing towards the ligands are higher by Δ = 10Dq (fig. 3) a parameter that is commonly fitted to an experimental spectrum. A diluted aqueous solution of NiCl2 contains Ni(H2O)62+ ions and a solution of CuSO4 contains Cu(H2O)42+ ions. The water molecules turn the negative oxygen atom towards the positively charged metal ion. The spectrum of the NiO crystal is similar to that of Ni(H2O)62+. The Ni atom in NiO is also surrounded by six oxygen atoms in an octahedral geometry. These oxygen atoms can be considered as O2− ions in both the NiO crystal and in a dissolved Ni(H2O)62+ complex in water. Reality is more complicated than this, however. CuO would by the same reasoning be expected to be greenish - blue like Cu(H2O)42+ in water, but is black. This is due to other

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types of excitations which are not included in the CFM. The black colour of CuO may be due to inter-metal coupling of band structure type, superposed on the CFT spectrum. Energi

3

eg 3d

Δ=10Dq

1

1

D

t2g

T2

E

3 3

F

A2 Δ

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Figure 3. CFT orbital energy diagram (left) and total energy diagram (right) for octahedral field. Δ=10Dq is the CFT gap corresponding to the energy splitting between the t2g and eg 3d atomic orbitals. The spin orbital occupation corresponds to Ni2+ high spin (3d8). 3F is the ground state of a free Ni2+ atom (Δ=0) and 3A2 the ground state of Ni2+ in a crystal or in aq. solution.

Strong field and weak field. Metal complex are of two types, “strong field” and “weak field”. The strong field case causes a large Δ splitting compared to a Hund’s rule exchange integral. A strong field is obtained if the charge on the transition metal ion is larger than or equal to +3 (for example Co3+). A high charge pulls the ligands closer to the metal ion, whereby Δ increases. Most metal ions with +2 or lower charge form weak field metal complexes where Hund’s rule wins and the maximum possible total spin is obtained in the ground state. The Mott insulators are typical examples. Δ depends, however, on both metals and ligands [82]. Ammonium (NH3) ligands have a 25% larger Δ than water. Cl− and Br− have a 25% smaller Δ than water. The tendency for a strong field is higher in the 4d and 5d series than in the 3d series. In the calculation of the energy levels the field is considered as a perturbation on the multiplet structure in the weak field case. In the strong field case multiplet stucture is treated as a perturbation after the ligand field splitting has been taken into account. Ligand field theory (LFT). Upgrading of CFT to an MO model leads to ligand field theory (LFT) which is a simplified MO theory that involves the metal d-orbitals and ligand valence orbitals (for example O 2p for an oxygen ion, fig. 4). The 3d splitting is caused primarily by MO formation between a metal 3d orbital (χM) and a ligand orbital (χL). The metal orbitals that are directed towards the ligands have a greater overlap with the ligand orbitals than those that are directed in between the ligands. Therefore a larger energy splitting between bonding and antibonding orbitals is obtained in the former case than in the latter. This leads to an energy splitting between the antibonding orbitals (= the 3d orbitals in CFT). The lower, bonding MO is normally dominated by the ligand valence orbitals (fig. 4). All MO have both ligand and metal character (not shown in fig. 4).

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One exception to the simple rule is when the ligands are diatomic molecules. In that case the unoccupied orbitals of, say N2 or CO, are also taking part in the interaction. In the case of transition metal oxides and salts we are usually dealing with ligands which can be assumed to be atomic ions. For example the H2O ligand does not have any unoccupied MO’s which can be used as acceptor MO’s and therefore behaves as O2− in LFT. Higher levels

M →L

Metal levels

4s

b1g*

eg* HOM O-LUMO

M →M

M →M

3d L→M

Shake-up

Ligand valence levels b1g octahedral

square planar

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Figure 4. MO’s in a transition metal complex with a single metal atom (LFT model). M → M is the energy gap corresponding to a ligand field transition between the t2g* and eg* in octahedral geometry (Oh) on the same meta ion M. In the right part the splittings in a square planar complex (D4h) are shown.

The octahedral geometry in the spin triplet Ni2+ complex is a consequence of the occupation by eight electrons. The most antibonding MO (eg) is two-fold degenerate and is occupied by two electrons with equal spins according to Hund’s rule, forming together a triplet state. The resulting charge density may be obtained by writing down the expressions for the two degenerate MO’s, denoted 3d(z2) and 3d(x2 – y2), and singly occupied. Let us assume that they are pure 3d-orbitals. The charge density is: 3 d ( z 2 ) = R 32 ( r ) ⋅

1 5 2z2 − x 2 − y 2 ⋅ ⋅ 2 4π r2

(14)

1 15 x 2 − y 2 ⋅ ⋅ 2 4π r2

(15)

3 d ( x 2 − y 2 ) = R 32 ( r ) ⋅

The sum of the squares add up to a charge density ρ: ρ( x , y , z ) = R 232 ( r )

5 16 π 2 r 4

(

⋅ x 4 + y 4 + z4 − x 2 y 2 − y 2 z2 − z2 x 2

)

(16)

We see that the charge density ρ arising from both of these two singly occupied orbitals is higher along the coordinate axes x, y, and z than in between. Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

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Co3+ complexes are also of the strong field type. The six 3d-electrons all enter into the three possible t2g orbitals which point in between the ligands. Jahn-Teller effect. The 3d8 configuration of Cu3+ complexes is a strong field complex. Still one would expect one electron in each of the eg orbitals since the latter are degenerate. Occupation with opposite spin of the two degenerate orbitals, producing a spin singlet state also has to be examined, however. In that case the Jahn-Teller theorem comes into play, suggesting that the octahedral local symmetry can no longer remain [83,84]. The geometry has to be square-planar, either with an elongated or compressed bond along the z-axis. In fact the oxides of both Cu3+ (d8) and Cu2+ (d9) are found to be elongated or having no ligand at all on the z-axis. The empty MO is the one which is the most antibonding one, that is the one with 3d(x2−y2) character. The Cu3+ complex therefore is a spin singlet state. The Jahn-Teller effect has often been suggested as a component in the explanation of high-TC superconductivity. In the case of the cuprates the involvement is clear from the fact that all elongated axes are parallel, perpendicular to the CuO2 plane. Aside from this important effect, the Jahn-Teller theorem is unimportant for superconductivity in the cuprates and very likely all other such superconducting systems, for example K3C60. For a given oxidation state (or charge on C60) there may be many possible Jahn-Teller configurations, but since all have the same charge, the latter cannot be related to conductivity phenomena.

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M→L charge donation and transferred hyperfine interaction. Mixing of metal and ligand orbitals accounts for covalency. Covalency is often incorrectly referred to as “hybridization”. The term hybridization should be reserved for mixing of orbitals on the same atom with different ℓ quantum numbers. In LFT a bonding and an antibonding MO with the same symmetry indices are formed, as is already mentioned. This linear combination may be written as (0 ≤ θ ≤ π/2). Bonding: Antibonding:

φ1 = cosθ χL + sinθ χM φ2 = −sinθ χL + cosθ χM

(17)

This form of the MO assumes that the atomic orbitals χL and χM are orthonormalized. This is done preferably using a method called symmetric orthonormalization [85] in which the atomic character of the orbitals is maintained as much as possible. In eq.(17) trigonometric coefficients (cosθ and sinθ) are used to conserve normalization. CFT corresponds to the case θ = 0, where only the 3d, 4d, or 5d orbitals are treated. Notice that if both φ1 and φ2 are fully occupied there is no charge or spin transfer contribution by the fact that cos2θ + sin2θ = 1. Eq.(17) has been verified in several calculations without restrictions on form [86]. θ may be determined using hyperfine structure, spectroscopy or calculations, provided, of course, that the orbital pair in eq.(14) is not fully occupied. As eq.(17) shows, charge flow between ligands and metal ion is allowed in ligand field theory. If only the lower orbital φ1 is occupied, the amount of 3d character is sin2θ arising from interaction of one of the 3d orbitals with the ligand orbital with the same symmetry label. This is referred to as ligand to metal (L → M) charge donation. In the normal case the ligand energy levels are at a much lower energy than the metal orbitals and therefore the

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mixing charge coefficient sin2θ is small (a few %). The “effective number of 3d electrons” is larger in LFT than in CFT due to L → M charge donation.

bonding

antibonding

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Figure 5. The φ1 = b1g (bonding) and φ2 = b1g* (antibonding) pair of MO with the symmetry x2−y2. The latter MO is unoccupied in the Cu3+ 3d8-singlet ground state (the Zhang-Rice singlet [87]).

L → M charge donation is large for cuprates and this has unfortunately lead to confusion in the assignment of electronic states and valence states to metal sites. The majority of chemists prefer to use the state label used in the CFT model (for example 3d8 in the case of the Zhang-Rice singlet [87]). In part of the physics literature one instead counts the L → M charge donation to the metal charge. In the cuprates where the L → M charge donation is close to a full electron, one may refer to the Zhang-Rice singlet as a 3d9 system. Detailed calculations were carried out on two complexes of KCuO2: CuO45− and Cu3O87− [88] using the CASSCF model. In the smaller system there are 25 inactive orbitals. The active space contained essentially the ligand valence and metal 3d levels in fig.3. The number of configurations was 45 for the smallest active space and 825 for the largest. In the larger system the active space had to be reduced as much as possible. This choice generated 4950 configurations. In the CI expansion of the ground state wave function four of these configurations have an expansion coefficient with an absolute value larger than 0.1. The calculated L → M charge donation is consistent with experimental data [89]. Other d8 (more correct: b1g* −2) systems are also well-known in chemistry, for example 8 5d systems (Au3+, Pt2+). In the mixed valence Au3+/Au+ system (fjg. 1) the Au+ site does not have any CFT spectrum since all O2p - Cu3d orbitals are occupied. The ligand field spectrum of the Au3+ sites is overshadowed by the mixed-valence spectrum as analysed by Hush [8]. Data on structure has been obtained either by calculations [90] or crystallography [91] and show that at increased pressure the Au sites tend to become identical and octahedral, corresponding to valence state +2. At the same time the conductivity is increasing very much with pressure, in agreement with existing theory [21]. Satellite structure in photoelectron spectra. UPS and XPS have been widely used to study electron structure in transition metal complexes, not least in cuprates. The probe is excited by monochromatic ultraviolet or X-ray radiation of energy hν0. The kinetic energy (mv2/2) of the ejected photoelectron is measured. The following equation holds:

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Correlation Effects and Superconductivity in Cuprates: A Critical Account ⎛ mv 2 h ν0 = Ii + ⎜ ⎜ 2 ⎝

⎞ ⎛ mv 2 ⎟ = Ei ⎜ − + E N N −1 ⎟ ⎜ 2 ⎠i ⎝

⎞ ⎟ ⎟ ⎠i

17

(18)

Hüfner has presented well-resolved XPS spectra for copper metal and compounds in the oxidation states +1, +2, and +3 [89, particularly fig. 5.86]. In copper metal and compounds where Cu has the oxidation state +1, only a single 2p3/2 emission line is seen, consistent with the one-electron picture. In the case of the two higher oxidation states of copper there is a satellite at a lower kinetic energy (higher binding energy). Very similar 2p3/2 spectra appear for CuO and for the cuprates (fig. 6). The intensities of satellite and main line are almost the same.

Intensity

Main line

Satellite

Kinetic energy (eV)

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Figure 6. Typical Cu 2p3/2 spectrum of CuO and cuprates.

Emission of photoelectrons corresponding to a higher binding energy than expected is normal and referred to as “shake-up”, suggestive of energy loss of the leaving photo-electron. Shake up is due to contraction of the charge density when the photo-electron leaves. The intensity is given in the “sudden approximation”, derived by Manne and Åberg [92]: ∞

ΨR ( N − 1 ) =

∑ Ψi

ΨR Ψi ( N − 1 )

(19)

i =0

Ψi are eigenfunctions of the (N−1)-particle Hamiltonian after ionization. The squares of the coefficients,

Ψi ΨR

2

, are proportional to satellite intensity. The unrelaxed remainder

ground state immediately after ionization of the 2p3/2 electron, is given by the Slater determinant, ΨR(N−1), where a 2p3/2 spin orbital is missing from Ψ. Normally the still present spin orbitals in ΨR(N−1) change very little after ionization, implying only one coefficient different from zero in eq.(19) and hence negligible satellites. The unusually large copper satellite may be interpreted in the following way [93]. The main line, corresponds to the ground state: 2p3/2−1b1g*−1. The valence electrons see a much more attractive copper atom after core ionization and hence the greatest 3d component now

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Sven Larsson

appears in the bonding orbital in eq.(17). The antibonding, half occupied b1g* orbital has instead a much larger ligand component. In other words the electronic density in the valence shell has increased on copper in the ground state. For copper the θ value in eq.(17) is more sensitive to changes in core attraction than in other transition metal atoms and the change is in a single MO. sin2θ may change from 0.5 to 0.9 or more, and such a great change can only be satisfied in eq.(19) if more than a single term is used in the right member. A satellite appears. In the case when the 3d shell is full there can be no charge transfer of this type between ligand and copper, and therefore there are no satellites. In the 3d8 (more correct: b1g*-2) case only the lower MO of b1g symmetry in eq.(17) is occupied. The contribution to the charge distribution, according to LFT, is then (ignoring the mixed term):

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ρ = 2φ12 = 2(cosθ χL + sinθ χM)2 = 2cos2θ χL2 + 2sin2θ χM2

(20)

If sin2θ = 0.6 the contribution to the effective number of 3d electrons is 1.2 electrons (increasing to almost two after ionization) while the other mixed orbitals contribute with eight 3d electrons (the same after ionization). Hence there are effectively 9.2 3d electrons. The satellite in NaCuO2 has a lower intensity than the one in CuO and in cuprate(II) [89]. This is expected since the charge increase after ionization is distributed over two orbitals [92]. The main 2p3/2 peak corresponds to a case where the effective number of 3d electrons is close to 10. Hüfner [89] prefers to call this state a 3d9L−1 state whereas the notation 3d8 is preferred here for reasons mentioned above. The latter notation is consistent with the fact that we are dealing with a spin singlet state LFT state. Doped, superconducting cuprates have essentially the same XPS spectrum as ordinary cuprates and CuO. The smaller satellite intensity reflects the fact that hole doping introduces Cu3+ sites among the Cu2+ sites [94]. One might think that XPS would be an excellent technique to probe the presence of localized as opposed to delocalized mixed valence, since the ionization energy depends on the oxidation state. A mixed valence state should give two different XPS peaks for 2p3/2 (or 2p1/2) ionization, due to the different environment of the two cores. In the case with delocalized valences there would be a single peak corresponding to the average valence state. This is incorrect since there is instead a main line and a satellite with about the same intensity. As shown by Hush et al., the two cases cannot be easily distinguished [95].

4. Metal – Metal Interaction and the MV-2 Model In the 1950’s it was noticed using isotope labeling that metal ions in different valence states exchange electrons in aqueous solution: *

Fe2+ + Fe3+ ↔ *Fe3+ + Fe2+

(21)

The reaction rates for a number of similar reactions with different pairs of identical metal ions vary over more than ten orders of magnitude. This is primarily due to differences in the reorganization energy λ, which is a measure of how much the structure changes when the

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Correlation Effects and Superconductivity in Cuprates: A Critical Account

19

number of electrons is changed. If the metal – ligand bond lengths is much increased when an electron is added (or decreased when removed), λ is large and the electron exchange is slow. The activation barrier in the Marcus model is λ/4 if the coupling is negligible. Good agreement with the experiments is obtained when the Marcus model is applied [7].

Q

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Figure 7. Half-breathing mode. The bond lengths are increased at one metal site and decreased at the adjacent one (the Q coordinate).

If the coupling between the metal ions is of the same magnitude as λ, it cannot be neglected when the activation barrier is calculated. There is every reason to believe that transport of electrons in a crystal lattice with localized electrons, where there are additional electrons on some sites, follows the same principles for motion between the sites. Large λ compared to coupling implies localized transport with thermal activation energy, while large coupling compared to reorganization energy implies delocalized states with absent thermal activation energy. The former type of activation energy is typical for a localized system, while activation over the band gap requires a delocalized system. In the Holstein model [9] the site with more or fewer electrons is called a polaron. The Holstein model is founded on the same physical principles as the Marcus model but is intended for systems with translational invariance, for example it defines an effective mass to replace, in a sense, the activation energy. We are going to use the Marcus model particularly for electron pair transfer (see the next section). Reorganization energy (λ). If Fe2+ is oxidized to Fe3+ in an octahedral complex with water ligands, the average bond length is increased by about 0.15 Å when the complex is reduced (one electron is added). The contribution to the reorganization energy is called bond reorganization energy (BRE). The most important motion to induce an electron leap from one site to the next is thus shortening of bond distance on the Fe2+ site and lengthening of the bond length on the Fe3+ adjacent metal site at the same time. The corresponding vibrational mode is the breathing mode or the half breathing mode. The solvent reorganization energy (SRE) cannot be described as a motion between two equilibria. One reason is that there are many possible equilibria within a small energy range. Statistical theory has to be used as in the original Marcus model. Nevertheless we assume a many-dimensional reaction pathway for both bond and solvent coordinates.

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Sven Larsson Energi

PES0

λ0

PES−

λ−

Q Q0−

Q0

Figure 8. Calculation of the reorganization energy. When an electron is swapped between two identical subsystems the geometry is changed from Q0 to Q0− and from Q0− to Q0. The reorganization energy λ = λ 0 + λ −.

The reorganization energy is the increase of the energy assuming the harmonic approximation as we move away from the equilibrium point qi0 along M different coordinates (bonds and angles) to the new equilibrium point: λ=

k

∑ 2 (q i0 − q i )2

(22)

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i

Alternatively λ may be calculated in total energy calculations for the subsystem (fig. 8). Potential energy surfaces (PES). We construct a reaction path so that all motion can be carried out simultaneously from an energy minimum Q = −Q0 to a new minimum Q along this reaction path. To derive an energy expression along breathing modes or half-breathing modes we use mixed valence theory and use Q to measure the advance along the reaction path. The origin is defined so that the equilibrium geometry is –Q0 in the precursor state and Q0 in the successor state, after swapping of the oxidation states. In the MV-2 case the energies of the diabatic states (M+M and MM+, without any interactions between) are described by the following equations for PES: H11 = ½k (Q + Q0)2 H22 = ½k (Q – Q0)2

(23)

The reorganization energy is the energy increase if the equilibrium coordinates are substituted in the PES which has the other minimum. Thus insert Q = –Q0 in the equation for H22 or insert Q = Q0 in the equation for H11. In both cases we obtain λ = 2kQ02.

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Correlation Effects and Superconductivity in Cuprates: A Critical Account

21

The PES after introduction of an interaction (H12) are determined from the following secular equation: H 11 − E H 12 =0 H 12 H 22 − E

(24)

where the interaction matrix element H12 is chosen to fit the actual energy difference Δ between the PES when R1 = R2. This corresponds to H11 = H22 and hence Δ = 2H12. The coupling is in fact quite independent of Q, and is therefore assumed to be constant. If quantum chemical calculations are used we may thus use H12 = Δ/2 for Q=0. The solution of eq.(24) is given by: 2

E± =

H 11 + H 22 ⎛ H − H 22 ⎞ 2 ± ⎜ 11 ⎟ + H 12 2 2 ⎝ ⎠

=

(

)

k 2 Q 2 + Q 20 ± k 2 Q 20 Q 2 + H 12 2

(25)

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Energy

The intersection of the two parabolas H11 and H22 occurs for Q = 0. This point corresponds to the case when both complexes have moved half-way towards the other equilibrium geometry so that the geometries are the same. At this point the resulting PES’s go through an avoided crossing. In a crystal the sites are within a small distance therefore there is usually a quite large interaction between the sites.

λ Δ

−Q0

0

Q

Q0 N

l

di

t

Figure 9. Definition of the parameters used in the Marcus model.

For example if we are dealing with two MnO6 – complexes with one oxygen ion in common and seven 3d electrons, corresponding to Mn3+ (3d4) and Mn4+ (3d3) the overlapping 3d(z2) orbitals, connecting the two sites, have quite a large interaction. This interaction shows up as the splitting Δ = 2H12 between the PES at Q = 0 (fig. 9). The value of Δ depends on the interaction between the metal complexes, particularly on the distance between them. The electronic wave functions may be calculated in principle. At the avoided crossing, corresponding to a totally symmetric system (the same bond lengths in the left and right

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22

Sven Larsson

molecule) the wave function for the lower active orbital is left-right symmetric while the upper wave function is antisymmetric. We are usually interested in only the ground state PES (E–). The location of the two minima may be obtained if we set the derivative of E– [eq.(25)] equal to 0. If kQ02 >⏐H12⏐ there are two minima: 1/ 2

⎛ H2 ⎞ Q = ±Q 0 ⎜ 1 − 2 12 4 ⎟ ⎜ k Q 0 ⎟⎠ ⎝

(26)

The minima correspond to the two equilibrium points. If kQ02 λ, as follows from eq.(27) in the finite case. There is no similar relationship that involves the ratio U/t, to the author’s knowledge. It is therefore strange that U/t is commonly used as a condition for delocalization in the crystal case. The Hush delocalization condition, based on the ratio λ/t may be used as a delocalization condition in the infinite case too [34,35] as was shown by Klimkāns and the present author. If an electron or an electron hole is created in a crystal either delocalization or localization is possible. If the coupling between the sites is negligible, the electron (or hole) localizes itself (“digs a hole”) with a stabilization energy essentially equal to the reorganization energy λ. If the coupling to the adjacent sits is switched on, and if this coupling is sufficiently large, some of the charge will automatically delocalize to the latter sites. The charge on the original site will be less and the tendency for localization less since λ is decreased, due to the fact that the charge causing the new structure is less. A new eigenvalue problem is set up which leads to a new charge distribution. Calculations of this kind tend to lead to a converged result that is either localized or delocalized. If a parameter is gradually changed, for example the reorganization energy λ for a single hole, there is a very sharp metal-insulator phase transition [34,35] (fig. 10). Since valency is a property of a single metal ion, mixed-valency (MV) systems, finite or infinite, are by most people understood as systems with localized valencies. However, the nature of mixed valency is such that delocalization appears in certain ranges of the parameter values (2H12>λ) [33-35]. Therefore delocalisation over many metal ions is contained in mixed valence theory. In infinite, crystalline systems, delocalization leads to a metallic spectrum and metallic conductivity. The band model for electrons should only be used in the delocalized case (except as a computation model).

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C12 1 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

λ (eV)

Figure 10. Delocalization as a function of λ.

GMR. In LaMnO3 Mn has the oxidation state Mn3+. Doping to La1-xSrxMnO3, needs compensation by lifting x Mn3+ to Mn4+ and thus a mixed valence system is obtained of type MV-2 [105]. Mn4+ is a very stable d3 ion. x=1 thus corresponds to only 3d↑3 configurations,

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Correlation Effects and Superconductivity in Cuprates: A Critical Account

25

while x=0 corresponds to only 3d↑4 configurations. The additional 3d electron goes to an axial 3d orbital, thereby extending the distance from the metal to the O ligand common with the next site. The transfer of an electron to the next site is an MV-2 problem. If delocalization occurs because of a large coupling, the spins are automatically ordered. Therefore an applied magnetic field may be sufficient to cause a sudden delocalization with a much higher metallic conductivity. If we go the other way and cause localization in the metallic system, the resistance suddenly increases consistent with fig. 10 [105]. In both systems decrease of the activation barrier for electron transfer (ET) between two adjacent sites is consistent with the MV-2 model. There are anomalies in the bond stretching vibrations [43]. In the delocalized system the barrier has disappeared since the coupling is larger than the reorganization energy λ [eq.(26)]. The conductivity is metallic. The conductivity increases as the temperature is lowered, but superconductivity is not achieved.

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5. Mott Problem Involves Three Oxidation States (MV-3) In pure NiO the Fermi level passes through the energy band created by the upper 3d levels (fig. 4). According to the band model NiO would be a metal. This is inconsistent with the fact that NiO may be considered as formed by Ni2+ ions surrounded octahedrally by O2− ions. Spectrum and magnetic properties support this localized picture rather well. B. Johansson has shown that the band model is a good model for only for TiO and VO in the series of 3d-transition metal monoxides [31]. He pointed out that in these two cases the 3d orbitals are fairly extended [31]. The inter-metal coupling for t = H12 is then also large and we may expect a large ratio coupling-to-reorganization energy, consistent with the discussion of delocalization in the previous section. The Bloch model is a good description of the ground state only then. If on the other hand the polarization forces are large, the reorganization energy λ also tends to be large and the trend towards localization wins. Since antiferromagnetic ordering takes place, there is apparently a ground state below the band solution in energy that cannot be described by a single Slater determinant. A method has been mentioned above [34,35] whereby it is possible to check whether the system is localized or delocalized, i.e. whether the band model is useful. The term “Mott problem” is used here for problems associated with the absence of a band gap at the Fermi level, particularly in the case of the “Mott insulators” MnO, FeO, CoO, and CuO. According to Mott the system is localized but the band model may still be used provided that the energy U, needed to move one electron to the next site, is added to the unoccupied bands. In the first approximation U is essentially the self-energy of a 3d electron, equal to several eV. The Mott-Hubbard solution [30,106,107] amounts to inserting an artificial band gap =U at the Fermi level, thereby saving the band model, since NiO this way becomes an insulator. The extended model explains insulation but hardly antiferromagnetism [17]. The single Slater determinant wave function is still inappropriate. Anderson pointed out [17] that there was another group of metal oxides which seemed to demand a negative U. He also sketched how such a negative U may come about (see below in this section). Interestingly superconductivity was discovered in the “negative U” compounds in 1975 [15,16] and later (1986) in cuprates. The pairing phenomenon appeared easier to solve if U would be negative. Unfortunately a negative U is not sufficient as a solution to the

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problem. As already remarked in the introduction, the problem is to achieve low effective mass rather than negative pairing forces. In the original definition Mott-Hubbard-U is the energy to move one electron from one site to the next, for example from one to another Ni2+ site, creating Ni+ and Ni3+ sites. In chemistry a process of the type: 2M → M+ + M− (32) is referred to as a disproportionation reaction, A CDW phase may be considered as a disproportionated system since the charges are alternating. Here we will first study the Marcus (Holstein) problem involving electron pair transfer. As we will see the effective mass for pair transfer will be infinite unless three consecutive oxidation states of the metal ion are involved. PES for three valence states. As before, we centre quadratic PES (H11 and H22) in each of two equilibrium points, Q = −Q0 and Q = Q0. Such a surface is called a diabatic PES and corresponds to localization of an electron pair at two different but equivalent sites (M+M−, M− M+). The third diabatic PES (H33) corresponds to a state with one electron at each site (MM), and is assumed to have an energy minimum in between the two previous ones, for Q = 0. We assume that the force constants are the same and that H33 is placed U above the minimum of the others, consistent with the definition of Mott-Hubbard U:

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H11 = ½k (Q + Q0)2 H22 = ½k (Q – Q0)2 H33 = ½k Q2 + U

(33)

The eigenvalues corresponding to the PES’s are determined from the following secular equation: H 11 − E 0 H 13 0 H 22 − E H 13 = 0 (34) H 13 H 13 H 33 − E where the interaction matrix element H13 is the coupling between the M+M− and MM states (or the M−M+ and MM states). The direct coupling between H11 and H22 vanishes in the MV-3 case [72]. The solution of eq.(34) is shown in fig. 11. For a small coupling H13 the solution to the 3x3 problem may be obtained as the solution to the 2x2 problem, where U is added and Q0 exchanged by 0 in the right curve. We may choose H13 and Q0 to fit previously calculated PES along a breathing or half-breathing mode. The Mott-Hubbard parameter [106,107] is the vertical excitation for the equilibrium geometry of the SDW state (Q=0). Of the greatest interest is that there is a phase interaction region if the CDW and SDW diabatic energy surfaces cross. In this region the quantum states have large components of both electronic and vibrational wave functions. For most cuprates we know that the ground state is a SDW state with ΔG0 0 in eq.(32) is the actual condition for the existence of a negative-U compound according to fig. 11. In other words, the optimization of the bond distance is of decisive importance for a correct judgement of the existence of the CDW and SDW phases. U may be positive if measured at the equilibrium geometry of SDW but negative if measured at the equilibrium geometry of CDW. Creation of the negative-U phase does not require a negative U but a negative ΔG0. Using fig. 11 the activation energy for transfer from the CDW to the SDW state (or vice versa) may be taken as activation energy for conductivity. This energy is a fraction of an eV rather than several eV’s. The “Mott delocalisation” condition may be taken as ΔG0≈0, but this does not mean that the system is delocalized in the sense of Hush [8,33,108,109]. In cuprates U appears to be 1-2 eV according to the calculated example in fig. 14 below. The activation energy for conductivity is one quarter of this value, about 0.25 eV. However, the actual activation energy is considerably less, since a finite coupling reduces activation energy in eq.(34). ΔG0 is very small according to the calculation in fig. 14. Thus if the

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Sven Larsson

difference in bond distances for different oxidation states is properly taken into account there is no problem to explain why Mott insulators may become superconductors at a low temperature. Intermetal coupling in MV-3. The coupling matrix element H12 between the two symmetrized CDW states M+M− + M−M+ and M+M− − M−M+ is normally equal to zero [72]. The interaction matrix element H13 in eq.(34) between each of the two CDW states and the equal charge state MM (usually a SDW state) is finite and provides effective coupling between the two CDW states. This makes it possible for electron pairs to be transferred between two adjacent sites. If on the other hand Uλ/4, there is a much smaller effective coupling or no coupling at all, the gap is very small, and the pairs transfer with difficulty. A strong interaction between SDW and CDW is therefore essential for superconductivity. It should also be remembered that one of the phases may be hidden from experimental discovery due to high energy. In the cuprates evidence of a CDW has been absent. U > λ/4 implies SDW in fig.(11), and this happens in most cases. To obtain a CDW phase it is necessary that ΔG0 < 0. There is a span of U values which leads to superconductivity rather than a CDW. The wave functions for the CDW ground state may be written as: ΨCDW = [cosϕ (a2 + b2) + sinϕ (a2 − b2)] /√2

(35)

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In fig. 11 ϕ = π/4 (cosϕ = sinϕ) and ΨCDW = a2 at the left minimum and hence the wave function proportional to a2 (all charge is on one of the sites. In the other minimum ϕ = −π/4 (cosϕ = −sinϕ) and ΨCDW = b2 in eq. (35). For Q=0 ΨCDW = (a2 + b2)/√2 in the ground state. The SDW wave function MM is almost independent of the nuclear geometry: ΨSDW = (a+b)(a+b)/2. The coupling between SDW and CDW (H13) is hence calculated as: H 13 = ( a + b )( a + b ) / 2 H ΨCDW .

(36)

Intervalence spectrum. The coupling between metal ions gives a new source for spectral lines in the electronic spectrum, in many cases in the visible spectrum. In the MV-2 case the vertical excitation in fig. 9 is an intervalence transition, close to λ in magnitude if ⏐H12⏐ is small [7,9-12]. These are allowed transitions which usually give strong colors to the compound. Ink, containing Fe2+/Fe3+ mixed valence, is a good example. One may show that the transition moment in the localized case may be written as: μx = e

RH 12 λ

(37)

R 2

(38)

and in the delocalized case as: μx = e

The intensity and shape of the electronic absorption spectrum tells us whether the system is localized or delocalized [110,111]. In particular Reimers and Hush [112] have treated the case of the Creutz-Taube complex containing two metal ions. In this case the electrons are

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delocalized in the sense that the localization of the electrons is only weakly depending on the position of the nuclei. A number of experiments support this view. In the MV-3 case the spectrum is a charge transfer transition from the left minimum in fig. 11, corresponding to the M+M− state, to the SDW (MM) energy surface. The spectrum is thus an intervalence spectrum of similar type as in the MV-2 case. Very little intensity arises from the a charge transfer transition brtween M+M− and M−M+ because of the small effective coupling. The intervalence spectrum often gives a strong color to the complex. In ink the color is due to transfer of one electron from the metal ion with the lower oxidation state to the one with the higher oxidation state. The band model cannot be used in the localized case. The bands corresponding to the lower oxidation state are higher in energy than the bands corresponding to the higher oxidation state. Therefore the charge transfer energy of the type we are talking about should have a negative energy in the band model. In fact the existence of intervalence transitions shows that the Q-coordinate in figs. 9 and 11 is very important and cannot be ignored in a reasonable description. The band model can only be used in delocalized systems. MV-3 delocalization problem. A CDW system may be localized or delocalized. BaBiO3 is a typical CDW system, as is evidenced by alternant Bi sites, which suitably may be labelled as Bi3+ sites and Bi5+ sites [19,20]. The BiO bond distances are different by as much as 0.18 A. Doping with Pb or K, according to the compositions BaPbxBi1-xO3 and Ba1-xKxBiO3, respectively, leads to smaller distance differences [16,113]. Since the matrix element H12 is negligible, this can only be explained as being due to an effective H12 due to interaction with the Bi4+Bi4+ state. Doping appears to lower the energy of the SDW state relative to CDW states and bring the states into interaction. This leads first of all to delocalization since the coupling between the two charge distributions M−M+ and M+M− increases. This type of delocalization is not of the same type, depending on the λ/t ratio, as in the MV-2 systems. If the doping continues to increase, the delocalized phase becomes superconducting [16,113]. A superconducting system has to be expected to be fully delocalized, meaning that the alternation in geometry is not visible in experiments. The ground state is a vibronic state where the wave function is correlated: The electronic probability amplitude depends on the positions of the nuclei. The electronic and nuclear motion cannot be separated. Another well-defined MV-3 system is Cs2Au2X6 (X=Cl, Br) [21]. The metal ions are Au+ and Au3+ (U kQ02/2), the lowest vibronic state is below the CDW states. There is activation energy to move an electron to an adjacent site, i.e. to form a CDW state and this is consistent with the semiconducting properties of CuO. We conclude at this point that it is possible to construct SDW and CDW states from the electrons involved in the van Hove singularity. Pair currents arise from quantum mechanical superposition of SDW and CDW as we have seen in sect. 7. In the case of half-breathing motion, H13 ≠ 0, and hence there is a gap between the lowest vibronic states. In the case of the full-breathing mode, H13 = 0, and hence the final gap is also equal to zero. Doping may change occupations or energy levels. In CuO the two sites with different oxidation states, Cu3+ and Cu2+, may exchange single electrons according to the MV-2 mechanism. The other possibility is that the CDW (Cu+/Cu3+) state is lowered in energy enough to interfere with the SDW state to provide a pair exchanging MV-3 system.

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Isotope effects. The most direct way to test the involvement of atomic nuclei in a phenomenon such as superconductivity is to exchange isotopes. In the model for superconductivity that emerges here, the oxygen atoms clearly are responsible for the isotope effects. Experimentally isotope effects have been discovered in all cuprates. The fact that the oxygens of the CuO2 planes are responsible and not the axial oxygens may be taken as further support for the theory. Isotope effects appear show in a very direct way that nuclear dynamics is involved in a phenomenon. In the present case the theory of the present section shows this very clearly. The total wave function. The final electronic wave function consists of a closed subshell electronic wave function combined with an antisymmetrized geminal composed of the pair functions discussed in the previous section. The former describes an ordinary wave function with a static charge that contains everything except 3d(x2-y2) components. The latter is strongly coupled to the nuclear coordinates. With the help of Sasaki’s theorem [130] we may ignore the former part. As mentioned above the latter wave function has to be combined with a vibrational wave function, in fact the vibrational wave functions of the minima in fig. 11. The vibronic wave function is thus [103]:

∑ C i χ i (Q )

Ξ k = A 2 Φ(1,2 ; Q )

(63)

i

A sum over all geminals is formed: Ξ=

∑ Ξk k

Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

(64)

44

Sven Larsson The wave function is expanded as a product of singlet geminal (APSG) [131-133]. ΨAPSG = A N [Ξ(1,2 )Ξ(3 ,4 ).... Ξ(N − 1, N )]

(65)

A gas of electrons satisfies Fermi-Dirac statistics and a gas of electron pairs BoseEinstein statistics in the limit of no interactions between electrons and electron pairs, respectively. These two types of statistics become visible as “condensation” in certain limit cases in ordinary systems as well. In the case of superconducting cuprates the interaction is specifically a coupled motion on adjacent sites involving two electrons and leading to a finite gap between the ground state and first excited state of the full system. The electron pairs, coupled with vibrations, behave as a gas of Bose-Einstein particles as T → 0. In the Fermi-Dirac case the corresponding interaction involves a single electron and is possible between near and distant sites with arbitrary magnitude of the coupling. There is no pairing and no gap and the behavior as T→ 0 according to the laws of Fermi-Dirac condensation. Delocalization of an MV-2 system leads to an ordinary metal.

9. Phonon Softening

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The term “softening” refers to decrease of a force constant compared to “standard”. In a phonon spectrum of a cuprate one usually refers to a doped case compared to the undoped system or copper oxide. The probability is found to decrease for some higher frequency and increase for some lower frequency. In the standard harmonic, adiabatic treatment the second derivative of the total energy is calculated [134]: ∂2 E = Φ( α, β, l − l´, k , k ' ) , ∂u α , l , k ∂u β, l' , k '

(66)

where α and β are x, y, or z, coordinates of nuclei l and k (l’,k’). The frequencies are determined from a secular equation. Phonon softening may be discovered in connection with catalysis when a chemical bond is softened due to occupation of antibonding MO, for example by the electrons of the atoms on a metal surface. Mixed valence systems are excellent examples since electrons may enter a bond orbital in ET. PES for different valence state interact, PES becomes anharmonic. The modes involved are breathing modes, half or full breathing modes (figs. 7, 13). Connections are known between softening of phonon modes and superconductivity. B. Matthias early drew attention to some drastic changes in the Debye temperature which may be connected to phonon softening [135]. Most important work in recent years has been carried out by Pintschovius and Braden [40-44] and others [136-137] using inelastic neutron scattering (INS) in cuprates and other MV systems. Phonon softening has been found in inelastic neutron scattering (INS) and inelastic X-ray scattering (IXS) in several mixed valence systems, including bismuthates and cuprates. There are hardly general ways to treat anharmonicities and non-adiabaticities of the type double

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Correlation Effects and Superconductivity in Cuprates: A Critical Account

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well potential, although the problem has been discussed, not least in connection with MV-3 theory [73,137-138]. Non-adiabaticity has been included in the MV-2 case [103]. The reference system might be the undoped system such as La2NiO4 or La2CuO4 while the doped systems are La2-xSrxNiO4 or La2-xSrxCuO4, with x>0. In a localized case sites are introduced where the oxidation state of Ni and Cu, respectively, is raised from +2 to +3. Great changes in the phonon spectrum have been associated with double well potentials as well as to superconductivity. In NiO the Ni2+ ion is a high spin 3d8 ion with (t2g6eg↑2) configuration. The adjacent valence state Ni+ is an unstable (t2g6eg3 ) system. Ni3+ is a stable (t2g6eg1) system. The latter is formed as a localized site if Li2O is used as a dopant [98]. As an MV-2 system (Ni2+/Ni3+) the coupling should be quite large, since the interacting orbitlas are eg orbitals, but the reorganization energy also high, and for the same reason. The Li2O doped state is close to being delocalized with a black colour but with finite activation energy for conductivity [98]. The insulator to metal transition is at about 500 K, indicating that the CDW system is at a high energy. Thus we do not expect MV-3 behaviour in Ni2+ oxides. In the nickelates the mother compound is La2NiO4 where Ni has the valence state +2. Some La3+ ions may be replaced by Sr2+. In La2-xSrxNiO4 Ni obviously has the average valence state +(2+x) thus a mixture of Ni2+ and Ni3+ sites. Ordering (stripe phase) occurs in diagonal directions in an alternant manner [43,44]. Softening is observed in the bond-stretching dispersions as compared to those of non-doped La2NiO4. Tranquada et.al. [42,139] point out that there is no softening in the [1 1 0] direction (valencies are the same), but in the [1 0 0] direction (mixed-valency). Exchange of valence is possible in the [1 0 0] direction according to the mixed valence model (fig. 14) The cuprates have a superconducting phase if some La3+ is replaced by Sr2+ in La2CuO4 (La2-xSrxCuO4) and a maximum Tc for x = 0.125. There is softening in the half-breathing mode but not in the full-breathing mode, clearly related to the accidental vanishing of the coupling (H13) between (Cu3+/Cu+) and (Cu2+/Cu2+) in the case of the full-breathing mode, as we have seen in section 6. In this and other cuprates stripes are formed. Doping provides Cu3+ sites and stabilizes the (Cu3+/Cu+) CDW state, lowering U. The softening in the half-breathing mode appears to be a mixed MV-2 and MV-3 effect. Reznik et al. [44] have pointed out that “dynamic charge inhomogeneity” may explain the giant electron-phonon anomaly discovered by them. The CDW diabatic state appears to be responsible and is in fact directly related to the mechanism of superconductivity [18].

10. ARPES Angular resolved photo-electron spectroscopy (ARPES) is the experimental technique which comes closest to a “measurement” of one-electron energies. In this technique monochromatic ultraviolet radiation is used to excite a probe and thereby cause emission of photo-electrons, whose velocity v is measured. The obvious energy equation is: hν = Ii + mv2/2

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If the ionization energy of electron i, Ii, is set equal to −εi according to Koopmans’ theorem, the role of PES as a one-electron spectroscopy becomes clear. However, since Koopmans’ theorem is not an exact theorem, the correct definition of the ionization energy is: Ii = Ei(N−1) − E(N)

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The errors in Koopmans’ theorem are due to missing relaxation of the wave function in the ionized state, and a greater correlation error in E(N), the ground state energy before ionization, than in Ei(N−1), the remainder state after ionization. By experience these to errors tend to cancel each other. Total energies are thus measured in ARPES. The task for the theoretician is to calculate the correlation corrected total energies of the many-electron states rather than just orbital energies [92]. In the ARPES technique the directions of the in-coming electron and the outgoing photoelectron are measured, and this gives information about the momentum distribution of the electrons before emission. Together with Inverse Photo-emission spectroscopy, measuring the orbital energies of the empty orbitals in the same sense as ARPES is measuring the energies of the occupied orbitals, i.e. as total energy differences, it has been possible to measure the superconducting gap in cuprates and other superconductors [38,39,140]. This has given important evidence of d-wave pairing. The gap to the first excited state exists along the bond direction, but not to any measurable extent in the diagonal direction of the CuO2 plane. It remains to show that the results of the ARPES measurements are consistent with the delocalized electron pair transfer model as outlined above. In the construction of states in momentum space E(N) is subject to correlation of the type important for superconductors only when the states are formed along the bond directions. Thus there is a gap in the kdirection equal to Δ(k), with a maximum corresponding to the bond direction. The spectrum is therefore of the type: (x2 − y2)

Δ(k) = Δ0[cos(kxa) − cos(kya)]

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The superconducting gap between the ground state and the first excited state is now replaced by a band gap. An artificial band gap is introduced.

11. Discussion and Conclusion A planar grid with one active electron per site has two possibilities to form an insulating phase: CDW and SDW. Good examples are copper oxides and cuprates and gold oxides and aurates, where the metal ions both have the average oxidation state +2. Although copper and gold belong to the same group in the periodic table, the ground state of cuprates has alternant spins (SDW) and the ground state of Au2+ alternant charges (CDW) [91,141]. In the latter gold compound the sites are alternating between Au+ and Au3+. Gold in fact exists in oxidation state +2 only at a very high pressure and disproportionates to Au+ and Au3+ along the displacement coordinate Q (fig. 1).

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Mott-Hubbard U may be defined as the vertical energy from the SDW minimum to the potential energy surface of the disproportionated phase. If U is large, disproportionation is prevented. In a CDW compound U cannot be too large but may remain positive, since the relaxation of the structure to the CDW energy minimum along Q is connected to a very significant lowering of the free energy. The collective Q-coordinate that connects the minima of the SDW and CDW phases is the mode for electron pair transfer. We find that if the SDW and CDW free energy minima are approximately the same, there will be an intermediate phase, since the coupling along the half-breathing mode is very significant. The intermediate valence state mediates the interaction that permits the electron pair to move from one site to the next. The SDW state may be regarded as the “glue” that one is looking for. However, it acts as a “glue” just because of vibronic interactions (electron – phonon coupling). The result is an energy gap between the ground state and the first excited state. We also found that the corresponding electron pairs are free to move without resistance and may be set in motion by external forces and create a persistent macroscopic current (below the critical temperature). The complex wave function necessary to describe a pair current is a simple superposition of wave functions typical for CDW and SDW. This is essentially where we find the “strong correlation effects”. In the new phase there is delocalization, and thus validation of the k-space model (BCS). In the BCS model it is argued that some attractive interaction (no matter how small) is necessary to create attraction between the electrons in the Cooper pair [142]. This may hold true if we want to create a pair in a gas of delocalized electrons. In ordinary chemical systems the electron pair is a lone pair, a bonding pair, or any other pair. In cuprates the pair consists of two electrons in a b1g* MO or two electrons in half of the Bi 6s orbitals in BaBiO3. This type of pairing is common and there seems to be little reason to start worrying about the stability of such a pair. It simply follows from the presence of a strongly attractive nucleus. What is important is that the effective mass of the moving pair is small. A fundamental criterion is that the activation energy for pair transfer should disappear, and this leads automatically to the interaction picture between SDW and CDW. The correlation between oxidation state and geometry is the crucial observation in a real space treatment of superconductivity. The localization of the electrons is very sensitive to the geometry of the nuclei. Superconducting electrons are transported between sites in a crystal like the steam in a steam engine, correlated with the movements of the pistons. The motion of electron pairs in a crystal is similarly correlated with the motion of the nuclei around their equilibrium positions. In the quantum version no energy whatsoever is needed for this transport. All motions are correlated in a perfect way to a macroscopic current. The latter current is started by external forces and remains as an electronic ground state (more correctly: a vibronic ground state) with conserved angular momentum. Possibly it was the unsuccessful fate of Ogg’s pairs [48] that made people concerned about the stability of the electron pair. The search for a pairing mechanism is futile since it is hidden in obscure chemistry. It is only necessary to show that a pair of electrons can move between sites without any expenditure of energy (without activation energy). The latter pairs satisfy the conditions of Bose-Einstein statistics like all other electron pairs. The characteristic behavior below TC becomes visible as “Bose-Einstein condensation” if the pairs are moving as free pairs. There is a finite gap to the first excited state. Excitation over this gap breaks the superconductivity.

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In this chapter well-known and well established theories in physics and chemistry theories have been presented. In fact it is hard to see what else could possibly be necessary to explain superconductivity. Coupling to the phonons is the most conspicuous feature of MV theory, evident experimentally as electron – phonon coupling. Fundamentally this depends on the reorganization energy (λ), equal to a few eV in cuprates. Delocalization occurs if λ < 2H12. It still remains to check that there is quantitative agreement between theory and experiment such as INS and IXS [40-44] and in a number of other experiments. This is not an easy task but can be carried out, particularly if accurate calculation of the electronic structure can be performed along the relevant PES suggested here. MV-3 theory for superconductivity is clearly verifiable. Other systems mentioned in the introduction disproportionate in the same way as the cuprates, for example 2Ti3+ ↔ Ti2+ + Ti4+; 2W5+ ↔ W4+ + W6+. In the first place the pair is a Ti 3d pair and in the second case a W 6d pair. Other examples may be found in large molecular π systems, particularly the C60 molecule which may accept 1-6 additional electrons. K3C60 is superconducting below 30 K. The probable disproportionation is 2C603− ↔ C602− + C604−. This is consistent with the absence of paramagnetism in C603−. A reasonable hypothesis is that ordinary conductivity arises in MV-2 systems while superconductivity arises in MV-3 systems. In the latter case there is a unique electron pair transfer process with identical gap. In the MV-2 case, there is no need for an intermediate state and hence transfer processes involving a range of energies possible.

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[109] Hush, N.S. Mixed-Valence Compounds in Chemistry, Physics and Biology, NATO ASI Series, ed. D.B. Brown, D. Reidel Publishing Company, Dodrecht, The Netherlands, 1980, p. 151. [110] Creutz, C. Progr. Inorg. Chem. 1983, 30, 1-73. [111] Creutz C.; Taube, H. J. Am. Chem. Soc. 1969, 91, 3988; 1973, 95, 1086. [112] Reimers, J.R., Hush, N.S. Chem. Phys. 2004, 299, 79-82. [113] Sleight, A.W. Acc. Chem. Research 1995, 28, 103. [114] Girerd, J.J.; Launay, J.-P. Chem. Phys. 1983, 74, 217-226. [115] Borshch, S.A.; Kotov, I.N. Chem. Phys. Letters 1991, 187, 149-152. [116] Guo, Y.; Langlois, J.-M.; Goddard III, W.A. Science 1988, 239, 896-899. [117] Martin, R.L.; Saxe, P.W. Int. J. Quant. Chem. Symp. 1988, 22, 237-244; Moreira, I.D.P.R.; Illas, F.; Martin, R.L. Phys. Rev. B 2002, 65, 155102. [118] Kato, T.; Kondo, M.; Tachibana M.; Yamabe, T.; Yoshizawa, K. Chem. Phys. 2001, 271, 31-39. [119] Bersier, C.; Renold, S.; Stoll, E.P.; Meier, P.F. Phys. Cond. Matter 2006, 18, 74817495. [120] Chen, H; Callaway, J.; Misra, P.K. Phys. Rev. B 1987, 36, 8863-8865. [121] Klimkāns, A, thesis, Chalmers, Department of Chemistry, Göteborg, Sept. 2001. [122] Pierloot, K.; Dumez, B, Widmark, P.-O.; Roos, B.O. Theor. Chem. Acc. (Theor. Chim. Acta) 1995, 90, 87; Andersson, K.; Roos, B.O. Chem.Phys.Lett. 1992, 191, 507. [123] Barandiarán, Z.; Seijo, L.; Huzinaga, S. J. Chem. Phys. 1990, 93, 5843. [124] Orgel, L.E. J. Chem. Soc. 1958, 4186. [125] Bancroft, G.M., Chan, T.; Puddephatt, R.J.; Tse, J.S. Inorg. Chem. 1982, 2946-2949. [126] Rodríguez-Monge, L.; Larsson, S. J. Phys. Chem. 1996, 100, 6298-6303. [127] Ohashi, Y.; Momoi, T. J. Phys. Soc. Japan 1996, 65, 3254-3259. [128] Newton MD, Sutin N. Ann. Rev. Phys. Chem. 1984, 35, 437. [129] Landau, L. Phys. Z. Sow. 1932, 2, 46; Zener, C. Proc. Roy. Soc. A 1932, 137, 696. [130] Sasaki, F. Technical Note 77, Uppsala Quantum Chemistry Group 1962; see also Coleman, A.J. Revs. Modern Phys. 1963, 35, 668-687. [131] Yang, C.N. Revs. Modern Phys. 1962, 34, 694. [132] Coleman, A.J. Can. J. Phys. 1967, 45,1271;. [133] Kutzelnigg, W. J. Chem. Phys. 1964, 40, 3640-3647. [134] Born, M. and Huang, K. 1954, Dynamical theories of crystal lattices, Oxford, Clarendon. [135] Matthias, B.T.; Stewart, G.R.; Giorgi, A.L.; Smith, J.L.; Fisk, Z.; Barz, H. Science 1980, 208, 401-402. [136] Braden M.; Reichardt, W.; Sidis, Y.; Mao, Z.; Maeno, Y. 2007, Phys. Rev. B 76, 014505. [137] Liu, K., Moritomo, Y., Nakamura, A., Kojima, N. J. Phys. Soc. Japan 1999, 68, 31343137. [138] Zacher, R.A. Phys. Rev. B 1987, 36, 7115-7117; Hardy, J.R. and Flocken, J.W. Phys. Rev. Letters 1988, 60, 2191-2193. [139] Tranquada, J.M.; Sternlieb, B.J.; Axe, J.D.; Nakamura, Y.; Uchida, S Nature 1995, 375, 561-563.

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[140] Shen, Z.-X.; Dessau, D.S.; Wells, B.O.; King, D.M.; Spicer, W.E.; Arko, A.J.; Marshall, D.; Lombardo, L.W.; Kapitulnik, A.; Dickinson, P.; Doniach, S.; DiCarlo, J.; Loeser, A.G.; Park, C.H. Phys. Rev. Letters. 1993, 70, 1553-1556. [141] Larsson, S. Chem. Eur. J. 2004, 10, 5276-5283. [142] Cooper, L.N. Phys. Rev. 1956, 104, 1189-1190.

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In: Superconducting Magnets and Superconductivity… ISBN: 978-1-60741-017-1 Editors: H. Tovar and J. Fortier, pp. 55-76 © 2009 Nova Science Publishers, Inc.

Chapter 2

IMPROVED FLUX PINNING PROPERTIES FOR RE123 SUPERCONDUCTORS BY CHEMICAL METHODS Yui Ishii, Jun-ichi Shimoyama and Hiraku Ogino Department of Applied Chemistry, University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

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Abstract Effects of dilute impurity doping to Y123 single crystals and Y123 melt-solidified bulks on their flux pinning properties were systematically studied. It has been already reported in our previous papers that the dilute Sr-doping to Ba site improved the critical current properties of Y123 crystals, and the dilute Lu-doping to Y site was also found to increase critical current density, Jc. In addition, the dilute impurity doping to Cu in the CuO-chain was found to dramatically enhance Jc under magnetic field without large decreases in Tc in contrast to the case of dilute impurity doping to the superconducting CuO2-plane by Zn or Ni. Furthermore, we have attempted to extend this method to Dy123 melt-solidifed bulks, and found that the dilute impurity doping to Cu in the CuO-chain certainly improved Jc, and the dilute Sr-doping is also effective particularly below 50 K when a small amount of Dy/Ba substitution occurred in the Dy123 phase. Additions of Tb-containing compounds were also attempted to improve Jc properties of Dy123 melt-solidified bulks. Although CeO2 addition has been well known to enhance Jc by reducing size of RE211 particles, it always accompanies slight decrease in Tc. Our newly developed method, introduction of fine BaTbO3 precipitates, was found to be more effective for enhancement of Jc without such decrease in Tc. In particular, Tb4O7 added Dy123 meltsolidified bulks exhibited remarkably improved Jc due to the generation of fine BaTbO3 precipitates with 0.1 ~ 0.2 μm in size in the Dy123 matrix. This result means Tb4O7 is a more effective additive than CeO2.

1. Introduction Shortly after the discovery of REBa2Cu3Oy (RE123; RE = rare earth elements) superconductors [1], the melt-solidification process was attempted to overcome weak-link problem of RE123 sintered bulks. Its critical current density, Jc, at 77 K was firstly improved

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up to 4,000 A cm-2 by this process [2], while the sintered bulk exhibited only poor critical current density. This method has two important advantages; improvement of intergrain coupling and introduction of non-superconducting precipitates, to the large RE123 crystals. Both factors are essentially important for development of practically applicable RE123 bulks with high Jc. In particular, the RE2BaCuO5 (RE211) precipitates are recognized as one of the most popular and predominant pinning sites, because they are easily introduced through a peritectic reaction from RE211+liquid phase to RE123 when RE211-rich compositions are adopted. Needless to say, the flux pinning phenomenon has been widely investigated in terms of both practical and physical interests in the last half-century. The specific pinning sites were found to be effective and useful for each substance; for example, non-superconducting α-Ti lamellae are appropriate for Nb-Ti, and grain boundaries are predominant pinning sites of Nb3Sn and V3Ga. In the case of the RE123 superconductors, one of the most effective pinning sites is the super/non-super (S/N) interfaces at the surface of the RE211 particles as mentioned above. A proportional relationship between the S/N interface density and the Jc is generally known [3]. It is still controversial whether they act as effective pins only under low magnetic fields or up to high magnetic fields. However, Jc in low magnetic fields are remarkably increased by the refinement of RE211 particles. Another effective pinning site in the RE123 compounds is due to the oxygen deficiency in the CuO-chain. The oxygen deficiencies are categorized to the 3 dimensional point-defectlike pinning sites and considered that they bring the broad secondary peak effect in magnetization hysteresis loops, i.e., Jc-H curves, which is characteristic for the RE123 [4]. Similar peak effect also appears in the samples having RE1+xBa2-xCu3Oy-type solid solution, in which the RE-rich regions are known to act as effective pins under wide range of magnetic fields [5]. Since the discovery of the high-Tc superconductors (HTSCs), effects of the various impurity substitutions on the superconducting properties were widely investigated and enormous volume of knowledge came to be shared. Since the main purpose of such studies was the clarification of the mechanisms of high-Tc superconductivity, the substitution levels were usually larger than 1 % of the target site and they always decreased the Tc. Among these studies, substitution of Zn for Cu concentrated the physical interests, because it largely decreased Tc by its selective substitution for Cu in the CuO2-plane. From a practical viewpoint, these impurity substitutions were considered to be unfavorable for maintaining the inherent high Tc, and simultaneously achieving the high Jc properties of the materials. However, it was recently reported that the Zn-doping to the Cu site in the CuO2-plane for Y123 melt-solidified bulks was effective to enhance Jc properties in magnetic fields when the doping level was quite low [6]. According to Krabbes et al., this effect is due to the locally suppressed superconductivity in the surrounding of the Zn ions. Similar effects by the dilute impurity doping to the CuO2 plane were also found in Bi(Pb)2212 single crystals [7]. These results indicate that the introduction of dilute impurity for the CuO2 plane is universally effective to enhance the Jc properties of HTSCs. Subsequently, dilute RE3+ substitution for the Ca2+ site in Bi(Pb)2212 single crystals was discovered to dramatically improve their Jc characteristics without a large decrease in Tc [8]. This result is noteworthy in respect that the impurity substitutions for the particular site other than Cu in the CuO2-plane significantly affected the critical current properties. In this case, locally lattice distorted regions around the substituted RE3+ are considered to act as effective

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pins without large decrease in bulk Tc. Of course, similar improvement can be expected for the RE123 and slight improvement has been observed for the dilute Sr substituted Y123 meltsolidified bulks [9]. It should be noted that the actual doping levels of impurities are far less than 1 % in all the above successful studies. This indicates that it is essentially important to control the mean intervals of doped impurities in the doped layer parallel to the ab-plane longer than the double of superconducting coherence length 2ξab. The lowest electromagnetic anisotropy (γ 2 ≡ mc* / mab* = 25~50) is the unique feature for RE123 among HTSC compounds. According to a report by Tallon et al. [10], the superconducting carriers exist on the CuO-chain as well as the CuO2-plane, which contributes the relatively large ξc resulting in the large pinning volume and the strong pinning energy. This indicates that the low level impurity substitution to the each atomic site except for the Cu in the CuO2-plane can locally reduce the superconducting condensation energy of RE123 corresponding to the local lattice deformation. Especially to Cu in the CuO-chain, large enhancement of the flux pinning properties is expected, because the doped impurities strongly affect the local superconducting order parameter without decreasing Tc. With regard to the additives for the RE123 melt-solidified bulks, CeO2 and Pt addition are widely investigated thus far. These additives are effective for decreasing particle size of RE211 in the melt-solidified RE123 bulks, resulting in high Jc. In particular, CeO2 addition is known to suppress coarsening of the RE211 precipitates during the melt-solidification process [11] and its mechanism was widely investigated. Among the RE elements, Ce and Tb are known not to form the RE123 phase because their tetravalent ionic state is stable and tend to form perovskite-type oxides with Ba, such as BaCeO3 and BaTbO3. Generation of such perovskite phase may suppress the grain growth of RE211, however, effects of Tb addition have not been well understood thus far. Based on these backgrounds, we have studied effects of various chemical doping on the critical current properties of RE123 melt-solidified bulks and single crystals. In the next chapter, improved Jc characteristics of RE123 by dilute impurity doping and low level RE substitution for the Ba-site are shown.

2. Dilute Impurity Doping Effects The crystal structure of the REBa2Cu3Oy (RE123) was shown in Fig. 2-1. The constituent elements of the RE123 are the three cations, RE, Ba and Cu, and nonstoichiometric oxygen. The ionic radii [12] of the important elements including the three cations in the RE123 are listed in Table 2-1. This ionic radii and its valence approximately determine the site selectivity of impurity ions. And the difference in the ionic radii between the impurity element and the host site would generate the local lattice distortions. We first examined the changes in the superconducting properties of Y123 melt-solidified bulks by dilute Lu-doping for Y or Co-doping for Cu in the CuO chain, because Lu3+ has the smallest ionic radius among the all RE3+ ions and configuration of oxygen in CuO chain is locally changed by Co doping. After these studies, the dilute doping effects of Co and Sr, which has already reported for Y123 crystals [9], are applied for the Dy123 melt-solidified bulks including (Sr,Co)-codoping effects for Cu(1) and Ba sites.

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Figure 2.1. Crystal structure of REBa2Cu3Oy (RE123). Various cation substitutions examined in this study are also indicated.

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Table 2.1. Effective ionic radii for RE elements, Ba, Sr, Cu and doped impurities [12]

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2.1. Dilute Lu-Doping Effect for Y123 Melt-solidified Bulk Y123 melt-solidified bulks were prepared by the cold-seeding method starting from Y1Lu Ba x 2Cu3Oz (x = 0, 0.01, 0.03, 0.05) pre-calcined powder. They were mixed with the Y211 x pre-calcined powder with the molar ratio of 7 : 3 and 0.5 wt% of Pt was added. Each powder mixture was pelletized with dimensions of 20 mmφ × ~8 mm by uniaxial pressing, and then melt-solidified in air using a Nd123 seed crystal. The temperature parameters during the meltsolidification are summarized in Table 2-2 and Figure 2-2 together with those of the other impurities doped samples described in the following section. Samples composed of a single domain were successfully obtained for all compositions. The rectangular samples with typical dimensions of 1.7 × 1.7 × 0.7 (// c) mm3 cut from 1 mm below the seed crystal (in the c-growth region) were used for the magnetization measurements throughout this section. The superconducting properties were measured by SQUID magnetometer (Quantum Design MPMS-XL5s) under fields applied parallel to the c-axis. Jc was calculated from the magnetization hysteresis using the extended Bean model [13].

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Table 2.2. Temperature parameters applied for the melt-solidification process of RE123

Figure 2.2. The temperature profile of the melt-solidification process performed for RE123 bulks.

The SEM observation and the EDS local compositional analysis revealed that the most of Lu existed in the Y211 precipitates of the Lu-doped Y123 melt-solidified bulks. As shown in Figure 2-3(a), Tc was not changed by Lu-doping reflecting its low substitution level in the Y123 phase. In the Jc-H curves shown in Figure 2-3(b), only a 5% Lu-doped sample exhibited the secondary peak effect at 77 K probably due to the dilute Lu substitution for Y-site.

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(a)

(b)

Figure 2.3. (a) ZFC magnetization curves and (b) Jc–H curves at 77 K for Y1-xLuxBa2Cu3Oz meltsolidified bulks annealed at 400ºC in flowing oxygen.

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2.2. Effect of Dilute Impurity Doping to CuO-Chain In order to prevent the contribution of the RE211 precipitates on the Jc characteristics, undoped and the Co-doped Y123 single crystals were prepared by the self-flux method using the BaZrO3 crucibles. The flux compositions adopted for the crystal growth were Y : Ba : Cu : Co = 1 : 30 : 60 : 0, 1 : 30 : 60 : 0.3 and 1: 30 : 60 : 0.6. Compositional analysis for resulting crystals was performed by the inductively coupled plasma method. The analyzed Co composition of the single crystals prepared from the latter two compositions were y = 0.008 and 0.024 in the chemical formula of YBa2Cu3-yCoyOz, respectively. The typical sample size of obtained single crystals was 1×1×0.25~0.5(//c) mm3. Post annealing was done for these crystals at 450ºC for 100 h in flowing oxygen. Magnetic susceptibility curves of Co-doped Y123 single crystals were shown in Figure 24(a). All the samples exhibited high Tc’s above 90 K and Tc for the Co-doped sample with y = 0.008 kept Tc ~92 K which is almost the maximum Tc for the pure Y123. The slight difference in Tc is thought to originate from the small difference in carrier doping levels. The Jc-H curves at 77 K for the samples of y = 0, 0.008 and 0.024 are shown in Figure 24(b). Co-doped Y123 single crystals with y = 0.008 and 0.024 exhibited the largely enhanced secondary peaks, while the pure Y123 single crystal showed a small peak probably due to the oxygen vacancies. Similar improvement of Jc in magnetic field was also observed in the dilute Co-doped Y123 melt-solidified bulks. Details of the sample preparation procedures will be found in elsewhere [14]. According to the magnetic susceptibility measurements, high Tc above 90 K was maintained in contrast to the cases of Zn-doped and Ni-doped Y123, in which Tc largely decreases even if the doping level was low below a few % for Cu[15]. Jc-H curves for Codoped Y123 melt-solidified bulks were shown in Figure 2-5. The Jc under magnetic fields systematically increased with an increase of the nominal Co doping level y, similar to the behaviors of Co-doped Y123 single crystals.

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Figure 2.4. (a) ZFC magnetization curves and (b) Jc–H curves at 77 K for YBa2Cu3-yCoyOz single crystals annealed at 450ºC in flowing oxygen.

Figure 2.5. Jc–H curves at 77 K for YBa2Cu3-yCoyOz melt-solidified bulks annealed at 450ºC for 100 h in flowing oxygen. Samples were cut from the c-growth region of the bulks.

Effects of impurity doping to Cu in the CuO-chain on Jc-H characteristics were examined for various impurity elements. Fe or Ga doped Y123 melt-solidified bulks were prepared by the same method. For comparison, Zn-doped Y123 melt-solidified bulks were also prepared. The nominal doping level was fixed to be y = 0.02 in the chemical formula of YBa2Cu3-yMyOz (M = Fe, Co, Zn, Ga). All the melt-solidified samples were composed of the single domain. The rectangular samples cut from these bulks were annealed at 450ºC for 100 h in flowing oxygen. The microstructural observations by SEM revealed the homogeneous dispersion of Y211 particles with few micrometers in size in the Y123 matrix. The volume fraction, V211~10%, and the grain size, d211, of the Y211 precipitates did not change by the impurity doping independent of the impurity doping levels. The EDS analysis revealed that doped Co preferentially existed in the Y123 phase. Figure 2-6 represents the temperature dependence of

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the magnetic susceptibility for these impurities doped samples. The Fe-, Co- or Ga-doped samples exhibited higher Tc than the Zn-doped sample, suggesting that the Fe, Co or Ga substitute for Cu in the CuO-chain.

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Figure 2.6. ZFC magnetization curves for YBa2Cu2.98M0.02Oz (M= Fe, Co, Ga, Zn) and undoped YBa2Cu3Oz melt-solidified bulks annealed at 450ºC for 100 h in flowing oxygen.

Figure 2.7. Dependence of Tc on the oxygen annealing temperature of YBa2Cu2.98M0.02Oz (M= Fe, Co, Ga, Zn) and undoped YBa2Cu3Oz melt-solidified bulks.

Since substitution of the high valence ions, such as Fe, Co and Ga, for Cu, leads a decrease in hole carrier concentration, oxygen annealing temperature to obtain carrier optimally doped samples is generally reduced. Figure 2-7 shows the relationship between the annealing temperature and resulting Tc of YBa2Cu2.98M0.02Oz (M= Fe, Co, Ga, Zn) and undoped YBa2Cu3Oz melt-solidified bulks. The annealing temperature giving the optimal Tc

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for the undoped sample was approximately 500ºC, and it was not changed by the Zn-doping because it did not affect the carrier doping state. On the other hand, the annealing temperatures giving the optimal Tc for Fe-, Co- or Ga doped samples were apparently low (400~450ºC) due to substantially reduced hole concentrations. The Jc-H curves for these impurities doped samples annealed at 450ºC for 100 h were displayed in Figure 2-8. As previously reported, the Zn-doped sample exhibited higher Jc than the undoped Y123 sample. However, more prominent improvements in Jc-H characteristics were observed for Fe-, Co- or Ga-doped Y123 samples. Through TEM observations performed for undoped and Co-doped (y = 0.02) samples, it was found that the twin structures were not changed by Co-doping, while substitutions by these high valence metals were reported to increase density of twins when their substitution levels were high, y~0.05 [16]. In addition, homogeneous distribution of the doped impurities was confirmed by TEM-EDS analysis performed for a Co-doped sample with y = 0.1.

Figure 2.8. Enhanced Jc at 77 K of Y123 melt-solidified bulks by dilute impurity doping for Cu site in the CuO chain by Fe, Co and Ga.

The closed symbols in the Figure 2-8 represent the Jc for the samples annealed at 300ºC for 384 h in flowing oxygen. The Jc for the undoped and the Zn-doped samples decreased under whole magnetic fields after the low temperature post annealing due to decreases in oxygen defects and Tc. On the contrary, a large secondary peak was remained for the Gadoped sample, although they were almost fully oxygenated. In addition, Jc increased under high fields by the low temperature oxygen annealing. Similar behaviors were also observed for the Fe and the Co-doped samples. Therefore, dramatic enhancements in Jc by Fe-, Co- or Ga-doping are not originated from the change in twin density or the oxygen defects. These results indicate that observed large secondary peaks were mainly originated from the impurities doped to the CuO-chain, which locally affect lattice distortions and electronic state. Figure 2-9 shows the pinning force, Fp, at 77 K for the YBa2Cu2.98M0.02Oz (M= Fe, Co, Ga, Zn) and undoped YBa2Cu3Oz melt-solidified bulks. It is clear that the Fe-, Co-, and Gadoped samples exhibited larger Fp(max) than that of the Zn-doped one, and their Fp in high

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fields largely enhanced by oxygen annealing at 300ºC. This suggests that the increase of oxygen content, i.e., carrier doping strengthens the elemental flux pinning force due to the doped impurity ions. In other words, this result may indicate that spatial gradient of the superconducting order parameter was increased by the development of superconductivity also at the CuO-chain in the carrier overdoped state [10].

Figure 2.9. Fp–H curves at 77 K for YBa2Cu2.98M0.02Oz (M= Fe, Co, Ga, Zn) and undoped YBa2Cu3Oz melt-solidified bulks annealed at 450ºC for 100 h in flowing oxygen.

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2.3. Dilute Impurity Doping Effects for Dy123 Melt-Solidified Bulks The melt-solidified bulks with the nominal compositions of Dy(Ba1-xSrx)2Cu3-yCoyOz (x = 0~0.0045, y = 0~0.025) were prepared by the same procedure described above. Each obtained bulk was composed of a single domain. Homogeneous dispersion of Dy211 particles in the Dy123 with 1~3 μm in size was confirmed by microstructural observation using SEM. The volume fraction, V211, and grain size, d211, of the Dy211 precipitates were not affected by Sr or Co-doping. Although Co incorporation to the Dy123 matrix was quantitatively confirmed by EDS analysis, Sr could not be detected possibly due to its quite low substitution level. The EDS analysis also revealed the selective substitution of Co only for the Dy123 phase not for the Dy211 precipitates. As shown in Figures 2-10(a) and (b), both Sr- and Co-doping improved the Jc–H characteristics of the Dy123 melt-solidified bulk at 77 K. The Co-doped sample with y = 0.015 exhibited ~60,000 A cm-2 at the second peak field, while the Jc was slightly increased by of dilute Sr-doping. This difference suggested that dilute impurity doping to Cu in the CuO chain is more effective to enhance pinning force than that for Ba-site at 77 K. Nonetheless, dilute Sr-doping to Ba-site becomes quite effective for enhancement of Jc at 77 K in the slightly Dy-rich Dy123 bulks and it also improves Jc at low temperatures. Details of effects by each dopant on the critical current properties of Dy123 will be discussed in Sec. 2.4.

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Figure 2.10. Jc–H curves for (a) the Sr-doped, (b) the Co-doped and (c) the (Sr,Co)-co-doped Dy123 melt-solidified bulks annealed at 400ºC for 120 h in flowing oxygen.

Here we attempted the further improvement of Jc for Dy123 melt-solidified bulk by codoping of Co and Sr. Bulks with single domain were obtained even for co-doped ones. As shown in Figure 2-11, Tc’s of the (Sr,Co)-co-doped samples were slightly and systematically decreased with an increase of the Co-doping level, indicating that actual Co- and Sr-doping levels of these bulks were systematically controlled by changing starting compositions. Although the Jc for the Dy123 sample was certainly increased by the Sr or Co-doping, further improvement of Jc was achieved by (Sr,Co)-co-doping as shown in Figure 2-10(c). The Jc at each second peak in the Jc-H curves, Jc(peak), and Fp(max) at 77 K for (Sr,Co)-codoped samples with various doping levels are summarized in Figure 2-12. It should be noted that their Jc’s were increased by co-doping independent of the Co-doping level. In other word, the Jc for Dy(Ba0.997Sr0.003)2Cu2.98Co0.02Oz sample exhibited higher Jc than the Sr-free samples although its Co-doping level, y = 0.02, was higher than the optimum Co-doping level, y = 0.015. This result can not be simply explained by an increase of the pinning site density in the Dy123 matrix, corresponding to local regions substituted by Sr and Co ions. In the chemical

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viewpoint, Co ions were known to increase the oxygen coordination number resulting in smaller space at the Ba site. Since the Sr ion has a smaller ionic radius than the Ba ion, it can be considered that Sr ions tend to locate near the Co ions, resulting in large spatial gradient of the superconducting order parameter.

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Figure 2.11. The nominal doping level dependence of Tc for the Sr-doped, the Co-doped and the (Sr,Co)-co-doped Dy123 melt-solidified bulks annealed at 400ºC for 120 h in flowing oxygen.

Figure 2.12. The nominal doping level dependence of the Jc(peak) and the Fp(max) for the Dy(Ba1-xSrx)2Cu3(x = 0, 0.003, y = 0~0.02) melt-solidified bulks at 77 K.

yCoyOz

2.4. Changes in Pinning Effect by Dilute Chemical Doping to Different Cation Sites In this section, effects of partial substitution of RE for the Ba site was taken into account. Among the RE elements, La, Nd, Sm, Eu and Gd can partially substitute for the Ba site in the RE123 phase due to their relatively large ionic size. Such trivalent RE substitutions for

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divalent Ba increase oxygen content at O(5) site and hence the macroscopic orthorhombicity decreases as in the case of Co-doping. This means that crystal structure would include strains with an increase of the RE3+ substitution level. According to TEM observations, such RE concentration is fluctuated in spatial size of ~50 nm [17]. This fluctuation was also observed in STM study as the difference in local conductance [18]. Therefore, the relatively RE-rich regions are considered to act as effective pins under magnetic fields. The RE3+ substitution level is affected by its ionic radii, the starting composition and also the oxygen partial pressure, PO2, during the phase preparation. In the case of sintered bulks starting from stoichiometric composition and reacted in air, Gd123 can form the Gd123 single phase with Tc well above 93 K after oxygen annealing, while moderately reduced atmosphere is needed in the synthesis of La123, Nd123 and Sm123 to prevent the RE substitution for Ba, which largely decreases Tc. Then Gd123 can be considered as one of the most promising substances for the practical bulk applications in terms of Tc and Jc. It has been generally considered that Dy and Ho do not substitute for the Ba site because of their small ionic radii. However, it was recently reported that the secondary peaks in Jc-H curves at 77 K for the Dy123 and the Ho123 melt-solidified bulks was systematically enhanced with increasing the PO2 during the crystal growth [19]. These suggests the small amount of Dy/Ba and Ho/Ba substitution occurs and their RE-rich regions are also the main origin of the peak effect for Dy123 and Ho123 melt-solidified bulks besides the oxygen deficiencies. Figure 2-13(a) displays a bright-field TEM image of Dy123 melt-solidified bulk prepared in flowing oxygen gas. Strong contrast in the image was originated from the lattice distortion corresponding to the inhomogeneous Dy-concentration. Furthermore, this fluctuation-originated contrast was observed in a Dy123 melt-solidified bulk grown in air. Figures 2-13(b) and (c) show the bright-field TEM images for the Dy123 melt-solidified bulks grown in air with cooling rates of 0.8ºC h-1 and 0.4ºC h-1, respectively. Both samples were annealed at 400ºC in flowing oxygen for 120 h. In the sample grown under the low cooling rate of 0.4ºC h-1, the similar structures to the Fig. 2-13(a) was observed. Besides, this sample did not have twins though it was properly oxygenated. These results support that the Dy/Ba substitution certainly occurred and its substitution level is also influenced by the cooling rate as well as the atmosphere during the crystallization, whereas its mechanism has not been well understood yet [20, 21].

Figure 2.13. The bright field TEM images for the Dy123 melt-solidified bulks prepared (a) in flowing oxygen, (b) in air with 0.8ºC h-1 of cooling rate and (c) in air with 0.4ºC h-1 of cooling rate.

Figure 2-14 shows the magnetic susceptibility curves and Jc-H properties at 77 K for Dy123 and Gd123 melt-solidified bulks grown under various cooling rates of 0.4 ~ 0.8ºC h-1 in air. Although the Tc for the Gd123 samples were systematically increased from 92.5 K to

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94.5 K with increasing the cooling rate from 0.4°C h-1 to 0.8°C h-1, it was almost unchanged for the Dy123 samples possibly due to quite low substitution level of Dy at Ba-site. In addition, their Jc at the peak fields decreased with increasing the cooling rate, indicating that the high cooling rate decreased effective pinning sites corresponding to partial Dy- and Gdsubstitution for the Ba site.

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Figure 2.14. (a) ZFC magnetization curves and (b) Jc–H curves at 77 K for REBa2Cu3Oz (RE = Gd, Dy) melt-solidified bulks prepared at various cooling rates.

Figure 2.15. Jc–H curves at 77 K for (a) the Co doped (y = 0.02) and (b) the Sr-doped (x = 0.003) Dy(Ba1-xSrx)2Cu3-yCoyOz melt-solidified bulks prepared at various cooling rates.

It was found that the pinning effects by the dilute Sr- and Co-doping described in Sec. 2.3 was considerably influenced by this RE/Ba substitution level, and that the doping effects were different to each other. Figures 2-15(a) and (b) represent the Jc-H curves at 77 K for the dilute Co-doped and the dilute Sr-doped Dy123 melt-solidified bulks prepared in air with various cooling rate of 0.6 ~ 0.9ºC h-1. The Jc for the dilute Co-doped samples with y = 0.02 systematically increased with an increase of the cooling rate, while the Tc did not almost

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change. Similar improvement was also confirmed for Ga-doped Dy123 bulks with y = 0.02. Since the samples grown under the high cooling rate have less amount of Dy-incorporation for the Ba site, the improved Jc observed for the Co-doped samples can be explained by the deep pinning potential generated around Co ions in inherently strong superconducting matrix, that is, the stoichiometric Dy123 superconducting phase. On the contrary, the Jc for the Srdoped sample decreased with increasing the cooling rate. In other words, this result indicates that the Jc for the Sr-doped sample can be enhanced by an increase of Dy/Ba substitution level.

Figure 2-16. Jc–H properties at various temperatures for the (a) undoped (cooling rate; r = 0.6ºC h-1), (b) the Sr-doped (x = 0.003, r = 0.6ºC h-1), (c) the Co-doped (y = 0.02, r = 0.8ºC h-1) Dy(Ba1-xSrx)2Cu3yCoyOz melt-solidified bulks.

Our recent experiments revealed the excellence of the dilute Sr-doping technique for developing high-Jc RE123 materials operating at low temperatures. Figure 2-16 shows the JcH properties at various temperatures for the undoped, the Co-doped and the Sr-doped Dy123 melt-solidified bulks. The cooling rates were tuned so as to achieve enough high Jc for each impurity-doped system in accordance with the Fig. 2-14 and Fig. 2-15; i.e., 0.6ºC h-1 for the undoped and the Sr-doped samples, and 0.8ºC h-1 for the Co-doped sample. The Jc for Sr-

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doped sample was lower than that of the Co-doped one at 77 K. However, Jc at the second peak field of the Sr-doped sample abruptly increased with decreasing temperature and it became higher than that of the Co-doped sample below 50 K. In addition, the relaxation rate of magnetization at 50 K for the Sr-doped sample below its peak field was very slow corresponding to its large second peak effect. These results mean that the Sr-doping is one of the promising techniques particularly for the low temperature applications. Further studies are necessary to understand the essential dilute impurity doping effect discussed all in Sec. 2 on their flux pinning mechanisms.

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3. New Effective Pinning Site: BaTbO3 Precipitates In Sec. 2, various dilute doping techniques effective for improving Jc of RE123 were described. These techniques were effective mainly for in-field Jc of RE123 single crystals and the c-growth region of RE123 melt-solidified bulks. On the other hand, considerable efforts have been paid for achieving ideal microstructures of RE123 materials, such as meltsolidified bulks and coated conductors, thus far. Through those studies, improvement of inplane orientation and introduction of the non-superconducting fine particles or nanorods were well recognized approaches for improving inter- and intra-Jc properties, respectively. In the case of RE123 melt-solidified bulks, the a-growth regions, which is a major region in the typical disk-shaped RE123 bulk magnets, has the relatively large volume fraction of the RE211 precipitates, resulting in high Jc mainly under low magnetic fields, for example below 1 T at 77 K. In this section, we describe newly discovered non-superconducting precipitates which largely improve Jc under low magnetic fields of the a-growth regions. Among the RE elements, the effects of Tb-addition on Jc of the RE123 melt-solidified bulks have not been studied well as mentioned before in the Sec. 1. In fact, Tb4O7-addtion in the synthesis of RE123 powder starting from Tb4O7, RE2O3, BaCO3 and CuO always form BaTbO3 besides RE123 phase [22]. Note that any apparent changes in both lattice parameters and superconducting properties of RE123 were not observed by Tb4O7-addtion. This means that almost all the doped Tb was consumed for generation of BaTbO3. Based on this result, effects of BaTbO3-addition on critical current properties of Dy123 melt-solidified bulk were examined. Furthermore, we have attempted to decrease size of BaTbO3 precipitates, Tb4O7 powder was added to the powder mixture of Dy123 and Dy211 with a molar ratio of 7 : 3 before the melt-solidification.

3.1. BaTbO3 Addition The BaTbO3 powder with the average diameter of ~0.48 μm was prepared by the solidstate reaction and the following mechanical ball milling process. Obtained fine BaTbO3 powder was added to the powder mixture with composition of Dy123 : Dy211 = 7 : 3. In this study, 0.5 wt% of Pt and 2 wt% of CeO2 were also added. The starting compositions and the temperature patterns for the melt-solidification were summarized in Table 3-1 and Table 3-2, respectively, together with the Tb4O7-added samples described in the next section. The characteristic temperatures, t1, t2 and t3, in Table 3-1 correspond to those in Fig. 2-2. For the sample A (BaTbO3-free) and the sample B (BaTbO3-added), bulks composed of a single

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domain were obtained. The rectangular samples for the measurements with the typical dimension of ~1.7×1.7×0.7(//c) mm3 were cut from the a-growth region located just below the surface of each bulk. Hereafter the distance between the seed crystal and samples was expressed as La. Table 3.1. Starting compositions for BaTbO3 and Tb4O7 added Dy123 melt-solidified bulks.

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Table 3.2. Temperature parameters applied for the melt-solidification of BaTbO3 and Tb4O7 added Dy123 melt-solidified bulks.

Figure 3.1. Temperature dependence of magnetic susceptibility for Dy123 melt-solidified bulks; BaTbO3-free (sample A) and BaTbO3-added (sample B). La represents the distance from the seed crystal.

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As shown in Figure 3-1, the Tc(onset)’s for the BaTbO3-added sample were almost independent of the La and they were well above 90 K, while CeO2 addition which is a common method to decrease particle size of RE211 always accompanies decreases in Tc by 2~3 K. SEM observation and EDS compositional analysis revealed that the added fine BaTbO3 particles were successfully introduced into the Dy123 matrix. It should be noted that Ce was incorporated into the cubic perovskite phase and formed Ba(Tb,Ce)O3. The average diameter of the BaTbO3 precipitate was ~0.5 μm, which is almost unchanged from the initial powder, meaning that BaTbO3 particle does not grow during the melt-solidification, as in the cases of BaZrO3 [23] and BaSnO3 [24]. The Jc-H properties at 77 K for BaTbO3-added and the BaTbO3-free Dy123 samples are summarized in Figure 3-2. The BaTbO3-added sample exhibited high Jc in low magnetic fields exceeding 1 × 105 A cm-2 at 77 K under 0.5 kOe.

Figure 3.2. Magnetic fields dependence of Jc for Dy123 melt-solidified bulks; BaTbO3-free (sample A) and BaTbO3-added (sample B).

3.2. Tb4O7 Addition Tb4O7-addition just before the melt-solidification of Dy123 was found to be effective for decreasing size of resulting BaTbO3 precipitates. In the synthesis process, 0~3 wt% of Tb4O7 powder was added to the powder mixture with the composition of Dy123 : Dy211 = 7 : 3, while CeO2 was not added. All the prepared samples were composed of a single domain. Figure 3-3 displays the microstructure for the 2 wt% of Tb4O7-added sample (sample C). One can see the very small precipitates dispersed in the Dy123 matrix. Local compositional analysis performed on these small precipitates, indicated by arrows in Fig. 3-3, revealed that they are composed of Ba : Tb = 1 : 1 (molar ratio), indicating generation of BaTbO3. The typical size of these small

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precipitates was found to be ~0.1 μm, which is much smaller than the ball-milled BaTbO3 particle described in the previous section.

Figure 3.3. (a) FE-SEM image of Tb4O7 2 wt% added Dy123 melt-solidified bulk (sample C, La = 1 mm).(b) A typical EDS spectrum for the fine particles indicated by arrow.

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The magnetic susceptibility curves and the Jc-H properties at 77 K for these samples are shown in Figures 3-4(a) and (b), respectively. Although the Pt- and the CeO2-co-added sample (A) slightly decreased Tc, the Pt- and the Tb4O7-co-added samples (D, E, F) maintained the high Tc equal to the Pt-added and CeO2-free sample (C). In the Jc-H curves, the Tb4O7-added sample exhibited obviously higher Jc than the Tb4O7-free samples under the whole magnetic fields. Particularly, Jc for the 2 wt% of Tb4O7 added sample (E) reached ~1×105 A / cm2 under external field Hex = 0.5 kOe.

Figure 3.4. (a) Temperature dependence of magnetic susceptibility and (b) Jc–H curves for Dy123 meltsolidified bulks with/without CeO2 or Tb4O7.

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Figure 3-5 is the summary of these BaTbO3-addition techniques described in this section. In this figure, the Jc at 0.5 kOe and 77 K for the BaTbO3-added sample, described in the Sec. 3-1, the Tb4O7-free and the Tb4O7-added samples were plotted against the effective interface density of the Dy123 matrix and the non-superconducting particles, Vf/d. In the conventional CeO2-addition technique, one can see that the Jc did not largely increase even though the Vf/d increased by the refinement of precipitates. However, in the BaTbO3-added samples exhibited the high Jc corresponding to both the increase of Vf/d and the high Tc. This result strongly suggests that the BaTbO3- or Tb4O7-addition is the more excellent way for the enhancement of Jc under low magnetic fields than the well-known CeO2 addition.

Figure 3.5. Relationship between Jc in 0.5 kOe at 77 K and interface density (Vf/d) of the a-growth region with La = 1 mm of the Dy123 melt-solidified bulks with various additives.

As described above, additions of BaTbO3 or Tb4O7 are effective for improving Jc of the a-growth region of the Dy123 melt-solidified bulks. However, these additions were found to be ineffective for improving Jc of the c-growth region. Microstructural observation by SEM revealed that there are very few non-superconducting precipitates, such as BaTbO3 and Dy211 in the c-growth region. Although the reason for this phenomenon has not been explicitly understood, slight change in viscosity of the melt is considered to affect the pushing and trapping behaviors for non-superconducting particles during crystal growth.

4. Conclusion We have attempted the enhancement of Jc for RE123 melt-solidified bulks by introduction of new pinning sites based on the chemical backgrounds. We found that the Ludoping to the RE site also effective for enhancement of Jc at 77 K for Y123. Furthermore, Jc was dramatically enhanced when a small amount of Fe, Co, or Ga was doped. In the Dy123

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system, which can have partial Dy/Ba substitution, it was found that the Co-doping was most effective when the Dy/Ba substitution was suppressed; while the Sr-doping was most effective when the Dy/Ba substitution was proceeded, vise versa. In addition, Jc for the Srdoped Dy123 melt-solidified bulk increased dramatically with decreasing temperature. Studies on detailed flux pinning properties are now undergoing in order to clarify the essential mechanisms for these drastic improvements in Jc both for impurity doping to CuO-chain and Sr-doping. With respect to the Jc in low magnetic fields in the a-growth regions, BaTbO3 was found to be new effective non-superconducting precipitates besides RE211. Addition of Tb4O7 was quite effective to improve the Jc particularly in low magnetic fields at 77 K by generation of finely dispersed BaTbO3 particles. It is noteworthy that addition of Tb4O7 does not decrease Tc contrary to the CeO2 addition. We believe all the methods introduced in this paper will improve the trapping field performance of RE123 melt-solidified bulks and will give effective hints for further improvements in Jc characteristics of RE123 coated conductors.

Acknowledgements The authors sincerely appreciate Mr. Tazaki who engaged in the whole experiments associated to the Lu-doping and the BaTbO3 and the Tb4O7 additions. This study is partly supported by Grants-in-Aid for scientific research of Ministry of Education, Culture, Sports, Science and Technology, by the Sasakawa Scientific Research Grant from The Japan Scientific Society, and by the JSPS Research Fellowship.

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References [1] M. K.Wu, J. R. Ashburn, C. N. Torng, P. H. Hor, R. L. Meng, L. Gao, Z. J. Huang, Y. Q. Wang and C. W. Chu; Phys. Rev. Lett. 58, 908 (1987). [2] S. Jin, R. C. Sherwood, T. H. Tiefel, R. B. van Dover, D. W. Johnson and G. S. Gader; Appl. Phys. Lett. 51, 855-857 (1987); S. Jin, T. H. Tiefel, R. C. Sherwood, M. E. Davis, R. B. van Dover, G. W. Kammlott, R. A. Fastnacht and H. D. Keith; Appl. Phys. Lett. 52, 2074 (1988). [3] M. Murakami, K. Yamaguchi, H. Fujimoto, N. Nakamura, T. Taguchi, N. Kashizuka and S. Tanaka; Cryogenics 32, 930 (1992). [4] M. Daeumling, J. M. Seuntjens and D. C. Larbalestier; Nature 346, 332 (1990). [5] For example, S. I. Yoo, N. Sakai, H. Takaichi and M. Murakami; Appl. Phys. Lett. 65, 633 (1994). [6] G. Krabbes. G. Fuchs. P. Schätzle, S. Gruß, J. W. Park, F. Hardinghaus, G. Stöver, R. Hayn, S. –L. Drechsler and T. Fahr; Physica C 330, 181 (2000). [7] M. Shigemori, T. Okabe, S. Uchida, T. Sugioka, J. Shimoyama, S. Horii and K. Kishio; Physica C 408-410, 40 (2004). [8] S. Uchida, J. Shimoyama, T. Makise, S. Horii and K. Kishio; J. Phys.: Conf. Ser. 43, 231 (2006).

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[9] J. Shimoyama, T. Maruyama, M. Shigemori, S. Uchida, S. Ueda, A. Yamamoto, Y. Katsura, S. Horii and K. Kishio; IEEE Trans. Appl. Supercond. Part 3 15, 3778 (2005). [10] J. L. Tallon, C. Bernhard, U. Binninger, A. Hofer, G. V. M. Williams. E. J. Ansaldo, J. I. Budnick and C. Niedermayer; Phys. Rev. Lett. 74, 1008 (1995).; C. Bernhard, C. Niedermayer, U. Binninger, A. Hofer, C. Wenger, J. L. Tallon, G. V. M. Williams, E. J. Ansaldo, J. I. Budnick, C. E. Stronach, D. R. Noakes and M. A. Blankson-Mills; Phys. Rev. B 52, 10488 (1995). [11] C. J. Kim K. B. Kim, D. Y. Won, H. C. Moon, D. S. Suhr, S. H. Lai, P. J. McGinn; J. Mater. Res. 9, 1952 (1994). [12] R. D. Shannon; Acta Cryst. A 32, 751 (1976). [13] E. M. Gyorgy, R. B. van Dover, K. A. Jackson, L. F. Schneemeyer, and J. V. Waszczak; Appl. Phys. Lett. 55 (1989) 283. [14] Y. Ishii, J. Shimoyama, Y. Tazaki, T. Nakashima, S. Horii and K. Kishio; Appl. Phys. Lett. 89, 202514 (2006). [15] N. Kobayashi, T. Sato, T. Nishizaki, K. Shibata, M. Maki and T. Sasaki; J. Low Temp. Phys. 131, 925 (2003). [16] Y. Xu, M. Suenaga, J. Tafto, R. L. Sabatini, A. R. Moodenbaugh, P. Zolliker; Phys. Rev. B 39, 6667 (1989). [17] T. Egi, J. G. Wen, K. Kuroda, H. Unoki, and N. Koshizuka; Appl. Phys. Lett. 67, 2406 (1995). [18] W. Ting, T. Egi, K. Kuroda, N. Koshizuka and S. Tanaka; Appl. Phys. Lett. 70, 770 (1997). [19] T. Nakashima, Y. Tazaki, Y. Ishii, S. Horii, J. Shimoyama and K. Kishio; CryoPrague 2006 (Proceedings of ICMC '06 and 9th Cryogenics) Vol. 2, 109 (2007). [20] Hu, N. Sakai and M. Murakami; Appl. Phys. Lett.78, 2539 (2002). [21] K. Iida, J. Yoshioka, M. Murakami; Physica C, 357-360, 677 (2001). [22] K. S. Knight, N. Bonanos; Mater. Res. Bull. 30, 347 (1995). [23] T. Oka, Y. Itoh, Y. Yanagi, H. Tanaka, S. Takashima, Y. Yamada and U. Mizutani; Physica C 200, 55 (1992). [24] J. Shimoyama, J. Kase, S. Kondoh, E. Yanagisawa, T. Matsubara, M. Suzuki and T. Morimoto; Jpn. J. Appl. Phys. 29, L1999 (1990).

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Chapter 3

MECHANICAL CHARACTERIZATION AT NANOMETRIC SCALE OF CERAMIC SUPERCONDUCTOR COMPOSITES J.J. Roa∗, X.G. Capdevila and M. Segarra Departamento de Ciencia de los Materiales e Ingeniería Metalúrgica, Facultad de Química, Universidad de Barcelona, C/ Martí i Franqués, 1; 08028, Barcelona (Spain)

Abstract

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The nanoindentation or indenter testing technique (ITT) is a functional and fast technique that can give us a lot of information about the mechanical properties of different materials at nanometric scale, from soft materials, such as copper, to brittle materials, such as ceramics. The principle of the technique is the evaluation of the response of a material to an applied load. In a composite material, if the size of the residual imprint resulting from a certain load is lower than the size of the studied phase, then is possible to determine its mechanical properties, and therefore its contribution to the global mechanical properties of the composite. Depending on the tipped indenter used, different equations should be applied to study the response of the material and calculate stress-strain curves and parameters such as hardness, Young’s modulus, toughness, yield strength and shear stress. These equations are related to the different deformation mechanisms (elastic, plastic or elastoplastic) that the material undergoes. In the case of most of the ceramic composites, when a spherical tipped nanoindenter is used, elastic deformation takes place, and Hertz equations can be used to calculate the yield stress, shear stress and the strain-stress curves. On the other hand, when a Berckovich indenter is used, plastic deformation takes place, then Oliver and Pahrr equations must be applied to evaluate the hardness, Young’s modulus and toughness. Nevertheless, in the hardness study, Indentation Size Effect (ISE) must be considered. In this work, the mechanical properties of a ceramic superconductor material have been studied. YBa2Cu3O7-δ (YBCO or Y-123) textured by Bridgman and Top Seeding Melt Growth (TSMG) techniques have been obtained and their mechanical properties studied by ITT. This material presents a phase transition from tetragonal to orthorhombic that promotes a change in ∗

E-mail address: [email protected]

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J.J. Roa, X.G. Capdevila and M. Segarra its electrical properties, from insulating to superconductor, and that can be achieved by partially oxygenating the material. On the other hand, the structure of the textured material is heterogeneous, and two different phases are present: a Y-123 as a matrix and Y2BaCuO5 (Y211) spherical inclusions. Moreover, the texture process induces an anisotropic structure, thus being the ab planes the ones that transport the superconductor properties while the c axis remains insulating. The purpose of this study is the characterization of the mechanical properties, in elastic and plastic range, of orthorhombic phases of YBCO samples textured by Bridgman and TSMG technique. With the ITT technique, the oxygenation process can be followed and its kinetics established.

1. Introduction High Temperature Superconducting (HTSC) materials can be use in a wide range of applications: (i). (ii). (iii).

related to energy storage as flywheels, current transport devices as cables or fault current limiters, magnetic field related applications as squids, etc.

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The magnetic properties of such materials are well known but the structural or mechanical properties that will allow the construction of real devices are not still completely described. The purpose of this work is to determine this set of parameters and characterize the material behaviour under determined stress conditions. This study will help us to define the optimal dimension of the material pieces in order to resist the stress suffered during its operating life.

1.1. High-Temperature Superconductors (HTSC) Since about 1962 it has become universally recognized that there exists a whole new class of superconductors, type II, which are characterized by the fact that they exhibit a new type of reversible magnetic behaviour. This discovery has made it possible to understand many of the previously unexplained superconducting properties of a number of elements and alloys. Furthermore, it has led to the recognition of the existence of a new thermodynamic state, the mixed state, which is only shown by type II superconductors. In addition to their intrinsic scientific interest they have a technological importance: niobium-zirconium alloys and the compound Nb3Sn have been used in the construction of superconducting solenoids capable of producing steady fields of 50 or 100 kOe. In 1986 Bednoz and Muller [1 and 2], discovered a new type of superconducting materials, now known as high temperature superconductors (HTSCs), which drastically improved the superconducting transition temperature (Tc). Further, YBa2Cu3O7-δ (YBCO or Y-123) was discovered by a group of researchers at the University of Houston in 1987 [3]. The Tc of YBCO material exceeded the boiling point of liquid nitrogen (92 K vs. 77 K). As a consequence, the superconducting products were expected to be operable using liquid nitrogen, which is cheaper and easier to handle than

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Helium; this is one of the most important advantages of HTSCs [4]. Moreover, the next year new superconducting materials were discovered in the Bi (Pb)-Sr-Ca-Cu-O system [5] with a Tc of 110 K and in the Tl-Ba-Ca-Cu-O system [6] with a Tc of 125 K. In 1993 HgBa2Ca2Cu3O9, with a Tc of 134 K was discovered [7], and at present, Tc has reached 164 K, under high pressure [8]. HTSC materials present structural features of ionic crystals. They contain CuO2 planes in their crystal structure. The layers between the CuO2 planes are called the charge reservoir layers. Features of the crystal structure of HTSCs include the following: Crystal structure is layered and the CuO2 layer and charge reservoir layer are stacked periodically. b. The parent material is the antiferromagnetic insulator. By doping electrons or holes to the CuO2 plane from the charge reservoir layer, the CuO2 plane becomes metallic and the superconductivity appears. c. At least one CuO2 plane, where the superconductivity current flows, must be included in the unit cell. d. When the superconductivity appears, the number of oxigens in the CuO2 plane for the number of copper ions is 0.15 to 0.20, and then the effect of the antiferromagnetism is strongly exhibited.

a.

The crystal structure, space group, and lattice constants of YBCO HTSC are summarized in Table 1. Table 1. Crystal data of YBCO HTSC [9].

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Material YBa2Cu3O7-δ (YBCO or Y-123)

Crystal Structure Orthorhombic

a (nm)

b (nm)

c (nm)

0.38177

0.38836

1.16827

Space Group Pmmm

HTSC have unique features such as a high upper critical field, high anisotropy, and an extremely short coherence length, ξ, in addition to the Tc being very high. The physical properties of YBCO are shown in Table 2. Table 2. Physical Properties of YBCO [9]. Material name Critical temperature, Tc (K) Upper critical field, Bc2 (at 0K) [T] Lower critical field, Bc1 (at 0K) [T] Carrier concentration n [cm-3] Coherence length, ξGL (at 0K) [nm] Penetration depth, λ (at 0K) [nm]

YBCO or Y-123 92 674 (//ab) 122 (//c) 0.025 (//ab) 0.085 (//c) 1.5·1022 1.15 (//ab) 0.15 (//c) 142 (//ab) >700 (//c)

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Melt and growth processes for producing Y-123 superconductive oxide [10, 11 and 12] are considered to be an effective process to obtain high critical current densities. Many investigations [13, 14 and 15] were performed to clarify the growth mechanism of Y-123 crystal from the partial molten state where Y2BaCuO5 (Y-211) and liquid phase coexist. Recently, it was found that the peritectic reaction of the Y-123 phase formation proceeded by the solute diffusion between Y-211 particles dispersed in the liquid phase and the growing Y123 interface [13, 14 and 16]. The growth models proposed assuming the mass transfer limiting were suggested, and the growth rate was found to be affected by an interface undercooling as the driving force for solute diffusion [13, 14, 17 and 18]. Therefore, the undercooling is a principle parameter in controlling the growth of Y-123 single crystals.

Figure 1. Phseudoternary phase diagram of YO1.5-BaO-CuO system at 900ºC and oxygen partial pressure of 0.21 atm (From Chemical Processing of Ceramics, Second Edition, edited by Burtrand Lee, Sridhar komarneni; Chapter 23. Synthesis and Processing of High-Temperature Superconductors, page 604)

The inclusion of Y-211 particles has several advantages to the growth and properties of bulk YBCO superconductors: a. Preventing the liquid flow so as to decrease the amounts of holes. b. Shorten space between the Y-211 particles to help growing the Y-123 crystal as well as creating more Y-123/Y-211 boundaries which are the effective flux pinning site.

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Volume fraction of Y-211 particles is approximately constant; the current density, Jc, value is an inverse measure of the mean size of Y-211 phase [19 and 20]. High critical current density values can be achieved by controlling volume fraction and particle size of the Y-211 phase.

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Recent studies of YBCO materials pointed to two major issues: creating pinning centers and eliminating weak-lines between grain boundaries. It has been reported that some defects may act as pinning centers. Pinning strength and critical current density were increased by the introduction of fine Y-211 inclusions [21 and 22]. Flux pinning may be effective by two ways: first, the defects around the Y-211/Y-123 boundary, such as dislocations or stacking faults, and second, the magnetic pinning caused by the different induction generated in Y-123 superconducting matrix and Y-211 non-superconducting phase [23]. However, the weak-link can be caused by impurity phases, micro-cracks, or high-angle misalignment of the crystals [24]. Figure 1 shows the phseudoternary phase diagram of YO1.5-BaO-CuO system at 900ºC and oxygen partial pressure of 0.21 atm. [9]. There are four types of quaternary compounds – YBa2Cu3O7 (Y-123), Y2BaCuO5 (Y-211), Y2Ba8Cu6O18 (Y-143), YBa6Cu3O11 (Y-163) - and five types of ternary compounds – Y2BaO4 (Y-210), Y3Ba3O9 (Y-340), Ba2CuO3 (Y-021), BaCuO2 (Y-011), and Y2Cu2O5 (Y-101).

BaO:YO1.5 1:5

mol% YO1.5

BaO:CuO 3:5

Figure 2. Vertical cross-sectional diagram including Y-123 and Y-211 compounds in pseudoternary phase diagram (From Chemical Processing of Ceramics, Second Edition, edited by Burtrand Lee, Sridhar komarneni; Chapter 23. Synthesis and Processing of High-Temperature Superconductors, page 604).

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Figure 2, shows the vertical cross-sectional diagram including Y-123 and Y-211 compounds [9]. The Y-123 phase is produced by the following ternary peritectic reaction at 1010 ºC: L+Y-211+Y-143 → Y-123

(1)

The peritectic transformation of Y-211 + L → Y-123 proceeds with the solute diffusion through liquid between Y-211 particles and Y-123 interface, and the undercooling acts as a driving force of the diffusion [14 and 17]. All texturing techniques make use of the peritectic reaction where Y-123 phase grows from the Ba- and Cu-rich liquid (BaCuO2 + CuO) and the solid Y-211 phase [25]. From the viewpoint of solute diffusion, the large amount of Y-211 phase can supply more yttrium solute to the growing interface. During heating, the YBCO bulk to a high temperature, CuO reacts with Y-123 to form Y-211 and liquid. CuO + Y-123 Æ Y-211 + L

(2)

The solidification temperature of the liquid is lower than that of the peritectic liquid. Thus no Y-123 grain nucleates at the sample surfaces. Another way to prevent the surface nucleation is to coat the bulk surface with CeO2 powder. CeO2 was reported to react with barium of the Y-123 phase to form BaCeO3 [26] and the reaction is given by:

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CeO2 + Y-123 Æ BaCeO3 + Y-211 + CuO

(3)

As by-products, CuO and Y-211 form. The CuO reacts with Y-211 to produce lower melting point liquid in the same manner as the CuO addition, and hence surface nucleation is suppressed. The peritectic growth of Y-123 phase at typically low growth rates, coupled with the relatively large amount of highly reactive and viscous liquid phase generated during peritectic decomposition, makes it difficult to support bulk samples processed by directional solidification [27]. Melt processing technique of (RE)BCO generally involves slow solidification of a Y-211 and a Ba-Cu-O liquid phase mixture typically between 30 and 40 ºC below the Tp. A crystal that grows with faceted interfaces such as Y-123 crystal needs the driving force of interface kinetics for growth; i.e., it needs the saturation. The saturation or the undercooling kinetics, σ, can be re-written as:

σ=

(C

i

−C L ,Y −123 ) C L ,Y −123

(4)

where Ci and CL,Y-123 are the Y-123 concentration in the growth interface and in the liquid, respectively. The kinetic undercooling, is the difference between the chemical potential of a crystal and its surroundings. This concept is schematically shown in Figure 3.

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a)

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b)

Figure 3. Schematic ilustration of (a) the yttrium concentration profile between the Y-211 particle and the Y-123 interface, and (b) the phase diagram showing the relation between the concentrations and the undercoolings (From Nakamura et al., J. Mater. Res., Vol.11, No. 5, May 1996).

The growth of atomically flat face takes place by a step flow mechanism, the so-called lateral growth. Some probable mechanisms are schematically shown in Figure 4. The Y-211 particle is the yttrium source for Y-123 growth, and the yttrium concentration near the particle is higher than that near the Y-123 (Figure 3). This higher concentration near the Y211 particle causes the concentration on the growing interface to be higher by approaching close to the interface and promotes the two-dimensional nucleation (Figure 4.a). The boundary between the Y-124 crystal and the Y-211 particle on the growing interface can act as the heterogeneous two-dimensional nucleation sites (Figure 4.b), and the entrapment of Y211 particle may field the misfit dislocations behind it (Figure 4.c). These nucleations and/or dislocations act as the step sources. These sites may become dominant when more Y-211 particles are entrapped. Therefore, the undercooling dependence of growth rate of the sample with excess Y-211 phase becomes linear [16].

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a) Y-211

Y-123

b) Y-211

Y-123

c)

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Y-211

Y-123

Figure 4. Schematic illustration showing some possible step sources on the Y-123 interface. a) A twodimensional nucleation resulting from higher concentration with approaching Y-211 close to the interface. b) Heterogeneous nucleation at the Y-211/Y-123 interface and c) A misfit dislocation caused by the entrapment of Y-211 particle. (From Nakamura et al., J. Mater. Res., Vol.11, No. 5, May 1996)

1.1.1. Solidification and Microstructure of YBCO Bulk Materials Fabrication of single crystals usually involves the building of a structure, adding atoms layer by layer. Techniques to produce large monodomains included slowly dragging a rotating seed out of a molten bath of feeder material (as in Crochralski process and the Bridgman technique). Previously to the thermal treatment, we must to have a perform with shape and dimensions close to the final ones. These green bodies are achieved after pressing and sintering stages. The application of pressure varies within methods. It could be in uniaxial form or with an isostatic pressure field, etc. By the other hand, the sintering step could be realised previous to texturing process, such in the Bridgman technique or included in it, such in the TSMG method. Melt processing has been shown to be a suitable technique for the fabrication of bulk YBCO high-temperature superconductors with good flux pinning properties and high critical current densities [28, 29 and 30]. Several variations of melt texturing technique under a variety of names [27] have been developed based on this principle with a view to improving

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either features of the melt process or the quality of the product bulk material. The bulk textured material can be produced using the top-seeded Melt-Growth (TSMG) technique and the Bridgman technique with the purpose to obtain single crystals or monodomains. A monodomain can be described as a monocrystal (no grain boundaries) with numerous defects like secondary phases and mosaicity. Details on the texturation process can be found elsewhere [31]. The final microstructure in both techniques shows a homogeneous distribution of Y-211 particles in the textured Y-123 matrix. It is well-known that the Y-211 inclusions play an important role for the pinning of vortices [25].

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a. Top-Seeded Melt-Growth (TSMG) Technique Top-seeded melt growth (TSMG) is known as the most effective process to fabricate block-type simples used for example in energy storage application, such a superconducting flywheel system, and others. In this technique, too long isothermal step and so great processing times is regarded as one of the most significant weak points. In the case of melt texturing with a temperature gradient, processing time is shorter, but the sample shape and size are restricted. The TSMG technique has been widely used to grow YBa2Cu3OX crystals as large as several centimeters [32], but these crystals were intended for applications such as magnetic levitation. The size of single-domain is generally about several centimeters in diameter, and limited to 10 cm for high quality YBCO bulk up to now, because of the grains mis-orientation during the melt growth process [33]. The resulting cubic centimeter-size YBCO crystals are further annealed to obtain the oxygen-ordered orthorombic phase (x = 6.5). Uniquely, the TSMG process yield large, single grains of approximately the dimensions of the green body [34, 35 and 36]. The TSMG technique has become the preferred method for the fabrication of bulk (RE)BCO superconductors and is used routinely in the processing of single-grain cylindrical/square shape samples of up to 50 mm in diameter [37]. TSMG processing is classified into two types by seeding method; cold seeding and hot seeding. Cold and hot seeding are named for the moment when the seed crystal is placed on the powder compact. In handling the sample, the cold seeding method is easier than hot seeding, because the seeding is performed at room temperature [38]. In this study the seeding method used is cold seeding. In order to obtain large-domain-sized YBCO and to control the growth orientation, seeding effects should be taken into consideration sufficiently [39]. The seed should not only have a similar structure and lattice constant to those of as-grown YBCO bulks, but also control the orientation and ensure single-domain growth [40]. The seed crystal initiates the nucleation and growth of the Y-123 phase in the incongruent melt, which subsequently solidifies into a single grain during controlled cooling. A variety of seed have so far been applied for the melt-textured (MT) growth of YBCO bulks [39], which can be classified into three major categories: • • •

Non-superconductors, such as MgO. Bulk superconductors, such as RE123 MT bulks or single crystal, such as Nd123, Sm-123, and others. RE123 thin films [41].

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In principle, the seed materials should not melt during the texture process, because the maximum process temperature is below their melting points. But the seed crystals were observed to dissolve frequently when they were in contact with the Ba-Cu-O liquid that was formed as a result of the incongruent melting of Y-123 compact during a high temperature holding period [42]. As the seed dissolve during processing, it no longer acts as a seed. In the case of dissolution formed, decreasing the levitation force and trapped magnetic field by reducing the size of the shield current loop. For this reason, the growth mode of YBCO grains is significantly dependent on seed thickness [43]. To obtain the green bodies, the well-mixed powder fabricated by Alcohol polyvinyl method (PVA method) with 1 wt. % CeO2, was uniaxially pressed into pellets with a diameter of 25 mm and thickness of 10-20 mm [44]. Stoichiometric quantities of the oxides and carbonates are weighted to give 69% w/w Y-123, 30 %w/w Y-211 and 1% w/w CeO2. This proportion has been demonstrated to maximize critical current density [45]. The NdBa2Cu3O7 (NdBCO or Nd-123) single crystals were manufactured with the Bridgman technique [49] with a temperature gradient 1-4 ºC/m in vertical direction. The (001) Nd-123 was placed on the top of sample. The Y-123 crystal grew from the Nd-123 seed crystal epitaxially with a squared sharp interface. The growth distance was defined as the length from the edge of the seed crystal to the solid-liquid interface. Figure 5 shows the heat treatment pattern followed in this study. The optimization of the temperature profile is crucial for the production of largesingle-domain YBCO monoliths. The TSMG YBCO samples were fabricated by the following process: The YBCO samples were heated up to 1055ºC at a rate of 300 ºC/h, and held for 2 h for homogenous melting. After that, the samples were cooled to about 1010 ºC at a rate of 30 ºC/h, and further cooled to 1008 ºC at a rate of about 0.02 ºC/h. Then the samples were cooled to room temperature at a rate of 300 ºC/h. Figure 6 shows YBCO samples with tetragonal structure obtained with the thermal treatment explained over (figure a, shows a single crystal and b, shows a poly-crystal). Finally, the as-grown samples were annealed at 450ºC for 300 h (In this step occurs the transformation from tetragonal to orthorhombic phase). Texture process Oxigenation process

Y-211 + L Temperature, ºC

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86

TNucleation = 1010ºC ΔT

Tf,Nucleation = 1008 ºC TOxigenation = 450ºC

Time, h Figure 5. Heat pattern of the experimental procedure in the manufacture of YBCO samples by TSMG technique.

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To define the degree of undercooling, the equilibrium peritectic temperature of Y-123 phase is needed. The temperature has been reported to be 1010 ± 10ºC by many researchers [16].

a)

b)

1 cm

1 cm

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Figure 6. Photograph of the top view of the typical seeded grown sample, a) Single crystal and b) polysingle-crystal.

During the oxygenation process, cracks in the (a,b) planes appeared as a major drawback for the c-axis elements. The occurrence and propagation of cracks was found to be directly related to the oxygenation of the material. The oxygen uptake results in a decrease of the cell unit in the c-direction. The low oxygen diffusion rate yields large oxygen gradient during a classical annealing treatment. The cracks are then an easy diffusion path and the shrinkage of the unit cell along the c-axis at the crack tip provides the driving force for the cracks propagation. Ceramic superconductors exhibit an extensive defect zonology ranging from point-like defects of various natures, such as chain oxygen vacancies or anisette defects in some RE substituted Y-123 compounds, to a variety of extended defects. In addition, Y-123 compounds typically display ferroelectric transition which takes place well above room temperature and results in extensive twin formation. Dislocations, as well as their dissociated configurations are confined onto a prominent glide plane (001), although dislocation lines out of this plane may be frequently found building grain boundaries or as non-assembled segments that have climbed out the glide plane. This high anisotropy of dislocation configurations is consistent with the anisotropy of the crystal structure, the prominent glide plane corresponding to the largest lattice spacing. The melt processed ceramic composites contain a dense population of fine peritectic (non-reacted) particles, which drastically affects the microstructure acting as nucleation sites for dislocations and stacking faults as well as microcrak stoppers thus enhancing the mechanical toughness of the composite [46]. For TSMG techniques, under isostatic pressing conditions, plastic deformation is only possible if there is some kind of anisotropy in the material. In melt textured Y-123

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composites, two kinds of anisotropy exist: elastic anisotropy [47] in the matrix; and plastic anisotropy between the peritectic inclusions and the matrix [48].

a)

b)

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c)

Figure 7. a) TEM micrograph close to the (001)plane of a Y-123 sample submitted to CIP, showing the selective expansion of stacking faults in a particular twin variant. Dislocation loops are indicated by arrows. The selective dissociation of a dislocation is one twin variant is indicated by A. Selective reorganization of a partial loop orientated at the interface of a Y-211 particle is indicated by B, b) Schematic drawing of the reorganization of a partial loop driven by CuO fluxes from twin variant 2 to twin variant 1 as indicated by arrows, and c) Interaction of a gliding perfect dislocations with a partial loop and resulting selective dissociation in twin variant 1. (From Sandiumenge et al., Advanced Materials, 12, No.5, 2000)

The main microstructural features are shown in figure 7a. The two dark areas in the left side and the upper right corner are Y-211 peritectic inclusions. Several dislocations oriented along (001) appear attached to the left hand side inclusions. It can be observed that the lower

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dislocations (indicated by A) is decorated with stacking faults appearing at every other twin domain. Similarly, stacking faults associated to the Y-211 particle located at the upper right corner of the micrograph (indicated by B) are observed to expend selectively on one twin domain. This distinction is only relevant to the orthorhombic lattice. Figures 7b and 7c are schematic drawings of the basic mechanism associated to the observed microstructural modifications. Another way to obtain green bodies used for TSMG is High Oxygen Pressure Processing. In this case, TEM observations have revealed a strong increase in the density of stacking faults nucleated at the interface of the Y-211 particles. Typical observations are presented in Figure 8a and 8b. The main distinguish features is the development of dendritic-like morphologies of the stacking faults.

a)

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b)

Y-211

Figure 8. a) Typical TEM image of dendritic-like stacking faults generated at the interface of Y-211 particles. Imaging conditions are such that the stacking faults appear dark in order to emphasize their irregular shape. b) Enlarge view where the stacking fault contrast is avoided showing the high density of partial dislocation associated to the Y-211 interface (From Puig et al., Applied Physics Letters, Vol. 75, No. 13, 1999, 1952-1954).

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b. Bridgman Technique The Bridgman technique is a method single crystal ingots or boules. It is a popular method of producing certain semiconductor crystals, such as gallium arsenide, II-V Crystals (ZnSe, CdS, CdTe) and BGO, where the Crochralski process is more difficult. The method involves heating polycrystalline material in a container above its melting point and slowly cooling it from one end where a seed crystal is located. Single crystal material is progressively formed along the length of the container. The process can be carried out in a horizontal or vertical geometry. In this study a modified Bridgman technique have been used. In this case, the sample was introduced into the furnace at one determined thermal gradient and a external Superconducting YBCO bars were prepared by a modified Bridgman process. After heating the presintered YBCO bars well above the peritectic temperature, Tp, the semisolid bars were displaced at a constant rate of 1 or 2 mm·h-1 through a region having an axial temperature gradient of 20 K·cm-1 at Tp until the full length of the bars was cooled down to 900ºC. Details of the sample preparation are given in [49]. The thermal treatment can be observed in figure 9.

Temperatue, ºC

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T2

Tp = 1010 ºC T1

To to

Time, h

Figure 9. Thermal treatment of bulk superconductor samples textured by Bridgman technique.

The two main problems found when obtaining YBCO samples with this technique are: ¾ The first one is related to the different liquid properties. The optimal viscosity have to be found between minimize the sample flowing and maximize atomic diffusion. ¾ It is well known that the liquid phase generally migrates to the cold zone. In this case, the bar loss the correct stochiometry and in the end of the bar have a rich Y-211 zone.

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After the texture process, the YBCO bars were introduced in an oxygenation furnace at 450ºC for 240 h. Figure 10 shows a schematic drawing and optical micrograph (OM) of YBCO bars textured by Bridgman technique. By means of polarized light microscopy an initial region can be observed where polynucleation and growth competition phenomena take place. This competition region has a length which is inversely proportional to the processing rate, thus indicativy that the nucleation is promoted by an enhanced undercooling [50]. In the single domain region the c-axis of Y-123 has a tilt angle of 45º respect to the long axis of the bars.

Syngle-Crystal (45º)

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a)

b)

300 μm Figure 10. a) OM and b) schematic drawing showing from bottom to top, a multinucleation region, a growth competitive region and a single domain region of a Bridgman melt-grown sample. (Figure 10.b, from Ullrich et al., Materials Science & Engineering B, 53, 1998, page 143-148).

Figure 11 shows a TEM micrograph of a Bridgman sample. In this picture, a high quality dislocations can be observed in the matrix of this material. The dislocation substructure of Y-123 is highly anisotropic, being confined to the (001) plane. Lubenets et al. reported that strong covalent and ionic bonds create high Peierls barriers, which constrain the dislocation mobility in YBCO single crystals [51].

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Dislocation

200 nm

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Figure 11. TEM micrograph of a Bridgman sample observed of a region containing dislocations in the maximum anisotropy plane (001 plane, ab).

Residual Stress

Twins

1 μm Figure 12. TEM micrograph of Bridgman sample of a region containing several precipitates with different sizes. Note that several twins and residual stress produced during the texture process are present into the precipitates.

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The micrograph presented in Figure 12 also shows lower amounts of dislocations placed preferentially in the grain boundary between Y-123 and Y-211. Moreover, the trajectories of the dislocations appear to be either unaffected or changed across the twin boundaries. On the other hand, this micrograph presented a higher amount of twins inside the precipitates. These effects could be due to two different factors: the compressive strain during the cooling treatment in the texture process and the different thermal expansion coefficients between the matrix and the particles (from 20ºC to 900ºC are 1.24·10-3 K-1 and 1.70·10-3 K-1 for Y-211 and Y-123, respectively [46]). Finally, we observe twins and residual stress inside the Y-211 particles. Stresses associated to the Y-211/Y-123 interface can arise from two different mechanisms. The first takes into account the thermal expansion and elastic modulus mismatch between the two phases [46].Secondly, stress is thought to result from the incorporation of Y-211 decomposition products into the matrix. Below the peritectic temperature, there is a thermodynamically driving force for solid-state dissolution of precipitates embedded in the matrix. Under uniaxially pressing conditions, plastic deformation is only possible if there is some kind of anisotropy in the material. In melt textured Y-123 composites, two kinds of anisotropy exist: elastic anisotropy [46] in the matrix, and plastic anisotropy between the peritectic inclusions and the matrix.

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Dislocation

Residual Stress

0.5 μm Figure 13. TEM micrograph of a sample textured by Bridgman technique shows a precipitate with residual stress and dislocation. Both defects are due to the compressive strain produced during the texture process of material.

In Figure 13, inside the Y-211 particle a dislocation can be observed, as well as in the middle of the particle a microcrack and a residual stress at the corners of the particle produced during the texture process. At the grain boundary between Y-211 and Y-123 phases, a fine dislocation line surrounding the particle can be observed. If we observe the matrix we will find a great amount of dislocations.

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1.1.2. Oxygenation Process

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Although in the standard melt-grown YBCO microcracks also appear in the vicinity of Y211 particles due to thermal expansion difference of Y-211 and Y-123 [52], these microcraks do not propagate across the Y-211 precipitates, and the length of such microcraks is limited. The Y-211 particles are under compression when the material is cooled and can therefore break the Y-123 matrix along ab-planes [53]. These are the intrinsic microcraks, which are generated during the tetragonal-orthorombic phase transformation. Large secondary phase inclusions (unreacted liquid and Y-211) can also create macrocraks due to thermal expansion mismatch. Oxygen annealing, necessary to make Y-123 samples superconducting, is reported to be responsible for further macro and microcracking [53]. That is why oxygen concentration gradients lead to large mechanical stress in the material. The oxygenation diffusion coefficient in the ab-plane is about 104-106 times larger than in c-direction [53]. Figure 14 shows a micrograph of Y-123 observed with scanning transmission electron microscopy of a YBCO samples textured by Bridgman technique. We can observe a twin produced during the oxygenation process.

Twin

0.2 μm Figure 14. TEM micrograph of YBCO sample textured by Bridgman technique of a twin present in the matrix (Y-123) in the maximum anisotropy plane (001).

Figure 15 shows a high resolution TEM (HRTEM) micrograph viewed along the grain boundary between Y-123 and Y-211 phases. In this micrograph we can observe, along the (001) plane, the transition from a well ordered region (Y-123) to a highly defective one (Y-211). The distance between crystallographic planes is 7 Ǻ. We can also observe the high anisotropy in the Y-211 phase. In the case of growing the matrix (Y-123) on Yttrium substrates, the copper oxide rich liquid reacts with the substrate to form precipitates of Y-211

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at the enriched interface [54], such that subsequent nucleation of the Y-211 liquid interface [55]. The mechanism would explain the anisotropic orientation distribution of Y-211 observed in figure 14. On the other hand, for Y-123 growing directionally from the melt, the particle engulfment process is governed by the velocity of the advantaging interface, melt viscosity and particle size. For anisotropic materials, such as Y-123 and Y-211 crystals, we can expect the above process to be governed by a complex relation, between orientation and size of the particle. These considerations suggest that the incorporation of Y-211 particles into the bulk Y-123 would be favoured for particular orientations if the growth rate is parallel to ab planes.

Y211 Y123/Y211

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Y123 5 nm Figure 15. HRTEM image of a Y-211/Y-123 interface viewed along the [001] direction showing a transition from ordered zone (Y-123) to a highly defective zone (Y-211). Distance between cristalographic planes in the Y-123 zone is 7 Å.

YBCO pellets textured with TSMG and/or Bridgman technique have different cracking mechanisms [53]: a) Mechanical stress Macrocraks in textured pellets appear during the final cooling stage to room temperature. Three major causes for large mechanical stresses, schematically illustrated in Figure 16, are considered at this stage: •

An increasing Y-211 concentration is observed with increasing resistance from the seed in the c-growth sector while it remains relatively homogeneous in the ab-growth sector. Thermal expansion coefficients (in ab- and c- direction) of an Y-123/Y-211 composite are a function of the Y-211 content. The c-axis stress due to Y-211

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• •

inhomogenities described above can reach 50-100 MPa in the center of the pellet and will tend to open ab-plane macrocracks. During cooling a thermal gradient builds up with a colder surface and a hotter bulk due to relatively low thermal conductivity of Y-123. During to cooling to room temperature it takes up oxygen mostly in ab-direction and in vicinity of the surface.

a)

b)

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c)

Figure 16. Schematic of mechanical (compressive or tensile) stresses associated with a) the Y-211 distribution into a pyramid pattern, b) thermal gradient during cooling and c) oxygen gradient ( From Isfort et al, Physica C 390, 2003, 341-355).

b) Propagation of cracks during oxygenation [56] Cracking due to the oxygenation uptake of the surface does not seem to damage the sample severely, as the crazing is limited to the surface. The mechanism can be described as follows: In a first stage stress is building up at the exterior surface which creates the regular crazing pattern. Oxygen will then penetrate in the cracks and oxygenate their walls. The contraction in c-direction due to the oxygen uptake will then create a stress field around the crack tip that tends to make the crack progress inside the material (See Figure 17).

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σσ σ σ σ σσ σ σ YBCO σσσσσ O2

σσσ σσ σ σ σ σ σσ σ σ σ Tensión Stress

Figure 17. Schematic of mechanism of propagation of cracks during oxygenation (From J.J. Roa et al., Análes de Mecánica de la Fractura, 25, 2008, Spain, ISSN-0213-3725).

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1.1.3. Applications At present, many applications using melt-textured YBCO are currently discussed [57]. The first demostrators, e.g. motors, flywheels, and others, have already been built. The HTSC capability to transport electric current without any losses, together with low thermal conductivity, suggest the application of HTSC for high current transport to low temperature SC devices such as magnets. The fault current limiter (FCL) seems to be the most promising superconducting power device that will be installed in the electric power networks. The higher levitation force of the YBCO bulk means that it can be used for various applications, such as non-contacted superconducting bearing [58], flywheel [59], magnetic levitation transport system and motors [60]. The application is mainly dependent on the physical properties of the YBCO bulk, such as levitation force and others. Although YBCO compound is one of the most widely studied superconducting materials, bulk YBCO superconductors are brittle and exhibit poor mechanical properties (strength and fracture toughness) [61]. Bulk textured Y-123 has to be considered as a brittle composite material due to the presence of micro-sized Y-211 inclusions in the Y-123 matrix. The most important problem of the superconductor materials is their poor mechanical properties. However, can become as important if one takes into account the stresses appearing in practical service due to the mechanical action caused by magnetic and/or thermal cycling between room and liquid nitrogen temperatures [57]. These poor and unknown mechanical properties limit the performance of melt-textured YBCO in manufacture applications. The purpose of this study is the characterization of the mechanical properties, in elastic and plastic range, of orthorhombic phases of YBCO samples textured by Bridgman and TSMG technique. With the ITT technique, the oxygenation process can be followed and its kinetics established.

1.2. Indentation Testing Technique Indentation or hardness testing has long been used for characterization and quality control of materials, but the results are not absolute and depend on the test method. In general,

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traditional hardness test consist of the application of a single static force and corresponding well time with a specified tip shape and tip material, resulting in a hardness impression that has dimensions on the order of nanometers or micrometers depending on the applied load. The output of these hardness testers is typically a single indentation hardness value that is a measure of the relative penetration depth of the indentation tip into the sample. Actually, there are a lot of durometers which are used to characterize the mechanical resistance of materials, such as in polymer materials, that have different spring constants and either a flat conical tip, a sharp conical tip, or a spherical tip, as specified in ASTM D 2240, Standard Test Method for Rubber Property-Durometer Hardness. The deformation of materials occurs via two distinct processes: elastic (reversible) and plastic (irreversible) deformation. Since elastic formation is a reversible process, and is governed by angstrom scale (10-10 m), interaction parameters such as the crystallographic lattice constants, elastic deformation of materials exhibit virtually no size dependence unless a large population of preexisting deffects is involved [62]. The plastic deformation response, which occurs as a result of the generation, annihilation, and motion of deffects such as dislocations, displays marked size effects when those material dimensions are in the range of microns or below. Instrumented indentation testing (ITT), also known as depth-sensing indentation, continuous-recording indentation, ultra-low-load indentation, and nanoindentation, is a relatively new form of mechanical testing that significantly expands on the capabilities of traditional hardness testing. The past two decades, ITT employs high-resolution instrumentation to continuously control and monitor the loads and displacements of an indenter. The method was introduced in 1992 for measuring hardness and elastic modulus by instrumented ITT and has widely been adopted and used in the characterization of mechanical behaviour of materials at nanometric scales [63 and 64]. The principal advantage of this technique is that the mechanical properties can be determined directly from indentation load and displacement measurements or also from load-unload curves without the need to image the hardness impression. For this reason, this method has become a primary technique for the determination of the mechanical properties of thin films and small structural features [65, 66 and 67]. ITT provides information about composite materials when the particles in the matrix have a lower size than the residual indentation imprints. This technique, also, permits the study of the mechanical properties of monolithic samples [68]. During the past decade, ITT has introduced several important changes to the method that both improve its accuracy and extend its real of application. The changes have been developed both through experience in testing a large number of materials and by improvements to testing equipment and techniques. Some of this changes were [68]: new methods for calibrating indenter area functions and load frame compliance, the measurement of contact stiffness by dynamic techniques allowing continuous measurement of properties as a function of depth, and others. A nanoindenter is a type of an indentation instrument tester, which means that load and total depth of penetration are measured as function of time during loading and unloading. Depending on the details of the specific testing system, loads as small as 1 nN can be applied, and displacements of 0.1 nm (1 Å) can be measured. Mechanical properties, such as: hardness, Young’s modulus, toughness, yield strength, shear stress,…, can be obtained with the load-displacement data.

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The technique most frequently employed measures the hardness, and the elastic modulus or Young’s modulus [63 and 69]. This technique, also permits evaluating the yield stress and strain-hardening curves of metals [70], characteristic parameters of damping and internal friction in polymers, such as the storage and loss modulus and the activation energy and stress exponent for creep [71 and 72]. Mechanical properties are routinately measured from submicron indentations, and with careful technique, properties have even been determined from indentating only a few nanometers deep. Many ITT testing systems are equipped with automated specimen manipulation stages. In these systems, the spatial distribution of the near-surface mechanical properties can be mapped on a point-to-point basis along the surface in a fully automated way.

1.2.1. Testing Equipment (Instrumentation) Many instrumented indentation systems can be generalized in terms of the schematic illustration shown in figure 18 [73].

Load application device

Springs

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Probe Tip Displacement Sensor

Sample Load frame

Figure 18. Schematic illustration of an instrumented indentation system.

As shown in the figure, equipment for performing instrumented indentation tests consists of three components: a) an indenter of specific geometry usually mounted to a rigid column through which the force is transmitted, b) an actuator for applying the force, and c) a sensor for measuring the indenter displacements. However, to date, most ITT development has been performed using instruments specifically designed for small-scale work. Advances in instrumentation have been driven by

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technologies that demand accurate mechanical properties at the micron and submicron levels, such as the microelectronic and magnetic storage industries.

1.2.2. Nanoindenter’s Tips A variety of indenters made from a variety of materials are used in ITT testing. Diamond is probably the most frequently used material because its high hardness and elastic modulus minimize its contribution to the measured displacement from the indenter. Indenters can be made of other less-stiff materials, such as sapphire, tungsten carbide, or hardened steel. The indenters can be classificated in four different groups: •

Pyramidal indenters

The most frequently sharp indenter in nanoindentation technique is the Berkovich indenter. This indenter presents a three-sided pyramid with the same depth-toarea relation as the four-sided Vickers pyramid used commonly in microhardness work. With this indenter the hardness and the Young’s modulus can be determined.

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•

Spherical indenters

For spherical indenters, the contact stress is initially small and produces only elastic deformation. As the spherical indenter is driven into the surface, a transition from elastic to plastic deformation occurs, which can theoretically be used to examine yielding and work hardening, and to recreate the entire uniaxial stressstrain curve from data obtained in a single test [74 and 75]. At the micron scale, the use of spherical indenters has been impeded by difficulties in obtaining high-quality spheres made from hard, rigid materials. •

Cube-Corner Indenters

A three-sided pyramid with mutually perpendicular faces arranged in geometry like the corner of a cube. The center-line-to-face angle for this indenter is 34.3º whereas for the Berkovich indenter it is 65.3º. The sharper cube corner produces much higher stress and strains in the vicinity of the contact, which is useful, for example, in producing very small, well defined cracks around hardness impressions in brittle materials; such cracks can be used to estimate the fracture toughness at small scales [76]. Also, the toughness of brittle materials can be determined with a Berkovich indenter. •

Conical Indenters

The conical indenter is also attractive because the complications associated with the stress concentrations at the sharp edges of the indenter are absent. Curiously, very little ITT testing has been conducted with cones. The main reason is that it

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is difficult to manufacture conical diamonds with sharp tips, making them of little use in nanoindentation technique [77]. The most important indenters used in nanoindentation technique are the Berkovich and Spherical indenter. It allows to characterize the plastic (hardness, Young’s modulus and toughness fracture) and the elastic (yield strength, mean contact pressure, shear stress and stress-strain curves) deformation. A typical Berkovich and Spherical indenter tip are shown in figure 19.

200 μm

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a)

500 μm b) Figure 19. High magnification SEM scan of the most important indenters used in nanoindentation technique, a) Berkovich indenter and b) Spherical indenter.

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1.2.3. Good Experimental Practice As in any experimental work, accurate measurements can be obtained only with good experimental technique and practice. The most important factors that can be controlled in a nanoindentation test are shown below: a)

Choosing an appropriate indenter

Choosing an appropriate indenter requires consideration of a number of factors. One consideration is the strain the tip imposes on the test material. Although the indentation process produces a complex strain field beneath the indenter, it has been proven to be useful to quantify the field with a single quantity, often termed the characteristic strain, ε [78]. There are problems, however, in obtaining accurate measurements of hardness and elastic modulus with cube-corner indenters [78]. Although not entirely understood, the problems appear to have two separate origins. First, as the angle of the indenter decreases, friction in the specimen-indenter interface and its influence on the contact mechanics becomes increasingly important. Second, the relation among the contact stiffness, contact area and effective elastic modulus. Corrections are required, and the magnitude of the correction factor depends on the angle of the indenter. The spherical indenter can be used when one wishes to take advantage of the continuously changing strain. In principle, one can determine the elastic modulus, yield stress, and strain-hardening behaviour of a material all in one test.

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b)

Environmental control

To take full advantage of the fine displacement resolution available in most ITT testing system, several precautions must be taken in choosing and preparing the testing environment. Uncertainties and errors in measured displacements arise from two separate environmental sources: vibration and variation in temperature that cause thermal expansion and contraction of the sample and testing system. To minimize vibration, testing systems should be located on quiet, solid foundation (ground floors) and mounted on vibration-isolation system. Thermal stability can be provided by enclosing the testing apparatus in an insulated cabinet to thermally buffer it from its surroundings and by controlling room temperature to within ± 0.5ºC. c)

Surface preparation

Surface roughness is extremely important in instrumented indentation testing because the contact areas, from which mechanical properties are deduced, and calculated from the contact depth and area function, on the presumption that the surface roughness depends on the anticipated magnitude of the measured displacements, and the tolerance for uncertainty in the contact area. The greatest problems are encountered when the characteristic wavelength of the roughness is comparable to the contact diameter.

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Testing procedure

To avoid interference, successive indentations should be separated by at least 20 to 30 times the maximum depth when using a Berkovich or Vickers indenter. For other geometries, the rule is 7 to 10 times the maximum contact radius. The importance of frequently testing a standard material can not be over emphasized. e)

Detecting the surface

One very important part of any good IIT testing procedure is accurate identification of the location of the specimen’s surface. This is especially important for any small contacts, in other words, when the applied load is very low, for which small errors in surface location can produce relatively large errors in penetration depth. For hard and stiff materials, such as hardened metals and ceramics, the load and/or contact stiffness, both of which increase upon contact, are often used. However, for soft, compliant materials, like polymers and biological tissues, the rate of increase in load and contact stiffness is often too small to allow for accurate surface identification. In these situations, a better method is sometimes offered by dynamic stiffness measurement [79 and 80].

1.2.4. Experimental Techniques

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a. Hardness and elastic modulus measurements The analysis of force-displacement or load-unload curves produced by instrumented indentation system is often based on work by Doerner and Nix [81] and Oliver and Pharr [73]. The two mechanical properties measured most frequently by ITT methods are hardness and elastic modulus or Young’s modulus with a Berckovih indenter. For materials that do not experience pile-up, which includes most ceramics, hard materials, and soft metals that work harden, these mechanical properties can be determined generally within ± 10 %, sometimes better. The method was developed to measure the hardness and elastic modulus of a material from indentation load-displacement data obtained during one cycle of loading and unloading. Although it was originally indented for applications with sharp indenter, like Berckovich. A schematic representation of a typical data set obtained with a Berkovich indenter can be observed in figure 20, where P designates the load, and h the displacement relative to the initial undeformed surface. The load-displacement curve shows the elastic/plastic behaviour of each sample. From the difference between total indentation depth at maximum indented load (ht) and depth of residual impression upon loading (hf), the elastic recovery can be calculated [82]. In figure 20, there are four important quantities that must be measured: • •

the maximum load, Pmax, the maximum displacement, hmax,

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•

the elastic unloading stiffness, S = dP/dh, defined as the slope of the upper portion of the unloading curve during the initial stages of unloading. The parameter S has the dimensions of force per unit distance, and is known as the elastic contact stiffness, or more simply, the contact stiffness, and the final depth, hf.

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Figure 20. Schematic illustration of indentation load-displacement data showing important measured parameters (From J. J. Roa et al., Nanotechnology, 18, 2007, 385701/1-385701/6).

The characteristics of figure 20 and such experimental results are summarized as follows [62]: The initial load-displacement (P-h) response is elastic and can be described by continuum level contact mechanics. The first departure from this elastic response occurs when the local maximum shear stress level sustained by the indented material is on the order of the theoretical shear strength of material. Subsequent to this initial plastic event, a series of similar discontinuities in the P-h response occurs. Although the fundamental mechanisms responsible for the experimentally observed discrete deformation processes under this nanoscale contact are still debated in the literature. The loading response of the material will show, therefore, whether if the indentation probe is blunt o sharp, producing an elastic initial response or an elasto-plastic response. Once the loading is sufficiently high, both indenters will produce similar elasto-plastic response in the material. It has to be taken into account that all real sharp indenters have a tip curvature which translates into a blunt indentation at the initial contact depths. The analysis of the P-h curves depends if it is a loading curve (which can be elastic or elasto-plastic) or a unloading curve, where the material is usually deformed.

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The accuracy of the mechanical properties in plastic deformation range (hardness and Young’s modulus) depends on how well these three parameters can be experimentally measured. The last parameter is the permanent depth of penetration after the indenter is fully unload; in other words, with this value the plastic work necessary to deform the material of the study can be calculated. The exact procedure used to measure H and E is based on the unloading process shown schematically in figure 21, in which it is assumed that the behaviour of the Berckovich indenter can be modelled by a conical indenter.

Figure 21. Schematic illustration of the unloading process showing parameters characterizing the contact geometry (From J. J. Roa et al., Nanotechnology, 18, 2007, 385701/1-385701/6).

The fundamental relations from which hardness and Effective Young’s modulus or reduced Young’s modulus are determined as follows:

H=

Pmax A

(5)

where Pmax is the maximum applied load and A is the projected contact area at that load. The hardness valued is a measure of the load-bearing capacity of the contact computed by dividing the applied load by the projected area of contact under load. This should not be confused with the more traditional definition of hardness, the load divided by the projected area of contact of the residual hardness impressions. These two different definitions of hardness yield similar values when the plastic deformation process dominates and a fully plastic permanent impression is formed. However, they give very different values when contact is predominantly elastic, because for purely elastic contact, the residual contact is vanishingly small, giving an infinite hardness based on the traditional definition. The Effective Young’s modulus can be determined as a function of S, A and a constant which depends on the geometry of the indenter.

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E eff = f (S , A, β )

(6)

π S ⋅ 2⋅β A

(7)

Equation 6, can be re-written as:

E eff =

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Equation 7 is found in elastic contact theory [86] and holds for any indenter that can be described as a body revolution of a smooth function [73]. Because this equation was derived for an axysimetric indenter, it formally applied only to circular contacts, for which the indenter geometry parameter is β = 1. However, it has been shown that the equation works equally well when the geometry is not axysimetmetric, provided that different values of β are used [83, 84 and 85]. For indenters with square cross sections like the Vickers pyramid, β = 1.012, for triangular cross sections like the Berkovich and the cube-corner indenters, β = 1.034 [84]. This factor plays a very important role when accurate property measurements are desired. This constant affects not only the elastic modulus calculated from the contact stiffness by means of equation 7, but the hardness as well because procedures for determining the indenter area function are also based on equation 7, and area functions can be mistaken if the wrong value of β is used. An effective modulus, Eeff, is used in equation 7 to account for the fact that elastic displacements occur in both the indenter and the sample. The elastic modulus of the studied material, E, is obtained from Eeff, using the next equation:

1 1 − ν 2 1 − ν i2 = + E eff E Ei

(8)

where ν is the Poisson’s ratio for the test material, and E is the Young’s modulus. The subindex i denote the values of the indenter. For diamond indenter, the elastic constants Ei =1141 GPa and νi = 0.07 are often used [63 and 68]. While it may seem counterintuitive that one must know the Poisson’s ratio of the studied material in order to calculate its Young’s modulus using the equation 8, even a rough estimate, say ν = 0.25 ± 0.1, produces only about a 5% uncertainty in the calculated value of E for most materials. The most important thing is to determine the contact stiffness and the contact area as well as possible. From equation 5 and 7, it is clear that, in order to calculate the hardness and elastic modulus of the studied material from indentation load-displacement curves, an accurate measurement of the S and the A must be performed. One of the main distinctions between IIT and conventional hardness testing is the way to obtain the contact area. Rather than by imaging, the area is established from an analysis of the indentation load-displacement data. The analysis used to determine the hardness, H, and the elastic modulus, E, is essentially an extension of the method proposed by Doener and Nix [69] that accounts for the fact that unloading curves are distinctly curved in a manner that cannot be accounted for by that flat punch approximation. Doener and Nix consider that the contact area remains constant as the

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indenter is withdraw, and the resulting unloading curve is linear. The Oliver & Pahrr method begins by fitting the unloading portion of the load-displacement curve to the power-law relation:

P = B ⋅ A ⋅ (h − h f

)

m

(9)

where Β and m are power law fitting constants [63], and hf is the final displacement after complete unloading, also determined from the curve fit. The S is established by analytically differentiating equation 9; and evaluating the results at the maximum depth of penetration, h = hmax, that is:

S=

dP dh

(10) h = h max

Differentiating equation 9, the S can be obtained with the following equation:

S = B ⋅ m ⋅ (hmax − h f )

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m −1

(11)

It is thus prudent, when the S has been calculated with equation 11, to fit only the upper portion of the unloading curve; moreover, the value of S determined from this fit should be checked by comparing the curve fit to the data. Fitting the upper 25 to 50% of the data is usually sufficient. Now, the contact depth, hc, has been calculated, which for elastic contact is less than the total depth of penetration (hmax) as illustrated in the figure 21. The basic assumption is that the contact periphery sinks in a manner that can be described by models of indentation of a flat elastic half-space by rigid punches of simply geometry [86]. This assumption limits the applicability of the method because it does not account for the pile-up of material at the contact periphery that occurs on some elastic-plastic materials. Assuming, however, that pileup is negligible, the elastic models show that the amount of sink-in, hs, is given by the next equation:

hs = ε ⋅

Pmax S

(12)

where ε is a constant that depends on the geometry of the indenter. Typical values are: 0.72 for a conical punch indenter, 0.75 for a parabolic of revolution and 1 for flat punch [68]. From the geometry of figure 21, the depth along contact is made between the indenter and the specimen, hc, can be calculated by:

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hc = hmax − hs

(13)

Using equation 12 and 13, the contact between the indenter and the specimen is:

hc = hmax − ε ⋅

Pmax S

(14)

The projected contact area is calculated by evaluating an empirically determined indenter area as a function of A = f (d) [68] at the contact depth hc; that can be re-writen as:

A = f (hc )

(15)

The area function, also known as the shape function or tip function must be carefully calibrated by independent measurements, so that deviation from non-ideal indenter geometry are taken into account. These deviations can be quite severe near the tip of the Berkovich indenter, where some rounding inevitably occurs during the grinding process.

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b. Spherical indentation (Determination of the stress-strain curves) In the past 20 years, instrumented nanoindentation experiments have emerged as a powerful tool in understanding the mechanical behaviour of solids in general, and single crystals and thin films in particular. However, since the majority of the work has been carried out using Berkovich indenters, the emphasis has been put on extracting the hardness and the Young’s modulus of the material [68]. Berkovich indenter is quite sharp and results in plastic deformation almost instantly; consequently, much of the information about the purely elastic region and, as important, the elastic to plastic transition is lost, a fact that has long been appreciate. Given that the conversion of load-displacement curves to indentation stress-strain curves is almost as old as the technique of using indentations to probe the mechanical properties of solids, it is somewhat surprising that this conversion is not much more common than it is. This comment notwithstanding, there have been a number of papers in which spherical nanoindenters have been used [87]. Roughly a decade ago, Field and Swain suggested a method to extract indentation stress-strain curves from load-displacement curves [74 and 75]. Before acquiring the Continuous Stiffness Measurement (CSM) option, Field and Swain method was used to convert load-displacement results obtained on Ti3SiC2 [88], and single crystals of mica [89] and graphite [90], loaded parallel to the c-axis to indentation stressstrain curve. Typically, a nanoindentation test results in the load, P, and displacement into surface, ht, data. Additionally, the CSM attachment provides the harmonic contact stiffness, S, values over the entire range of loading. The elastic contact between two bodies was first described by Hertz in 1882, and the equations describing this response are usually known as Hertzian equations. Hertzian indentation has been applied to ceramic systems with heterogeneous microstructures, where an intermediate form of damage is observed.

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The vast majority of spherical nanoindentation analysis is based on the Hertz equation in the elastic region [68, 74 and 75].

P=

3 1 3 ⋅ E eff ⋅ R 2 ⋅ he 2 4

(16)

where R is the radius of the indenter, he is the elastic distance into the surface (see figure 22), and Eeff is the effective modulus given by equation 8.

Figure 22. Schematic representation of spherical indentation (From S. Basu et al., J. Mater. Res., 2006, Vol. 21, No. 10, page 2628-2637).

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For a rigid spherical indenter, Sneddon [86] showed that the elastic displacements of a plane surface above and below the contact circle are equal, and given by:

a3 he = ht = R

(17)

where a is the contact radius during the indentation, see figure 22. Combining equation 16 and 17, next equation can be obtained:

po =

3 ⎛a⎞ ⋅ Eeff ⋅ ⎜ ⎟ 4 ⋅π ⎝R⎠

(18)

The left side of equation 18 represents the indentation stress or mean contact pressure, also referred to as the Meyer hardness [91]. The expression in parentheses or a/R on the right side represents the indentation strain [91]. With the spherical indenter two different mechanisms can be studied: the elastic and elasto-plastic mechanism.

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Elastic regime

Both the Oliver and Pharr [68] and Field and Swain [75] methods use the slopes of the initial portions of the unloading curves, dP/dh, to calculate he. Differentiating equation 16 with respect to h can yield: 3 1 dP = 2 ⋅ E eff ⋅ R 2 ⋅ he 2 dh

(19)

which when substituted this equation in equation 16, results in

P=

2 dP ⋅ ⋅ he 3 dh

(20)

he =

dh 3 ⋅P⋅ dP 2

(21)

Therefore,

Since dP/dh is nothing but the stiffness, S*, of the system composed by the specimen and the load frame, the stiffness of the material itself can be calculated from next equation:

1 1 1 = *− S S Sf

(22)

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where Sf is the load-frame stiffness. This value is obtained from the manufacturer of the instrument. ii.

Elasto-plastic regime

Again following the Oliver and Pharr [68] and Field and Swain [75] methods, it can be assumed that the contact depth, hc, can be defined as the distance from the circle of contact to the maximum penetration depth, see figure 22, to be given by the next equation:

he 2

(23)

3 P ⋅ 4 S

(24)

hc ≈ ht − Combining equations 22 and 23 yields:

hc = ht −

Equation 24, can be modified and re-written as follows:

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Mechanical Characterization at Nanometric Scale…

hc = ht −

3 P ⋅ +δ 4 S

111

(25)

where δ is an adjustable parameter of the order of a few nm needed to obtain the correct elastic moduli. Once hc is known, a can be calculated with the next equation:

a = 2 ⋅ R ⋅ hc − hc2 ≈ 2 ⋅ R ⋅ hc

(26)

Equation 26 is only valid for hc he/2 it follows that hc ≈ ht (equation 23) [91].

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c.

Fracture toughness measurement

Ceramics are generally brittle and prone to generation of cracks when indented. Toughness estimation by microfracture Vickers indentation is a well-known and broadly employed technique in ceramic materials. This technique consist of the application of a Vickers or Berkovich indenter at a given load to the material sufficiently high to nucleate cracks at the corners of the imprint, and further measure the crack lengths produced at the corners of the imprint, c, in order to evaluate the fracture toughness of the studied material. The fracture toughness of brittle bulk materials, an important measure of the resistance of these materials to fracture and crack propagation, can be evaluated through conventional microindentation using a microindenter. However, because of the bluntness of the microindenter tip, large forces are necessary to produce the cracks needed for analysis. Hardness and modulus of thin films or particles in a matrix can not be measured with microindentation without tedious work in correcting the substrate effect or the contribution of the matrix on the results. So, the fracture toughness of thin film or ceramic composite, such as YBCO, cannot be easily determined with the microindentation method. Fracture toughness at small scales can be measured by ultra-low load indentation using techniques similar to those developed for microindentation testing [92 ]. Nanoindentation can be used to evaluate the fracture toughness of material and interfaces in a similar manner to that conventionally used in large scale testing. During loading, tensile stresses are induced in the specimen material as the radius of the plastic zone increases. Upon unloading, the additional stresses arise as the elastically strained material outside the plastic zone attempts to resume its original shape but is prevented from doing so by the permanent deformation associated with the plastic zone. There exists a large body of literature on the subject of indentation cracking with Vickers and other sharp indenters, such as Berkovich indenter. Recently, several methods for fracture toughness estimation without visualization of the crack length have been proposed, for example the works of Field et al. [93], where they relate

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the crack length produced to the pop-in behaviour during loading or the work of Dahami et al. [94] where they related the crack length with the total load penetration depth for fused silica. Generally, there are three types of crack. A scheme of it can be observed in figure 23: Radial crack are vertical half penny type cracks that occur on the surface of the specimen outside the plastic zone and at the corners of the residual impression at the indentation site. These radial cracks are formed by a hoop stress and extend downward into the speciment, but are usually quite shallow. See Figure 23a. Lateral cracks are horitzontal cracks that occurs beneath the surface and are symetric with the load axis. They are produced by a tensile stress and often extend to the surface, resulting in a surface ring that may lead to chipping of the surface of the specimen. See Figure 23b. Median cracks are vertical circular penny cracks that form beneath the surface along the axis of symmetry and have a direction aligned with the corners of the residual impression. Depending on the loading conditions, median cracks may extend upward and join with surface radial cracks, thus forming two half-penny cracks that intersect the surface. See Figure 23c.

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a)

b)

c) Figure 23. Scheme of the different types of cracks, a) radial crack, b) lateral cracks and c) median cracks.

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In addition, other fracture events, such as delamination or radial cracking at the interface, may be activated in case that the indentation field comprises both materials. As in the case of radial cracking, the delamination event will only be detected in an instrumented indentation test for certain materials. Fracture mechanics treatment of these types of cracks seek to provide a measure of fracture toughness based on the length of the radial surface cracks. Attention is usually given to the length of the radial cracks as measured from the corner of the indentation and then radially outward along the specimen surface as shown in figure 24.

l c

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a

Figure 24. Crack parameters for Berkovich indenter. Crack length c is measured from the center of contact to end of crack at the specimen surface.

Conventional indentation toughness methods were initially developed for monolithic bulk materials tested by microindentation when well-developed radial cracks form (for example Marshall and Lawn [95], Anstis et al. [96]). The toughness KIC is related to the applied load P, H, E and the cracks dimension, c. The fracture toughness can be obtained with the next equation: 1

K IC

⎛E⎞ 2 P = χ ⋅⎜ ⎟ ⋅ 3 ⎝H⎠ c 2

(27)

where E and H are the Young’s modulus and the Hardness of the material; these values are obtained with nanoindentation technique using the Oliver and Pahrr approach. For Berkovich and Vickers indenters, χ = 0.016. The values obtained by this method will depend on the residual stress in the coating since equation 27 is strictly only valid in the absence of internal stresses. At higher loads using a Berkovich indenter, two crack systems are observed [97]. Initially, radial cracks are observed along the edge of the indenter. These are followed by picture-frame cracks at the edge of the impression once sufficient bending has occurred.

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J.J. Roa, X.G. Capdevila and M. Segarra

For Vickers and Berkovich indenters, cracking threshold in most ceramic materials are about 250 mN or more [98], and since the indentation produced at these loads are relatively large, the cracking thresholds place severe restrictions on the spatial resolution which can potentially be achieved. It should be noted that the cracking threshold depends on the condition of the indenter tip, generally being higher for tips that have been blunted by wear. At a given load, the cube-corner and Berkovich diamonds should, to a first approximation, penetrate the material to produce approximately equal projected contact areas, such as the hardness measured with the two indenters should be about the same. Given that the nucleation and propagation of indentation cracks are promoted by large stress and strains, one would then qualitatively expect a reduction in the threshold for the sharper indenter.

1.2.5. The Effective Indenter Shape The effective indenter shape is outlined in figure 25. The basic principles are derived from observation gleaned from finite element simulations of indentation of elastic-plastic materials by rigid conical indenter with a half included angle of 70.3º [68]. During the initial loading of the indenter (figure 25a), both elastic and plastic deformation processes occur, and the indenter conforms perfectly to the shape of the hardness impression. However, during unloading, figure 25b, elastic recovery causes the hardness impression to change its shape. A key observation is that the unload shape is not perfectly conical, but exhibits a subtle convex curvature that has been exaggerated in figure 25b, the contact area increases gradually and continuously until full load is again achieved, a process which must be the reverse of what happens during unloading because both processes are elastic.

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a)

b)

Load

Unload

c)

Reload

P

P Z = U(r)

elastic Elastic/ Plastic

elastic Z

P P

PP Z = U(r)

r Effectiv indenter shape Effective indenter r

shape

r

Figure 25. Concepts used to understand and define the effective indenter shape, a) loaded, b) unloaded and c) reloaded. (From W. C. Oliver et al., J. Mater. Res., Vol. 19, No. 1, 2004, pages 3-20).

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It is this continuous change in contact area that produces the nonlinear unloading curves. Furthermore, the relevant elastic contact problem is not that of conical indenter on a flat surface, but a conical indenter pressed into a surface that has been distorted by the formation of the hardness impression.

1.2.6. Errors due to Pile-up and Sinking-in In an indentation into an elastic material, the surface of the specimen is typically drawn inwards and downwards underneath the indenter, and sinking-in occurs. When the contact involves plastic deformation, the material may either sink in, or pile up around the indenter. In the fully plastic regime, the behaviour is seen to be dependent on the ratio E/σys, where σys is the yield strength of the material, and the strain-hardening properties of the material. The mechanical nature of a typical specimen can be described by a conventional stress-strain relationship that includes a strain-hardening exponent:

σ = E ⋅ ε when ε ≤ σ = k ⋅ ε x when ε ≥

σ ys E

σ ys E

(28) (29)

where k is equal to:

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⎡ E ⎤ k = σ ys ⋅ ⎢ ⎥ ⎢⎣σ ys ⎥⎦

x

(30)

Such a complication arises from the pile-up or sink-in of the material around the indenter, which is primarily affected by the plastic properties of the material [99]. In a low-strainhardening alloy, plastically displaced material tends to flow up to (and pile-up against) the faces of the indenter due to the incompressibility of plastic deformation. The result is a barrelshaped impression due to pile-up around the sharp-indenter. In high strain hardening materials, the plasticity deformed region is pushed out from the indenter with the imprint sinking below the initial surface level. The result is a pin-cushion like impression around the sharp indenter, as shown in figure 26. As a consequence of pile-up or sink-in, large differences may arise between the true contact area and the apparent contact area which is usually observed after indentation. One significant problem with the determination of the contact area with equations from 12 to 15, is that this method does not account for pile-up of material around the contact impression. When pile-up occurs, the contact area is greater than that predicted by these equations, and both the hardness estimated from equation 5 and the effective modulus from equation 7 are overestimated, sometimes by as much as 50% [100]. Pile-up is large only when the relation between hf/hmax is close to 1 and the degree of work hardening is small. It should also be noted that when this relation is lower than 0.7, very little pile-up can be observed no matter what the work-hardening behaviour of the material. In this case, the contact area given by this method match very well with the true contact area obtained.

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J.J. Roa, X.G. Capdevila and M. Segarra Apparent contact diameter

Apparent contact diameter

Pile-up

True contact diameter

True contact diameter

Sink-in

True contact perimeter Indenter Side Surface

a)

b)

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Figure 26. Schematic illustration of a) pile-up and b) sink-in around a sharp indenter.

Furthermore, when this relation is higher than 0.7, the accuracy of the method depends on the amount of work-hardening in the material. For Berkovich indenters, indentations with a large amount of pile-up can be identified by the distinct bowing out at the edges of the contact impression [68]. If pile-up is large, accurate measurements of H and E cannot be obtained using the contact area deduced from the loaddisplacement curve. In this case, the most useful method to correct these problems is known as Cheng and Cheng method [101 and 102]. These equations are function of the total work of indentation (Wtot), and the work recovered during unloading (Wu). The method that they proposed to account for pile-up is based on the work of indentation [102], which measure from the areas under indentation loading and unloading curves. The relation proposed by Cheng and Cheng can be observed below:

Wtot − Wu H ≅ 1− 5⋅ Wtot E eff

(31)

Combining equation 5 and 7 and considering β as 1. Another equation involving H and Eeff can be obtained:

4 Pmax H ⋅ 2 = 2 π S E eff

(32)

where Wtot, Wu, Pmax, and S are all measurable from load-displacement curves. Spherical indentation differs from conical or pyramidal indentation in that there is no elastic singularity at the tip of the indenter to produce large stresses. Stress-strain curves can be approximated from indentation data using the classical approach of Tabor [68]. Field and

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Swain [103 and 104] have applied Tabor’s approach to instrumented indentation and have developed a method that uses the indentation load-displacement data to approximate the stress-strain curve and the work-hardening exponent The approach requires the deformation to be fully plastic. Pile-up geometry can change considerably during the course of spherical indentation and it is therefore not possible to predict the pile-up based on the mechanical properties of the material alone, even when fully plastic deformation is achieved.

1.2.7. Indentation Size Effect, ISE The indentation size effect (ISE) is extensively studied in literature and the mechanisms driving it remain irresolute. Examples of suggested mechanisms include inadequate measurement capabilities of extremely small indents, presence of oxides or chemical contamination on the surface, indenter-specimen friction, and increased dominance of edge effects with shallow indents [105]. In conclusion, ISE produces a greater hardness at a smaller applied load [106]. Depth-sensing indentation with loads higher than 1 N was proposed to avoid the inaccuracy due to the ISE effect [107]. The Indentation Size Effect (ISE) can be classified in two different types:

•

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•

Intrinsic: From interference of the size (or thickness) of the sample, the plastic region of the sample, L, or the autonomous plastic regions of the structure (the grains), D, with some of the characteristic microstructural lengths. Extrinsic: Associated to deformation gradients. Size effects in the resistance to plastic flow appear in metals when the dimensions of the specimen or of the zone subjected to plastic deformation are in the range of μm. The heterogeneity of the field of plastic deformation implies differences in the dislocation fluxes across a crystalline volume element (internal storage of “geometrically necessary dislocations, GND”). The analysis of this phenomenon is important because: i) the crystal deformation and fracture processes take place at this length scale and ii) implications on the development of micro-electro-mechanical systems (MEMS) and in the micro-electronics industry.

Recently, it is has become possible to perform indentation test at dimensions of tens to hundreds of nanometers using nano- and microindentation methods. At these small indentations depths classic plasticity theory predicts constants hardness using a geometrically self-similar indenter on a homogenous material. Nevertheless a strong size dependent indentation hardness result is well known. ISE is characterized by an increasing hardness as the indentation depth is reduced to the order of microns or submicrons. This phenomenon was interpreted by Nix and Gao (1998) [108], De Guzman et al. (1993) [109], Poole et al. (1996) [110], McElhaney et al. (1998) [111] and Fleck et al. (1994) [112] who proposed a strain gradient theory to explain the presence of the ISE. Nix and Gao [108] showed that the ISE for crystalline materials can be explained using the concept of geometrically necessary dislocations, which leads to a strain gradient plasticity law. Swadener et al. [113] extended this model for the case of spherical tip indenters.

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J.J. Roa, X.G. Capdevila and M. Segarra

For loads up to and exceeding that to initiate pop-in at the beginning of the elastic/plastic part of the loading curve all indents can be described by an approximately spherical contact. They were analyzed using the approach of Swader et al. [113], which proposes that spherical indenters show a dependence of hardness on the indenter radius rather than on the depth of the penetration. The load dependence of the hardness is referred to as the ISE. The ISE can be seen remarkably under extremely low loads such as nanoindentation test [114]. Up to low various factors such as strain-hardening and friction effects have been reported to explain the phenomenon of the ISE [115]. This theory assumes that the stress flow of metals depends on the density of statistically stored and geometrically necessary dislocation. The density of the statistically stored dislocation varies with the effective strain whereas the density of the geometrically necessary dislocations depends on the strain gradient. In crystalline solids (such as ceramics), dislocations are responsible for sustained plastic deformation and dislocations impedes the motion of new dislocations. Dislocations are generated during plastic deformation (when the applied load is higher than yield strength of the studied material). The dislocations are then moved, and stored. Storage of dislocations abets strain hardening. It is postulated that dislocations become stored because they either accumulate by randomly trapping each other or they are required for compatible deformation of various parts of the material. When they randomly trap each other, they are often known as the statistically stored dislocation [105], whereas when they are required for compatibility purposes, they are often called geometrically necessary dislocation and they are related to the gradient of plastic shear strain in a material [105]. ISE have been observed since the early days of indentation testing [116]. In many cases, this effect is largely due to the rising uncertainties involved in making and measuring small indentations. There has been voluminous literature devoted to study the origin of the ISE. Consequently, several empirical or semi-empirical equations widely applied in ceramic materials can be applied to solve this problem, such as Meyer’s law [117], the Hays-Kendall approach [118], the elastic recovery model [117], the energy-balance approach [119], the proportional specimen resistance model [120], and others, that have been proposed for describing the variation of the indentation size with the applied test load. a. Meyer’s law The most widely used empirical equation for describing the ISE is the Meyer’s law, which correlates the test load and the resultant indentation size using a simple power law [117],

Pmax = A ⋅ hcn

(33)

where A and n are constants that can be derived directly from the curve fitting of the experimental data. n is known as Meyer’s index and it is usually considered as a measure of the ISE. To obtain the A and n values, the values obtained from the nanoindentation data are plotted in an ln Pmax – ln hc scale. Each set of the data shows an excellent linear relationship,

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implying that the traditional Meyer’s law is suitable for describing the nanoindentation data. Trough linear fitting analyses, the best fit values of the A and n were obtained. b. Hays-Kendall approach When examining the ISE in the Knoop hardness testing of a number of metals, Hays and Kendall [118] advanced that there exists a minimum level of the applied test load, W, named the test-speciment resistance, below which permanent deformation due to indentation does not initiate, but only elastic deformation occurs. An effective indentation load has been introduced, Peff = Pmax – W, and proposed the following relationship,

Peff = A1 ⋅ hc2 = Pmax − W

(34)

where W and A1 are constants independent of the test load for a given material. Equation 34 predicts that a plot of Pmax versus hc2 would yield a straight line. This equation provides a satisfactory description of the nanoindentation data for the studied material. c.

Elastic recovery model or Elastic/Plastic deformation model

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In microhardness test, the indentation size is measured after the indenter is removed from the specimen surface. Note that the elastic recovery would occur in the vicinity of the remaining indentation impression after the indenter is removed so that the indentation size would shorten to a certain degree [121]. Considering this effect, Tarkanian et al. [117] suggested that the measured indentation size should be corrected with a revised term in order to obtain the true hardness. The true hardness can be calculated as follows:

Ho = k ⋅

P (d + d o )2

(35)

where do is the correction in the indentation size d due to the elastic recovery and k is a constant dependent on the indenter geometry. Furthermore, several authors have pointed out that, for calculating the hardness from the recorded indentation test such as the nanoindentation, similar correction in indentation size should be considered because of the elastic recovery associated with the new bands of plastic deformation [122] and /or the blunting of the indenter tip [123]. Thus it is necessary to check if equation 35 is suitable for describing the nanoindentation data obtained for ceramics. To analyze the nanoindentation data, equation 35 may be re-written in the next form: 1

Pmax2 = χ

1

2

⋅ hc + χ

1

2

⋅ ho

(36)

where ho is the correction in hc and χ = Ho/k is a constant related to the true hardness. Equation 36 allows to determine ho and χ from the plots of Pmax1/2 against hc.

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J.J. Roa, X.G. Capdevila and M. Segarra There are two ways to calculate the true hardness based on the elastic recovery model:

− −

Directly use equation 35, or Obtain χ.

d. Proportional specimen resistance model or PSR model Proportional specimen resistance (PSR) model was proposed by Li and Bradt [120]. This model can be considered as a modified Hays-Kendal approach. In this model, the testspecimen resistance to permanent deformation is assumed not to be a constant, but this factor increases linearly with the indentation size. This phenomena is governed by the next equation:

W = a1 ⋅ hc

(37)

To a first approximation, the equation 37 can be considered to be similar to the elastic resistance of a spring with the opposite sign to the applied test load. The effective indentation load and the indentation dimension can be related as follows:

Peff = Pmax − W = Pmax − a1 ⋅ hc = a 2 ⋅ hc2

(38)

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where a1 and a2 are constants for a given material. According to the analysis of Li and Bradt [120], the parameters a1 and a2 can be related to the elastic and the plastic properties of the test material. a2 was suggested to be a measure of the socalled true hardness, Ho. For the nanoindentation test with a Berkovich tip indenter, Ho can be calculated directly from a2 with:

Ho =

Peff 24.5 ⋅ h

2 c

=

Pmax − a1 ⋅ hc a = 2 2 24.5 ⋅ hc 24.5

(39)

Equation 38 can be rearranged as;

Pmax = a1 + a 2 ⋅ hc hc

(40)

which enables to determine both a1 and a2 from the plot of Pmax/hc against hc. An alternative explanation for the physical meaning of equation 40 was proposed by Flöhlich et al. [124] based on energy-balanced analysis. According to the energy balance consideration, the parameters a1 and a2 in equation 40 are related to the energies dissipated for creating a new surface of a unit area and for producing the permanent deformation of a unit volume, respectively. Also, a2 is a measure of the true hardness. e,

The modified PSR model

Examining the load-dependence of the microhardness of some ceramics measured in a wide load range, Gong et al. [125], found that the resultant P/d-d (where d is a half-length of the Vickers indentation) curves exhibit significant nonlinearity and argued that the PSR model

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mentioned above may only be used to represent the experimental data measured in a narrower range of applied loads. Gong et al. [125] suggested that the PSR model should be modified as follows:

Pmax = a o + a1 ⋅ hc + a 2 ⋅ hc2

(41)

where a0 is a constant related to the surface residual stress associated with the surface machining and polishing and a1 and a2 are the same parameters as those in equation 40. An equation with the same form as 41 has been also deduced based on a modified energy-balanced analysis [120]. The modified PSR model provides two different ways to obtain the true-hardness:

Ho 1 =

Pmax − a o − a1 ⋅ hc 24.5 ⋅ hc2

(42)

a2 24.5

(43)

Ho

2

=

2. Mechanical Properties

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2.1. State of the Art of Mechanical Properties of YBCO Samples The mechanical properties of YBCO samples have been studied during the last years. The most important properties studied have been the hardness (at micrometric and nanometric scale), the Young’s modulus and the toughness fracture. Some authors studied the mechanical properties at room or at cryogenic temperatures also known as work temperature. The techniques used to perform these studies have been: microindentation or nanoindentation, to obtain the hardness of the YBCO samples; the bending, X-ray diffraction and others, to obtain the Young’s modulus; and microhardness to obtain the toughness fracture. Reported values of Young’s modulus, hardness and fracture toughness of YBCO composite obtained using different experimental techniques are summarized in tables 3, 4 and 5, respectively. Table 3. Literature values of Young’s modulus for YBCO with different techniques Author Joo. et al. [139] Lucas et al. [126] Soifer et al. [127]

Material YBCO YBCO + 5% vol. Ag YBCO + 10% vol. Ag YBCO + 15% vol. Ag Y-123 YBCO film

Young’s modulus (GPa) 110 103 103 97 154.30 ± 16.34 210

Method Pulse echo technique Indentation Nanoindentation

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122

J.J. Roa, X.G. Capdevila and M. Segarra Table 3. Continued Author

Material

Young’s modulus (GPa)

Ledbetter et al. [129] Sheahen et al.[130] Goyal et al. [131] Soifer et al. [132]

YBCO polycrystalline, 50K YBCO polycrystalline, 160K YBCO polycrystalline, 180K YBCO polycrystalline, 293K YBCO polycrystalline Syntherized with Ag Syngle crystal Thin film

90.8-101.8 75-120 220 ± 20 210

Goyal et al. [133] Goyal et al. [133] Reddy et al. [134]

Y-211 Texturized Texturized

213 182 95.89

Güçlü et al. [128]

Method

47.20 29.63 28.47

Vickers indentation

9.39 Ultrasonic Ultrasonic Nanoindentation with AFM Nanoindentation Nanoindentation Ultrasonic

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Table 4. Literature values of Hardness for YBCO with different techniques Author

Material

Hardness (GPa)

Lucas et al. [126]

Lim et al. [138] Cook et al. [143] Goyal et al. [131] Goyal et al. [133] Soifer et al. [132]

Y-123 YBCO, MTG-1100ºC, 5 min YBCO, MTG-1100ºC, 10 min YBCO, MTG-1100ºC, 15 min YBCO, Solid state reaction YBCO film YBCO 40K YBCO 293 K YBCO polycrystalline, 50K YBCO polycrystalline, 160K YBCO polycrystalline, 180K YBCO polycrystalline, 293K YBCO single crystal YBCO Textures Textures Thin film

10.28 ± 1.67 5.4 5.0 5.1 4.7 8.5 18 ± 2.5 5.2 ± 0.5 3.58 1.03 0.95 0.53 7.81 ± 0.23 8.7 6.7 10.8 8.5

Goyal et al. [133]

Y-211

14.0

Li et al. [135] Soifer et al. [127] Yoshino et al. [136] Güçlü et al. [137]

Method Indentation Vickers indentation Nanoindentation Vickers indentation Vickers indentation Nanoindentation Vickers indenter Vickers indenter Nanoindentation Indentation with AFM Nanoindentation

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Table 5. Literature values of fracture toughness for YBCO with different techniques Author

Material

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YBCO YBCO + 5% vol. Ag Joo. et al. [139] YBCO + 10% vol. Ag YBCO + 15% vol. Ag Lenblond-Harnois et YBCO al. [140] YBCO + 5% wt Ag YBCO, MTG-1100ºC, 5 min YBCO, MTG-1100ºC, 10 min Li et al. [135] YBCO, MTG-1100ºC, 15 min YBCO, Solid state reaction YBCO 40K Yoshino et al. [136] YBCO 293 K Joo et al. [141] YBCO

Fracture Toughness (MPa·m1/2) 1.60 2.10 2.50 2.80 1.53 1.88 1.9

Method Single-edge-notch beam Vickers indentation method Vickers indentation

1.7 1.7 1.3 0.4 1.3 1.6

YBCO + 5 vol% Ag

2.2

YBCO + 10 vol% Ag

2.6

Vickers indentation Single-edge-notch beam

Leenders et al [142] YBCO + 30 mol% Y-211

1.01

YBCO + 60 mol% Y-211

1.44

Cook et al. [143]

YBCO

1.1

Vickers indenter

Sheahen et al. [130]

YBCO

0.8-1.0

Bending

Sheahen et al. [130]

YBCO textured

1.6

Bending

Fujitomo et al. [144]

YBCO textured

0.99-1.20

Vickers indenter

1.6

Bending

3.8

Bending

1.6-2.1

Vickers indenter

Sheahen et al. [130] YBCO with 5, 15 and 25 % of Ag textured Sheahen et al. [130] YBCO with 20% Ag Fujitomo et al. [144]

YBCO with Ag

Vickers indenter

From last tables it can be observed the high scattered differences between the different studies carried out last century. Now, we have performed a study of the mechanical properties of YBCO samples textured by Bridgman and TSMG techniques by Nanoindentation. Figure 27a to 27c, a representation of Young’s modulus, Hardness and fracture toughness respect the technique can be observed.

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124

J.J. Roa, X.G. Capdevila and M. Segarra 250 220

210 213 200

210

182

E (GPa)

154,3 150 110 96,3

100

95,98

103 103

97

47,2

50

29,6 28,4 9,3

0

Indentación Ultrasonic

Nanoindentación AFM

Vickers Pulse echo technique

a) 20 18

18 16

14

H (GPa)

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14 12

10,8

10,28 10

8,7

8 6

8,5

8,5

7,81

6,7 5,4 5,1 4,7

4

5,2 3,58

2

1,03 0,95 0,53

0

Indentación

Vickers

Nanoindentation

AFM

b) Figure 27. Continued on next page.

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Mechanical Characterization at Nanometric Scale…

125

4

3,8

3,5

KIC (MPa·m1/2)

3

2,8 2,6

2,5

2,5

2,2

2,1

1,88 1,9

2 1,6

1,6

1,85

1,7 1,7

1,6 1,6

1,53

1,5

1,3

1,3

1,44 1,01

1

1,1 1,05

0,9

0,4

0,5 0

Single edge notch beam

Vickers

Bending

c)

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Figure 27. Representation of the bibliographic mechanical property versus the different technique for a) Young’s modulus, b) Hardness and c) Fracture toughness.

2.2. Mechanical Properties of YBCO Samples Textured by Bridgman and TSMG Technique by Nanoindentation 2.2.1. Plastic Deformation Experimental Conditions The nanoindentation technique was performed by a Nano Indenter® XP Systems equipped with Test Works 4 Professional level software. Nanoindentation imprints were observed with optical microscope, with an AFM NanosCope III-A atomic force microscope, and with a field emission Hitachi H-4100 scanning electron microscope. The experiments were performed on the (001) plane at room temperature for the different indenters used (Berkovich and Spherical tip indenters). First of all, a plastic study was carried out with a Berkovich indenter. In this case, the hardness, Young’s modulus and fracture toughness were studied at different applied loads: 5, 10, 30 and 100 mN. The loading/unloading time was selected to be constant for all indentations: 15 s. Table 6 shows fixed test parameters to perform measurements of nanoindentation when a Berkovich indenter was used.

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J.J. Roa, X.G. Capdevila and M. Segarra Table 6. Test inputs of nanoindentation Name Allowable drift rate Load rate multiple for unload rate Maximum load Number of times to load Peak hold time Per cent to unload Time to load

Value 0.05 1 5, 10, 30 and 100 5 30 90 15

Units nm·s-1 mN s % s

The small nanoindentations were made by a three-sided pyramid Berkovich diamond indenter, see figure 19.a. The displacement or also known as penetration depth, was continuously monitored and load-time history of indentation recorded. Each hardness, Young’s modulus, fracture toughness and other values listed in this section, are an average of 40 measurements performed on two different samples in order to achieve statistical significance. The values obtained by the Oliver and Pharr equation (equations 5, 7 and 8) have been corroborated by FE-SEM. After that, we studied the Young’s modulus evolution (from 0 to 700 mN of applied load) with continuous stiffness measurement, CSM, and with a spherical tip indenter. CSM supplies information about the first steps of the indentation and allows to know the evolution of the Young’s modulus with the penetration depth. The radii of this tips was 25 μm.

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Experimental Curves The load/unload curves can give a qualitative information of the hardness of each phase of the study when the applied load is lower than 10 mN. For these loads, the mechanical properties of each phase can be isolated. Figure 28, shows the load/unload curves for YBCO samples textured by Bridgman technique when the applied load was 5 mN. Figure 28a, shows the qualitative manner to predict the hardness of each phase. The hardness of one material is a function of two different parameters, the maximum applied load and the contact area. The last parameter is a function of contact penetration and this one of the maximum penetration. For this reason, when one phase presents a high penetration depth, this one has a low hardness value; one material has a high hardness value when the penetration depth is low. From figure 28a, the distribution of hardness for each phase is: HY-123 < HY-123/Y211 < HY-211. From figure 28b, the same effect that in the figure 28b can be observed. In this case, the difference between the different phases is less than in the figure 28a. The YBCO samples textured by TSMG technique present the same relation: HY-123 < HY-123/Y-211 < HY-211. This technique presents a little difference in the hardness value at ultra low load such as 5 mN, because after the oxygenation process, the orthorhombic phase presents a high porosity in the ab-plane.

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5,0 4,5 4,0 3,5

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3,5 3,0 2,5 2,0 1,5

Y-123 Y-211 Y-123/Y-211

1,0 0,5 0,0 0

50

100

150

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h (nm) b) Figure 28. Load/Unload curves for YBCO samples when the applied load was 5 mN textured by a) Bridgman technique and b) TSMG technique.

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Figure 29, shows the load-unload curve for YBCO samples textured by Bridgman and TSMG technique recorded by CSM when the applied load was 700 mN. 700 600

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h (nm) Figure 29. P-h curve for YBCO samples textured by Bridgman and TSMG technique recorded by CSM when the applied load was 700 mN.

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Figure 29, shows that for the two different techniques of study the first steps of nanoindentation present the same tendency. The scattered occurs mainly in the unload curve, for this reason the Young’s moduli will be different. Characterization Imprints All imprints realise in the samples of study have been observed with Optical Microscopy and FE-SEM. In order to know the correct value of the hardness, Young’s modulus and fracture toughness of Y-123, Y-211 and Y-123/Y-211 in the case that the applied load permit isolate each mechanical property. Figure 30 shows indentation imprints performed by applying 30 and 100 mN on the ab plane of the monodomain, the Y-123 matrix and Y-211 inclusions. In this figure, the particles or inclusions can be observed homogeneously distributed in the textured samples, so that they can be easily identified but not indented separately. It is important to highlight that the size of Y-211 inclusions (from 1 to 5 μm, approximately) is smaller than the nanoindentation imprints performed at these loads, so that we can only measure the mechanical properties of the composite (Y-123+Y-211) when the applied load was higher than 10 mN. In Figure 30b, it can be observed that the residual indentations are highly affected by the superficial porosity. Every residual imprint shows a crack in its corner. Thus indicating, give us that YBCO is a brittle material.

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50 μm

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a)

30 mN

100 mN

50 μm

b) Figure 30. Optical microscope micrographs of nanohardness impressions developed on the surface of a sample of YBCO orthorhombic phase (ab plane) at room temperature, a) for Bridgman textured sample (From Roa et al. Nanotechnology, 18, 2007, page 38571-1 to 38571/1-38571/6) and b) for TSMG textured sample, when the applied load was 30 and 100.

When the applied loads are higher than 10 mN (such as 30 and 100 mN), only the mechanical properties of composite can be determined and we cannot isolate the contribution of each phase. Figure 31 shows Y-123, Y-211 and YBCO composite imprints for samples textured by Bridgman technique. Figure 31b, shows propagation of the cracks following the corners of the indentation, for Y-211 precipitate. Nanohardness of Y-211 is twice as high as the Y-123 matrix (see table 7); for this reason the Y-211 phase presents a brittle fracture. The analysis of SEM images can be used to determine the profile impressions with nanometer resolution and to provide information about the shape change on unloading. As can be seen in this figure, the indentation exhibit triangular geometry.

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1.2 μm a)

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b)

1.2 μm c) Figure 31. Micrograph of nanoindentation imprints obtained by FE-SEM when the applied load was 10 mN. a) matrix Y-123; b) precipitate, Y-211 and c) Y-23/Y-211 composite (From Roa et al. Nanotechnology, 18, 2007, page 38571/1-38571/6).

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Figure 32 shows Y-123, Y-211 and YBCO composite imprints at 10 mN for samples textured by TSMG technique.

1.50 μm

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a)

1.50 μm

b)

1.50 μm

c) Figure 32. Micrograph of nanoindentation imprints obtained by FE-SEM when the applied load was 10 mN. a) matrix Y-123; b) precipitated, Y-211 and c) Y-23/Y-211 composite.

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In figure 32b, a crack at the corners of the imprint and another fracture mechanism known as chipping, can be observed. When the applied load is higher than 10 mN, the mechanical properties of each phase cannot be isolated because the size of the residual imprint is lower than the size of Y-211. Figure 33, shows the nanoindentation imprints obtained by FE-SEM when the applied load was 30 and 100 mN for samples textured by Bridgman technique.

2.0 μm

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a)

3.0 μm μm 3.0 b) Figure 33. Micrograph of nanoindentation imprints obtained by FE-SEM of samples textured by Bridgman technique at applied load of, a) 30 mN, and b) 100 mN.

Figure 33a, shows cracks at the corners of the imprints; in this case the fracture toughness can be calculated. Inside de imprints, a radial cracks can also be observed. The fracture mechanism in this case will be further discussed. Figure 33b, also shows cracks, radial ones inside the imprint and length ones at two of the three corners of the imprint. The missing crack at one corner has been stopped by a Y-211 particle. In this case, the fracture toughness cannot be calculated.

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In the case of the samples textured by TSMG technique, when the applied load was 100 mN, we can observe a lot of porosity and sink-in besides the residual imprints (Figure 34).

30 μm Figure 34. Micrograph of nanoindentation imprints obtained by FE-SEM of samples textured by TSMG technique.

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Figure 35, shows residual nanoindentation imprints at applied loads of 30 and 100 mN. When the load applied was 30 mN (figure 35a), cracks at the corners of the imprints cannot be observed. No cracks can be observed at the corners of imprints, but only radial cracks. Note that when the applied load was 100mN, all imprints show a high porosity beside the imprint for both texturing technique, see figure 34 and 35b.

2.00 μm

a) Figure 35. Continued on next page.

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5.00 μm

b) Figure 35. Nanoindentation imprints obtained by FE-SEM of samples textured by TSMG technique, a) 30 mN of applied load and b) 100 mN of applied load.

Hardness

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Table 7 shows the calculated nanohardness values of the orthorhombic phase of Y-123, Y-211 and Y-123/Y-211 composite for the different monodomains samples textured by Bridgman technique, when the applied loads were 5, 10, 30 and 100 mN. Table 7. Nanohardness of orthorhombic phases of YBCO textured by Bridgman (From Roa et al. Nanotechnology, 18, 2007, page 38571-1 to 38571/1-38571/-6) and TSMG technique for applied loads of 5, 10, 30 and 100 mN.

Monodomain Applied Load (mN) Sample 1 5 Sample 2

Sample 1 10 Sample 2

Y 123 Y 211 Y123/Y211composite Y 123 Y 211 Y123/Y211composite

Hardness; H (GPa) Bridgman TSMG 11.0 ± 0.5 11.41 ± 0.51 20.0 ± 1.0 15.0 ± 1.05 15.2 ± 0.3 14.86 ± 1.05 9.8 ± 0.4 10.59 ± 0.45 18.1 ± 0.5 15.35 ± 0.58 14.4 ± 0.7 14.92 ± 0.98

Y 123 Y 211 Y123/Y211composite

11.4 ± 0.4 17.1 ± 0.5 15.3 ± 0.3

11.12 ± 0.96 17.14 ± 0.25 14.58 ± 0.78

Y 123 Y 211 Y123/Y211composite

11.0 ± 0.3 16.7 ± 0.6 14.9 ± 0.2

9.59 ± 0.76 16.89 ± 0.86 14.86 ± 0.95

Phase

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Table 7. Continued.

Monodomain

Applied Load (mN)

Phase

30

Y123/Y211 composite

Sample 1

Hardness; H (GPa) Bridgman TSMG 11.2 ± 0.3

8.71 ± 0.95

Sample 2

11.0 ± 0.3

8.26 ± 0.79

Sample 1

8.8 ± 0.2

7.96 ± 0.72

9.1 ± 0.1

7.88 ± 0.89

100 Sample 2

Y123/Y211 composite

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Nanohardness and Young’s modulus of each phase (Y-123 matrix and Y-211 inclusions) can be determined when the applied load was ultra-low (less 10 mN), each phase can be indented and the respective mechanical properties can be isolated. For higher loads, the mechanical properties of the interaction of both phases can be obtained the YBCO composite or Y-123/Y-211. Values of Y-211 phase were higher than for Y-123 in both cases of study, for the samples. Nanohardness of Y-211 was about twice as high as for Y-123. This fact could be due to different reasons: (i). Ionic bond of Y-211 is stronger than the Y-123 bond, (ii). High anisotropy of dislocation confined onto a (001) plane [145], and/or (iii). the melt processed ceramic composites contain a dense population of fine peritectic inclussion embebbed in a matrix, which drastically affects the microstructure acting as nucleation sites for dislocation [145]. When the applied load is 100 mN, overall nanohardness is very similar for both samples (for Bridgman technique 8.90 GPa, and TSMG technique 7.92 GPa) but is lower than the nanohardness of the Y-123 phase. Therefore, these results are in agreement with a previous work reported by Lim and Chaudhri [138] which studied the hardness of YBCO single crystal by Nanoindentation technique and the hardness value obtained was 7.81 GPa. This value is in agreement with the result obtained for samples textured by TSMG technique. On the other hand, the hardness value obtained for samples textured by Bridgman technique is in agreement with a previous work reported by Soifer et al. [132] and Cook et al. [143]. When the applied load is higher than 10 mN we are working within the microindentation range and we can observe microcraks at the corners of imprints (see figure 33a), thus causing the reduction of the hardness value (H100mN < H30mN). This effect is known as indentation size effect, ISE. Verdyan et al [146] reported a nanohardness for orthorhombic YBCO thin film can be around 8.5 GPa, when the applied load varies between 0.1 and 9mN, which are similar values to those found in the present study for the orthorombic phases. Values of the Young’s

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modulus are also comparable to our results for the orthorhombic composite, studied at loads of 30 and 100 mN. If we compare our data for Hardness with data reported of YBCO (see table 7), we conclude that the broad distribution of hardness observed in table 4 can be attributed both to the different measuring techniques and to different quality of the studied YBCO samples (grain structure, porosity, texturing process, etc). Young’s Modulus Table 8 shows the calculated nanohardness and Young’s modulus values of the orthorhombic of Y-123, Y-211 and Y-123/Y-211 composite for the different monodomains samples studied when the applied loads were 5, 10, 30 and 100 mN textured by TSMG technique. Table 8. Young’s modulus of orthorhombic phases of YBCO textured by Bridgman (From Roa et al. Nanotechnology, 18, 2007, page 38571-1 to 38571/1-38571/-6) and TSMG technique for applied loads of 5, 10, 30 and 100 mN. Applied Load (mN)

Phase

5

Y 123 Y 211 Y123/Y211composite Y 123 Y 211 Y123/Y211composite

Young’s modulus; E (GPa) Bridgman TSMG 193 ± 7.60 176.44 ± 15.31 199 ± 10.20 224.35 ± 20.45 204 ± 7.12 207.33 ± 21.65 200 ± 6.53 175.89 ± 15.39 189 ± 8.56 223.54 ± 16.23 198 ± 7.02 208.26 ± 19.52

Y 123 Y 211 Y123/Y211composite

185 ± 3.16 209 ± 4.29 201 ± 6.92

173.62 ± 18.20 207.27 ± 10.58 189.61 ± 17.87

Y 123 Y 211 Y123/Y211composite

192 ± 4.21 203 ± 3.56 206 ± 5.49

174.59 ± 15.35 206.58 ± 9.59 190.12 ± 15.2

179 ± 5.49

140.09 ± 14.21

Sample 2

181 ± 4.35

139.85 ± 13.25

Sample 1

171 ± 2.51

128.88 ± 5.54

175 ± 3.87

130.37 ± 5.69

Monodomain Sample 1

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Sample 2

Sample 1 10 Sample 2

Sample 1 30

100 Sample 2

Y123/Y211 composite

Y123/Y211 composite

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When the applied load is 100 mN, overall Young’s modulus are very similar for both samples and techniques (for Bridgman technique 173 GPa and TSMG technique 129 GPa) but is lower than the Young’s modulus of the Y-123 phase. Therefore, the results obtained by Bridgman technique are in agreement with a previous work reported by Goyal et al. [133] which studied the Young’s modulus of texturized YBCO by Nanoindentation technique and the Young’s modulus value obtained was 182 GPa. The Young’s modulus value obtained for samples textured by TSMG technique is in agreement with a previous work reported by Joo et al. [139] which value had been obtained by pulse echo technique. Goyal et al. [133] studied the Young’s modulus of Y-211 phase present in YBCO samples by Nanoindentation; this value was 213 GPa. This value obtained by Goya et al. [133] is in agreement with a value obtained by Nanoindentaion for YBCO samples textured by TSMG technique when the applied load was 10 mN and the Y-211mechanical properties have been obtained. Several measurements of Young’s modulus for Y-123 have resulted in values scattered within the range E=40-200 GPa. Most probably, this scatter is caused by residual porosity and bad contacts between the grains [147]. Sakai et al [148] reported a value of 370 GPa for 5 mm3 cubic specimens cut from a single-crystal Y-123 prepared by TSMG using a single domain of Sm-123 as a seed. The authors attribute the difference of Young’s modulus to the 40% excess of Y-211 phase present in the material. Obtained results of Young’s modulus are in agreement with Johansen et al. [147] and Alford et al [149] when the applied loads were 30 and 100 mN. If we compare our data for Young’s modulus with data reported for YBCO (see table 8) we also conclude that the broad distribution of Young’s moduli observed in table 3 can be attributed both to the different measuring techniques and to the different quality of the studied YBCO samples (grain structure, texture, and others). The tendency of the Young’s modulus versus penetration depth obtained with a spherical tip indenter and CSM recorded for each one of these texturing techniques is presented in the next figures. Figure 36a, shows that the Young’s modulus increases until a constant value, which is around 300 nm of penetration depth. When the penetration depth is lower than 300nm, the spherical contact area could be smaller than the size of Y-211 particles. So, at first stages, we could measure each phase or interface separately. This can explain the highly scattered values that we can see in figure 36a. When the residual imprints are higher than the superficial defects, the Young’s modulus value of the samples textured by Bridgman technique present only a constant value with the penetration depth. Figure 36b, shows two different tendencies of the Young’s modulus: until 300 nm a high dependence of the Young’s modulus with the penetration depth can be seen. On the other hand when the penetration depth is higher than 300 nm, this trend decreases. In this case, a constant value of Young’s modulus cannot be achieved. The difference sample response between Bridgman and TSMG monodomain could be related to the fragilization of the material due to the high porosity density present in the TSMG samples. Even for penetrations depth lower than 300 nm, we cannot isolate the porosity effect over the Young’s modulus value. This is reflected in figure 36b by the high homogeneity on the measurements. For higher applied loads, in the Bridgman samples with lower porosity the Young’s modulus tends over the Y-123/Y-211 value. In TSMG samples, the inner porosity not allows this kind of stabilization.

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E (GPa)

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100 80 60 40 20 0 0

100

200

300

400

500

600

h (nm)

b) Figure 36. Evolution of the Young’s modulus versus the penetration depth for YBCO samples at maximum applied load for a spherical tip indenter for samples, a) textured by Bridgman technique and b) textured by TSMG technique.

Table 9, shows the Young’s modulus values for the YBCO samples obtained with a spherical tip indenter at 300 nm of penetration depth using CSM.

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Table 9. Young’s modulus of YBCO samples obtained by spherical indentation at 300 nm of penetration depth using CSM. Material YBCO

Technique Bridgman TSMG

Young’s modulus (GPa) 120 ± 20 108 ± 8

The Young’s modulus of YBCO samples textured by Bridgman technique obtained with spherical tip indenter are similar to the values obtained with Berkovich indenter, see table 8. In Figure 36 and Table 9, the values present a difference tendency before and after than 300 nm of penetration depth. When the sample was textured by TSMG technique at 300 nm of penetration depth, the different curves of Young’s modulus versus penetration depth present a lower scattered value. In this case, when the penetration depth is lower than 300 nm the sample present a high homogeneity between Y-123 and Y-211 particles. For penetration depth higher than 300 nm, the Young’s modulus does not present a constant value. In the other hand, for samples textured by Bridgman technique when the penetration depth is lower than 300 nm, the material presented a high interaction with the surface’s defects; for this reason the Young’s modulus present high scattered values. For penetrations depths higher than 300 nm, the material presents a constant value of Young’s modulus.

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c.

Indentation Size Effect

The indentation size effect is generally attributed to train gradient plasticity that generates geometrically necessary dislocations; for ceramic compounds where the plasticity is limited allow loads, the elastic recovery can be significant. When the indentation size is smaller, the density of geometrically necessary dislocations decreases and, as a result, hardness becomes higher at small loads [150]. In fact, from the present calculations of the indentation height before and after removal of the load, it is found that 10% of total work done during the indentation at loads of 5, 10, 30 and 100 mN, is due to elastic deformation. While most of the methodologies of analysis of instrumented indentation test assume that materials are homogeneous and continuous this is not the case for real materials, as the continuous broke at a given length (size effect). However, nanohardness can be strongly affected by the presence of defects and impurities that can cause almost no change in dislocation movement, which would affect the hardness. This phenomenon is known as ISE. Instrumented indentation is, a priori, a good technique for local evaluation of the residual stresses, and, for that purpose, several methods have been developed, specially for metals with relatively low yield stress and that do not work harden appreciably when indented with sharp indenters. However, these tests work well when the measured residual stress is in the order of magnitude of the order of the hardness of the material. In YBCO materials, this is often the case because of the high hardness of these materials. In this material, the differences obtained in the load-unload curve from a stressed to an unstressed material are too small to qualitatively extract a value of residual stress. The load-displacement curves obtained for various loads and techniques are shown in figure 37. For each material, all the data points of ten measurements are given in this plot when the applied load was 100 mN. The same effect also can be observed at different studied

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applied loads, such as: 5, 10 and 30 mN. For YBCO samples textured by Bridgman and TSMG techniques, slight scatters exist among the different load-displacement curves. These scatters may be attributed to the intrinsic microstructural inhomogeneity of these two different techniques to produce single crystals. 100 90 80

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P (mN)

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b) Figure 37. Load/unload curves at 100 mN of applied load on samples textured by, a) Bridgman technique and b) TSMG technique.

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Figure 37. a and b, shows the load-unload curves for YBCO samples textured by Bridgman and TSMG techniques at 100 mN of applied load, respectively. At this load, the hardness and Young’s modulus of YBCO sample has been studied because the residual indentation imprint is higher than the size of Y-211 particles. If the hardness of YBCO sample is the same for all indentations performed on the ab-plane, then we only have to look at one load-unload curves. Nevertheless, in this figure we can observe a high scattered loadunload curves; this effect may be probably due to the ISE. When the applied load is increased, their hardness value is decreasing due to the same factor. Some authors have studied the ISE effect and proposed some equations to solve these problems, such as: Meyer’s lawn, HaysKendall approach, elastic recovery model, proportional specimen resistance model or PSR model and the modified PSR model (see equations 33 to 43). Figure 38 shows the experimentally determined nanoindentation hardness, H, as a function of the peak load, Pmax, for the tested materials. Until 30 mN of applied load, the high tendency is due to the existence of two different phases embedded in the samples of study. When the applied load is 10 mN lower, the mechanical properties of different phases can be studied (Y-123, Y-211, Y-123/Y-211). In figure 38, a high dependency at low applied loads can be observed. This effect is due to the difference between the different phases present in YBCO samples. The ISE can be used for isotropic materials. In fact, YBCO samples present high anisotropy properties in the ab-plane. The different models proposed to study the ISE for ceramics, present some problems to apply in YBCO samples, such as: o

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o

These approaches do not calculate or evaluate the area factor, and the contact area can be strongly affected at low applied loads. This effect produces and overestimated hardness value. The ISE is only affected by the dislocation movement. YBCO samples and a lot of ceramic compounds, present another mechanism such as radial cracks, fracture mechanisms, etc. If the material presents this mechanism, the equations proposed for this study cannot be applied.

In conclusion, YBCO samples textured by Bridgman and TSMG techniques may present ISE. These phenomena’s cannot be solved because the different equations for ceramics compounds do not a present the different fracture mechanism. If someone has to evaluate ISE for ceramic compounds, the best thing to do is perform a high amount of nanoindentation imprints, and apply the Weibull approach in order to have an average of measurements. Investigations have confirmed that hardness numbers calculated with the Oliver & Pharr method show load dependency as the manner of ISE depicted in figure 38. In order to describe the ISE behaviour, several relationships between applied indentation test load, P, and penetration depth, h, have been presented in the literature [151]. In the same figure it, can be observed that at small loads (less than 10 mN), the Bridgman and TSMG samples have the same Hardness. When the applied load increase, the difference between the two different methods of texture increase. It is due to the high porosity present in TSMG sample, see figure 39.

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16

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TSMG technique

Hardness, H (GPa)

14 13 12 11 10 9 8 7 0

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20

30

40

50

60

70

80

90

100

Peak load, Pmax (mN)

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Figure 38. Variation of nanoindentation hardness with the peak load for the tested materials.

2 μm Figure 39. Optical micrograph of YBCO samples textured by TSMG technique.

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d. Pile-up and Sink-in Problems Another common problem in the determination of the hardness by Nanoindetation technique, is the pile-up and sink-in, for more detail see figure 26. The most common effect in YBCO samples is the sink-in effect. The pile-up or sink-in can be obtained from the rate between the final and the maximum penetration depth. If this relation is higher than 1.0, YBCO may present a pile-up effect. On the other hand, if the relation is lower than 1.0, the sample could presented sink-in effect. This study has been performed in both samples textured by these two different methods. Next table summarizes the relation hf/hmax for each phase, load and technique. Every value presented in the table is an average of eighty indentations performed in two different single crystals.

Table 10. Relation hf/hmax for each phase, load and technique. Phase

Peak load (mN)

Bridgman

TSMG

0.72 ± 0.02

0.66 ± 0.04

0.65 ± 0.02

0.63 ± 0.02

Y-123/Y-211

0.69 ± 0.01

0.65 ± 0.01

Y-123

0.72 ± 0.02

0.69 ± 0.03

0.67 ± 0.01

0.65 ± 0.01

0.70 ± 0.01

0.66 ± 0.04

Y-123 Y-211

Y-211

5

10

Y-123/Y-211

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Technique

Y-123/Y-211

30

0.73 ± 0.02

0.70 ± 0.02

Y-123/Y-211

100

0.76 ± 0.01

0.70 ± 0.01

When the applied load is lower than 10 mN, the relationship can be isolated for each phase. At 5 and 10 mN of applied load, the Y-211 and Y-123/Y-211 phases do not present pile-up effect. On the other hand, Y-123 can present sink-in effect. In this case, the area of the residual imprint can be overestimated and produce a reduction of the hardness of these phases; this effect strongly affected the samples textured by Bridgman technique. Whereas it cannot be observed on samples textured by TSMG technique. In conclusion, the pile-up/sinkin can be related to the hardness of each phase. When the applied load is higher than 10 mN, we cannot isolate this relation for each phase present in the studied material; in this case, only the existence of the sink-in in the Y-123/Y-211 can be determined. When a residual imprint present pile-up or sink-in effect, principally for samples textured by Bridgman technique, two different things can be performed: ƒ

Observe all the residual imprints with scanning electron microscopy (SEM) and calculate the contact area. Compare the contact area obtained with SEM and the area supplied by the Nanoindenter. If the areas have the same value, the pile-up effect will not exist, but a fracture mechanism takes place.

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In the case that the two areas have a high scattered value, the Cheng and Cheng approach must be used, and the hardness corrected.

In this study the Cheng and Cheng equations have not be used to correct the contact area, because the size of the imprints obtained by SEM are very similar to the size obtained by the Indenter. This effect has been widely observed in the residual imprints when the YBCO was textured by TSMG technique and the applied load was higher than 10 mN. This will be discussed widely under next heading related to fracture toughness.

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5.00 μm Figure 40. Sink-in effect besides a residual imprint due to the uniaxial compression focused under the Berkovich indenter.

Fracture Toughness Ceramics are generally brittle, and prone to generation of cracks when indented. Toughness estimation by micro fracture Vickers indentation is a well-known and broadly employed technique in ceramic materials. This technique consists of the application of a Vickers (or Berkovich) indenter at a given load, onto the material, sufficiently high to nucleate cracks at the corners of the imprint, and further measure the crack length produced at the corner of the imprint, c, in order to evaluate the fracture toughness, KIC. In case that very small volumes of material are going to be evaluated obtuse indenters are required (like Berkovich), which do not produce cracks large enough for a correct estimation of toughness fracture. YBCO samples present different mechanisms of fracture, such as crack at the corners of the imprints, chipping, radial cracks inside the imprint, sink-in, and others. When the sample was textured by Bridgman technique method the following observations have been made:

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Y-211: at applied loads lower than 10 mN, have lateral cracks producing chipping; this effect can be observed in figure 31b. Y-123: when the applied load is lower than 10 mN any fracture mechanism can be observed (see figure 31a). Y-123/Y-211: at 10 mN of applied load, a propagation of crack is only observed at one of the corners; in this case, the fracture toughness cannot be calculated (see figure 31 c). When the applied load was 30 mN, the toughness of YBCO samples textured by Bridgman technique can be obtained. Figure 33a, shows a crack at the corner of the nanoindentation imprints performed at 30 mN. With the crack length and the Palmqvist equation (equation 27), the toughness can be calculated for an applied load of 30 mN. The obtained value is 2.85 ± 0.11 MPa·m1/2. When the applied load was 100 mN, porosity besides the imprint can be observed due to the localized deformation during the nanoindentation test, and also radial cracks inside the imprint (figure 33b). The radial cracks are vertically halfpenny type cracks that occur on the surface of the specimen outside the plastic deformation zone, and at the corners of the residual impression at the indentation site. These radial cracks are formed by a hoop stress and extended downward into the indentation; an example of the radial cracks can be observed in figure 33b.

When the sample was textured by TSMG technique method the following can be observed:

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ƒ

ƒ

ƒ

When the applied load was 10 mN, for Y-211 particles, the same fracture mechanism has been observed. In this case, chipping is the most important mechanism but also exists another mechanism that reduce the toughness of the studied sample; this factor is the porosity near the imprints. For Y-123 and Y-123/Y-211 at the same load, any fracture mechanism has been observed. At this applied load, the fracture toughness has not been calculated (see figure 32). When the applied load was 30 mN, any crack propagation have been observed; only one fracture mechanism could be detected, in this case; in the residual imprint internal cracks can be observed, perpendicular to the surface (see figure 32a). Figure 32a, shows that the Y-211 are harder than Y-123, it is due to Y-211 particles, which after the unloading are not deformed. When the applied load was 100 mN, any crack propagation in the corners of the imprint can be observed. In this case a sink-in and porosity near the indentation imprint can be observed, see figure 35b. The sink-in had been predicted with the relation between the final and the maximum penetration depth; this result is in accordance with the value obtained and presented in table 15. Nest figure, shows cracks inside the nanoindentation imprint, and a sin-in near the imprint.

In conclusion, YBCO samples textured by Bridgman and TSMG technique are brittle materials. The Y-123 phase is a soft phase because absorb all the plastic deformation during the nanoindentation imprint. On the other hand, Y-211 is a hard material and it is broken during the indentation. When the indenter was performed at the interface, any

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fracture mechanisms have been detected. When the applied load was higher than 10 mN, the fracture toughness can be calculated for the samples textured by Bridgman technique. Nevertheless, for TSMG technique, internal perpendicular cracks have been detected. At maximum applied load, 100 mN, a sink-in effect is produced and porosity can be created due to it.

5.00 μm

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Figure 41. Residual imprint with different fracture mechanism when the applied load was 100 mN and the material textured by TSMG technique.

The fracture toughness obtained for YBCO samples are in concordance with the values published in the bibliography, concretely with Joo et al. [139 and 141] for samples with 15% w/w of Ag which value is 2.80 MPa·m1/2, or 10% w/w Ag which present a value of 2.60 MPa·m1/2. Both values have been obtained with single-edge-notch beam. Sheahen et al. [130] present a fracture toughness value of 3.80 MPa·m1/2 for a sample with 10% w/w Ag.

2.2.2. Elastic Deformation Experimental Conditions The nanoindenter (XP System, MTS) used in this work was equipped with a CSM attachment. The harmonic displacement for the CSM was 2 nm with a frequency of 45 Hz. The test was carried out on two different samples for each technique of texture studied. Figure 42 shows indentation imprints performed when the maximum load was 700 mN on the ab-plane of the monodomain with a spherical tip nanoindenter. Also in this figure also, it can be observed that the Y-211 particles are homogeneously distributed in the textured sample, so that they can be easily identified but not indented separately. It is due to the size of the Y-211.

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50 μm Figure 42. OM micrographs of residual impressions performed on the surface of YBCO orthorhombic phase (ab-plane) at room temperature, when the applied load was 700 mN and the sample textured by Bridgman technique.

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Stiffness versus Displacement into SURFACES Before plotting the indentation stress-strain curves it is crucial to determine the effective elastic stiffness of the YBCO material examined. Plot of S versus displacement into surface, allows to determine if the contact point indenter-sample is good or not. In that case a straight line can be observed, whereas a non linear relation is found when the contact is not good enough. Figure 43, shows the tendence of stiffness with the penetration depth. This figure gives us information about which stress-strain curves will be suitable or present high scattered values. When S versus h is a straight line, the contact point between indenter and sample is right and therefore the stress-strain curve will be suitable. This implies that S is not affected by pop-ins or plastic deformation. A bad contact point can give us an overestimated strain value, and therefore inaccurate yield strength is obtained. In this case, MTS software must be used in order to correct experimental values. Next figure shows the experimental curves and the same corrected with MTS software, for both samples studied.

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b) Figure 43. Plot of harmonic stiffness versus penetration depth as determined from spherical nanoindentation for YBCO materials textured by a) Bridgman technique and b) TSMG technique.

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b) Figure 44. Continued on next page.

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d) Figure 44. Contact point for samples textured by, a) Bridgman technique with bad contact point, b) TSMG technique with bad contact point, c) Bridgman technique with good contact point, and d) TSMG technique with good contact point.

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25

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Displacement (nm) Figure 45. Load/depth-of-penetration results for YBCO monodomain ab-planes, with a 25 μm spherical indenter, for Bridgman technique at 100 nm of penetration depth.

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50 P (mN)

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60

Pop-in

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Displacement (nm) Figure 46. Load/depth-of-penetration results for YBCO monodomain ab-planes, with a 25 μm spherical indenter, for Bridgman technique at 200 nm of penetration depth.

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Determination of the Elasto-plastic Transition The typical load-displacement curves obtained when the YBCO ab-plane is indented with a 25 μm spherical indenter, figure 46, are characterized by the elasto-plastic transition or commonly known as pop-in, indicated in the figure as a change in the slope of the loading curve.

2.2 μm

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(a)

2,2 μm (b) Figure 47. AFM image at different penetration depths for YBCO monodomains, textured by Bridgman technique, a) 200 nm and b) 800nm.

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If we observe the surface of the imprints with AFM, residual imprint is found, for penetrations depths lower than 150 nm which correlates with the change in the slope of the loading curve (pop-in effect). For higher penetration depth, different residual imprints can be observed by AFM (Figure 47). During the first steps of nanoindentation (elastic regime), the load curve can be adjusted a Hertzian contact (P = C·h3/2); when the applied load is higher than the elasto-plastic transition, a residual imprint can be observed, and the load curves can be adjusted as P = C·h2. In both cases, C is the slope of the load curve in each regime.

Stress-Strain Curves In the case of spherical indentation, by plotting po (mean contact pressure) against a/R (contact radium/indenter radium) it is possible to determine the point at which there is the transition from elastic to elasto-plastic regime, shown in Figure 46 as a change in the slope for the studied samples. These values are around 3.5 GPa and 3.0 GPa for Bridgman and TSMG, respectively. To obtain the yield strength, next equation has been used:

1 2

σ ys = ⋅ (1 − 2υ ) ⋅ po

(44)

Elastic (Hertzian theory)

6 5

Stress, po (GPa)

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where, σys is the yield strength, υ is the Poisson ratio of the material and po is the mean contact pressure at the point of elastic to elasto-plastic transition (obtained from data in Figure 48). Values obtained for Bridgman and TSMG samples are 700 MPa and 600 MPa respectively. The difference between both techniques can be attributed to the different amount of defects, as well as correlated to data for hardness, Young’s moduli and fracture toughness previously reported.

4 3 2 1 0 0

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0.15

Strain, a/R (Dimensionless)

a) Figure 48. Continued on next page.

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Elastic (Hertz theory)

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Strain, a/R (dimensionless)

b) Figure 48. Indentation stress/strain curves for YBCO textured by a) Bridgman, and b) TSMG techniques.

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2.2.3. Kinetic Study by Nanoindentation Technique of TSMG Samples along the c-axis at 450ºC An important phenomenon in bulk superconductors obtained by TSMG, is the formation of cracks due to the inherent brittleness of the Y-123 phase matrix. These cracks form during the texturing of the superconducting monolite and play an important role in the limitation of current flow. As-grown Y-123/Y-211 bulk superconductors prepared by the TSMG process must be further oxygenated in order to transform from tetragonal to orthorhombic phase. Oxygenation at temperatures around 400-450ºC could take between one to two weeks for a 2 cm diameters and 4 mm height sample. Oxygen diffuses through cracks previously formed during cooling in the texturing process. Two types of macrocraks are formed: (i). a/b-macrocraks parallel to the a/b-plane and (ii). c-macrocraks parallel to the a/c-plane. While the a/b-macrocraks extend over almost the whole sample, the c-macrocrak length is limited by the a/b-macrocraks spacing [152]. c-axis macrocrack network represents a mesoscopic defect in bulk superconductors obtained by TSMG, and limits the local current density in the sample.

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Reddy and Rajasekharan [153] studied the inner structure of partially oxygenated Y-123 bulk melt-textured material. During the oxygenation process microcraks parallel to the a/bplanes (a/b-microcracks) appear in the melt processed YBCO (dark lines in Figure 49). Y-211 phase particles (dark), and twins can also be seen.

a,b c

10µm

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Figure 49. The microcracks parallel to the a/b-planes (a/b-microcracks) in the melt processed YBCO (dark lines). Y-211 phase particles (dark) and twins can also be seen. (From Diko et al., Superconductor Sci. Technol, 16, 2003, pages 90-93).

Experimental Conditions Samples used to perform this study were obtained by the TSMG technique. A small NdBCO melt textured grain was used to seed the melt growth process; it was placed at the centre on top of the basal pellet surface before heating and the melt-growth process was applied. From textured sample, a 20 mm x 20mm x 4mm monodomain was obtained that was further cut into four pieces of 10 mm x 10 mm x 4 mm. Afterwards, these four pieces were introduced in a horizontal furnace for an oxygen annealing (99.999% purity) at 450ºC for different oxygenation times, 24 h, 48 h, 96 h and 187 h. After oxygenation each piece has been cut following next schema. The applied load used to determine the mechanical properties was 10 mN. At this load, the mechanical properties of Y-123 can be isolated, and the hardness evolution with the oxygenation time can be obtained, as oxygen concentration gradients lead to large mechanical stress in the material. The oxygenation diffusion coefficient in the ab-plane is about 104-106 times larger than in c-direction [154].

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The structure of all samples was observed by optical microscopy in polarized light after polishing and etching in 1 w/w % HCl in ethyl alcohol. Nanoindentation imprints were observed by FE-SEM.

Figure 50. Preparation samples scheme.

Oxygenation Defects and Macro-microckracking in Melt-textured YBCO Bulks First, the c-axis section of the oxygenated samples was investigated in order to locate regions of tetragonal phase which can be clearly distinguishable, under polarized light as darker or brighter regions depending on the orientation of the Y-123 crystal axes to the vector of polarized light and/or the mutual position of the analyses and polarizer [155]. The macrocracking and the porosity were the dominant structural changes in the samples after oxygenation. Observation at optical microscope under polarized light revealed a high density of porosity formed during the oxygenation process. Figure 51, shows the optical micrographs of c-axis YBCO samples textured by TSMG technique at different oxygenation times.

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Figure 51. Optical micrographs of the c-axis at different oxygenation times, a) 24 h, b) 48 h, c) 96 h and d) 187 h. O, orthorhombic phase and T, tetragonal phase.

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Different defects produced after the oxygenation process can be seen in Figure 52. If we observe in detail a porosity, we will see a microcrack (Figure 52a). Another problem during this process is the presence of macrocracks,.that have been generated during the cooling in the texture process due to the different expansion coefficients between Y-123 and Y-211, and that growth during the oxygenation process, see figure 52b. Both defects produce an embrittlement of the material of study.

7.49 μm

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a)

15 μm b) Figure 52. Continued on next page.

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66.7 μm

c)

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Figure 52. Defects observed by FE-SEM after the oxygenation process, a) formation of microcracks around the porosity, b) generation of macrocracks and c) micrograph of the c-axis after oxygenation process.

Macrocracks can easily form in regions with low Y-211 concentration [152 and 156], because the fracture toughness increases with the concentration of Y-211 particles [157 and 158]. Moreover, Y-211 phase has a lower thermal expansion coefficient than Y-123 phase in the c-direction [159], and Y-211 is therefore under tension during cooling down, and a/bmacrocracking in the Y-211 is enhanced. Length and the spacing of a/b- and c-macrocracks, increase continuously with increasing the oxygenation temperature. At higher oxygenation temperature the a/b- and especially the cmacrocracks spacing is significantly increasing. In contrast, at lower oxygenation temperature (such as 450ºC) the c-macrocracks are very fine and their density is very high. Tensile stress in the c- and a/b- direction are induced by shrinkage of the lattice parameters of the Y-123 phase with oxygen content. These stresses are then responsible for a/b- and c-crack formation in the oxygenated layer. Oxygen may then flow along the c-cracks to the a/c-surfaces, oxygenate the c-crack’s surface and c-crack’s tip and further c-crack propagation. It can be observed in figure 52c, that in the Y-123 material, the network of pores, dispersed in the Y-123/Y-211 skeleton, also supports the orthorhombic transformation. This means that the region around the pores is oxygenated by fast diffusion in the a/b-direction. All these microstructural items are important during the process of oxygenation of YBCO bulk materials.

Determination of the Kinetics of Oxygenation by Nanoindentation The hardness of YBCO bulk material has been studied from the oxygenation surface oxygenation until the bottom of the sample along the c-axis. For this study the thickness of the samples used were 4 mm. Figure 55, shows the different nanoindentation imprints along the c-axis of the samples studied.

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Figure 53. Nanoindentation imprints along the c-axis in YBCO samples textured by TSMG technique.

Figure 53, shows a matrix of indentations being the distance between each are 40 μm. The representation of hardness versus the position from the upper surface of the sample, can give us information about the oxygenation process.

Tetragonal

11.0

24 h 48 h 96 h 187 h

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10.5 10.0 9.5 9.0 8.5

Orthormhombic

8.0 7.5 7.0 0

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Height from the upper surface along the c-axis (μm) Figure 54. Hardness versus height for YBCO samples textured by TSMG technique.

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Figure 54, shows the tetragonal-orthorhommbic transition in YBCO samples textured by TSMG technique. In this figure, it can be observed that the transition is function of the oxygenation time. When the sample is oxygenated during 187h, has a little portion of tetragonal phase. The high disperssion presented in this figure is due to superficial defects produced during the oxygenation process. At the end of oxygenation, all samples present a tetragonal structure; in this case, the hardness value is higher than for orthorhombic structure, which has a macro- and microcraks that produce a reduction of the hardness of the YBCO samples. When the oxygenation time was 187h, the transformation has taken place in the first 3000 μm, remaining the rest of the sample with a structure not purely tetragonal. In this case, an interface between tetragonal and orthorhombic phase could be formed. For samples with a height of 4 mm, the oxygenation time needed to complete the transformation is greater than 187 h.

Prediction of the Oxygenation Time in YBCO Bulk Materials

Distance from surface along the c-axis (mm)

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To predict the oxygenation form a bigger sample, we represent the distance from the surface that has undergone the transition versus the time needed to achieve it.

4000 3500 3000 2500 2000 1500 1000 500 0 0

50

100 150 200 250 300 350 400 time (h)

Figure 55. Distance from the surface along the c-axis transformed into orthorhombic phase versus time of oxygenation for YBCO samples textured by TSMG technique.

As can be seen in this figure the time necessary to complete oxygenate a sample with a height of 4 mm is around 380h. Moreover, figure 55 shows two different tendends, a quick oxygenation process through macrocracks formed during cooling in the texture process, and a slower oxygenation process that take place when microcracks are formed, induced by oxygen.

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In: Superconducting Magnets and Superconductivity… ISBN: 978-1-60741-017-1 Editors: H. Tovar and J. Fortier, pp. 169-196 © 2009 Nova Science Publishers, Inc.

Chapter 4

UNDERSTANDING THE ROLES OF HEAVY ION AND GAMMA-IRRADIATIONS ON THE MAGNETIC AND TRANSPORT PROPERTIES OF SUPERCONDUCTORS I.M. Obaidat1,∗, B.A. Albiss2 and M.K. Hasan (Qaseer)2 1

Department of Physics, United Arab Emirates University, Al-Ain 17551, United Arab Emirates 2 Superconductivity & Magnetic measurements Lab, Department of Physics, Jordan University of Science and Technology, Irbid 22110, Jordan

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Abstract The roles of point defects created by γ-irradiation and chemical doping along with the roles of extended defects such as grain boundary regions and columnar defects created by Pb-ion irradiation on the behavior of the critical current density, Jc have been investigated in several superconductors. The critical current density in YBa2Cu3O7-δ (YBCO) polycrystalline sample doped with Mn was found to be significantly enhanced. The enhancement of Jc was found to be more significant at the lower temperatures. Magnetization measurements were carried out on YBa2(Cu1-xBx)3O7-δ (x = 0, 0.05, 0.1) polycrystalline samples. Considerable increase in the hysterisis width of the magnetization M versus the applied magnetic field H with increasing boron concentration was observed in these samples suggesting an enhancement of the vortex pinning forces in these samples. A limitation of the role of point and columnar defects on enhancing Jc was found in all our samples. We report an explanation of the effect of γirradiation and Pb-ion irradiation doses on Jc in these materials. The explanation is based on combining several competing mechanisms of irradiation which we believe to take place mainly in the regions of the grain boundaries. The influences of these mechanisms were found to vary with the irradiation dose level. The effect of γ-irradiation on the normal state resistivity of MgB2 polycrystalline superconducting specimens was also investigated. An increase in the normal state resistivity and a broadening of the resistive transition to the superconducting state were observed with increasing γ-irradiation dose. Different temperature ∗

E-mail address: [email protected]. Tel. 00971(3) 7134510, Fax. 00971(3)7671291 (Corresponding author).

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170

I.M. Obaidat, B.A. Albiss and M.K. Hasan (Qaseer) dependence of normal-state resistivity and different residual resistivity ratios, RRR were obtained for different doses.

Keywords: MgB2, gamma irradiation, Pb-ion irradiation, residual resistivity ratio, nanoparticles doping, critical current density.

1. Introduction

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1.1. Normal State Resistivity of MgB2 Bulk Sample Since their first discovery in 2001 [1], magnesium diboride MgB2, superconductors have become the subject of an enormous interest. This is the case because of it is being a simple inter-metallic binary compound with negligible grain boundary effect and small anisotropy and thus its suitability for device applications. The mechanism of superconductivity in MgB2 was found to be electron phonon mediated [2, 3]. MgB2 superconductors were also found to have two superconducting gaps [4, 5]. Compared to high temperature superconductors (HTSCs), MgB2 has a lower critical temperature Tc ∼ 39 K, a simple hexagonal structure (probable absence of problems associated with weak links in HTSCs, a lower anisotropy, a relatively low cost of production and a high critical current density, Jc. All these properties make MgB2 promising material for applications in the temperature range 20 - 30 K. The underlying mechanism of superconductivity in this system is still an open question [6, 7]. Several groups [8-11] studied the resistivity in the normal state of MgB2. However, the results reported were found to be different from group to group. The reported resistivity at room temperature ranges from 9.6 to 100 µΩcm and that at 40 K from 0.38 to 21 µΩcm [8, 9]. Some groups [8, 11], reported that the resistivity in the normal state follows T3 behavior while others [9, 10] claimed a T2 behavior. The residual resistivity ratio, RRR which is the ratio of the resistivity at room temperature to that at Tc, RRR = ρ(297 K)/ρ(39 K), was also observed to vary with samples. The RRR reported by in [11] is about 25 while other groups [9, 10] reported RRR values between 2 and 3 even for epitaxial thin films [12]. The superconducting transition temperatures in those studies were found to be nearly the same although their properties in the normal state were found to be quite different. Contradictory results on the effect of ion irradiation on the critical temperature and on the normal state resistivity in MgB2 were also reported by several groups [13, 14].

1.2. Enhancing the Critical Current Density in HTSCs HTSCs in magnetic fields of strength higher than Hc1 and smaller than Hc2 are known to be in the mixed state, where magnetic field penetrates the superconductors in form of quantized magnetic vortices. When a transport current density, J is applied to HTSC in the mixed state it will produce a Lorentz force, FL on the vortices trying to move them. The Lorentz force is given by; FL= (1/c) J×Φo where Φo is the quantized magnetic flux per vortex. If the Lorentz force is not opposed by pinning forces, an electric field, E = (-1/c) B×v will be induced in the sample causing energy to be dissipated, were v is the velocity of vortices transverse direction to both J and B, and B is the applied magnetic field.

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Understanding the Roles of Heavy Ion and Gamma-Irradiations on the Magnetic… 171 Unfortunately, the pinning of vortices in HTSCs is fairly weak, especially at high temperatures [15]. The vortices can be effectively pinned by any structural inhomogeneities of the material. A lot of work has been done to increase the critical current density Jc that can be sustained by these materials. Defects created in HTSCs using several types of energetic radiations were found to be very successful in increasing the vortex pinning forces [16-19]. Other techniques have been used during the sample preparation by changing the heat treatment where good quality samples might be obtained to carry high currents [20, 21]. Introducing nano-particles in bulk superconductors has been also used to increase the pinning forces [22, 23] and thus to enhance Jc. Although great progress has been made to prepare HTSCs with high critical current density, some key problems are still unsolved so far. Jc values in HTSCs are usually drops rapidly with the increase of applied magnetic field, especially at high temperatures. Interestingly, chemical doping [24, 25] was confirmed to be an effective method in improving the behavior of the critical current density under the applied magnetic field. Introducing chemical doping is proven to effectively enhance the critical current density in HTSCs [22, 23]. When doping YBCO with Manganese, neutron diffraction experiments [26] revealed that Mn takes the Cu site which constitutes a CuO chain along the b-axis. It was also reported [27] that a reduction in the diamagnetic signal and a broadening of the transition occur beyond Mn concentration of 1.0 %. A reduction in the superconducting fraction and a broadening of the transition after Mn doping was also reported [28]. In Ref. [29], DC magnetization measurements on YBa2(Cu1-xMnx)3O7-δ polycrystalline samples with various Mn concentrations were reported. A monotonic suppression of the zero field critical current density was reported up to Mn concentration of 2.5 % followed by saturation for the larger concentrations. But interestingly, an anomalous increase of the vortex pinning forces and the zero field critical current density was reported at 77 K for x = 2.0 %. This anomalous behavior was attributed to the matching field condition at that concentration. The effect of thermal neutron irradiation on the critical current density in boron-doped melt-textured YBCO superconductors was investigated [30]. The critical current density was found to be enhanced in the irradiated born-doped samples. But the effect of boron alone on the critical current density was not reported.

1.3. Effects of Heavy Ion - and γ -Irradiations on Superconductors There are two types of disorder that can appear in HTSC materials; microscopic and macroscopic. The microscopic disorder, which can consist of things such as impurities, vacancies is associated with perturbation of crystal lattice on the atomic scale. The macroscopic disorder such as grain boundaries is associated with structural inhomogeneities such as granular structure. In the Ginzburg-Landau (GL) theory [31], the superconducting wavefunction (the order parameter) Ψ is characterized by its amplitudeψ and phase ϕ iϕ

through Ψ = ψe . The microscopic disorder suppresses superconductivity by reducing the amplitude of the order parameter. The macroscopic disorder on the other hand suppresses superconductivity by suppressing the phase coherence of the superconducting electrons between weakly coupled superconducting grains. Both of these mechanisms can operate simultaneously in the HTSC polycrystalline sample.

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The interaction of the energetic γ-rays with the HTSC material causes two major effects; atomic ionization [32] and atomic disorder [33]. When the energetic γ-photon collides with the atoms of the material, a Compton electron and a photon with lesser energy will be created. In addition to that, some of the energy will imparted to the atoms by momentum conservation. If the recoil energy is higher than the displacement threshold energy, atomic displacement will occur in the material. The atomic disorders created by γ-irradiation are mainly of the microscopic type of disorder such as, vacancies and interstitials (point-like defects) of small separations and randomly distributed through the sample. The vortex pinning forces of these point-like defects are not expected to be significant in the region of the grains or the grain boundaries. Because of their low value of displacement threshold energy of about 10 eV, oxygen displacements from the CuO2 planes (into vacant oxygen sites along the a-axis and baxis) are the most common atomic displacement. The energetic Compton electrons are known to produce such inhomogeneous distribution of oxygen atoms [34, 35]. The removal of oxygen atoms from the CuO2 planes creates disorder in the lattice and also causes the depletion of holes in the CuO2 planes. This depletion of holes results in a reduction of the carrier concentration which leads to suppression of the superconducting properties [36, 37]. This effect will be more pronounced in the regions of the grain boundaries since they are usually depleted from oxygen. The photon which was created after the γ-photon collides with the atoms of the material will dissipate its energy producing further ionizations. On the other hand, the energetic Compton electron ejected after the collision of γ-photon with the atoms of the material, mostly will collide with the atoms of the material causing excitations of the electrons to the valance band, producing holes and energetic phonons. Thus, the created photon and Compton electron produce further ionization. Atomic ionization enhances the superconducting properties of the whole material especially those with low density of charge carrier. But because the density of charge carrier in the regions of the grains is relatively high [38], the γ-ray fluxes used can not produce a significant enhancement of the superconducting properties in them. On the other hand, since the grain boundaries and the regions around them are strongly depleted from charge carriers [39], the atomic ionizations created by γ-rays will have a significant effect on enhancing the superconducting properties. Also the regions of grain boundaries are known to produce effective vortex pinning forces, preventing vortex motion perpendicular to the grains orientation [40], and leading to an enhancement in the transport properties. But at high γ-irradiation doses, a well pronounced and opposite effect might take place in the regions of the grain boundaries. The weak links between the regions of the grains might be damaged by the high doses of γ-rays. According to GL theory, this damage of weak links suppresses the phase coherence of the superconducting wave functions between the grains in polycrystalline HTSCs which leads to a suppression of the transport properties [41, 42].

1.4. Aims of the Article There are four goals of this article: The first goal is to report the results of a study on the effect γ-irradiation on the nature of the resistive behavior in the normal state of MgB2 bulk samples. Other than the broadening of the transition, we have found that, γ-irradiation has very slight effect on the transition temperature. We have also found that γ-irradiation has a

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Understanding the Roles of Heavy Ion and Gamma-Irradiations on the Magnetic… 173 significant effect on the temperature-dependences of the normal state resistivity, and on the residual resistivity ratio (RRR). The second goal of the article is to report on the behavior of critical current density in YBCO polycrystalline samples doped with Manganese and Boron. This study was done at high temperatures and under several applied magnetic field values. Our I-V measurements on polycrystalline YBa2(Cu1-xMnx)3O7-δ samples with concentrations x = 0 and x = 0.02 show a significant enhancement of the effective pinning forces in the Mn-doped sample. We discuss the results in terms of the microstructure of these samples. We also report on the effect of boron addition on the pinning properties of YBCO polycrystals as function of magnetic field at high temperatures. Our I-V measurements on polycrystalline YBa2(Cu1-xBx)3O7-δ samples of with x = 0, 0.05, 0.10 show a clear enhancement in the critical current density Jc, with boron doping. This enhancement was found to be more significant at small applied magnetic fields. The third goal of this article is to report on the limitations of γ-irradiations and heavy ion irradiation on enhancing the critical current density in MgB2 and other HTSCs. The fourth and most important goal of this article is to try to understand and explain the mechanisms that might cause this limitation effect of irradiations.

2. Experimental Procedures Seven different superconducting samples were used in this study. These are MgB2 polycrystalline sample, YBa2Cu3O7-δ crystalline sample, YBa2Cu3O7-δ polycrystalline sample, Mn-doped YBa2Cu3O7-δ polycrystalline sample, B-doped YBa2Cu3O7-δ polycrystalline sample, Polycrystalline Bi1.6Pb0.4Sr2Ca2Cu3O10 sample, and Tl2Ba2Ca2Cu3O10 tape. Now we shall explain the experimental methods used to prepare and characterize these samples.

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2.1.1. The MgB2 Polycrystalline Sample Stoichiometric mixture of Mg and B was made from Mg turnings and finely powdered boron. The mixture was grinded and compacted under a pressure of 100 MPa into a pellet with 10 mm in diameter and 1 mm in thickness. The pellet was then wrapped in a tantalum foil and heat treated in a programmable three-zone tubular furnace in flow of a commercially available gas mixture of 92% Ar and 8% H2. The pellet was held at 600 ºC for 2 hours, followed by 800 ºC for 2 hours and finally at 950 ºC for one hour, then it is cooled down to room temperature. The produced pellet was grinded for 1 hour, pressed again into a pellet under a pressure of 200 MPa. This pellet is heated at 950 ºC for 2 hours followed by a slow cooling to room temperature at the rate of 1 ºC/min. The resulting pellets, all with nominal composition of ration 1:2 were cut into rectangular specimens approximately 1mm thick with a diamond saw. Based on X-ray diffraction studies, the samples used in the present study were found to be nearly single hexagonal MgB2 crystalline phase with insignificant traces of secondary phases and small amounts of MgO impurities possibly due to the use of commercial gas mixtures. The resistivity versus temperature, ρ(T) curves were obtained using a standard four point dc-method. The sample was cooled in a closed cycle helium cryostat (Leybold 320) under a vacuum of 10-3 mbar. The temperature was measured with a calibrated platinum resistance sensor with an accuracy of ± 0.1 K. The electrical probes were attached to a rectangular-shaped samples (~ 8×4×1 mm3) using silver paste. Gamma irradiation of the

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samples was carried out at room temperature by a Co60 gamma ray source at a dose rate of 0.5 MR/h.

2.1.2. The YBa2Cu3O7-δ Crystalline Sample Our YBCO crystalline samples were circular disks, 2 mm in diameter, 0.1 mm thick, which had been cut from a melt-textured YBCO platelet. The crystallographic c axes of the samples were aligned normal to the disk planes. One sample was left without irradiation. Two other samples were irradiated with 208Pb56+ ions along both opposite directions along the c axis, since the penetration depth of the irradiation was only about 30 µm. The density of the columnar defect created by ion irradiation is frequently measured in terms of a dose matching field BΦ, which denotes the columnar defect density in terms of the vortex density corresponding to a particular magnetic field value. Knowing that the total magnetic flux (Φ) in the sample is given by Φ = nΦ o (where Φo = 2.0679×10-15 T m2 and n is the number of vortices per unit area), we can estimate the density of columnar defects created by each irradiation dose. The density of columnar defects will be equal to the density of vortices corresponding to each dose matching field. For example, at BΦ = 1 T, the number of columnar defects will be ~ 4.836 × 1014/m2 (or ~ 4.836/λ2). In our YBCO crystalline samples, the irradiation corresponds to dose matching fields of BΦ = 1 T in one sample and of BΦ = 2 T in the other one. The ion beam was initiated at 1.38 GeV from the Argonne Tandem Linear Accelerator System (ATLAS) and was passed through a thin gold foil to achieve a uniform beam distribution and through a 4.8 mm diameter collimator before impinging on the sample.

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2.1.3. The YBa2Cu3O7-δ Polycrystalline Sample The polycrystalline YBa2Cu3O7-δ sample used in this study was prepared by the conventional solid-state reaction method from stoichiometrically mixed high purity (99 %) powders of BaCO3, (99.99 %) powders of Y2O3, and (99.95 %) powders of CuO. The powders were thoroughly mixed and ground in agate mortar for 2 h to get a powder of uniform gray color. The powder was then pressed into a pellet by using stainless steel molt. The pellet was then placed in a three-zone furnace where the temperature was raised from room temperature to 500 oC and held at this temperature for 20 h. The sample was then cooled down to room temperature and ground again using agate mortar for 2 h. The mixture was then pressed under 10 ton pressure. The pellet was then placed in a three-zone furnace where the temperature was raised from room temperature to 930 oC and held at this temperature for 20 h after which it was cooled to room temperature. The pellet was then ground again and pressed with 10 tons of pressure and heated at 930 oC for 10 h. The sample was then cooled down at a rate of 5 K/min to 500 oC at which it was sintered and annealed in flowing oxygen for 5 h. The sample was finally cooled down to room temperature and cut in a rectangular shape.

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Understanding the Roles of Heavy Ion and Gamma-Irradiations on the Magnetic… 175

2.1.4. The Mn Doped YBa2Cu3O7-δ Polycrystalline Samples The Mn-doped polycrystalline YBa2(Cu1-xMnx)3O7-δ sample with x = 0.02 used in this study was prepared by the conventional solid-state reaction method from stoichiometrically mixed high purity (99 %) powders of BaCO3, (99.99 %) powders of Y2O3, (99.95 %) powders of CuO, and high purity powder of MnO2.

2.1.5. The B Doped YBa2Cu3O7-δ Polycrystalline Samples Samples of YBa2(Cu1-xBx)3O7-δ with x = 0, 0.05, 0.10 has been prepared. The samples were grinding for different times and reheated three times. The powders were die preset and a disk shaped was made of radius 5 mm in diameter and approximately of 1 mm thick. Three pellets were made each for different x. The sample structures were studied using the x-ray diffraction and the average grain size was estimated from the scanning electron microscopy SEM morphology of sample surface.

2.1.6. The Bi1.6Pb0.4Sr2Ca2Cu3O10 Polycrystalline Sample

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The polycrystalline Bi1.6Pb0.4Sr2Ca2Cu3O10 sample was prepared by solid-state reaction technique from stoichiometrically mixed powders of Bi2O3, PbO, SrCO3, CaCO3 and CuO. The oxides and carbonates were thoroughly mixed and ground for 2 h followed by a threestep calcinations treatment each of 20 h in air with intermediate grinding and pressing between steps. The first step was performed at 800 oC. Subsequent steps were performed at temperatures up to 850 oC in flowing oxygen. Then the calcined product was ground and pressed to form 10 mm-diameter disks, and then sintered for 50 h at 850 oC in air. The sample was cut in a rectangular shape with dimensions 3mm×2mm×1mm.

2.1.7. The Tl2Ba2Ca2Cu3O10 Tape In preparing the Tl2Ba2Ca2Cu3O10 tape appropriate amounts of high purity Tl2O3, BaO2, CaO and CuO were thoroughly mixed and ground for 2 h. The mixed powder was then rapped in a gold foil and subjected to a three-step heat treatment, each of 5 h at 890 oC in air with intermediate grinding and pressing between steps. The resulting powder was pressed into pellets and annealed at 910 oC in flowing oxygen. The powder was then packed into a silver tube with 4 mm outer and 2.4 mm inner diameters. The tube was then rolled to produce a 0.2 mm thick and 4 mm wide tape. The tape was then annealed at 850 oC in air for 5 h followed by a uniaxial pressing at 500 MPa. The annealing cycle was repeated to produce a nearly single phase and textured Tl2Ba2Ca2Cu3O10 tape.

2.2. Gamma Irradiation Source The samples were irradiated at room temperature by a 60Co γ-ray source at a dose rate of 0.5 MR/h. The irradiation of the samples was conducted in a gamma research irradiator instrument (Gamma-facility PX-γ-30, Issledovatelj, Techsnabexport, Moskivia) in Jordan

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Atomic Energy Commission, Amman, Jordan. The instrument is spherically shaped and has a sample capacity of 4400 cm3 and uses 60Co as an irradiation source.

2.3. Magnetic Measurements In determining the strength of the vortex pinning in the YBCO crystals, a vibratingsample magnetometer (VSM) was used to measure the saturation remanent magnetization MR. According to the critical state model [24], the critical current density, Jc in a thin slab superconductor is proportional to the width of the magnetization hysterisis loop, ΔM of the decreasing and increasing field branches. In our measurements, this ΔM is directly proportional to MR. Thus MR is directly related to Jc, which is proportional to the strength of the pinning. Our planned procedure was to first establish the field cooled (FC) MR at 4.2 K along the c axis or the ab plane. For example, in establishing the (FC) MR at 4.2 K along the c axis, the maximum value of the magnetic field was applied along the c axis at room temperature. The sample was then cooled down to 4.2 K under the applied magnetic field after which the magnetic field was reduced to zero. The remaining nonzero magnetization value is MR along the c axis. The variations of MR for each sample are then measured as the temperature is gradually raised in zero magnetic fields. For our YBCO crystals, we measured the remanent MR at 4.2 and 77 K after field cooling to 4.2 K in our maximum field (≈ 14 kOe). In the magnetization measurements, a vibrating-sample magnetometer (VSM) was used to determining the strength of the vortex pinning from the width of the magnetization hysterisis loops.

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2.4. Voltage-Current Measurements In obtaining the critical current density in the polycrystalline samples, we conducted voltage-current (V-I) measurements using the standard dc four-probe method. The probes were fixed to the samples with silver paste. The sample was cooled using a closed cycle helium cryostat (Leybold 320) under a vacuum of 10-2 mbar. The voltage was measured using a sensitive nano-voltmeter (Keithley 2182) with accuracy better than 10 nV. The current was measured using a programmable constant current source (Keithley 2245) with accuracy of 1 µA. The current density used for electrical resistance measurement was 1 mA/cm2. The temperature was measured with a calibrated platinum resistance sensor with an accuracy of 0.1 K. In all measurements, the sample was first zero-field cooled down to 74 K, and then the temperature was raised slowly to the required temperature at which the V-I characteristics were measured. The V-I characteristic curves were obtained at all temperatures for all applied magnetic fields parallel to the surface of the samples. From these V-I characteristic curves we obtained the critical current density at each applied field and temperature using an electric field criterion of 1μV/cm.

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Understanding the Roles of Heavy Ion and Gamma-Irradiations on the Magnetic… 177

3. Results and Discussion 3.1. Normal State Resistivity of the MgB2 Sample Figure 1 shows the temperature dependence of resistivity for all samples, before and after irradiation with γ-rays with different doses. It can be seen that, γ-irradiation has a slight effect on the superconducting transition where the transition temperature is found to range between 38 and 39 K. The figure also shows a small broadening of the transition region due to γirradiation. The broadening region becomes larger as the irradiation dose is increased. It can be seen from figure 1 that our normal resistivity results at all temperatures are larger than those reported by some groups [8-11], but they are very comparable to those reported others [31,43].

10 non-irradiated 10 MR 20 MR 100 MR

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ρ (mΩ cm)

8

B = 0.0 Tesla

6

4

2

0

0

50

100

150

200

250

300

T( K ) Figure 1. The temperature dependence of resistivity, ρ(T) at zero applied field for all MgB2 samples, before and after γ-irradiation with different doses.

Another interesting effect of γ-rays on the temperature dependence of the resistivity is the downward curvature observed just above Tc as γ-irradiation dose increases. This downward curvature was also observed in VN films [32,44] as the disorder increases. The downward

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curvature of the temperature dependence of the resistivity can be explained by the superconducting fluctuations. Figure 2 shows the values of normal resistivity at the transition temperature ρ(~ 39 K), taking into account the broadening regions after irradiation. We see that the resistivity increases rapidly and nearly linear with the γ-irradiation dose up to 20 MR. At the dose of 100 MR, the resistivity increased slightly above that of 20 MR and nearly reaches a plateau. These results are in excellent agreement with those reported by Sekkina et al. [31,43] were they found that normal state resistivity increases linearly with γ-irradiation doses up to 20 MR, after which ( between 20 MR and 40 MR) the rate of increase of resistivity slows down.

3.4 B = 0.0 Tesla 3.2

2.8

n

ρ (mΩ cm)

3

2.6

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2.4 2.2

0

20

40

60

80

100

Irradiation Dose (MR) Figure 2. The normal resistivity values at the transition temperature ρn(T), taking into account the broadening regions after irradiation.

To analyze the temperature dependence of ρ(T), the normal-state resistivity values (for T between 39 and 297 K) were fitted to the expression

ρ (T ) = ρ o + aT n ,

(1)

rather than a Block-Grüneisen formula. The experimental data could be fitted very well as shown in figure 3 for the temperature range between Tc and 297 K and for all samples, before and after irradiation. Figure 3 shows the temperature dependence of the normalized resistivity for all samples. We can clearly see the changes to the behavior of resistivity as the dose increases. For the

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Understanding the Roles of Heavy Ion and Gamma-Irradiations on the Magnetic… 179 non-irradiated sample, figure 3 shows that the parameter values obtained from this fit are ρo = 2.139 mΩcm, a = 3.5 × 10-4 mΩcm/Kn, and n = 2.1. This value of the exponent is in between the reported results n = 2 [9] and n = 3 [8]. Figure 3 also shows that for the sample irradiated with 10 MR, the parameter values obtained from this fit are ρo = 2.784 mΩcm, a = 6.6 × 10-4 mΩcm/Kn, and n = 2.0. We also see in figure 3 that for the sample irradiated with 20 MR, the parameter values obtained from this fit are ρo = 2.654 mΩcm, a = 1.19 × 10-3 mΩcm/Kn, and n = 1.5. For the sample irradiated with 100 MR, figure 3 shows that the parameter values obtained from this fit are ρo = 2.908 mΩcm, a = 2.03 × 10-3 mΩcm/Kn, and n = 1.41. The residual resistivity ratio, RRR is also shown for each case in figure 3. For the non-irradiated sample, RRR = 3.26. For the sample irradiated with 10 MR, RRR = 2.96. For the sample irradiated with 20 MR, RRR = 2.78. For the sample irradiated with 100 MR, RRR = 2.64.

1 Nonirradiated 10 MR 20 MR 100 MR

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ρ(T) / ρ(297K)

0.8

RRR = 2.64 n = 1.41 0.6

RRR = 2.78 n = 1.5 RRR = 3.26 n = 2.1 RRR = 2.96 n=2

0.4

0.2

0

50

100

150

200

250

300

T (K) Figure 3. The temperature dependence of the normalized resistivity for all MgB2 samples. The rsistivity is fitted to the expression ρ

(T ) = ρ o + aT n . For the non-irradiated sample, ρ

o=

2.139 (mΩ cm) and

2.1

a = 0.000035 (mΩ cm / K ). For the sample irradiated with 10 MR, ρo = 2.784 (mΩ cm) and a = 0.000066 (mΩ cm / K2.0). For the sample irradiated with 20 MR, ρo = 2.654 (mΩ cm) and a = 0.00119 (mΩ cm / K1.5). For the sample irradiated with 100 MR, ρo = 2.908 (mΩ cm) and a = 0.00203 (mΩ cm / K1.41).

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In figure 4, we plotted the RRR and n values versus the γ-irradiation dose. This figure represents the clear correlation between RRR and n values where both the RRR and n values decrease as the irradiation dose increases. The rate of decrease of both values is very similar, with it is fast up to the dose of 20 MR and then becomes very slow after that.

2.2

3.3 n

2.1 2

3.2 3.1

ρ = ρo + aTn

1.9

RRR = ρ(297 K)/ρ(39 K)

3

1.8

RRR

n

RRR

2.9 1.7 2.8

1.6

2.7

1.5

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1.4

0

20

40

60

80

2.6 100

Gamma Dose (MR) Figure 4. The RRR and n values versus γ-irradiation dose.

The power-law behavior in resistivity changes from n = 2.1 for the non-irradiated sample to 1.41 for the sample irradiated with the dose of 100 MR. Similar behavior of the power-law in resistivity and of the RRR value was observed in VN films [32,44] and in MgB2 bulk samples [33] [45]. In [32,44], RRR for the VN films was found to drop from 8.4 for the wellordered stoichiometric sample to 1.14 for the sample after irradiation with α-particles. For these samples, the fitting parameter, n for low-temperature dependent resistivity between 10 and 30 K drops from 4 to 2.25. This drop in n value was attributed to the increase in disorder caused by irradiation. In [33,45], the resistivity of MgB2 samples, with different nominal composition was studied. The samples were found to have different temperature dependence of normal-state resistivity and residual resistance ratio while having nearly similar Tc values. Also a correlation between the power-law dependence of resistivity and the RRR values was found. The RRR and n values were found to be large for the sample with the large Mg ratio. The RRR and n values were found to decrease as Mg ratio was decreased. In order to relate their results

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Understanding the Roles of Heavy Ion and Gamma-Irradiations on the Magnetic… 181 [33,45] with those found in [32,44], the authors suggested that the Mg excess causes less disorder leading to the large RRR and n values whereas Mg loss causes more disorder leading the observed low RRR and n values. The changes in RRR and n values in our results can be attributed to the disorder created by γ-rays. High γ-irradiation doses cause large disorder which leads to the small RRR and n values. While low γ-irradiation doses cause less disorder which leads to the large RRR and n values.

3.2. Critical Current Density in the MgB2 Sample There are two main effects that might be caused by γ-rays in MgB2 polycrystalline samples. The first effect is the migration of MgO inside the MgB2 grains and the other effect is oxygen segregation in the grain boundaries. It was reported that the transport properties of MgB2 are dominated by oxygen related defects which arise from the existence of MgO inside the MgB2 grains [34] [46]. Oxygen, unlike some other elements, is present in MgB2 as an unintentional impurity due to its high reactivity with MgB2. Typically, oxygen rich precipitates form in the bulk of MgB2 where they enhance vortex flux pinning, enhancing the critical current density [35] [47]. Moreover, it was found that oxygen can segregate at grain boundaries of MgB2 where it contributes significantly to flux pinning [36] [48].

2000 B = 0.0 Tesla

T = 28 K T = 29 K T = 30 K

2

J (A/cm )

T = 31 K T = 32 K

1000

c

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1500

500

0

0

20

40

60

80

100

Dose (MR) Figure 5. The critical current density in the MgB2 sample as a function of γ-irradiation dose at zero applied magnetic fields and at different temperatures.

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Figure 5 is a plot of Jc in the MgB2 sample as a function of γ-irradiation dose at different temperatures. As can be seen in the figure, there is a significant increase in the value of Jc after an irradiation dose of 10 MR at all temperatures. After an irradiation dose of 20 MR, Jc was found to have slight further enhancement above that caused by the γ-irradiation dose of 10 MR. After an irradiation dose of 100 MR, Jc was found to have nearly no significant further enhancement above that caused by the γ-irradiation dose of 20 MR. We also see that at high temperatures, the effect of γ-irradiation on Jc becomes insignificant. At γ-irradiation dose of 10 MR, we believe that both migration of MgO inside the MgB2 grains and oxygen segregation in the grain boundaries took place simultaneously, leading to a significant enhancement of Jc. As the dose of γ-irradiation increases, the amount of MgO decreases leading to a less effective role of enhancing Jc. On the other hand the amount of oxygen segregation increases which leads to enhancement in Jc. But as the dose increases further, a point is reached where the MgO migration becomes minimal leading to very slight effect on enhancing Jc. It is expected that at γ-irradiation doses larger than those used in this study, a point will be reached where large amounts of oxygen segregation will damage the grain connectivity leading to a suppression of Jc.

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3.3. Magnetization of the Pb-Ion Irradiated YBa2Cu3O7-δ Crystal Figure 6 displays our measurements of the FC MR values along the c axis and along the ab plane. We can see that for each sample the values of MR along the c axis are much larger than the corresponding values along the ab plane. This difference between the MR values along the c axis and those along the ab plane reflects the large crystallographic anisotropy of the pinning process. This anisotropy effect probably derives in large part from the columnar defects produced parallel to the c axis by the ion beam. The observed anisotropy of MR (and thus of Jc) is attributed to the expected strong pinning force along the whole length of the columnar defect compared to that perpendicular to columnar defect. We can also see that MR values of the second irradiated sample (with the larger dose) are smaller than those for the first irradiated sample (with the smaller dose). This result is in agreement with some previous findings [37-40] [49-52]. The suppressed values of MR (and correspondingly of Jc) indicates that there is a limiting effect on Jc at very high dosages of Pb-ion irradiation of YBCO. It is also noticed in figure 6 that MR values of the irradiated samples are always much larger than those for the non-irradiated sample which evidences quantitatively that the vortex pinning is greatly enhanced by the Pb-ion irradiation. Moreover, the MR values for the irradiated samples at 77 K, though much smaller than those for the same samples at 4.2 K, are only moderately smaller than the MR values for the non-irradiated sample at 4.2 K. Hence, the enhancing effect of the Pb-ion irradiation on the pinning nearly cancels the suppressing effect resulted from the thermal factor at high temperatures. The enhancement of Jc at ion irradiation dose of BΦ = 1 T and the suppression of it at the higher dose of BΦ = 2 T can be explained to be due to a combination of competing two factors [38] [50]. The first factor is the enhancing factor of ion irradiation at low doses. This enhancing factor is attributed to the columnar defects created by ion irradiation. As stated before, these columnar defects act as effective vortex pinning forces at all temperatures. At high temperatures the superconducting coherence length nearly matches with the diameter of

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Understanding the Roles of Heavy Ion and Gamma-Irradiations on the Magnetic… 183 25 4.2 K 77 K

(a)

3

M (kemu/cm )

20

Along the c axis 15

R

Melt-texured YBCO crystal 10

5

0

0

1 B (T)

2

φ

4 4.2 K 77 K

(b)

3

M (kemu/cm )

In the ab plane 2

R

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3

Melt-texured YBCO crystal 1

0

0

1

2

B (T) φ

Figure 6. The remanent magnetization in YBCO crystal, MR at T = 4.2 and 77 K after FC measured (a) along the c axis and (b) along the ab plane.

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the columnar defect, producing significant pinning forces. The second (suppressing) factor takes place at higher irradiation doses. Large irradiation doses increases the damage of the sample. Thus, the non-superconducting volume of the sample increases while the superconducting flow volume decreases. As the irradiation dose increases, the decrease of the superconducting volume becomes large enough to put a limit on the current flow, effectively suppressing Jc. At low irradiation doses, the enhancing role of these columnar defects is larger than the suppressing role. At high irradiation doses, the suppressing factor might become larger than the enhancing effects. In general, a third (suppressing) factor can exist under the application of a magnetic field. This factor could be attributed to the reduction of the effective pinning of vortices [41] [53] or to the negative pinning effect of vortices [42] [54] that are surrounded by vortices in direct interaction with pinning centers.

3.4. Critical Current Density in the Mn-Doped YBa2Cu3O7-δ Sample Figure 7 displays the critical current density, Jc as function of temperature for H = 0 Oe, 300 Oe and 500 Oe. The figure shows Jc for the both the undoped and the Mn-doped YBCO samples. It is very important to notice that in figure 7, Jc values reported for our samples are much smaller than those reported for single-domain YBCO bulks where Jc is of the order of 106 A/cm2 at 77 K. The low values of Jc in our polycrystalline samples indicate that their properties are apparently governed by the grain-boundary characteristics. In figure 7, we can

190 undoped, 0 Oe Mn doped, 0 Oe undoped, 300 Oe Mn doped, 300 Oe undoped, 500 Oe Mn doped, 500 Oe

170

c

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J (mA/cm )

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160 150 140 130 120

70

75

80

85

90

T (K) Figure 7. The critical current density, Jc versus temperature for both the undoped and the Mn-doped samples at all applied fields.

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Understanding the Roles of Heavy Ion and Gamma-Irradiations on the Magnetic… 185 see that Jc in the doped sample is significantly larger than Jc in the undoped sample at all the applied fields used in the study. It is clear from figure 7 that Jc decreases with increasing the temperature in both samples. The rate of decrease of Jc with increasing temperature in the doped sample is larger than the rate of decrease of Jc in the undoped sample. Figure 8 shows Jc as function of H at several temperatures. We can see that Jc in the doped sample is significantly larger than Jc in the undoped sample at all temperatures. We can also see that Jc decreases with increasing the applied magnetic field at all temperatures. The rate of decrease of Jc is nearly the same in both samples. undoped, 74 K Mn doped, 74 K undoped, 77K Mn doped, 77 K undoped, 84 K Mn doped, 84 K undoped, 88 K Mn doped, 88 K

190 180

2

J (mA/cm )

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c

160 150

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140 130 120

0

100

200

300

400

500

600

H (Oe) Figure 8. The critical current density, Jc versus the applied magnetic field, H, at all temperatures for both the undoped and the Mn-doped samples.

Figure 9 displays the relative critical current density, Jrc as function of temperatures at all applied fields. For a particular applied field and temperature, Jrc is given by J rc =

(J

d c

− J cu ) d u , where J c is the critical current density in the doped sample and J c is u Jc

the critical current density in the undoped sample. We can see that Jrc is always positive, which indicates that there is an enhancement of Jc in doped sample at all temperatures and fields. The values of Jrc at low temperatures are significantly larger than those at the higher temperatures at all applied fields. This indicates that the enhancement of Jc is larger at low temperatures as would be expected. Even though, the Jc values at H = 500 Oe are lower than

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those at 300 Oe and 0 Oe as illustrated in figure 9, the Jrc values are higher at 500 Oe. This indicates that the enhancement of Jc at 500 Oe is larger than the enhancement of Jc at 300 Oe and at 0 Oe. Also we can see in figure 8 that the Jc values at 300 Oe are lower than those at 0 Oe. But as shown in figure 9, Jrc values at 300 Oe are larger than those at 0 Oe. This indicates that the enhancement of Jc at 300 Oe is larger the enhancement of Jc at 0 Oe. Although, the Jc values obtained for our sample are considered to be low, our results clearly display the significant enhancement of the critical current density in the Mn-doped sample compared to the undoped one. This increase in Jc after doping is a clear evidence of an enhancement in the vortex pinning forces in the doped sample. 0.25 H = 0 Oe H = 300 Oe H = 500 Oe

0.2

Jrc

0.15

0.1

0.05

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0 70

75

80

85

90

T (K) Figure 9. The relative critical current density, Jrc versus temperature at all applied fields.

Contrary to the results of the DC magnetization measurements reported in Ref. [29] for Mn concentration of 2.0 %, the results of our I-V measurements reveal that the enhancement of Jc was obtained at all high temperatures and at all applied fields used in this study. To investigate the effect of Mn-doping on the grain microstructure of the samples, we have used SEM to look at the surfaces of the samples. Figure 10 displays the SEM micrographs of the surfaces of both samples. It is shown in this figure that the average grain size in the doped sample is smaller than that in the undoped sample. The grain size in undoped sample varies nearly from 2 μm to 4 μm while the grain size in the Mn-doped sample varies nearly from 2 μm to 9 μm. The reduction of the average grain size in the doped sample is opposite to what was expected, where it is well known that a small amount of impurity usually decreases the melting point of the material and thereby promotes the grain growth. The reduction of the average grain size in the Mn-doped sample results in larger regions of the grain boundaries. The larger grain boundary regions in Mn-doped sample produce an increase in the effective vortex pinning forces which leads to larger Jc values.

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Understanding the Roles of Heavy Ion and Gamma-Irradiations on the Magnetic… 187

Figure 10. The SEM micrographs of the surfaces; the upper one for the undoped sample and the lower one for the Mn-doped sample.

3.5. Magnetization of the B-Doped YBa2Cu3O7-δ Sample The structure of the three samples of YBa2(Cu1-xBx)3O7-δ with x = 0, 0.05, 0.1 was identified using the x-ray diffractometer. The x-ray diffraction results showed an orthorhombic structure with a small increase in the a-axis lattice parameter with increasing the boron concentration x. The average grain size was estimated from the SEM morphology of sample surface and showed that the grain size is larger in the samples that were doped with boron compared to those in the undoped samples.

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I.M. Obaidat, B.A. Albiss and M.K. Hasan (Qaseer) 2.80 YBa (Cu B ) O

2.60

2

1-x x 3

7-δ

V = 0.900 (μm) o

2.40

V/V

o

2.20 2.00 1.80 1.60 1.40 1.20 0.00

0.05

0.10

0.15

0.20

0.25

x Figure 11. The normalized average grain size (V/Vo) in the YBa2(Cu1-xBx)3O7-δ samples versus the boron concentration x. Vo is the average grain size in the undoped sample.

800 YBa (Cu B ) O 2

1-x x 3

7-δ

T = 80 K

600 500 c1

H (Oe)

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700

400 300 200 100 0

0

0.02

0.04

0.06 x

0.08

0.1

0.12

Figure 12. The lower critical magnetic field Hc1(Oe) in the YBa2(Cu1-xBx)3O7-δ samples as function of boron concentration x at T = 80 K.

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Understanding the Roles of Heavy Ion and Gamma-Irradiations on the Magnetic… 189 Figure 11 shows the variations of the average grain size with x for sample with x = 0.05, 0.1 and 0.2. It is clear from the figure that the average grain size starts from a bigger value compared to the grain size in the undoped sample. Thus the average grain size in the boron doped samples is larger than that in the undoped sample. The figure also shows that the average grain size in the doped sample with x = 0.1 is smaller than that in the doped sample with x = 0.05. The transition temperatures for different boron concentration x of these samples were determined using the DC magnetization measurements by applying a small magnetic field of 5 Oe. The transition temperature for the all the samples was found to be nearly the same and equal to 89 K. To study the effect of the boron concentration x, on the vortex pinning forces, we have measured the magnetization M as the applied magnetic field H is varied at 80 K. We have found that the lower critical field Hc1 was significantly increased with x from approximately 42 Oe for x = 0 to 750 Oe for x = 0.1. Figure 12 shows the lower critical magnetic field Hc1 as function of boron concentration x. It is clearly seen from the figure that Hc1 increases with increasing the boron concentration x. We have studied the variations of the width of hysterisis loop with the applied field H. 1.2 YBa2(Cu1-xBx)3O7-δ T = 80 K

1

x = 0.00 x = 0.05 x = 0.10

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ΔΜ/ΔΜ

o

0.8 0.6 0.4 0.2 0

0

200

400

600

800

1000

H(Oe) Figure 13. The normalized hysterisis width ∆M/∆Mo in all the YBa2(Cu1-xBx)3O7-δ samples versus the applied magnetic field H (Oe) at T = 80 K.

Figure 13 shows ∆M/∆Mo as function of the applied magnetic field H for the undoped sample and the two boron doped samples with x = 0.05 and x = 0.1. The measurements were conducted at 80 K and ∆Mo is the value of ∆M at zero applied magnetic fields. The values of ∆Mo were as follows; ∆Mo = 4.6541 (emu/cm3) for x = 0.00, ∆Mo = 3.2407 (emu/cm3) for x = 0.05, and ∆Mo = 3.1482 (emu/cm3) for x = 0.1. Figure 11 shows a clear effect of boron on ∆M and thus on the critical current density Jc which is directly proportional to ∆M.

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There are three interesting results that can be obtained from figure 13. The first one is that the critical current density Jc is larger in the boron doped samples nearly at all applied magnetic fields. The second result is that the rate of decrease of Jc with increasing the applied magnetic field is much smaller in the doped samples than the rate of decrease of Jc in the undoped sample. The third result is that Jc is almost the same in both doped samples with x = 0.05 and x = 0.1 except at the largest applied magnetic field values used in this study. At the largest applied magnetic fields used, Jc drops sharply in the sample with x = 0.05, compared to that rate of decrease of Jc in the sample with x = 0.1. The reason for the larger values of ∆M in the undoped sample at very low applied magnetic field values is currently under investigating. The results in figures 10 and 11 clearly suggest that the boron doping caused an enhancement of the vortex pinning forces in the samples which is very important for practical applications of these superconductors.

3.6. Critical Current Density in the γ-irradiated YBa2Cu3O7-δ Sample Figure 14 shows the behavior of the critical current density in the YBa2Cu3O7 polycrystal as function of γ-irradiation dose at T = 80 and 90 K and at zero applied magnetic field, Happlied = 0. As can be seen in the figure, Jc was enhanced significantly over its non-irradiated value after irradiation doses up to 20 MR, above which Jc nearly has a plateau.

4000

c

2

J (A/cm )

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3000

2000

YBa Cu O Polycrystal 2

3

7

B = 0.0 Tesla T = 80 K T = 90 K

1000

0

0

20

40

60

80

100

Radiation Dose (MR) Figure 14. The critical current density in the YBa2Cu3O7 polycrystal as function of γ-irradiation dose at T = 80 and 90 K and at zero applied field.

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Understanding the Roles of Heavy Ion and Gamma-Irradiations on the Magnetic… 191

3.7. Critical Current Density in the γ-irradiated Tl2Ba2Ca2Cu3O10 Tape Figure 15 shows the behavior of the critical current density in the Tl2Ba2Ca2Cu3O10 tape as a function of γ-irradiation dose at T = 80 and 90 K and at Happlied = 0. We can see that Jc was enhanced significantly after irradiation dose up to 200 MR, beyond which it nearly has a plateau. These results are in a very good qualitative agreement with the results of our previously reported numerical simulations on the effect of the density of pinning sites on the critical current density [55]. In both Figures 14 and 15, we can see that Jc values at 80 K are always larger than those at 90 K for all irradiation doses.

1400 1200

2

J (A/cm )

1000

c

800 600 Tl Ba Ca Cu O Tape

400

2

2

2

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

200 0

3

10

B = 0.0 Tesla T = 80 K T = 90 K

0

100

200

300

400

Radiation Dose (MR) Figure 15. The critical current density in the Tl2Ba2Ca2Cu3O10 tape as a function of γ-irradiation dose at zero applied magnetic fields and at T = 80 and 90 K.

3.8. Critical Current Density in the γ-irradiated Bi1.6Pb0.4Sr2Ca2Cu3O10 Sample Figure 16 shows the behavior of the critical current density in the Bi1.6Pb0.4Sr2Ca2Cu3O10 sample as a function of γ-irradiation dose at Happlied = 0 and at T = 90 K. We can see that Jc was enhanced significantly after irradiation at a dose of 50 MR, and was strongly suppressed at the dose of 300 MR.

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5

2

J (A/cm )

4

c

Bi1.6Pb 0.4Sr2Ca2Cu3O10 Polycrystal B=0T T = 90 K

3

2

0

100

200

300

Gamma Dose (MR)

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Figure 16. The critical current density in the Bi1.6Pb0.4Sr2Ca2Cu3O10 sample as a function of γ-irradiation dose at zero applied magnetic fields at T = 90 K.

3.9. Mechanisms of γ-Rays in the Grain Boundary Regions We believe that the granular structure of polycrystalline HTSCs plays a significant role in determining the superconducting transport properties in theses materials [56, 57]. These polycrystalline HTSCs are generally composed of superconducting grains separated by highly disordered non-superconducting or even dielectric grain boundaries [39]. As discussed in the introduction, the regions of the grains are largely insensitive to γ-rays, while the regions of the grain boundaries of the polycrystalline HTSC material are expected to be very sensitive to γirradiation doses, causing either a net enhancement at low doses and might cause a net suppression of the transport properties (mostly at high doses). This result has been confirmed by experiments on HTSCs polycrystals [41] and crystals [58, 59]. Thus, we may conclude that the major changes to the critical current density observed in our results are mainly due to the nature of the regions of the grain boundaries combined with their high sensitivity to γrays. The coexistence of the enhancing and suppressing mechanisms in the grain boundaries and their varying influences with the dose level is the logical way towards explaining our observed results. At low γ-irradiation doses, the enhancing mechanism is dominant, leading to the noticeable the observed enhancement of Jc. As the dose of γ-irradiation increases, the role

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Understanding the Roles of Heavy Ion and Gamma-Irradiations on the Magnetic… 193 of the suppressing mechanism increases, leading to the slow rate of the observed increase of Jc. At some dose value, the role of the suppressing mechanism becomes comparable to the role the enhancing mechanism leading to the observed plateau of Jc. At the highest dose, the role of the suppressing mechanism (due to damage of weak links) becomes larger than the role of the enhancing mechanism leading to a decrease in Jc. This competing effect of γirradiation dose on the superconducting properties of HTSCs has been previously reported [42, 60].

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4. Conclusion The resistivity in the normal state has been investigated for MgB2 samples before and after irradiation with several doses of γ-rays. We have found that γ-irradiation has a significant effect on the temperature dependence of resistivity. We also have found that there is a clear correlation between the residual resistance ratio, RRR and the power law dependence of resistivity, n as the irradiation dose increases. This correlation indicates that the electron-phonon interaction is important in these samples. The changes of RRR and values were attributed to the different disorder created by different doses of γ-rays. Although the transport properties of the samples were found to be different, their superconducting transition temperatures are nearly the same with a slight broadening of the transition region. The effect of Pb-ion irradiation on the critical current density, Jc in YBCO crystal and the effect of γ-irradiation doses on the behavior of Jc in MgB2 have shown that there is a limitation of these irradiation mechanisms in enhancing the critical current density. At some doses, the critical current density was even suppressed. We have suggested that these results might be attributed to the competing roles of opposite mechanisms of irradiation which we believe to take place mainly in the regions of the grain boundaries. The influences of these mechanisms vary with the irradiation dose level. Transport measurements were conducted on undoped and Mn-doped YBCO samples. The critical current density was found to be significantly enhanced in the doped sample at all temperatures and applied fields. The enhancement of the critical current density was found to be larger at the lower temperatures. The larger Jc values in the Mn-doped sample are attributed to the large grain boundary regions compared to grain boundary regions in the undoped sample. Magnetization measurements were made on YBa2(Cu1-xBx)3O7-δ polycrystalline samples with x = 0, 0.05, 0.10. The lower critical magnetic field Hc1 was found to be enhanced with increasing x. We have found that except at very low applied magnetic fields, the critical current density Jc, is significantly enhanced with the boron doping. Also the rate of decrease of Jc as the applied magnetic field is increased was found to be much smaller in the boron doped samples. At high magnetic fields, Jc was found to be larger in the sample with the larger boron concentration x. Therefore, the boron is suggested to enhance the vortex pinning forces in the samples. The x-ray analysis showed an increase in the lattice parameter a, with an orthorhombic structure. The SEM techniques showed that the average grain size in the boron doped samples is larger than that in the undoped sample. The average grain size was found to be smaller in the doped sample with x = 0.10 than that in the doped sample with x = 0.05.

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Acknowledgment This work was financially supported by the Research Affairs at the UAE University under a contract no. 03-02-2-11/07 and Jordan University of Science and Technology under grant no. 2006-99.

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Understanding the Roles of Heavy Ion and Gamma-Irradiations on the Magnetic… 195 [22] Sun-li Huang, D. Dew-Hughes, D.N. Zheng and R Jenkins, Supercond. Sci. Technol. 9, 358 (1996). [23] I.E. Agranovski, A.Y. Llyushechkin, I.S. Altman, T.E. Bostrom and M. Choi, Physica C 434, 115 (2006). [24] X.F. Rui, Y. Zhao, Y.Y. Xu, L. Zhang, X.F. Sun, Y.Z. Wang, H. Zhang, Supercond. Sci. Technol. 17 (2004) 689. [25] H.L. Xu, Y. Feng, Z. Xu, G. Yan, C.S. Li, E. Mossang, A. Sulpice, Physica C 443, 5 (2006). [26] N. L. Saini, K. B. Garg, H. Rajagopal, and A. Sequeira, Solid State Commun. 82, 895 (1992). [27] Dhingra, S. C. Kashyap, and B. K. Das, J. Mater. Res. 9, 2771 (1994). [28] N. L. Saini, P. Srivastava, B. R. Sekhar, and K. B. Garg, Int. J. Modern Phys. B 6, 3575 (1992). [29] E Isaac Samuel, V Seshu Bai, N Harish Kumar and S K Malik, Supercond. Sci. Technol. 14, 429 (2001). [30] Ugur Topal, Lev Dorosinskii, Husnu Ozkan, Hasbi Yavuz, Physica C 388-389, 401 (2003). [31] V. L. Ginsburg, L. D. Landau: Zh. Eksp. Teor. Fiz. 20, 1044 (1950). [32] H. A. Bethe and J. Askin, in Experimental Nuclear Physics, Vol. 1, edited by E. Segre (John Wiley & Sons. New York, 1953); G. J. Dienes and G. H. Vineyard, Radiation effects in Solids (Interscience Publishers, New York, 1957). [33] G. J. Dienes and G. H. Vineyard, Radiation effects in Solids (Interscience Publishers, New York, 1957). [34] J. Yu, Y. Chen, Y. Wang, J. Nucl. Mater 233-237, 771 (1996); M. Faiz, and N. M. Hamdan, Journal of electron spectroscopy and related Phenomena 114-116, 427 (2001). [35] Kumar, P. Kumar, M. R. Tripathy, A. K. Arora, R. P. Tandon, Materials Chemestry and Physics 97, 230 (2006). [36] Carrington, D. J. C. Walker, A. P. Mackenzie, and J. R. Cooper, Phys. Rev. B 48, 13051 (1993); J. Ruvalds, Supercond. Sci. Technol. 9, 905 (1996). [37] J. Jorgensen, B. W. Veal, W. K. Kwok, G. W. Crabtree, A. Umezawa, L. J. Nowicki, and P. Paulikas, Phys. Rev. B 36, 5731 (1987); P. J. Ford, and G. A. Saunders, Contemp. Phys. 38, 63 (1997). [38] M. V. Sadovskii, Phys. Rep. 282, 255 (1998). [39] V. S. Boyko, J. Malinsky, N. Abdellatif, and V. V. Boyko, Phys. Lett. A 244, 561 (1998). [40] E. V. Thuneberg, J. Kurkijärvi, D. Rainer, Phys. Rev. B 29, 3913 (1984). [41] J. Clark, A. D. Marwick, F. Legoues, R. B. Laibowitz, R. Koch, and P. [42] Madakson, Nucl. Instr. Meth. in Phys. Res. B 32, 405 (1988). [43] [42] B.I. Belevtsev, I.V. Volchok, N.V. Dalakova, V.I. Dotsenko, L.G. Ivanchenko, A.V. Kuznichenko, I.I. Logvinov, Physica status solidi (a) 181, 437 (2000). [44] M. M. A. Sekkina, K. M. Elsabawy, Physica C 377, 411 (2002). [45] J. F. Zasadzinski, A. Saggese, K. E. Gray, R. T. Kampwirth and R. Vaglio, Phys. Rev. B 38, 5065 (1988). [46] X. H. Chen, Y. S. Wang, Y. Y. Xue, R. L. Meng, Y. Q. Wang, and C. W. Chu, Phys. Rev. B 65, 024502 (2001).

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[47] Y. Y. Xue, R. L. Meng, B. Lorenz, J. K. Meen, Y. Y. Sun, C. W. Chu, Physica C 377, 7 (2002). [48] X. X. Liao et al., J. Appl. Phys. 93, 6208 (2003). [49] Larbalestier et al., nature (London) 414, 368 (2001). [50] A.V. Samoilov, M.V. Feigel_man, M.Konczykow ski, F.Holtzberg, Phys. Rev. Lett. 76 (1996) 2798. [51] Alberto Gandini, Roy Weinstein, Drew Parks, Ravi P. Sawh, and Shi Xue Dou, IEEE Transactions on Applied Superconductivity, Vol. 13, No. 2, 2003. [52] Hijiri Kito et al Physica C 392-396, 181 (2003). [53] Noriko Chikumoto, Ayako Yamamoto, Marcin Konczykowski and Masato Murakami, Physica C 388-389, 167 (2003). [54] Reichhardt, J. Groth, C.J. Olson, S.B. Field, F. Nori, Phys. Rev. B 54 16108 (1996). [55] H. Brandt, Rep. Prog. Phys. 58, 1465 (1995). [56] M. Obaidat, U. Al Khawaja and M. Benkraouda, Supercond. Sci. Technol. 18, 1380 (2005). [57] R. Gross, Physica C 432, 105 (2005). [58] H. Hilgenkamp and J. Mannhart, Rev. Mod. Phys. 74, 485 (2002). [59] T. Kato, M. Watanabe, Y. Kazumata, H. Naramoto, T. Iwata, Y. Ikeda, H. Maekawa, and T. Nakamura, Jap. J. Appl. Phys. 27, L2097 (1988). [60] R. Rangel, D. H. Galvan, E. Adem, P. Bartolo-Perez, and M. B. Maple, Supercond. Sci. Technol. 11, 550 (1998). [61] N. M. Hamdan, A. S. Al-Harthi, and M. F. Choudhary, Physica C 282-287, 2273 (1997).

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In: Superconducting Magnets and Superconductivity… ISBN: 978-1-60741-017-1 Editors: H. Tovar and J. Fortier, pp. 197-214 © 2009 Nova Science Publishers, Inc.

Chapter 5

ON THE MELT PROCESSING OF BI-2223 HIGH-TC SUPERCONDUCTOR CHALLENGES AND PERSPECTIVES A. Polasek1,∗, E.T. Serra1 and F.C. Rizzo2 1

CEPEL – Electric Power Research Center – Av. Horacio Macedo, 354 - Cidade Universitaria - 21941-911 Rio de Janeiro - RJ - Brazil 2 Department of Materials Science and Metallurgy, Pontifical Catholic University of Rio de Janeiro – Rua Marquês de São Vicente, 225 - Gavea - 22453-900, Rio de Janeiro - RJ - Brazil

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Abstract Although the production of high performance (Bi,Pb)2Sr2Ca2Cu3O10+x (Bi-2223) silver sheathed superconducting tapes has already reached industrial level, there is still room for further improvements. On the other hand, sintered Bi-2223 bulk parts frequently exhibit high porosity and poor electrical properties. Melt processing has been for long attempted as an alternative to sintering, in order to improve density, texture and microstructure of tapes and bulk parts based on this compound. However, recrystallizing Bi-2223 from the melt is challenging, due to the lack of knowledge of the Bi-2223/melt equilibrium, the phase narrow stability range and sluggish formation kinetics. In fact, Bi-2223 decomposes peritectically into liquid and solid phases, but most of the attempts to revert such decomposition reaction have failed, generating mainly Bi2Sr2CaCu2O8+x (Bi-2212) and Bi2Sr2CuO6+x (Bi-2201), together with secondary phases such as (Ca,Sr)2CuO3, CuO and Ca2PbO4. This may be partially attributed to kinetical reasons, since Bi-2212 and Bi-2201 form faster than Bi-2223. In addition, Ca and Cu segregation as well as Pb volatilization take place above the solidus line, hampering Bi-2223 recrystallization from the melt. Nonetheless, promising results have been reported in the literature, particularly in the case of partial peritectic decomposition. Other works indicated that Bi-2223 can be partially recovered after complete peritectic decomposition, as well as after full melting via the glass ceramic route. Therefore, the investigation of Bi-2223 melting and recrystallization may provide results of high scientifical and technological relevance. In the present work, a general review of this theme is undertaken with the aim of shedding light on this complex issue. ∗

E-mail address: [email protected]. Contacting author: Alexander Polasek.

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Key words: high-Tc superconductor, Bi-2223, melt processing, peritectic decomposition, recrystallization

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1. Introduction The development of materials and applications based on high critical temperature superconductors (HTSC’s) has been experiencing remarkable progress [1,2,3]. Real scale prototypes have been successfully demonstrated and market projections indicate that HTSC equipments and devices will become commercially available in the next decade [1-4]. However, further improvements in processing and performance of HTSC materials are crucial for the future of HTSC applications. In spite of the great progress that has been achieved with such materials, low cost/benefit, high reliability and performance reproducibility are still needed. Among these materials, Bi2Sr2Ca2Cu3O10+x (Bi-2223) silver sheathed tapes are the first HTSC wires to be produced in industrial scale [5,6]. Kilometers of Bi-2223 tapes are routinely produced, with critical current densities (Jc) exceeding 50 kA/cm2 and critical currents (Ic) of 180-210 A, at 77 K [6, 7]. Therefore, such tapes have been the main choice for most of the large-scale electric power prototypes, such as the Long Island 110 kV superconductor cable, as well as superconductor ship propulsion motors [2,5,8]. In spite of the rapid advance and high performance of second generation Y-Ba-Cu-O (YBCO) wires which are promising to reach the DOE cost/benefit goal for HTSC wires (US$ 10,00 / kA.m) [9] - they are still in the pilot-scale production stage [10,11]. They cannot be considered as a mature product yet, since higher reproducibility and better Ic homogeneity over long lengths are still needed. The Bi-2223 phase belongs to the Bi-Sr-Ca-Cu-O (BSCCO) system, which also includes the Bi2Sr2CuO6 (Bi-2201, Tc < 20 K) and Bi2Sr2CaCu2O8 (Bi-2212, Tc < 96 K) superconducting phases [12,13]. Due to their very high current transport anisotropy, Bi-2201, Bi-2212 and Bi-2223 are practically “two-dimensional superconductors”. BSCCO grains have a plate-like morphology with high aspect ratios [14-16]. Basically, Bi-2223 wires are made by the so-called oxide-powder-in-tube (OPIT) method, where a pre-reacted precursor powder is inserted into a silver or silver alloy tube that is further drawn into a round wire which is then rolled into a flat tape; this composite tape is subjected to a long term heat treatment process (sintering), in order to convert the silver sheathed precursor into Bi-2223 [16]. Intermediate cold-rolling steps are performed to improve reaction and grain alignment. For two decades the processing methods have been optimized, improving critical current densities and mechanical properties of these composite tapes [3,5,6,12,16-18]. In spite of the remarkable progress achieved, the cost/benefit remains still quite high for large-scale commercialization, mainly because of the silver sheath. This metal is employed due to its suitable equilibrium properties at the processing temperatures, high ductility and oxygen permeability [16]. In addition, silver is thought to enhance 2223 phase formation and grain alignment [19]. On the other hand it is believed that the current transport properties can be further improved, bringing the cost/benefit down to US$20,00/kA.m [6]. It was shown that the local current transport of Bi-2223/silver tapes is extremely inhomogeneous, with local critical currents reaching up to five times the overall current density of the entire superconductor, indicating that their microstructure is far from perfect [20,21].

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On the Melt Processing of Bi-2223 High-Tc Superconductor Challenges…

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Conventionally, the 2223 phase is synthesized by long term sintering at temperatures close to the melting [12,13]. Nevertheless, the fully sintered ceramic frequently presents high porosity and several cracks, which decreases the current transport capacity [22,23,27]. Another hampering factor is the complex multiphase equilibrium involved, leading to the formation of secondary-phases that disturb the alignment and coupling of the superconducting platelets [24,25]. It is worth of notice that high quality long tapes with a bulk density of almost 100% are being routinely fabricated through the controlled overpressure process (CTOP) [2,6,26]. Despite of the optimum microstructure and performance of CT-OP tapes, this process involves a sintering step under an atmospheric pressure of 300 atm, in order to densify the ceramic core, improving grain connectivity [6]. Such method requires a sophisticated furnace for hot isostatic pressing, which contributes to the complexity and cost of the whole process. The Bi-2223 superconductor can also be manufactured in bulk form, such as rings, rods and tubes. The consolidation and shaping techniques can be separated in two groups [28]: room temperature methods, such as cold uniaxial pressing and cold isostatic pressing (CIP); high temperature methods, such as hot isostatic pressing (HIP) and sinter forging. The main problem for sintering bulk Bi-2223 is related to its extremely anisotropic plate-like grains, which results in poor texture and density reduction along sintering. Although this is also a problem for making Bi-2223 tapes, this density reduction is more critical for bulk parts. In the case of consolidation and shaping at room-temperature, intermediate pressing can be used to improving density [28]. However, intermediate cold pressing can induce the formation and propagation of cracks, which is deleterious for current transport. An alternative approach involves the use of organic additives to facilitate the extrusion of rods [28]. In general, high temperature pressing leads to better results than cold temperature pressing and can shorten the total heat treatment time (usually > 100 h). Bars with a density of 85-95 % and Jc of 104 A/cm2 can be produced by hot pressing or sinter forging [28 - 30]. Alternatively, a melt-processing route may also lead to high bulk densities, high texture and long superconducting plates. In addition, it may simplify the manufacture of tapes by avoiding intermediate rolling steps; in the case of bulk samples, a melt-processing route would be cheaper and more scalable than hot pressing and sinter forging. The melt-processing consists of full or partial melting of a precursor powder, with subsequent slow-cooling and further annealing. Full melting takes place above 1000oC [13,31,32], whereas partial melting consists of Bi-2223 peritectic decomposition into liquid and solid phases at 870-950oC in air [32-35]. Complete or full melting is the total transformation of the solid sample into liquid. There is some confusion in the literature with respect to the nomenclature “partial melting”; sometimes “partial melting” designates the Bi2223 peritectic decomposition into liquid and solid phases [33-37], but “partial melting” can also mean partial Bi-2223 peritectic decomposition, that is, a considerable fraction of Bi-2223 remains after such partial decomposition [38,39]. In order to avoid this ambiguity, “partial melting” will not be used in the present work; instead of “partial melting” we will use “complete peritectic decomposition” or “partial peritectic decomposition” referring to processes taking place below 1000oC. Both Bi-2212 and YBCO superconductors are routinely obtained via melt-processing, being peritectically decomposed and then recrystallized through the inverse peritectic reaction by further slow cooling [3]. Melt processed Bi-2212 tapes and bulk parts are characterized by a very low porosity and large crystals of 2212 compared to the 2223 crystals of OPIT tapes.

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Melt-processed 2212 bulk parts have been successfully employed in, e.g., fault current limiter prototypes [3]. However, Bi-2212 typically presents a critical temperature of 80-90K, whereas the critical temperature of Bi-2223 is about 105-110K. Attempts to grow Bi-2223 from the melt have been done since the early years of the discovery of this superconducting phase, but its narrow stability range and sluggish formation kinetics prevent its crystallization from the melt [33-43]. Besides, there is a lack of knowledge on the concentration region where 2223 is in equilibrium with liquid, but some works indicated the feasibility of the 2223 crystallization from the melt [35-39]. Giannini et al. [38,39] observed Bi-2223 reformation in situ directly from a reaction involving melt and solid phases, though in samples where the precursor Bi-2223 phase was not fully decomposed. Our group developed the Continuous Cooling Sintering method for tapes processing, which involves a Bi-2223 partial decomposition step [44-48]. This method may be interesting for large scale production, since it allows a relatively wide heat treatment thermal window, when compared to more conventional OPIT routes, where Ic and Jc values are very sensitive to sintering temperature [44-48]. Afterwards, we started investigations on the melt-processing of bulk samples with complete peritectic decomposition of the Bi-2223 phase [33-36]. Through this work we have obtained promising results, showing significant Bi-2223 fractions after complete decomposition and subsequent slow cooling [35,36]. The present work provides a general review on Bi-2223 melt processing with emphasis on peritectic decomposition methods, taking into account the main challenges and perspectives involved.

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2. The Bi-2223 Phase and the BSCCO System The Bi-Sr-Ca-Cu-O (BSCCO) system contains three main superconducting phases: Bi2Sr2CuO6+x (Bi-2201, Tc < 20 K), Bi2Sr2CaCu2O8+x (Bi-2212, Tc < 96 K) and Bi2Sr2Ca2Cu3O10+x (Bi-2223, Tc = 110 K) [12,13]. In fact, they are solid solutions, forming with compositions deviating from the stoichiometric ones, which can be better described by the general formula Bi2+x+zSr2−x−yCan−1−z+yCun−yO4+2n+d, showing that: (i) Bi substitutes for Sr and Ca, (ii) Sr substitutes for Ca and vice versa, (iii) Cu is slightly deficient, and (iv) oxygen is in excess [13]. The superconducting Bi-2201, also known as Raveau-phase, can be better represented as Bi11Sr9CaxCu5Ox (119x5) and is frequently confused with the nonsuperconducting 2201 phase, which has a different crystalline structure [14]. For the sake of simplicity, both will be referred as 2201 throughout the present work. While Bi-2201 and Bi-2212 are stable within relatively large composition and thermal ranges, Bi-2223 has a very narrow concentration region and sluggish formation kinetics [12,13]. A schematic BSCCO phase diagram is shown in fig. 1. Phase stability and formation kinetics of Bi-2223 are enhanced by means of Pb doping, where Pb atoms partially substitute Bi atoms [12]. Hence, this phase is normally Pb-doped, being described as (Bi,Pb)2Sr2Ca2Cu3O10+x. Synthesizing Pb-free Bi-2223 is a hard task, but it has already been done by sintering within a very narrow temperature range close to the peritectic decomposition onset (Tsolidus) [15]. Both Pb-doped and Pb-free Bi-2223 are usually obtained through long term sintering with intermediate grinding steps that improve the powder homogeneity and lower the total sintering time.

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Figure 1. Temperature versus concentration diagram (schematically) within the range between Bi2Sr2CuO6 and Bi2Sr2Ca2.6Cu3.6O11.2 [13].

3. Melt Processing of Bi-2223 A Bi-2223 melt-processing route has for long been sought. However, Bi-2223 meltprocessing is a challenging task, due to the above mentioned narrow stability range and sluggish formation kinetics of this phase. On the other hand, promising results have already been achieved such as the partial recovery of Bi-2223 from the melt [35-39, 41-43]. The melt-processing can be divided in two main groups: complete melting and peritectic decomposition. Complete melting takes place above 1000oC in air and is usually carried out with Pb-free compositions, due to the high volatility of lead [31,32,41,49]. Nevertheless, bismuth and oxygen losses, as well as crucible reaction are also critical issues around 1000oC [31,32]. The melt is so corrosive that even zirconia and platinum crucibles can react with the liquid [31]. Glass-ceramic routes may overcome these problems by using short melting times, followed by very fast quenching, in order to obtain an amorphous product, which is then annealed for recrystallizing Bi-2223 [31, 41, 50-54]. Peritectic decomposition, also known as “partial melting”, is carried out at 850-930oC, either in air or in 7.5%O2 / N2 (optimum atmosphere for sintering Pb-doped Bi-2223), followed by slow-cooling, in order to recrystallize Bi-2223 upon cooling [33-40, 42-48].

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Crucible reaction, as well as oxygen and lead losses are also critical for peritectic decomposition routes, but in a lower extent than in complete melting.

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3.1. The Glass-Ceramic Route Processing via glass ceramic route can improve density and homogeneity, reduce phase segregation and extend solid solubility of the material [31]. The glass-ceramic route is based on the obtention of a glassy precursor that is further annealed in order to crystallize BSCCO phases. The process starts after mixing and calcining a mixture of suitable reagents, such as oxides and carbonates [31, 41, 50-54]. The melting step is undertaken at 1000-1300oC for dwell times not exceeding 30 minutes, followed by very fast cooling procedures such as splat quenching, melt casting or roller quenching, in order to produce a glassy precursor [31, 41, 50-54]. Shi et al [50] investigated the crystallization of glasses with nominal compositions Bi2Sr2Ca2Cu3Ox, Bi2Sr2Ca3Cu4Ox and Bi2Sr2Ca4Cu5Ox. They observed eutectic crystallization producing Bi-2212 (Tc = 85 K) plus calcium- and copper-rich phases in all glasses. Subsequent long-term annealing of several days crystallized Bi-2223 with Tc = 105 K, but high amounts of secondary phases remained in the final products [50]. Moon et al. [52] reported the formation of Bi-2223 by “rapid thermal melt processing”. Previously sintered samples were melted for 2 minutes at 1200oC, followed by fast cooling. Further annealing at 865oC / 120 h gave rise to Bi-2223, as revealed by dc resistance and magnetic susceptibility measurements. However, the fraction of Bi-2223 formed was not reported and any XRD pattern was provided, so that it is not clear whether they produced an intermediary amorphous phase, as well as whether they succeeded in obtaining a significant fraction of Bi-2223. On the other hand, the density of the fully processed samples was considerably higher than the density measured before the melting step [52]. Amorphous Bi1.6Pb0.4Sr2Ca2+2xCu3+xOy ceramics were prepared by melt-quenching [53]. Activation energies of 468-477 kJ/mol were calculated for the amorphous phase. Further annealing crystallized the amorphous phase in the following sequence: 1) Bi-2201, 2) Bi2212 and 3) Bi-2223. Although relatively high fractions of Bi-2223 crystallized by annealing at 840oC / 200 h, Bi-2212 was still the predominant phase [53]. Nilsson et al. [31] performed a thoroughly investigation on the critical aspects involved in the preparation of Bi-2223 by glass ceramic route, emphasizing the evolution of the actual chemical composition of the samples. The effect of the nominal composition on the actual chemical composition of the amorphous phase was investigated: relatively high Bi/Ca ratios reduced the deviation of the glass composition from the nominal one [31]. Glassy precursors with nominal compositions Bi2Sr2CaCu2O8 and Bi4Sr3Ca3Cu4O16 became completely amorphous, with slight deviations from their respective nominal compositions. However, glassy precursors with nominal composition Bi2Sr2Ca2Cu3Ox showed significant chemical deviations in the glassy state, mainly Ca deficit in comparison to the nominal composition. The question arised whether the Ca-deficit could be related to non-homogeneous melting, even at 1300oC [31]. Another critical issue for Bi-2223 crystallization is the oxygen releasing along melting. This oxygen loss takes place according to 2CuO = Cu2O + ½ O2. The oxygen loss must be balanced during further crystallization, in order to form high fractions of Bi2223.

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3.2. Melt-Processing by Peritectic Decomposition

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3.2.1. Peritectic Melting Quite comprehensive work has been done to study the peritectic melting of 2223 [32, 5559]. Phase diagram studies showed that, between 850°C and 880-890°C, Bi-2223 is in equilibrium with a melt [59]. The melting initiation temperature (Tsolidus) depends strongly on factors like the nominal composition, Pb-doping and silver addition [32, 55-59]. High temperature XRD of Pb-free samples showed Bi-2223 disappearing at an initial temperature of 880–890 oC (without Ag) and 860–870oC (with Ag), together with the appearance of (Ca,Sr)14Cu24O41 (14:24) followed by (Ca,Sr)2CuO3 (2:1) [32]. Lead doping also decreases Tsolidus [59]. The exact nature of the liquid in equilibrium with the 2223 phase remains to be elucidated, with some works pointing to a nearly Ca-rich 2201 melt and others suggesting a composition closer to that of the 2212 phase. Schulze et al. [60] pointed out that the melt should be rich in bismuth, having approximately a Bi2(Sr,Ca)1.5CuOx (Ca-rich 2201) composition, since this phase melts congruently and crystallizes directly from the liquid, differently from the 2212 and 2223 phases. According to Majewski [59], with increasing temperatures from below to above the melting of 2212 and 2223, the liquid first becomes more Ca-, Cu- and Bi-rich, with concentration of Sr increasing later; at about 850 oC in air the liquid phase ranges from the Bi-rich concentration region to a composition around Bi2SrCaCuOx. The composition of the melt in equilibrium with 2212 and 2223 was determined to be about Bi2Sr1-1.3Ca0.4-1CuOx and Bi2Sr0.8Ca0.9CuOx, respectively [59-61]. However, other studies revealed a melt with compositions closer to that of the 2212 phase [32,62]. Park et al. [32] found a Bi26.8Sr22.6Ca12.4Cu31.1Ox melt forming at 860-870oC in air, in precursors mixed with silver. Styve et al. [62] observed similar compositions. It was reported that the Bi-2201 and Bi-2212 phases are not equilibrium products of the melting, but both may coprecipitate upon cooling, suggesting that the real melt composition falls between the Bi-2212 and Bi-2201 compositions [63]. In fact, the melt in equilibrium with the 2212 phase possibly falls closer to the Ca-rich 2201 primary phase-field, whereas the melt in equilibrium with the 2223 phase might have a 2212-like composition [33,34]. This becomes clear by recalling the main solid-state reactions forming 2212 and 2223 in the subsolidus region [64]: 2201 + 1/2Ca2CuO3 + 1/2CuO = 2212

(1)

2212 + 1/2Ca2CuO3 + 1/2CuO = 2223

(2)

Considering the above reactions in the decomposition direction allows the conclusion that, above the peritectic melting temperatures, 2212 may decompose into a 2201-like liquid, 2:1 and CuO, whereas 2223 would decompose into a 2212-like liquid, 2:1 and CuO. Moreover, the liquid in equilibrium with 2223 could move from the 2212-like composition towards the 2201-like one, becoming richer in Bi by loosing Ca and Cu due to the precipitation of Ca-Cu-O phases, as a function of the sample composition, the temperature and the cooling rate. Consequently, a deviation from the 2212-like melt to the 2201-like melt would occur favoring thus the crystallization of 2201, instead of 2223. Indeed, a slight

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decrease in the Ca and Cu contents of the melt with increasing temperature has been shown [62]. Figure 2 shows a Pb-free sample with Bi-rich 2223 nominal composition that was partially molten at 855oC, in 7.5% O2/N2 and subsequently quenched [34]. The EDS analysis revealed that the matrix had a Ca-rich 2201 composition, while the gray “plates” exhibited Bi25-28Sr17-21Ca19-23Cu29-33Ox compositions, falling in the vicinity of the Bi-2223 primary phase-field. However, the corresponding XRD pattern revealed the presence of Bi-2212 instead of Bi-2223 [34]. There is still an interesting feature in Figure 2, namely the presence of narrow white regions in the vicinity of the “plates”. Although the MEV/EDS resolution was not able to distinguish these regions from the matrix, their white tonality suggests a Carich 2201 composition with a lower Ca content than that of the matrix. Considering the Bi-2223 primary phase field [13,59], the liquid coexisting with this phase may also occupy a very small equilibrium volume surrounded by very flat but elongated multiphase regions pertaining to the Bi-Sr-Ca-Cu-O system. This might explain why it is so challenging to find the concentration region where 2223 is the only Bi-Sr-Ca-Cu-O phase coexisting with the liquid. The question remains whether it is feasible to develop a successful melt processing route for synthesizing Bi-2223.

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5 μm

Figure 2. SEM image showing a partially molten sample with Pb-free / Bi-rich 2223 nominal composition that was quenched from 855oC, 7.5% O2/N2. Gray plates: Bi-2212; Light-gray matrix and white regions: Ca-rich 2201 [34].

3.2.2. Bi-2223 Crystallization from the Peritectic Melt Efforts have been made to synthesize 2223 out of a peritectic melt or in the presence of a peritectic melt [33-40, 42-48, 60, 61, 63, 65] and phase diagram studies showed that, at about 850°C up to 890°C, Bi-2223 is in equilibrium with a melt [12,13,59]. The main necessity to develop an efficient peritectic melting route is to define the concentration region within the quaternary system Bi2O3-SrO-CaO-CuO in which 2223 is in thermodynamic equilibrium with a melt.

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Giannini et al. [38,39] showed the reversibility of the 2223 melting through high temperature neutron diffraction, observing 2223 decomposition and further reformation as follows: (Bi,Pb)-2223 = (Sr,Ca)2CuO3 + 14:24 + liquid = (Bi,Pb)-2223

(3)

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This outstanding result proved that Bi-2223 can be recovered directly from the liquid, without the previous solidification of Bi-2212. Since the formation of Bi-2223 by sintering in the subsolidus region needs the previous formation of Bi-2212 as a precursor phase that is further converted to Bi-2223 (equation 2), it was believed that in the case of melt-processing the Bi-2212 phase would always crystallize before Bi-2223, being then converted to the latter by suitable annealing. On the other hand, it should be noticed that the reaction showed by the left side of equation (3) was a partial decomposition, i.e., a significant amount of Bi-2223 remained after such melting [38,39]; this residual Bi-2223 may have acted as a substrate for the nucleation and growth of recrystallized Bi-2223. To our knowledge, the Bi-2223 recovery directly from the melt after a complete Bi-2223 decomposition has never been unambiguously proved. Polasek and Majewski investigated the melt processing of Pb-free 2223 [65]. Samples with Bi-rich compositions were partially molten at T > 860oC in 7.5%O2/N2. Bi-2223 formed in the presence of liquid after long-term annealing above the peritectic melting temperature (Tsolidus). Large Bi-2223 crystals exceeding about 500 µm in a,b-direction could be seen at broken surfaces of the sample without a microscope. Figure 3 shows a SEM image of a quenched sample. The presence of Bi-2223 was confirmed by magnetic susceptibility analysis (Figure 4) and XRD. However considerable fractions of Bi-2212 and (Ca,Sr)CuO2 were also present.

10 μm crystallized melt 2223 + 2212

CuO

(Ca,Sr)CuO2

Figure 3. SEM image of a quenched sample with Bi2.4Sr1.9Ca2.1Cu3Ox nominal composition after long term annealing above Tsolidus. Crystallized melt: Ca-rich 2201 [65].

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Susceptibility [a.u.]

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0

10

20

30

40

50

60

70

80

90

100 110 120

Temperature [K]

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Figure 4. AC Susceptibility vs. temperature of a sample with Bi2.4Sr1.9Ca2.1Cu3Ox nominal composition after long term annealing above Tsolidus. The first transition at about 105 K corresponds to Bi-2223, whereas the second transition at about 86 K corresponds to Bi-2212 [65].

The recovery of considerable Bi-2223 fractions after complete peritectic decompotision was achieved [35, 36]. Figure 5 shows the XRD pattern of a Bi1.84Pb0.32Sr1.84Ca1.97Cu3.00Ox sample sealed in a thin silver foil. The sample was melted at 862oC/1h, slow cooled (0,1oC/min) to 838oC, further cooled to 400oC and subsequently quenched [35]. However, these works could not demonstrate whether Bi-2223 formed directly from the peritectic reaction between liquid and solid phases, since any in situ results had been given. Since high fractions of Bi-2212 were found in the quenched samples, it is possible that Bi-2212 crystallized firstly during slow cooling, being then partially converted to Bi-2223. Nonetheless, dense and textured samples with long Bi-2223 platelets were obtained [35,36], which brings out the conclusion that melt-processing may be advantageous even when Bi2212 crystalllizes before Bi-2223. Furhter studies are still nedeed to convert all residual Bi2212 into Bi-2223 by a suitable annealing procedure. Both Bi-2201 and Bi-2212 may coprecipitate upon cooling [63]. These phases form faster than Bi-2223, specially Bi-2201, which is known to precipitate very fast from the melt, even in quenched samples [33,34]. Hence, very slow cooling rates ( < 10oC / h) are used to crystallize Bi-2212 and Bi-2223 [33-39, 43-48]. Calorimetric studies using DTA and XRD results indicate that the Bi-2201 crystallization precedes Bi-2212 and Bi-2223 reformation upon cooling [37, 66]. As mentioned in the previous section, the precipitated Bi-2201 presents a Ca-rich composition. Oxygen and lead loss during peritectic melting are critical issues. As mentioned in section 3.1, oxygen content is lowered by melting, being further uptaken through crystallization. This loss should be balanced by crystallization, in order to promote Bi-2223 formation. Besides, Tc of BSCCO phases is very dependent on oxygen content [13]. Lead loss deviates the system from the Bi-2223 primary phase field. The TG curve of a silver packed Bi-2223 sample, with nominal composition Bi1.84Pb0.32Sr1.84Ca1.97Cu3.00Ox (Figure 6)

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exhibited considerable mass loss with increasing temperature, but the net mass loss was about 0.55% [36]. Above 860oC the mass loss can be ascribed to both oxygen and lead releasing. On the other hand, sample mass increased upon cooling, due to oxygen absorption during Bi2212 and Bi-2223 solidification. Still, it can be noticed that, at 650oC, the mass recuperation is not complete. Supposing that the oxygen loss was totally compensated upon slow cooling, the mass difference between the heating and cooling cycles can be mainly attributed to Pb volatilization (about 0.1% weight loss, corresponding to 1.6 wt % of the total Pb content of the sample). If the 2223/Ag composite were heated to 10-20oC above Tsolidus, the Pb losses could have increased. In this sense, the thermal schedule was designed to minimize Pb-losses, since the maximum temperature was only about 5oC above 2223 Tsolidus. In addition, the silver foil possibly reduced Pb-loss, since silver decreases Tsolidus. Another way to reduce the effect of Pb-loss is to start with a mixture containing excess of Pb [37]. However, this can lead to precipitation of lead rich phases such as Ca2PbO4. By applying high isostatic pressure during heat-treatment (Ptotal = 100 bar), Lomello-Taffin et al. [67] prevented Pb evaporation above the decomposition temperature and kept the actual stoichiometry close to the nominal one. The decomposition temperature changed with the isostatic pressure, which was interpreted by these authors in terms of temperature-dependent Pb-solubility in the Bi-2223 single phase region. 3+2 500

3

3+1

400

Int. [a.u.]

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3 2

3 2

1 100

2

3

3

2 2:1

1 3 32

Pb

3

1 3

2 2

1

2

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300

1 Ag

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0 5

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2 Theta Figure 5. XRD pattern of a Bi-2223/silver quenched specimen that was previously molten and slowly cooled. 1- 2201; 2- 2212; 3- 2223; 2:1- Ca2CuO3; 14:24; Pb- Ca2PbO4 [35].

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H2O O2 Pb + O2

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Figure 6. TGA of silver sealed Bi-2223, in air [36].

Heating at excessively high temperatures (well above Tsolidus) and/or for long dwell times may induce more Ca and Cu segregation from the melt, forming large 2:1 and CuO precipitates [34]. Such precipitates tend to remain stable over further annealing, which hampers the recovering of substantial Bi-2223 amounts. Yamada et al. [42] heated Bi-2223 silver sheathed tapes with the nominal composition Bi1.72Pb0.34Sr1.83Ca1.97Cu3.13Ox up to 930oC, in 7.7 % O2 / N2. Although 930oC lies well above the Tsolidus range, the samples were rapidly cooled (600oC/h) down to 860oC and subsequently slow cooled at rates between 0.2 and 10oC / h to 827oC, being further annealed for 100 h at this temperature and finally cooled at 200oC/h to room temperature. The rapidly cooling step from the maximum processing temperature could be a way to minimize oxygen and lead losses, as well as to reduce calcium and copper segregation into secondary phases. Magnetic Susceptibility of the obtained melt textured tapes presented Tc = 109 K, and XRD showed over 80% of well textured Bi-2223 [42]. However, it should be noticed that this XRD result did not give any information from the bulk core of the tape, since it was taken from the surface of the material. Besides, texture may lead to Bi-2223 fractions apparently higher than the real value. Bulk samples that were heat treated in a similar way presented only about 10% vol. of Bi-2223, even after prolonged annealing for 300 h [42]. The Ca2CuO3 segregation was considered the main obstacle to obtain high fractions of Bi-2223 [42], as confirmed in other works employing distinct nominal compositions and heat-treatments schedules [33-37]. Melt-processing via Bi-2223 partial peritectic decomposition can enhance properties of tapes and bulk samples [44-48, 68-70]. Li et al. [69] studied the phase evolution in Bi2223/silver tapes during partial decomposition and recrystallization. Their results are in good agreement with the works about complete decomposition and recrystallization mentioned above. In quenched samples, they observed that Bi-2201 becomes gradually the predominant phase with temperature increasing and that 2:1 was the main secondary phase [69]. They concluded that (Bi,Pb)-2223 decomposes into a 2212-like liquid and (Ca,Sr)-cuprates, as also

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pointed out in other works [34, 63]. Silver sheathed (Bi,Pb)-2223 tapes processed by a “controlled melt process” were heated up to 860oC/20 min in air and reached Jc = 36.8 kA/cm2 at 77 K, whereas tapes with the same composition that were just sintered exhibited Jc = 24.5 kA/cm2 [68]. Xia et al. [44-48] developed the Continuous Cooling Sintering (CCS) method, involving a partial peritectic decomposition step, so that a relatively large fraction of liquid is formed in order to reduce porosity, heal cracks and improve grain connectivity of sintered tapes. The CCS method enhanced the transport properties of 2223 tapes, significantly reduced the processing time, and increased the processing thermal window. Figure 7 shows the CCS thermal treatment regime; the CCS method was employed in the second heat treatment step, with Tmax = 845-860oC, under 7.5% O2/N2 [44-48]. In the second heat treatment, 2223 partially decomposes when Tmax ≥ 845oC. Some liquid flew out from the ends of tapes quenched from 845-860oC. The quenched tapes exhibited some amounts of Bi2Sr2CaCu2Ox (2212), Bi2Sr2CuOx (2201), (Sr,Ca)2CuO3 and (Sr,Ca)14Cu24O41 [44-48]. By slow cooling to Tfinal, the phases formed at Tmax were significantly reacted to form 2223. However, a too high Tmax leads to strong phase segregation, and a relatively fast cooling rate leads to excessive 2201 precipitation, preventing 2223 reformation [44,47]. The presence of Lead-rich precipitates was observed [44], which can be attributed to a low Pb solubility in the 2223 phase at high temperatures [59]. The CCS tapes exhibited higher densities and longer 2223 plates than the tapes sintered at T = 810-835oC [47].

Figure 7. Thermal treatment regime of the Continuos Cooling Method (CCS) [47].

The influence of Tmax and Tfinal on Jc of CCS tapes is shown in Figure 8. It can be seen that Jc is not strongly sensitive to Tmax and Tfinal within the range Tmax = 820-850oC, which is a relatively large thermal window in comparison with conventional sintering process. Figure 9 shows Jc as a function of the duration of the second heat treatment (HT2). The results indicate just a slight variation of Jc with HT2 duration for t > 10 h. Since the first sintering step (HT1) takes about 30 h, the total time required in the whole thermal processing (HT1 + HT2) is 4050 hours. In addition, the partial decomposition method also leaded to a better Jc ~ H//ab behavior (Figure 10), indicating an improved grain connectivity [47]. The use of applied magnetic field is a possible way to promote Bi-2223 texturization during solidification. Bulk textured (Bi,Pb)-2223 was solidified under high magnetic field [70]. Pellets with nominal composition Bi1.8Pb0.4Sr2Ca2.2Cu3Ox were sintered at 840oC/90h in air and melted at 855-900oC with subsequent slow-cooling (2oC/h) down to 800oC under a magnetic field of 8 T. The maximum Jc value of 1450 A/cm2 at 77 K was reached by the

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pellet melted at 860oC in air [70]. This temperature is within the range of partial (Bi,Pb)-2223 decomposition.

35 77 K, 0 T

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30 25 20 15

o

T final=770 C o

T final=790 C

10 810

820

830

840

850

860 o

Heating temperature T m ax ( C) Figure 8. Critical current density (Jc) versus the maximum heating temperature Tmax, in the CCS process, for different Tfinal values [47].

30

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J c (kA/cm )

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15 77 K, 0 T

10 0

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HT2 sintering time (h) Figure 9. Jc as a function of the second heat treatment duration [47].

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1

77 K

J c/J c0

H // ab

H // c o

0.1

T m ax = 860 C o

T m ax = 850 C o

T m ax = 840 C o

T m ax = 815 C

0

200

400

600

800

1000

Applied magnetic field (mT) Figure 10. Normalized Jc versus applied magnetic field of Bi-2223/Ag CCS tapes heated to different Tmax values and slowly cooled to Tfinal = 770 oC [48].

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4. Summary The development of an optimum melt-processing route for producing Bi-2223 tapes and bulk parts has long been sought. In the present work, a general review of this theme was undertaken with the aim of shedding light on this complex issue. Several investigations and attempts to obtain high Bi-2223 fractions via melt-processing have been made. Some of these works succeeded in crystallizing considerable fractions of textured Bi-2223 via different melt-processing routes. Both complete melting (glass ceramic routes) and partial melting have been studied. In fact, the partial melting is the peritectic decomposition of Bi-2223 into liquid and solid phases. This decompostion can be a total decomposition or a partial decomposition. In the case of partial decomposition, significant Bi-2223 fractions remain after peritectic melting. The reversibility of the peritectic decomposition has been shown and the best results have been achieved by partial decomposition routes. Although full melting (glass ceramic routes) and complete peritectic decomposition methods usually provide high Bi-2212 and/or Bi-2201 fractions, some works have shown that further annealing can increase the Bi-2223 amount. In this case, Bi-2223 forms by the reaction of Bi-2212 with secondary phases like Ca2CuO3 and CuO. Nonetheless, the use of melt-processing methods can enhance density, texture and intergrain connectivity of the final product, regardless of the way Bi-2223 forms (directly from the melt or by post-annealing the melt-processed material). However, the

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achievement of an optimum melt-processing route still depends on the elucidation of the melting behavior as well as of the nature of the Bi-2223/melt equilibrium.

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References [1] W. Hassenzahl et al., Proceedings of the IEEE, Vol. 92, No. 10, October 2004, pp. 1655-1674 [2] K.-I. Sato, Sumitomo Electric Industries, Sei Technical Review, Number 66, APRIL 2008, pp. 55-67. [3] P. Malozemoff and J. Bock, Working group SC D1.15 Materials Section – Superconductors, CIGRÈ, 2007 [4] American Superconductor Corporation, press release, Hauppauge, NY, April 30, 2008, www.amsc.com, accessed on June 01, 2008. [5] L.J. Masur et al., European Conference on Applied Superconductivity - EUCAS 2003, Sorrento, Italy, September 15-18, 2003 [6] N. Ayai et al., Sumitomo Electric Industries, Sei Technical Review, Number 63, December 2006, pp. 58-64 [7] Sumitomo Electric Industries, 2007 Annual Report, http://www.sei.co.jp/iv_e/annual/07 index.html, accessed on May 29, 2008 [8] G. Snitchler, B. Gamble and S.S. Kalsi, IEEE Transactions on Applied Superconductivity, Volume: 15, Issue: 2, Part 2, June 2005, pp. 2206- 2209 [9] M.P. Paranthaman and T. Izumi, Mrs Bulletin, August 2004, pp. 533-535 [10] A P Malozemoff et al, Supercond. Sci. Technol. 21, February 20, 2008 [11] Y. Chen et al., Presentation at 2008 Spring Meeting of the Materials Research Society (MRS), March 24-28, 2008, San Francisco, CA, USA [12] Flükiger, R.; Grasso, G.; Grivel, J.C.; Marti, F.; Dhallé, M.; Huang, Y. Supercond. Sci. Technol., v. 10, p. A68, 1997. [13] P. Majewski, Supercond. Sci. Technol., 10 (1997), 453-467. [14] Roth, R.S.; Rawn, C.J.; Burton, B.P.; Beech, F. J. Res. NIST, v. 95, p.291, 1990. [15] Polasek et al., IEEE Transactions on Applied Superconductivity, v. 15, n. 2, p. 31413144, 2005. [16] Kitaguchi, H.; Kumakura, H. Mrs Bulletin, v. 26, n. 2, p. 121, February / 2001. [17] Grant, P.M. IEEE Transactions on Appl. Supercond., v. 7, n. 2, p. 112, 1997 [18] R.M. Scanlan et al., Proceedings of the IEEE, Vol. 92, No. 10, October 2004, p. 1639. [19] Feng, Y. et al. Physica C, v. 192, p. 293, 1992 [20] Grasso, G.; Hensel, B.; Jeremie, A.; Flükiger, R. Physica C, v. 241, p. 45, 1994. [21] Patnaik S, et al, IEEE Trans. Appl. Supercond. 13, 2930 (2003) [22] Jiang, J. et al., IEEE Trans. on Appl. Supercond., v. 11, n. 1, p. 3561, 2001. [23] Cai, X.Y.; Polyanskii, A.; Li, Q., Riley Jr., G.N.; Larbalestier, D.C. Nature, v. 392, p. 30, 1998. [24] Wang, W.G.; Horvat, J.; Li, J.N.; Liu, H.K.; Dou, S.X. Physica C, v. 297, p. 1, 1998. [25] Lisboa, M.B.; Soares, G.A.; Serra, E.T.; Polasek, A.; Xia, S.K. Materials Characterization, v. 46, p. 75, 2001. [26] S. Kobayashi et al., Ieee Transactions On Applied Superconductivity, vol. 15, no. 2, june 2005, 2534-2537

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[27] J Jiang et al, Supercond Sci Technol, 14 (2001) 548-556 [28] J. S. Abell and T. W. Button, Sintering Techniques for BSCCO, in Handbook of Superconducting Materials, ed. David A. Cardwell and David S. Ginley, Institute of Physics Publishing, 2003, p. 259 [29] J.G. Noudem, J. Beille, D. Bourgault , A. Sulpice, R. Tournier, Physica C 230 (1994) 42-50. [30] Tampieri, G. Calestani, G. Celotti, R. Masinic, S. Lesca, PhysicaC 306 1998 21–33 [31] Nilsson, W. Gruner, Jörg Acker, Klaus Weitzig, Journal of Non-Crystalline Solids, 354 (2008), n. 10-11, pp. 839-847. [32] Park, C.; Wong-Ng, W.; Cook, L.P.; Snyder, R.L.; Sastry, P.V.P.S.S.; West, A.R. Physica C, v. 304, p. 265, 1998. [33] Polasek A, Majewski P, Serra E T, Rizzo F., Physica C. 2004; 408-410, p. 860. [34] Polasek A, Majewski P, Serra ET, Rizzo F, Aldinger F. Materials Research, vol. 7, n. 3, 2004, p. 393. [35] MARINKOVIC, Bojan ; JARDIM, P. M. ; D. Medeiros ; T. Chehuan ; POLASEK, A. ; ASSUNÇÃO, Fernando Cosme Rizzo . Journal of Physics: Conference Series, v. 43, p. 59-62, 2006. [36] B.A. Marinkovic, P.M. Jardim, D. Medeiros, T. Chehuan and F. Rizzo, Materials Letters, Volume 60, Issue 19, August 2006, Pages 2366-2370 [37] Bispo, e.r.; Polasek, a.; Neves, m.a.; Rizzo, F., Revista Matéria, vol.13, no.1, 2008, pp. 238 – 245. [38] Giannini, E.; Passerini, R.; Toulemonde, P.; Walker, E.; Lomello-Tafin, M.; Sheptyakov, D.; Flükiger, R. Physica C, v. 372-376, p. 895, 2002. [39] Giannini, E., et al., Superconductor Science Technology, v. 15, pp. 1577-1586, October 2002. [40] Oka, Y.; Yamamoto, N.; Kitaguchi, H.; Oda, K.; Takada, K. Jpn. J. Appl. Phys., v. 28, n. 2, L213, 1989. [41] Bock, J.; Preisler, E. Proceedings of ICMC ’ 90 Topical-Conference on Materials Aspects of High-Temperature Superconductors, Garmisch-Partenkirchen, Germany, p. 215, 1990. [42] Yamada, Y.; Graf, T.; Seibt, E.; Flükiger, R. IEEE Trans. Magn., v. 27, p. 1495, 1991. [43] Flükiger, R.; Giannini, E.; Lomello-Tafin, M.; Dhallé, M.; Walker, E. IEEE Trans. Appl. Supercond., v. 11, n. 1, p. 3393, 2001. [44] S K Xia, M B Lisboa, E T Serra and F Rizzo, Supercond. Sci. Technol. 14 (2001) 103– 108. [45] S K Xia, M B Lisboa, E T Serra and F Rizzo, Physica C 354 (2001) 463-466. [46] S K Xia, E T Serra and F Rizzo, Physica C 361 (2001) 175-180. [47] Xia, S.K.; Serra, E.T. Studies of High Temperature Superconductors, ed. A. Narlikar, Nova Science Publisher, New York, USA, v. 43, p. 63, 2002. [48] S. K. Xia, A. Polasek, L. A. Saléh, B. Marinkovic, F. Rizzo, and E. T. Serra Physica C, Volumes 408-410, 2004, pp. 911-912. [49] Strobel, P.; Tolédano, J.C.; Morin, D.; Schneck, J.; Vacquier, G.; Monnereau, O.; Primot, J.; Fournier, T. Physica C, v. 201, pp. 27-42, 1992. [50] D. Shi et al., Physical Review B, vol. 40, n. 4, August 1989, pp. 2247-2253. [51] M. Xu et al., Appl. Phys. Lett. 55 (21), 20 November 1989, pp. 2236-2238. [52] B.M. Moon et al., Appl. Phys. Lett. 55 (14), 2 October 1989, pp. 1466-1468.

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[68] [69] [70]

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In: Superconducting Magnets and Superconductivity... ISBN 978-1-60741-017-1 c 2009 Nova Science Publishers, Inc. Editors: H. Tovar and J. Fortier, pp. 215-256

Chapter 6

VORTEX T HEORY OF I NHOMOGENEOUS S UPERCONDUCTORS B.J. Yuan∗ Department of Physics, YuXi Normal University, China

Abstract

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Combining a number of well-known theories of conventional superconductors [1–6], which include Abrikosov vortex theory, Gorkov formalism, Extended GinzbergLandau theory, and de Gennes-Werthamer proximity coupling theory, a general vortex theory for inhomogeneous superconductors is proposed [7–9]. This general vortex theory is applied to three characteristic and simple geometries, a bulk superconductor with finite size, a thin film in parallel applied filed and a special two-components superlattice, respectively. In each of the three cases, spatial form of order parameter and magnetic field in the geometry are derived, which lead to an analytical expression of Gibbs free energy. For a finite bulk superconductor, the eigenvalue spectrum in central region of the bulk geometry is in similar to the case of an infinite geometry with lower symmetry. A vortex lattice in the central region prefers a hexagonal structure if only minimum free energy is concerned, however, when effects of repulsion between flux lines or arrayed defects are included, symmetry of a vortex lattice varies. For a thin film geometry in a parallel field, the solution shows that the inhomogeneity caused by limited thickness of a film enhances extended GL parameter, κ2 , which proves that thin film superconductor tends to be type II superconductor. For a two-components (NS) superlattice in a parallel applied filed, we find that, in addition to a slowly spatial varying pair amplitude nucleating in N layers, a highly condensed pair amplitude confined in the other component layers (S) can also simultaneously satisfy both a minimum eigenenergy requirement and a minimum free energy requirement. The formation of a vortex lattice either in N or S layers is determined by the competition of the two mechanisms. The analysis to different geometries shows that there are two energy criteria in forming a vortex lattice. Both energy criteria are affected explicitly or implicitly by the inhomogeneities of a superconductor. ∗

The author is a adjunct researcher in this university. The author himself is living in the United States and working in a non-academic organization. Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

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1.

B.J. Yuan

Introduction

Vortex lattices in a superconductor have been an interesting research subject since it was predicted in theory by Abrikosov [2] and was first found in an experiment by Essmann and Trauble [10] a few decades ago. With advances of material preparation and experimental techniques, abundant vortex lattices of various spatial structures have been discovered in different superconductors or under varying external physical conditions. These recently discovered vortex lattices, when in their steady states, usually possess spatial structures deviating from a hexagonal symmetry predicted by traditional vortex theory [2]. The consensus to various vortex lattices observed in experiments is that superconductors may have inhomogeneities caused by impurities, point defects or planar interfaces etc., these inhomogeneities affect the spatial structures of vortex lattices through different mechanisms. Traditional vortex theory proposed by Abrikosov [2] applies a simple criterion that a minimum free energy of a superconductor determines both existence and symmetry of a vortex lattice. It is difficult, within the existing formula of the Abrikosov vortex theory [2], to study possible effects of inhomogeneities of a superconductor on the properties of a vortex lattice. Based on previous research works [7–9], we construct a vortex theory for inhomogeneous superconductors by combining a number of well-known theories, which include Abrikosov vortex theory [2], Gorkov theory [3], Extended Ginzberg-Landau theory [1, 4], and de Gennes-Werthamer proximity coupling theory [5, 6]. The Gibbs free energy of this new vortex theory is in a similar form to that of Abrikosov’s vortex theory, nevertheless, it is able to accommodate order parameters and internal magnetic fields in a inhomogeneous superconductor. The order parameters and internal magnetic fields may be obtained by solving appropriate differential equations together with appropriate boundary conditions. In next section, we analyze the Abrikosov vortex theory to find out what mathematical nature of the theory impedes its application to an inhomogeneous superconductor. We then reform the format of its key formulae to have the theory be able to accommodate more complicated order parameter and internal magnetic field found in inhomogeneous superconductors. In Section 3., we apply the new vortex theory to a bulk superconductor in a shape of rectangular prism [9]. The eigenvalue spectrum shows that a finite sized bulk geometry exhibits different magnetic properties in different region. Near the edge of the geometry, there is a surface layer which excludes flux lines, while in the central region of the geometry, the vortex lattices formed are in similar to that of an infinite superconductor. However, in addition to the minimum free energy requirement, the vortex lattices in the central region may be affected by other factors such as repulsion force among flux lines or arrayed defects, hence, varying symmetry of vortex lattices may exist. We examine this type of vortex lattices by comparing the theoretical results with experimental data in V3 Si [12,14], Nb [15] and NbSe2 [17]. In Section 4., we apply the vortex theory [7] to a superconducting film in parallel applied field. The critical field of this geometry was previously studied in detail by Saint-James and de Gennes [16] within the framework of linearized GL theory [1]. Our work shows that there are two energy criteria governing a superconducting state in the film. The minimum eigenenergy requirement determines a pair amplitude’s position within the geometry and

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Vortex Theory of Inhomogeneous Superconductors

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magnitude of upper critical field. The Gibbs free energy is evaluated by using an order parameter consisting of degenerate pair amplitudes at the minima of eigenenergy spectrum. The theoretical results in this film geometry suggest that the minimum eigenenergy requirement imposed on a pair amplitude surpasses the minimum free energy requirement in determining a vortex state. This conclusion is consistent with the experimental results reported earlier [18, 20]. In Section 5., the vortex theory [8] of inhomogeneous superconductor is applied to a two-components superlattice (N/S) in a parallel applied field. This type of superlattice may be realized by multilayered materials such as Nb/NbZr [21, 22] or NB/NbTi [23] in experiments. We perform a theoretical calculation targeted on Kuwasawa et al.’s superlattice(ρ = 0.2586) [21]. The calculation shows a periodical eigenenergy spectrum for a given strength of applied field. The eigenenergy spectrum shows that the nucleation center appears at the center of a N-type layer at low applied field and at the center of a Stype layer at high field. A pair amplitude positional shift is found at certain value of applied field, which produces Takahashi-Tachiki effect [24]. The vortex state in this superlattice is calculated based on the degenerate pair amplitudes found in periodical minima of the spectrum. Again, we find two energy criteria competing with each other in determining a vortex state.

2.

Vortex Theory for Inhomogeneous Superconductors

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In this section, we analyze Abrikosov’s vortex theory and modify some parts of it, if necessary, in order to develop a vortex theory for inhomogeneous superconductors. Since there are a number of theories involved in deriving new formulae, we use a set of notations mainly adopted from Fetter and Hohenberg’s work [25].

2.1.

Solution of Linearized Ginzburg-Landau Equation

The differential form of Ginzburg-Landau (GL) theory [1] is specified by the following equations 2 1 e∗ −i¯ h ∇ − A Ψ + aΨ + b|Ψ|2 Ψ = 0 , 2m∗ c e∗ ¯h e∗ 2 c ∗ ∗ ∇×h= (Ψ ∇Ψ − Ψ∇Ψ ) − |Ψ|2 A = J , 4π 2m∗ i m∗ c





(1) (2)

and boundary conditions at the surface of a superconductor e∗ n · −i¯h∇ − A Ψ = 0 , c S n × (h − H)|S = 0 . 



(3) (4)

Abrikosov’s vortex theory starts from solving a linearized GL equation, e∗ 1 −i¯ h ∇ − A 2m∗ c 

2

Ψ1 + aΨ1 = 0 ,

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218

B.J. Yuan

where Ψ1 denotes a order parameter governed by linearized GL equation, A(x) is a vector potential related to applied field, Ha , through A(x) = Ha (0, x, 0) ,

(6)

Substituting vector potential into Eq. (5) gives rise to 2 #

∂2 ∂2 ∂ e∗ ¯2 h − − + −i − Ha x 2m∗ ∂x2 ∂z 2 ∂y h ¯c "



Ψ1 = −aΨ1 .

(7)

There is an intrinsic length and an energy unit in this equation, the intrinsic length is defined as s |e∗ | 2|e| 2πHa φ0 −2 ξH = Ha = 2πHa = → ξH = , (8) ¯hc hc φ0 2πHa where φ0 = hc/2|e| is a flux quanta. The energy unit is defined as ǫ=

−2 ¯ 2 ξH h , 2m∗

(9)

hence, the eigenenergy in Eq. (7) is given by −2 ¯ 2 ξH h 2m∗

−a = ηǫ = η

!

.

(10)

Using the intrinsic length, ξH , to measure every dimension of coordinate space, x→

x y z , y→ , z→ . ξH ξH ξH

(11)

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Eq. (7) is simplified to "

2 #

∂2 ∂2 ∂ − 2− 2+ i −x ∂x ∂z ∂y 

Ψ1 (x) = ηΨ1 (x) .

(12)

We write the order parameter in a form that Ψ1 (x) = e−iky f (x) cos pz = ψ(x, y) cos pz .

(13)

Eq. (7) is reduced to "

#

d2 − 2 + (x − k)2 f (x) = ηf (x) , dx

(14)

where p2 = 0 is chosen since we search only the minimum eigenvalue. Obviously, a normalizable order parameter in an infinite geometry requires a boundary condition lim f (x) → 0 .

x→∞

(15)

Eq. (14) and (15) define an one dimensional eigenvalue problem, its ground state solution is given by 1 2 (16) f (x) = e− 2 (x−k) , η = 1 . Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

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f (x) is a Gaussian wave-packet centered at a position k. This solution possesses an infinitesimal translational invariance which reflects the characteristic of a uniform infinite superconductor. The ground state solution satisfies an important mathematical relation1 that 

(x − k) +

1 d 2 e− 2 (x−k) = 0 . dx



(17)

The eigen equation, Eq. (14), can generate a ground state solution to satisfy Eq. (17), only with the boundary condition, Eq. (15). If Eq. (14) remains unchanged while the boundary condition takes different form, the ground state solution does not satisfy Eq. (17). However, it is based on this property that Abrikosov’s vortex solution is derived [25], hence, Abrikosov’s vortex theory has to be modified in order to study vortex states in inhomogeneous superconductors. Even though we do not have the ground state solution yet, we still can write a general order parameter with a set of degenerate pair amplitudes, {ψj }

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Ψ1 (x) = F(x, y) =

X

cj ψj (x, y) =

{kj }

X

{kj }

cj f (x − kj )e−ikj y ,

(18)

where {kj } denotes a discrete set of positions( nucleation center), f (x − kj ) is a function governed by Eq. (14) and centered at position kj , ψj = f (x − kj )e−ikj y is a pair amplitude belongs to the degenerate set {ψj }. The detail of f (x − kj ) is unknown without a boundary condition. In fact, this is a key assumption which allows us to include inhomogeneity into a vortex solution through appropriate boundary conditions. cj = |cj |e−iαj is a complex number. Considering each pair amplitude is equally weighted in the linear superposition and the quantities we calculate is independent of the absolute magnitude of the order parameter, we choose |cj | = 1.

2.2.

Vortex Solution

The general form of order parameter, Eq. (18), allows us to construct a current density c 2 i (F∇F ∗ − F ∗ ∇F) − x|F|2 ey J(x) = Z 4π ξH 2 



(19)

∗

h ¯ where Z = 2πe m∗ c and the coordinates in the expression are all dimensionless. In order to solve the second GL equation, Eq. (2), −1 ξH ∇ × h(x, y) =

4π J, c

(20)

to obtain a vortex solution, h(x, y), we introduce a transformation to Eq. (14) that (x − kj )fj2 = 1

1 d 2 [(x − kj )2 fj2 − ηfj2 − f ′ j ] , 2 dx

It may be written as a|0i = 0, where a is annihilation operator and |0i defines a ground state.

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B.J. Yuan

where fj = f (x − kj ). Defining two functions that 2

Wj (x) = [(x − kj )2 fj2 − ηfj2 − f ′ j ] ,

Wij (x) =

[fi fj′

−

(22)

fj fi′ ] ,

(23)

the current density can be written as a differential form 



i6=j

X Wij Zc ∂ X Wj + cos(kij y + αij ) Jx = 4πξH ∂y j k ij i,j

  i6=j X X −Zc ∂  Wij Jy = Wj + cos(kij y + αij )

4πξH ∂x

i,j

j

kij

(24)

(25)

where kij = (ki − kj ) denotes a distance between the geometric center of ψi and ψj , αij = (αi − αj ) denotes a difference of initial phase between ψi and ψj , while ψi and ψj are defined by ψl = ψ(x − kl , y) = f (x − kl ) e−ikl y , l = i, j . (26) The Second GL equation, Eq. (20), may be written as follows ∂ 4π ∂ ˆy ˆx −e hz = ξH (Jx + Jy ) . e ∂y ∂x c





(27)

Comparing both sides of Eq. (27), we obtain vortex solution hz = Ha − Zbz = Ha + hsz ,

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where bz = −

 X 

Wj +

j

i6=j X Wij i,j

kij

cos[kij y + αij ]

(28)  

.

(29)



We note that, in deriving this vortex solution, all we required is a transformation of eigen equation, Eq. (21), and a degenerate set of pair amplitudes, {ψj }. This indicates that the form of the vortex solution, Eq. (28), does not depend on any type of boundary condition. 1 2 It is straightforward to show that, if fj = e− 2 (x−kj ) is substituted into Eq. (29), Eq. (28) reduces to Abrikosov’s vortex solution, hz = Ha − Z|Ψ1 |2 ,

2.3.

(bz → |Ψ1 |2 ) .

(30)

Gibbs Free Energy of Vortex Solution

Before we continue to derive Gibbs free energy based on the general vortex solution, Eq. (28), we introduce certain theoretical results of extended GL theory in dirty limit [3–6, 26–28]. Using these results, the derived vortex theory is valid along the entire upper critical field, Hu (T ) [25]. Defining an operator Π = −i∇ −

e∗ A, ¯hc

Π2 = Π · Π ,

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221

and introduce three operator functions, X(Π2 ), Y (Π2 ) and Z(Π2 ). In GL theory, the three operator functions are expressed as follows X(Π2 ) = (¯ h2 /2m∗ )Π2 − |a| ,

(32)

2

Y (Π ) = b , 2

∗

(33)

∗

Z(Π ) = 2πe ¯h/m c .

(34)

Using the three operator functions, the GL differential equations may be expressed as X(Π2 )Ψ + Y (Π2 )|Ψ|2 Ψ = 0 , (35) c J= Z(Π2 )[Ψ∗ ΠΨ + (ΠΨ)∗ Ψ] . (36) 4π Considering the approximation of linearizing GL equation, we specify the vector potential in a form of A1 = Ha (0, x, 0), which leads to a approximated operator, Π1 , Π1 = −i∇ −

e∗ A1 . ¯hc

(37)

The linearized GL equation becomes X(Π21 )Ψ1 = 0 .

(38)

Assuming the order parameter, Ψ1 , satisfies an eigen equation, Π21 Ψ1 = q12 Ψ1 ,

(39)

where q12 is an eigenvalue,

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q12 = η

2πHu e∗ Hu −2 =η = η ξH . u ¯hc φ0

(40)

In above expression, η = 1 for an infinite large uniform superconductor, in general, η 6= 1 and η needs to be determined by solving a eigenvalue problem. Once the eigenvalue, q12 , is available, the linearized GL equation provides an implicit relation in determining the upper critical field, Hu ,   2πHu X(q12 ) = X η = 0. (41) φ0 Obviously, if η = 1, this equation determines Hu = Hc2 , if η 6= 1, the upper critical field differs from Hc2 2 . In the extended GL theory in dirty limit(l ≪ ξ0 ) [3–6,26–28], the form of linearized differential equation, Eq. (39), does not change, however, the three operator functions subject to a algebraic modification that X(q12 ) Y

(q12 )

Z(q12 ) 2

!



T + ln Tc



= 0,

(42)

=

1 f1 8(πkB T )2

¯ Dq12 h 2πkB T

!

,

(43)

=

eτ N g mckB T

¯ Dq12 h = χ 2πkB T

¯ Dq12 h 2πkB T

!

,

For example, if η = 0.59, Hu = Hc3 (surface effect).

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B.J. Yuan

where D is diffusion constant and ¯hD replaces ¯h2 /(2m∗ ) in the eigenenergy unit, Eq. (9). Functions χ(z),f1 (z) and g(z) are given by 1 1 1 χ(z) = ψ( + z) − ψ( ) , 2 2 2 X 1 f1 (z) = 1 1 3 , m (m + 2 + 2 z) X 1 g(z) = 1 1 2 , m (m + 2 + 2 z)

(45) (46) (47)

respectively, and ψ(z) is a digamma function By introducing three operator functions, X(Π2 )Y (Π2 ) and Z(Π2 ), we have unified expressions of original GL theory and extended GL theory, hence, we are able to derive a free energy of a vortex state based on both original and extended GL theory near the upper critical field. The integral form of the non-linear GL equation is given by hΨ|X(Π2 )|Ψi + hΨ|Y (Π2 ) |Ψ|2 |Ψi = 0 ,

(48)

where we have defined spatial average of an operator, O, as hOi =

1 V

Z

d3 x O ,

V

hΨ|O|Ψi =

1 V

Z

d3 x Ψ∗ OΨ .

(49)

V

Expanding Ψ in Eq. (48) around the order parameter, Ψ1 [7, 25], we obtain a formula,

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X ′ (q12 )

2πη 2πη (Hu − Ha )h|Ψ1 |2 i = X ′ (q12 ) hΨ1 |hsz |Ψ1 i + Y (q12 )h|Ψ1 |4 i . φo φo

(50)

Substituting the solution for an infinite homogeneous superconductor into Eq. (50), i.e., Ψ1 = F, η = 1, Hu = Hc2 and bz = |F|2 , we recover the second identity of Abrikosov’s theory X ′ (q12 )

2π 2π (Hc2 − Ha )h|F|2 i = −X ′ (q12 ) Z(q12 ) + Y (q12 ) h|F|4 i . φo φo 



(51)

To derive a free energy for inhomogeneous superconductors based on Eq. (50), we define a generalized geometric factor, βB , βB =

hF|bz |Fi , hbz ih|F|2 i

(52)

and a generalized GL parameter, κ ˜2, 2˜ κ22 =

Y (q12 )φo h|F|4 i h|F|4 i 2 = 2κ . 2 2 2 ηhF|bz |Fi Z(q1 )X ′ (q1 )2π ηhF|bz |Fi

(53)

Substituting βB and κ ˜ 2 into Eq. (50), we obtain Z(q12 )hbz i =

(Hu − Ha ) . [2˜ κ22 − 1]βB

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The thermodynamical magnetic flux is obtained by spatial average of internal magnetic field, hz , B = hhz i = Ha − Z(q12 )hbz i Hu − Ha  = Ha −  2 . 2˜ κ2 − 1 βB

(55)

The thermodynamic relation,



∂G ∂H



T

=−

B , 4π

(56)

then gives rise to the Gibbs free energy for inhomogeneous superconductor, Gm (Ha , T ) = Gm (Hu , T ) +

1 1 (Hu − Ha )2  2  (Hu2 − Ha2 ) − , 8π 8π 2˜ κ2 − 1 βB

(57)

where subscript m denotes a mixed state (vortex state). An alternative expression of this formula is the difference of Gibbs free energy between a vortex state and a normal state, Gm (Ha , T ) − Gn (Ha , T ) = −

1 (Hu − Ha )2  2  . 8π 2˜ κ2 − 1 βB

(58)

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The vortex theory of inhomogeneous superconductors derived above is general, which contains the vortex theory of homogeneous superconductor as a special case. If we substitute bz = |F|2 and η = 1 into Eq. (58)we recover Abrikosov’s geometric factor, βA , βB |bz →|F |2 =

h|F|4 i = βA , h|F|2 i2

(59)

and Maki’s GL parameter [4], κ2 ,

2κ ˜ 22

bz →|F |2 ,η→1

→ 2κ22 =

Y (q12 )φo . Z(q12 )X ′ (q12 )2π

(60)

This limit indicates how the vortex theory of inhomogeneous superconductors approaches the vortex theory of homogeneous superconductors within the framework of extended GL theory in dirty limit(l ≪ ξo ). Near the transition temperature, Tc , Maki’s GL parameter reduces to original GL parameter κ, lim κ2 → κ ,

T →Tc

(61)

so the Abrikosov vortex theory is totally recovered at T ∼ Tc . In the following sections, we apply the vortex theory developed in this section to a few types of inhomogeneous superconductors, the inhomogeneity of superconductors is specified through appropriate boundary conditions at surfaces or continuity conditions at interfaces [26].

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3.

B.J. Yuan

Vortex State in a Rectangular Prism

Sosolik et al. [12] reported that in a rectangular prism of V3 Si, a vortex lattice exhibits a hexagonal symmetry at Ha = 1 Tesla, however, the symmetry of the vortex lattice varies as the strength of the applied field increases, when the applied field reaches 5 Tesla, a square vortex lattice is observed. Based on a series of images of vortex lattices presented by Sosolik et al., it is reasonable to believe that the vortex lattice undergo a hexagonal-tosquare continuous symmetry variation as the applied field is increased from 1 to 5 Tesla. The V3 Si geometry is illutrsated in Fig. 1.

z

6









Lx 0

6Ha 





       Lz  Ly -

x

Figure 1. This rectangular prism is denoted by Lx = 2mm, Ly = 7mm and Lz = 2mm. Applied field Ha is in z direction. The vector potential is given by A = Ha (0, x, 0).

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3.1.

Eigenenergy Spectrum

Scaling coordinates by the intrinsic length ξH , applying energy unit ǫ and substituting the vector potential into the gauge derivative, the eigenvalue problem for the rectangular prism given in Fig. 1 is simplified to the following equations and boundary condition, "

2 ∂2 ∂ ∂2 −x − 2− 2+ i Ψ(x) = ηΨ(x) , ∂x ∂z ∂y ∂ Ψ(x)|x=± 1 lx = 0 , 2 ∂x     ¯hD Tc 2πHu η T χ + ln = 0, 2πkB Tc T φ0 Tc



 #

(62) (63) (64)

where we have employed dirty limit theory so that the upper critical field is determined by Werthamer relation, Eq. (64). Assuming Ψ(x) = e−iky f (x) cos pz = ψ(x, y) cos pz ,

(65)

and choosing p = 0, the eigenvalue problem reduces to −

d2 f (x) + (x − k)2 f (x) = ηf (x) , dx2

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1.1 1

η(k) 0.9

S

0.8

S



-

R

0.7 0.6 0.5 -20

-15

-10

-5

0

k

5

10

15

20

Figure 2. Eigenvalue spectrum η(k) in a geometry with a width of Lx = 40ξH . S denotes a surface region, which causes surface superconductivity. R denotes a flat central region with η(k) being almost a constant, η(k) ≈ 1, which reproduces a scenario of a bulk uniform superconductor. df dx x=± 21 lx

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= 0.

(67)

Solving this eigenvalue problem numerically for a geometry of Lx = 40ξH , we obtain an eigenvalue spectrum as shown in Fig. 2. This eigenvalue spectrum is characterized by two types of regions. Near the surfaces, η(k) varies rapidly and reaches a minimum of η = 0.59. This minimum eigenvalue indicates surface superconductivity and corresponds to upper critical field Hc3 [16]. Away from surfaces about 4ξH , the eigenvalue is nearly a constant, η ≈ 1.0. The flat region is marked by R and reproduces a scenario of an infinite uniform superconductor(η = 1). A width of 2mm may be converted to a width about 105 ξH at Ha = 1 Tesla, it can be verified that the eigenvalue spectrum for a 105 ξH width bulk superconductor is similar to the one presented in Fig. 2. In addition to a measurable Hc3 , the S region itself is observable. In an experiment for a rectangular prism of NbSe2 , Olsen et al. [17] observed a vortex free band near the edge of the rectangular geometry, the width of the band is about 5µm at a applied field, Ha = 0.2mT. Corresponding to this applied filed, ξH is about a length of 1.28µm, the width of the band is thus about 3.88ξH which is close to the width of S-layer in Fig. 2. Moreover, Olsen et al. noticed that, as the applied field increases, the vortex free band remains but becomes a bit thinner, which is consistent with a shrink ξH as Ha increasing. The eigenvalue spectrum marked by R in Fig. 2 is nearly a constant, η ≈ 1. It indicates that an Abrikosov vortex lattice with certain symmetry may be formed in R-region.

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3.2.

B.J. Yuan

Vortex Solution and Varying Symmetry

Within R region, the general solution of an order parameter, F(x), can be constructed by a linear superposition of a set of degenerate pair amplitudes, ψj (x), F(x) =

X

cj ψj (x) =

X

{kj }

{kj }

cj f (x − kj )e−ikj y ,

(68)

where {kj } denotes a discrete set of positions within R region and cj = |cj |e−iαj is a complex number with unit module, |cj | = 1. Following the steps described in subsection 2.2., we obtain a vortex solution hz = Ha − Zbz with bz = −

 X 

Wj +

j

i6=j X Wij i,j

kij

(69)

cos[kij y + αij ]

 

.

(70)



Note that, we do not apply the mathematical property of Eq. (17) in deriving the vortex solution, hz , nevertheless, the eigenvalue of f (x − kj ) is close to 1 up to six significant 1 2 figures, hence, it is a valid approximation to replace f (x − kj ) by e− 2 (x−kj ) if a function value needs to be evaluated. With this approximation, we have 2

Wj (x) ≈ −e−(x−kj ) ,

(71)

− 12 (x−ki )2

Wij (x) ≈ −kij e

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and bz =

 X 

j

2

e−(x−kj ) +

i6=j X

− 21 (x−ki )2

e

− 21 (x−kj )2

e

− 21 (x−kj )2

e

,

cos[kij y + αij ]

i,j

(72)  

.

(73)



The key to form a flux line lattice(FLL) in R-region is that an array of pair wave functions should form a lattice of holes (zeros of order parameter |F|, ZL) in R-region through phase coherence. The ZL allows flux lines to pass through the superconductor. A spatial pattern of a ZL formed by phase coherence of pair functions depends on the relative distances among these pair functions, {ψj }, as well as the choice of initial phases, {αj }. We study first the case presented in Fig. 3. In Fig. 3, we show an array of wave functions, {ψj }, which reside at an array of positions of {kj }. The initial phase of a ψj is taken alternatively αj = 0 or π2 [29]. Due to the choice of the initial phase, there are two types of arrays of holes in y direction denoted by A and B, respectively. The positions of the holes, where the flux lines pass through, may be described by the following formula π cos[(2n + 1)2ky ± (−1)n ] = −1 , n = 0, 1, 2, · · · 2

(74)

where + and − correspond to array B and A, respectively, n denotes the pairs of wave functions which contributes to the interference. E.g. for array A, pair (ψ−1 , ψ0 ) corresponds to n = 0, pair (ψ−2 , ψ1 ) corresponds to n = 1, and so on. Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

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227

y 6 -

x

A

B 2sy

2k

γ 2sx

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ψ−3 ψ−2 ψ−1 ψ0 ψ1 ψ2 ψ3 Figure 3. An array of pair wave functions, ψj = e−iα e−ikj y f (x − kj ), aligned along x direction. Distance between peaks of neighboring wave functions is denoted by a dimensionless length, sx . The wave functions {ψj } drawn in solid line corresponding to initial phase αj = 0 and dash line corresponding to αj = π2 . 2k denotes a difference of wave vectors or a distance of the peaks between neighboring pair amplitudes. A and B denote two arrays of holes along y direction, respectively. The cell denoted by 2sx × 2sy is chosen to establish a relation between sx and γ.

Requiring conservation of magnetic flux passing through the rectangular cell as shown in Fig. 3, we have Ha (2Sx )(2Sy ) = 2φo →

2πHa −2 Sx Sy = π → ξH Sx Sy = π, φo

(75)

which, in terms of dimensionless lengths sx = Sx /ξH and sy = Sy /ξH , may be written as sx sy = π → sx =

r

π . tan γ

(76)

The symmetry of a FLL ( or a ZL) can be described by γ. γ and sx are one to one correspondent, that is, all the hexagonal FLL (γ = 60o ) observed in experiments correspond to the same dimensionless length, sx = 1.3468. This theoretical result is examined against data of a few experiments as listed in Table 1. We find that, for the experiments of Essmann and Trauble [10] and Hess et al. [30], the theoretical results matches their experiment data precisely. For Vinnikov et al.’s experiment [31], there is no explicit data about the spacing between flux lines, however, estimated value of 2Sx based on the image in their article seems to confirm the theoretical result, too. It is interesting to note that these experiments are performed on different superconductors, therefore, the universality of sx suggests that the true physical spacing between flux lines,

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Table 1. The universality of sx = 1.3468 is examined against a few experimental results about hexagonal FLL. The third experiment does not report the spacing between flux lines explicitly in the reference. Ref. [10] [30] [31]

Material Pb-4at%In NbSe2 MgB2

γ 60o 60o 60o

sx 1.3468 1.3468 1.3468

Ha (Gauss) 195 10,410 200

˚ ξH (A) 1,299 177.8 1,283

˚ 2Sx (A) 3,499 479 3,455

˚ Exp.(A) 3,500 479 NA

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2Sx = 2 sx ξH , or spacing between the peaks of neighboring wave functions, Sx = sx ξH , maight reflect certain material property of a homogeneous superconductor. The finite width of R-region can accommodate limited number of pair amplitudes, as applied field increases, the number of flux lines increases, so the number of pair amplitudes has to increase in order to create enough holes through phase coherence among pair amplitudes to accommodate more flux lines, therefore, the physical spacing between neighboring pair amplitudes, Sx , reduces. If a superconductor is quite uniform and minimum free energy is the dominant factor in determining the symmetry of a FLL, then a hexagonal lattice characterized by γ = 60o should be formed [2, 29]. If there is no other disturbance, as the applied field increases, the physical spacing, Sx , decreases at the same rate as ξH decreases so that γ remains unchanged in terms of Eq. (76). This usually happens at low applied filed [12, 14], at which flux lines are well separated.

Figure 4. Four contour plots indicate four images of a FLL. From left to right, the four images show a continuous variation of symmetry of a flux line lattice as applied field varies at 1, 2, 3 and 5 Tesla, respectively. This hexagonal-to-square symmetry change of a FLL has been observed in a V3 Si superconductor by Sosolik et al. [12]. However, if we assume that there is a kind of mechanism which impedes two neighboring flux lines or two neighboring pair amplitudes approaching each other too close, then Sx may not be able to keep the same rate of decreasing as ξH does, which may cause γ change. The cell drawn in Fig. 3 indicates that γ can change while maintain flux conservation in the cell and our calculation, in Fig. 6, will show that there is a possible path for a continuous variation of γ from 60o to 45o , which only causes a tiny raise of free energy. Hence, under the assumption of existing certain repulsion between flux lines or pari amplitudes, a

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continuous symmetry variation of a FLL is possible. Table 2. Spacing between flux lines in x direction, 2Sx , is listed corresponding to four images of FLLs in V3 Si [12]. γ 60o 58.3o 51.2o 45o

sx 1.3468 1.3929 1.5893 1.7725

Ha (T) 1 2 3 5

˚ ξH (A) 181 128 105 81

˚ 2Sx (A) 488 357 334 287

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At a high applied field, the total number of flux lines in a restrictive area increases significantly, which squeezes physical distance, Sx , either between neighboring flux lines or neighboring pair amplitudes. As the decrease of Sx is slower than that of ξH as Ha further increases, γ varies [12, 14]. The varying γ of a FLL has been observed in a a number of experiments, we particularly focus on the experiments performed on A3 Si by Sosolik et al. [12] and Yethiraj et al. [14]. Our calculation is based on a rectangular prism which Sosolik et al. [12] prepared. We present four calculated images of a FLL in dimensionless space corresponding to applied filed at 1,2,3 and 5 Tesla in Fig. 4, respectively. It shows a hexagonal-to-square symmetry variation of the FLL as applied field increases. These images are nearly the same as what observed in Sosolik et al.’s experiment [12]. Quantitative results of the four images are listed in Table 2, in which physical spacing 2Sx are calculated. The calculated data are very close to the ones measured from the images presented in Ref. [12].

y

6

-

A

B

x sy 2k

γ sx

ψ−3 ψ−2 ψ−1 ψ0 ψ1 ψ2 Figure 5. Vortex lattice corresponds to the choice of initial phase, αi = 0. All the pair amplitudes are denoted by solid lines. In an earlier experiment, Yethiraj et al. [13] observed a hexagonal-to-square symmetry

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variation of a FLL in V3 Si of a cylindrical geometry. They found that, the hexagonal-tosquare symmetry variation of a FLL occurs in a very narrow range of applied field, around 1 T. A similar measurement on a very clean crystal of V3 Si is performed recently by Yethiraj et al. [14], the result also supports a rapid hexagonal-to-square symmetry variation. To our point of view, it seems that, while the cleanness of a V3 Si specimen does play a role in the symmetry variation of a FLL, the effect of different geometry of a specimen may not be excluded. In addition to the vortex lattice determined by the pair wave functions in Fig. 3, another class of possible vortex structures may be formed by choosing {αj = 0}, as shown in Fig. 5. The conservation of magnetic flux in the cell is expressed as Sx Sy Ha = φo → sx =

s

2π . tan γ

(77)

The positions of zeros along y direction is given by cos[(2n + 1)2ky] = −1 , n = 0, 1, 2, · · · .

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3.3.

(78)

Thermodynamical Analysis

In last subsection, we found that it is possible to form FLLs with different symmetry in the central region of a finite sized bulk superconductor. The final observed symmetry of a FLL may be resultant from a balance of multiple controlling factors. Now, we examine the thermodynamical constraint on a vortex lattice. In calculating a free energy, pair wave functions in R region may be approximated by 1 2 f (x − kj ) ≈ e− 2 (x−kj ) , which simplifies the free energy calculation nearly to the same as Abrikosov [2] and Kleiner et al. [29] did earlier. Using η = 1 and bz = |F|2 , we obtain 2˜ κ22 = βB =

Y (q12 )φ0 h|F|4 i Y (q12 )φ0 2 → 2κ = , 2 X ′ 2πη ZhF|bsz |Fi X ′ 2πZ h|F|4 i h|F|4 i → β = . A hbz ih|F|2 i h|F|2 i2

(79) (80)

The Gibbs free energy is given in its dirty limit form that Gm (Ha , T ) − Gn (Ha , T ) = −

1 (Hu − Ha )2 . 8π [2κ22 − 1]βA

(81)

We calculate βB (βB ≈ βA ) for both initial phase choices as shown in Fig. 3 and Fig. 5, respectively, and the results are shown in Fig. 6. The solid line in Fig. 6 reproduces Abrikosov’s result [2] at γ = 45o . The dash line reproduces Kleiner et al.’s result [29], in which a hexagonal FLL at γ = 60o is the thermodynamically favored structure. Nevertheless, the minimum of βB ⋄ at γ = 60o is quite shallow, which differs from βB ⋄ at γ = 45o about 2%. Besides, there can be a continuous variation of βB ⋄ from γ = 60o to γ = 45o , which is the possible reason that the repulsion between flux lines is able to change the symmetry of a FLL continuously as observed in V3 Si [12].

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Table 3. Some important values of βB⋄ and βB2 . initial phase αi = 0 αi = 0, π2 αi = 0, π2

γ 45o 45o 60o

sx 2.51 1.77 1.35

βB 1.18034060(βB2 ) 1.18474904(βB⋄ ) 1.15958750(βB⋄ )

We note that both βB⋄ and βB2 in Fig. 6 are symmetric about γ = 45o , which reflects a symmetry of the rectangular geometry (see Fig. 1) that a 90o rotation of xoy plane along z-axis does not change the eigenvalue problem. Four particular values of βB⋄ corresponding to symmetry variation in Sosolik et al’s experiment, γ = 60o , 58.3o , 51.2o and 45o , are marked on the dash line in Fig. 6. The data at γ = 30o for βB⋄ marks a symmetric minimum with respect to γ = 60o . We find that βB⋄ and βB2 are almost the same at γ = 45o , however, in a rectangular geometry, the two square lattices are distinguishable. The square lattice having its sides parallel to the sides of the rectangular geometry corresponds to βB2 , while the square lattice has its sides tilted a 45o relative to the sides of the rectangular geometry corresponds to βB⋄ . A few special numerical values of βB⋄ and βB2 are listed in Table 3. 2 1.9 1.8 1.7

βB2

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1.6 1.5 1.4

βB⋄

1.3 1.2 1.1 0

10

20

30

40

50

60

70

80

90

γ (deg.) Figure 6. βB⋄ and βB2 are calculated according to the vortex lattices displayed in Fig. 3 and Fig. 5, respectively.

3.4.

Effect of Artificial Arrayed Pining Centers

The new vortex theory, together with the repulsion force between flux lines, is able to describe symmetry variation of a FLL in a finite sized uniform geometry. If a superconductor contains defects, the vortex theory is in general not applicable, nevertheless, in certain Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

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circumstances, the theory can work together with defects. Harada et al. [15] finds that the regular arrays of artificial defects in Nb thin film is able to pin a vortex lattices and participate in determining the symmetry of a vortex lattice. We describe how Harada et al.’s observation can be explained within the current theoretical framework. On the surface of a Nb film, Harada et al. deposited a square lattice of defects (DL). ˚ A defect can trap The spacing between the neighboring defects is d = 0.83µm = 8300A. a flux line, when defects and flux lines establish a one-to-one matching, the applied field is at H1 = φo /d2 = 30G. This matching between a flux line lattice and a defect lattice is illustrated in Fig. 7(a). The symbol, 2, in Fig. 7 denotes defects, the symbol, ×, in Fig. 7 denotes flux lines. In terms of our theory, we draw an array of pair amplitudes at the bottom of Fig. 7(a), the solid lines indicates the pair amplitudes are in phase, {αi = 0}. The phase coherence among the pair amplitudes generates a square lattice of holes (ZL), which matches the defects lattice (DL) and the flux line lattice (FLL). A white circle denotes a hole of ZL, however, it will only be drawn when it neither meets a defect nor a flux line. (a)

(b)

(c)

(d)

(e)

(f )

Figure 7. Lattice matching illustrations. (a) Perfect matching of ZL, FLL and DL at H = H1 . All the pair amplitudes in phase, αi = 0. (b) ZL matches FLL at H = 12 H1 , both are sublattices of DL. Alternative phase of pair amplitudes denoted by solid and dash lines, respectively. (c) ZL matches FLL H = 14 H1 , both are sublattices of DL. (d) At H = 32 H1 , DL fills half of ZL while FLL fills three fourth of ZL. In an experiment, if one measures both flux lines and order parameters, the holes represented by the white circles should be found. (e) ZL matches FLL at H = 2H1 , DL is a sublattice of both ZL and FLL. (f) ZL matches FLL at H = 4H1 , DL is a sublattice of both ZL and FLL. As the applied field reduces to 12 H1 , the number of flux lines reduces fifty percent, the flux lattice is illustrated in Fig. 7(b), where the pair amplitudes at the bottom takes

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alternatively the initial phase, αi = 0 or π2 , which is denoted by solid and dash lines, respectively. In this case, the ZL matches the FLL, they both become sublattices of DL. Further reducing the applied field to 14 H1 , the matching scenario among the three lattices is illustrated in Fig. 7(c). We note that, while the defects spacing, d, is not changing, the variation of γ is compensated by variation of applied field as specified in Eq. (76). In the case of increasing applied field to 2H1 , the number of flux lines is twice as much as the number of defects. In order to accommodate the increased flux lines within the restrictive area, the number of pair amplitudes is doubled as shown in Fig. 7(e), again the solid and dash line denote initial phases of 0 and π2 , respectively. This is a case that the ZL matches the FLL and the DL becomes a sublattice of ZL or FLL.

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It is now interesting to investigate the case that the applied field is 1.5H1 , which is illustrated in Fig. 7(d). In our vortex theory, increasing a single pair amplitude in the Rregion brings about an array of holes along y direction, while a flux line may be added or removed as a single object. Therefore, in principle, number mismatch between ZL and FLL exists. However, in a uniform superconductor, a small fraction of number mismatch between ZL and FLL may be neglected, while a large portion of number mismatch between ZL and FLL may not survive since the symmetry of a FLL and the distance in neighboring flux lines may be spontaneously adjusted to balance out the large number mismatch. In Harada et al.’s experiment, the existence of the DL pins a part or all of flux lines, which limits the possible ways of adjusting symmetry and spacing of a FLL. In fact, the DL serves as a selection rule that, amount all the possible configurations of a FLL can take in Rregion, only a FLL which includes the DL as a sublattice or becomes a sublattice of the DL exists. When comparing Fig. 7(d) with (a), one finds that, in order to accommodate 50% more flux lines, the number of pair amplitudes is doubled so the holes are increased 100%, therefore, there are unfilled holes in the area. The matching of ZL, FLL and DL at H = 4H1 is shown in Fig. 7(f), which is in similar to the case illustrated in Fig. 7(f). In fact, there are two more scenarios reported for H = 2.5H1 and H = 3H1 in Harada et al.’s experiment. Both scenarios correspond to partial filling of flux lines on a ZL shown in Fig. 7(f).

4.

Vortex State in a Superconducting Film

In this section, a film geometry in a parallel applied field is under investigation, the theoretical analysis will show that, as the size of a pair amplitude is comparable with the thickness of a superconducting film , the eigenenergy associated with the shape of the pair amplitude is affected. The eigenenergy serves as a second energy criterion in determining a vortex state in the film geometry besides the minimum free energy requirement. The film geometry in a parallel applied field is illustrated in Fig. 8.

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234

B.J. Yuan z

Ha 6

6

-

−dr

dr x

0

Figure 8. A film in a parallel applied field. The thickness of the film is 2dr .

4.1.

Eigen Problem

The eigenvalue problem for a film in parallel applied field is the same as the one for a bulk superconductor as studied in last section. We specify the vector potential by A = Ha (0, x, 0) ,

(82)

the eigen problem in a dimensionless coordinate space is given by

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"

2 ∂2 ∂ ∂2 −x Ψ(x) = ηΨ(x) , − 2− 2+ i ∂x ∂z ∂y ∂ Ψ(x)|x=±d = 0 , ∂x     ¯hD Tc 2πHu η T χ + ln = 0, 2πkB Tc T φ0 Tc



 #

(83) (84) (85)

where d = dr /ξH . This eigen problem is in the same form as that of a bulk finite superconductor studied in last section, what we will deal with in this film case is that the eigenvalue, η, is far away from the special case, η ≈ 1. Assuming the order parameter is in a form that Ψ(x) = e−iky f (x) cos pz = ψ(x, y) cos pz ,

(86)

the eigen problem reduces to find the unknown function f , which satisfies the Weber equation, −

d2 f (x) + (x − k)2 f (x) = ηf (x) , dx2 df = 0. dx x=±d

(87) (88)

In solving this eigenvalue problem, one thing to note is that the thickness, d, is a ratio, d = dr /ξH . That means that a small d may be realized by either a very thin film in a modulate applied field or a not-so-thin film in a weak applied field. Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

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c

η

1

c

0.75

5

0.5

cc 4

0.25

−dr

0

c

3

c

0.2

h = d2

0.4

k

2

0.6 0.8 1.0

dr

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Figure 9. A surface plot of dimensionless eigenenergy spectrum, η(k, h), or an effective potential. k denotes a position of the geometric center of a pair amplitude within the film geometry. At the minimum of η(k), k ∗ is called a nucleation center, i.e., a pair amplitude with minimum eigenenergy centered at this position. Each circle denotes a “particle” associated with a pair amplitude, which is a alternative view of the pair amplitude. In Fig. 9, we assume a fixed film thickness, 2dr , and the varying d is caused by varying applied field, Ha . The numerical results of the eigenvalue spectrum is displayed as a surface 2 , to denote the plot, η(k, d2 ). We introduce a reduced applied field, h = d2 = d2r /ξH variation of applied field. For each given h ∼ Ha , there exists a line of η(k), k denotes a position of the geometric center of a pair amplitude in the film geometry. At the minimum of the eigenvalue, η(k), k ∗ is called a nucleation center. A pair amplitude nucleating at k ∗ may be mapped to a “particle”, which is denoted by a circle (a small ball) on the surface plot. It is interesting to see that the varying pair amplitude in a superconducting film with a varying applied field can be described by a classical model of a “particle” moving in an effective potential, η(k). A charged particle’s motion in a given applied field is constrained by a parabolic potential in Eq. (87) and a confinement condition of Eq. (88). Combining both effects, the “particle” is moving in a effective potential, η(k, h). At low applied field, the shape of an eigenenergy spectrum, η(k), is similar to a parabolic well, a particle stays at the bottom of the well. As applied field is increased slightly, the shape of the eigenenergy well is changed slightly and the bottom of the well is raised, however, the shape of the well remains parabolic-like and the particle stays at the center with a higher eigenenergy. At h = hc ≈ 1.63, two minima is gradually developed at the bottom of the well and two particles occupy the minima respectively. As h further increases, the two minima of the well are separated by a eigenenergy barrier and approaches two surfaces, respectively. If h keeps increase, the eigenenergy barrier in the center of the film will extend and becomes a flat region with η ≈ 1, which is the bulk superconductor

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case we studied in last section. Since the nucleation center, k ∗ , determines the position of a pair amplitude and the eigenenergy, η(k), determines the shape of a pair amplitude, they are both important for understanding a vortex state, which is derived from pair amplitudes. The Werthamer relation, Eq. (85), maps the minimum eigenvalue to a upper critical field, Hu (T ), which serves as an independent experimental examination for theoretical results in addition to the images of vortex lattice. The minimum eigenenergy obtained from this eigenenergy spectrum is drawn in Fig. 10(a) as a function of h and the upper critical field Hu (T ) calculated from the minimum eigenvalue is drawn in Fig. 10(b). In fact, this part of the calculation was performed earlier by Saint-James and de Gennes [16] using linearized GL theory [1] and the theoretical results were successfully verified in experiments carried out by Burger et al. and Hart et al. [18–20]. Not only the minimum eigenenergy is mapped to upper critical field, but it is also intuitive to see from Fig. 9 that the minimum or minima of the eigenenergy spectrum provide a “pinning” mechanism, which pins pair amplitudes at nucleation centers. In this film geometry, the inhomogeneity which affects superconductivity of a geometry, is caused by the reduced thickness of the film. When the thickness is small or comparable with the size of a pair amplitude, the film squeezes the pair amplitude and force it to stay in the center of the film. As the thickness increases, the two physical surfaces moving away from each other, near each single surface, a minimum of eigenenergy is developed, which pins a pair amplitude and forms a surface superconducting state. In next subsection, we explore vortex states in the film geometry based on those pined pair amplitudes. 0.8

η

1.8 1.6

(a)

0.7

(b)

Hu /Hc2

1.4

0.6

1.2 0.5

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1 0.4 0.8 0.3 0.6 0.2

0.4

0.1

0.2

0 0

0.5

1

1.5

2

2.5

3

3.5

h

4

4.5

5

5.5

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

Figure 10. Solid line in (a) describes the variation of the minimum eigenvalue, η, with reduced field, h. As h increases, η approaches a constant, 0.59, which causes a Hc3 of Hu surface superconductivity. Solid line in (b) denotes the reduced upper critical field, H as c2 T a function of reduced temperature, t = Tc . Near the upper critical field, a few dots and a dash line marks the value of H and t, at which vortex states are calculated.

4.2.

Vortex State in a Film

There are only two types of pair amplitudes exist in a film under parallel field, a single pair amplitude resides at the center of a film (when h < hc ) or two degenerate pair ampliSuperconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

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237

tudes reside at two minima of the eigenenergy spectrum (when h > hc ), respectively. The general order parameter may be expressed as follows h

F(x, y) = eik

∗y

fη∗ (k∗ ) (x) + e−ik

∗y

i

fη∗ (−k∗ ) (x) ,

(89)

where k ∗ denotes the nucleation center and η ∗ (k ∗ ) denotes minimum eigenenergy. For h < hc , k ∗ = 0, there is a single pair amplitude; for h > hc , k ∗ 6= 0, there are two degenerate pair amplitudes. Applying the identity, Eq. (21), we define three functions Wp (x) = Wm (x) = Wc (x) =

i 1h ′ (x − k ∗ )2 fη2∗ (k∗ ) (x) − η ∗ fη2∗ (k∗ ) (x) − fη2∗ (k∗ ) (x) , 2 i 1h ′ (x + k ∗ )2 fη2∗ (−k∗ ) (x) − η ∗ fη2∗ (−k∗ ) (x) − fη2∗ (−k∗ ) (x) , 2 i −1 h ′ ′ ∗ (−k ∗ ) (x) − fη ∗ (k ∗ ) (x) f ∗ f (x) f (x) , ∗ ∗ ∗ η η (−k ) 2k ∗ η (k )

(90) (91) (92)

which satisfy the following relations d Wp (x) = (x − k ∗ ) fη2∗ (k∗ ) (x) , dx d Wm (x) = (x + k ∗ ) fη2∗ (−k∗ ) (x) , dx d Wc (x) = 2 x fη∗ (k∗ ) (x) fη∗ (−k∗ ) (x) , dx

(93) (94) (95)

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respectively. The current density may be written in components form c 2 ∂ Z(q12 ) [cos(2k ∗ y) Wc (x)] , 4π ξH ∂y c 2 ∂ = − Z(q12 ) [Wp (x) + Wm (x) + cos(2k ∗ y) Wc (x)] , 4π ξH ∂x

Jx =

(96)

Jy

(97)

while the second GL equation, Eq. (2), may be written as ∂ ∂ ˆy −e hz = ∂y ∂x   ∂ ∂ ˆy ˆx 2Z(q12 ) e −e [Wp (x) + Wm (x) + cos(2k ∗ y) Wc (x)] , ∂y ∂x 

ˆx e



(98)

which gives rise to a solution of magnetic field hz (x, y) = 2Z(q12 ) [Wp (x) + Wm (x) + Wc (x) cos(2k ∗ y)] + C .

(99)

The boundary condition of the magnetic field, Eq. (4), requires hz (d, y) = Ha ,

(100)

which determines the constant, C, and the vortex solution is given by hz (x, y) = Ha + hsz (x, y) = Ha − Z(q12 )bz (x, y) , Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

(101)

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B.J. Yuan

where bz (x, y) = 2 {[Wp (d) − Wp (x)] + [Wm (d) − Wm (x)] − Wc (x) cos(2k ∗ y)} .

(102)

Using these solutions, order parameter and internal magnetic field corresponding to a few characteristic cases marked in Fig. 10(b) can be plotted.

|F(x)|2 |F(0)|2

−bz

1

0

0.9

-0.1

0.8 -0.2 0.7

h

h (a)

(b)

Figure 11. For h < hc = 1.63, the module squared order parameter, (a), the spatial variation of the magnetic field, −bz , is plotted in (b).

|F (x)|2 , |F (0)|2

is plotted in

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Fig. 11(a) shows that, for h < hc = 1.63, the order parameter always nucleates in the center of the film, its spatial shape varies from nearly a constant to a bell-shape as h increases. Fig. 11(b) shows the spatial variation of the magnetic field for h < hc = 1.63, however, since the normalization constant of a pair amplitude is not determined, this plot shows a qualitative result about the variation of the magnetic field.

−bz

|F (x)|2

0 1.5

-0.25 0.25

1

-0.5

0.5

-0.75 -1

0

y

y (a)

(b)

Figure 12. At h = hc ≈ 1.63, a laminar vortex state is formed. For h = hc ≈ 1.63, the eigenenergy spectrum develops two minima close to each other (see Fig. 9), the coherence of these two pair amplitudes generates a laminar structure for both module squared order parameter and magnetic field, the results are shown in Fig. 12(a) and (b), respectively.

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Further increasing the reduced applied filed, h, the two pair amplitudes near two surfaces are well separated, the coherence between the two pair amplitudes forms a single array of flux lines in the center of the film geometry. The order parameter and magnetic field are plotted in Fig. 13(a) and (b), respectively.

−bz

|F (x)|2

0 0.6 -0.05 0.05 0.4

-0.1 -0.15

0.2

-0.2

0

y (a)

(b)

y

Figure 13. At h = 5, two well separated pair amplitudes form a single array of vortex lattice through phase coherence.

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4.3.

Thermodynamical Properties of the Vortex State in a Film

We have obtained the vortex states in a superconducting film based on the solutions of pair amplitudes selected by minimum eigenenergy requirement, now we calculate the minimum free energy corresponding to these vortex states. The mathematical analysis performed in section 2.2. enables us to calculate two important factors in a Gibbs free energy, that is βB and κ ˜2,

βB = 2˜ κ22 =

hF|bsz |Fih|F|2 i hF|bsz |Fi = β , A hbsz ih|F|2 i hbsz ih|F|4 i

Y (q12 )φo h|F|4 i h|F|4 i 2 = 2κ = 2κ22 T 2 . 2 η ∗ hF|bsz |Fi Z(q12 )X ′ (q12 )2π η ∗ hF|bsz |Fi

(103)

(104)

For an infinite superconductor, βA is a critical factor which uniquely determines the symmetry of a vortex lattice. In a finite sized bulk superconductor, βB determines different class of symmetry through choices of initial phase, {αi }. In Fig.6, βB2 is larger than βB⋄ except for γ = 45o , which means that {αi = 0, π2 } is a thermodynamically favored choice than {αi = 0}. In the film geometry, the eigenvalue spectrum possesses a non-trivial spatial form, the minimum eigenenergy provides a pinning mechanism of pair amplitudes, which determines a vortex structure. The weight of βB in determining a vortex structure is significantly reduced. On the other hand, the inhomogeneity of a superconductor brings about a geometric enhancement factor, T 2 , to the GL parameter, which affects categorizing a superconductor as type I or II.

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Based on the vortex state obtained in Eq. (101), we calculate both βB and T 2 in a range of h ≤ 5. Both results are plotted in Fig. 14. 1.6

1000

1.5

(a)

1.4

(b)

βB

T2

100

1.3 1.2 1.1

10

1

βA

0.9

1

0.8 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

0

0.5

1

1.5

h

2

2.5

3

3.5

4

4.5

5

5.5

h

Figure 14. Plots of geometric factor, βA and βB , in (a) and plot of squared enhancement factor, T 2 , in logarithmic scale in (b). 4

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

h|F |i (marked by 3) is calculated in terms of the definition In Fig. 14 (a), βA = h|F |2 i 2 of Abrikosov’s geometric factor, it is plotted as a comparison with βB in order to show the difference caused by inhomogeneity of a film. βB (marked by +) exhibits a jump at h = hc ≈ 1.63, which corresponds to the appearance of laminar structure in the film as shown in Fig. 12. For h > hc , βB undergoes a non-monotonic variation. It decreases as the two minima separate from each other and reaches a minimum at h ≈ 3.0. βB then increases as the overlap between two surface solutions gradually disappear and the increasing βB indicates surface superconductivity. The enhancement factor, T ,

T =

h|F|4 i ∗ η hF|bsz |Fi

!1 2

,

(105)

also reflecting the inhomogeneity of a film geometry, is plotted in Fig. 14(b). The numerical result shows that the squared enhancement factor is greater than one for any magnitude of h, therefore, in the expression of the Gibbs free energy, it always enhances Maki’s GL parameter, κ2 . Particularly towards the thin film case(small h), T 2 increases rapidly, which indicates that any type I superconductor may become type II in a film geometry under parallel applied field.

5.

Vortex State in a Superlattice

In either Section 3. or Section 4., a superconductor under study is uniform in material property, the inhomogeneity introduced into the theory is mainly by limiting the geometry size. In this section, we consider a two components superlattice denoted by NS [8], which bears a spatially modulated diffusion constant, D(x). The structure of a NS superlattice is illustrated in Fig. 15.

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241

H

z



dN

-

dS

6

6

-

x

0 S

N

S

N

S

N

S

Figure 15. A superlattice consists of two components which marked by N and S, respectively. The N and S possess a same transition temperature, TcS = TcN , same density of states at Fermi surface NS (0) = NS (0) and same BCS coupling constant, VS = VN . However, they differ by diffusion constants, DS 6= DN . In this work, we assume the thickness of N and S layer are the same, d = dN = dS . It is assumed for the two components that the densities of states at Fermi surface, the BCS [32] coupling constants and the transition temperatures are all the same, however, they differ in diffusion constants, DS 6= DN . This type of superlattice have been materialized in a number of experiments, e.g. Nb/NbZr and Nb/NbTi [21–23], in which the diffusion constants are determined by the properties of a bulk superconductor of each component, respectively.

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5.1.

Eigen Problem in a Superlattice

Within the framework of the vortex theory of inhomogeneous superconductor, the eigenvalue problem may be expressed by the following equations and continuity conditions. Taking the London gauge, A = Ha (0, x, 0), and introducing intrinsic length, ξH , we assume a pair amplitude at position k is in a form F(x, y, z) = cos(pz) eiky f (x) ,

(p = 0) .

The differential equation governing the unknown function, f (x), in the superlattice is given by # " d2 2 (106) D(x) − 2 + (x − k) f (x) = η(k) f (x) , x ∈ S , N dx where η is no longer a dimensionless eigenvalue since the modulate function of diffusion constant appears in the equation. The continuity conditions at the interfaces of a NS superlattice are given by de Gennes [5] that Ds

d d f (x ∈ S → xi ) = DN f (x ∈ N → xi ) , dx dx f (x ∈ S → xi ) = f (x ∈ N → xi ) ,

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where xi denotes ith interface of the superlattice. Requiring a normalizable order parameter in an infinite superlattice, we have lim f (x) → 0 .

(107)

x→±∞

Again, the theoretical calculation includes two parts, searching the minimum eigenvalue to obtain the upper critical field and finding the vortex states in the superlattice. Solving the eigenvalue problem numerically, we obtain a periodical energy potential which combines the effect of a parabolic potential and a spatially modulated geometry formed by alternative material parameter, D(x). The plot for the eigenenergy potential in

b 0.75

η(k, d)

2.4 0.5 1.8

d

b

0

1.2 1

k

2 0.6 3

4

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Figure 16. Eigenvalue, η, is plotted as a function of nucleation center, k, and dimensionless layer thickness, d. The ratio of Ds and DN is taken ρ = DS /DN = 0.2586 [21].

Fig. 16 acrosses four NS cells. In a point of view of classical mechanics, there is a particle rests at the minimum of the energy potential and tracing the minimum as the applied field increases. At low applied field, the particle stays at the center of N-layer, at high field, the particle stays at the center of S-layer. By tracing the eigenenergy at the center of N and S layer, respectively, we find an eigenenergy crossover at d ≈ 2.01, which causes a discontinuous slope in upper critical field [24]. Using Werthamer relation, Eq.(85), we calculate the upper critical fields for the superlattice. The eigenvalues and upper critical fields are plotted in Fig.17 (a) and (b), respectively. The numerical calculation performed in this work is aimed at Kuwasawa et al.’s experiment on a superlattice of Nb/NbZr [21] characterized by ρ = DS /DN = 0.2586 and ˚ The entire upper critical field curve, Hc2 (t), in Fig.17(b) agrees with dN = dS = 250A. Kuwasawa et al.’s experimental measurement. It is then interesting to find out what makes a nucleation center shift in eigenenergy spectrum, or equivalently, a discontinuous slope in upper critical field to occur. The plots of the eigen functions corresponding to the eigenvalue in Fig. 17(a) answers this question.

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Vortex Theory of Inhomogeneous Superconductors 0.75

η

243

3.5

(a)

0.7 0.65

ηS

0.6

3

N center S center

2.5

0.55

2

0.5

1.5

0.45

ηN

0.4

(b)

Hc2 (t)

N center S center

(t∗ ,H ∗ )

1 0.5

0.35 0.3 0

0.5

1

1.5

2

d

2.5

3

0 0.6

0.65

0.7

0.75

0.8

t

0.85

0.9

0.95

1

Figure 17. Eigenvalue in center of N(ηN ) and S(ηS ) are plotted in (a), at d ≈ 2.01, eigenvalue shows a crossover, which indicates a switch of minimum energy from N-center to S-center. The upper critical field calculated based on the eigenvalue in (a) is plotted in (b). The upturn at (t∗ , H ∗ ) indicates a T-T effect [24].

fN (x)

fS (x) 1

0.75 2.2

0.75

2.2

0.5 0.5 0.25

1.8

0

1.8

0.25 0

1.4 -1.25 0

(a)

d

1.0

1.4

(b)

-1.25 0

1.25

1.25 2.5

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d

1.0

2.5

Figure 18. Eigenfunctions nucleating in center of N-layer denoted by fN (x) in (a). Eigenfunctions nucleating in center of S-layer denoted by fS (x) in (b). Through the eigen equation, an eigenvalue relates to every point of the eigenfunction, we can simply take the central point of an eigenfunction to determine the eigenvalue, ηN ηS

′′ (0) fN , fN (0) f ′′ (0) = DS S , fS (0)

= DN

x∈N, x∈S,

(108) (109)

where ηN and ηS denote the eigenvalue in center of N-layer and S-layer, respectively. The expressions above show that there are two factors affect the magnitude of an eigenvalue, the ′′ curvature at the central point of the eigenfunction, f (0), and the value of the eigenfunction at the central point, f (0). The smaller the curvature, the smaller the eigenvalue, and the larger the function value, the smaller the eigenvalue.

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Comparing the shape of the eigenfunctions plotted in Fig. 18, we see that the eigenfunctions nucleating at center of N-layer possesses a smaller curvature, while the eigenfunctions nucleating at center of S-layer possesses a larger function value. A function with smaller curvature associating with a smaller energy is a well accepted assumption as we do most of the time when we write a trial function in variational method, nevertheless, now we find that a wave function highly condensed within a layer may also associate with a smaller energy. The eigenenergy crossover in the calculation is precisely the result of competition between a smaller curvature and a larger function value. Therefore, the upturn shown in the upper critical field indicates that, for t > t∗ , the pair amplitude centered at center of N-layer extends across several layers and, for t < t∗ , the pair amplitude centered at center of S-layer highly confined within just that S-layer and has little tail in neighboring layer.

5.2.

Vortex States in a Superlattice

Either the nucleation occurs at the center of N-layer or S-layer, we can construct a general order parameter by linear superposition of those degenerated pair amplitudes F(x, y) =

X m

Cm f (x − km ) eikm y ,

(110)

where f (x−km ) denotes mth pair amplitude, Cm denotes a complex coefficient, km = m∧ while ∧ denotes dimensionless length of a NS cell, ∧ → ∧/ξH = (dS + dN )/ξH . The periodicity of F(x, y) in x direction may be assumed by

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Cm = Cm+M .

(111)

In following calculations, we assume M = 2, hence, there are only two independent coefficients, C0 and C1 . Further, assuming the two coefficients differ by a constant phase, α, |C1 | = e2iα |C0 | , (112) the order parameter is thus expressed as F(x, y) = C0 eiα

X m

fm exp[i(m ∧ y − (−)m α] ,

(113)

where fm = f (x − m∧). As mentioned earlier, the calculation of thermodynamical quantities of a vortex state is independent of the coefficient of the order parameter, so we assume C0 eiα = 1. The current density is given by J=

c 2 ¯ i 2π ZD(x) [F∇F ∗ − F ∗ ∇F] − ξH A|F|2 4π ξH 2 φo 



,

(114)

where, in order to maintain continuity of the current density at the interfaces, we extract the diffusion constant out of the Z function in Eq. (44), and Z¯ is defined as Eg 1 3eN (0) g Z¯ = 2 2πkB Tc t mvF ckB T 



,

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(115)

Vortex Theory of Inhomogeneous Superconductors

245

where N (0) is the density of state at Fermi surface, vF denotes Fermi velocity, g is define by Eq. (47). Defining Wn and Wmn as follows 1 [(x − n∧)2 fn2 − ηfn2 − fn′2 ] , 2 ′ = fm fn′ − fn fm ,

Wn = Wmn

(116) (117)

we derive two identities from Weber euqation, Eq. (106), d Wn = (x − n∧) fn2 , dx d Wmn = [2x − (n + m)∧](m − n) ∧ fm fn . dx

(118) (119)

Using these two identities, the current density may be written into a form that, Jy =

c 2Z¯ ∂ X D(x) Wmn cos[Λmn y − Smn α] , 4π ξH ∂y k 6=k 2Λmn n

m

   X −c 2Z¯ ∂ X D(x) Jx = Wn D(x) + cos[Λmn y − Smn α] , Wmn  4π ξH ∂x  2Λmn n

(120)

(121)

n6=m

where Λmn and Smn are defined by

Λmn = (m − n)∧ , m

(122) n

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Smn = (−1) − (−1) ,

(123)

respectively. Substituting the current density into second GL equation, Eq. (2), we obtain 

ˆx e

∂ ∂ ∂ ∂ ˆy ˆy ˆx −e hz = 2Z¯ e −e ∂y ∂x ∂y ∂x 





  X X D(x) ×  Wn D(x) + Wmn cos[Λmn y − Smn α] . n

n6=m

2Λmn

(124)

The analytical solution of the magnetic field is in a form ¯ z, hz = Ha − Zb where

  X X D(x) Wmn bz = −2  Wn D(x) + cos[Λmn y − Smn α] . n

n6=m

2Λmn

(125)

(126)

The factors related in free energy calculation include Maki’s factor M=

s

f1 (z)g 2 (0) , f1 (0)g 2 (z)

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(127)

246

B.J. Yuan

the enhancement factor, T =

s

1 h|F|4 i , η hF|bz |Fi

(128)

hF|bz |Fi , h|F|2 ihbz i

(129)

and the geometric factor, βB =

where functions f1 (z) and g(z) are given by Eq. (46) and Eq. (47), respectively. The Gibbs free energy of a vortex state is given by Gm (Ha , T ) − Gn (Ha , T ) = −

1 1 (Hu − Ha )2 , , 8π Υ

(130)

where Υ = [2˜ κ22 − 1]βB ,

(131)

κ ˜ 22 = κ2N S M2 T 2 .

(132)

and κN S is the effective GL parameter for a NS type superlattice, the value of which needs to be determined by experiment data. The minimum free energy is determined by the minimum of Υ. In order to calculate Υ, we evaluate two important factors in nearest neighbor approximation, h|F|4 i = FA + FB cos(4α) ,

hF|bz |Fi = BA + BB cos(4α) .

(133) (134)

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Substituting these two expressions into Υ leads to Υ(α) =

1 [(2κ22 FA − BA ) + (2κ22 FB − BB ) cos(4α)] , hbz ih|F|i2

(135)

where FA , FB , BA and BB are given in appendix, respectively. Selection of α may be determined by the following condition ∂ Υ(α) = 0 , ∂α which leads to

(136)

π , (137) 4 which provides two choices of initial phases for an array of pair amplitudes the same as the choices in the case of a bulk superconductor, i.e., {αi = 0} or {2αi = 0, π2 }. Differing from a bulk homogeneous superconductor analyzed in Sec. 2., the pinning force provided by the minima of eigenvalue spectrum (effective potential) on the pair amplitudes does not allow a continuous variation of the physical spacing in x direction between the peaks of neighboring pair amplitudes, hence, for a given symmetry, it can only be realized under a definite magnitude of magnetic coherence length, ξH (or applied field). α=0

or

α=

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247

16

1.18

(a)

1.16

14

(b)

1.14 12

1.12

TN2 2

TN2 △

2 TS2

2 TS△

M2N

1.1

10

1.08

M2S

1.06

8

1.04 6

1.02 1 0

0.5

1

d

1.5

2

2.5

4 0.8

1

1.2

1.4

1.6

d

1.8

2

2.2

2.4

2.6

Figure 19. For M2 in (a), the solid line corresponds to the nucleation of pair amplitudes in N centers, while dash line for nucleation in S centers, respectively. M2 = min[M2N , M2S ]. T 2 in (b) is the enhancement factor caused by inhomogeneity of a superlattice. For curves of T 2 , TN2 2 is calculated with {αi = 0} and nucleation in N center, T 2 , TN2 2 is denoted by 2 is calculated with {α = 0} and nucleation in S center, T 2 is denoted by dash line, TS2 i S2 solid line, TN2 △ is calculated with {αi = 0, π2 } and nucleation in N center, TN2 △ is denoted 2 is calculated with {α = 0, π } and nucleation in S center, T 2 is denoted by +. by +, TS△ i S△ 2

For example, considering a square lattice for {αi = 0}, ξH may be obtained by applying Eq. (77), s ∧ 2π φo sx = → ξH = √ → H = 2 , for γ = 45o , (138) tan γ ∧ 2π

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where ∧ = dS + dN is the length of a NS cell. For {2αi = 0, π2 }, we have sx =

r

∧ 1 φo π → ξH = √ → H = , tan γ π 2 ∧2

for γ = 45o ,

(139)

Therefore, in a given superlattice, we can expect that, at what magnitude of applied field, a square lattice can be observed. In addition to the choices of the initial phases, M and T needs to be calculated in order to determine the unknown GL parameter, κN S . M2 and T 2 are calculated in a nearest neighbor approximation and the results are plotted in Fig. 19. It is seen that M2 , M2 = min[M2N , M2S ] in Fig. 19(a), is varying between 1 and 1.08. This value causes a minor enhancement to κN S . The magnitude of the enhancement factor, T 2 , is much larger than M2 , therefore, T 2 enhances κN S significantly. The large value of T 2 means that inhomogeneity of a superlattice favors a type II superconductor. From Eq. (130), we learn that the Gibbs free energy is determined by Υ, however, in addition to the factors closely related to the inhomogeneity of a superlattice, Υ contains a unknown GL parameter of the superlattice, κN S . On the other hand, we also learn that the discontinuous slope of the upper critical field is observed at (t∗ , Hu∗ ) in the experiemnt [21], which indicates an eigenenergy crossover. Therefore, by requiring a free energy crossover occurs at the same time as the eigenenergy crossover, the unknown GL parameter is determined, κN S = 0.218. The discrepancy that eigenenergy crossover occurs at d ≈ 2.01 and

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248

B.J. Yuan 2.4 2.2

0.8

(a)

2 1.8 1.6

0.6

β2S β2N β⊲S β⊲N

(b)

0.4 0.2 0 -0.2

1.4

-0.6

1 0.8 0.8

Υ2S Υ2N Υ⊲S Υ⊲N

-0.4

1.2

-0.8

1

1.2

1.4

1.6

1.8

d

2

2.2

2.4

2.6

-1 0.8

1

1.2

1.4

1.6

d

1.8

2

2.2

2.4

2.6

Figure 20. βB is plotted in (a) for four scenarios, i.e., the nucleation center in N and S layer and the choice of {αi = 0} and {2αi = 0, π2 }. Υ is also plotted in the same four scenarios. The unknown GL parameter of a NS superlattice is fitted by a Gibbs free energy crossover at d ≈ 2.1, around which an upturn of upper critical field corresponding to nucleation center shift has been observed in the experiment [21].

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that the free energy crossover occurs at d ≈ 2.1 may be caused partly by the numerical calculation. The results of βB and Υ are plotted in Fig. 20(a) and (b), respectively. Fig. 20(b) also shows that, for a weak applied field or a small reduced thickness, d < 1, Υ is negative so the Gibbs free energy seems to favor a normal state rather than a superconducting state. Based on our previous calculation about the semi-infinite superlattice [33,34], we know that a surface layer effect is inevitable at a small d, therefore, an infinite superlattice model may not appropriate for studying vortex state. Hence, we consider only the vortex states for d > 1.2.

5.3.

Vortex Lattice in a Superlattice

In this subsection, we calculate vortex lattices in a superlattice alongnear the upper critical field. Three points on the upper critical field (see 17(b)) are chosen, which correspond to d = 1.2 d = 1.8 and d = 2.4, respectively. In Fig. 21(a), the module squared order parameter, |F(x)|2 , is plotted across two NS cells with N layer in the center. The spatial variation part of the internal magnetic field, bz , is plotted in in Fig. 21(b) across one NS cell also with N layer in the center. Both |F(x)|2 and bz are calculated at d = 1.2 (t ≈ 0.92, H ≈ 0.91T) with initial phases {αi = 0}. The plot for bz -field does not show clear flux lines in S-layers, instead, it seems to be more close to a laminar structure with some of the flux pass through the interfaces between N and S layers. For initial phase of {2αi = 0, π2 }, |F(x)|2 and bz -field are plotted in Fig. 22(a) and (b), respectively. The symmetry of either |F(x)|2 or bz -field is changed as expected. The calculation of Υ shows that these two types of vortex structure corresponding to {αi = 0} and {2αi = 0, π2 } contribute nearly the same to the Gibbs free energy, therefore, there is not a preferential choice of a vortex structure in the current case.

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Vortex Theory of Inhomogeneous Superconductors

|F (x)|2

249

bz 0.2

0.4 0.1

0.2 0

0

y

y (a)

(b)

Figure 21. Plots for module squared order parameter and spatial variation part of the internal magnetic field at d = 1.2, which corresponds to the point at upper critical field (t ≈ 0.92, H ≈ 0.91T). The initial phase selection is {αi = 0}.

|F (x)|2

bz 0.15

0.4 0.1

0.2

0.05 0

0

y

y

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

(a)

(b)

Figure 22. Plots for module squared order parameter and spatial variation part of the internal magnetic field at d = 1.2, which corresponds to the point at upper critical field (t ≈ 0.92, H ≈ 0.91). The initial phase selection is {2αi = 0, π2 }. For d = 1.8 (t ≈ 0.82, H ≈ 1.7T ), the overlap between the neighboring pair amplitudes are very weak, the module squared order parameter is nearly condensed in N-layers for both choices of initial phase as shown in Fig. 23(a) and Fig. 24(a). bz -field plots in either Fig. 23(b) and Fig. 24(b) show a flux confinement in S-layers. It seems that the interfaces between NS layers are preferential position for flux passing through, however, the pair amplitudes at interfaces are not smooth, and the numerical calculation includes derivatives of pair amplitudes at interfaces, therefore, it is not certain that the flux condensed area near the interfaces has to exist. Fig. 25 plots both module squared order parameter and bz -field at d = 2.4 (Ha ≈ 3.04 T, t∗ ≈ 0.69). The order parameter condensed in S-layer and bz -field in N-layer. At the center of S-layer, it seems to form a flux layer in parallel with layers of the superlattice.

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B.J. Yuan

|F (x)|2

bz

0.4 0.1

0.3 0.2

0 0.1 0

y

y (a)

(b)

Figure 23. Plots for module squared order parameter and spatial variation part of the internal magnetic field at d = 1.8, which corresponds to the point at upper critical field (t ≈ 0.82, H ≈ 1.7T ). The initial phase selection is {αi = 0}.

|F (x)|2

bz

0.4 0.3

0.1

0.2 0

0.1

y

0

y

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

(a)

(b)

Figure 24. Plots for module squared order parameter and spatial variation part of the internal magnetic field at d = 1.8, which corresponds to the point at upper critical field (t ≈ 0.82, H ≈ 1.7T ). The initial phase selection is {2αi = 0, π2 }.

6.

Conclusion

Based on previous research works [7–9], we propose a vortex theory for inhomogeneous superconductors. Applying the identity of Weber equation, Eq. (21), we transform current density, J, into differential forms, Eqs. (24) and (25), and derive an analytical vortex solution, hz in Eq. (28), without being limited by boundary conditions imposed on a pair amplitude. We further derive the Gibbs free energy, Eq. (58), based on formal solutions of vortex structure, hz , and order parameter, F. The Gibbs free energy, Eq. (58), is in a similar form to Abrikosov’s theory [2], however, the GL parameter, κ ˜ 2 , and geometric factor, βB , are dependent on both order parameter and internal magnetic field. By selecting the geometry of a superconductor and specifying boundary conditions or continuity conditions, the inhomogeneity of a superconductor is included into the theory

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Vortex Theory of Inhomogeneous Superconductors |F|2

251

0.8 0.7 0.6 0.5 0.4

bz

0.3 0.2 0.1 0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

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Figure 25. Since the overlap between the neighboring pair amplitudes is almost negligible at d = 2.4 (Ha ≈ 3.04 T, t∗ ≈ 0.69), there is no vortex lattice formed. The module squared order parameter (solid line) and spatial variation part of the magnetic field (doted line) are plotted in this graph. and manifested in the eigenenergy spectrum, or the effective potential, of a pair amplitude. The order parameter, F, is in general a linear superposition of a degenerate set of pair amplitudes, {ψj }, which is selected from the eigenenergy spectrum based on minimum eigenenergy requirement. Therefore, the order parameter carries the effects of inhomogeneity of a superconductor. When the order parameter is plugged into the derived analytical solutions of internal magnetic field, hz , and the Gibbs free energy, Eq. (58), effects of inhomogeneity are passed to vortex structure, the generalized GL parameter and the geometric factor, βB . The most significant aspect of the new vortex theory is the joint role of two energy criteria, the eigenenergy for each individual pair amplitude and the free energy for a vortex state. A large amount of observed phenomena of vortex states can be explained by studying the competition and coordination between the two energy criteria. We apply the new vortex theory to three simple but characteristic geometries. In Sec. 3., we show an eigenenergy spectrum for a rectangular prism in Fig. 2. The S layer of the eigenenergy brings about two observable phenomena, the minimum of η ∗ = 0.59 causes a Hc3 upper critical field [16] and the width of the S layer causes a vortex free band [17], both are observed in experiments. The R region of the spectrum is characterized by a almost constant eigenvalue, η ≈ 1.0, which reproduces the case of an infinite superconductor, η = 1. The upper critical field Hc2 for a bulk superconductor is determined by η ≈ 1.0. A vortex state is derived from an order parameter, F, which consists of a degenerate set of pair amplitudes, {ψj }. The undetermined phase factors of the degenerate set, {ψj }, in a vortex state is then determined by minimum free energy requirement. Nevertheless, there are other factors, which are not included in the theory, may work together with the thermodynamical constraints to determine the symmetry of a flux line lattice. In the case of V3 Si, we introduce repulsion between flux lines to explain how the the thermodynamical state at a minimum of free energy may be slightly pushed, which causes a continuous symmetry variation of flux line lattice from hexagonal to square as applied field increases [12, 14]. In the case Harada et al.’s experiment [15], we show how the solution can work with a regular array of artificial defects, which pins flux lines, to

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create various flux line lattices the same as observed in a Nb film. In Sec. 4., we study vortex states of a film in a parallel field. It differs from the bulk superconductor in Sec. 3. in that the eigenenergy spectrum in a film does not have a flat Rregion(see Fig. 9). The eigenenergy spectrum, or effective potential, varies within the film at different positions. The minimum eigenenergy itself determines the upper critical field [16], while the nucleation center, at which the minimum eigenenergy is found, determines the position of a ground state pair amplitude. Since the pair amplitudes are the building blocks of a vortex lattice, the role of the eigenenergy spectrum in the film case is much more critical than the R-region in bulk case when η ≈ 1. We see from Fig. 9 that there are at most two degenerate pair amplitudes, the difference of constant phase factors between the two pair amplitudes can be treated as a selection of origin in y direction, therefore, the spatial form of a vortex lattice is completely determined by the information provided by the eigenenergy spectrum. We also conclude that the minima of the eigenenergy spectrum provides a pinning mechanism which nails a pair amplitude instead of a flux line. A classical point of view to this pinning mechanism may be expressed as that an effective potential traps a little ball at its minima, as illustrated in Fig. 9. The calculation of κ ˜ 2 indicates that, in a parallel applied field, thin film of a superconductor tends to be a type II superconducto due to the enhanced κ ˜2. For the superlattice in Sec. 5., we study the effects of another type of inhomogeneity on the vortex lattices. The inhomogeneity is caused by the different diffusion constants in different component layers of a multilayer specimen modelled as a superlattice. The inhomogeneity enters the theoretical model through de Gennes continuity condition [5]. The combined effects of an external magnetic field on an electron-pair and the spatially modulated diffusion constant generate a periodical effective potential (see Fig. 16), or eigenenergy spectrum, through de Gennes-Werthamer theory [5, 6]. For this particular superlattice, the theoretical results of minimum eigenenergy calculation have been verified by Kuwasawa et al.’s [21] experiment. The pinning force of a vortex lattice in this superlattice is provided by the minima of the effective potential and the pinning force is acting on a pair amplitude not a flux line. The minimum eigenenergy crossover, which indicates a nucleation center shift from center of a N layer to center of a S layer, manifests itself through Takahashi-Tachiki effect [24] of upper critical field, accordingly, there should be a vortex lattice shift at nearly the same temperature and applied field. We can classify the inhomogeneities, which the current theory handles, into three categories based on how they appear in the new vortex theory. In category I is the inhomogeneity caused by randomly distributed impurities in a superconductor. This type of impurity may appear in a poorly prepared superconductor or through a designed procedure of doping. The component, NbZr, in NB/NbZr superlattice is an example of designed doping [21]. The de Gennes-Werthamer(DW) theory [5,6], when it is derived from Gorkov theory [3], deals with electron scattering from randomly distributed impurity potentials, which results in a diffusion constant, D = 13 vF l [35]. The diffusion constant appears in the eigen equation of DW theory, it may be viewed as a replacement of h2 ¯ hD in the linearized GL equation, Eq. (5). In the dimensionless space, D appears 2m∗ with ¯ −2 h2 −2 ¯ only in the energy unit, ǫ = 2m hDξH , it does not affect the shape and the position ∗ ξH = ¯ of a pair amplitude, therefore, the randomly distributed impurities does not have direct impact on vortex lattices. The new vortex theory includes DW theory as a major component,

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Vortex Theory of Inhomogeneous Superconductors

253

therefore, it handles this type of impurity induced inhomogeneity. Within the DW theoretical framework, the diffusion constant, D, is measurable by the slope of upper critical field at transition temperature,

hD ¯ φ0 kB

=

4 π2

dHc2 −1 dT .

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In category II, the inhomogeneity is caused by boundary conditions at surfaces or continuity conditions at interfaces. The inhomogeneity coming with a boundary condition is associated with the fact that every piece of superconductor is in a definite geometry and surfaces of a geometry always exist. The inhomogeneity coming with the interfaces may be caused by a designed process, e.g. a synthesized superlattice, or a naturally layered material (some HTSC may fall in this category). The new vortex theory of inhomogeneous superconductor is mainly targeted on handling this type of inhomogeneity. Through the applications of the new vortex theory to several geometries, we find that this type of inhomogeneity affects vortex lattices the most. The inhomogeneity appearing in the theory through boundary conditions, which works together with eigen equation to determine the eigenenergy spectrum (the effective potential), the minimum eigenenergy then determines the shape and position of pair amplitudes, which limits the construction of a order parameter and the order parameter passes the effect of inhomogeneity to vortex solutions. Although we only calculate vortex structures explicitly in a few simple geometries, however, we did calculate more eigenenergy spectra for more complicated geometries. We calculated eigenenergy spectra for multilayer superconductors [33,34] with physical lengthes ranging from nano to meso scale, which includes both boundary conditions and interface continuity conditions. The theoretical results show that the effects of surfaces, interfaces, thickness of surface layers and thickness of each individual layer in a multilayer material, are all reflected in the eigenenergy spectra, or the effective potential. By tracing a virtual ball’s move along the varying minimum of the effective potential( see Fig. 9), we are able to imagine how the complex inhomogeneity affects a vortex structure under varying physical conditions. In the third category are the inhomogeneities caused by certain type of defects which is not included in the vortex theory. However, it works together with the vortex theory in determining a vortex lattice. The regular arrays of artificial point defects in NB film [15] prepared by Harada et al. falls into this category. As shown in Fig. 7, the square lattice of point defects serves as a selection rule, which selects a subset of vortex lattices from the vortex solutions derived for a bulk uniform superconductor. Only this subset of the vortex lattices can be realized in experiments. This type of inhomogeneity differs from that in category II is that, it provides a pinning force mainly acting on flux lines while the inhomogeneity in category II provides a pining force mainly nails the pair amplitudes.

Acknowledgement The author sincerely thanks Professor Zong-Chao Yan and Dr. YiXun Xue for their help on reference materials. The author thanks XueYan for her encouragement and discussion in completing this work. The author gratefully thanks Professor M. B. Brodsky for his help in publishing a research work, which leads to a authoring of this article. Communications with Professor Liantao Wang and Dr. Changhai Lu are appreciated.

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Appendix 1 ∧

h|F|i2xy = FA =

1 ∧

Z

∧ 2

−∧ 2

Z

∧ 2

1 FB = ∧

Z

−2 ∧

Z

−2 ∧

−∧ 2

2 dx [f−1 + f02 + f12 ]

Z

∧ 2

−∧ 2

−∧ 2

dx

−∧ 2

(2)

(3)

dx D(x)[W−1 + W0 + W1 ]

(4)

∧ 2

−∧ 2

 

D(x)





∧ 2

(1)

2 dx [4f−1 f02 f1 + f−1 f12 ]

1 X

fn2

n=−1

1 X

Wm

m=−1

D(x) 1 W0,−1 f0 f1 + W1,−1 f−1 f1 − W0,1 f1 f0 ∧ 4

+ −2 BB = ∧

∧ 2

4 2 2 dx [f−1 + f04 + f14 ] + [4f−1 f02 + 4f12 f02 + 3f−1 f12 ]

hbz ixy =

BA =

Z



1 dx D(x) −W0,1 f0 f−1 + W1,−1 f−1 f1 + W0,−1 f1 f0 4 

(5) 

(6)

References Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

[1] V. L. Ginzburg and L. D. Landau. Zh. Eksperim. i Teor. Fiz., 20:1064, 1950. [2] A. A. Abrikosov. Soviet Phys. JETP, 5:1174, 1957. [3] L. P. Gor’kov. Sov. Phys. JETP, 9:1364, 1959. [4] Kazumi Maki. Physics, 1:21, 1964. [5] P. G. De Gennes. Rev. Mod. Phys., 36:225, 1964. [6] N. R. Werthamer. Phys. Rev., 132:2440, 1963. [7] B. J. Yuan Z. Physik B 96 165, (1994). [8] B. J. Yuan Z. Physik B 98 457, (1995). [9] B. J. Yuan Physica C , 454:48, (2007). [10] U. Essmann and H. Trauble. Phys. Lett., 24A 526, (1967). [11] Chorng-Haur Sow, Ken Harada, Akira Yonomura, George Crabtree and David G. Grier Phys. Rev. Lett., 80 2693, (1998). Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

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[12] C. E. Sosolik, Joseph A. Stroscio, M. D. Stiles, E. W. Hudson, S. R. Blankenship, A. P. Fein, and R. J. Celotta Phys. Rev. , B:68, 1140503(2003). [13] M. Yethiraj, D .K .Christen, D. McK. Paul, P. Miranovic and J. R. Thompson Phys. Rev. Lett., 82:25, 5115(1999). [14] M. Yethiraj, D .K .Christen, A. A. Gapud, D. McK. Paul,S. J. Crowe,C. D. Dewhurst,R. Cubitt,L. Porcar and A. Gurevich Phys. Rev. , B:72, 060504(R)(2005). [15] K. Harada, O. Kamimura, H. Kasai, T. Matsuda, A. Tonomura and V. V. Moshchalkov Science, 274:1167, 1996. [16] D. Saint-James and P. G. de Gennes. Phys. Lett., 7:306, 1963. ˚ A.F.Olsen, H. Hauglin, T.H.Johansen, P.E.Goa, and D. Shantsev Physica C , 408[17] A. 410:537, (2004). [18] J. P. Burger, G. Deutscher, and E. Guyon et A. Martinet. Phys. Lett., 16:220, 1965. [19] J. P. Burger, G. Deutscher, and E. Guyon et A. Martinet. Phys. Rev., 137:A853, 1965. [20] H. R. Hart Jr. and P. S. Swartz. Phys. Lett., 10:40, 1964. [21] Y. Kuwasawa, U. Hayano, T. Tosaka, S. Nakano, and S.Matuda. Physica C, 165:173, 1990. [22] W. Maj and J. Aarts. Phys. Rev. B, 44:7745, 1991.

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[23] M. G. Karkut, V. Matiasevic, L. Antognazza, J. M. Triscone, N. Missert, M. R. Beasley, and Ø. Fischer. Phys. Rev. Lett., 60:1751, 1988. [24] S. Takahashi and M. Tachiki. Phys. Rev. B, 34:3162, 1986. [25] A. L. Fetter and P. C. Hohenberg. Superconductivity Vol. 2 Chapter 14, edited by R. D. Parks. Marcel Dekker, Inc, New York, (1969). [26] P. G. De Gennes. Physik Kondensierten Materie, 3:79, 1964. [27] E. Helfand and N. R. Werthamer. Phys. Rev. Letters, 13:686, 1964. [28] E. Helfand and N. R. Werthamer. Phys. Rev., 147:288, 1966. [29] W. H. Kleiner, L. M. Roth and S. H. Autler. Phys. Rev. 133A 1226, (1964). [30] H. F. Hess, R. B. Robinson, R. C. Dynes, J. M. Valles, Jr., and J. V. Waszczak. Phys. Rev. Lett., 62 214, (1989). [31] L. Y. Vinnikov, J. Karpinski, S. M. Kazakov, J. Jun, J. Anderegg, S. L. Bud’ko and P. C. Canfield Phys. Rev. B, 67, 092512, (2003). [32] J. Bardeen, L. N. Cooper, and J. R. Schrieffer. Phys. Rev., 108:1175, 1957. [33] B. J. Yuan and J. P. Whitehead Phys. Rev. B, 44 6943, (1991). Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

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[34] B. J. Yuan and J. P. Whitehead Phys. Rev. B, 47 3308, (1993).

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[35] Bojiong Yuan. Upper Critical Field of Inhomogeneous Superconductors Department of Physics. Memorial University of Newfoundland, Canada, 1994.

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In: Superconducting Magnets and Superconductivity… ISBN: 978-1-60741-017-1 Editors: H. Tovar and J. Fortier, pp. 257-274 © 2009 Nova Science Publishers, Inc.

Chapter 7

FABRICATION OF PYROCHLORE-BASED BUFFER LAYERS FOR COATED CONDUCTORS VIA CHEMICAL SOLUTION DEPOSITION Hechang Lei, Xuebin Zhu∗, Yuping Sun and Wenhai Song Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei 230031, People’s Republic of China

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1. Introduction Many efforts have been devoted to fabricating YBa2Cu3O7-δ (YBCO) coated conductors in the last decades, because it has the higher critical current density in the high magnetic field than Bi-based HTS tapes [1-3]. For high critical current density, highly biaxial textured YBCO films are essential. Now there are two methods to fabricate YBCO coated conductors: one is ion-beam assisted deposition (IBAD) [4] and other is rolling-assisted biaxially textured substrates (RABiTS) [5]. Based on RABiTS method, chemical solution deposition (CSD) method is a cost efficient way to fabricate YBCO coated conductors [6-10]. This approach can grow not only superconducting layers but also buffer layers. If both layers prepared by CSD method, it can be called all CSD or all metal organic deposition (MOD) method, which may be one of the most economy routes [11]. Comparing to vacuum deposition methods, the CSD approach requires less equipment and is easier to be scaled up for long lengths. Furthermore, it offers several other advantages, such as controlling stoichiometry of precursor solution excellently and easily gaining homogeneous large-area films. In RABiTS approach, buffer layers have twofold functions: one is blocking the diffusion of oxygen to metallic substrates and Ni to YBCO layers; second is transfering the texture of metallic substrates to the YBCO layers. The most prominent buffer layers consist with CeO2, Y2O3-stabilized ZrO2 (YSZ) and CeO2 [12]. For all CSD method, however, this kind of sandwich structure is too complicate ∗

E-mail address: [email protected]

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and grows difficultly. In the last few years, the pyrochlore-based La2Zr2O7 (LZO) buffer layers have been developed [13]. LZO buffer layers have the superior block property with thinner thickness and have better lattice match. The lattice mismatch to Ni substrates is about 8%, and to YBCO layers is about 0.5% and 1.8% (for a, b axial parameters respectively). Growing YBCO layers via trifluoroacetate metal organic deposition (TFA-MOD) on CeO2/LZO has been researched [14, 15]. However, there is existing reaction between YBCO and CeO2, and it will decrease the critical current Ic. So it is necessary to find out another buffer or cap layer on LZO buffer layers. In this chapter, pyrochlore-based buffer layers will be prepared using chemical solution deposition (CSD) method on NiW (200) substrates. Firstly, the effects of annealing temperature and seed layers on LZO buffer layers will be discussed; secondly, the influences of films thickness and precursor solution concentration on the orientation of LZO buffer layers will be studied; thirdly, we will present the results of YBCO deposited on CeO2/LZO/LZO seed layer/NiW substrates; finally, a new pyrochlore-based Y2Ti2O7 (YTO) buffer layer will be reported and the effects of orientation of underlying LZO on YTO will also be discussed.

2. Effects of Annealing Temperature and Seed Layers on the LZO Buffer Layers LZO buffer layers were fabricated using La-acetate (99.9%, Alfa Aesar) and Zr-(IV) npropoxide (70% w/w in n-propanol, Alfa Aesar). Firstly, La-acetate was dissolved into

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D

propionic acid at 70 C and continuously stirred under a magnetic stirrer for 20 minutes, and then Zr-(IV) n-propoxide (70% w/w in n-propanol) was added to the solution, and stirred at room temperature for more than 8 hours in order to obtain well mixed solutions. The final concentration of solution is 0.4M in cations. For fabricating LZO seed layer, the LZO solutions were diluted by propionic acid to 0.05M in cations. After aging the solution for 24 hours, the seed layers were deposited on NiW substrates using spin coating processing with the rotation speed of 4000 rpm and times of 60 seconds. D

Then, the as-deposited seed layers were dried at 400 C for 30 minutes under flowing 4% H2/N2 atmosphere. The dried films were then annealed at different temperatures for 2 hours under the atmosphere similar to that of the drying possessing. The subsequent LZO buffer layers were then fabricated on NiW substrates with and without seed layer using the same procedures described above, and the difference is just the concentration of precursor solution which was 0.4M. In order to obtain thicker LZO buffer layers, above-mentioned processing was repeated to the desired numbers. In order to research the effects of annealing temperature, the LZO buffer layers of D

different thickness without seed layer were annealed from 800 to 1000 C . The XRD θ-2θ D

results are shown in Fig. 1. It can be seen that the 800 C -annealed one layer LZO buffer layer is not crystallized and the intensity of the LZO (400) diffraction peak is obviously D

increased with the annealing temperature above 900 C . For one layer LZO, it is highly cD

axial orientation whatever the annealing temperature is either 900 or 1000 C . However, when increasing the thickness of LZO buffer layers, the diffraction intensity of LZO (222) Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

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259

D

can be clearly observed in 900 C -annealed sample but this diffraction peak can not presents D

in 1000 C -annealed sample. It can be concluded that higher annealing temperature is good for obtaining better c-axial orientation. The orientation improvement of LZO buffer layer with the increase of annealing temperature can be attributed to the enhancement of driving force of heterogeneous nucleation as well as grain growth at interface [16]. In the subsequent paragraph, except where otherwise noted, all mentioned LZO buffer layers were fabricated D

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using the annealing temperature of 1000 C . We further examine the in-plane orientation using XRD Φ-scanning which is shown in inset of Fig. 1. From this inset, it indicates that there exist some LZO grains which are 45° rotated in a-b plane related to the dominant grains in buffer layers and they decrease the biaxial texture of buffer layers and are undesired.

C -annealed one layer; (b) 900 DC -annealed one D D D layer; (c) 1000 C -annealed one layer; (d) 900 C -annealed two layers and (e) 1000 C -annealed D two layers. Inset shows the Φ-scanning result of the 1000 C -annealed LZO with two layers. Figure 1. XRD θ-2θ results of LZO/NiW: (a) 800

D

There are several papers indicating seed layer can improve the orientation of films [17, 18]. In order to improve the in-plane orientation and eliminate the undesired rotated grains in LZO buffer layers, the effects of seed layer are studied. Fig. 2a shows the XRD θ-2θ results D

of 1000 C -annealed two layers LZO deposition on NiW substrates with and without seed layer. From the XRD θ-2θ patterns, It can be seen that the two samples are highly (h00)oriented. Fig. 2b shows the Φ-scanning results of LZO buffer layers with and without seed layers. Although there are also two series of diffraction peaks existing, the intensity ratio between the four dominant diffraction peaks and the four rotated peaks is increased when a seed layer is inserted. It indicates that the seed layer is effective for eliminating the undesired LZO grains. So, from the above results, it can be concluded that it is necessary to deposit one

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seed layer before depositing subsequent buffer layers in order to improve the in-plane orientation. Generally speaking, seed layers can reduce the interf1acial energy by elimination of high-energy misoriented grains [19] and act as nucleation sites for homo-epitaxial growth of oriented films that subsequently deposited on the substrate.

Figure 2. (a) XRD θ-2θ results of the samples with and without seed layer; (b) Φ-scanning results of the corresponding samples.

3. Effects of Film Thickness and Solution Concentration on the LZO Buffer Layers In order to reach desired block property, the thickness of LZO buffer layers should be more than 70nm [13]. On the other hand, each layer thickness for once coating using the

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solution of 0.4M in cations is about 30nm. Multi-coating is necessary and it is worth of studying the effect of thickness to the texture of buffer layers.

Figure 3. (a) XRD θ-2θ results of LZO/seed layer/NiW with different thickness; (b) Φ-scanning results of LZO with different thickness on seed layer/NiW.

Fig. 3a shows the XRD θ-2θ results of 1 to 4 layer(s) LZO/seed layer/NiW substrates using the solution with a concentration of 0.4 M in cations. It is easily seen that all samples are highly (h00)-orientated without LZO (222) diffraction peaks. The corresponding Φscanning results are shown in the Fig. 3b. It can be seen that the full width at half maximum (FWHM) of the Φ-scanning results is almost same for all samples (~8°). Furthermore, it can be seen that there are also two series of diffraction peaks in the LZO buffer layers and it is observed that with the increase of thickness the diffraction intensity ratio in the Φ-scanning result between the primary peaks and the rotated peaks is increased, which indicates that the increase of thickness can effectively eliminate the undesired LZO grains.

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D

Figure 4. FE-SEM results of 1000 C -annealed LZO/seed layer/NiW prepared by 0.4M solution: (a)1 layer; (b) 2 layers; (c) 3 layers and (d) 4 layers.

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Fig. 4 present the FE-SEM pictures of 1-4 layer(s) LZO/seed layer/NiW derived from 0.4M solution. From these images, it can be seen that the surface of LZO is flat and the grain size of buffer layers is homogenous. Furthermore, it is rather dense despite of some small pores existing on the surface and there are not any microcracks which should be avoided in the fabrication process.

Figure 5. XRD θ-2θ results of 1000 different thickness.

D

C -annealed LZO/seed layer/NiW prepared by 0.8M solution with

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For improving fabrication efficiency, it is necessary to reduce the preparation procedure of LZO buffer layers. A possible method is increasing the precursor solution concentration in order to increase the thickness of single layer. Fig. 5 shows the XRD θ-2θ results of one and two layer(s) LZO/seed layer/NiW buffer layers prepared using the LZO solution concentration of 0.8 M in cations (about 50 nm for each layer). It can be seen that the LZO (222) diffraction peaks are obviously observed due to increasing the bulk nucleation and the ratio of LZO(222)/LZO(400) is increasing with thickness increasing, which can be attributed to enhanced interface nucleation of LZO(222) due to degenerating c-axis orientation of first buffer layer comparing with NiW substrates. It can be concluded that enhancing single layer thickness via higher solution concentration is not good for buffer layer orientation.

Figure 6. (a) Φ-scanning results of 4 layers LZO/seed layer/NiW derived from 0.4M solution and 6 layers samples derived from 0.2M solution; (b) Ф-scanning results LZO/seed layer/NiW derived from solution with 0.2M in cations with different thickness (1-8 layer(s) from bottom to top).

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On the other hand, lower solution concentration is avail of improving the texture of LZO buffer layers. The Φ-scanning results of the 4 layers and 8 layers of LZO (both about 120 nm)/seed layer/NiW fabricated using the solution concentration of 0.4 M and 0.2M in cations respectively is shown in the Fig. 6a, and it can be seen that the undesired LZO grains have been completely eliminated. The results suggest that it is an effective way to eliminate the undesired LZO grains by simply using diluted solution. Fig. 6b present the Φ-scanning results of LZO buffer layers with different thickness on the seed layer/NiW prepared using the 0.2M solution. It is seen that the FWHMs of Φ-scanning results of all buffer layers are about 7° and undesired 45° rotated grains disappear at all because thinner thickness will depress the bulk nucleation and improve the epitaxy of buffer layers. In a word, using low concentration solution, we can prepare LZO buffer layers which have rather highly biaxially textured. But it is worth noting that it may be not appropriate for preparing LZO buffer layers using too low concentration solution because it will decrease the fabrication efficiency and raise the cost. The FE-SEM results of LZO buffer layers prepared by the 0.2M solution are shown in Fig. 7 and it can be seen that the surface of LZO is similar with that of LZO fabricated using 0.4M solution: the surface is flat without microcracks and the grain size of buffer layers is homogenous. But it is seen that the grain size of 8 layers LZO (about 120nm) prepared using 0.2M solution is smaller than that of 4 layers LZO (about 120nm), which may be attributed to confining the grain growth in virtue of thinning thickness of each layer.

Figure 7. FE-SEM results of LZO/seed layer derived from 0.2M: (a) 2 layers; (b) 4 layers; (c) 6 layers and (d) 8 layers.

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4. The Property of YBCO/CeO2/LZO/NiW Substrates In order to check the qualities of the derived LZO buffer layers, the CeO2 cap layer is deposited on the LZO (120 nm)/NiW using CSD method. Fig. 8 shows the XRD θ-2θ, FESEM and AFM results of CeO2/LZO/NiW buffer layers. Because lattice parameter of LZO D

D

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(10.808 A , PDF card 73-0444) is close to twice times that of CeO2 (5.411 A , PDF card 040593), It can be seen that it is impossible to distinguish the LZO (400) and CeO2 (200) diffraction peaks. However, any diffraction peak of CeO2 (111) can not be traceable which indicates that the CeO2 film is highly (200)-oriented. From the FE-SEM and AFM, it can be seen that the CeO2 is relatively dense and smooth with root-mean-square (rms) roughness 7.8 nm, which is suitable for depositing YBCO films.

Figure 8. Continued on next page.

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Figure 8. Results of CeO2/4 layers LZO/LZO seed layer/NiW: (a) XRD θ-2θ pattern; (b) FE-SEM image and (c) AFM image.

Figure 9. XRD θ-2θ results of one and two layers of YBCO on buffered NiW substrates.

YBCO films are deposited on CeO2/LZO/NiW buffer layers using the TFA-MOD method as described elsewhere [20, 21]. Briefly, the TFA precursor solution for fabrication of YBCO was prepared by dissolving the TFA salts of Y, Ba and Cu with 1:2:3 cation ratio into sufficient methanol. The TFA solution was controlled to have the total metal ion concentration of 1.5mol/l and was coated onto buffered substrates by spin-coating method for 60 second at 4000 rpm. The heat treatment of the coated film was applied in two stages. In the first calcination stage, the film coated the TFA solution was decomposed to the amorphous D

precursor film by slowly heating up to 400 C in a humid oxygen atmosphere. In the case of multi-coating, the spin coating and calcination were repeated at several times after the first calcination. In the second calcination stage, the amorphous precursor film was heated up to D

800 C and then held for 70 min in a humid nitrogen and oxygen gas. Then the furnace atmosphere was switched from humid to dry nitrogen and oxygen gas mixture atmosphere

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D

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and cooled to 450 C . Finally, films are oxygenated at 450 C under a flow of O2 for 2 h to obtain the superconducting phase. Fig. 9 shows the XRD θ-2θ results of YBCO/CeO2/LZO/NiW with one and two layer(s) of YBCO. It can be easily seen that the YBCO is highly c-axis oriented. However, there exist clear diffraction peaks about BaCeO3, which is similar to the previous reports [22, 23]. BaCeO3 is usually found in the films which YBCO layers are prepared by chemical method rather than vacuum method [24-27]. Maybe the reason is that MOD-YBCO has a higher reactive activity than PLD-YBCO. The FE-SEM results of the two samples are also shown in Fig. 10. It can be seen that the YBCO films are dense. When the thickness of the YBCO is increased, however, the a-axis grains appear. It is worth noting that none of NiO or NiWO4 can be detected in XRD pattern and it is indicated that 120nm LZO buffer layers prepared by MOD method have a superior capability to block the diffusion of oxygen.

Figure 10. FE-SEM results of YBCO on buffered NiW substrates with (a) one layer and (b) two layers.

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Figure 11. (a) Temperature dependence of resistance for one and two layer(s) YBCO/CeO2/LZO/NiW, (b) Temperature dependence of resistance under different fields for two layers YBCO/CeO2/LZO/NiW.

Fig. 11a shows the temperature dependence of resistance results of one and two layer(s) YBCO/CeO2/LZO/NiW. It can be seen that the critical temperature (TC0) of the samples with one layer and two layers of YBCO is 86 and 90 K, respectively. Moreover, the linear fitting of the normal state resistivity results shows that the residual resistance of the sample with two layers YBCO is smaller than that of the sample with one layer, indicating that the connection

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of the YBCO grains is increased when the thickness of YBCO is increased. And the reaction between first YBCO layer and CeO2 cap layer maybe also increase the normal state resistivity of YBCO. Fig. 11b shows the temperature dependence of resistance under different applied fields for two layers YBCO. It can be seen that the TC0 is about 75 K under 7 T, which is similar to the theoretical results. The critical current of two layers YBCO at 77 K under selffield is about 12 A/cm-width. The above results suggest that CeO2/LZO buffer layers prepared using CSD method are suitable for coated conductors.

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5. YTO/LZO Composite Pyrochlore-Based Buffer Layers Derived by CSD Method In section 4, we have shown that CeO2/LZO buffer layers are suitable buffer layers for YBCO coated conductors prepared by all MOD method. However, the reaction problem between YBCO and CeO2 should be solved because the formed reaction layer will not affect the Jc but decrease the thickness of YBCO layer to result in Ic reducing. It is necessary to deposit another buffer or a cap layer on LZO buffer layers and it is chemical compatible with YBCO and LZO. On the other hand, until now, buffer layer with pyrochlore structure, only zirconium-based materials such as LZO and GZO are explored [13, 28]. Because buffer layers with this structure have superior block property, it is worth searching other materials with pyrochlore structure as buffer or cap layers. Recently, we have developed a new pyrochlore-based buffer layer consisting with Y2Ti2O7 (YTO) for coated conductors by CSD method [29]. The reasons we selected YTO are as follows: firstly, YTO has low lattice mismatch with Ni and YBCO. YTO is cubic structure and its lattice parameter is a=10.09Å (PDF card 85-1584), which is close to that of Ni (a=3.535Å, PDF card 65-0380, lattice mismatch 0.9%.), and YBCO (a=3.821Å, b=3.885Å, PDF card 85-1877, lattice mismatch 6.6% for a axis); secondly, it is observed that the SrTiO3 does not react with the YBCO films indicating no chemical reaction between Ti ions and YBCO [30]. But YTO buffer layers growing on NiW substrates directly are high (222) orientation, which is not suitable for YBCO layers deposition [29]. It is well known that the inserted template can obviously change the orientation of the continuous layer [31], and LZO maybe a suitable choice, because of similar structure and closed lattice parameter (a=10.786Å, PDF card 71-2363, lattice mismatch 6.5%.) with YTO. In this section, we will discussion about YTO/LZO composite pyrochlore-based buffer layers prepared using the CSD method on NiW substrates. The results show that YTO can be considered as one of candidates for buffer or cap layers in the all CSD route for coated conductors. YTO buffer layers were fabricated using Y-acetate (99.99%, Alfa Aesar) and Ti (IV) nD

butoxide (99%, Alfa Aesar). Firstly, the Y-acetate was dissolved into propionic acid at 70 C and continuously stirred under a magnetic stirrer for 20 minutes, and then the Ti (IV) nbutoxide was added to the solution, and stirred at room temperature for more than 8 hours in order to obtain well mixed solutions. The final concentrations of solution are 0.05M and 0.4M in cations for seed layer and subsequent layers respectively. The YTO seed layer was deposited on 4 layers LZO/LZO seed layer/NiW substrates using spin coating processing with the rotation speed of 4000 rpm and times of 60 seconds. Then, the as-deposited films were D

dried at 400 C for 30 min under flowing 4% H2/N2 atmosphere. The dried films were then Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

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annealed at 900 C for 2 hours under the atmosphere similar to that of the drying possessing. And then, YTO subsequent layers were coated, heat treatment as mentioned above. The thickness of the YTO seed and subsequent layer for once coating is about 6 and 15 nm, respectively. Thickness is also determined by the ellipsometer.

D

C -annealed LZO/NiW.

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Figure 12. XRD θ-2θ result of YTO/900

Figure 13. Results of (a) XRD θ-2θ, (b) Ω-scanning and (c) Ф-scanning of YTO grown on 1000 annealed LZO/NiW.

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C-

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In order to check the LZO/NiW template effects on the orientation of the YTO layer, we D

deposited the YTO on the 900 C -annealed LZO buffered NiW substrates and the XRD θ-2θ result is shown in Fig. 12. It can be seen that the YTO (222) is clearly observed. The results D

suggest that on the 900 C -annealed LZO/NiW substrate, the YTO cannot grow in the (400) orientation, which is attributed to the remaining (222)-oriented LZO grains in the LZO buffer layers. In order to improve the orientation of the LZO buffer layer as well as to study the influence of LZO orientation improvement on the YTO orientation, YTO buffer layers were D

deposited on the 1000 C -annealed LZO buffer layers which were highly c-axis orientation. The XRD θ-2θ result is shown in Fig. 13a. It is observed that the YTO is highly (400)oriented and the YTO (222) peak can not be detected. These results suggest that the orientation of the YTO layer can be effectively tuned by the orientation of the LZO layer. For epitaxial film, the orientation is determined by several factors: the energy of mismatch εmis, which commonly increases with enlargement of mismatch between substrate and epitaxial film; Surface free energy σsurf, which is decided by the interface energy of epitaxial film and atmosphere; Interface energy between epitaxial film and substrates γinter. As like discussion Ref [29], for YTO deposited on NiW(200) substrates, the energy of σsurf(222)+γinter(222)+εmis(222) is less than σsurf(400)+γinter(400)+εmis(400), so the orientation relation between YTO and NiW is YTO(222)||NiW(h00). On the other hand, when YTO D

deposited on LZO buffer layers annealed at 900 C , which has small (222) orientation, σsurf+γinter+εmis of YTO(222)||LZO(222) is so small that once (222)-oriented LZO phase appearing in LZO buffer layers, YTO(222) phase will grow on LZO(222) grains epitaxially and YTO(222) phase grows quickly which consumes most materials of amorphous precursors resulting in the growth of (400)-oriented YTO being suppressed. When YTO deposited on

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D

highly (h00)-orientated LZO annealed at 1000 C , however, there is an obvious different case. Because LZO (222) phase absents in LZO buffer layers and YTO (222) growing on LZO (400) need more energy than YTO (400) growing on LZO (400), the growth of YTO (400) will be easier than that of YTO (222). So we obtain highly (400)-oriented epitaxial YTO/LZO buffer layers on NiW substrates. The out-of-plane and in-plane orientations of YTO were detected using Ω and Φscanning respectively (Fig. 13b and c). It was determined that the FWHMs of the Ω and Φscanning are about 6° and 6.5° respectively. Moreover, it can be seen that in the Φ-scanning result, there are four symmetric diffraction peaks with relatively large diffraction intensities concomitanted with four other symmetric diffraction peaks which are relatively low diffraction intensities. The in-plane orientation result suggests that the YTO (400) grows on LZO (400) with cube-on-cube mode, and the 45° rotated grains in YTO can be attributed to the effect of remaining 45° rotated grains in LZO buffer layers. Surface morphologies of YTO buffer layers are investigated by FE-SEM and AFM (Fig.14). The rms roughness determined from the AFM result is only 6 nm in 1×1μm area which indicates that the surfaces of buffer layers are smooth. From FE-SEM results, it’s concluded that YTO buffer layers are dense, continuous and crack-free with smooth surfaces, which are consisting with the AFM results. These characteristics are important for YBCO deposition in order to enable perfect growth without introduction of defects from YTO buffer layers and also essential to block the interdiffusion between YBCO and NiW substrates.

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Figure 14. (a) AFM image and (b) FE-SEM image of YTO/1000

D

C -annealed LZO/NiW.

6. Conclusion In summary, pyrochlore-based buffer layers LZO and YTO/LZO were prepared using CSD method on NiW substrates. Firstly, the results show that increasing annealing temperature and inserting seed layer can obviously improve the crystallization and biaxial texture of the LZO buffer layers; secondly, the in-plane orientation of LZO improves with increasing the film thickness although undesired 45° rotated grains still exist; thirdly, the effects of solution concentration are studied and the results show that when the concentration increasing, the orientation of LZO decreases. On the contrary, texture can be improved obviously and the 45° rotated grains could be eliminated completely with decreasing the solution concentration; fourthly, YBCO was prepared on the smooth CeO2/LZO/NiW substrates using TFA-MOD method and the critical current of two layers YBCO at 77 K under self-field is about 12 A/cm-width. The results indicate that CeO2/LZO buffer layers

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prepared using CSD method are suitable for coated conductors; finally, a kind of new pyrochlore-based materials YTO was developed and the orientation of YTO/LZO/NiW are studied. The results show that the orientation of the YTO can be tuned through the orientation of the LZO buffer layers. When there is an LZO (222) diffraction peak, the YTO is (222)oriented; however, when the LZO is highly (400)-oriented, the YTO is highly (400)-oriented. YTO could be considered as a kind of potential buffer or cap layers for coated conductors.

Acknowledgement This work was supported by the National Key Basic Research Program of China under contract No. 2006CB601005, the National Nature Science Foundation of China under contract No. 50802096, Director’s Fund of Hefei Institutes of Physical Science.

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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

D. Larblestier et al., Nature, 2001, vol. 414, 368-377. S. B. Kim et al., IEEE Trans. Appl. Supercond., 2005, vol. 15, 2645-2648. M. Paranthaman et al., IEEE Trans. Appl. Supercond., 2001, vol. 11, 3146-3149. Y. Iijima et al., Physica C, 2001, vol. 357-360, 952-958. Goyal et al., Appl. Phys. Lett., 1996, vol. 69, 1795-1797. T. Araki et al., Supercond. Sci. Technol., 2003, vol. 16, R71-R94. X. Obradors et al., Supercond. Sci. Technol., 2004, vol. 17, 1055-1064. Sheth et al., Supercond. Sci. Technol., 2003, vol. 16, 322-328. X. B. Zhu et al., Physica C, 2004, vol. 415, 57-61. X. B. Zhu et al., Physica C, 2005, vol. 418, 59-62. X. Obradors et al., Supercond. Sci. Technol., 2006, vol. 19, S13-S26. Y. Akin et al., IEEE Trans. Appl. Supercond., 2003, vol. 13, 2673-2676. T. G. Chirayil et al., Physica C, 2000, vol. 336, 63-69. M. P. Paranthaman et al., IEEE Trans. Appl. Supercond., 2005, vol. 15, 2632-2634. S. Sathyamurthy et al., J. Mater. Res., 2006, vol. 21, 910-914. R. W. Schwartz, Chem. Mater., 1997, vol. 9, 2325-2340. X. B. Zhu et al., Scripta Mater., 2004, vol. 51, 659-663. Z. Fu et al., Chem. Mater., 2006, vol. 18, 3343-3350. Seifert et al., J. Mater. Res., 1996, vol. 11, 1470-1482. M. W. Rupich et al., MRS Bull., 2004, vol. 29, 572-578. P. C. Mclntyre et al., J. Appl. Phys., 1992, vol. 71, 1868-1877. D. E. Wesolowski et al., J. Mater. Res., 2007, vol. 21, 1-4. T. Kato et al., Physica C, 2002, vol. 378-381, 1028-1032. J. S. Matsuda et al., Physica C, 2005, vol. 426-431, 1051-1055. Ichinose et al., Physica C, 2004, vol. 412-414, 1321-1325. L. Molina et al., Physica C, 2007, vol. 460-462, 1407-1408. Y. Tokunaga et al., Physica C, 2003, vol. 392-396, 909-912. Y. X. Zhou et al., IEEE Trans. Appl. Supercond., 2005, vol. 15, 2711-2714. X. B. Zhu, et al., Physica C, 2006, vol. 433, 154-159.

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[30] M. P. Siegal, et al., Appl. Phys. Lett., 2002, vol. 80, 2710-2712. [31] X. D. Wu, et al., Appl. Phys. Lett., 1992, vol. 60, 1381-1383.

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In: Superconducting Magnets and Superconductivity… ISBN: 978-1-60741-017-1 Editors: H. Tovar and J. Fortier, pp. 275-291 © 2009 Nova Science Publishers, Inc.

Chapter 8

RADIATION SHIELDING SCHEMES AND ADVANCED FABRICATION TECHNIQUES FOR SUPERCONDUCTING MAGNETS 1

Laila A. El-Guebaly1,∗ and Lester M. Waganer2,♣ University of Wisconsin, Fusion Technology Institute, Madison, WI, USA 2 Consultant for the Boeing Company, USA

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Abstract The tokamak has been perceived as the most promising magnetic confinement concept to achieve fusion energy. For economic reasons, superconducting rather than normal magnets will be used in commercial magnetic fusion plants to significantly reduce the circulating power fraction. A number of superconducting magnet designs have been developed over the past 50 years with both low and high temperature superconductors. For a practical fusion facility, these magnets must be well shielded from energetic fusion neutrons to achieve the predicted performance and service lifetime. With innovative manufacturing techniques, these magnets can be fabricated faster, cheaper, and more practically than conventional processes currently indicate.

I. Introduction During half a century of fusion research (1958-2008), the principles of the magnetic confinement have been refined with the tokamak evolving to be the most promising magnetic confinement approach. The tokamak has years of research and development throughout the world with increasing size and fidelity of experimental facilities. However, there remains the other half of the solution, namely, generation of economical power. It has been recognized that using normal magnets would require excessive recirculating power (300-400 MWe) for a nominal 1000 MWe power plant. There is general agreement that ∗ ♣

E-mail address: [email protected] E-mail address: [email protected]

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superconducting (S/C) magnets are the technology of choice for large fusion power generating facilities. Even long-pulse experiments such as ITER (ITER Project) find it worthwhile to employ superconducting magnet systems. While the current level of understanding is sufficient to build and operate S/C magnets for experimental devices, designing magnets for fusion power plants require significant extrapolation from current magnet designs as well as vigorous research and development program to achieve efficient and reliable magnet systems (Greenwald 2007). The ARIES advanced tokamak studies (ARIES Project) have proposed both lowtemperature (LT) S/C magnets (operating at 2-4 K) for tokamaks and stellarators and hightemperature (HT) S/C magnets (operating at 70-80 K) for tokamaks (Bromberg 2001, Bromberg 2007). These magnet systems featured structural simplicity, resilient steel structures, radiation-resistant electrical insulators, and low-cost fabrication processes. Increasing the overall compactness of the power core is a major cost reduction incentive and is a primary feature of the ARIES fusion power plant designs. These designs were achieved by high-fidelity characterization of the radiation environment throughout the core resulting in a well-optimized radial build for all in-vessel components. These components, in addition to their primary function, provide an integrated shielding function, primarily for the magnets as well as other plant systems inside and outside the central power core. The magnet radiation tolerance strongly influences the composition and dimensions of the vacuum vessel in particular. The predominant radiation limits are the fast neutron fluence to the superconductor and the deposited nuclear heat in the magnet. The cryogenic power requirement for nitrogen-cooled HT magnets is significantly less than the liquid heliumcooled LT magnets. An emerging engineering issue is the manufacturing of S/C magnet systems and the maturity of the technology to fabricate low-cost, yet highly reliable magnets. Current S/C fabrication technologies are still evolving with little incentive to mature and incorporate innovative, lower cost, and higher reliability designs as only a few experimental machines (for fusion and accelerators magnets) are being constructed worldwide per decade or two. Although there are a large number of S/C magnets sold each year for medical applications, the magnets are relatively small part of the overall cost of these systems, with little effort on developing revolutionary techniques for these applications. Innovative fabrication processes include additive fabrication of structural components, highly automated conductor winding, direct deposition of the superconducting, insulating, and normal conducting materials, and tailoring of the structure, superconductor, and copper to local structural and electromagnetic loads. Demonstration, pilot, and/or prototype plants will affirm the maturity and applicability of these technologies for future fusion power plants. The objective of this chapter is to briefly review and assess the feasibility of three main topics that received considerable attention during several in-depth fusion conceptual design studies (ARIES Project): magnet radiation limits that control the radial standoff distance, radiation shielding schemes developed specifically for advanced fusion designs, and innovative fabrication techniques that offer lower cost and higher reliability designs. To establish the context for this assessment, a brief overview is first provided for the main magnet system constituents and their current state of developmental efforts.

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II. Magnet System Constituents Accompanying chapters in this book explain in detail the superconductivity phenomenon observed in metals and ceramics when cooled to specific critical temperatures (ranging from 2 to 100 K). When these materials reach below their critical temperatures, their electrical resistance decreases to zero and no ohmic losses are created. Thus, they can carry large electric currents for indefinite periods of time when supercritical. There are critical current (Jc) and critical magnetic field (Bc) parameters associated with the critical temperature (Tc). This means certain metal alloys and ceramics are in S/C state only within a specific domain in the T-B-J space. The boundary of this space is material specific. If any of the three parameters (T, B, and J) exceeds its critical value, the S/C characteristic ceases and the material reverts back to a normal, non-S/C state, causing the entire magnet to quench (i.e., become normally conductive). The HT and LT S/C materials will support high current density in the magnets and thus can generate intense magnetic fields. Since the current density is high, S/C magnets can be compact and occupy small volumes as compared to normal magnets. Figure 1 displays typical cross sections of LT and HT magnets. Multi-layered electric insulators electrically isolate the conductors from the surrounding support structure. Because of the demanding cooling requirement for the magnets to remain superconducting, liquid helium (LHe) is used for LT S/C magnets with Tc around 4 K while liquid nitrogen (LN) can be used for HT materials with their higher Tc criteria. Typical winding pack compositions are 10% conductor, 20% structure, 50% stabilizer, 10% insulator, and 10% LHe for a LT magnet. For the HT magnet, the compositions are 9% conductor, 70% structure, 20% insulator, and 1 % Ag. The silver component covers the conductor strand (see Fig. 1). The HT magnet overall composition is dominated by the steel structure that can operate at high fields (> 16 T) without a quench requirement. The surrounding LN-cooled coil cases provide conduction cooling for HT winding packs. The cryogenic power requirement for HT magnets is significantly lower as compared to LHe-cooled LT magnets. Thus, the allowable nuclear heating deposited in the HT magnets could be higher with minimal concern about the cryogenic heat load. Superconducting Wires Helium Channel Sliver

Structure

YBCO Insulator

Structure

LT Winding Pack

HT Winding Pack

Figure 1. Cross section of ITER’s LT S/C magnet (ITER Project) and schematic of HT S/C winding pack.

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II.1. Superconducting Materials Niobium tin (Nb3Sn) and niobium titanium (NbTi) are well characterized superconducting materials, operating at temperatures of 4 K and 9K, respectively. Nb3Sn can support higher field strength than NbTi. A superconducting coil is wound with many fine filaments. The NbTi or Nb3Sn conducting filaments are embedded in a Cu matrix to protect the magnet hardware in the event of a conductor quench. The Nb3Sn conductor is brittle, difficult to wind, and more expensive than NbTi. However, fabrication of ITER coils will enhance the industrial experience of Nb3Sn production since the required ITER quantity of Nb3Sn strands exceeds 30 tonnes. Certain semiconducting compounds become superconductive at higher temperatures. Yittrium barium copper oxide (abbreviated YBCO for YBa2Cu3O7) becomes superconducting above the nitrogen boiling point (77 K). Other S/C compounds include bismuth strontium calcium copper oxide (BSCCO with Tc > 90 K) and magnesium diboride (MgB2 with Tc of 39 K). The significance of the HT S/C materials is in the high magnetic fields that they generate (up to 22 T), low refrigeration requirement for cooling with LN below Tc, low material unit cost, and inexpensive construction techniques. Since the discovery of HT S/C materials in 1986, progress in basic research and fabrication techniques have substantially improved. In the long term, it is anticipated that HT S/C magnets will play an essential role in fusion applications as well as many other commercial applications, such as power transmission.

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II.2. Structural Materials The structural material in the magnet system should be compatible with the superconductor to prevent strains due to thermal contraction between heat treatment temperature and low operating temperature. The iron-nickel base steel superalloy (Incoloy908), used extensively in the US, is thermally compatible (Morra 1992) by having a thermal contraction coefficient similar to those of the superconductors. An alternative low-carbon, low-boron alloy (JK2LB) was recently developed in Japan (Nakajima 2004) as a S/C structural material with enhanced features (Wang 2008). Three main benefits for JK2LB as compared to Incoloy-908 are: • • •

JK2LB is less expensive that Incoloy-908 because of its high Ni content (Wang 2008) JK2LB is environmentally more attractive than the high Nb content (3 wt%) Incoloy908 (El-Guebaly 2008). JK2LB exhibits superior recycling and clearance characteristics providing a strong incentive to replace Incoloy-908 in future magnet designs (El-Guebaly 2008).

II.3. Insulating Materials Both ceramic and organic insulators are considered for use in fusion magnets. Normally, woven ceramic insulators or S-glass tapes with organic insulators (such as kapton, mylar, epoxy, and polyimide) are wrapped around the conductors. For the woven ceramic insulators,

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the insulation process could be preformed during the winding process before the heat treatment of the conductor. The ceramic tape can survive the heat treatment process while the glass tape cannot (Bromberg 2007, Wang 2008). During the last decade, advanced radiationresistant ceramic-based and organic-based insulators with improved mechanical properties have been proposed for fusion applications (Puigsegur 2004, Fabian 2002, Humer 2006).

II.4. Stabilizing Materials To provide an alternate current path if the superconductor becomes normal by any instability or temperature fluctuation, a low-resistance material, such as Cu, is bonded to the S/C filaments. During operation of the fusion experiment or plant, neutrons that reach the normal conducting materials produce point defects in those materials that increase their resistivity (Sawan 1986). For Cu, partial recovery (80-90%) of the radiation-induced defects can be achieved through room temperature annealing (Sawan 1984). For HT S/C magnets, there is no need for a substantial fraction of stabilizer, in contrast to LT S/C magnets (Bromberg 2001). The only normal conducting materials required for HT magnets is whatever needed to manufacture the superconductor (e.g., Ni tapes for YBCO and silver matrix for BSCCO).

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III. Radiation Limits In fusion devices, the magnet system is subjected to neutrons and gamma radiation that will degrade the physical and mechanical properties of magnet materials (conductor, insulator, and stabilizer). This is an issue of great concern for the long-term operation of S/C magnets. As such, the magnet designers carefully define a set of radiation limits to assure the proper long-term performance of the magnet in the fusion environment. The magnet radiation effects are interrelated and determined by the radiation flux level and spectrum throughout the magnet (Sawan 1986). As an example of the necessary requirements discussed in the prior paragraph, the radiation limits for ARIES magnets are given in Table I. At the end of ARIES power plant operation, the fast neutron fluence (En > 0.1 MeV) at the magnet should not exceed a certain limit, see Table 1 below, to avoid degradation of the superconductor critical properties (Jc, Bc, and Tc). The neutron-induced atomic displacement to the Cu-stabilizer should remain below 0.006 dpa (displacements per atom) to avoid a significant increase in the Cu resistivity. Magnet annealing could reverse most of the stabilizer damage, but the process may be economically prohibitive due to the time to accomplish. The absorbed radiation dose to insulators should be restricted to avoid degradation of mechanical properties. It is undesirable to deposit > 50 kW of nuclear heating in all magnets to avoid excessively high cryogenic power load to the cryoplant. The specific heating should be restricted because of local impact on magnet protection (Sawan 1986). For HT magnets, the limits on the stabilizer damage and cryogenic heat load can be relaxed (Bromberg 2007) and the peak nuclear heating could reach 5 mW/cm3. The radiation tolerance for YBCO is at least as good as that of Nb3Sn (Bromberg 2001).

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Laila A. El-Guebaly and Lester M. Waganer Table I. Radiation limits for ARIES magnets (Bromberg 2001, Bromberg 2007) Peak fast neutron fluence to: Nb3Sn and YBCO superconductors NbTi superconductor Peak atomic displacement to Cu stabilizer Peak dose to glass-fiber-filled polyimide Peak nuclear heating: HT magnets LT magnets Total nuclear heating in all magnets

1019 n/cm2 3x1018 n/cm2 0.006 dpa 1011 rad 2 mW/cm3 5 mW/cm3 < 50 kW

Most S/C magnets are currently manufactured with a fiber-reinforced epoxy insulator that imposes a relatively low limit of 108 rad on the allowable radiation dose. Polyimide-based insulators can withstand higher radiation doses of 109 or more, depending on the magnet design. The lower 109 rad limit is for the case where the insulator needs to withstand substantial shearing forces. It is possible to increase the insulator dose limit to 1011 rad in the absence of interlaminar shear and if the magnet is designed with the insulator loaded in compression only (Sawan 1986, Bromberg 2001). At present, the newly developed insulators (cyanate ester and hybrid epoxy/inorganic) have demonstrated outstanding mechanical and electric properties when tested at 2.5 x 1010 rad (Fabian 2002).

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IV. Radiation Shielding Schemes The magnet radiation limits determine the size of in-vessel components that, in turn, influence the radial build, machine size, and overall cost of electricity. The internals of any DT fuelled fusion device are the standard power core components, namely the blanket, shield, vacuum vessel (VV), and magnet. The primary function of the blanket is to convert the nuclear heating of the blanket materials into high-grade thermal energy and breed needed tritium for fueling the plasma. All inner components (blanket, shield, and VV) provide a shielding function for the magnet. The composition of the outermost component closest to the magnet (i.e., the VV) is strongly influenced by the magnet radiation limits. In power plants with anticipated operating lifetimes of 40-50 years, the predominant radiation limit is the fast neutron fluence for the conductor followed by the peak nuclear heating. The latter peak nuclear heating limit pertains only to LT magnets. The shielding optimization process helps define the design space for any fusion device and eliminates unnecessary radioactive waste generated by non-optimized radial builds. The following examples highlight the radiation shielding schemes for two leading magnetic concepts: the tokamaks (toroidal (donut) configuration with a D-shaped plasma) and stellarators (toroidal configuration with a periodic multi-shape plasma): •

In tokamaks, such as the more recent ARIES-RS (El-Guebaly 1997) and ARIES-AT (El-Guebaly 2006), the shielding effort focused on the inboard region where a better shielding performance made a notable difference to the overall machine size. Less emphasis is placed on the outboard and divertor areas where the shielding space is

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generally unconstrained and no economic or design enhancements are gained with the use of high-performance, compact shields in these regions. In stellarators, such as ARIES-CS (El-Guebaly 2008), the minimum plasma-coil distance is the most influential parameter for the overall size and capital cost of the machine. This parameter is a measure of how close the magnet is located to the plasma edge. As this parameter is reduced, the space available for blanket, shield, and vacuum vessel is increasingly constrained. An innovative approach was developed specifically for ARIES-CS to reduce the blanket thickness at these critical locations and utilize a highly efficient shield/VV to protect the magnet (thus trading tritium production for shielding capability).

1

35

Blanket

Thickness (cm)

Plasma

24

Gap

Vacuum Vessel

2

Shield

40

Gap

53

Coil Case

17

Gap + Th. Insulation

14

Winding Pack

Figures 2 and 3 depict the radial builds for ARIES-AT advanced tokamak with HT magnets and ARIES-CS compact stellarator with LT magnets. In both designs, the blanket, a replaceable component, protects the shield such that it will last for the plant life (40-50 years). The shield and VV protect the magnets and external systems. The VV is the closest component to the magnet and its composition plays an important role in controlling the radiation damage level at the magnet. Unlike the blanket and shield, the VV is not a power production component as its temperature will be kept below 200oC. Thus, the VV will have its own cooling system and employ a coolant (e.g., water) with a good shielding performance to help reduce the radial build. In most ARIES designs, the skeleton of the double-walled VV with internal ribs is filled with shielding bulk materials and is optimized to achieve the necessary shielding requirements for magnet protection. Tradeoff analyses of the water and filler materials are conducted for the optimized VV for each individual design. Two candidate fillers are considered for the VV: high-performance tungsten carbide (WC) and borated ferritic steel (B-FS; FS with 3 wt% B). The WC-based VV is thinner with much reduced nuclear heating at the magnet. Typical volume fractions would be 75/25 WC/H2O and 30/70 B-FS/H2O. An optimal combination for tokamaks is WC in the inboard section of the VV to reduce the size of the overall machine and B-FS in the divertor and outboard VV sections. In compact stellarators, WC is limited to the shield and VV in the critical areas where the magnet is closer to the plasma (refer to Fig. 3).

Coil Case

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•

281

Figure 2. Inboard radial build of ARIES-AT (El-Guebaly 2006).

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Vacuum Vessel Mag ne

Gap

35 32

LiPb & He Manifolds FS-Shield Back Wall

5

Full Blanket

54 4

Divertor

25 c mL o ca

20

t

l Sh ield

26 He

Tube

2 28 2

WC-Shield

34

Non-uniform Blanket

FS-Sh ield

First Wall

14 25 4 cm

Plasma Figure 3. Toroidal cross section through ARIES-CS blanket, shield, VV, and magnet (El-Guebaly 2008).

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V. Advanced Fabrication of Superconducting Magnet Systems For one-of-a-kind experimental facilities, independent of size, the magnet coil support structures can be fabricated to conform to the magnet shape. The conventional approach of designing tokamak and stellarator magnetic coil structures for single use experiments employs a strong metal case surrounding the superconducting coils of the proper shape (round or deformed elliptic) either planar (for tokamaks) or non-planar (for stellarators). Additional structural elements are added inside (toward the center of the power core) as bucking cylinders or elements to resist the inward toroidal field (TF) coil electro-magnetic (EM) centering forces and limit the coil distortions to acceptable levels. Other structures are needed between the TF coils to resist overturning EM forces both during normal operation and offnormal conditions. These structures will likely operate at cryogenic temperatures, close to those required for the superconductor. Structural supports for gravity loads on all coils must transition from cryogenic temperature to normal temperatures, using appropriate materials and cross section/length in order to provide high thermal resistance. Placement of the coil structures will be dictated by the location of maintenance access and auxiliary ports. The tokamak TF coils would be a variation on the classical D-shape surrounding the plasma. This geometry variation relates to the inner coil legs being as close to the plasma as possible while the outer midline coil region is extended outward to provide maintenance access between the outer legs of the coils. This modified D-shape increases the EM bending stresses that are resisted by the coil structure. The poloidal field (PF) coils would be circular coils encompassing the entire power core in the toroidal direction, just outside the TF coils. The modular stellarator coils are non-planar with a periodic repeating set of coil geometries (Najmabadi 2008). These stellarator coils have some areas with tight bend radii in the

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poloidal direction. The superconducting cables are wound in grooves on the coil structures that have the appropriate shape for the magnetic coils. A compact stellarator geometry is so access-limited that sector maintenance may not be possible, thus modular maintenance of the power core would have to be accomplished through a few dedicated maintenance ports. As the design for experiments evolve toward those of commercial power plant facilities, economics begin to play an ever increasing role. For a power plant, maintenance access is vitally important to achieve rapid and efficient maintenance and replacement of the power core. The requirement for increased access to the core elements demands larger power cores, larger openings between coils, and more innovative coil design and fabrication solutions. Conceptual design studies (Najmabadi 1997, Najmabadi 2006) of commercial tokamak power plants have shown the outer legs of the TF D-shaped coils are increased in radius, as shown in Fig. 4, to allow full power core sectors to be removed radially out as a single unit. This replaceable unit contains the first wall, blanket, shield, divertor and supporting hot structure. The design of the poloidal coil set typically allows the outer midplane PF coils to be permanently relocated above and below the midplane to permit the radial removal of the power core sectors.

Figure 4. ARIES-AT isometric view.

The TF coils for all power plants must be highly reliable to assure that there is never a major failure of a coil that would require replacement. The PF coils could be replaced, but it would be a very costly, lengthy, and onerous repair process, especially for the ones underneath the power core. If these lower PF coils beneath the power core would fail, it would be prudent to provide spares in place. The failed coils could be disassembled and the in-place spares could then be raised into position. This places an additional incentive to

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develop coil superconductor cable fabrication and winding procedures to be highly automated and process controlled, which should enhance the coil reliability. The requirement for economic operation of a fusion power plant also suggests higher plasma density power cores, which increase the maximum magnetic fields in the plasma and at the magnet. NbTi superconducting magnets cooled by liquid helium at 4.2 K are adequate up to about 8-9 Tesla for experiments, but higher fields are necessary for commercial plants. Fortunately, Nb3Sn superconductors can extend the magnetic field up to about 16 Tesla. The Nb3Sn materials are more brittle and present design and manufacturing challenges. If higher magnetic field strengths than 16 Tesla are needed, then higher temperature superconductors are required. These HT materials also affect the design and fabrication of the supporting coil structures and installation processes. These requirements form the design and fabrication guidelines for commercial power plant magnet structures.

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V.1. Conventional Coil Fabrication Approaches The conventional approach to fabricate superconducting coil structures is to rough form steel panels in the approximate final shape. Certain steel alloys are selected for low temperature material properties, and compatibility with high magnetic fields and operational conditions, primarily fatigue. ITER (Mitchell 2006) is planning to forge the coil plates to rough shape and then machine them to near final dimensions. Internal stresses from forging will relax during machining and thus distort the machined dimensions. These machined panels are then welded into a much larger assembly to form an approximate final net shape. This complete assembly will then be final machined to desired tolerances. Superconducting NbTi or Nb3Sn cable will be fabricated off-site and delivered to the construction site. The superconducting cables will be compacted, insulated and wound onto the coil structure. Cooling provisions for the cables will determine the coolant flow channel through the coil and/or around the coil. Bridging structures to carry TF centering and out-of-plane loads will be bolted and/or welded structures. These structures would have to accommodate all the access requirements for the power core heating, current drive, fueling, diagnostic, thermal management, and maintenance equipment. This conventional approach of forging, welding, and machining is very labor intensive with a poor material yield for the final product. These many fabrication steps require a lot of time for tooling, setup on several machines, fabrication process time, and quality checks at each step. As common magnet designs are standardized and applied to several production plants, multiple uses of the tooling fixtures and process improvements can result in significant learning improvements. As automation is further utilized, this will speed up the processes initially, but learning progress on subsequent units will be less due to the use of automation. This process of subtractive fabrication (starting with large forgings or plates and machining away all material that is not necessary) results in an inefficient use of time and material, even if materials are reprocessed and recycled. The final component cost would be ten to twenty times the material cost used.

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V.2. Innovative Structural Fabrication Approaches

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The proposed process of additive fabrication or rapid manufacturing process is a much more efficient use of time, labor, and material. The parts can be fabricated using a near raw form of the metal (powder or wire). The part would be fabricated directly from the CAD definition by melting a metal powder or wire feedstock and depositing the molten metal in layers up to the near net shape required. A variety of heating sources may be used, such as electric arc, plasma torches, and lasers. Gas shielding will prevent contaminants in the melt layer. Annealing the formed part may be required to reduce stresses and limit future deformation. Any dimensional errors or voids can be corrected by machining and/or redepositing new material. Other metals or materials can be added or adjusted during the part build up with varying density and material composition throughout the component, to match local requirements. Intricate shapes can be reproduced that might not be possible with conventional fabrication processes. The accuracy of the fabricated components is continuing to improve as the fabrication processes mature. Many components may not need final machining, much like some casting processes. Most parts will require a light final machining to reach final dimensions and finish, especially on mating parts. Presently, an electron beam melting process is capable of producing fully functional components in widely accepted production grade materials, such as Ti-6Al-4V (Arcam). Material properties are comparable to bar stock material with corresponding performance for: • • • • • • •

Static strength Fatigue strength Fracture toughness Crack propagation rate Notch sensitivity Corrosion resistance Stress corrosion resistance.

The above approaches are representative for several other rapid manufacturing and freeform fabrication processes and companies, with a variety of high strength materials. These processes were initially used for fabricating small prototypes. Then the additive fabrication began to be used for small production runs of smaller parts. Recently, the process has been extended to larger parts and higher production quantities. The advantage of rapidly manufacturing directly from CAD definition to intricate final assemblies is very appealing, time saving, and cost effective. The present technology typically uses a single fabrication device, but as the component size is increased, multiple fabrication elements can be employed to speed up the fabrication process. The conceptual designs for commercial tokamak power plants have not fully embraced innovative designs that would utilize rapid manufacturing processes. Instead, they have been evolving from the conventional design approaches for single use magnetic confinement experiments. The ARIES-AT is representative of advanced TF coil structures designs, as shown in Fig. 5. However, the ARIES-CS (Najmabadi 2008, Waganer 2008) compact stellarator coil structure is a very complex shape and would be very difficult to build with conventional fabrication processes. As a consequence, the ARIES-CS power plant study took an innovative approach and designed a monolithic TF coil structure for each of the three field

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periods. A continuous convoluted toroidal tube, shown in Fig. 6, supports the modular winding pack within grooves on the internal surface of the tube. The structure is an integral part of the coil casing. The tube is thickened on the sides and outside of the coils, providing additional structural material where needed. The part thickness between coils is appropriately sized for the local stresses and deflections. Thus, the volume and mass of the toroidal field coil structure is optimized precisely for the local EM forces. Additional tailoring of the structure is accomplished for the necessary access ports and support features. This tailoring and optimal use of materials also results in a lower cost. This coil structure is a monolithic assembly weighing approximately 1000 tonnes per one of the three field periods. The final part is constructed on site with multiple additive fabrication devices operating continuously for several months. Low temperature, long exposure heat treatment methods are used to remove residual stresses in the field period structure after the material deposition process. There is no need of thermal aging of the material for increased strength.

Figure 5. ARIES-AT TF coil structure.

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Figure 6. ARIES-CS magnet structure with coils embedded in internal grooves.

Most interior and exterior surfaces of these components will probably not require any finish machining as there are no mating surfaces or interface requirements. The few mating surfaces and interfaces will be final machined to the required finish and tolerance. The inner coil grooves are created by the additive fabrication process to near final dimensions. Automated milling machines will be guided by rails attached near the sides of grooves. These guide rails follow the contour of the grooves and the inner toroid surface and allow the milling machine to follow the guide rails. The machine path is determined by the CAD definition. Local fiducial reference marks or datums will achieve the desired accuracy of grooves. The near net shape of the groove structure will allow shallow machine cuts on the groove walls and floor with low milling forces. The same rails will be used to guide the winding and positioning of the insulated coil cables into the grooves. After the coils and insulation are positioned and secured, guided welding machines will weld cover plates over the coil cables. Technology advances might allow direct deposition of higher temperature superconducting materials directly in the coil grooves at some future time. Currently, direct deposition of the superconducting materials is not possible to produce continuous, thin superconducting filaments for HT magnets. The additive technique could be applied to non-superconducting magnets as well. In the design of the ARIES-ST spherical torus power plant (Najmabadi 2003), the central conductor is constructed of normally-conducting copper. It is a 30-m long cylinder comprised of an upper tapered section to electrically mate with the outer shell, a straight cylindrical section with 1.6 m diameter, and a larger cylindrical section at the bottom (3.35 m diameter) as shown in Fig. 7. This centerpost, weighing ~850 tonnes, is constructed of high-conductivity copper and cooled with 15% water in a single pass flowing through internal water passages. The conventional method of constructing the Cu centerpost assembly with internal water passages would be to fabricate cylindrical wedge segments with grooves for the coolant and then weld or hot isostatic press the entire assembly in a fixture to minimize distortion (Waganer 2003). This difficult and costly technique would involve labor-intensive welding and inspection. Obtaining a completely leak-tight assembly would be difficult. The unit cost

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to conventionally fabricate this part is well over $80/kg, thus this large assembly would likely cost $70M to $100M or more. This component must be replaced every three years due to activation considerations and this will impact the cost of electricity as a significant operating expense. The additive fabrication, laser forming process, described earlier, is ideally suited to construct the ARIES-ST centerpost. It is a simple geometry of cylinders and cones with multiple, continuous coolant passages from the top to the bottom of the part. An initial blank or preform will be used to start the process at the bottom. First, the lower 3.35-m-diameter cylinder, refer to Fig. 7, would be constructed with the integral coolant passages. Continuing upward, the laser heads would form the tapered cylinder with the coolant passages being transitioned to those in the central 1.6-m-diameter cylinder and the top tapered section. It is estimated that the 850 tonne centerpost could be completed in roughly eight months and cost $6.9M, or a fabricated unit cost of $8.09/kg (Waganer 2003). This value is only a few times the material cost of $2.86/kg. This significant cost reduction is possible because of the drastic reduction in labor in the fabrication process and minimal material wastage.

Figure 7. ARIES-ST power core illustrating the copper centerpost and aluminum outer shell of TF coil.

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The outer part of the ARIES-ST TF coils is the aluminum shell that could also be constructed with a different advanced fabrication approach. The shell has three distinct parts, all water-cooled. An upper shell extends from the top of the centerpost to the midplane, where it is connected to one of the power supply busbar leads. The middle shell extends from midplane, where the other busbar connection is made, down to a maintenance break. The removable third shell extends from the maintenance break down to the lower connection of the centerpost. Aluminum was chosen as the material for the outer return TF shell as it is conductive, lighter, easier to fabricate, and less expensive. Conceivably, the outer shell could be constructed with the same laser forming process as mentioned above. But because the weight of an aluminum TF shell is more than three times as much as the copper centerpost, the time and cost to construct would be correspondingly higher. Spray casting of a molten metal involves holding the metal just above the melting temperature, atomizing it, and spraying it onto a preform structure. This would enable all three shells to be spray cast in less than six months. Some final machining will be required locally at the flanges and joints with the interfacing hardware, such as the busbars and vacuum pump flanges. This process requires a preform structure to initiate the process. This is a desirable feature for the TF shell since the inner preform will serve as the power core vacuum vessel. This perform structure would be fabricated in the conventional manner. After the spray cast process commences, the vacuum vessel will be an integral part of the TF shell. The weight of the vacuum vessel and spray cast shell is 1560 tonnes, assuming 85% aluminum and 15% water. A wastage allowance of 5% is allowed for the spray cast process. It is not advisable for the water coolant to be in direct contact with the aluminum. Therefore, stainless steel tubes are embedded in the cast aluminum shell to contain and distribute the water coolant. As the shell is being fabricated, the tubes can be placed in position. The aluminum is spray-cast around the tubes, embedding them in the alumimum structure. Due to the size of the finished parts, it is anticipated that both shell and centerpost will be fabricated and inspected on the plant site (Waganer 2003).

V.3. Summary The definition of the advanced fabrication processes will be directly developed from the CAD definition. They are fast, automated, repeatable, and highly reliable processes. The approach used in the ARIES-CS stellarator example can be adapted for use on tokamak coil designs with significant improvement in material, time, and cost. The structural design will be more efficient as the material placement is tailored for the allowable structural stresses and deflections. Since tooling is minimized in this approach, design changes can be accepted at later stages in the design process with lesser cost or schedule impact. These innovative design and fabrication approaches are still being matured, but they have inherent design, cost, and schedule advantages over conventional approaches.

VI. Conclusion Superconducting magnets have been perceived as an essential element of magnetic fusion plasma confinement. The ability of the in-vessel components of any fusion device to protect

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its superconducting magnets and satisfy the radiation allowables draws heavily on the optimization process of these components. This process helps enhance the economics through minimizing the power core radial thickness and reducing the overall size of the machine. A number of past design studies indicated a minimum of one-meter thick, well-optimized invessel components could adequately protect the magnet against radiation and control the deposition of nuclear heating. The US power plant designs are strongly driven by economics as this parameter is felt to be essential to achieve a viable commercial product. This mandates new innovations in fabricating the magnet and other fusion components rather than relying on more costly, labor intensive approaches. The proposed additive fabrication technique offers significant cost savings with efficient use of time, labor, and materials. The magnets can be fabricated directly from the CAD definition. Moreover, adjustments can be made during the part build up process to accommodate final design changes. Such a fabrication technology should be matured on demonstration, pilot, and/or prototype plants to be applied on future fusion power plants with minimal technical and program risk.

Acknowledgements Partial funding support for this work came from the US Department of Energy, Office of Fusion Energy Sciences.

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References Arcam AB publications, Krokslätts Fabriker 27A, SE-431 37 Mölndal, Sweden. ARIES Project. The ARIES project: http://aries.ucsd.edu/ARIES/. Bromberg, 2001. L. Bromberg, M. Tekula, L. El-Guebaly, and R. Miller, “Options for the Use of High Temperature Superconductor in Tokamak Fusion Power Plant,” Fusion Engineering and Design 54 (2001) 167-180. Bromberg, 2007. L. Bromberg, J.H. Schulz, L. El-Guebaly, and L. Waganer, “High Performance Superconducting Options for ARIES Compact Stellarator,” Fusion Science and Technology 52, No. 3 (2007) 422-426. El-Guebaly, 1997. L.A. El-Guebaly, “Overview of ARIES-RS Neutronics and Radiation Shielding: Key Issues and Main Conclusions,” Fusion Engineering and Design 38 (1997) 139-158. El-Guebaly, 2006. L.A. El-Guebaly, “Nuclear Performance Assessment of ARIES-AT,” Fusion Engineering and Design 80 (2006) 99-110. El-Guebaly, 2008. L. El-Guebaly, P. Wilson, D. Henderson, M. Sawan, G. Sviatoslavsky, T. Tautges et al., “Designing ARIES-CS Compact Radial Build and Nuclear System: Neutronics, Shielding, and Activation,” Fusion Science and Technology 54, No. 3 (2008) 747-770. Fabian, 2002. P.E. Fabian, J.A. Rice, N.A. Munshi, K. Humer, H.W. Weber, “Novel Radiation-Resistant Insulation Systems for Fusion Magnets,” Fusion Engineering and Design 61-62 (2002) 795-799.

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Greenwald, 2007. M. Greenwald et al., “Priorities, Gaps and Opportunities: Towards A LongRange Strategic Plan For Magnetic Fusion Energy,” Report to the Fusion Energy Sciences Advisory Committee, page 70 (October 2007). Available at http://www.science.doe.gov/ofes/fesac.shtml Humer, 2006. K. Humer, K. Bittner-Rohrhofer, H. Fillunger, R.K. Maix, R. Prokopec, and H.W. Weber, “Innovative Insulation Systems for Superconducting Fusion Magnets,” Superconducting Science and Technology 19 (2006) 96-101. ITER Project. The ITER Project: http://www.iter.org/ Mitchell, 2006. N. Mitchell, “Quality Control in the Design, Fabrication and Operation of the ITER Magnets,” Fusion Engineering and Design 81 (2006) 2325-2339. Morra, 1992. M.M. Morra, R.G. Ballinger, and I.S. Hwang, “INCOLOY 908, a low coefficient of expansion alloy for high-Strength cryogenic applications: Part I. Physical metallurgy,” Metallurgical and Materials Transaction A 23, No. 12 (1992) 3177. Najmabadi, 1997. F. Najmabadi et al., “Overview of the ARIES-RS Reversed-Shear Tokamak Power Plant Study,” Fusion Engineering and Design 38 (1997) 3-25. Najmabadi, 2003. F. Najmabadi and The ARIES Team, “Spherical Torus Concept as Power Plants–the ARIES-ST Study,” Fusion Engineering and Design 65 (2) (2003) 143-164. Najmabadi, 2006. F. Najmabadi et al., “Overview of the ARIES-AT Advanced Tokamak, Advanced Technology Power Plant Study,” Fusion Engineering and Design 80 (2006) 323. Najmabadi, 2008. F. Najmabadi and the ARIES Team, “The ARIES-CS Compact Stellarator Fusion Power Plant,” Fusion Science and Technology 54, No. 3 (2008) 655-672. Nakajima, 2004. H. Nakajima, K. Hamada, K. Takano, and N. Fujitsuna, “Development of Low Carbon and Boron Added 22Mn-13Cr-9Ni-1Mo-0.24N Steel (JK2LB) for Jacket Which Undergoes Nb3Sn Heat Treatment,” IEEE Transactions on Applied Superconductivity 14 (2004) 1145-1148. Puigsegur, 2004. A. Puigsegur, F. Rondeaux, E. Prouzet et al., “Development of an Innovative Insulation for Nb3Sn Wind and React Coils,” Advances in Cryogenic Engineering ICMC 771 (2004) 266-272. Sawan, 1984. M Sawan, “Charts for Specifying Limits on Copper Stabilizer Damage Rate,” Journal of Nuclear Materials 122-123 (1984) 1376. Sawan, 1986. Mohamed Sawan and Peter Walstrom, “Sperconducting Magnet Radiation Effects in Fusion Reactors,” Fusion Technology 10/3 (1986) 741-746. Waganer, 2003. L.M. Waganer, D.A. Deuser, K.T. Slattery, G.W. Wille, F. Arcella, B. Cleveland, “Ultra-Low Cost Coil Fabrication Approach for ARIES-ST,” Fusion Engineering Design 65, No. 2 (2003) 339-352. Waganer, 2008. Lester M. Waganer, Kevin T. Slattery, John C. Waldrop III, and the ARIES Team, “ARIES-CS Coil Structures Advanced Fabrication Approaches,” Fusion Science and Technology 54, No. 3 (2008) 878-889. Wang, 2008. X.R. Wang, A.R. Raffray, L. Bromberg, J.H. Schultz, L.P. Ku, J.F. Lyon, S. Malang, L. Waganer, L. El-Guebaly, and C. Martin, “ARIES-CS Magnet Conductor and Structure Evaluation,” Fusion Science and Technology 54, No. 3 (2008) 818-837.

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In: Superconducting Magnets and Superconductivity… ISBN: 978-1-60741-017-1 Editors: H. Tovar and J. Fortier, pp. 293-399 © 2009 Nova Science Publishers, Inc.

Chapter 9

THERMAL STABILITY CHARACTERISTICS OF HIGH TEMPERATURE SUPERCONDUCTING COMPOSITES V.R. Romanovskii1 and K. Watanabe2 1

2

Russian Research Center ‘Kurchatov Institute’, Moscow 123182, Russia High Field Laboratory for Superconducting Materials, Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan

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Abstract The thermal stability characteristics of the high-Tc superconducting composites like multifilamentary current-carrying elements of superconducting magnets are discussed in the framework of the macroscopic continuum approximation. The performed analysis was based on the zero- and one-dimensional static and transient thermo-electric models. Various types of the voltage-current characteristic of a superconductor are considered. The thermal runaway conditions of the superconducting composite are investigated under the conditions cooled by cryocoolers with various operating temperatures, liquid helium or hydrogen coolants. The linear and nonlinear temperature dependences of the critical current were used. As a result, the evolution peculiarities of the stable and unstable thermal and electric modes as a function of sweep rate, volume fraction of the superconductor in a composite, and its cross section are formulated for the partially and fully penetrated states under the different cooling conditions. It was shown that permissible stable values of the current and electric field might be both lower and higher than those determined by use of the standard critical current criterion. The reasons leading to these regimes are discussed. Consequently, the unavoidable temperature rise of the superconducting composite before its transition to the normal state takes place. The latter depends on a broad shape of the voltage-current characteristic of the high-Tc superconductor and the current sharing between a superconducting core and a matrix. In the limiting case, a stable value of the temperature of a composite may equal the critical temperature of a superconductor. For these operating modes, the criterion of the complete thermal stability condition is written when the charging current will flow stably only in a matrix. It is also validated that there exists the thermal degradation mechanism of the currentcarrying capacity of a superconducting composite. According to this mechanism, the quench currents do not increase proportionally to the increase of the critical current of a composite. The performed analysis reveals also the connection between different criteria used to determine the thermal runaway conditions. In the framework of the nonlinear temperature dependence of the critical current, it is shown that the operating regimes may have manyvalued stable and unstable branches appearing in accordance with the nontrivial variation of

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V.R. Romanovskii and K. Watanabe the differential resistivity of a composite. These states exist, first of all, due to the temperature change of the quantity ∂Jc/∂T and are accompanied by the jump-like current-sharing mechanism. The thermal runaway parameters are numerically derived as a function of operating temperature accounting for the additional stable branches of the voltage-current characteristics. The formation peculiarities of these phenomena are discussed.

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1. Introduction A magnetic field, which is one of the important physical parameters, can change the thermodynamic energies of materials. The materials such as semiconductors, metals and superconductors except insulators show the clear responses to a magnetic field. Physical properties of magnetoresistance, Hall effect, cyclotron resonance, de Haas-Van Alfen effect and so on are obtained in magnetic fields. Recently, high magnetic fields up to 20 T have been easily obtained by a superconducting magnet, and even a conduction-cooled superconducting magnet is now available in fields up to 18 T for a long term experiment without any labour for charging liquid helium [1]. The conduction-cooled superconducting magnet has opened a new field in science, especially in the material processing technology. A high field technique is progressing rapidly and static high fields over 20 T are required for research of high-Tc superconductors. High-Tc superconductors offer the truly new phenomena and the important characteristics. The critical temperature of high-Tc superconductors is very high in comparison with low-Tc superconductors like NbTi and Nb3Sn, and, as a result, the coherence length of high-Tc superconductors is very short. Since the fluctuation related with the short coherence length extremely increases at high temperature, it is very difficult to observe the clear superconducting phase transition at liquid nitrogen temperature due to the thermal effect. In addition, high-Tc superconductors exhibit large anisotropy in the superconducting properties, because the electrical conduction plane is two-dimensional due to the crystal structure. These features are disadvantageous from a viewpoint of the superconducting application. The practical applications using high-Tc superconductors are the most promising for the superconducting technology. In order to develop high-Tc superconductors, the critical surface consisting of the critical current density, critical field, and critical temperature have to be determined. That is why the power application of superconductors is carried out within the critical surface. The high-Tc superconductors maintain their superconducting properties in very high fields exceeding 30 T at low temperature [2]. Although a conduction-cooled superconducting magnet consisting of high-Tc superconducting current leads has been developed, a high-Tc superconducting magnet is not practical yet. A high-Tc superconducting magnet is a representative demonstration of a practical superconducting power application. The static high magnetic fields over 30 T will push the frontier research forward, and it is expected that a new phenomenon will be discovered by the extreme condition combined with high pressure and ultra low temperature in high fields. A long length superconducting wire is necessary for fabricating a practical superconducting magnet. It is an important technical issue to know how a high-Tc superconductor can be manufactured as a long length wire and be wound in a coil shape. Fortunately, km-length high-Tc superconducting wires of Bi2Sr2CaCu2O8 (Bi2212) and Bi2Sr2Ca2Cu3O10 (Bi2223) can be fabricated by use of a powder-in-tube (PIT) method. Until now, a primitive design of a high field superconducting magnet which can exceed 20 T at 4.2 K and a conduction-cooled superconducting magnet operating at 20 K started to be

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demonstrated by using Ag-sheathed Bi-system high-Tc superconducting wires. Moreover, high-Tc superconducting bulk materials are already used for actual current leads, and in particular, Institute for Materials Research, Tohoku University has firstly demonstrated a practical conduction-cooled superconducting magnet using high-Tc superconducting current leads [3]. In addition, high-Tc bulk materials are expected for bulk magnets such as a magnetic levitation carrier device, a magnetic bearing, a flywheel, and a motor. Recently, we have demonstrated that CVD-processed YBa2Cu3O7 (Y123) high-Tc superconductors can surely overcome huge hoop stress over 1000 MPa using a 250 mm widebore test coil in an external magnetic field of 11 T [4]. From these results, we intended to design a 30 T superconducting magnet as compactly as possible, consisting of an Y123 insert coil. An Y123 insert coil will add 16 T at a 236 A operation current in the 14 T background field. It is found that a 30 T – 52 mm room temperature bore superconducting magnet with coil parameters of 80 mm inner diameter, 860 mm outer diameter, 1000 mm coil height and a magnetic stored energy 32 MJ can be sufficiently constructed. It is possible to fabricate a 16 T Y123 insert coil with a size of 80 mm inner diameter and 410 mm outer diameter, employing 5 mm wide and 33 km long Y123 coated conductor tapes. On the other hand, the critical current may be determined by using a resistivity criterion of 10-11 Ω·cm, for instance. This value is three orders of magnitude smaller than the resistivity for copper for which ρ = 10-8 Ω·cm at 4.2 K. That is why the resistivity criterion of 10-11Ω·cm is adopted as almost zero resistance for measurements [5]. Since the phase transition from the superconducting to the normal state is very sharp for conventional low-Tc superconductors, the full critical current cannot be utilized stably for practical applications. In the case of lowTc superconductors, the 90 % value of the critical current lies in a very low resistivity region below 10-14 Ω·cm, which corresponds to further three orders of magnitude as low as the critical current criterion. Namely, this means that the superconducting power application is established in a quite low resistivity state of less than 10-14Ω·cm. The phase transition from the superconducting to the normal state for conventional practical superconductors with copper stabilizer occurs promptly, and the resistivity change at 4.2 K is ranging from below 10-14Ω·cm to 10-8Ω·cm. The high-Tc superconductors have attracted a strong attention due to the high critical temperature above liquid nitrogen for practical use. However, some problems encountered for high-Tc superconducting power applications remain unsettled. In particular, the thermal stability investigation of a superconducting composite (superconducting filaments inside the normal metal matrix) is one of the most fundamental issues. This research is important for the characterization of the operating performances of the superconducting magnets and the understanding of the mechanisms limiting their capability. Therefore, the increasing use of high-Tc superconductors for many practically important applications has generated much interest in this study. The instabilities in superconducting magnets may be caused by perturbations of different nature [6]. Despite rather high thermal stability of high-Tc superconducting composites with respect to the external thermal disturbances, the current instability, known as thermal runaway phenomenon, may be observed when a magnet is operated under the high current load [7-15]. This phenomenon is practically similar to the quench process in low-Tc composite superconductors and may be accompanied with their irreversible temperature rise up to the critical temperature.

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To estimate the limiting currents that can stably flow in a superconductor without its transition into the normal state, the voltage-current characteristics are often used. Consequently, one measures the voltage-current characteristic (VCC) and then defines the critical current density Jc at a given operating temperature and applied magnetic field. The determination of Jc-value may be based on various criteria. As a rule, this quantity is defined by a fixed electric field criterion. This technique is based on Bean’s model, which omits the nonlinear part of the voltage-current characteristic. It is a good approximation for low-Tc superconductors with sufficiently steep voltage-current characteristics. However, the voltagecurrent characteristics of high-Tc superconductors have a broad shape. The nonlinear dependence of the electric field E on the current density J of high-Tc superconductors is owing to many reasons: pinning heterogeneity, vortex structure defects, thermal activation of flux, etc. There exist various theoretical models explaining the observed voltage-current characteristics of high-Tc superconductors. In particular, the phenomenological power and exponential equations are extensively used to describe the macroscopic electromagnetic properties of high-Tc superconductors. Numerous studies (see, for example, [16-24]) show that the following equations E=Ec(J/Jc)n

(1)

E=Ecexp[(J-Jc)/Jδ]

(2)

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and

can be used in many practical cases. Here, Jc is the current density at the a priori defined critical electric field Ec (in the framework of our investigation this quantity is defined as Ec=10-6 V/cm); n is the creep exponent of VCC; Jδ is the creep current density. As known, the power equation corresponds to a logarithmic current dependence of the potential barrier when the flux creep is determined by numerous spatial defects of the superconductor. The thermally activated model with a linear current dependence of the potential barrier lies at the basis of the exponential VCC. This model describes the flux-creep state of the superconductor with point defects of its structure. There are also some macroscopic reasons leading to an exponential form of the VCC. In particular, it may result from the bulk heterogeneity of superconducting properties inside the sample. In addition to the bulk heterogeneity of critical parameters the superconductor may have the longitudinal heterogeneity. However, the VCC of such superconductors are also approximated satisfactorily by the power equation. Using the voltage-current characteristics described by equations (1) or (2), it was proved that the macroscopic electrodynamics behavior of high-Tc superconductors has some particular peculiarities that cannot explain in the framework of Bean’s model (see, e.g., [2531] and references cited in therein). Moreover, it appears that the Bean model does not allow, in principle, to explain the current instability phenomenon even in low-Tc superconductors during fully penetrated current regimes [21]. Therefore, below, we discuss the key static features that are initially characteristic for the thermal runaway problem, and formulate the general criteria indicating the influence of the parameters of a superconductor and a matrix on its stable thermal and current properties. Various types of high-Tc superconductors have been developed. At present, the most promising high-Tc superconductors are the Bi-based cuprates. In particular, Bi2Sr2CaCu2O8 is

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already superior to low-Tc superconductor at low cooling bath temperature, which allows one to make a new generation of superconducting magnets that are the conduction-cooled highfield magnets [32, 33]. From this point of view, the peculiarities of the stable regimes of Agsheathed Bi2Sr2CaCu2O8 composite will be discussed below considering with the voltagecurrent characteristics described by equations (1) or (2).

2. Sub- and Overcritical Static Stable Regimes of Superconducting Composites with Different Voltage-Current Characterictics 2.1. Static Zero-Dimensional Thermo-electric Model In general, the evolution of the temperature and electric field inside the superconducting magnet obeys the multi-dimensional Fourier and Maxwell equations, respectively [34-38]. Such approximation allows one to account for the geometry of a conductor, location of the superconducting filaments, nonlinear temperature-field-dependent properties of a superconductor and a matrix, etc. But this description is mathematically sophisticated. To understand, first of all, the basic physical peculiarities of the thermal runaway problem and to evaluate the stability conditions without the large volume of computations, let us consider an infinitely long superconducting composite. In order to simplify the analysis, let us assume that −

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− − − −

− − −

the applied magnetic induction B is constant and the twist pitch of a superconductor is not very small; the longitudinal magnetic field variation is negligible; the applied current is changed at an infinitely low sweep rate (dI/dt→ 0) in the fully penetrated regime and its self-field is less than the external magnetic field; the size of the superconducting filament is relatively small and magnetic instability is absent; the superconductor is evenly distributed over a cross section of a composite with the volume fraction η (0 < η < 1) and the macroscopic continuum estimate can be applied; the conduction heat exchange between the composite and the refrigerator occurs on the surface; the transverse conduction heat flux essentially exceeds the heat flux to the coolant; the voltage–current characteristic of a superconductor may be described by equations (1) or (2).

Under this static approximation, the distribution of the electric field E, current density J, and temperature T in the cross section of a composite are approximately uniform. Then the heat balance equation with the Kirchoff and Ohm laws describe these quantities. Therefore, in terms of the zero-dimensional model [6], the temperature of the superconducting composite is given by equation

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EJ =

hp (T − T0 ) S

(3)

Here, h is the heat transfer coefficient; p is the cooling perimeter; S is the cross section of a composite; T0 is the operating temperature. The total transport current density is equal to the sum of currents in the superconducting core Js and matrix Jm and is defined as follows

J = η J s + (1 − η ) J m

(4)

The steady electric field is generated by the parallel circuit on a superconducting core and a matrix according to the relations n

⎡ Js ⎤ E = Ec ⎢ ⎥ = J m ρ m (T , B ) ⎣ J c (T , B) ⎦

(5)

for the superconductor with the power VCC and

⎡ J − J c (T , B) ⎤ E = Ec exp ⎢ s ⎥ = J m ρ m (T , B ) Jδ ⎣ ⎦

(6)

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for the superconductor with the exponential VCC. Here, ρm is the matrix resistivity. The critical current density of a superconductor depends on the temperature and magnetic field. To describe Jc-dependence let us consider the following relations. First, to simplify analysis, let us use the well-known temperature linear approximation

J c (T , B) = J c 0 ( B)

TcB ( B) − T TcB ( B) − T0

(7)

Here, the current density Jc0 and temperature TcB are the constant at a given value of the applied magnetic induction. Second, let us calculate the critical current density of Bi-based superconductors according to the nonlinear model proposed in [39] γ

⎛ T ⎞ ⎡ ⎛ ⎞⎤ B0 βB + χ exp ⎜ − J c (T , B) = J 0 ⎜ 1 − ⎟ ⎢ (1 − χ ) ⎟⎥ B0 + B ⎝ Tc ⎠ ⎣ ⎝ Bc 0 exp(−α T / Tc ) ⎠ ⎦

(8)

summarizing the results presented in [40, 41]. This formula considers the huge flux-creep states of Bi-based superconductors in high magnetic fields, which leads to the strong temperature degradation of the critical current. Formula (8) may be used also to calculate the effective values of Jc0 and TcB by the linear fitting to the corresponding nonlinear curves Jc(T,B). As an illustration, Figs.1 and 2 show the corresponding comparison between our

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experimental data and calculations, which were made for Ag/Bi2212 tape having S=0.012 cm2 and η=0.263. The next constants Tc=87.1 K, J0=1.1×106 A/cm2, Bc0=465 T, B0=0.0075 T, α=10.3, β=5, γ=1.73, χ=0.2

(9)

were used, which were adapted from our experiment and data presented in [35]. The resistivity of silver as a function of the temperature and the magnetic field (Kohler’s rule) was approached using the dependences proposed in [42, 43]. The characteristic values of the residual resistivity ratio RRR=ρm(273 K)/ρm(4.2 K) were used during the simulations at ρm(273K)=1.48×10-6 Ω⋅cm according to [42]. The typical temperature dependences of silver resistivity are presented in Figure 3.

2.2. Static Thermal Runaway Conditions and the Current Sharing Effect on the Stationary Fully Penetrated Current Regimes Let us reduce equations (3)-(6). They may be rewritten as

1 −η ⎤ ⎡ ⎢ J − E ρ (T , B ) ⎥ m ⎥ E = Ec ⎢ ⎢ η J c (T , B ) ⎥ ⎢⎣ ⎥⎦

n

(10)

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for a superconductor with the power VCC and

1 −η ⎡ ⎤ ⎢ J − ρ (T , B ) E − η J c (T , B) ⎥ m ⎥ E = Ec exp ⎢ η Jδ ⎢ ⎥ ⎢⎣ ⎥⎦

(11)

for a superconductor with the exponential VCC. These equations are transformed to the simple formulas when the linear Jc-equation (7) is used and assuming that ρm(T,B)≈ρm(T0,B)=const. The latter approximation is reasonable in a temperature range up to 20 K (Figure 3). In this case, eliminating temperature from equations (10) or (11), the relevant dependence of the current flowing in a composite on the electric field is expressed by the following analytical expressions 1/ n

⎛ E⎞ 1 −η η Jc0 ⎜ ⎟ + E Ec ⎠ ρm ⎝ J= 1/ n η J c 0 SE ⎛ E ⎞ 1+ ⎜ ⎟ hp(TcB − T0 ) ⎝ Ec ⎠ for a superconductor with the power VCC and

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(12)

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E 1 −η + E ρm Ec η J c 0 SE 1+ hp (TcB − T0 )

η J c 0 + η J δ ln J=

(13)

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Figure 1. Critical currents of Ag-sheathed Bi2Sr2CaCu2O8 superconductor versus applied magnetic field: (x) - experiment, (⎯⎯) - fit calculations

Figure 2. Temperature dependence of the critical current of a Ag-sheathed Bi2Sr2CaCu2O8 at B=10 T: (+) - experiment, (⎯⎯) - fit calculation, (- - - -) - linear approximation

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Figure 3. Resistivity of silver as a function of temperature.

for a superconductor with the exponential VCC. They may be rewritten as

( E / Ec ) + E / E1 , J = η J c0 1/ n 1 + ( E / E2 )( E / Ec ) 1/ n

J = η J c0

1 + ( J δ / J c 0 ) ln ( E / Ec ) + E / E1 1 + E / E2

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for the superconductor with the power and exponential voltage-current characteristics, respectively. Here,

E1 =

η J c 0 ρm hp(TcB − T0 ) , E2 = η Jc0 S 1 −η

which are the characteristic values of the electric field rooted the VCC-formation of a superconducting composite. Indeed, the main part of the applied current stably flows in the superconducting core (ηJs >> (1-η)Jm) under the condition of

E 0 are described by the following inequalities 1−

2

⎛ η Jc0 ⎞ ⎡ ⎤ S ⎜ 1/ n ⎟ ⎢ ⎥ ⎝ Ec ⎠ ⎣ hp(TcB − T0 ) ⎦

1 n

1+

⎛ ρm ⎞ ⎜ ⎟ ⎝ 1 −η ⎠

1 n

< 1

and

1−

Jδ Jc0


1 takes place in many experiments. Therefore, it corresponds to such thermal states of superconducting composite at which its quench overheating may occur due to the temperature rise of a superconductor. Thus, the formulas (14) - (19) allow one to get the estimates of the current sharing and temperature effects during the stable increase of the applied current. However, they do not describe the boundary of the stable and unstable current regimes. To define the thermal runaway parameters, let us find the necessary dependences, which allow one to analyze the

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slope of its voltage-current and temperature-current characteristics. According to (12) and (13), they are equal to 1/ n E1 ⎡ E ⎛ E⎞ ⎤ ⎢1 + ⎜ ⎟ ⎥ η J c 0 ⎢ E2 ⎝ Ec ⎠ ⎥ ⎣ ⎦

∂E = ∂J 1+

1 n

2

2 n

E1 ⎛ E ⎞ E1 ⎛ E ⎞ E ⎛ E⎞ ⎜ ⎟ − ⎜ ⎟ − ⎜ ⎟ nE ⎝ Ec ⎠ E2 ⎝ Ec ⎠ nE2 ⎝ Ec ⎠

1 n

(20)

for the superconductor with the power VCC and

∂E = ∂J

E1 ⎛ E ⎞ ⎜1 + ⎟ η J c 0 ⎝ E2 ⎠

2

J E E ⎛ J J E⎞ 1 + δ 1 − 1 ⎜1 − δ + δ ln ⎟ J c 0 E E2 ⎝ J c 0 J c 0 Ec ⎠

(21)

for the superconductor with the exponential VCC. Besides, the following connection between ∂E/∂J and ∂T/∂J takes place

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∂T S ⎛ ∂E ⎞ = ⎜E+ J ⎟ ∂J hp ⎝ ∂J ⎠

(22)

It follows from equation (3). These formulas indicate that the quantities E(J) and T(J) may have both positive and negative values in the dependence on the sign of the denominator. Consequently, in the static zero-dimensional approximation under consideration, the voltage-current and temperaturecurrent characteristics of a superconducting composite with positive derivatives correspond to the stable regime and their negative values determine unstable states. Therefore, the boundary between the stable and unstable operating regimes determines by the condition

∂Ε/∂J→∞

(23)

as it was formulated for the first time in [20] for low-Tc superconductors. According to the (22), the stability condition may be formulated also as

∂Τ/∂J→∞

(24)

They define the limiting current-carrying capacity of a superconducting composite and the quench temperature before the thermal runaway in the static approximation at the given applied magnetic field, cooling condition, etc.

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Using equation (4) one can write the connection between the differential resistivity of the composite and the differential resistivity of the superconducting core described by the quantity ∂Ε/∂Js. It has the form

∂E 1 ∂E / ∂J s = ∂J η 1 + ∂E 1 − η ∂J s ρ m

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This relation allows one to understand the effect of the matrix on the current distribution in a superconducting composite.

Figure 4. Effect of the cooling conditions and matrix resistivity on the stable and unstable formation of the static voltage-current and the temperature-current characteristics: 1- RRR=1000, h=1 W/(cm2K); 1' - RRR=10, h=1W/(cm2K); 2-RRR=1000, h=10-2W/(cm2K); 3 - RRR=1000, h=10-3W/(cm2K); 4 RRR=10, h=10-3W/(cm2K)

Firstly, the operating regime of the superconducting composite may be stable even when the current flowing in the superconducting core of the composite is not stable. Indeed, one can see that the inequality ∂Ε/∂J > 0 may exist when ∂Ε/∂Js < 0. During these states the current sharing will be an intensive. They satisfy the relation

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∂J s 1 = 0, ∂Τ/∂J>0) both without an irreversible transition of the superconductor into the normal stated (curves 1, 1' and 2) and before the superconducting-to-normal transition (curves 3 and 4). The calculations were made at the effective cooling condition close to the perfect cooling (curves 1 and 1') and at the heat transfer coefficient that close to conduction-cooled conditions. The corresponding temperature curves shown in the insets have both stable (∂Τ/∂J>0) and unstable (∂Τ/∂J Ec) at the conduction-cooled condition. Figure 5 presents the current redistribution modes that occur inside the composite with the VCC depicted in Figure 4 by curves 1 and 3. They refer to the current flowing in the superconducting core and low resistivity matrix as a function of the total current for two limiting cooling conditions. Under the "good" cooling condition (Figure 5(a)), the thermal runaway is absent and the current sharing has the stable character. At "bad" cooling condition (Figure 5(b)), the thermal runaway happens because the current sharing has the unstable behavior according to the conditions (25) and (26). Comparing Figs. 4 and 5 it is easy to see that below the critical current Ic defined by the magnitude of Ec=10-6V/cm the main part of the transport current flows in the superconductor and the temperature of the composite close to the cooling temperature. When the current exceeds Ic, the noticeable part of the transport current starts to flow in the non-superconducting matrix. As a result, the current flowing in the matrix may be equal to the current in the superconducting core. Moreover, the superconducting composite may stably operate at the current far exceeding Ic. In these overcritical modes, the heating of the superconducting core will play an important role in the formation of the modes preceding the thermal runaway. The relevant temperature rise of the

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composite describes by formulae (18). If the cooling condition is "bad", then the influence of the temperature on the current repartition is a visible. Accordingly, the Joule losses inside the composite cause an intensive increase of its temperature at ∂Ε/∂Js < 0 that decreases the current in the superconductor and results in the thermal runaway (Figure 5(b)).

Figure 5. Current sharing in a superconducting composite at RRR=1000 during the stable mode (a h=1 W/(cm2K) ) and before and after thermal runaway (b - h=10-3W/(cm2K)).

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Figure 6 indicates the peculiarity of the thermal runaway onset in the superconducting composite conditioned by change of the differential resistivity of the superconducting core. Solid curves are calculated according to equation (12) and display the stable and unstable parts of the voltage-current characteristics of the composite with the high (curves 1) and low (curves 2) resistivity of the matrix. Dashed curves correspond to the E(Js)-dependences. The boundaries between stable and unstable states are marked. They denote the points on the unstable parts of E(Js)-traces describing the corresponding negative values of the differential resistivity of superconducting core. As mentioned above, although the superconducting core becomes unstable the superconducting composite may be stable over a long period of the current charging. These states existing due to the current sharing mechanism depending on the matrix resistivity are readily illustrated by Figure 5.

Figure 6. Formation of the stable and unstable parts of E(J) (⎯⎯) and E(Js) (− − − −) dependences at h=10-3W/(cm2K): 1 - RRR=10, 2 - RRR=1000.

2.3. Qualitative Analysis of the Thermal Runaway Conditions Thus, the specific current exists after which the irreversible transition of a superconductor into the normal state will occur. It is not equal to the corresponding value of the critical current. Under the conditions (23) or (24), it is easy to find that within the framework of the temperature-linear Jc-model defined by equation (7) and assuming that ρm(T,B)≈ρm(T0,B)=const the boundary of the stable current regimes are determined by the following quench values of the electric field Eq, current Iq and temperature Tq, which are described by expressions

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η J c0 n Ec1/ n

1− n n q

E

309

2

2 1+ n ⎤ ⎛ J c0 ⎞ S 1 − η ⎡η J c 0 S n Eq = Eq n − 1⎥ − ⎜η 1/ n ⎟ ⎢ 1/ n ρ m ⎣ n Ec hp(TcB − T0 ) ⎝ Ec ⎠ hp(TcB − T0 ) ⎦ J 1 −η η 1/c 0n Eq1/ n + E Ec ρm q Iq = S, (28) n +1 Jc0 ηS Eq n 1 + 1/ n Ec hp (TcB − T0 )

1 − η Ec1/ n ( n −1) / n E ρm J c 0 q Tq = T0 + (TcB − T0 ) hp(TcB − T0 ) Ec1/ n − ( n +1) / n Eq η+ S J c0

η+

for a superconductor with the power VCC and

1+

η J c 0 SEq ⎛ J c 0 E ⎞ 1 −η Eq = − 1 + ln q ⎟ ⎜ η Jδ ρm hp (TcB − T0 ) ⎝ J δ Ec ⎠ Iq =

⎞ hp (TcB − T0 ) Jδ ⎛ 1 −η Eq ⎟ ⎜1 + J c 0 ⎝ η Jδ ρm ⎠ SEq

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Tq = T0 +

(29)

⎞ Jδ ⎛ 1 −η Eq ⎟ (TcB − T0 ) ⎜1 + J c0 ⎝ η Jδ ρm ⎠

for a superconductor with the exponential VCC. Let us introduce the dimensionless variables εq=Eq/Ec, iq=Jq/(ηJc0), θq=(Tq-T0)/(TcB-T0). Then the relations (28) and (29) may be rewritten as

1 εq nε 2

1+

1 n

+

ε1 ε ε − 1ε n ε2 2 n q

1 −1 n q

= 1, iq =

ε q1/ n + ε q / ε1 1+

1 n

1+

, θq =

1+ εq / ε2

1

ε q n + ε q2 / ε1 1+

1 n

(30)

εq + ε2

εq ε ε ε ε + δ = q (1 − δ + δ ln ε q ), iq = δ 2 + 2 , θ q = δ + q ε1 ε2 ε q ε1 ε1

(31)

for superconductors with the power and exponential voltage-current characteristics, respectively. Here,

δ=

Jδ E E , ε1 = 1 , ε 2 = 2 Jc0 Ec Ec

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These formulas show that the allowable temperature rise of a superconducting composite before the thermal runaway, which is absent in the Bean approximation, is always finite because there exist two mechanisms leading to its unavoidable overheating. Firstly, it is a broad shape of the VCC of the high-Tc superconductors: the lower n or higher δ, the higher the temperature rise of a composite. Secondly, the temperature of a composite depends on the current sharing, which is related to the superconductor and matrix properties. The latter is described by the dimensionless parameter ε1: the higher this value, the lower the current flowing in a matrix and the lower the stable temperature of a composite. Therefore, the formation of stable current modes is a result of relevant collective electric and thermal behavior of a superconductor and a matrix. Using formulas (30) and (31), it is easy to find the conditions describing the boundary between stable values of the electric field and current, which might be lower (sub-critical regimes) or higher (over-critical stable regimes) than those determined by the critical voltage criterion. Let us put εq=1. Then the boundary between the sub-critical and over-critical values of the quench electric field is defined by the quantity

⎧⎪(1 + nε1 ) / ( ε1 + n ) ε2 = ⎨ ⎪⎩ε1 (1 − δ ) / (1 + δε1 ) for superconductors with the power or exponential voltage-current characteristics, respectively. Under this value of the quantity ε2, the relevant quench currents is equal to

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1 ⎧ n + ⎪ iq = ⎨ n + 1 (n + 1)ε1 ⎪1 − δ ⎩ i.e., they are less then the critical current of a superconductor. Therefore, the allowable values of the electric field and the current before the thermal runaway are sub-critical (Eq < Ec, Iq < Ic), if the operating parameters are satisfied the inequality

η J c 0 Ec S hp(TcB − T0 )

>

ηρ m J c 0 + n(1 − η ) Ec nηρ m J c 0 + (1 − η ) Ec

for a superconductor with the power VCC and

η 2 J c20 Ec S ρ m hp(TcB − T0 )

>

ηρ m Jδ + (1 − η ) Ec 1 − Jδ / J c0

for a superconductor with the exponential VCC. Using the thermal stability parameter, the latter inequalities may be rewritten as

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α > 1+

311

ε12 − 1 1 + ε1 δ , α > 1+ ε 1n + 1 1− δ

for superconductors with the power and exponential voltage-current characteristics, respectively. Taking into consideration that ε1 >>1 these criteria will have the form

α > 1+

ε1 n

, α > 1 + ε1

δ 1− δ

So, there exists the character value of the thermal stability parameter, which will lead to the sub-critical operating regimes. The latter depends on the shape of the VCC. If these conditions are not valid, then the quench electric field exceeds the voltage criterion Ec. However, in these cases, the quench current may be both sub-critical and overcritical. The later (iq > 1) exist, if εq > εf, where the value of εf satisfies the relation

ε1 = ε 1/f n + ε f − αε 1f+ (1/ n ) for a superconductor with the power VCC and equals

εf =

ε1δ α −1

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for a superconductor with the exponential VCC. Consequently, the possible stable operating modes of the high-Tc superconducting composites will have the over-critical values of the quench electric field and the sub-critical values of the quench current when 1 < εq < εf and

η J c 0 Ec S hp (TcB − T0 )


1. These sub-critical modes depending on the quantity ε2 are characterized by the finite quench temperature because θq→ 0 only at n→ ∞ and δ → 0 when εq→ 0 and iq→ 1. This overheating of a superconductor is an unavoidable, first of all, due to a broad shape of the voltage-current characteristic, as discussed above, and becomes noticeable when voltagecurrent characteristics correspond to the strong flux-creep states (n < 10, δ > 0.1) even at subcritical regimes. This result shows that the sample may have stable overheating and the current modes may be unstable below critical point {Ec, Jc0} during measurements of the critical currents. The importance of this conclusion should be emphasized because it is usually believe that the temperature of a sample equals the cooling bath temperature and the operating regime of a composite is stable as the charging current and induced voltage do not exceed the boundary defined by the parameters {Ec, Jc0} used. At the same time, the proper current-carrying capacity of a composite may not satisfy such experimental observation for the reason that it is sub-critical. The effect of the parameter ε2 on the sub-critical and over-critical regimes is depicted in Figure 8. It indicates the existence of two characteristic regions of the quench currents. First, the iq(ε2)-dependence has an area where the sub-critical quench currents depend essentially on the increasing ε2. These current modes are observed at small quantities of ε2. In other words, this peculiarity will be detected at non-intensive cooling conditions or when a superconducting composite is massive. Besides, as calculations show, this iq-value range depends also on the matrix resistivity. It is larger when a matrix has lower resistivity. Second, the quench current has an area where their values change weakly with increasing quantity ε2. Under these regimes, the operating modes become over-critical. They will be observed at intensive cooling conditions or in the case of composite with a small cross section. In the limiting case, which is characterized by high voltage occurring at

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Thermal Stability Characteristics of High Temperature Superconducting Composites

Figure 7. Continued on next page.

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Figure 7. Sub-critical thermal runaway boundary of the electric field (a), current (b) and temperature (c) versus smoothness parameters.

⎧⎪ n−1 ε q >> ⎨ε1 ⎪⎩ε1δ Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

n

the observed steep electric field rise can be estimated as n ⎧ 2 ⎛ ⎞ ε ⎪ 2 ⎜ ⎟ ⎪ ε ε q ~ ⎨⎝ 1 ⎠ ⎪ ⎞ 1 ⎛ ε2 ⎪exp ⎜ + δ − 1⎟ δ ⎝ ε1 ⎠ ⎩

(34)

for the superconductor with power or exponential voltage-current characteristics, respectively. This sharp allowable increase of the electric field will lead to the corresponding high temperature rise of a composite that may affect its differential resistivity and determination of the stability condition. This peculiarity is discussed below. As a whole, the written estimates and Figs. 7 and 8 prove that the superconducting composites with the power and exponential voltage-current characteristics will have practically the same current stability conditions in the weak creep range (n > 10, δ < 0.1). In the meantime, the difference increases with decreasing quantity ε2 due to the relevant

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temperature effect on the operating current modes of a composite, as discussed above and follows from equations (28) and (29). As a result, the superconducting composite with power VCC is more stable than that with exponential VCC during sub-critical regimes. However, this tendency depends on the quantity ε2 (Figure 8).

Figure 8. Quench electric field (a) and current (b) as a function of dimensionless electric field ε2 at n=1/δ=10: ——— - exponential VCC; - - - - power VCC.

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2.4. Quantitative Analysis of the Current Instability Conditions at Low Operating Temperature (T0=4.2 K)

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Let us discuss the possible variation of the thermal runaway conditions of the Agsheathed Bi2Sr2CaCu2O8 superconductor under various operating parameters for the regimes close to the conduction-cooling power. Figures 9-11 show the quench values of the electric field, temperature and current as a function of the heat transfer coefficient, resistivity of a matrix and volume fraction of a superconductor in a composite with the power VCC in the field of 10 T. In the calculation, the used parameters were set according to the relationship (27).

Figure 9. Continued on next page.

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317

Figure 9. Quench temperature (a), electric field (b) and current (c) versus heat transfer coefficient: 1 RRR=1000; 2 - RRR=100; 3 - RRR=10; - - - - η=0.2; ——— - η=0.5.

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As follows from Figure 9, the quench parameters increase practically linearly with increasing heat transfer coefficient in the range under consideration. However, the slope of the Iq(h)-curves depends on the matrix resistivity and volume fraction of a superconductor. It is higher when RRR is higher or η is smaller. Therefore, the influence of the cooling condition on the thermal runaway parameters will be larger for a smaller amount of a superconductor in composite, when its matrix has low resistivity. In this case, the allowable overheating of a superconductor has a high value and Tq may approach TcB. As a result, the quench electric field may be very high (Eq > 10-5 V/cm) in “bad” cooling conditions.

Figure 10. Continued on next page.

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Figure 10. Temperature (a), electric field (b) and current (c) before thermal runaway versus residual resistivity ratio: 1 - h=10-2W/(cm2K); 2 - h=3×10-3W/(cm2K);3 - h=10-3W/(cm2K); - - - - η=0.2; ——— - η=0.5.

Figure 10 depicts the effect of the matrix resistivity on the thermal runaway parameters for some values of the heat transfer coefficient. It shows the existence of two characteristic stability regions. The first characteristic area is the area where the thermal runaway conditions depend essentially on the increasing RRR. Secondly, the thermal runaway parameters have an area where these quantities change weakly with increasing RRR. The values of RRR where these areas exist depend on the heat transfer coefficient and volume fraction of a

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superconductor. In particular, the quench temperature is high and Eq becomes much larger than the typical electric magnitude of 10-6 V/cm when the composite has a relatively small amount of a superconductor and is cooled by refrigerant with the relatively high heat transfer coefficient. The thermal runaway parameters as a function of the volume fraction of Bi2212 in composite under consideration with parameters (27) are plotted in Figure 11 for two limiting values of the heat transfer coefficient and RRR-parameter. Linear and nonlinear temperature models of Jc(T) were used. Curves 1-4 were defined according to the analytical formulae (28). Curves 3' and 4' correspond to the numerical simulation based on the model described by equations (3)-(5), (8), (9) and condition (23). In this case, the resistivity of silver has taken into account its dependence on the temperature and magnetic field.

Figure 11. Continued on next page.

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Figure 11. Quench electric field (a), temperature (b) and current (c) versus volume fraction of Bi2212 in composite: 1 - RRR=10, h=10-3W/(cm2K); 2 - RRR=10, h=10-2W/(cm2K); 3, 3' - RRR=1000, h=103 W/(cm2K); 4, 4' - RRR=1000, h=10-2W/(cm2K)

The results presented in Figs. 11(a) and 11(b) show that the thermal runaway values of the electric field and temperature decrease with the increasing η and two characteristic stability regions exist. Firstly, these quench parameters have an area where their values change weakly at the variation of η, if composite has relatively high values of η. In these modes, its overheating mainly is a result of the smoothness nature of VCC. The second characteristic area is the region where the change of η influences on the quench parameters essentially. The amount of the superconductor where these areas exist depends on the heat transfer coefficient and matrix resistivity. As a whole, the dominant role plays the second area at the relatively low value of η. In these cases, allowable electric field and temperature before the thermal runaway are high. In particular, Eq > 10-5V/cm and the quench temperature may exceed 10 K. Such over-critical operating states having high values of the allowable electric field and overheating were experimentally observed in [11-13]. Moreover, in the framework of linear Jc-model, quench temperature may come close to the temperature TcB. This tendency is valid for the calculations based on the nonlinear Jc-model. However, the performed simulations demonstrate some quantitative disagreement with similar calculations made under temperature-linear Jc-model. The most influence of the temperature dependence of Jc on the quench parameters will take place in the cases when the composite has the relatively low amount of a superconductor. For such superconducting composite the allowable electric field and temperature defined by the relationship (28) have not only higher values but show more drastic increasing. At the same time, if the temperature rise of the superconducting composite before the thermal runaway is relatively low, then both Jc(T)-models lead to the similar results. The curves plotted in Figure 11(c) show the η-dependence of the quench currents. It may have the minimum at relatively intensive cooling conditions. But the volume fraction ηmin, at which the minimum Iq exists, depends on the matrix resistivity: the lower resistivity, the

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higher minimum value of η. Therefore, the value of ηmin displaces to the area of lower values if composite has high resistivity matrix. The existence of the minimum value of Iq depends on both the current sharing and its intensity. Figure 12 shows the peculiarity of these mechanisms. The presented curves describe the current sharing during modes depicted in Figure 11 by curves 3' and 4'. It is seen that the characteristic value of η, where slope of Iq(η) changes, corresponds to the state when the current in the superconducting core equals to the current in the matrix (points A and B). However, the minimum value appears if the current sharing has intense character (point B). As also follows from Figure 12, the main part of the transport current will flow in the matrix at η < ηmin. That is why in this case the allowable temperature rise and electric field before the thermal runaway may be high.

Figure 12. Current sharing before the thermal runaway at RRR=1000: 3'- h=10-3W/(cm2K), 4'- h=102 W/(cm2K).

The discussed dependence of the quench electric field, current and temperature on the amount of a superconductor is explained by the current sharing behavior. The latter, first of all, depends on the matrix resistivity and amount of the superconductor in a composite, as follows from the criterion (14). Figure 13 shows the corresponding tendencies that prove the existence of two limiting cases leading to the different current-carrying capacity of a superconducting composite in dependence on η. They were obtained under the linear Jcmodel. It is seen that if the composite with the low volume fraction of a superconductor has a relatively low resistivity matrix, the characteristic value of which depends on the cooling conditions, then the main part of the transport current before thermal runaway will flow in the matrix (Figure 13(a)). In these cases, the current sharing essentially depends on the matrix properties and heat transfer coefficient. As a result, the limiting current in the matrix determines the thermal runaway current Iq. Then the allowable temperature and electric field before thermal runaway may be high (Figure 10).However, at high value of η, the currentcarrying capacity of the superconducting composite depends mainly on the properties of the superconductor, since almost all the transport current flows in the superconductor (Figure

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13(b)) according to the condition (14). Owing to this, Tq and Eq are limited to the relatively low values (Figure 10).

Figure 13. Current sharing before the thermal runaway at η=0.2 (a) and η=0.5 (b): 1 - h=103 W/(cm2K), 2 - h=3×10-3W/(cm2K); 3 - h=10-2W/(cm2K).

The thermal runaway parameters calculated for the Ag/Bi2212 composite with power and exponential voltage-current characteristics with various values of Jc0 are compared in Figure 14. The simulation was made at h=10-3 W/(cm2K), S=0.0123 cm2, p=0.47 cm, n=1/δ=11 for different values of RRR. Linear Jc(T)-dependence was used. The curves presented were defined according to the written analytical formulas (28) and (29). The depicted results show that the sub-critical modes may appear, first of all, at high values of the volume fraction coefficient, if a superconducting composite having the high critical current density is in the external magnetic field with high induction. That is why, for the operating modes presented in

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Figure 14, the allowable rise of the temperature and electric field before the thermal runaway are high in the wide range of η-variation. Besides, Figure 14(c) denotes that the resistivity of silver does not practically change the quench currents of a superconducting composite with the high critical current under the conduction-cooled conditions. Therefore, the Iq(η)dependence increases nearly linear with increasing a volume fraction of a superconductor in accordance with the estimates (32) and (33). However, the critical current density of a superconductor affects the region where sud- and over-critical currents exist. As a result, the difference between the critical and quench currents increases with increasing Jc0 according to estimates (16) and (17). Figure 14 also affirms the above-discussed tendency, which underlies

Figure 14. Continued on next page.

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Figure 14. Quench electric field (a), temperature (b) and current (c) versus volume fraction of a superconductor with various type of the voltage-current characteristics (⎯⎯ , - - - - – power VCC, + – exponential VCC) at B=10 T: 1 - Jc0 =104A/cm2; 2 - Jc0 =3×104A/cm2; 3 - Jc0 =105A/cm2.

the possible difference between the stability conditions of composites with the power and exponential voltage-current characteristics. In particular, it is seen that difference between allowable increase in temperature of such composite may become more visible in the range of small value of η. This regularity follows from formulas (18) and (19), which show that the difference in the thermal regimes of superconducting composites with power and exponential voltage-current characteristics will be observed in the high voltage range. The given conclusion should be taken into account when VCC-equation is recovered from the experiments because to write this relation correctly, the isothermal mode must be kept. Figure 15 demonstrates the magnetic field effect on the existence of the sub-critical and over-critical regimes of a superconducting composite just under consideration, but with the power VCC. The nonlinear Jc(T)-dependences described by relations (8) and (9) were used. The latter is depicted in Figure 15(b). The results plotted correspond to the numerical computations based on equations (3)-(5) and condition (23). In this case, the dependence on the temperature and magnetic field of the matrix resistivity has taken into account. The results presented show that the higher the magnetic field, the higher the range of η where the subcritical electric fields exist. Therefore, the possible quench currents have the tendency to be lower then the corresponding value of the critical currents in a wide range of the η-variation. As a whole, the change in the depicted Iq(η)-curves are due to the following reasons. First, the high critical current density of the superconducting composite considered leads to such current modes at which the current sharing occurs only at small values of the volume fraction coefficient, according to the estimate (14). Owing to this peculiarity, the quench currents may exceed the critical ones, as discussed above. Second, the overheating of the composite is not zero even at relatively high value of η due to the broad shape of the VCC. Thus, the temperature of a superconducting composite before the thermal runaway is not equal to the cooling bath temperature. It depends on many operating parameters. This

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unavoidable overheating leads to the thermal degradation of the quench current. Indeed, as follows from results depicted in Figure 15(b), the corresponding quench currents at B=10 T are equal to I′q=424.6A at η′=0.2 and I′′q=1512.7A at η′′=0.8, respectively. Thereby, the volume fraction of a superconductor increases as η′′/η′=4 but the increase in the quench current is less and is equal to I′′q/I′q=3.56. So, the quench currents do not increase proportionally to the proportional increase of the volume fraction of a superconductor: the higher the volume fraction coefficient, the higher difference between the critical and quench currents.

Figure 15. Stable boundary of the electric field (a) and current (b) as a function of volume fraction of a superconductor at various values of the applied magnetic field.

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Figure 16. Cryocooled variable temperature cryostat and coil sample.

Figure 17. Experimentally defined thermal runaway parameters of Ag/Bi2212 tape

As a whole, this feature follows from non-isothermal behaviour of the voltage-current characteristic of a superconducting composite. Its formation depends on the term 1/ n

⎛ E⎞ ⎜ ⎟ hp (TcB − T0 ) ⎝ Ec ⎠

η J c 0 SE

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in the case of a superconductor with the exponential VCC. Therefore, the thermal degradation will increase with increasing the volume fraction of a superconductor, critical current density Jc0, cross section of a composite, operating temperature at the same cooling conditions. But it decreases with increasing the critical temperature of a superconductor or the temperature margin defined as (TcB-T0). This non-isothermal feature has general character and shows that the thermal degradation of the quench current in the cases of low-Tc superconductors may be more essential than that in high-Tc superconductors. As a whole, this degradation of the current-carrying capacity of composite depicts directly the influence of the unavoidable overheating of a composite on the stable values of charging current that has to be considered in the thermal runaway analysis of superconducting composites.

η J c 0 SE hp(TcB − T0 ) Table. Ag-sheathed multifilamentary Bi2Sr2CaCu2O8 tape parameters __________________________

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Tape width (mm) 4.9 Tape thickness (mm) 0.38 Number of filaments 54 Ag/Bi2212 ratio 4/1 Tc (K) 87 Operating temperature (K) 4.2 __________________________ In order to prove the existence of the over-critical regimes, which characterized by high allowable temperature rise, the experiments were done under the conduction-cooling power. The Ag-sheathed Bi2Sr2CaCu2O8 tape used in the experiments was fabricated by Hitachi. Its parameters are listed in table. Figure 16 shows the cryocooled variable temperature cryostat and coil sample. In these experiments, an AIN bobbin holder was used for accurate control of the sample temperature. This is because the thermal propagation behaviour largely depends on the thermal conductivity, and the AIN material has good conductivity. However, the thermal contraction of AIN is almost zero from room temperature to 4.2 K. Since the Agsheathed Bi2Sr2CaCu2O8 tape has a thermal contraction of 1.0%, a tensile strain of 1% is yield for the Ag-sheathed Bi2Sr2CaCu2O8 tape coil wound onto the AIN bobbin at 4.2 K. As a result, such a large tensile strain for the Ag-sheathed Bi2Sr2CaCu2O8 tape causes a severe degradation of the critical current. Figure 17 shows the thermal runaway properties of this tape. It is seen that the tape demonstrates a very high quench parameters under the cryocooling method. Namely, the allowable quench currents drastically exceed the corresponding values of the critical current. For example, the quench current of Iq=174 A and quench temperature of Tq=23 K at B=10 T. Therefore, there exists a stable current level before the thermal runaway even when the sample temperature rise is about 20 K due to the stable Joule heating. This behaviour is in good agreement with the theoretical results

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discussed above. In particular, according to the results presented in Figure 10, the quench electric field and the quench temperature are calculated to be Eq=10-4 V/cm and Tq=23 K at h∼0.002 W/(cm2K) in the case of RRR=100. These estimates are reasonable to the experimental results. They are associated with the main feature of peculiar voltage-current characteristic for the Ag-sheathed Bi2Sr2CaCu2O8 tape, which comes from its high critical temperature.

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2.5. Conclusion The results of this paragraph show that the possible stable increase in temperature of high-Tc superconducting composite should be taken into account for correct investigation of their critical currents and thermal runaway conditions. As a result, the following features take place. The critical current density, the volume fraction coefficient, the cross section of a composite or smoothness character of the voltage-current characteristic affect the range of the allowable rise in the electric field at which the operating states of a composite are practically isothermal. The operating regimes of high-Tc superconducting composite having power and exponential voltage-current characteristics may not be equivalent. The noticeable difference between the thermal runaway conditions will be seen in the strong creep states (n < 10, Jδ /Jc0 > 0.1) due to the corresponding difference in the stable increase in temperature of a superconductor. The thermal runaway conditions of high-Tc superconducting composite may be both subcritical and over-critical. The sub-critical electric fields that do not exceed the fixed electric field criterion are more probable in the high magnetic fields or when a superconducting composite with high-resistivity matrix has a relatively high value of the volume fraction of a superconductor. Such current modes become unstable below the critical point {Ec, Jc0} because their stability boundary, first of all, is a result of the unavoidable overheating of a superconductor owing to a broad shape of the voltage-current characteristic. During overcritical regimes, the stable values of the electric field change in a wide range. In this case, the stable value of the current flowing in a superconducting composite may be lower or higher than those defined by the critical voltage criterion. These modes depend on the currentsharing mechanism between superconductor and matrix. Thereby, the corresponding unavoidable overheating of superconductor before the thermal runaway may be noticeable, in particular, when the superconducting composite has the low-resistivity matrix. The existence of the sub-critical and over-critical regimes and the possible temperature rise of high-Tc composite superconductors should be taken into account to recover correctly their critical parameters. The unavoidable overheating of the superconducting composite before its transition to the normal state may be noticeable, in particular, when the composite has a low-resistivity matrix or at a relatively small value of the volume fraction coefficient. In these cases, the current sharing underlies the current-carrying capacity of a composite that may lead to minimum quench currents. The overheating peculiarity takes place as well at sub-critical regimes owing to a broad shape of the voltage-current characteristic, as mentioned above. This overheating becomes essential when the superconducting composite has strong flux-creep parameters.

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Therefore, the temperature of a superconducting composite is not equal to the cooling bath temperature before the thermal runaway at both sub-critical and over-critical modes. The unavoidable overheating of a superconducting composite leads to the existence of the thermal degradation mechanism, which results to the relevant decrease in the current-carrying capacity of a composite. As a result, the quench currents do not increase proportionally to the proportional increase in the critical current of a superconductor.

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3. Multi-Stable Static Modes of High Temperature Superconducting Composites with Nonlinear Jc(T)Dependence at Different Operating Regimes As it was shown above, the heating of the current-carrying elements of high temperature superconducting magnets may become an important factor in their stability investigation. From this point of view, the analysis of the thermal modes is the necessary tool describing correctly the operating regimes of high temperature superconducting magnets before and after thermal runaway. On this account the temperature change of their physical properties should be considered. In particular, the critical current density of a high temperature superconductor has the noticeable temperature-decreasing dependence. Usually, this dependence is described by the simplified linear relationship. However, Bi2212 superconductors have nontrivial Jcrelations because at temperature above about 20 K their performances may strongly degrade due to the flux creep (Figure 2). In general, this temperature degradation depends on the magnetic field. Taking these facts into consideration, this paragraph is focused on the mechanism of the thermal runaway onset in the Ag/Bi2212 composite with the nonlinear temperature dependence of the critical current and its influence on the quench parameters. To make this study, we will use early formulated zero-dimensional thermo-electric model described by equations (3)-(5) and (8). Thereby, the thermal runaway phenomenon in Ag/Bi2212 composite with power VCC will be discussed taking into consideration the nonlinear temperature and magnetic field dependences of their critical current density and matrix resistivity. In this case, the simulation was based on the following parameters Tc=87.1 K, J0=5.9×104 A/cm2, Bc0=465.5 T, B0=1 T, α=10.33, β=6.76, γ=1.73, χ=0.27 (35) that adapted the experimental data published in [12]. The temperature and magnetic properties of a silver matrix were calculated using the residual resistivity ratio, as it was mentioned in item 2.1. The influence of the heat transfer coefficient, operating temperature, applied magnetic field, matrix resistivity on the static thermal and electric modes is discussed below. The corresponding thermal runaway parameters are numerically derived accounting for the specific formation of the voltage-current characteristic of a high temperature superconducting composite that may be observed in high voltage range.

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3.1. Static Differential Resistivity of a Superconducting Composite with Arbitrary Temperature Dependences of Its Properties To analyse the slope of VCC, consider the general case when the quantities Jc, ρm, h and n are the arbitrary functions of temperature. Omitting the intermediate transformations, the temperature-dependent differential resistivity of a composite for the given applied magnetic field may be written as

1+

ES γ 1 + γ 2 + γ 3 γ4 h(T ) p

∂E (T ) = 1/ n ∂J 1 − η η J c (T ) ⎛ E ⎞ JS γ 1 + γ 2 + γ 3 + ⎜ ⎟ − ρ m (T ) n(T ) E ⎝ Ec ⎠ γ4 h(T ) p

(36)

according to zero-dimensional thermo-electric model (3)-(5). Here 1/ n

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∂J ⎛ E ⎞ γ1 = η c ⎜ ⎟ ∂T ⎝ Ec ⎠

1/ n

1 −η d ρm dn ⎛ E ⎞ , γ2 = E 2 , γ 3 = η Jc ⎜ ⎟ dT ⎝ Ec ⎠ ρ m dT

1/ n 2

⎛ E⎞ ln ⎜ ⎟ ⎝ Ec ⎠

, γ 4 = 1+

EJS dh h 2 p dT

This relation permits the qualitative understanding of the thermal runaway onset. In principle, the thermal runaway happens because the ∂Jc/∂T-term is negative. The temperature dependence of the matrix resistivity also leads to the more unstable states. As a whole, the quantities |∂Jc/∂T| and ∂ρm/∂T have the destabilizing effect on the boundary of the stable current modes: the higher ones, the earlier and intensely the thermal runaway. At the same time, the stability conditions are improved when the quantity |∂Jc/∂T| decreases. Fig.18 shows the curves describing the temperature derivative of the critical current as a function of temperature (Ic=ηJc(T)S, S=0.01862 cm2, η=0.2). It demonstrates their non-monotonic temperature changes in the high-field range. As a result, there exist three characteristic regions of the potential variation of ∂Ic/∂T that follows from formula (8). Firstly, the values |∂Ic/∂T| increase with increasing temperature in a low temperature range. Secondly, |∂Ic/∂T| decreases constantly in the intermediate temperature area. The temperature when the degradation of the critical current starts depends on the magnetic field. For example, at B=10 T it takes places when the temperature exceeds about 15 K. Finally, the slope of |∂Ic/∂T|curves decreases essentially and the quantity ∂Ic/∂T has very small values at relatively high temperature. Moreover, they are practically close to zero at very high fields. As a consequence, the differential resistivity of a superconducting composite will have nontrivial behavior because the sign of ∂E/∂J depending on the quantity |∂Jc/∂T| may be both positive and negative in the superconducting temperature range. Therefore, the voltage-current characteristic of Bi2212 composite may change the well-known form. As a result, this will lead to the relevant transformation of the thermal runaway conditions, especially, when a relatively high allowable temperature rise of Ag/Bi2212 may be observed before the thermal runaway even at low cooling bath temperature.

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Besides, as follows from (36), there exists the specific stabilizing role of the temperaturedependent heat transfer coefficient associated with the quantity ∂h/∂T. The latter may be both positive and negative. Therefore, the condition of ∂h/∂T > 0 is more stable in comparison with ∂h/∂T < 0. Moreover, the function γ4(h) may be close to zero at ∂h/∂T < 0. As a result, such regimes approach extremely unstable states. In this way, the temperature-decreasing dependence of h(T) trends to induce the more unstable states. It should be also noted the possible temperature effect of the n-value of VCC on the allowable variation of the differential resistivity of composite. As follows from formula (36), ∂E/∂J depends not only on dn/dT, which is negative at high applied magnetic fields, but also the term ln(E/Ec). The latter has negative values at E < Ec and becomes positive at E > Ec. Therefore, the real temperature variation of the exponent n will play destabilizing role at E < Ec and stabilizing role at E > Ec. As a result, the appearance possibility of the VCC with the positive slope in the high electric field range (E > Ec) will increase due to temperaturedecreasing behavior of n(T). In other words, the superconducting composite may have stable regimes in the over-critical voltage ranges because an additional stable branch may exist due to temperature-deceasing behavior of n(T).

Figure 18. The temperature derivatives of the critical current versus temperature at a different applied magnetic field.

Thus, the differential resistivity of high temperature superconducting composites may have nontrivial temperature behavior, and their voltage-current characteristics may change the well-known form having one stable and unstable branches at a relatively high allowable temperature rise.

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3.2. Nontrivial Steady Electric and Thermal Regimes of Conduction-Cooled Ag/Bi2212 Composite The theoretical investigation of the phenomenon discussed is considerably complicated if the temperature dependences of all parameters will be taken into account. Such model does not allow one to make the step - by - step explanation of the appearance of possible manyvalued modes. So, firstly, let us consider the potential regularities in the case of the conduction-cooled conditions assuming that h=10-3 W/(cm2K). Secondly, also in order to simplify the discussion of the mechanisms based the formation of the voltage-current characteristics, we will consider the cases where the n-value does not depend on temperature. Similar approach is usually used in many numerical simulations of the operating regimes of both superconducting wires and magnets (see, for example, [10-14, 34-38]). In the framework of this approximation based on the temperature dependences of Jc and ρm, the differential resistivity of composite is equal to

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⎡ ∂J ⎛ E ⎞1/ n 1 − η ∂ρ m ⎤ ES ⎥ 1 + ⎢η c ⎜ ⎟ + E 2 ρ m ∂T ⎥ hp ⎢⎣ ∂T ⎝ Ec ⎠ ∂E ⎦ (T ) = 1/ n 1/ n ∂J ⎡ ∂J ⎛ E ⎞ 1 −η η J c ⎛ E ⎞ 1 − η ∂ρ m ⎤ JS c ⎥ + ⎜ ⎟ − ⎢η ⎜ ⎟ +E 2 ρ m n E ⎝ Ec ⎠ ρ m ∂T ⎥ hp ⎢⎣ ∂T ⎝ Ec ⎠ ⎦

(37)

This simplified theoretical approximation provides the principal basis for experimental analysis, which may be made to observe the main odd peculiarities of the voltage-current characteristics of Bi-based superconducting composites discussed below. Figures 19(a) and 19(b) present the voltage-current and temperature-current characteristics of Ag/Bi2212 composite with low resistivity matrix (RRR=1000). Stable and unstable parts of these static characteristics are shown by the solid and dashed curves, respectively. The stable current distribution before thermal runaway in the superconducting core and the matrix is plotted in Figure 19(c). The calculations were made for the composite under consideration at n=10, B=10 T, p=0.49 cm and different operating temperatures. The numerical calculations accounted for the dependence of the matrix resistivity on the temperature and magnetic field. It is seen that the E(I) and T(I) curves have well-known form with one stable and unstable branches. These modes are due to the fact that the main part of the transport current flows in the matrix before thermal runaway (Figure 19(c)). At the same time, the current sharing mechanism depends on the resistivity of the matrix. As it was shown in paragraph 2, the current sharing does not play an essential role before the thermal runaway onset, when the matrix has high resistivity. Under this condition, the thermal runaway occurs when practically all transport current flows in a superconductor. However, as it was discussed above, the degradation of the critical current is observed, if the stable temperature rise of composite happens in the intermediate temperature range. In these cases, the differential resistivity of a composite will be correspondingly changed. The possible modification of the voltage-current and temperature-current characteristics of a composite with high-resistivity matrix (RRR=10) is shown in Figs. 20(a) and 20(b). The corresponding peculiarity of the stable current sharing is depicted in Figure 20(c).

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Figure 19. Effect of the operating temperature on the static electric and thermal states of Ag/Bi2212 composite with the high residual resistivity ratio: 1 – T0=16 K; 2 – T0=18 K; 3 – T0=20 K

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Figure 20. Stable (——) and unstable (- - - -) branches of the voltage-current (a), temperature-current (b) characteristics of Ag/Bi2212 composite and the jump current redistribution (c) at B=10 T and low residual resistivity ratio: 1 – T0=10 K; 2 – T0=15 K; 3 – T0=16 K; 4 – T0=18 K; 5 – T0=20 K; 6 – T0=25 K; 7 – T0=26 K

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It is seen that a new additional stable current mode may appear as a result of the relevant differential resistivity change of a composite, when the operating temperature exceeds some characteristic value. As a result, the following possible types of the voltage-current and temperature-current characteristics of Ag/Bi2212 composites exist:

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1. Characteristics with one stable and unstable branches (Figure 19). 2. Characteristics with two stable and unstable parts, when the stable currents corresponding to the second stable part do not exceed the current in the first stable part (curves 1 and 2 in Figure 20). 3. Characteristics with two stable and unstable branches, when the allowable currents in the second stable part of VCC exceed the relevant values in the first stable part (curves 3 and 4 in Figure 20). 4. Characteristics with one large stable branches when the allowable temperature rise of a composite is high but less then the critical temperature of the superconductor (curves 5 and 6 in Figure 20). 5. Characteristics with one stable branch existing from the operating temperature up to the critical temperature of the superconductor (curve 7 in Figure 20). This diversity of the voltage-current and temperature-current characteristics is based on the two-coupled mechanisms. The first of them is any stable temperature variation of a composite affecting, first of all, the quantity |∂Jc/∂T|. The second is defined by the peculiarities of the current sharing between the superconducting core and matrix determining the induced temperature and electric field. In this case, the higher the quench current, the smaller the allowable stable electric field and temperature. This coupled variation of the current, electric field and temperature defines the different signs of denominator in formula (37) and, thus, results in relevant non-monotonic variation of the differential resistivity of a composite. As a result, the currents in the second stable mode may be both less and higher than that in the first stable mode despite the high electric field and temperature. Besides, the stable thermal and electric regimes may be observed in the wide temperature range changing from the operating temperature to the critical temperature of a superconductor. This is due to the fact that at a certain condition, for example, at relatively high operating temperature, the quantity |∂Jc/∂T| becomes very small. The additional stable parts of the voltage-current and temperature-current characteristics lead to the various evolution mechanisms of the thermal runaway. Firstly, if the maximum current on the first stable branch is higher than that on the second stable branch (curves 1 and 2 in Figure 20(a)), the thermal runaway leads to the well-known irreversible temperature rise of the superconducting composite. In this case, the possible temperature rise before thermal runaway is not appreciable and the first stable branch limits the current-carrying capacity of a composite. Secondly, the regimes with the maximum current on the second stable part of the voltage-current characteristics exceeding the relevant value on the first stable part lead to the static jump behavior of the thermal and electric modes (curves 3 and 4 in Figure 20(a) and 20(b)). According to this peculiarity, the main part of the transport current flows in the superconducting core before the jump and stably flows in the matrix after the jump (Figure 20(c)). The allowable temperature of the superconducting composite noticeably increases after the jump. The further increase of the current will lead to the thermal runaway, when the current exceeds the boundary of the second stable part of the voltage-current characteristics.

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Figure 21. Stable static electric (a) and thermal (b) modes of Ag/Bi2212 composite during stable current sharing (c) at relatively high operating temperature (RRR=10, T0=30 K)

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Figure 22. Stable (——) and unstable (- - - -) branches of the voltage-current (a) and temperaturecurrent (b) characteristics of Ag/Bi2212 at B=10 T: 1 - h=10-3 W/(cm2K), 2 - h=3×10-3 W/(cm2K), 3 h=10-2 W/(cm2K).

In other words, the current charging into the composite under these cooling conditions will accompany the jump current redistribution, which will have stable (the first jump without the transition into the normal state) and unstable (the second jump with collapse of the superconductivity) temperature and electric field variation. Thirdly, the two stable parts of E(I) and T(I) curves may merge (curves 5 and 6 in Figure 20). Then the thermal runaway has a behavior similar to the first case discussed above. During these modes the composite has a high stable temperature rise. Finally, as a limiting case of lasts modes, the unstable parts of

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the voltage-current characteristics are absent, when the operating temperature is relatively high (curve 7 in Figure 20). In this case, the current charging will have a stable character and the temperature of a superconductor will stably increase up to the critical one (Figure 21). It should be underlined that during these modes the transport current stably flows both in superconductor and a matrix at T < TcB. It is important result because the existence of such regimes is not based on the stability property of the heat capacity of both a superconductor and a matrix, but has the static nature. Figure 22 presents the possible modification of the voltage-current and temperaturecurrent characteristics of a composite under consideration caused at T0=4.2 K and RRR=10 by the variation of the heat transfer coefficient, which are close to the conduction-cooled conditions. Stable parts of these characteristics are shown by the solid curves and unstable parts are depicted by the dashed curves. It is seen that the voltage-current and temperaturecurrent characteristics have the standard form with one stable and unstable branches at h=10-3 W/(cm2K) (curve 1) because the stable temperature rise is relatively small and does not exceed 10 K. However, the more noticeable stable temperature rise modifies this shape of the VCC at h=3×10-3 W/(cm2K) and h=10-2 W/(cm2K). In particular, they may have the additional stable and unstable branches (curve 2) or increase essentially the stable part of the voltage-current characteristics (curve 3). Such transformation occurs because the stable value of the temperature before the thermal runaway becomes more than 20 K. Under this condition, the quantity |∂Jc/∂T| has the small value, as follows from Fig.18. According to (37), the existence of these additional stable states depends also on the value of the transport current flowing in Ag/Bi2212 before thermal runaway. The latter is the function of the parameters of Ag/Bi2212. For example, the stable value of the current decreases when the volume fraction of the superconductor in Ag/Bi2212 decreases, as it was shown in paragraph 2. Because of this, the appearance of the additional stable branches on the voltage-current and temperature-current characteristics is the most probably for the cases when η is relatively small. The nontrivial form of the voltage-current and temperature-current characteristics of the high-Tc superconducting composites, which can have additional stable and unstable parts, may be caused by the variation of the applied magnetic field. Figure 23 shows the possible modification of the voltage-current and temperature-current characteristics of the abovediscussed Ag/Bi2212 composite with the low matrix resistivity (RRR=1000) calculated at T0=4.2 K, h=10-3 W/(cm2K). These curves demonstrate the existence of the two characteristic values of an external magnetic field that may modify the E-J and T-J traces. Firstly, the second stable part appears, when the external magnetic field exceeds the first characteristic value. It is equal to 14.45 T for the case under consideration. This part increases quickly when the applied magnetic field increases. Similar to the states, which take place below 14.45 T, the voltage-current and temperature-current characteristics will have only one stable and unstable part, when the applied magnetic field exceeds the second characteristic value (about 16 T). The existence of such modes depends on the operating parameters. As a result, the voltage-current and temperature-current characteristics having only one stable branch may be observed in the large temperature range, as it is shown in Fig.24. The curves presented were calculated as a function of the cooling bath temperature at RRR=10, h=10-3 W/(cm2K), B=20 T.

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Figure 23. Stable (——) and unstable (- - - -) branches of the voltage-current (a) and temperaturecurrent (b) characteristics of Ag/Bi2212 for various applied magnetic fields at T0=4.2 K.

The possible existence of additional stable states may also be explained by the relevant variation of the temperature dependence of the Joule losses. Figure 25 shows the static heat releases in the superconducting composite superconductor as a function of the temperature. The curves presented were calculated for the above-considered conductor at I=32A and two values of the volume fraction of the superconductor in composite. The solid curve shows the total heat generation in composite calculated as G=JSE. The dashed curves depict the energy

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dissipation in the superconducting core (Gs=ηJsSE) and matrix (Gm=(1-η)JmSE). The dasheddotted curves display the possible heat removal under the assumption that the heat transfer coefficient is constant. The results of these calculations demonstrate the following peculiarities of the Joule heat release in the high-Tc superconducting composite with nonlinear Jc(T)-dependence. Firstly, it may have the non-monotonic temperature dependence.

Figure 24. Stable electric (a) and thermal (b) modes of Ag/Bi2212 composite at high external magnetic field (B=20 T) and various operating temperatures: 1 – T0=17 K; 2 – T0=15 K; 3 – T0=13 K; 4 – T0=11 K; 5 – T0=9 K; 6 – T0=7 K.

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Figure 25. Static Joule heating (1 - in composite, 2 - in the superconducting core, 3 - in the matrix) and possible heat removal at the different heat transfer coefficient and operating temperature (4, 5) versus temperature: a - η=0.2, b - η=0.5. Here A, A' and C are the stable points; B, B' and D are the unstable points.

Therefore, the curves describing the heat generation and cooling power capacity may have more than two balance points where these curves are intersected and, thus, may define new stationary states. Secondly, the non-monotonic behavior of G(T) is due to the current-sharing mechanism, which is accompanied with the energy dissipation in the superconducting core and matrix. Figure 25(a) shows that G(T) has the visible non-monotonic character just above

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20 K, when the current sharing starts for the case under consideration and the visible increase of Gm(T) with the temperature takes place. For arbitrary operating regimes, it means that in new stable state the main part of the transport current flows in the matrix and the superconductor has relatively a high temperature rise without the transition into the normal state. From this point of view, it is easy to understand that the formation of the additional stable state depends also on the resistivity of a matrix and amount of a superconductor. As an illustration of this conclusion, Figure 25(b) demonstrates that the behavior of G(T) becomes more monotonic, when the volume fraction of the superconductor in composite increases. The energy explanation allow one to see from another point of view the reason, which leads to the nontrivial form of the voltage-current characteristics having the multi-stable branches. The presence of a new stable balance point (point C in Figure 25(a) with the wellknown stable points A and A' in Figure 25(a) and Figure 25(b)) depends strongly on the coupled values of the heat transfer coefficient and operating temperature for a given value of η. As a result, the possibility of the appearance of a new stable balance point increases with increasing T0 and decreasing η under the condition that the cooling capability is relatively low. So, in spite of the possible non-monotonic character of G(T), the VCC of Ag/Bi2212 will not have multi-stable form, when the cooling capacity is high (Figure 22, curve 3). In this case, an additional stable balance point within the critical temperature will be absent. Therefore, the multi-stable branches of the VCC of Ag/Bi2212 are the result of the additional heat balance between the heat generation and low cooling power capacity at certain operating temperature and parameters of a superconducting composite. This additional balance point will be characterized by the high value of the temperature, which Ag/Bi2212 composite will have without the thermal runaway. The reasons leading to the additional stable static regimes (Fig.20, curve 7 and Fig.21) should be underlined. Usually the improved conditions of the thermal stabilization of hightemperature superconducting composites are connected to the dynamics stabilization role of their heat capacity. However, there exists the additional static mechanism of the thermal stabilization of a high temperature superconducting composites. It is based on the static nonlinear temperature change of the quantity |∂Jc/∂T| in the over-critical operating regime due to high allowable temperature rise, which is characterized by small but finite value of |∂Jc/∂T|. That is why many-valued stable regions are absent, when the critical current density of a superconductor has the linear temperature Jc-dependence when the quantity |∂Jc/∂T| is large and constant. Moreover, it is seen that at certain conditions the voltage-current characteristics of superconducting composite may have only one stable branch, when the temperature of a composite stably increases from the operating temperature up to the critical one. Such states take place when the operating temperature exceeds 25−30 K for the composite under consideration when the critical current has visible degradation. In other words, during these regimes Ag/Bi2212 composite will have the complete stable current regimes and their transition to the normal state will be a result of the trivial overheating exceeding the critical temperature of the superconductor by the current charging. These peculiarities demonstrate an important role of the correct definition of the permissible temperature variation of the Jcproperties of a high temperature superconductor in the high voltage range. As a consequence, the thermal runaway conditions defined in accordance with the existence of the multi-stable regimes allows one to find the essential increase of the thermal runaway current that may

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really exceed a priory determined critical current of a superconductor. Let us discuss the corresponding variation of the quench parameters when the multi-stable regimes exist.

3.3. Thermal Runaway Conditions of a Superconducting Composite with Multi-stable Current Modes

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Let us determine the quench parameters of Ag/Bi2212 composite as the parameters leading to its maximum current-carrying capacity without spontaneous transition to the normal state considering the multi-stable branches of the voltage-current characteristic and using the static stability condition (23) or (24).

Figure 26. Continued on next page.

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Figure 26. Electric field (a), temperature (b) and current (c) before thermal runaway versus applied magnetic field: 1 - RRR=1000, 2 - RRR=10

Figure 26 indicates the simulation results of the quench parameters as a function of the applied magnetic field. They were obtained at T0=4.2 K, h=10-3 W/(cm2K) for the composite examined above. The numerical calculations accounted for the dependence of the matrix resistivity on the temperature and magnetic field. Two characteristic values of the residual resistivity ratio were used. They correspond to the matrix with both the low and high resistivity. The relevant critical current dependence on the magnetic field is also plotted. Currents in the superconducting core and matrix before the thermal runaway are depicted in Figure 27. It demonstrates the origin of the current-sharing jump leading to the existence of an additional stable mode. The presented results show that the allowable temperature change of Ag/Bi2212 depending on the matrix resistivity decreases when the applied magnetic field increases in the range preceding the current-sharing jump. In the meantime, the jump occurs. After the jump the quench electric field and temperature are the increasing functions of the applied magnetic field due to the corresponding current redistribution from superconductor to matrix (Figure 27). These thermal and electric states of Ag/Bi2212 are the result of the magnetic influence on the critical current density of superconductor and matrix resistivity. They take place in spite of the allowable temperature decrease of Ag/Bi2212 before the current jump. This behavior also follows from the formula (37). Indeed, it is seen that not only the temperature increase may lead to the positive value of the denominator. It appears that the condition ∂Ε/∂J > 0 may also exist, when the applied magnetic field increases. Therefore, the magnetic field effect on the properties of the superconductor and matrix is the next reason, which may lead to the appearance of the additional stable branches on the voltage-current characteristics, among the possible high temperature rise of Ag/Bi2212. In accordance with these peculiarities, the magnetic field dependence of the quench current takes place. The jump transition to the additional stable state leads to the kink in the Iq(B)-dependence. The difference in the thermal and electric behavior of Ag/Bi2212 with various resistivity of the matrix also leads to the various quench currents, which are higher than the corresponding critical current. It is interesting to note that the quench current defined for Ag/Bi2212 with the low resistivity matrix is higher than that in the case of the matrix with high resistivity. This

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result shows not only the importance of the current-sharing mechanism in the description of the thermal runaway phenomena but also the role of the matrix in the thermal runaway onset. The latter may be important for the optimization of the thermal stability conditions of developed high-field magnets.

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Figure 27. Currents in the Bi2212 core and matrix before thermal runaway versus applied magnetic field

Figure 28. Continued on next page.

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Figure 28. Electric field (a), temperature (b) and current (c) before thermal runaway versus operating temperature: (– – –) – model based on the linear temperature Jc – dependence and constant value ρm; (——) – model based on the nonlinear temperature dependences of Jc and ρm; (– ⋅ – ) – model based on the linear temperature approximation of Jc and the nonlinear temperature dependence of ρm.

Figure 28 presents the numerical simulation results of the quench parameters as a function of the operating temperature made at h=10-3 W/(cm2K) and B=10T. The calculations taking into consideration the dependence of the matrix resistivity on temperature were done under both the nonlinear and linear Jc-relations. Linear-fitted critical temperature and current density were defined in accordance with (27). The constant values of the matrix resistivity (ρm(T,B)=ρm(T0,B)) were also used. The relevant critical current dependence on the operating temperature is plotted in Fig 28(c). Currents in the superconducting core and matrix before

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the thermal runaway are depicted in Figure 29, which demonstrates the role of the matrix in the jump-formation of the stability conditions.

Figure 29. Current sharing just before thermal runaway versus operating temperature: a-RRR = 1000, bRRR =10.

The performed calculations reveal the following operating temperature regularities of the quench parameter depending on the matrix resistivity. As it has been discussed, the dependences Eq(T0) and Tq(T0) have the breaking behavior due to the jump current sharing for the composite with high-resistivity matrix that occurs in the intermediate operating

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temperature area. This leads to the corresponding kink in the Iq(T0)-dependence. The quench current monotonically decreases, if the matrix has low resistivity due to the monotonic decreasing current in a superconductor (Figure 29). In these cases, the values of Iq(T0) for the matrix with both the low and high resistivity are higher than the relevant values of the critical current corresponding to the criterion Ec=10-6 V/cm. In the meantime, the variance in the thermal and electric behavior of Ag/Bi2212 composites with various resistivities of the matrix leads to the different currents initiated thermal runaway. It is seen that the quench current determined for the composite with the high-resistivity matrix is lower than that in the case of the matrix with low resistivity. This indicates that the current-carrying capacity of Bi2212 composite sheathed by silver with low resistance is used more effectively than that with high resistance of a silver matrix. The noticeable difference between quench parameters is observed in the operating temperature ranging from 15 K to 25 K for the conductor under consideration. In this case, the temperature rise of the composite with “bad” matrix leads not only to the more essential quench current degradation, but to the temperature increase of the composite up to the critical temperature of a superconductor. Therefore, the superconductivity of Ag/Bi2212 sheathed by the high-resistivity matrix collapses at operating temperatures above about 30 K. It should note the dependence of the quench parameters on the temperature change of the critical current and matrix resistivity. As follows from Figure 28, the model based on the linear Jc -equation may describe the quench parameters exactly enough, if the latter considers the corresponding temperature dependence of the matrix resistivity and the current jump is absent. Models using constant resistivity of a matrix (ρm(T,B)=ρm(T0,B)) may be used only at the operating temperatures, which close to 4.2 K. As a whole, the model with nonlinear temperature dependence of the critical current and constant resistivity of the matrix cannot define the quench parameters exactly enough. So, the corresponding temperature description of Jc(T,B) and ρm(T,B), may play an essential role in the determination of the stability conditions.

3.4. Conclusion The analysis of the thermal runaway problem performed for the Ag/Bi2212 composite with the nonlinear temperature and magnetic induction dependences of the critical current density of superconductor and matrix resistivity shows the possible existence of the multistable static modes of the high-Tc superconducting composites. They are conditioned by the modification of the voltage-current and temperature-current characteristics, which may differ from the well-known characteristics having only one stable and unstable branches. As a result, the E-J and T-J dependences in the static approximation may have the discontinuous parts leading to the jumps of the electric field, current and temperature without the transition of the superconducting composite into the normal state. The essential increase of the stable branch of the voltage-current characteristics may be observed, as a result of the junction of the many-valued stable parts. According to this peculiarity, the stable current charging may be observed during which an allowable temperature of superconducting composites stably increases up to the critical temperature of a superconductor at conduction-cooled conditions. The existence of these peculiarities is not based on the transient stability mechanism of the heat capacity of superconducting composite. Such behavior is due to the corresponding

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static change of its differential resistivity. The latter depends on the temperature and magnetic field variations of the quantities Jc(T) and ρm(T), |∂Jc/∂T| and ∂ρm/∂T. They may lead to the relevant non-monotonic change of the Joule heating in Ag/Bi2212 and, thus, may define a new static mode. Their existence depends on the current-sharing mechanism that influence on the respective allowable temperature change of Ag/Bi2212. In particular, the current sharing is more intrinsic property of a superconducting composite with a relatively small amount of a superconductor then that in a composite with high volume fraction of a superconductor. Therefore, the additional stable and unstable states with high allowable temperature rise will be more possible in superconducting tapes or wires with relatively small volume fraction of a superconductor. According to these peculiarities, the stability conditions are shaped. Thereby, the quench electric field and temperature may have a jump-like behavior due to the jump mechanism of the current sharing. In this case, the transition to the additional stable mode may lead to the kink in the dependence of the quench current on the operating temperature or the induction of the applied magnetic field. Thus, the thermal runaway phenomena in the high-Tc superconducting composites may have nontrivial features defined by the nonlinear dependences of the critical current density of a superconductor and matrix resistivity on the temperature and magnetic field. Correspondingly, the thermal runaway condition defined according to the existence of the multi-stable modes allows one to find the essential increase of the current stability range, which may really exceed the critical current of a superconductor a priory determined by fixed electric field.

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4. Thermal Runaway Phenomenon in Superconducting Composites Cooled by Liquid Coolant The results discussed in the previous paragraphs indicate that the thermal runaway conditions of high-Tc superconducting composite may differ from those in low-Tc ones. Mainly, this is due to the broad shape of the voltage-current characteristics of high-Tc superconductors and their wide temperature margin between the operating and critical temperatures. Therefore, in applications, the stable self-heating of a superconducting currentcarrying element will precede the thermal runaway phenomenon. As it was shown above, the possible allowable increase in temperature of a superconducting composite depends on the properties of a superconductor, matrix, coolant and etc. In particular, under the aboveinvestigated cooling conditions, which are the characteristic regimes of the conduction-cooled or gas-cooled high-Tc superconducting magnets, the stable overheating may be essential. However, the possible stable temperature rise of a composite may be limited by the coolant properties, if the liquid coolant cools the composite. In these cases, the heat flux from the composite to the coolant depends on the temperature difference between the composite surface and the coolant. This is due to the existence of the nucleate and film boiling modes of the liquid coolant. As a result, this peculiarity of the heat transfer conditions will influence on the stability conditions of high-Tc superconducting composite. To understand these features, the effects of the liquid helium and liquid hydrogen cooling properties on the thermal runaway conditions of Ag/Bi2212 composite are studied below.

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4.1. Thermo-electric and Cooling Models Consider an infinitely long superconducting composite with a power law of the voltagecurrent characteristic under the applied current charged with an infinitely low rate. Let us it is cooled by liquid coolant having the saturation temperature T0, and the distribution of the charging current is uniform over the cross section of the composite. In general, the cooling model based on the time-dependent heat transfer consideration should describe the properties of a liquid coolant. Besides, it has to take into account the surface properties, geometry of a conductor, its orientation, cooling channel form and many other factors. However, these dependences are adequately known only in some particular cases. To discuss the basic trends, which influence the thermal runaway conditions of high-Tc superconducting composite cooled by liquid coolant, let us use the typical steady-state relationship between the cooling heat flux and conductor temperature. Namely, for heating up to the critical overheating ΔTcr, depending on the coolant, the heat transfer occurs in the nucleate boiling mode. Beyond this value, the cooling condition enters the film boiling mode. In the first region, the heat is rapidly dissipated at the relatively small overheating of composite. The second region is characterized by a relatively low rate of cooling at the relatively high overheating. In the case under consideration, the zero-dimensional model is defined by Kirchoff and Ohm equations (4) and (5) and heat balance equation rewritten as

EJ = W (T ) p / S

(38)

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Here, W(T)=h0(T-T0)υ is the heat flux to the coolant (h0 and υ are known constants). The relationships of the heat flux during nucleate and film boiling regimes in the cases of liquid helium or hydrogen have been discussed in detail in [44]. Using these results, the boiling curves were fitted by the following temperature-dependent relations

⎧⎪2.15(T − T0 )1.5 W / cm 2 , T − T0 < 0.6 K W (T ) = ⎨ 0.82 2 ⎪⎩0.06(T − T0 ) W / cm , T − T0 ≥ 0.6 K

(39)

when the composite is cooled by liquid helium and

⎧⎪0.66(T − T0 ) 2.6 W / cm 2 , T − T0 < 3K W (T ) = ⎨ 1.1 2 ⎪⎩0.024(T − T0 ) W / cm , T − T0 ≥ 3K

(40)

if the coolant is liquid hydrogen. Let us describe the critical current density of a superconductor by the expression (8) under the parameters (9) and n=15, S=1.2×10-2 cm2, p=0.47 cm. The matrix resistivity as a function of the temperature and magnetic field was calculated in accordance with the assumptions formulated in item 2.1.

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4.2. Quench Characteristics of Ag/Bi2212 Composite under Nucleate and Film Boiling Regimes of Coolant The numerical method was used in solving the problem investigated. The results presented in Figs. 30 – 38 were obtained at B=10 T. Figure 30 shows the typical steady variation of the electric field and the temperature induced by the applied current in the superconducting composite cooled by liquid helium (T0=4.2 K). Two types of the calculations were made. First, the solid curves describe the growth of the electric field and the temperature in the nucleate boiling regime. These states were defined only by the first term in relationship (39). The temperature boundary Tcr=T0+ΔTcr and the corresponding current Iq of this regime are indicated by the dotted curves. Second, the dashed curves correspond to the static increase in the electric field and temperature obtained under the assumption that only the film boiling regime would exist. These states were described by the second term in relationship (39) in the whole operating mode. The marked ends of these curves (If,q) correspond to the current stability boundary taking place only in the film boiling regime. The quantity If,q appears as the border between stable and unstable parts of the voltage-current or temperature-current characteristics of a composite taking place according to the static stability conditions (23) or (24). The depicted curves indicate the following characteristic peculiarities of the thermal runaway onset in the superconducting composite cooled by liquid helium. For quite understandable reason, the operating mode in the nucleate boiling regime is more stable than that in the film boiling one. Therefore, the jump transition from the nucleate to the film boiling modes occurring at I>Iq corresponds to the transition from the stable to unstable current modes. In other words, the thermal runaway in the superconducting composite cooled by liquid helium is the result of its elementary self-heating exceeding the quantity ΔTcr at I>Iq. In this transition, the temperature of a composite will increase above the critical temperature of the superconductor. However, there exists the current range (I> a. At the same time, the thickness influences on the thermal runaway conditions. Indeed, according to the results presented in Figure 49(a), the following relationships take place: I''q/I'q=1.74 at a''/a'=1.9 and I'''q/I'q=3.21 at a'''/a'=3.8 under the

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condition η=const. This degradation of the current-carrying capacity of a superconductor is due to its unavoidable overheating when the temperature of a superconductor before thermal runaway is not equal to the coolant temperature, as discussed above. Accordingly, in the framework of the zero-dimensional approximation, this effect depends on the term

Figure 49. Electric field (a) and temperature (b) evolution on the surface of the superconducting slab with various thickness (1-a=0.01 cm, 2-a=0.019 cm, 3-2a=0.038 cm): 1, 2, 3 – static zero-dimensional model; 1', 2', 3' – transient zero-dimensional model; 1'', 2'', 3'' – diffusion model; 4 – scaling approximation.

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1/ n

aη J c 0 Eq ⎛ Eq ⎞ ⎜ ⎟ h(TcB − T0 ) ⎝ Ec ⎠

As a whole, the thermal degradation of the current-carrying capacity of superconducting slab will be caused by the variation of the thickness and the quench electric field, if the volume fraction coefficient is constant. Therefore, the quench currents will not increase proportionally to the corresponding increase of the thickness of a slab. Discussed regularities depict that the superconducting tapes having the same cross section but different sizes will not have the identical quench currents. To illustrate this conclusion, Figure 50 shows the evolution of the electric field and the temperature, which takes place at dI/dt=10 A/s and parameters formulated early, in three superconducting slabs when their sizes were changed so that their cross sections were kept constant. Here, the group of the traces 1 were calculated at a=9.5×10-3 cm, b=0.49 cm; the group of the traces 2-at a=1.9×10-2 cm, b=0.245 cm; the group of the traces 3-at a=3.8×10-2 cm, b=0.1225 cm. Dotted lines indicate the relevant quench quantities following from the static zero-dimensional model. Presented results demonstrate the following size-dependent formation of the thermoelectric modes, which will take place despite the constant cross section. First, the low-voltage boundary that allows one to find the critical current of a superconductor is equal to

Vb =

μ0 a 3 dI 8S dt

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i.e., becomes strongly sensitive to the thickness of a slab at S=const.

Figure 50. Continued on next page.

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Figure 50. Transient and static variation of the electric field and temperature on the surface of the superconducting slab having different width and thickness under the condition that the composite crosssectional area is constant. Used models: 1, 2, 3 – static zero-dimensional model; 1', 2', 3' – diffusion model.

Figure 51. Temperature and size effect on the currents in the electric field range closed to the critical value Ec. Used models: 1, 2, 3 – static zero-dimensional model; 1', 2', 3' – diffusion model, E(a,t); 1'', 2'', 3'' – diffusion model, E(0,t).

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Second, as discussed above and follows from Figure 50(b), an allowable temperature increase of the slab is not identical when its thickness increases under the condition S=const. As a result, the values of the stable currents decrease with increasing this quantity due to the thermal degradation mechanism. In this case, the variation of the quench temperature has an inverse tendency: the higher the thickness of a slab, the lower the allowable temperature increase. This result demonstrates the strong temperature effect on the condition of the thermal runaway onset. Under the considered size variation of the superconducting slab, thermal degradation of its current-carrying capacity assembles about 16%. Moreover, the stable temperature increase of a slab depending on its geometry may also influence on the value Ic even in the voltage range that is close to the usually used quantity 10-6 V/cm. This peculiarity is pictured in Figure 51 for the group of the traces discussed above. In these cases, Ic =56.7 A at T=4.2 K. Figure 51 shows that the higher the thickness of a slab, the higher the allowable temperature increase in the voltage range 10-6 V/cm and the lower the corresponding values of the currents Ic defined at Ec in the framework of different operating regimes. Figure 52 demonstrates the existence of possible difference between the transient temperature simulations based on the zero- and one-dimensional models that influence on the determination of the thermal runaway boundary. The calculations were made in the framework of the zero-dimensional (curves 1 - 6) and one-dimensional (curves 1' – 6') transient models. In this case, it was assumed that dI/dt=0 after the time, when the applied current equals a certain value I0. According to Figure 52, the difference becomes visible during over-critical modes and increases when the thickness of a slab decreases. Physically, this is owing to the effect of a thermal conductivity of composite: the lower the thickness, the higher the thermal conductivity effect. However, the presented results depict that the observed difference, as a whole, is not principle. Therefore, the transient zero-dimensional model allows one to describe the thermo-electric modes of high-Tc composite superconductors with satisfactory accuracy in many practical applications. The above-presented results allow us summarized the results of the usage of the proposed simulation models. The performed study shows that the electric field evolution may be described with acceptable accuracy in the framework of the simplified scaling model in the partially penetrated stage even under the conduction-cooled conditions. In the meantime, the description of the fully penetrated current modes depends, as a whole, on the governing models. In this case, the following trends exist. The current redistribution in the initial complete penetration regime may be simulated only by the diffusion model due to the nonuniform electric field distribution taking place during this mode. However, the relevant further stable modes, which take place after the transient period and are not close to the thermal runaway boundary, are described in an equivalent way by diffusion, zero-dimensional transient and static models. The analysis of the thermo-electric regimes occurring just before and after thermal runaway depend on the simulation methods because of unavoidable stable temperature increase of a composite before the onset of the thermal runaway. This thermal variation of the electric states is a function of the transverse geometry of a composite. Therefore, the development of the static and transient modes before and after thermal runaway is different even under the criterion ha/λ ti+Δt. Figure 57(a) shows the stable temperature variation before and after intensive heat pulse (Qv=5 W/cm) with the duration Δt=0.1 s calculated at h=3×10-3 W/(cm2K), RRR=10, B=10 T, dI/dt=10 A/s and T0=4.2 K. The charged currents I(ti) did not exceed the maximum current, which is equal to 64.462 A in the second stable part on the voltage-current characteristic depicted by curve 2 in Figure 22. The formation of the additional stable state by the heat pulses having different energies for the current I(ti)=64 A, which belongs to the mentionedabove second stable part of the voltage-current characteristic, is shown in Figure 57(b). The presented results demonstrate clearly the thermal stabilization role of the additional stable part of the voltage-current characteristic and its influence on the thermal stability conditions. In particular, the latter may have many - valued energy boundary leading to both the stable and unstable states.

5.6. Conclusion The analysis of transient electric and thermal modes of high-Tc superconducting composite indicates the existence of formation peculiarities of their voltage-current and temperature-current characteristics. The over-critical parts of the E(I) and T(I) curves depend on the current sweep rate and operating temperature during continuous fully penetrated current charging. This dependence becomes more visible with increasing dI/dt and T0. It is due to the allowable temperature rise of composite before thermal runaway onset, which may noticeably increase its heat capacity. The temperature dependence of the matrix resistivity also has an influence on the formation of E(I) and T(I) curves. As a result, the differential resistivity of a high-Tc superconducting composite has only positive value during continuous

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current charging, both before and after thermal runaway. Therefore, the slope of VCC decreases when the quantities dI/dt and T0 increase. These peculiarities form the basis of a proper understanding of the current charging method with break, which is usually used in experiments and allows one to determine the thermal runaway boundary. In this method, the final stable values of electric field and temperature are the result of their drift to the relevant static quantities described by the static approximation. The transient behaviour of this phenomenon is determined by the temperature dependence of the properties of composite as well as by the current sweep rate. The thermal runaway condition of the fully penetrated current may be defined in the framework of static as well as transient models. Both models lead to equivalent stability criteria, although they are based on different mechanisms: the static variation of the differential resistivity or the transient relationship between the cooling power and heat generation in superconducting composite. As a whole, these models allow to solve the thermal runaway problem for both low-Tc and high-Tc superconducting composite from the common point of view based on the energy balance analysis (steady or temporary) taking the physical peculiarities of the investigated phenomenon into consideration. It is shown that there exist both low and high voltage boundaries of the voltage-current characteristics of a high-Tc superconducting composite. When these boundaries do not take into consideration then one may get incorrect definition of the critical current during continuous current charging. These peculiarities become more visible with increasing current charging rate, coolant temperature, volume fraction of the superconductor in a composite or its thickness and width. In particular, the allowable low-voltage boundary shifts to higher values, if the thickness of a tape increases or its width decreases. The upper-voltage boundary depends on the permissible stable temperature variation of a tape that is function, for example, of its thickness if b >> a. As a result, the voltage criteria used to define the critical currents of high-Tc superconducting composite must be selected inside these boundaries. Under this condition, the operating modes of a composite are characterized by practically uniform distributions of the electric field, charged current and temperature in a composite. Besides, the main part of the charged current flows in the superconducting core and the temperature of a composite is close to the cooling bath temperature. However, even under this voltage criterion, the allowable temperature increase of a superconducting composite may change the isothermal mode of the electric field evolution inside superconductor and, therefore, influences on the value of the current defined during experiment according to the a priori chosen critical value of Ec. Unavoidable temperature variation of the operating modes depending, in particular, on the thickness of a tape is accompanied by the relevant degradation of its current-carrying capacity. As a result, the quench currents will not proportionally increase with increasing thickness of a tape. Thereby, the higher this quantity, the higher the thermal degradation. This feature should be taken into consideration when the operating parameters of the currentcarrying elements of superconducting magnets are prognosticated.

6. Resume In principle, the temperature of high-Tc superconducting composite is not equal to the cooling bath temperature before the thermal runaway. As a result, its stable overheating

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influences on the formation of the voltage-current and temperature-current characteristics of a composite changing their slope during transient regimes. That is why the transient voltagecurrent characteristics of high-Tc superconducting composite cannot determine the boundary of the thermal runaway. An unavoidable temperature rise of superconducting composite leads also to the thermal degradation mechanism due to which the relevant decrease in the quench current occurs. As a result, the quench currents do not increase proportionally to the proportional increase in the critical current of a composite. There exist low and upper voltage boundaries, inside which the critical current may be defined. They depend on the volume fraction of the superconductor in a composite, thickness of a superconducting composite and current charging rate. All of them effect on the stable variation of the temperature of composite during fully penetrated states. As a result, a priori fixed electric field criterion, for example, Ec=10-6 V/cm does not define the boundary of the stable current regimes of high-Tc superconducting composites. Moreover, their operating stable modes may be both sub-critical and over-critical. In particular, the charged current may be both sub-critical and over-critical when an acceptable electric field is over-critical. The sub-critical electric and current quantities are probably in the high magnetic field or when a composite with high-resistivity matrix has a relatively high value of the volume fraction of a superconductor. Such modes become unstable below the critical point {Ec, Jc} because the thermal runaway boundary is a result of the unavoidable overheating of a superconducting composite due to of the smooth voltage-current characteristic of a superconductor. During over-critical modes, the stable values of the electric field change in a wide range. Therefore, an unavoidable overheating of a superconductor may be noticeable. In particular, when the composite has the low-resistivity matrix it may exceed more than 10 K. In such cases, the current sharing underlies the current-carrying capacity of a composite. The thermal runaway phenomena in superconducting composites may be ensured by odd peculiarities depending on the temperature dependence of the critical current. Their existence is due to the coupled thermal and current-sharing mechanisms. As a result, the thermal runaway conditions may differ from the generally accepted stability conditions of the conventional superconducting materials. As a result, the many-valued stable operating states may appear. They are conditioned by the modification of the voltage-current and temperaturecurrent characteristics, which may differ from the standard characteristics having only one stable and unstable branches. As a result, in the approximation of the infinitely low sweep rate, the E-J and T-J traces may have the discontinuous parts describing the jumps of the electric field, current and temperature without the transition of the superconducting composite into the normal state. In the cases of the external thermal disturbances, the multi-stable regimes may be characterized by the multi-valued critical energies. Moreover, the essential increase of the stable branch of the voltage-current characteristics may be observed. The latter may lead to the stable current penetration into a composite during which its temperature stably increases up to the critical temperature of a superconductor even under the conductioncooled conditions. Thus, in the case of high-Tc superconducting composites, the conventional fixed electric field criterion and the corresponding value of the critical current do not have a meaning as the limiting current parameters. This feature must be considered during electrodynamics investigations of high-Tc superconductors, and first of all, when AC losses in high-Tc superconducting composites are analyzed.

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References [1] Nishijima, G.; Awaji, S.; Watanabe, K.; et al. Proc. Inter. Cryo. Eng. Conf. (ICEC) Prague 2007 261-264. [2] Watanabe, K.; Awaji, S.; Sci. Rep. RITU 1996 vol. A42, 371-373. [3] Watanabe, K.; Yamada, Y.; Sakuraba, J.; et al. Jpn. J. Appl. Phys. 1993 vol.32, L488L490. [4] Nishijima, G.; Oguro, H.; Awaji, S.; et al. IEEE Trans. Appl. Supercond. 2008 vol.18, in press. [5] Watanabe, K.; Motokawa, M. IEEE Trans. Appl. Supercond. 2001 vol.11, 2320-2323. [6] Wilson, M. N. Superconducting Magnets; Clarendon Press, Oxford, 1983. [7] Kalsi, S. S.; Aized, D.; Connor, B.; et al IEEE Trans Appl Supercond 1997 vol.7, 971975. [8] Kumakura, H.; Kitaguchi, H.; Togano, K. Cryogenics 1998 vol. 38, 163-167. [9] Kumakura, H.; Kitaguchi, H.; Togano, K. Cryogenics 1998 vol. 38, 639-643. [10] Kiss, T.; Vysotsky, V.S.; Yuge, H.; et al. Physica C 1998 vol. 310, 372–376. [11] Rakhmanov, A.L., Vysotsky, V.S.; Ilyin Yu. A.; et al. Cryogenics 2000 vol. 40, 19-27. [12] Seto, T.; Murase, S.; Shimamoto, S.; et al., Cryog Eng 2001 vol. 36, 60-67 (in Japanese). [13] Nishijima, G,; Awaji, S.; Murase, S. et al IEEE Trans Appl Supercond 2002 vol. 12, 1155-1158 . [14] Nishijima, G.; Awaji, S.; Watanabe, K. IEEE Trans Appl. Supercon 2003 vol. 13, 15761579. [15] Miyazaki, H.; Harada, S.; Iwakuma, M. IEEE Trans Appl Supercond 2005 vol.15, 16631666. [16] Anderson, P.W.; Kim, Y.B. Rev Mod Phys 1964 vol. 36, 39-43. [17] Beasely, M.R.; Labusch R.; Webb, W.W. Phys Rev 1969 vol. 181, 682-700. [18] Clark, A.F.; Ekin, J.W. IEEE Trans Mag 1977 vol. 13, 38-40. [19] Goodrich, L.F.; Fickett, F.R. Cryogenics 1982 vol. 22, 225-241. [20] Polak, M.; Hlasnik, I.; Krempasky, L. Cryogenics 1973 vol. 13, 702-711. [21] Edelman, H.S.; Larbalestier, D.C. J. Appl Phys 1993 vol. 74, 3312-3315. [22] Blatter, G.; Feigel’man, M.V.; Geshkenbein, V.B. et al Rev Mod Phys 1994 vol. 66, 1125-1388. [23] Yamafuji, K.; Kiss T. Physica C 1996 vol. 258, 197-212. [24] Caplin , A.D.; Bugoslavsky, Y.; Cohen, L.F.; Perkins, G.K. Physica C 2004 vol. 401, 16. [25] Rhyner, J. Physica C 1993 vol. 212, 292-300. [26] Gilchrist, J.; van der Beek, C.J. Physica C 1994 vol. 231, 147-156. [27] Gurevich, A. Int J of Modern Phys 1995 vol. B9, 1045-1065. [28] Brandt, E.H. Phys. Rev. 1996 vol. B54, 4246-4264. [29] Dresner, L. Cryogenics 1998 vol. 38, 205-209. [30] Romanovskii, V.R. Physica C 2003 vol. 384, 458-468. [31] Romanovskii, V.R. Eur Phys J 2003 vol. B33, 255-264. [32] Watanabe, K.; Awaji, S,; Motokawa, M.; et al. Jpn. J. Appl. Phys. 1998 vol.37, L1148L1150.

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Watanabe, K.; Awaji, S. J Low Temp Phys 2003 vol. 133, 17-30. Wetzko, M.; Zahn, M.; Reiss, H. Cryogenics 1995 vol. 35, 375-386. Lehtonen, J.; Mikkonen, R.; Paasi, J. Physica C, 1998 vol. 310, 340-344. Lehtonen, J.; Mikkonen, R.; Paasi, J. Supercond Sci Technol 2000 vol. 13, 251-258. Majoros, M.; Glowacki, B.A.; Campbell, A.M. Physica C, 2002 vol. 372-376, 919-922. Rettelbach, T.; Schmitz, G.J. Supercond Sci Technol 2003 vol. 16, 645-654. L. Bottura, Note-CRYO/02/027, CryoSoft library, CERN, 2002. van der Laan, D.C.; van Eck, H.J.N.; ten Haken, B.; et al. IEEE Trans Appl Supercond 2001 vol. 11, 3345-3348. Wesche, R. Physica C 1995 vol. 246, 186-194. Dresner, L. Cryogenics 1993 vol. 33, 900 -909. Lim, H.; Iwasa, Y. Cryogenics 1997 vol. 37, 789-799. Brentari, E. G.; Smith, R. V. Adv. Cryog. Eng. 1965 vol. 10, 325-341 Romanovskii, V.R. Cryogenics 2002 vol. 42, 29-37. Keilin, V.E.; Romanovskii V.R. Cryogenics 1993 vol. 33, 986-994.

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[41] [42] [43] [44] [45] [46]

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SHORT COMMUNICATION

Superconducting Magnets and Superconductivity: Research, Technology and Applications : Research, Technology and Applications, Nova Science

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In: Superconducting Magnets and Superconductivity… ISBN: 978-1-60741-017-1 Editors: H. Tovar and J. Fortier, pp. 403-414 © 2009 Nova Science Publishers, Inc.

THERMO-MECHANICAL PUMPS FOR A LARGE SUPERCONDUCTING MAGNET IN SPACE OPERATED BY USE OF SUPERFLUID HELIUM G. Kaiser, A. Binneberg and J. Klier Institut für Luft- und Kältetechnik gGmbH, Hauptbereich Kälte- und Tieftemperaturtechnik, Bertolt-Brecht-Allee 20, D-01309 Dresden, Germany

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Abstract In the Alpha Magnetic Spectrometer-02 experiment charged particles coming from cosmic radiation are separated in a large scale superconducting magnet. Two thermo-mechanical pumps (TMPs) are used to supply the current leads and the magnet coil with superfluid helium. These pumps, which use the Fountain-effect for operation, have been developed at the ILK Dresden. Due to the application the TMPs are required to pump a mass flow of 0.2 g/s. This mass flow rate is pumped from a superfluid helium reservoir at 1.8 K against a pressure head of 200 mbar. This equates to a temperature of 3.2 K in evaporation equilibrium. It is not possible to operate a TMP directly against an evaporator at a temperature higher than the lambda point (2.17 K). Therefore a special separator, operating by use of kinetic energy concurrence, was developed to interrupt the superfluid quantum state between the exits of the TMPs and the entrance of the evaporator. After introducing into basic principles essential for the function of the TMPs we will report about the development and tests of the TMPs in combination with the superfluid quantum state suppressor and the evaporator.

Introduction The Alpha Magnetic Spectrometer-02 (AMS-02) is a device, used in space to proof the fundaments of the Big Bang theory [1]. It containes a large superconducting magnet which operates at a temperature of 1.8 K, cooled by superfluid helium. The magnet is electrically supplied by current leads which transport the charge current to establish the magnetic field. The magnet power supply is at a temperature close to 300 K. The current leads for the magnet are actively cooled by a stream of helium with a mass flow rate of 200 mg/s. This mass flow

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rate evaporates at an absolute pressure of 200 mbar. The helium gas flows inside the heat exchanger of the current lead in high-temperature direction. The sensible heat is used to compensate the Joule losses of the electric current in order to keep the temperatures of the current lead in a stationary state. The reservoir for the helium supply of the current leads is a superfluid helium tank, which is at a temperature of 1.8 K and the corresponding vapor pressure of 16.3 mbar. In order to generate the entrance pressure of 200 mbar for the current lead evaporator a pump is required. In the AMS-02 experiment two thermo-mechanical pumps are used for this purpose. It is the second application of a TMP in space world wide. The first space operation of a TMP took place during the Superfluid Helium On Orbit Tranfer (SHOOT) mission [2]. The TMP was used to study the possibilities of the transfer of superfluid helium (SFH) in space by pumping between two separated vessels. The application of the TMP in the AMS-02 mission exceeded the aims of the SHOOT mission by far. During the first mission the SFH was transported against the relatively low flow resistance of the piping between the vessels. The required absolute pressure was small compared to the vapor pressure of helium at its lambda point (2.17 K, 49 mbar). Furthermore, in AMS-02 it is necessary to evaporate behind the TMP exit to generate the gas flow. In the SHOOT mission the SFH was kept in the liquid state all along the transport path. So, for AMS-02 two problems had to be solved: The TMP has to operate against an evaporating surface and the evaporation has to take place at a pressure which is higher than the vapor pressure achievable in the superfluid state. The first problem can cause for a rise in the power consumption of the TMP heater. The second problem is impossible to overcome with a closed superfluid column between the TMP exit and the evaporating surface. In order to overcome both problems with the same solution we developed a mean to interrupt the superfluid state locally between the TMP exit and the evaporating surface. After introducing into design and function of a TMP and the superfluid quantum state suppressor we present the results of our experimental work on TMP with a current lead thermal dummy.

Principle and Operation A schema of the TMP is shown in Figure 1. In its simplest case it consists of a tube containing a porous plug and an electric heater in the volume next to the exit of the TMP. SFH can be considered as a mixture of a superfluid and a normal fluid component with temperature dependent concentrations. A heat flow Q leads to a thermal non-equilibrium state and a part of the superfluid is converted to normal fluid. In order to re-establish the thermal equilibrium the superfluid component of the reservoir vessel crosses the porous plug. Normal helium reflux is suppressed by the micro-porous structure of the plug. In this way  is transported which can be expressed by: a mass flow m

m =

Q . ST

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(1)

Thermo-Mechanical Pumps for a Large Superconducting Magnet…

405

S and T are the specific entropy and the absolute temperature of the superfluid and normal fluid helium mixture, respectively. According to investigations by Nakai et al. [3] the equation (1) yields the so called laminar flow regime characterized by an upper critical value m c dependent on the fluid density ρ, the porosity ε, the cross sectional area A and the average pore diameter d:

m c = 2.4 × 10 −3 [ m 5 / 4 s −1 ]

ρεA 4

d

(2)

superfluid helium flow

porous plug separator

heater

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 1. Schematic drawing of a thermo-mechanical pump.

Under the influence of a pressure head Δp across the separator a temperature drop ΔT will be generated:

ΔT =

1 Δp ρS

. (3)

 r across the separator is given by basic In this case the normal helium reflux m considerations of fluid dynamics:

m r =

ερAd 2 Δp . 32ω 2ηl

(4)

η and ω represent the viscosity of the normal fluid component and the tortuosity of the separator, respectively. In order to separate the influence of the evaporator from the exit of the TMP a special type of a Gorter-Mellink channel was used. A Gorter-Mellink channel is usually a device used to generate a temperature drop across a column of not moving SFH. It uses the effect that the superfluid component of the SFH mixture cannot carry heat, so that the heat flow from the warm to the cold end of an SHF column is provided by a macroscopic mass flow of

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the normal fluid component. This mass flow carries the corresponding enthalpy flow. The normal flow does not interact with the superfluid flow, so that it can be considered as a separate flow with a certain flow velocity. The difference of the concentration of the normal component between the warm and the cold end ΔC N can be considered as a partial pressure difference Δp N of the normal flow, which is expressed as:

Δp N = pΔC N

(5),

with p as the absolute pressure. The flow velocities in the Gorter-Mellink channel are very small so that a laminar flow regime can be considered. The flow velocity v N therefore is: 2

vN =

rC Δp N 8ηl C

(6),

where rC and lC are radius and length of the Gorter-Mellink channel. In the special case of

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

the AMS-02 setup the Gorter-Mellink channel was modified in a certain way. The width of the channel was designed in a way that the mass flow, driven by the TMP and consisting of normal and superfluid components, propagates with a velocity at which the mass flow of the normal component is at least equal in both directions. In this case the heat flow across the Gorter-Mellink channel is interrupted and acts as a thermal insulator. Because of A. Hofmann (Karlsruhe Research Center), who encouraged us to follow our idea and verify it by use of experiments, we have called this special version of the GorterMellink channel “Hofmann channel” [4].

Explanation of the Experiment The fundamental problem and its solution to operate a TMP under evaporation equilibrium conditions at its exit is illustrated in Figure 2. Without the “Hofmann channel” the theoretical upper temperature limit at the exit of the TMP is 2.17 K, the lambda point. In order to operate a TMP against a larger absolute pressure and under conditions of evacuation equilibrium a mean to separate locally and functionally the evaporation from the TMP transport function is necessary. A “Hofmann channel” acts as a separator device. The evaporation takes place under normal fluid conditions at a pressure of 200 mbar and a temperature of 3 K. The TMP exit operates under sub-cooled SFH conditions at a temperature of 1.9 K and a pressure slightly higher than 200 mbar. Considering a temperature dependence of the specific entropy according to a T4-law the decrease of the TMP exit temperature from 2.17 K down to 1.9 K improves the pumping performance of the TMP according to equation (1) dramatically.

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Thermo-Mechanical Pumps for a Large Superconducting Magnet… TMP

a

407

load (current lead)

1.8 K

2.17 K (max)

16 mbar

49 mbar TMP

load (current lead)

Hofmann channel (vortex tube)

b

1.8 K

1.9 K

16 mbar

200 mbar +

3K 200 mbar

Figure 2. Operating conditions of a thermo-mechanical pump under evaporation equilibrium conditions without (a) and with a Hofmann channel (b)

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Experimental Investigations In order to get experimental experiences with the TMP a breadboard model of the TMP was designed, manufactured and tested. The sub-components of the breadboard TMP and the TMP after assembly are shown in Figure 3. The separator of the TMP (not visible in Figure 3) consists of a SIKA-R-0.5 micro-porous stainless steel element with a diameter of 20 mm and a thickness of 10 mm.

(a) Figure 3. Continued on next page.

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(b) Figure 3. Sub-components of the breadboard TMP (a) and the assembled TMP (b)

The breadboard TMP was tested according to the ascending tube method published by Murakami and Nakai [5]. The test setup is shown in Figure 4. P2 P1 safety valve

pressure regulation valve

bypass valve

to GHe supply

VP1 LD to LHe tank VP2 LN2 to HB to HR

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

to VP1

to VP2

Twin Power Supply

Vacuum Ruta 501

to P1 Test rod

to P2 to T1

to TB Ascending Tube to H1

Vacuum PT 150

LHe II

to H2

Kistler Monitor

Temperature Controller

Voltage

TMP Controller

Current

T1, T2, H1, H2 Legend

HB

Twin Power Supply

TMP

Test setup Test setup controls TB TMP controls Measuring instrumentation

Hx = heater Tx = thermometer

HR to LD

Leak Detector

Px = pressure transducer

Figure 4. Test setup for the experimental investigation of the breadboard TMP

The experimental results of the first breadboard TMP tests are shown in Figure 5. Although the separator of the TMP is designed to operate at a mass flow of 200 mg/s in laminar flow regime the TMP was not able to transport this mass flow in the experiments. Furthermore, the mass flow to heat flow ratio was not consistent with equation (1), this is a hint that heat is consumed by another process. This parasitic process has been identified as surface evaporation. A part of the heater power to drive the TMP gets transported by GorterMellink heat conduction to the surface of the SFH inside the ascending tube and is consumed

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409

by the latent heat of evaporation. This influence could be proven by opening the bypass valve to the pumping system of the cryostat in order to enforce the surface evaporation. Therfore the ascending tube method is useful under subcooled conditions inside the whole system only.

Mass Flow Rate [mg/s]

80

60

40 Bypass closed Bypass opened 20

0

0

250

500

750

Heater Power [mW]

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 5. Experimental results for the first tests of the breadboard TMP

Figure 6. Separator materials for the use with a thermo-mechanical pump (from left SIKA-R 0.5 (stainless steel), VitraPOR (borosilicate glas shrinked in stainless steel housing), Cellpore (PTFE)).

In order to improve the overall performance of the TMP system a comparison between different separator materials, shown in Figure 6, was calculated (see Table 1). The calculated results for the critical mass flow and the normal helium reflux for the materials which can provide 200 mbar are shown in Table 2.

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410

G. Kaiser, A. Binneberg and J. Klier Table 1. Comparison of the separator parameters Material SIKA-R 0.5 VitraPOR1 Cellpore2

Average pore diameter 3.5 µm ≤ 1 µm ≤ 1 µm

Porosity 53 % 30 % 40 %

Table 2. Calculated performance of the TMP separators

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Material VitraPOR Cellpore

Critical mass flow 1037 mg/s 1380 mg/s

Normal helium reflux 23 mg/s 39 mg/s

Figure 7. Qualification model of the thermo-mechanical pump.

Figure 7 shows the qualification model of the TMP, used for the performance tests with the “Hofmann channel”. The different tested “Hofmann channels” are shown in Figure 8. Because of the problem to estimate the fraction of the normal fluid helium after the TMP exit and after the “Hofmann channel” the optimization of the “Hofmann channel” was performed experimentally. The channel with 2.7 mm inner diameter turned out to be most suitable. The channel with 0.6 mm inner diameter showed a large pressure drop and the channel with 5 mm inner diameter was not capable to interrupt the superfluid state between the TMP exit and the evaporator.

1 2

VitraPOR is a registered trade mark of ROBU GmbH Cellpore is a registered trade mark of Esters GmbH

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Figure 8. Selection of Hofmann channels with a length of 40 mm and a diameter of 5 mm, 2.7 mm and 0.6 mm (from left).

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 9. Test rod assembly consisting of the evaporator, the Hofmann channel, and the TMP (from left). P2 P3 P1 safety valve

Current Lead Dummy H4 Nozzle VP3 T4

VP1

to P5

VacuumMonitor

P4 P5 to P3

VacuumMonitor

to P4

VacuumMonitor

to P2

VacuumMonitor

VP2 LN2 to H3 to H1

to VP3

Twin Power Supply

Vacuum Ruta7001

to H4

H4 Power Supply

to P1

VacuumMonitor

Vacuum Ruta 2001

to T1

to VP1

Temperature to T3 Controller to T4

to VP2

T3 Evaporator to H2 H3 Hofmann Channel

Vacuum PT 150

Legend

to T2

H1

Test setup Test setup controls TMP controls T1 Measuring instrumentation

H2, T2 TMP LHe II

Voltage

TMP Controller

Twin Power Supply

Temperature Controller

Current

Hx = heater Tx = thermometer Px = pressure transducer

Figure 10. Test setup for the experimental investigation of the TMP qualification model.

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G. Kaiser, A. Binneberg and J. Klier

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 9 shows the test rod assembly consisting of the evaporator, the “Hofmann channel”, and the TMP. The test setup for the performance tests is shown in Figures 10 and 11. The test rod with the TMP was inserted into a partially optical transparent glass dewar with a liquid nitrogen jacket and an inner dewar for SFH. The inner dewar was pumped by a 2500 m³/h roots pump with a rotary vane pre-pumping system. The exit of the evaporator was connected to a 3.6 m long heated aluminum tube with 9 mm inner diameter. This tube leaves the inner dewar at its top flange ending at an orifice flow meter with an inner diameter of 3.6 mm determining a choked mass flow of 200 mg helium at 200 mbar absolute pressure at 293 K. This aluminum tube represents the thermal dummy of the current lead for the superconducting magnet in the AMS-02 experiment. The exit of the orifice was connected to a 7500 m³/h roots pump with rotary vane pre-pumping system which simulates the exit to space.

Figure 11. Test setup for the performance tests of the thermo-mechanical pump.

The pressure was recorded in the dewar, at the exit of the TMP, at the exit of the evaporator as well as in front of and behind the orifice. The temperature was taken in the SFH of the inner dewar, at the exit of the TMP, inside the evaporator, and in front of the orifice.

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Channel ID = 2.7 mm L = 40 mm

Mass Flow Rate [mg/s]

300

200

100 evaporator 2.9 K evaporator 1.99 K evaporator 1.94 K 0

0

0.25

0.50

0.75

1.00

TMP Heater Power [W]

Figure 12. Mass flow of the TMP with Hofmann channel for different evaporator temperatures. Channel ID = 2.7 mm L = 40 mm 250

Pressure [mbar]

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

200

150

100 evaporator 2.9 K evaporator 1.99 K evaporator 1.94 K

50

0

0

0.25

0.50

0.75

1.00

TMP Heater Power [W]

Figure 13. Pressures inside the TMP test system (upper curve is the exit of the TMP, lower curve is the entrance of the orifice).

Figure 12 shows the experimental results for the mass flow. In the diagram it is visible, that it is important to operate the evaporator at a temperature where the whole mass flow transported by the TMP is evaporated immediately behind the “Hofmann channel”. If the evaporator capacity is not high enough to evaporate the mass flow transported by the TMP,

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G. Kaiser, A. Binneberg and J. Klier

the mass flow will be reduced with a further increase of the heater power of the TMP. This is because of the liquid helium floaded evaporator. The corresponding pressure drop is shown in Figure 13. The upper curve of each of the pairs shows the pressure directly behind the TMP exit, the lower curve is the pressure in front of the orifice.

Conclusion Within the frame of an international research project for the development of an experimental system to prove the Big Bang theory a TMP for the cryogenic supply of the current lead for a superconducting space magnet was calculated, designed, manufactured, assembled and tested. In order to overcome the problem of evaporation behind the exit of the TMP at a pressure larger than the vapor pressure of the superfluid transition, a special kind of Gorter-Melinck channel was developed. This channel separates the volumes before and behind itself by suppression of the superfluid state under the condition of a macroscopic mass flow into the direction of the higher temperature. The TMP was able to generate a mass flow of 200 mg/s at an absolute pressure of 200 mbar. This mass flow is sufficient for the cooling of the current lead.

Acknowledgement

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

The authors thank Dr. E. Ettlinger, LINDE, and Prof. A. Hofmann, FZ Karlsruhe for helpful discussion. This work is part of the AMS-02 mission. It is financially supported by the Eidgenössische Technische Hochschule Zürich.

References [1] http://ams.cern.ch [2] DiPirro, M.J., Shirron, P.J., Tuttle, J.G., Mass Gauging and Thermometry on the Superfluid Helium on Orbit Transfer Flight Demonstration, Advances in Cryogenic Engineering (1994), 39 129-135 [3] Nakai, H., Kimura, N., Murakami, M., Haruyama, T., Yamamoto, A. Transition to turbulent state of superfluid helium flow through thermomechanical pump element, Advances in Cryogenic Engineering (1996), 41 273-280 [4] Kaiser, G., Schumann, B., Stangl, R., Binneberg A., Wobst, E., New Achiewment of 200 mbar TMP exit pressure under evaporation, Presentation at the AMS-02 Technical Interchange Meeting, Houston, TX, September 2004 [5] Kimura, N., Nakai, H., Haruyama, T., Murakami, M., Yamamoto, A. Study of a porous element for a thermomechanical pump in superfluid helium, Advances in Cryogenic Engineering (1992), 37 133-138

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INDEX

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

A Abrikosov, ix, 215, 216, 217, 219, 220, 222, 223, 225, 230, 240, 250, 254 absorption, 28, 207 acceptor, 14, 22 access, 282, 283, 284, 286 accidental, 45 accounting, xi, 33, 294, 329, 368, 395 accuracy, 98, 105, 116, 173, 176, 285, 287, 369, 382, 387 achievement, 212 acid, 258, 269 activation, 19, 22, 23, 24, 25, 27, 37, 40, 42, 43, 45, 47, 99, 288 activation energy, 19, 23, 27, 37, 40, 42, 43, 45, 47, 99 additives, 57, 74, 199 adiabatic, 44 AFM, 122, 124, 125, 152, 153, 265, 266, 271, 272 aging, 258, 286 air, 59, 67, 68, 175, 199, 201, 203, 208, 209, 210 alkali, 5 alloys, 78, 277, 284 alternative, ix, 23, 120, 197, 199, 223, 235, 242, 278 aluminum, 288, 289, 412 ambient pressure, 29 ambiguity, 199 ammonia, 6 amorphous, 201, 202, 266, 271 amplitude, x, 29, 171, 215, 216, 217, 219, 233, 235, 236, 237, 238, 241, 244, 250, 251, 252 angular momentum, 37, 40, 47 anisotropy, vii, 3, 57, 79, 87, 88, 92, 93, 94, 135, 141, 170, 182, 198, 294 annealing, x, 60, 62, 63, 64, 67, 87, 94, 155, 175, 199, 202, 205, 206, 208, 211, 258, 259, 272, 279, 285 annihilation, 98, 219 anomalous, 171 antibonding, 13, 14, 15, 16, 18, 33, 35, 41, 44 antiferromagnetic, 5, 6, 9, 11, 23, 25, 36, 79

appendix, 246 application, xi, 84, 85, 97, 98, 99, 111, 144, 184, 216, 294, 295, 403, 404 APRIL, 212 aqueous solution, 12, 18 Arrhenius equation, 22 arsenide, 90 ASI, 52 aspect ratio, 198 assessment, 276 assignment, 16 assumptions, 350, 363 ASTM, 98 ATLAS, 174 atmosphere, 67, 201, 258, 266, 269, 270, 271 atmospheric pressure, 199 Atomic Energy Commission, 176 atomic force, 125 atomic force microscope, 125 atomic orbitals, 5, 8, 13, 15, 32 atoms, 4, 8, 9, 12, 18, 30, 31, 32, 33, 34, 35, 36, 37, 43, 44, 84, 172, 200 attachment, 108, 146 automation, 284

B band gap, 19, 46 barium, 82, 278 barrier, 19, 23, 24, 25, 33, 40, 42, 235, 296 barriers, 91 basic research, 278 basis set, 32 behavior, ix, 44, 47, 169, 170, 171, 172, 173, 178, 180, 190, 191, 193, 209, 212, 296, 301, 306, 310, 321, 330, 331, 335, 337, 341, 342, 344, 347, 348, 349, 359, 366 behaviours, 370 bending, 113, 121, 282 benefits, 278 biaxial, x, 257, 259, 272 Big Bang theory, 403, 414 binding, 17

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416

Index

binding energy, 17 bismuth, 201, 203, 278 Boeing, 275 boiling, 78, 278, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 362, 363, 365 bonding, 8, 13, 15, 16, 18, 33, 35, 41, 47 bonds, 11, 20, 35, 91 Boron, 173, 291 boron-doped, 171 Bose, 41, 44 Bose-Einstein, vii, 3, 44, 47 Boston, 214 boundary conditions, 216, 217, 219, 223, 250, 253, 367 Brazil, 197 breathing, vii, 3, 6, 19, 20, 26, 27, 30, 31, 41, 44 buffer, x, 102, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 269, 271, 272, 273 building blocks, 252 bulk materials, 111, 159, 281, 295 bypass, 409 by-products, 82

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

C Ca2+, 56 cables, 78, 283, 284, 287 CAD, 285, 287, 289, 290 calcium, 202, 208, 278 Canada, 256 candidates, 269 capacity, xi, 105, 176, 199, 293, 304, 312, 321, 327, 328, 329, 335, 338, 341, 342, 343, 348, 355, 362, 366, 367, 369, 374, 377, 382, 384, 385, 387, 388, 389, 390, 391, 393, 395, 396, 397, 413 capital cost, 281 carbide, 100, 281 Carbon, 165, 291 carbonates, 86, 175, 202 carrier, 60, 62, 63, 64, 172, 295 cast, 289 casting, 202, 285, 289 catalysis, 44 category a, 253 Catholic, 197 cation, 58, 266 cavities, 6 cell, 79, 87, 227, 228, 230, 244, 247, 248 ceramic, viii, ix, 77, 87, 108, 111, 114, 118, 135, 139, 141, 144, 197, 199, 202, 211, 278, 279 ceramics, viii, 77, 103, 118, 119, 120, 141, 202, 277 CERN, 399 channels, 410, 411 charge density, 3, 5, 6, 14, 17, 33, 41 charged particle, xi, 235, 403 chemical composition, 6, 202 China, 215, 257, 273 classical, 87, 116, 235, 242, 252, 282

classical mechanics, 242 clusters, 30 coherence, 57, 79, 171, 172, 182, 226, 228, 232, 238, 239, 246, 294 coil, xi, 277, 278, 282, 283, 284, 285, 286, 287, 288, 289, 294, 295, 326, 327, 403 colors, 28 combined effect, 252 commercialization, 198 compatibility, 118, 284 compensation, 24 competition, x, 91, 215, 244, 251 complexity, 199 compliance, 98, 369 complications, 100 components, 26, 43, 99, 237, 240, 241, 276, 280, 285, 287, 289, 290, 406 composites, viii, x, 77, 87, 88, 93, 135, 293, 295, 311, 314, 324, 327, 331, 332, 335, 338, 342, 348, 349, 362, 366, 374, 375, 377, 393, 397 composition, 6, 60, 67, 70, 72, 173, 180, 200, 202, 203, 204, 205, 206, 208, 209, 276, 277, 280, 281, 285 compounds, vii, viii, 3, 4, 5, 17, 25, 27, 49, 55, 56, 57, 81, 82, 87, 139, 141, 278 Compton electrons, 172 computation, 24 concentration, ix, x, 62, 67, 79, 82, 83, 84, 94, 95, 155, 159, 169, 171, 172, 186, 187, 188, 189, 193, 200, 201, 203, 204, 258, 261, 263, 264, 266, 272, 406 concordance, 146 condensation, 44, 47, 57 conductance, 67 conduction, 23, 42, 277, 294, 297, 397, 408 conductive, 277, 289 conductivity, 4, 5, 15, 16, 23, 24, 25, 27, 29, 37, 42, 45, 48, 96, 97, 327, 367, 369, 387 conductor, 276, 277, 278, 279, 280, 287, 295, 297, 339, 348, 350 configuration, 7, 11, 15, 32, 33, 45, 57, 280 confinement, x, 235, 249, 275, 285, 289 confusion, 5, 16, 199 connectivity, 182, 199, 209, 211 consensus, 216 conservation, 172, 227, 228, 230 consolidation, 199 constant rate, 90 constraints, 251 construction, 46, 78, 253, 278, 284 consumption, 404 contaminants, 285 contamination, 117 continuity, 223, 241, 244, 250, 252, 253 control, 57, 85, 98, 102, 276, 290, 327 conversion, 108 convex, 114 cooling, xi, 67, 68, 69, 85, 90, 93, 95, 96, 154, 158, 159, 161, 173, 176, 199, 200, 201, 202, 203, 206,

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Index 207, 208, 209, 277, 278, 281, 293, 297, 298, 302, 304, 305, 306, 307, 312, 317, 320, 321, 324, 327, 329, 330, 337, 338, 341, 342, 349, 350, 351, 353, 354, 355, 356, 357, 358, 365, 366, 367, 368, 369, 376, 377, 379, 380, 381, 388, 396, 414 Cooper pair, 47 coordination, 66, 251 copper, viii, 4, 6, 7, 17, 18, 30, 31, 32, 33, 34, 37, 44, 46, 77, 79, 94, 208, 276, 278, 287, 288, 289, 295 copper oxide, 4, 6, 44, 46, 94, 278 correlation, vii, 3, 4, 6, 7, 8, 9, 10, 11, 30, 31, 32, 34, 36, 40, 46, 47, 180, 193 corrosion, 285 corrosive, 201 Coulomb, 7, 11 coupling, vii, ix, 3, 4, 5, 6, 13, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 35, 40, 42, 44, 45, 47, 48, 56, 199, 215, 216, 241 coupling constants, 241 covalency, 5, 15 covalent, 91 crack, 87, 96, 111, 112, 113, 128, 132, 144, 145 cracking, 95, 111, 113, 114 creep, 99, 296, 314, 328, 329, 368 critical current, vii, ix, x, xi, 55, 56, 57, 64, 70, 80, 81, 86, 169, 170, 171, 173, 176, 181, 184, 185, 186, 189, 190, 191, 192, 193, 198, 257, 258, 269, 272, 277, 293, 294, 295, 296, 298, 300, 302, 306, 310, 312, 322, 323, 324, 327, 328, 329, 330, 331, 332, 342, 343, 344, 346, 348, 349, 350, 362, 363, 366, 367, 370, 382, 385, 388, 396, 397 Critical current density (Jc), vii, ix, x, 55, 81, 86, 169, 170, 171, 173, 176, 181, 184, 185, 186, 189, 190, 191, 192, 193, 210, 257, 294, 296, 298, 302, 322, 323, 324, 327, 328, 329, 342, 344, 349, 350, 367, 388 critical state, 176 critical temperature, 47, 170, 198, 200, 268, 277, 294, 295, 303, 327, 335, 342, 346, 348, 351, 359, 393, 397 critical value, 277, 396, 405 criticism, 7 cross-sectional, 81, 82 cryogenic, 121, 276, 277, 279, 282, 291, 414 crystal growth, 60, 67, 74 crystal lattice, 19, 52, 171 crystal structure, 57, 67, 79, 87, 294 crystalline, 24, 117, 118, 173, 174, 200 crystalline solids, 118 crystallization, 67, 200, 202, 203, 206, 272 crystals, vii, 5, 7, 55, 56, 57, 60, 61, 79, 81, 85, 86, 90, 95, 108, 176, 192, 199, 205 cuprate, 3, 18, 29, 33, 44 cuprates, vii, 3, 4, 5, 6, 7, 15, 16, 17, 18, 23, 25, 26, 27, 28, 30, 31, 32, 39, 42, 43, 44, 45, 46, 47, 48, 296 current limit, 78, 97, 200 cycles, 207 cycling, 97

417

cyclotron, 294

D damping, 99 data set, 103 de Gennes-Werthamer, ix, 215, 216, 252 decay, 370 decomposition, ix, 82, 93, 197, 198, 199, 200, 201, 202, 203, 205, 207, 208, 209, 210, 211 decomposition temperature, 207 defects, ix, x, 63, 81, 85, 87, 93, 137, 139, 153, 158, 161, 169, 172, 174, 181, 182, 184, 215, 216, 231, 232, 233, 251, 253, 271, 279, 296 deficiency, 56 deficit, 202 definition, 26, 46, 105, 240, 285, 287, 289, 290, 342, 370, 393, 396 deformation, viii, 57, 77, 87, 93, 98, 100, 101, 104, 105, 108, 111, 114, 115, 117, 118, 119, 120, 139, 145, 147, 285 degenerate, 7, 14, 15, 35, 36, 38, 217, 219, 220, 226, 236, 237, 251, 252 degradation, xi, 279, 293, 298, 325, 327, 329, 330, 332, 342, 348, 384, 385, 387, 396, 397 delocalization, 4, 5, 6, 22, 24, 25, 29, 40, 47 demand, 25, 100 density, vii, ix, x, 3, 6, 7, 18, 33, 55, 56, 63, 65, 74, 81, 86, 89, 118, 137, 139, 154, 156, 159, 169, 170, 171, 172, 173, 174, 176, 181, 184, 185, 186, 189, 190, 191, 192, 193, 197, 198, 199, 202, 210, 211, 219, 220, 237, 241, 244, 245, 250, 257, 277, 284, 285, 294, 296, 297, 298, 302, 322, 323, 324, 327, 328, 329, 342, 344, 346, 348, 349, 350, 352, 360, 362, 363, 364, 367, 373, 388, 405 density values, 81 Department of Energy, 290 deposition, x, 257, 258, 259, 269, 271, 276, 286, 287, 290 derivatives, 249, 304, 331 designers, 279 destruction, 351, 394 deviation, 108, 202, 203 DFT, 7, 9, 11, 12 diamond, 100, 106, 126, 173 diamonds, 101, 114 Dienes, 195 differential equations, 216, 221 diffraction, 171, 175, 187, 205, 258, 259, 261, 263, 265, 267, 271, 273 diffusion, 80, 82, 87, 90, 94, 155, 159, 222, 240, 241, 244, 252, 253, 257, 267, 368, 371, 372, 373, 382, 383, 384, 386, 387 dislocation, 84, 87, 88, 89, 91, 93, 117, 118, 135, 139, 141 dislocations, 81, 83, 87, 88, 89, 91, 92, 93, 98, 117, 118, 139 disorder, 171, 172, 177, 180, 181, 193

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418

Index

dispersion, 61, 64 displacement, 5, 46, 98, 100, 102, 103, 107, 108, 116, 126, 146, 147, 172, 279, 280 dissociation, 88 distortions, 57, 63, 282 distribution, 18, 24, 46, 63, 85, 95, 96, 99, 126, 136, 137, 172, 174, 297, 305, 332, 350, 367, 368, 369, 382, 387 diversity, 335 divertor, 280, 281, 283 dominance, 117 dopant, 45, 64 doped, ix, 5, 23, 27, 42, 44, 45, 57, 58, 59, 61, 62, 63, 64, 68, 70, 74, 169, 173, 184, 185, 186, 187, 189, 190, 193 doping, vii, viii, ix, 6, 18, 23, 26, 29, 41, 55, 56, 57, 60, 61, 63, 64, 65, 66, 68, 70, 75, 79, 169, 170, 171, 173, 186, 190, 193, 200, 203, 252 double bonds, 11 drying, 258, 270 ductility, 198 duration, 209, 210, 378, 393, 395

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E economics, 283, 290 eigenenergy, x, 215, 216, 217, 218, 222, 233, 235, 236, 237, 238, 239, 242, 244, 247, 251, 252, 253 eigenvector, 35, 37 elastic constants, 106 elastic deformation, viii, 77, 98, 119, 139 electric arc, 285 electric current, 97, 277, 404 electric field, xi, 170, 293, 296, 297, 298, 301, 302, 303, 306, 308, 310, 311, 312, 314, 315, 316, 317, 318, 320, 321, 323, 324, 325, 328, 331, 335, 337, 344, 348, 349, 351, 352, 354, 355, 357, 358, 359, 362, 363, 366, 367, 368, 369, 370, 371, 373, 377, 379, 382, 385, 386, 387, 388, 391, 392, 393, 396, 397 electric power, 97, 198 electrical properties, viii, ix, 78, 197 electrical resistance, 176 electricity, 280, 288 electromagnetic, 57, 276, 296 electromagnets, vii electron, vii, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 35, 36, 37, 38, 39, 40, 41, 43, 44, 46, 47, 48, 94, 125, 170, 172, 195, 252, 285 electron beam, 285 electron density, 33 electron microscopy, 143, 175 electron pairs, 4, 5, 6, 23, 28, 38, 44, 47 electronic structure, vii, 11, 12, 48 electron-phonon, 4, 45, 193

electrons, vii, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 21, 23, 24, 28, 29, 30, 32, 33, 35, 36, 37, 38, 39, 40, 41, 43, 44, 46, 47, 48, 79, 171, 172 emission, 17, 45, 46, 125 encouragement, 253 energy, vii, x, xi, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 32, 33, 37, 40, 41, 42, 43, 45, 46, 47, 48, 57, 78, 85, 120, 170, 172, 215, 216, 217, 218, 224, 233, 242, 243, 244, 251, 252, 260, 271, 275, 295, 339, 341, 342, 377, 379, 380, 394, 395, 396, 403 England, 50 enlargement, 271 entrapment, 83, 84 entropy, 405, 406 environment, 18, 102, 276, 279 epitaxy, 264 epoxy, 278, 280 equality, 311, 352, 383 equilibrium, ix, xi, 4, 9, 20, 21, 22, 26, 27, 30, 40, 47, 87, 197, 198, 199, 200, 203, 204, 212, 377, 403, 404, 406, 407 equipment, 98, 99, 257, 284 ester, 280 etching, 156 ethyl alcohol, 156 evacuation, 406 evaporation, xi, 207, 403, 404, 406, 407, 408, 409, 414 evolution, xi, 126, 155, 202, 208, 293, 297, 335, 367, 368, 369, 374, 377, 379, 380, 382, 383, 384, 385, 387, 388, 393, 396 excitation, 12, 26, 28 exponential functions, 8 exposure, 286 extrapolation, 276 extrusion, 199

F fabricate, vii, x, 85, 257, 276, 284, 287, 288, 289, 295 fabrication, 84, 85, 262, 263, 264, 266, 276, 278, 283, 284, 285, 286, 287, 288, 289, 290 failure, 11, 283 fatigue, 284 fault current limit, 78, 97, 200 faults, 81, 87, 88, 89 FCL, 97 February, 212 feedstock, 285 Fermi, 3, 5, 6, 11, 12, 25, 36, 38, 41, 241, 245 Fermi level, 5, 6, 11, 25, 36, 38 Fermi surface, 241, 245 Fermi-Dirac, vii, 44 ferromagnetic, 10 ferromagnetism, 7 fidelity, 275

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Index field theory, 5, 12, 13, 33 filament, 297 fillers, 281 film, x, 121, 122, 215, 216, 217, 232, 233, 234, 235, 236, 238, 239, 240, 252, 253, 265, 266, 271, 272, 349, 350, 351, 352, 353, 354, 356, 357, 358, 363, 365 film thickness, 235, 272 films, x, 177, 180, 257, 258, 259, 260, 265, 266, 267, 269 flow, xi, 15, 80, 83, 115, 117, 118, 154, 159, 173, 184, 267, 284, 293, 296, 302, 306, 321, 365, 393, 403, 404, 405, 406, 408, 409, 410, 412, 413, 414 flow rate, xi, 403 fluctuations, 178 fluid, 404, 405, 406, 410 flux pinning, vii, 55, 56, 57, 64, 70, 75, 80, 84, 181 force constants, 26 Ford, 195 Fourier, 297 fracture, 97, 100, 101, 111, 113, 117, 121, 123, 125, 126, 128, 129, 132, 141, 143, 144, 145, 146, 153, 159 fracture processes, 117 free energy, x, 47, 215, 216, 217, 220, 222, 223, 228, 230, 233, 239, 240, 245, 246, 247, 248, 250, 251, 271 friction, 99, 102, 117, 118 funding, 290 fusion, x, 275, 276, 278, 279, 280, 284, 289, 290 FWHM, 261

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G gallium, 90 Gamma, 173, 175, 180, 192 gamma radiation, 279 gas, 41, 44, 47, 67, 173, 266, 404 gas-cooled, 349 gauge, 224, 241 Gaussian, 219 generation, viii, 55, 70, 72, 75, 98, 111, 144, 159, 275, 297, 303, 339, 341, 342, 355, 376, 377, 379, 381, 394, 396 generators, 39 geology, 49 Germany, 213, 403 Gibbs, x, 215, 216, 217, 220, 223, 230, 239, 240, 246, 247, 248, 250, 251 Gibbs free energy, x, 215, 216, 217, 220, 223, 230, 239, 240, 246, 247, 248, 250, 251 Ginzberg-Landau, 216 glass, ix, 197, 202, 211, 279, 412 glasses, 27, 202 glassy state, 202 goals, 172 gold, 5, 32, 46, 174, 175 gold compound, 46

419

grain, ix, 56, 57, 61, 64, 81, 82, 85, 87, 93, 94, 136, 137, 155, 169, 170, 171, 172, 175, 181, 182, 186, 187, 188, 189, 192, 193, 198, 199, 209, 259, 262, 264 grain boundaries, ix, 56, 81, 85, 87, 169, 171, 172, 181, 182, 186, 192, 193 grains, 85, 86, 117, 137, 171, 172, 181, 182, 192, 198, 199, 259, 260, 261, 264, 267, 269, 271, 272 graph, 251 graphite, 108 gravity, 282 grids, 5 Ground state, 10, 40 ground state energy, 46 groups, 7, 100, 170, 177, 199, 201 growth, 57, 60, 67, 74, 80, 82, 83, 85, 86, 91, 95, 155, 158, 186, 205, 259, 260, 264, 271, 351, 373, 382 growth mechanism, 80 growth rate, 80, 82, 83, 95 guidelines, 284

H halogen, 5 Hamiltonian, 7, 17, 33, 35, 37 handling, 85, 253 hardness, viii, 77, 97, 98, 99, 100, 101, 102, 103, 105, 106, 108, 109, 114, 115, 117, 118, 119, 120, 121, 125, 126, 128, 135, 136, 139, 141, 142, 143, 144, 153, 155, 159, 160, 161 Hartree-Fock method, 7, 10, 11, 22 heat, 86, 171, 173, 175, 198, 199, 200, 208, 209, 210, 266, 270, 276, 277, 278, 279, 286, 297, 298, 303, 306, 312, 316, 317, 318, 319, 320, 321, 329, 331, 338, 339, 340, 341, 342, 348, 349, 350, 355, 362, 367, 368, 369, 373, 374, 376, 377, 379, 381, 382, 388, 389, 390, 391, 393, 395, 396, 404, 405, 406, 408, 409 heat capacity, 338, 342, 348, 355, 374, 377, 382, 388, 389, 390, 391, 395 heat release, 340, 373 heat removal, 340, 341 heat transfer, 298, 306, 312, 316, 317, 318, 319, 321, 329, 331, 338, 340, 341, 342, 349, 350, 367 heating, 82, 90, 155, 207, 210, 266, 277, 279, 280, 281, 284, 285, 290, 306, 327, 329, 341, 349, 350, 355, 379 height, 139, 154, 160, 161, 295 helium, xi, 173, 176, 276, 277, 284, 293, 294, 349, 350, 351, 353, 354, 355, 358, 359, 362, 365, 388, 403, 404, 405, 409, 410, 412, 414 heterogeneity, 117, 296 heterogeneous, viii, 78, 83, 108, 259 heuristic, 7 hexagonal lattice, 228 high pressure, 29, 46, 79, 294 high resolution, 94

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420

Index

high temperature, 23, 78, 82, 86, 170, 171, 173, 182, 186, 199, 208, 209, 275, 294, 314, 329, 331, 342, 344, 366 high-Tc, x, xi, 4, 56, 198, 293, 294, 295, 296, 310, 311, 327, 328, 338, 340, 348, 349, 350, 366, 373, 374, 375, 377, 381, 382, 387, 388, 393, 395, 396, 397 HIP, 199 Holland, 51 HOMO, 23 homogeneity, 137, 139, 198, 200, 202 homogenous, 86, 117, 262, 264 host, 57 hot pressing, 199 housing, 409 HRTEM, 94, 95 HTS, x, 257 hybrid, 280 hybridization, 15 hydrogen, xi, 7, 8, 11, 31, 293, 349, 350, 356, 357, 358, 359, 361, 363, 364, 365 hyperfine interaction, 15 hypothesis, 48 hysteresis, 56, 59 hysteresis loop, 56

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I identification, 103 identity, 222, 237, 250 images, 67, 129, 224, 228, 229, 236, 262 imaging, 106 impurities, 57, 58, 59, 62, 63, 139, 171, 173, 216, 252 in situ, 200, 206 in transition, 4, 16 inactive, 11, 16, 31 incentive, 276, 278, 283 inclusion, 80 increased access, 283 indices, 15 induction, 81, 297, 298, 322, 348, 349 industrial, ix, 197, 198, 278 industrial experience, 278 industry, 117 inelastic, vii, 3, 6, 7, 44 inequality, 303, 305, 310, 355, 365 infinite, x, 24, 26, 40, 105, 215, 216, 218, 219, 221, 222, 225, 239, 242, 248, 251 inhomogeneities, x, 171, 215, 216, 252, 253 inhomogeneity, x, 45, 140, 215, 219, 223, 236, 239, 240, 247, 250, 251, 252, 253 initiation, 203 inorganic, 280 INS, vii, 3, 6, 7, 44, 48 insight, 3 inspection, 32, 287 instabilities, 295

instability, 11, 279, 295, 296, 297, 306, 363, 365, 367 instruments, 99 insulation, 25, 279, 287 insulators, 6, 11, 13, 23, 25, 28, 276, 277, 278, 279, 280, 294 intensity, 17, 18, 28, 29, 40, 258, 259, 261, 321 interaction, vii, 3, 6, 7, 14, 15, 21, 26, 28, 29, 32, 33, 44, 47, 98, 135, 139, 172, 184, 193 interactions, 4, 20, 33, 34, 40, 44, 47 interface, 56, 74, 80, 82, 83, 84, 86, 88, 89, 93, 95, 102, 113, 137, 145, 161, 242, 253, 259, 263, 271, 287 interface energy, 271 interference, 103, 117, 226 interpretation, 7 interstitials, 172 intrinsic, 78, 94, 140, 218, 224, 241, 349 Investigations, 141, 407 ion beam, 174, 182 ionic, 9, 10, 57, 58, 66, 67, 79, 91 ionization, 8, 11, 12, 17, 18, 46, 172 ionization energy, 8, 18, 46 ions, 4, 5, 7, 10, 12, 13, 14, 18, 19, 24, 25, 28, 29, 32, 34, 41, 45, 46, 56, 57, 62, 64, 65, 66, 69, 78, 79, 174, 269, 288, 345, 395 irradiation, ix, 169, 170, 171, 173, 174, 175, 176, 177, 178, 180, 182, 184, 190, 191, 192, 193 irradiations, 173 isostatic pressing, 87, 199 isothermal, 85, 302, 303, 324, 328, 396 isotope, 6, 18, 43 isotopes, 43 isotropic, 141 Italy, 212 ITT, viii, 77, 78, 97, 98, 99, 100, 102, 103

J Japan, 52, 55, 75, 278, 293 Japanese, 398 joints, 289 Jordan, 169, 175, 176, 194 Joule heating, 327, 341, 349, 355, 379 Jun, 255 Jung, 167, 194

K kinetic energy, xi, 16, 17, 403 kinetics, viii, ix, 78, 82, 97, 197, 200, 201 King, 49, 53, 165

L labeling, 18

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Index labor, 284, 285, 288, 290 labor-intensive, 287 labour, 294 lambda, xi, 403, 404, 406 lamellae, 56 lamina, 238, 240, 248, 405, 406, 408 laminar, 238, 240, 248, 405, 406, 408 Landau theory, ix, 215 large-scale, 198 laser, 288, 289 lasers, 285 lattice, x, 9, 10, 56, 57, 63, 67, 70, 79, 85, 87, 89, 98, 159, 172, 187, 193, 215, 216, 224, 225, 226, 228, 229, 230, 231, 232, 236, 239, 241, 247, 248, 251, 252, 253, 258, 265, 269 lattice parameters, 70 lattices, 52, 216, 224, 231, 232, 233, 248, 252, 253 law, 107, 117, 118, 119, 193, 350, 367, 369, 377 laws, 44, 297, 369 lead, x, 16, 24, 94, 112, 155, 199, 201, 202, 206, 207, 208, 215, 311, 314, 320, 328, 330, 335, 344, 349, 351, 362, 363, 367, 377, 379, 388, 394, 396, 397, 404, 407, 412, 414 learning, 284 lifetime, x, 275 ligand, 5, 12, 13, 14, 15, 16, 18, 19, 25, 32, 33, 41 ligands, 3, 4, 5, 12, 13, 14, 15, 19, 31, 33 limitation, ix, 154, 169, 173, 193, 377 limitations, 173 linear, xi, 15, 29, 83, 107, 118, 119, 147, 178, 219, 226, 244, 251, 268, 293, 296, 298, 299, 300, 303, 320, 321, 323, 329, 342, 346, 348, 364, 367, 369, 382 linear model, 364 links, 170, 172, 193 liquid helium, xi, 276, 277, 284, 293, 294, 349, 350, 351, 353, 354, 355, 358, 359, 362, 365, 414 liquid hydrogen, 349, 350, 356, 357, 358, 359, 361, 363, 365 liquid nitrogen, 78, 97, 277, 294, 295 liquid phase, 56, 80, 82, 90, 203 localization, 24, 25, 26, 29, 47 location, 22, 103, 282, 297, 359 London, 6, 8, 50, 51, 196, 241 long period, 308 long-term, 202, 205, 279 losses, 97, 201, 202, 207, 208, 277, 307, 339, 397, 404 low temperatures, 64, 69, 185 low-temperature, 180 LUMO, 23

M machines, 276, 284, 287 Madison, 275 magnesium, 170, 278 magnesium diboride, 170, 278

421

magnet, x, xi, 275, 276, 277, 278, 279, 280, 281, 282, 284, 287, 290, 294, 295, 297, 377, 403, 412, 414 magnetic, vii, ix, x, 4, 25, 30, 37, 55, 56, 60, 62, 63, 67, 70, 71, 72, 73, 74, 75, 78, 81, 85, 86, 97, 100, 169, 170, 171, 173, 174, 176, 181, 184, 185, 188, 189, 190, 191, 192, 193, 202, 205, 209, 211, 215, 216, 223, 227, 230, 237, 238, 239, 245, 246, 248, 249, 250, 251, 252, 257, 258, 269, 275, 277, 278, 280, 282, 283, 284, 285, 289, 294, 295, 296, 297, 298, 299, 300, 304, 319, 322, 324, 325, 328, 329, 330, 331, 332, 338, 339, 340, 344, 345, 348, 349, 350, 363, 364, 366, 367, 369, 373, 397, 403 magnetic field, vii, ix, x, 25, 37, 55, 56, 60, 63, 67, 70, 73, 74, 75, 78, 86, 169, 170, 171, 173, 174, 176, 181, 184, 185, 188, 189, 190, 191, 192, 193, 209, 211, 215, 216, 237, 238, 245, 248, 249, 250, 251, 252, 257, 277, 278, 284, 294, 295, 296, 297, 298, 299, 304, 319, 322, 324, 325, 328, 329, 330, 331, 332, 338, 339, 344, 349, 350, 363, 364, 366, 367, 369, 373, 397, 403 magnetic field effect, 324 magnetic fusion, x, 275, 289 magnetic properties, 25, 78, 216 magnetization, ix, 56, 59, 60, 61, 62, 68, 70, 169, 171, 176, 183, 186, 189 magnetoresistance, 294 magnets, vii, x, 70, 97, 275, 276, 277, 278, 279, 280, 281, 284, 287, 289, 290, 293, 295, 297, 329, 332, 345, 349, 366, 396 main line, 17, 18 maintenance, 282, 283, 284, 289 management, 284 mandates, 290 Manganese, 171, 173 manipulation, 99 manufacturer, 110 manufacturing, x, 275, 276, 284, 285 market, 198 Mass Flow, 409, 413 mass loss, 207 mass transfer, 80 matrix, viii, xi, 21, 23, 26, 28, 29, 33, 34, 35, 37, 55, 61, 64, 65, 69, 72, 74, 78, 81, 85, 88, 91, 93, 94, 97, 98, 111, 128, 129, 130, 131, 135, 154, 160, 204, 278, 279, 293, 295, 296, 297, 298, 302, 303, 305, 306, 308, 310, 312, 316, 317, 318, 320, 321, 324, 328, 329, 330, 332, 335, 338, 340, 341, 342, 344, 345, 346, 347, 348, 349, 350, 351, 352, 355, 359, 360, 361, 362, 363, 365, 366, 369, 373, 374, 380, 388, 390, 395, 397 Maxwell equations, 297, 367 measurement, 45, 98, 103, 106, 111, 117, 126, 176, 230, 242 measures, 99, 232, 296 mechanical properties, viii, 77, 78, 97, 98, 99, 100, 102, 103, 105, 108, 117, 121, 123, 128, 129, 132, 135, 141, 155, 198, 279 mechanical stress, 94, 95

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422

Index

mechanical testing, 98 median, 112 melt, viii, ix, 55, 57, 59, 60, 70, 73, 74, 84, 85, 86, 87, 93, 95, 135, 155, 197, 198, 200, 201, 202, 203, 204, 205, 206, 208, 209, 211, 212, 285 melting, ix, 82, 86, 90, 186, 197, 199, 201, 202, 203, 204, 205, 206, 211, 212, 285, 289 melting temperature, 203, 205, 289 melts, 203 MEMS, 117 mesoscopic, 154 metal ions, 4, 5, 7, 12, 13, 18, 19, 24, 28, 29, 32, 41, 46 metal oxide, vii, 3, 4, 14, 25 metal oxides, vii, 3, 4, 14, 25 metallurgy, 291 metals, vii, 3, 5, 13, 41, 49, 63, 99, 103, 117, 118, 119, 139, 277, 285, 294 methanol, 266 MgB2, ix, 169, 170, 172, 173, 177, 179, 180, 181, 182, 193, 228, 278 mica, 108 microcracking, 94 microscope, 125, 129, 156, 205 microscopy, 91, 94, 156 microstructure, ix, 72, 85, 87, 135, 173, 186, 197, 198, 199 microstructures, 70, 108 migration, 181, 182 mining, 303 Ministry of Education, 75 misfit dislocations, 83 mixing, 15, 16, 33, 202 mobility, 7, 91 MOD, 257, 267, 269 models, vii, x, 3, 7, 12, 80, 107, 141, 293, 296, 319, 369, 371, 373, 380, 382, 383, 386, 387, 389, 396 modulus, viii, 77, 93, 98, 99, 100, 101, 102, 103, 105, 106, 108, 109, 111, 113, 115, 121, 122, 123, 125, 126, 128, 135, 136, 137, 138, 139, 141 molar ratio, 59, 70, 72 molecular orbitals, 8 molecules, 12, 14 momentum, 37, 40, 46, 47, 172 monolithic, 98, 113, 285, 286 Moon, 76, 163, 202, 213 morphology, 175, 187, 198 Moscow, 293 motion, vii, 3, 4, 5, 19, 20, 29, 40, 41, 43, 44, 47, 98, 118, 172, 235 motivation, 32 motors, 97, 198 movement, 139, 141 MRS, 212, 273 MTS, 146, 147

N nanoindentation, viii, 77, 98, 100, 101, 102, 108, 109, 113, 118, 119, 120, 121, 125, 126, 128, 130, 131, 132, 133, 141, 142, 145, 148, 153, 159 nanometer, 129 nanometers, 98, 99, 117 nanoparticles, 170 nanorods, 70 NATO, 52 natural, 32 neglect, 33 Netherlands, 52 network, 154, 159 neutrons, x, 275, 279 New York, 49, 51, 195, 213, 255 Newton, 52 NiO, 11, 12, 23, 25, 42, 45, 267 niobium, 278 NIST, 212 nitrogen, 78, 97, 266, 277, 278, 294, 295, 412 nonlinear, xi, 115, 222, 293, 296, 297, 298, 303, 319, 320, 324, 329, 340, 342, 346, 348, 349, 368, 388 nonstoichiometric, 57 normal, ix, x, xi, 3, 5, 15, 17, 169, 170, 172, 173, 174, 177, 178, 193, 223, 248, 268, 269, 275, 276, 277, 279, 282, 293, 295, 296, 306, 308, 328, 337, 342, 343, 348, 359, 394, 397, 404, 405, 406, 409, 410 normal conditions, 282 normalization, 15, 238 normalization constant, 238 nuclear, 6, 7, 23, 27, 28, 29, 33, 40, 41, 43, 276, 277, 279, 280, 281, 290 nucleation, 82, 83, 84, 85, 87, 91, 95, 114, 135, 205, 217, 219, 235, 236, 237, 242, 244, 247, 248, 252, 259, 260, 263, 264 nuclei, vii, 3, 4, 6, 8, 9, 22, 29, 40, 43, 44, 47 nucleus, 8, 47

O observations, 61, 63, 67, 89, 144 oil, 174, 175, 289 one dimension, 218 operator, 219, 220, 221, 222 optical, 91, 125, 156, 412 optical microscopy, 156 optimization, 27, 34, 86, 280, 290, 345, 410 organic, 33, 34, 199, 257, 258, 278 organization, 215 orientation, x, 12, 70, 85, 95, 156, 172, 258, 259, 260, 263, 269, 271, 272, 273, 350 orthorhombic, viii, 77, 78, 86, 89, 97, 126, 129, 134, 135, 136, 147, 154, 157, 159, 161, 187, 193 overdoped, 64

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Index oxidation, vii, 3, 4, 5, 7, 15, 17, 18, 20, 23, 24, 26, 28, 29, 30, 32, 40, 41, 43, 45, 46, 47 oxide, 11, 80 oxides, vii, 3, 14, 15, 45, 46, 57, 86, 117, 175, 202 oxygen, 3, 6, 12, 13, 21, 30, 31, 43, 56, 57, 60, 61, 62, 63, 64, 65, 66, 67, 80, 81, 87, 89, 94, 96, 154, 155, 159, 161, 172, 174, 175, 181, 182, 198, 200, 201, 202, 206, 207, 208, 257, 266, 267 oxygen absorption, 207 oxygenation, viii, 78, 87, 91, 94, 96, 97, 126, 155, 156, 157, 158, 159, 160, 161

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P pairing, 6, 25, 26, 44, 46, 47 paper, 3, 4, 6, 30, 75 parabolic, 41, 42, 107, 235, 242 paradoxical, 27 parameter, x, 6, 12, 24, 26, 27, 30, 57, 64, 66, 80, 104, 105, 106, 111, 126, 171, 179, 180, 187, 193, 215, 216, 217, 218, 219, 221, 222, 223, 226, 234, 237, 238, 239, 240, 242, 244, 246, 247, 248, 249, 250, 251, 253, 265, 269, 281, 290, 303, 310, 311, 312, 347 particles, viii, xi, 41, 44, 55, 56, 61, 64, 70, 72, 73, 74, 75, 80, 81, 82, 83, 85, 87, 89, 93, 94, 95, 98, 111, 128, 137, 139, 141, 145, 146, 155, 159, 235, 403 performance, ix, x, 75, 97, 197, 198, 199, 275, 279, 280, 281, 285, 406, 409, 410, 412 periodic, 46, 280, 282 periodic table, 46 periodicity, 244 permeability, 198 permit, 128, 283, 366, 373, 380 perovskite, 57, 72 perturbation, 12, 13, 171 perturbations, 295 phase diagram, 80, 81, 83, 200, 204 phase transformation, 94 phonon, vii, 3, 6, 23, 30, 40, 44, 45, 47, 48, 170 phonons, 48, 172 photon, 172 physical and mechanical properties, 279 physical properties, 329 physicists, 4, 5 physics, 4, 6, 7, 16, 48, 49, 164 pinning effect, 68, 184 pitch, 297 planar, 5, 11, 14, 23, 29, 46, 216, 282 planning, 284 plants, x, 275, 276, 283, 284, 290 plasma, 60, 280, 281, 282, 284, 285, 289 plastic, viii, 77, 78, 87, 88, 93, 97, 98, 100, 101, 103, 104, 105, 108, 111, 112, 114, 115, 117, 118, 119, 120, 125, 145, 147 plastic deformation, viii, 77, 87, 98, 100, 105, 108, 114, 115, 117, 118, 145, 147

423

plasticity, 115, 117, 139 platelet, 174 platelets, 199, 206 platinum, 173, 176, 201 play, 15, 85, 154, 230, 278, 283, 306, 331, 332, 348, 366 point defects, ix, 169, 216, 253, 279, 296 point-to-point, 99 Poisson, 106, 153 Poisson ratio, 153 polarization, 25 polarized light, 91, 156 polarized light microscopy, 91 polycrystalline, ix, 90, 122, 169, 171, 172, 173, 174, 175, 176, 181, 184, 192, 193 polyimide, 278, 280 polymer, 98 polymer materials, 98 polymers, 99, 103 poor, ix, 56, 97, 197, 199, 284 population, 87, 98, 135 pore, 405, 410 pores, 159, 262 porosity, ix, 126, 128, 133, 136, 137, 141, 145, 146, 156, 158, 159, 197, 199, 209, 405 porous, 404, 405, 414 ports, 282, 283, 286 potential energy, 9, 47 powder, 59, 70, 72, 82, 85, 86, 174, 175, 198, 199, 200, 285 powders, 174, 175 power, x, 97, 107, 118, 193, 198, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 287, 288, 289, 290, 294, 295, 296, 298, 299, 301, 302, 303, 304, 306, 309, 310, 311, 312, 314, 315, 316, 322, 324, 326, 327, 328, 329, 341, 342, 350, 365, 367, 376, 377, 379, 380, 381, 394, 395, 396, 403, 404, 408, 414 power plant, 275, 276, 279, 280, 283, 284, 285, 287, 290 power plants, 276, 280, 283, 285 power-law, 107, 180 precipitation, 203, 207, 209 precursor states, 11 pressure, xi, 6, 16, 29, 46, 67, 79, 80, 81, 84, 101, 109, 153, 173, 174, 199, 207, 294, 403, 404, 405, 406, 408, 410, 411, 412, 414 probability, 29, 40, 44, 365 probe, 16, 18, 45, 104, 108 process control, 284 production, ix, 86, 170, 197, 198, 200, 278, 281, 284, 285 program, 276, 290 promote, 37, 206, 209 propagation, 87, 97, 111, 114, 129, 145, 159, 199, 285, 327, 367 property, 24, 38, 106, 125, 128, 219, 226, 228, 240, 258, 260, 269, 338, 349, 366, 388 propionic acid, 258, 269

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424

Index

propulsion, 198 protection, 279, 281 prototype, 11, 276, 290 pseudo, 81 PTFE, 409 pulse, 137, 395 pulses, 395 pumping, 404, 406, 409 pumps, xi, 403, 404 PVA, 86 pyramidal, 116

Q quality control, 97 quanta, 218 quantum, xi, 4, 6, 15, 21, 26, 31, 43, 47, 403, 404 quantum chemical calculations, 21 quantum state, xi, 26, 403, 404

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R radiation, xi, 16, 45, 190, 191, 195, 276, 277, 279, 280, 281, 283, 285, 287, 289, 290, 291, 403 radiation damage, 281 radioactive waste, 280 radium, 153 radius, 57, 66, 103, 109, 111, 118, 175, 283, 406 range, viii, ix, 19, 48, 56, 78, 97, 98, 105, 108, 117, 120, 121, 135, 137, 170, 177, 178, 197, 200, 201, 208, 209, 210, 230, 240, 299, 302, 312, 314, 317, 323, 324, 328, 329, 330, 331, 332, 335, 338, 342, 344, 349, 351, 358, 359, 363, 366, 368, 370, 373, 374, 382, 386, 387, 388, 393, 395, 397 rare earth, 55 rare earth elements, 55 reaction rate, 18 reactivity, 181 reagents, 202 reasoning, 12 recalling, 203 recognition, 78 recovery, 103, 114, 118, 119, 120, 139, 141, 201, 205, 206, 279 recrystallization, ix, 197, 198, 208 recrystallized, 199, 205 recycling, 278 redistribution, 306, 334, 337, 344, 374, 382, 387 reduction, 114, 135, 143, 161, 171, 172, 184, 186, 199, 276, 288 reference system, 45 refrigerant, 319 refrigeration, 278, 358, 359, 365 regular, 96, 232, 251, 253 regulation, 408 relationship, 24, 32, 56, 62, 115, 118, 119, 143, 316, 320, 329, 350, 351, 367, 377, 396

relationships, 141, 350, 366, 367, 373, 383, 388 relaxation, 27, 46, 47, 70 relaxation rate, 70 relevance, ix, 23, 29, 197 reliability, 198, 276, 284 repair, 283 research, vii, 75, 175, 216, 250, 253, 258, 275, 276, 278, 294, 295, 414 research and development, 275, 276 researchers, 78, 87 reservoir, xi, 79, 403, 404 resistance, 4, 25, 47, 95, 98, 111, 117, 118, 119, 120, 141, 173, 176, 180, 193, 202, 268, 269, 277, 282, 285, 295, 348, 404 resistive, ix, 169, 172 resistivity, ix, xi, 169, 170, 173, 177, 178, 179, 180, 193, 268, 269, 279, 294, 295, 298, 299, 302, 305, 306, 308, 312, 314, 316, 317, 318, 319, 320, 321, 323, 324, 329, 330, 331, 332, 333, 334, 335, 338, 342, 344, 346, 347, 348, 349, 350, 355, 359, 360, 365, 366, 369, 370, 373, 374, 382, 388, 395, 396 resolution, 94, 102, 114, 129, 204 revolutionary, 276 rings, 199 risk, 290 rods, 199 rolling, 199 room temperature, 85, 86, 95, 96, 102, 125, 129, 147, 170, 173, 174, 175, 199, 208, 258, 269, 279, 295, 327 root-mean-square, 265 roughness, 102, 265, 271 Rubber, 98 runaway, x, xi, 293, 294, 295, 296, 297, 303, 304, 306, 307, 308, 310, 311, 312, 314, 316, 317, 318, 319, 320, 321, 322, 323, 324, 326, 327, 328, 329, 330, 332, 335, 337, 338, 342, 344, 345, 346, 347, 348, 349, 350, 351, 352, 354, 355, 357, 358, 359, 360, 363, 365, 366, 373, 374, 375, 376, 377, 380, 381, 382, 383, 384, 387, 388, 391, 394, 395, 396, 397 Russia, 293

S safety, 408, 411 salts, vii, 3, 4, 14, 22, 266 sample, ix, 59, 60, 62, 63, 64, 65, 67, 69, 70, 71, 72, 73, 74, 82, 83, 85, 86, 87, 88, 90, 91, 92, 93, 94, 96, 98, 102, 103, 106, 117, 129, 137, 139, 141, 143, 144, 145, 146, 147, 154, 155, 159, 160, 161, 169, 170, 171, 172, 173, 174, 175, 176, 179, 180, 181, 182, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 199, 203, 204, 205, 206, 207, 259, 268, 296, 303, 312, 326, 327 sapphire, 100 satellite, 17, 18 saturation, 82, 171, 176, 350

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Index savings, 290 scalable, 199 scaling, 368, 369, 370, 371, 382, 383, 384, 387 scanning electron microscopy, 59, 61, 64, 72, 74, 101, 129, 143, 144, 175, 186, 187, 193, 204, 205 scatter, 137 scattering, vii, 3, 6, 7, 44, 252 schema, 155, 404 Schmid, 165 SCs, 78 search, 47, 218 searching, 242, 269 second generation, 198 secular, 21, 26, 44 seed, x, 59, 71, 84, 85, 86, 90, 95, 137, 155, 258, 259, 260, 261, 262, 263, 264, 266, 269, 270, 272 seeding, 85 segregation, ix, 181, 182, 197, 202, 208, 209 selecting, 250 selectivity, 57 SEM micrographs, 186, 187 semiconductor, 90 semiconductors, 294 sensitivity, 4, 192, 285 separation, 8, 9 series, 11, 13, 25, 104, 224, 259, 261 shape, xi, 28, 84, 85, 89, 98, 108, 111, 114, 129, 174, 175, 216, 233, 235, 236, 238, 244, 252, 253, 282, 283, 284, 285, 287, 293, 294, 296, 310, 311, 312, 324, 328, 338, 349, 390 shaping, 199 sharing, xi, 293, 302, 303, 305, 306, 307, 308, 310, 321, 322, 324, 328, 332, 335, 336, 342, 347, 349, 351, 355, 360, 361, 362, 365, 366, 373, 397 shear, viii, 77, 98, 101, 104, 118, 280 shear strength, 104 sign, 11, 12, 35, 36, 37, 38, 120, 304, 330 signs, 335 silica, 112 silver, ix, 173, 175, 176, 197, 198, 203, 206, 207, 208, 277, 279, 299, 301, 319, 323, 329, 348, 369 simulation, 306, 311, 319, 322, 329, 344, 346, 371, 374, 387, 395 simulations, 114, 191, 299, 320, 332, 382, 387, 389 single crystals, vii, 55, 56, 57, 60, 70, 80, 84, 85, 86, 91, 140, 143 single test, 100 sintering, ix, 84, 197, 198, 199, 200, 201, 205, 209, 210 sites, 5, 6, 11, 16, 18, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30, 32, 33, 34, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 56, 57, 68, 74, 83, 87, 135, 172, 191, 260 skeleton, 159, 281 Slater determinants, 8, 9, 31, 32 smoothness, 302, 312, 314, 320, 328 software, 125, 147 solid phase, ix, 197, 199, 200, 206, 211 solid solutions, 200 solid state, 4, 7, 11

425

solidification, 59, 82, 205, 207, 209 solid-state, 93, 174, 175, 203 solubility, 202, 209 solutions, 30, 200, 238, 239, 240, 250, 251, 253, 258, 269, 283, 352 solvent, 19 Spain, 77, 97 spatial, x, 8, 36, 64, 66, 67, 99, 114, 215, 216, 222, 223, 226, 238, 239, 248, 249, 250, 251, 252, 296 specific heat, 279, 367, 369 spectroscopy, 15, 45, 46, 195 spectrum, x, 11, 12, 13, 16, 17, 18, 23, 24, 28, 29, 44, 45, 46, 73, 215, 216, 217, 225, 235, 236, 237, 238, 239, 242, 246, 251, 252, 253, 279 speed, 258, 269, 284, 285 spheres, 5, 30, 100 spin, vii, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 30, 32, 33, 35, 36, 39, 40, 45, 258, 266, 269 square lattice, 231, 232, 247, 253 SQUID, 59 stability, ix, x, xi, 4, 47, 102, 197, 200, 201, 293, 295, 297, 303, 304, 310, 311, 314, 318, 320, 324, 328, 329, 330, 338, 343, 345, 347, 348, 349, 351, 357, 359, 366, 375, 377, 380, 382, 394, 395, 396, 397 stabilization, 24, 137, 306, 342, 395 stabilize, 377 stable states, 338, 339, 363 stages, 84, 99, 104, 137, 266, 289, 370 stainless steel, 174, 289, 407, 409 statistics, 44, 47 steady state, 216, 393, 395 steel, 100, 174, 276, 277, 278, 281, 284, 289, 407, 409 stiffness, 98, 102, 103, 104, 106, 108, 110, 126, 147, 148 STM, 67 stock, 285 stoichiometry, 207, 257 storage, 78, 85, 99, 100, 117 strain, 93, 100, 102, 108, 109, 115, 117, 118, 147, 154, 327 strains, 67, 100, 114, 278 strength, viii, 12, 77, 81, 97, 98, 101, 104, 115, 118, 147, 153, 170, 176, 217, 224, 278, 285, 286 stress, viii, 77, 78, 92, 93, 94, 95, 96, 98, 99, 100, 101, 102, 104, 108, 109, 112, 113, 114, 118, 121, 139, 145, 154, 155, 159, 295 stress-strain curves, viii, 77, 101, 108, 147 stretching, 25 strong interaction, 28 strontium, 278 structural changes, 156 structural sensitivity, 4 substances, 67 substitutes, 200 substitution, viii, 55, 56, 57, 59, 62, 63, 64, 66, 67, 68, 69, 75

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426

Index

substrates, x, 94, 257, 258, 259, 261, 263, 266, 267, 269, 271, 272 Sun, 163, 194, 195, 196, 257, 258, 260, 262, 264, 266, 268, 270, 272, 274 superconducting, vii, ix, x, xi, 5, 6, 7, 15, 18, 23, 29, 40, 44, 45, 46, 48, 55, 56, 57, 59, 64, 66, 69, 70, 78, 79, 81, 85, 94, 97, 154, 169, 170, 171, 172, 173, 177, 178, 182, 184, 192, 193, 197, 198, 199, 200, 216, 233, 235, 236, 239, 248, 257, 267, 275, 276, 277, 278, 282, 283, 284, 287, 290, 293, 294, 295, 296, 297, 298, 301, 302, 303, 304, 305, 306, 307, 308, 310, 311, 312, 314, 315, 320, 321, 322, 323, 324, 326, 327, 328, 329, 330, 331, 332, 335, 338, 339, 340, 341, 342, 344, 346, 348, 349, 350, 351, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 372, 373, 374, 375, 377, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 393, 394, 395, 396, 397, 403, 412, 414 superconducting gap, 6, 23, 46, 170 superconducting magnets, vii, x, 284, 290, 293, 295, 297, 329, 349, 366, 396 superconducting materials, 79, 97, 278, 287, 377, 397 superconductivity, vii, 3, 4, 5, 6, 15, 23, 25, 28, 29, 33, 43, 44, 45, 47, 48, 56, 64, 79, 167, 170, 171, 225, 236, 240, 277, 337, 348, 351, 394 superconductor, viii, ix, x, xi, 6, 27, 77, 78, 90, 97, 176, 198, 199, 215, 216, 217, 219, 221, 222, 223, 225, 226, 228, 230, 231, 233, 234, 235, 239, 240, 241, 246, 247, 250, 251, 252, 253, 276, 278, 279, 280, 282, 284, 293, 294, 296, 297, 298, 299, 300, 301, 302, 303, 304, 306, 307, 308, 309, 310, 311, 312, 314, 316, 317, 319, 320, 321, 323, 324, 325, 326, 327, 328, 329, 332, 335, 338, 339, 342, 343, 344, 348, 349, 350, 351, 354, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 374, 377, 380, 382, 384, 385, 388, 390, 396, 397 superconductors, vii, ix, x, 3, 4, 6, 28, 30, 46, 55, 56, 78, 80, 84, 85, 87, 97, 154, 169, 170, 171, 190, 198, 199, 215, 216, 217, 219, 222, 223, 227, 250, 253, 275, 278, 280, 284, 294, 295, 296, 298, 304, 309, 310, 311, 327, 328, 329, 349, 366, 387, 397 supercritical, 277 superfluid, xi, 403, 404, 405, 406, 410, 414 superlattice, x, 215, 217, 240, 241, 242, 246, 247, 248, 249, 252, 253 superposition, 11, 33, 43, 47, 219, 226, 244, 251 supply, xi, 82, 289, 403, 404, 408, 414 suppression, 171, 172, 182, 192, 414 suppressor, xi, 403, 404 surface layer, 216, 248, 253 surface properties, 350 surface region, 225 surface roughness, 102 susceptibility, 60, 62, 67, 71, 73, 202, 205 Sweden, 3, 290 symbols, 63

symmetry, x, 4, 8, 15, 16, 18, 22, 31, 32, 33, 42, 112, 215, 216, 224, 225, 227, 228, 229, 230, 231, 232, 233, 239, 246, 248, 251 synthesis, 67, 70, 72 systems, vii, 3, 4, 5, 6, 7, 11, 15, 16, 19, 24, 25, 26, 29, 32, 40, 41, 44, 45, 47, 48, 49, 99, 102, 108, 113, 117, 276, 281

T tantalum, 173 technology, 276, 285, 290, 294 TEM, 63, 67, 88, 89, 91, 92, 93, 94 temperature, ix, x, xi, 23, 25, 28, 44, 47, 59, 61, 62, 63, 70, 75, 78, 79, 82, 85, 86, 87, 90, 93, 97, 102, 121, 159, 169, 170, 173, 174, 176, 177, 178, 179, 180, 184, 185, 186, 193, 198, 199, 200, 203, 204, 205, 206, 207, 208, 210, 236, 252, 258, 259, 268, 269, 272, 275, 276, 277, 278, 279, 281, 282, 284, 286, 287, 293, 294, 295, 296, 297, 298, 299, 301, 302, 303, 304, 306, 307, 308, 310, 312, 314, 315, 316, 317, 319, 320, 321, 323, 324, 326, 327, 328, 329, 330, 331, 332, 333, 335, 336, 337, 338, 339, 340, 341, 342, 344, 346, 347, 348, 349, 350, 351, 357, 358, 359, 362, 363, 364, 365, 366, 367, 368, 369, 372, 373, 374, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 392, 393, 394, 395, 396, 397, 403, 404, 405, 406, 412, 413, 414 temperature dependence, ix, xi, 61, 177, 178, 179, 180, 193, 268, 269, 293, 299, 320, 329, 332, 339, 340, 346, 348, 362, 363, 395, 397, 406 temperature gradient, 85, 86, 90 tensile, 96, 111, 112, 327 tensile stress, 112 tension, 159 Tesla, 177, 178, 181, 190, 191, 224, 228, 229, 284 textbooks, 12 TGA, 208 theory, vii, ix, 4, 5, 6, 7, 13, 15, 16, 19, 20, 24, 34, 36, 40, 43, 45, 48, 51, 106, 117, 118, 153, 154, 171, 172, 215, 216, 217, 219, 220, 221, 222, 223, 224, 231, 232, 233, 236, 240, 241, 250, 251, 252, 253, 403, 414 thermal activation, 19, 296 thermal activation energy, 19 thermal aging, 286 thermal degradation, xi, 293, 325, 327, 329, 385, 387, 396, 397 thermal energy, 280 thermal equilibrium, 404 thermal expansion, 93, 94, 102, 159 thermal resistance, 282 thermal stability, x, xi, 293, 295, 303, 310, 311, 345, 395 thermal treatment, 84, 86, 90, 209 thermodynamic, 78, 204, 223, 294 thermodynamic equilibrium, 204

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Index thermo-mechanical, 403, 404, 405, 407, 409, 410, 412 thin film, x, 85, 98, 108, 111, 135, 170, 215, 232, 234, 240, 252 thin films, 85, 98, 108, 111, 170 Thomson, 4 threshold, 114, 172 thresholds, 114 time, 16, 19, 22, 85, 98, 125, 126, 155, 161, 199, 200, 209, 210, 244, 247, 277, 279, 284, 285, 287, 289, 290, 301, 302, 304, 312, 320, 330, 332, 358, 363, 366, 369, 373, 377, 380, 383, 387, 395 tin, 278 titanium, 278 TMP, xi, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414 tokamak, x, 275, 276, 281, 282, 283, 285, 289 Tokyo, 55 tolerance, 102, 276, 279, 287 torus, 287 total energy, 13, 20, 44, 46 toughness, viii, 77, 87, 97, 98, 100, 101, 111, 113, 121, 123, 125, 126, 128, 132, 144, 145, 146, 153, 159, 285 trade, 410 trading, 281 trans, 104 transducer, 408, 411 transfer, vii, 3, 4, 15, 18, 19, 23, 25, 26, 27, 28, 29, 30, 33, 41, 46, 47, 48, 298, 306, 312, 316, 317, 318, 319, 320, 321, 329, 331, 338, 340, 341, 342, 349, 350, 367, 404 transformation, 82, 86, 94, 159, 161, 199, 219, 220, 330, 338 transformations, 330, 352 transition, vii, viii, ix, xi, 3, 4, 7, 11, 12, 13, 14, 18, 22, 24, 28, 29, 32, 40, 45, 77, 78, 87, 94, 95, 100, 108, 152, 153, 161, 169, 170, 171, 172, 177, 178, 189, 193, 206, 223, 241, 253, 282, 293, 294, 295, 296, 303, 306, 308, 328, 337, 342, 343, 344, 348, 349, 351, 352, 356, 357, 359, 363, 365, 392, 393, 394, 397, 414 transition metal, vii, 3, 4, 7, 12, 13, 14, 18, 22, 32 transition metal ions, 7, 12 transition temperature, 78, 170, 172, 177, 178, 189, 193, 223, 241, 253 transitions, 28, 29 translation, 12 translational, 4, 19, 219 transmission, 94, 278 transparent, 412 transport, viii, 19, 47, 78, 97, 170, 172, 181, 192, 193, 198, 199, 209, 298, 306, 321, 332, 335, 338, 342, 368, 376, 393, 403, 404, 406, 408 traps, 252 treatment methods, 286 trend, 25, 137, 362 trial, 244 trifluoroacetate, 258

427

tritium, 280, 281 tubular, 173 tungsten, 100, 281 tungsten carbide, 100, 281 turbulent, 414 twins, 63, 67, 92, 93, 155 two-dimensional, 83, 198, 294

U ultraviolet, 11, 16, 45 uncertainty, 102, 106 uniform, 174, 219, 221, 225, 228, 231, 233, 240, 253, 282, 297, 350, 368, 369, 373, 377, 382, 387, 396 unit cost, 287, 288 United Arab Emirates, 169, 194 United States, 215 universality, 227, 228

V vacancies, 60, 87, 171, 172 vacuum, 12, 173, 176, 257, 267, 276, 280, 281, 289 valence, vii, 4, 6, 8, 10, 13, 14, 16, 17, 18, 20, 24, 26, 27, 28, 29, 32, 42, 44, 45, 47, 57, 62, 63 validation, 47 validity, 302 values, xi, 24, 28, 30, 81, 105, 106, 107, 108, 113, 118, 119, 121, 122, 123, 126, 134, 135, 136, 137, 138, 139, 146, 147, 153, 170, 171, 173, 178, 179, 180, 181, 182, 184, 185, 186, 189, 190, 191, 193, 200, 210, 211, 231, 293, 298, 299, 301, 302, 304, 308, 310, 311, 312, 316, 318, 319, 320, 321, 322, 324, 325, 327, 328, 330, 331, 335, 338, 339, 342, 344, 346, 348, 351, 352, 354, 355, 356, 359, 360, 361, 364, 366, 367, 369, 370, 373, 377, 378, 379, 380, 382, 387, 390, 391, 393, 396, 397 vapor, 404, 414 variable, 326, 327 variables, 309 variance, 348 variation, xi, 22, 32, 42, 102, 118, 209, 224, 228, 229, 230, 231, 233, 235, 236, 238, 240, 246, 248, 249, 250, 251, 282, 293, 297, 302, 312, 316, 320, 330, 331, 335, 337, 338, 339, 342, 343, 351, 358, 359, 363, 365, 366, 368, 369, 373, 379, 380, 382, 385, 386, 387, 388, 392, 393, 394, 395, 396, 397 vector, 156, 218, 221, 224, 234 velocity, 45, 95, 170, 245, 406 vessels, 404 vibration, 102 viscosity, 74, 90, 95, 405 visible, 28, 29, 44, 47, 307, 324, 341, 342, 375, 387, 392, 395, 396, 407, 413 visualization, 111 voids, 285

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428

Index

volatility, 201 volatilization, ix, 197, 207 vortex, vii, ix, x, 169, 170, 171, 172, 174, 176, 181, 182, 186, 189, 190, 193, 215, 216, 217, 219, 220, 222, 223, 224, 225, 226, 230, 231, 232, 233, 236, 237, 238, 239, 240, 241, 242, 244, 246, 248, 250, 251, 252, 253, 296, 407 vortex pinning, ix, 171, 172, 176, 182, 186, 189, 190, 193 vortices, 85, 170, 171, 174, 184

W water, 12, 13, 19, 281, 287, 289 wave packet, 40 weak interaction, 11 wear, 114 Weibull, 141 weight loss, 207 welding, 284, 287 wind, 278 wires, 198, 294, 295, 332, 349 Wisconsin, 275 writing, 14

X XPS, 11, 16, 17, 18 x-ray diffraction, 187 X-ray diffraction, 187, 121, 173, 202, 203, 204, 205, 206, 207, 208, 258, 259, 260, 261, 262, 263, 265, 266, 267, 270, 271

Y YBCO, viii, ix, x, 77, 78, 79, 80, 81, 82, 84, 85, 86, 90, 91, 94, 95, 97, 111, 121, 122, 123, 125, 126, 127, 128, 129, 131, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 151, 152, 154, 155, 156, 159, 160, 161, 169, 171, 173, 174, 176, 182, 183, 184, 193, 198, 199, 257, 258, 265, 266, 267, 268, 269, 271, 272, 278, 279, 280 yield, viii, 77, 85, 98, 99, 101, 102, 105, 110, 115, 118, 119, 139, 147, 153, 284, 327 YSZ, 257 yttrium, 82, 83

Z

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Zener, 52 zirconia, 201

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