Superconductivity Research Developments

SUPERCONDUCTIVITY RESEARCH DEVELOPMENTS

SUPERCONDUCTIVITY RESEARCH DEVELOPMENTS

JAMES R. TOBIN Editor

Nova Science Publishers, Inc. New York

Copyright © 2008 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. 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 Superconductivity research developments / James R. Tobin (editor). p. cm. ISBN-13: 978-1-60692-762-5 1. Superconductivity--Research. I. Tobin, James R. QC611.96.S938 2008 537.6'23072--dc22 2007035862

Published by Nova Science Publishers, Inc.

New York

CONTENTS Preface

vii

Acknowledgements

xiii

Chapter 1

Optimization of Critical Current Density in MgB2 S. K. Chen and J. L. MacManus-Driscoll

Chapter 2

Magnetic Properties of Artificially Prepared Highly Ordered Two-Dimensional Shunted and Unshunted Nb–AlOx–Nb Josephson Junctions Arrays Fernando M. Araújo-Moreira and Sergei Sergeenkov

Chapter 3

Superconducting Noble Metal Diboride A. K. M. A. Islam and F. Parvin

Chapter 4

Perspectives of Superconducting Temperature Increase in HTSC Copper Oxides Svetlana G. Titova and John T. S. Irvine

1

27

63

93

Chapter 5

Effect of Apical Oxygen Ordering on Tc of Cuprate Superconductors H. Yang, Q. Q. Liu, F. Y. Li, C. Q. Jin and R. C. Yu

125

Chapter 6

Pairing Correlations in Copper Oxide Superconductors Rongchao Ma and Yuefei Ma

149

Chapter 7

Single Intrinsic Josephson Junction Fabricated from Bi2Sr2CaCu2O8+x Single Crystals L. X. You, A. Yurgens, D. Winkler and P. H. Wu

167

Chapter 8

Rare Earth Modified (Bi,Pb)-2212 Superconductors A. Biju and U. Syamaprasad

187

Chapter 9

Novel Approaches to Describe Stability and Quench of HTS Devices V. S. Vysotsky, A. L. Rakhmanov and Yu. A. Ilyin

223

Index

241

PREFACE Superconductivity is a phenomenon occurring in certain materials at extremely low temperatures, characterized by exactly zero electrical resistance and the exclusion of the interior magnetic field (the Meissner effect). The electrical resistivity of a metallic conductor decreases gradually as the temperature is lowered. However, in ordinary conductors such as copper and silver, impurities and other defects impose a lower limit. Even near absolute zero a real sample of copper shows a non-zero resistance. The resistance of a superconductor, on the other hand, drops abruptly to zero when the material is cooled below its "critical temperature", typically 20 kelvin or less. An electrical current flowing in a loop of superconducting wire can persist indefinitely with no power source. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum mechanical phenomenon. It cannot be understood simply as the idealization of "perfect conductivity" in classical physics. Superconductivity occurs in a wide variety of materials, including simple elements like tin and aluminium, various metallic alloys and some heavily-doped semiconductors. Superconductivity does not occur in noble metals like gold and silver, nor in most ferromagnetic metals. In 1986 the discovery of a family of cuprate-perovskite ceramic materials known as hightemperature superconductors, with critical temperatures in excess of 90 kelvin, spurred renewed interest and research in superconductivity for several reasons. As a topic of pure research, these materials represented a new phenomenon not explained by the current theory. And, because the superconducting state persists up to more manageable temperatures, more commercial applications are feasible, especially if materials with even higher critical temperatures could be discovered. This new book presents leading research from around the world in this dynamic field. Chapter 1 - This work focuses on the optimisation of critical current density, Jc of bulk polycrystalline MgB2 through studies of the influence of boron precursor powder, nominal Mg non-stoichiometry and by chemical modification. On the influence of the nature of the boron precursor on the superconducting properties of MgB2, Jc’s of samples made from crystalline boron powders are around an order of magnitude lower than those made from amorphous precursors. X-ray, superconducting transition temperature, Tc and resistivity studies indicate that this is as a result of reduced current cross section due to the formation of (Mg)B-O phases. The influence of Mg content was investigated in a series of samples with systematic variation of nominal Mg non-stoichiometry. Jc(H) was found to be influenced significantly with nominal Mg content while Tc remained unchanged. Mg deficient samples

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James R. Tobin

show a higher degree of disorder as inferred from the Raman spectroscopy, residual resistivity ratio and x-ray diffraction. The Mg-deficient samples also showed higher Hirr and Hc2 compared to samples with larger nominal Mg contents. Finally, GaN and Dy2O3 additions into MgB2 during the in situ reaction (owing to enhanced intragranular crystallinity and pinning, respectively) enhance Jc at 6K and 20K up to 5T without changing Tc appreciably. Chapter 2 - Josephson junction arrays (JJA) have been actively studied for decades. However, they continue to contribute to a wide variety of intriguing and peculiar phenomena. To name just a few recent examples, it suffice to mention the so-called paramagnetic Meissner effect and related reentrant temperature behavior of AC susceptibility, observed both in artificially prepared JJA and granular superconductors. Employing mutual-inductance measurements and using a high-sensitive home-made bridge, the authors have thoroughly investigated the temperature and magnetic field dependence of complex AC susceptibility of artificially prepared highly ordered (periodic) two-dimensional Josephson junction arrays (2D-JJA) of both shunted and unshunted Nb–AlOx–Nb tunnel junctions In this Chapter, the authors report on three phenomena related to the magnetic properties of 2D-JJA: (a) the influence of non-uniform critical current density profile on magnetic field behavior of AC susceptibility; (b) the origin of dynamic reentrance and the role of the Stewart-McCumber parameter, βC, in the observability of this phenomenon, and (c) the manifestation of novel geometric effects in temperature behavior of AC magnetic response. Firstly, the authors present evidences for the existence of local type non-uniformity in the periodic (globally uniform) unshunted 2D-JJA. Specifically, the authors found that in the mixed state region AC susceptibilityχ(T, hAC) can be rather well fitted by a single-plaquette approximation of the overdamped 2D-JJA model assuming a non-uniform distribution of the critical current density within a single junction. According to the current paradigm, paramagnetic Meissner effect (PME) can be related to the presence of π-junctions, either resulting from the presence of magnetic impurities in the junction or from unconventional pairing symmetry. Other possible explanations of this phenomenon are based on flux trapping and flux compression effects including also an important role of the surface of the sample. Besides, in the experiments with unshunted 2D-JJA, the authors have previously reported that PME manifests itself through a dynamic reentrance (DR) of the AC magnetic susceptibility as a function of temperature. Using an analytical expression the authors successfully fit the experimental data and demonstrate that the dynamic reentrance of AC susceptibility is directly linked to the value of βC. By simultaneously varying the parameter βL, a phase diagram βC-βL is plotted which demarcates the border between the reentrant and nonreentrant behavior. The authors show that only arrays with sufficiently large value of βC will exhibit the dynamic reentrance behavior and hence PME. The last topic reviewed in this Chapter is related to the step-like structure observed when the resolution of home-made mutual-inductance bridge is improved. That structure (with the number of steps n = 4 for all AC fields) has been observed in the temperature dependence of AC susceptibility in unshunted 2D-JJA with βL(4.2K) = 30. The authors were able to successfully fit their data assuming that steps are related to the geometric properties of the plaquette. The number of steps n corresponds to the number of flux quanta that can be screened by the maximum critical current of the junctions. The steps are predicted to manifest themselves in arrays with the inductance related parameter βL(T) matching a "quantization" condition βL(0)=2π(n+1).

Preface

ix

Chapter 3 - The discovery of superconductivity in MgB2 (Tc~39K) revived interest in non-oxides and initiated a search for superconductivity in related materials. Currently about 100 binary compounds with an AlB2-type structure is known to exist. The noble metal diborides, AgB2 and AuB2 (quasi 2D structure with AlB2-type, space group P6/mmm), which correspond to effectively hole doped systems, have been predicted to be potential candidates for high-Tc superconductors. This is due to their larger density of B2p σ-like states near EF and electron-phonon coupling constant λ than MgB2 and hence higher Tc. Despite difficulties successful synthesis of silver boride thin films was made with nominal composition AgB2 by a Japanese team. Experimental observation confirmed the superconductivity with Tc significantly lower than the theoretically predicted value. The observed value of Tc is comparable with those for some d-metal diborides: ZrB2 (5.5K), TaB2 (9.5K) and NbB2 (5.2K). The authors attempt to explain the discrepancy between the predicted and the observed Tc of AgB2 by invoking the possible role of spin-fluctuations in the system. A study of the mechanical and electronic properties of the noble metal diborides, in comparison with lighter metal (Mg) diboride, has been made using self-consistent density functional theory (DFT). The electronic band structure has also been analysed in order to shed further insight into the differences between the two groups of diborides. The study also includes a review of the existing literatures and indications for future direction of research. Chapter 4 - At the review part the influence of various structure parameters on the temperature of superconducting transition Tc are considered. The original results devoted to this topic are presented in the next chapter. Third part presents low temperature X-ray and neutron powder diffraction study for Bi- and Hg-based HTSC cuprates, where three different structure anomalies at temperatures T0~Tc+15 K, T1~160 K and T2~260 K are established and their origin is discussed. It is shown that the structural anomaly at T0, in vicinity of Tc, is connected with “quasi-ferroelectric” distortion of CuO2-planes and is a sign of presence of corresponding soft phonon mode. It is shown that T0 is linear function of Tc when optimally doped compounds for different systems are compared. This fact means that the mechanism of superconductivity of HTSC cuprates must involve the electron-phonon interaction. Systematic analysis of crystal structure features as function of temperature shows an enhancement of thermal atomic vibration amplitudes and compression of apical bond in the temperature interval T1-T2. The whole complex of observed data is interpreted as result of localization of part of charge carriers at participation of lattice deformation in temperature interval ~160-260 K. Independence of this interval from charge carrier concentration and even chemical composition of HTSC compound confirms this interpretation. Low temperature border of this interval, connected with delocalization of charge carriers, determines the maximal possible Tc value for HTSC cuprates. Chapter 5 - Chemical disorder introduced into the charge reservoir blocks by doping has been shown to be one of the parameters influencing the superconducting transition temperature (Tc) of high-Tc cuprate superconductors (HTSs). The authors address the question of whether the Tc is susceptible to the ordering of dopant atoms. The answer is found by studying the Sr2CuO3+δ superconducting system with K2NiF4-type structure in which oxygen atoms only partially occupy the apical sites next to the CuO2 planes and act as hole dopants. The numerous Tcs appearing in this system are revealed to arise from different

x

James R. Tobin

modulated phases that are formed just by the ordering of apical oxygen, and each of the superconducting modulated phases is associated with a distinct type of the ordering. The superconductivity differences for the modulated phases are revealed to result, mainly, from the ordering of apical oxygen. Chapter 6 - The copper oxide superconductors, or high-Tc superconductors, possess a number of unusual properties due to their complicated interplay of electronic, spin, and lattice degrees of freedom. The mechanism of high-Tc superconductivity is one of the most enduring and important problems in physics, and has never been solved explicitly in theories or clarified thoroughly in experiments, because the multi-layered crystal structures of the materials make the theoretical modelling extremely difficult and the search for the mechanism of high-Tc superconductivity is not successful so far. The main problem is how pairs arise in these materials at such higher temperatures. Lattice vibration (phonon) has long been implicated in conventional low-temperature superconductivity under the BCS theory, but in some sense, has been ignored in high-Tc superconductivity. This article provides a short review on the recent progress in high-Tc superconductivity research - the paring mechanisms which are supported by the recent experimental evidences. Here the authors underline the phonons again based on the recent experimental results that they could also have a supporting role in the high-Tc superconductivity. The explanations to some of the physical phenomena are also given. Chapter 7 - Due to the short superconducting coherent length of high temperature cuprate superconductor (HTS), intrinsic Josephson effect is an exclusive tunneling effect which can be observed with HTS till now. With conventional photolithography and precise control of Ar-ion etching, the authors have first successfully isolated a single intrinsic Josephson junction in two geometries: a U-shaped mesa on top of- and a zigzag structure inside a Bi2Sr2CaCu2O8+x single crystal. The refined fabrication methods are introduced and compared. The single intrinsic Josephson junction (SIJJ) in both structures shows typical single junction behavior, however, with some different characteristics at a high current bias. Both two methods are quite controllable and reproducible. Subgap structures are observed in SIJJ with U-shaped mesa structure. However, the heating effect is still evident. In the SIJJ/IJJs with double-sided zigzag structure, the heating effect is much less. The authors can observe a few upturn (peak) structures in I-V (dI/dV-V) curves, which may originate from the in-plane superconducting transition and/or the energy gap. The SIJJ/IJJs with double-sided zigzag structure are important for both fundamental research like macroscopic quantum tunneling and applications like HTS SQUID. Chapter 8 - This chapter deals with a new class of (Bi,Pb)-2212 based superconductors with highly enhanced superconducting properties by modifying the system with the addition of Rare earths (RE) such as La, Nd, Gd, Dy and Yb. The effect of stoichiometric addition of the REs on the structural, superconducting and flux pinning properties of the bulk superconductor was studied and presented in detail. The samples were prepared by solid state synthesis in polycrystalline form. The RE content in the samples were varied from 0.0 to 0.5 on a general stoichiometry of Bi1.7Pb0.4Sr2.0Ca1.1Cu2.1RExOy (where x = 0.0, 0.1, 0.2, 0.3, 0.4 and 0.5. RE = La, Nd, Gd, Dy and Yb). The samples were characterized using Differential thermal analysis (DTA), X-ray diffraction analysis (XRD), Scanning electron microscopy (SEM), Energy dispersive X-ray analysis (EDX) density measurements and R-T measurements. Superconducting parameters such as critical temperature (TC), critical current density (JC) in self field and applied field, at a comparatively higher temperature of 64 K, of

Preface

xi

the samples were also measured. It was found that, when RE ions are added to (Bi,Pb)-2212 system, they enter into the crystal structure replacing Sr and/or Ca with significant changes in the lattice parameters, microstructure, normal state resistivity, hole concentration and flux pinning strength of the system. Consequently the TC, JC and the field dependence of JC (JC-B characteristics) of the system enhance considerably for an optimum doping level. At higher doping levels these properties decrease from the maximum values. The enhancement in these properties are explained to be due to the substitution of RE3+ ions in place of Sr2+/Ca2+ ions with consequent change in charge carrier concentration (holes) in the Cu-O2 planes. The decrease in the number of charge carriers in (Bi,Pb)-2212 change the system from ‘overdoped’ to ‘optimally-doped’ condition. The substituted RE3+ ions also act as pinning centers as point like defects and improve the field dependence of JC and hence the flux pinning properties. There is a possibility of formation of nano-size secondary precipitates, which may also act as flux pinning centers. At higher levels of addition, the system again changes from ‘optimally-doped’ condition to ‘under-doped condition’. Further the chemical inhomogenity and secondary phase fraction increases at higher levels of RE in the system, which in turn brings down the superconducting properties. Chapter 9 - In R&D of HTS devices most researchers and designers still use the traditional approach for their stability and quench development analysis based on normal zone determination, and consideration of its appearance and propagation. On the other hand most peculiarities of HTS and their relatively high operating temperature make this traditional approach quite impractical and inconvenient. The novel approaches were developed that consider the HTS device as a cooled medium with non-linear parameters with no mentioning of “superconductivity” in the analysis. The approaches showed their effectiveness and convenience to analyze the stability and quench development in HTS devices. In this review the authors present these approaches being well confirmed and verified by the experiments as well as their development for long HTS objects like HTS cables where "blow-up" regimes may happen. The difference of HTS (1-st and 2-nd generations) from LTS is discussed that lead to the difference of their stability and quench development. The authors consider these approaches as very useful for any researchers and designers of modern HTS devices from both first and second generation HTS.

ACKNOWLEDGEMENTS We would like to express our sincere thanks to Dr. Karen A. Yates for her comments on the manuscript and her help with Raman spectroscopy, Dr. Xueyan Song and Dr. Ming Wei on TEM imaging, Mary Vickers on XRD analysis, Douglas Guthrie and Dr. John Cooper for the resistivity measurement, and Dr Adriana Serquis for high field transport measurement. We are also grateful to the Institute of Physics, U.K. for their permission to reuse some of the published figures. Funding from EPSRC and Universiti Putra Malaysia is acknowledged.

In: Superconductivity Research Developments Editor: James R. Tobin, pp. 1-26

ISBN: 978-1-60021-848-4 © 2008 Nova Science Publishers, Inc.

Chapter 1

OPTIMIZATION OF CRITICAL CURRENT DENSITY IN MGB2 S. K. Chen† and J. L. MacManus-Driscoll Department of Materials Science and Metallurgy, University of Cambridge, Pembroke Street, Cambridge CB2 3QZ, U.K.

Abstract This work focuses on the optimisation of critical current density, Jc of bulk polycrystalline MgB2 through studies of the influence of boron precursor powder, nominal Mg non-stoichiometry and by chemical modification. On the influence of the nature of the boron precursor on the superconducting properties of MgB2, Jc’s of samples made from crystalline boron powders are around an order of magnitude lower than those made from amorphous precursors. X-ray, superconducting transition temperature, Tc and resistivity studies indicate that this is as a result of reduced current cross section due to the formation of (Mg)B-O phases. The influence of Mg content was investigated in a series of samples with systematic variation of nominal Mg non-stoichiometry. Jc(H) was found to be influenced significantly with nominal Mg content while Tc remained unchanged. Mg deficient samples show a higher degree of disorder as inferred from the Raman spectroscopy, residual resistivity ratio and xray diffraction. The Mg-deficient samples also showed higher Hirr and Hc2 compared to samples with larger nominal Mg contents. Finally, GaN and Dy2O3 additions into MgB2 during the in situ reaction (owing to enhanced intragranular crystallinity and pinning, respectively) enhance Jc at 6K and 20K up to 5T without changing Tc appreciably.

1. Background MgB2 is a potential candidate for magnetic applications for two main reasons: i) it has a higher superconducting transition temperature, Tc than niobium based superconductors [1] and ii) it has reduced weak-link behaviour at grain boundaries [2]. This implies that grain alignment is not essential for high Jc in MgB2 compared to high temperature superconductors †

E-mail address: [email protected]. Current address: Physics Department, Faculty of Science, Universiti Putra Malaysia, 43400, Serdang, Selangor, Malaysia. (Corresponding author)

2

S. K. Chen and J. L. MacManus-Driscoll

(HTS). Therefore, application in the temperature window of 20 – 26K is of particular interest where conventional Nb-based superconductors can play no role. Higher Tc also indicates cheaper cooling costs which can be achieved by using liquid coolants such as liquid hydrogen (20K), liquid neon (27.2K) or closed-cycle refrigerators that can readily reach below 20K. The cheaper materials cost of magnesium and boron compared to niobium is also an additional advantage for MgB2. Numerous processing methods have been adopted to successfully increase Jc(H) in MgB2. Among these, simple and economically feasible doping has been shown to enhance Jc enormously with only a marginal decrease in Tc [3-5]. The two-gap nature of this material is also intriguing. Because of this, MgB2 can be alloyed with non-magnetic impurities to achieve high Hc2 for different applications [6]. Recent studies have shown that Hc2 can be increased so as to be higher than Nb3Sn at liquid helium temperature [7, 8]. Therefore, MgB2 may well be a replacement for Nb based superconductors for cheaper and high performance applications in the near future. One of the potential applications for MgB2 is open access medical magnetic resonance imaging (MRI) magnets [9, 10]. For such an application, it is in the low field region (< 3T) at ~ 20K that Jc must be improved. Our effort towards optimising Jc is to study and to understand how Jc versus field behaviour for pure MgB2 is influenced by: (i) The form and purity of the boron precursor powders (ii) Nominal Magnesium non-stoichiometry In order to make MgB2 technologically feasible, we use simple and cost effective routes to modify the properties of MgB2 by chemical additions aiming at enhancing Jc(H) without severely degrading Tc.

2. Composition and Microstructure 2.1. Sample Preparation 2.1.1. Starting Powders High purity starting powders contain a lower concentration of impurities which can ‘dirty’ the grain boundaries. The oxidation of MgB2 powder is likely to be severe when stored over a long period of time. Therefore, different qualities of commercial MgB2 powders from Alfa Aesar have been reported. The impurities in the commercial powder include MgO and MgB4 [11-13]. In fact, the starting powder of Mg or B is also contaminated with oxygen to some extent. Impurities, especially MgO, are very common in commercial Mg powder [14]. In boron powder, the impurities can be Mg and B2O3 which are probably the remnants from reduction of B2O3 with Mg to form amorphous B. Impurities such as Si, C, metal impurities [15-18] and H3BO3 [19] can also be found. Variations of Tc of ∼ 1K have been observed in a wide range of samples synthesised using different boron form and purity [16, 18, 20]. The more reactive amorphous boron powder over the crystalline form [17] and smaller particle size of B [18] or Mg [14] enhances the reaction rate. In addition, the resulting formation of smaller MgB2 grains increases Jc.

Optimization of Critical Current Density in MgB2

3

This is thought to be due to grain boundary pinning [14, 18, 21]. Similarly, higher Jc can also be obtained by using smaller particle size commercial MgB2 powder, after ball milling treatment [22].

2.1.2. Reaction Conditions Since Mg is highly sensitive to oxygen, synthesis of MgB2 must be conducted in inert gas or a vacuum atmosphere. The formation of MgO at some level is inevitable as very small amounts of residual oxygen exist in commercially available inert gases used for reaction, and MgO is readily formed even in extremely low oxygen partial pressure [23]. In addition, some starting powders are already oxidised during production and storage as discussed above. As pointed out by Sergey Lee [24], extremely low oxygen partial pressure of 10-60 atm is required to prevent the formation of MgO at the melting point of Mg and this may vary in the range of 10-50 – 10-25 atm at higher annealing temperature of 900°C – 1400°C [25]. Oxygen getters such as tantalum and zirconium, and reducing gas mixtures can be used to minimise the oxidation of Mg. Due to the relatively large difference in melting point between Mg and B (normal melting point of Mg: 650 °C and B: 2075 °C [26]), annealing below and above 650°C leads to solidsolid and liquid-solid reaction, respectively. The latter yields larger grain size than the former due to grain growth at high annealing temperatures [27, 28]. In single crystal synthesis, the crystal shape changes according to synthesis conditions. The formation of needle-like and plate-like crystals is favoured when using a fast heating mode while step heating mode promotes the formation of hexagonal-like crystals [24]. The heating rate has also been found to affect the field dependence of Jc [29].

2.2. Magnesium Non-Stoichiometry Because of the low melting point of magnesium and its high vapor pressure, Mg deficiency will always occur during annealing hence making the determination of exact composition of magnesium in the final product difficult. The high volatility of Mg during annealing and inevitable formation of MgO can also imply Mg deficiency in the reacted samples. In fact, Mg deficiency was found to exist in samples where secondary phases such as MgO or MgB4 were absent [30] or present [31]. An Mg deficient layer has also been observed on subsurface regions of MgB2 pellets [32]. In most cases, Mg deficiency leads to the formation of MgB4 [15, 16, 33, 34]. Phase separation between Mg vacancy rich (non superconducting) and Mg vacancy poor (superconducting) regions in Mg deficient samples has been observed [35, 36]. The amount of Mg vacancy varies depending on sample preparation conditions (1 - 10% Mg vacancy) [15, 24, 34, 37, 38]. Nominal Mg non-stoichiometry has been shown to have a significant effect on Jc [39, 40]. The residual resistivity ratio (RRR) value which is given by the ratio of room temperature resistivity over the resistivity just above Tc, was found to increase with nominal Mg content [16, 41]. Although the correlation between Mg content and lattice parameters was observed, i.e. a-axis decreases with Mg deficiency and the opposite effect for c-axis [33, 42], this can be complicated by lattice strain [15, 34]. Such a strain behavior is also different in Mg rich, Mg stoichiometric and Mg deficient regions [15]. The strain is suggested to be mainly caused by

4

S. K. Chen and J. L. MacManus-Driscoll

oxygen related defects [34] which reside in the grains as stacking faults [43]. The strain effects induced by Mg vacancies have also been shown to decrease Tc [38].

2.3. Grain Connectivity Formation of oxide phases at the grain boundaries leads to increased resistivity [44]. Electron energy loss spectroscopy (EELS) observations show oxygen enriched regions at the grain boundaries, the majority of which contain a BOx layer with typical thickness ∼ 2 nm (amorphous phase). Other regions contain larger areas of MgO ∼ 10 nm thick which is sandwiched between the BOx layers forming the BOx/MgO/BOx structure [45]. The MgO layer of ∼ 10 nm thickness is far beyond the limit for tunnelling to occur (as also discussed by Rowell [44]). The BOx layers (∼ 2 nm) could act as Josephson junctions. However, the size of MgO can vary between 10 nm and 500 nm [43]. Such oxide phases in the grain or other defects will also decrease the electron mean free path resulting in increased intragrain resistivity. Other possible precipitates can be unreacted B, B-O, Mg-B-O and MgB4 [15, 18, 45, 46]. Reacted products between chemical dopant and Mg or B can also decrease grain connectivity especially when they reside at the grain boundaries [47]. The grain connectivity can be further worsened by voids due to the high volatility of Mg. The obtained sample density is normally around half of theoretical density or less even in some doped samples [3]. At least two types of grain boundaries (GB) have been identified in an ex situ grown sample, i.e. (i) an amorphous phase between poorly coupled grains and (ii) structurally intact GB between well coupled grains [43]. The oxygen enriched amorphous GB layer ranges in size from 10 - 50 nm with some crystalline precipitates which are likely to be MgO. Excess Mg is also found in some structurally intact GB. Both clean crystalline and amorphous grain boundaries have been observed in in situ reacted pellets [30]. The amorphous GB with thickness ∼ 20 nm contains a crystalline layer which is less than 10 nm thick [30]. Although the source of these amorphous phases is unclear, they are likely to be from the reaction product rather than a residue of the starting high purity crystalline B and Mg [30]. The most intriguing finding perhaps is from Song et al. [30] showing that even clean samples (ρ(40K) ∼ 1 μΩ cm) contain around 30% amorphous phase at the grain boundaries. The insulating amorphous phase can be B or oxygen rich regions [43, 45]. Therefore, the total cross sectional area for current transport is less than 50% after taking into account of porosity [30]. By compacting the sample by using hot pressing (HP) [48-51] and high pressure sintering [52, 53], Jc can be increased as a result of enhanced grain connectivity or flux pinning by defects. In addition, Jc has also been shown to increase systematically with increased sample density indicating improved grain connectivity by removing voids [54]. A comparative study of the HPed and unHPed samples show that the former reduces RRR as ρ300K decreases but increases in ρ40K [48]. As shown in Table 1, the dense samples show low RRR value comparable to nominally stoichiometric MgB2 (“probably not so clean”) at ambient pressure (Table 2). Higher resistivity is expected if there exist insulating phases at the grain boundaries although the grains are in close contact with each other. Nevertheless, the contribution from grain boundaries is significant in samples with fine grain size. The highly resistive hot pressed samples in [55] (Table 1) is probably due to low temperature

Optimization of Critical Current Density in MgB2

5

Table 1. Resistivity properties of some dense polycrystalline samples (samples are prepared by hot pressing or high pressure sintering). Starting ρ40K powder (µΩ cm) MgB2 [52] (Furuuchi Chemical) 32.1 - 41.7 21 MgB2 [59] (Alfa Aesar) 5.2 MgB2 [60] (Alfa Aesar) Ball milled MgB2 [61] (Alfa Aesar) 11 MgB2 [48] (home made) MgB2 (home made) [44, 55] 480 550°C 1560 700°C Mg + 2B (In situ) [49] 6 850°C 9 900°C

ρ300K

Δρ300-40K

(µΩ cm) 50 18 -

(µΩ cm) 29 12.8 -

RRR (ρ300K/ρ40K) 2.4 3.46 2.7

34

23

3.1

610 2050

130 490

1.27 1.31

27.5 29.8

21.5 20.8

4.6 3.5

Table 2. Resistivity properties of some nominally stoichiometric polycrystalline samples. Samples

ρ40K

ρ300K

Δρ300-40K

(µΩ cm)

(µΩ cm)

(µΩ cm)

RRR (ρ300K/ρ40K)

18 13.5 4.6 65 39 90 300

55.8 41.9 14 140 128.7 189 522

37.8 28.4 9.4 75 89.7 99 222

3.1 3.1 3.03 2.15 3.3 2.1 1.74

MgB2 treated in Mg vapour [30]: Slow cooled Quenched MgB2 (in situ) [41] MgB2 (in situ) [62] MgB2 (in situ) [63] MgB2 (in situ) [57] MgB2 (in situ) added with 10 wt.% SiC [57]

Table 3. Resistivity properties of some “clean” samples (after [30]). Samples Polycrystalline bulk [30] (After correction for restricted connectivity) Polycrystalline bulk [64] Polycrystalline bulk [63] MgB2 filament [65] MgB2 film [66]

ρ40K

ρ300K

Δρ300-40K

(µΩ cm) 1

(µΩ cm) 14.7

(µΩ cm) 13.7

RRR (ρ300K/ρ40K) 14.7

(0.5) 1.0 0.55 0.38 0.28

(7.9) 19.7 8.4 9.6 8.4

(7.4) 18.7 7.85 9.22 8.12

(15.8) 19.7 15.3 25.3 30

6

S. K. Chen and J. L. MacManus-Driscoll Table 4. Resistivity properties of some single crystals and pure Magnesium. Samples

Single crystal [52] Single crystal [67] Single crystal [68] Single crystal [69] Pure Mg [56] * ρ273K/ρ40K

ρ40K

ρ300K

Δρ300-40K

(µΩ cm) 1.8 2.3 1 0.12

(µΩ cm) 9.9 18.2 4

(µΩ cm) 8.1 15.9 3.88

RRR (ρ300K/ρ40K) 5.5 7.9 5 5* 34

sintering and hence poor grain connectivity. Some “clean” samples (Table 3) show lower ρ40K but much higher RRR than single crystals (Table 4) or nominally stoichiomteric MgB2 (Table 2) though they are not fully dense (∼ 30% porosity [30]). The low ρ40K and ρ300K may be related to excess Mg [56] as shown in Table 4. By dirtying the samples by doping with SiC [57] or by Mg vapor treatment [30], RRR is reduced accompanied by increase of both ρ300K and ρ40K (Table 2). If the increase of resistivity is indeed due to reduction in cross sectional area that carries current, then Jc(H=0) ∝ 1/Δρ as the supercurrent is also carried by the same cross sectional area [44]. However, only some films have been demonstrated to obey this relation suggesting that the resistivity behaviour is not solely due to “reduced effective area” [58].

3. Influence of Boron Precursor Powders The boron precursor powders have been found to have a strong influence on the superconducting properties of MgB2 [18]. The four types of commercially available boron powder that were studied are from different sources with varying purity, form and particle size distribution. The details of the four boron precursor powders are summarized in Table 5. Table 5. Source, form and purity of the different boron powders with their particle size distribution [18]. Boron powder B-C98 B-C99 B-A9597 B-A9999

Source

Form

Purity (%)

Alfa Aesar FluoroChem Fluka Alfa Aesar

Crystalline Crystalline Amorphous Amorphous

98 99 95 – 97 99.99

Peak value(s) of particle size distribution (μm) 21.10 11.42, 0.56 a 0.56, 2.42 a 0.54

a

: Powder with dual particle size distribution. Note: The MgB2 samples made from the above B precursors are denoted as C98, C99, A9597 and A9999, respectively.

As indicated in Table 5, particle size analysis showed the existence of a bimodal particle size in the boron precursor powders B-C99 and B-A9597. A plot of particle size distribution is shown in Figure 1. The peaks belonging to B2O3 are obvious in the XRD patterns [18].

Optimization of Critical Current Density in MgB2

7

Boron powder B-A9597 contains both crystalline and amorphous phases despite being labeled as amorphous on the commercial package. 100 B-C98 B-C99 B-A9597 B-A9999

Percentage (%)

80 60 40 20 0 0.1

1

10

Particle size (μm) Figure 1. Particle size distributions of the four studied boron precursor powders.

From the XRD patterns, MgB2 is the dominant phase in all the in-situ reacted Mg + 2B pellets with the presence of MgO as a second phase [18]. Additional impurities such as Mg3(BO3)2, B2O and B13O2 can be indexed in samples made from the crystalline B powder. Besides, several peaks due to unidentified phases were also observed. Some un-reacted Mg was always found in sample C98. The difference in phase formation is attributed to the reactivity of the powders as the amorphous form of B powder is more reactive than the crystalline. In addition, the smaller particle size of amorphous B powder further enhances the reaction rate [17]. The boron precursor powder of B-A9999 with smallest particle size exhibited highest reactivity in forming MgB2. As indicated in Figure 2, MgB2 as well as MgO only formed in the sample made from B-A9999 when the annealing temperature was reduced to 600°C, while others only showed Mg peaks. The observed grain size of ∼ 100 – 200 nm and ∼ 1μm for A9999 and C98, respectively, as shown in Figures 3 and 4 is consistent with the SEM imaging [18]. The crystalline samples show larger grains of about a few hundred nm compared to the amorphous samples with a size of about ~ 100 nm and less [18]. The bimodal grain size of some samples is also obvious and consistent with the bimodal particle size distribution (refer to Table 5) measured in the precursor B powders. The TEM image of Figure 3(a) showed that there are possible grain boundary wetting phases which can be a current limiting factor at the nano scale in sample C98.

8

S. K. Chen and J. L. MacManus-Driscoll

1.2

Mg

o: MgO; +: Mg

1.0 Mg

0.8 0.6

Mg

Mg

u Mg

0.2

Mg

Mg Mg Mg

Mg

C99 C98

0.0

+

1.0

101

0.8

0.2

u

110

MgO Mg Mg

Mg

002

001

0.4

100

0.6

Mg Mg

111 200 201

Intensity (a.u.)

0.4

Mg Mg

A9999 A9597

Mg

0.0 20

30

40

50

60

70

80

2θ (degree) Figure 2. X-ray powder diffraction patterns of MgB2 samples reacted at 600°C for 30 min. “u” denotes unidentified impurity.

(a) Figure 3. TEM images at two different magnifications for sample C98.

(b)

Optimization of Critical Current Density in MgB2

(a)

9

(b)

MgB2

Figure 4. TEM images at two different magnifications for sample A9999. Inset: The corresponding electron diffraction. (TEM images by Xueyan Song, Department of Materials Science and Engineering, University of Wisconsin-Madison).

Figure 5. shows the temperature dependence of the resistivity normalised to 300K for samples prepared from the various boron precursor powders. A9999 exhibited the lowest residual resistivity ratio, RRR (ρ290K/ρ40K) of 2.06 among the samples whereas sample C98 showed comparable RRR to A9597. The resistivity properties is presented in Table 6. As shown in Table 6, another batch of samples was prepared to check the reproducibility of the ρ290K. A9597 showed the lowest ρ290K. The room temperature resistivity of samples C99 and A9999 were very close with a resistivity approaching that of C98.

Normalised resistivity

1.0 A9999

0.8 C99

0.6

A9597

C98 C99 A9597 A9999

0.4 0.2

C98

0.0 0

50

100

150

200

250

300

Temperature (K) Figure 5. Temperature dependence of normalized resistivity.

10

S. K. Chen and J. L. MacManus-Driscoll Table 6. Resistivity properties of MgB2 pellets made from the various boron precursors. Samples C98 C99 A9597 A9999

ρ40K

ρ290K

Δρ290-40K

(µΩ cm) 82.5 83.1 14.9 109.5

(µΩ cm) 344.1 (274.8)* 230.3 (242.6)* 60.7 (44.6)* 225.3 (178.4)*

(µΩ cm) 261.6 147.2 45.8 115.8

RRR(ρ290K/ρ40K) 4.17 2.77 4.07 2.06

()*: 2nd batch samples.

The susceptibility of the samples measured by using SQUID showed the variation of superconducting transition temperature, Tc ∼ 1K, i.e. 37.9K (C99) and 38.8K (C98) while both samples made from amorphous B showed the same Tc of 38.2K [18]. However, their Jc’s differ very much from one and another (Figure 6). In general, Jc’s of the samples prepared from crystalline B precursors are lower than the amorphous ones. As can be seen from Table 6 and Figure 6, the crystalline precursor samples (C98 and C99) show larger Δρ290-40K compared to samples prepared from amorphous B indicating that they have reduced current carrying cross section. This is due to greater sample inhomogeneity and / or poorer intergrain connectivity for the crystalline precursor samples as they contain more oxide phases (as shown from XRD). Sample C98 should have higher resistivity values than sample C99 as it has a larger amount of oxide phases and lower Jc. Owing to the presence of Mg, the room temperature resistivity in C98 is only marginally higher than for C99 (Table 6). The RRR value of sample C98 is much higher than C99, approaching that of A9597.

6K

5

10

4

C98 C99 A9597 A9999

(Acm-2-2)) JJcc (A.cm

10

3

10

5

20K

10

4

10

3

10

2

10

0

1

2

3

4

5

Applied field (T) Figure 6. Magnetic critical current densities versus applied magnetic field at 6K and 20K [18].

Optimization of Critical Current Density in MgB2

11

From Figure 6, it can be noticed that the Jc of samples A9597 is lower than A9999 by a factor of 3 or more even though they have similar x-ray diffraction patterns. Besides, the room temperature resistivity of A9999 is about a factor of 4 higher than A9597. The room temperature resistivities of sample A9999 is around 200 μΩ cm with Δρ290-40K > 100 μΩ cm, approaching the crystalline precursor samples. The higher resistivity values of A9999 can be due to the existence of cracks in this sample as it is brittle in nature. Two other possible reasons for the higher resistivities are: (a) larger grain boundary because of the very fine grain size, (b) presence of Mg non-stoichiometry in the MgB2 grains Jc enhancement can be contributed by grain boundary pinning due to (a) while Hc2 would be increased by enhanced intragrain scattering due to scenario (b). This may explain the better Jc(H) behaviour of samples A9999 despite higher room temperature resistivity. Scenario (b) is explored in more detail in the next section.

4. Influence of Nominal Magnesium Non-stoichiometry In order to study the influence of nominal Mg composition, a series of MgxB2 samples were prepared by in situ reaction of crystalline Mg and amorphous B powders [71]. Samples were prepared by conventional mixing, pressing into pellets which were then wrapped in Ta foil and reacted at 900ºC for 15 minutes in a reducing atmosphere of 2% H2-Ar. The heating and cooling rates used were 15 ºC/min. Table 7 shows the secondary phases present in the samples with different nominal x in MgxB2. MgB2 dominates with MgO as a secondary phase in all samples. MgB4 was also seen for x ≤ 1.0 and no Mg peak was detected for x > 1.0. Semi-quantitative analysis was performed using the Philips software, HighScore where the amount (%) of MgB2, MgB4 and MgO was calculated from the scale factor and RIR (Reference Intensity Ratio) values. The error for the calculated values can be as large as 10%. The intensity ratio of the MgB4 (121) to MgB2 (101) peak, IMgB4/IMgB2 is also presented for comparison in Table 7. It is obvious from both RIR and the intensity ratios that the MgB4 phase increase in quantity with Mg deficiency [15, 33]. MgB4 can be from two sources, i.e. (i) in situ formation concurrently during the formation of MgB2 as a result of Mg deficiency, and (ii) decomposition of MgB2 [72, 73]. In fact, the onset of significant Mg evaporation (at > 10-9 Torr) has been shown to occur at as low as 425°C [73]. The amount of MgO stayed almost constant with Mg deficiency although a larger fraction was found in sample Mg1.5B2. No Mg was detected from XRD in samples x > 1 compared to references [39, 40] suggesting that excess Mg was evaporated and/or oxidised. The microstructures of the samples taken using FEG-SEM are shown in Figure 7. No significant change in grain size was observed with varying x from 0.95 to 1.2. However, sample x = 1.5 had slightly larger grains of ∼ 200 – 300 nm compared to ∼ 100 nm for the rest of the samples. The presence of a large fraction of voids is clearly seen showing the high porosity in these samples. The sample density is around 1.2 gcm-3 (compared to the theoretical density of 2.6 gcm-3) across the sample range, independent of x.

12

S. K. Chen and J. L. MacManus-Driscoll Table 7. Secondary phases as detected from XRD. The calculated amount of phases (%) is compared with the intensity ratio of MgB4 peak of (121) and MgB2 peak of (101), IMgB4/IMgB2. x (MgxB2)

0.95 0.98 0.99 1.0 1.1 1.2 1.5

Secondary Phase MgO, MgB4 MgO, MgB4 MgO, MgB4 MgO, MgB4 MgO MgO MgO

Calculated values (± 10%) MgB2 MgB4 MgO 65 15 20 63 16 21 66 10 24 68 7 25 75 25 78 22 71 29

Mg0.95B2

MgB2

Mg1.2B2

Mg1.5B2

IMgB4/IMgB2 0.070 0.066 0.035 0.041 -

Figure 7. FEG-SEM images taken from the fracture cross section of samples with various nominal Mg contents.

Figure 8 shows the temperature dependence of resistivity, ρ(T) normalised to 300K for samples MgxB2 where x = 0.95, 0.98, 1.0, 1.2 and 1.5. Samples x = 0.95, 0.98 and 1.0 exhibited almost identical ρ(T) whereas samples x = 1.2 and 1.5 showed a stronger temperature dependence of resistivity indicative of more metallic behaviour with higher nominal Mg content. A summary of the resistivity data is given in Table 8.

Optimization of Critical Current Density in MgB2

13

As shown in Table 8, ρ40K decreased rapidly from x = 0.95 to 0.98 and from thereon it remained fairly flat. There is no significant difference in ρ40K for samples x = 0.98, 1.0 and 1.5 while this is slightly higher for sample x = 1.2. The high ρ40K in sample x = 0.95 could result from the increase of scattering [44, 74] due to the presence of MgB4. Sample x = 0.95 also had the highest room temperature resistivity, ρ290K suggestive of supercurrent path blocking by the second phase of MgB4, most likely at grain boundaries. Then, ρ290K decreased to ∼ 70 μΩ cm, i.e. almost four times lower in samples x = 0.98 - 1.0 before increasing to 122.4 μΩ cm and 115.7 μΩ cm in samples x = 1.2 and 1.5, again indicative of second phases, possibly from excess Mg(O) as shown in Table 7.

Normalised resistivity

1.0

Mg0.95B2

0.8

MgB2 Mg0.98B2

0.6 Mg0.95B2

0.4 Mg1.2B2 0.2

Mg0.98B2 MgB2

Mg1.5B2

Mg1.2B2 Mg1.5B2

0.0 0

50

100

150

200

250

300

Temperature (K) Figure 8. Temperature dependence of normalised resistivity for samples x = 0.95, 0.98, 1.0, 1.2 and 1.5 (MgxB2).

Sample x = 0.95 also showed the highest Δρ among samples. We recall that Δρ is also a measure of current carrying cross section if the high resistivity is due solely to lack of grain connectivity [44]. Δρ in samples x = 1.2 and 1.5 is higher compared to samples x = 0.98 and 1.0. Hence, the higher Δρ values for x = 0.95, 1.2 and 1.5 suggests the lower current carrying cross section for these samples. The RRR for samples x = 0.95, 0.98 and 1.0 are very close and this value is higher with increasing Mg content as observed for samples x = 1.2 and 1.5 (Table 8). According to Chen et al. [41], the decrease of the power-law behaviour in low temperature resistivity and RRR with decreasing Mg nominal content suggests that disorder is induced by Mg deficiency while analysis of magnetoresistance data for samples exhibiting decreasing RRR with decreasing Mg nominal content implies that the variation in normal state resistivity behaviour arises from

14

S. K. Chen and J. L. MacManus-Driscoll

the grain rather than the grain boundary. Lower RRR was also observed in dsordered samples after neutron irradiation and moderate irradiation enhanced Hc2 [75]. Estimation of nonuniform strain of the x-ray data using a Williamson-Hall plot [76] showed non-uniform strain (%) values of (0.185 ± 0.061), (0.149 ± 0.042) and (0.148 ± 0.046) for samples x = 0.95, 1.0 and 1.5, respectively. The RRR values increase to 3.57 in the most Mg rich sample (Mg1.5B2) indicating that the samples become cleaner (less scattering) with a higher Mg content. The range of RRR value obtained in our sample series is comparable to some nominally stoichiometric polycrystalline MgB2 which are believed to be rather “dirty” (Table 2) as well as dense polycrystalline samples (Table 1) and is much lower than the RRR of single crystal (Table 4) or clean samples (Table 3). Table 8. Resistivity properties for samples 0.95 ≤ x ≤ 1.5 (MgxB2). Sample (MgxB2) Mg0.95B2 Mg0.98B2 Mg0.99B2 MgB2 Mg1.1B2 Mg1.2B2 Mg1.5B2

ρ40K

ρ290K

Δρ = ρ290K-40K

(µΩ cm) 157.25 33.21 32.60 47.89 32.38

(µΩ cm) 324 71 68.3 63.9 122.4 115.7

(µΩ cm) 166.75 37.79 31.3 74.51 83.32

RRR (ρ290K/ρ40K) 2.06 2.07 1.96 2.56 3.57

25

Hc2 Hirr MgB2

Magnetic field (Τ)

20

Mg1.5B2 Clean MgB2 [57]

15

10

5

0 0

5

10

15

20

25

30

35

40

Temperature (K) Figure 9. Hirr and Hc2 versus temperature for sample MgB2 and Mg1.5B2.

Figure 9 shows the irreversibility field, Hirr and upper critical field, Hc2 determined from transport measurements. Hirr and Hc2 were determined as 10% and 90% from the resistive transition curve, respectively as described in [77] and references therein. Sample “MgB2”, i.e. x = 1.0, exhibits higher Hc2 and Hirr than Mg1.5B2 and the difference between them is more obvious below 30K. We recall from Table 7 that the “MgB2” sample contains MgB4, indicating that it is Mg deficient as for the MgxB2 samples where x < 1.0. Hirr and Hc2 values

Optimization of Critical Current Density in MgB2

15

extrapolated to 4.2K are about 12.5 T and 21 T for MgB2 compared to 10 T and 17 T for Mg1.5B2. Hc2 and Hirr of a “clean” polycrystalline MgB2 (RRR ∼ 14.7, ρ(40K) = 1 μΩ cm) made from stoichiometric mixture of isotopically 99.5% enriched crystalline 11B and pure Mg [57] is also included for comparison. The Hc2 there was obtained from transport measurement while Hirr was estimated magnetically using a 100 Acm-2 criterion [57].

Intensity/Intensity 600 cm

-1

1.0

Mg0.95B2 Mg0.98B2

0.9

Mg0.99B2 MgB2 Mg1.1B2 Mg1.2B2

0.8

Mg1.5B2 300

400

500

600

700

800

900

1000

-1

Raman Shift (cm ) Figure 10. Normalised Raman spectra at 600 cm-1 for 0.95 ≤ x ≤ 1.5 in MgxB2.

1.01 1.00

I730/I600

0.99 0.98 0.97 0.96 0.95 0.94 0.9

1.0

1.1

1.2

1.3

1.4

1.5

x (MgxB2) Figure 11. Intensity ratio of 730 cm-1 over 600 cm-1 Raman peak versus x (MgxB2). Dashed lines are guides for eye.

Figure 10 shows the Raman spectra obtained across the series x = 0.95 to 1.5 normalised to the intensity at 600 cm-1. The broad and asymmetric peak centred at around 600 cm-1

16

S. K. Chen and J. L. MacManus-Driscoll

dominates the Raman spectrum of MgB2 and has been ascribed to the E2g mode which is the only Raman active mode for the P6mmm space group [78]. In addition to the peak at 600 cm-1, two shoulders are clearly seen at around 400 cm-1 and 730 cm-1. This feature was also observed in Al doped samples and associated with structural disorder [79-81]. The appearance of more than one phonon peak further implies the violation of Raman selection rules induced by disorder resulting in a modified Raman spectrum [79, 80]. The degree of disorder as indicated by the ratio of the normalised intensity at 730 cm-1 over 600 cm-1 [82] is plotted at Figure 11. Within the noise levels, the ratio is fairly flat for all the points in the range of x = 0.98 – 1.1, and then decreases to 1.5 indicative of higher disorder at low x and lower disorder at high x. As shown in Figure 9, it is obvious that sample MgB2 with I730/I600 > 0.985 exhibited much higher Hirr and Hc2 (below 30K) than sample Mg1.5B2 with I730/I600 of 0.955. In addition, the low x value samples have similar ρ characteristics compared to the x = 1.2 and 1.5 samples which are more metallic.

0.0

χ(a.u.)

-0.2

Mg0.95B2

(a)

Mg0.98B2

-0.4

Mg0.99B2

-0.6

MgB2

-0.8 -1.0 0.0 -0.2 -0.4 -0.6

MgB2

(b)

Mg1.1B2 Mg1.2B2 Mg1.5B2

-0.8 -1.0 20

25

30

35

40

Temperature (K) Figure 12. Temperature dependence of normalised susceptibility for samples MgxB2 (a) 0.95 ≤ x ≤ 1.0 and (b) 1.0 ≤ x ≤ 1.5.

Optimization of Critical Current Density in MgB2

17

As shown in Figure 12, Tc was almost independent of x, i.e. 37.5 K for all the samples except for sample Mg1.5B2 which has a slightly higher Tc of 38 K. Also, no broadening in the transition curve was observed from Figure 12 in agreement with [15, 34, 40, 41]. Figure 13 shows the field dependence of Jc at 6K and 20K for 0.95 ≤ x ≤ 1.0. Jc decreased with Mg deficiency for x < 1.0 with sample Mg0.95B2 showing much lower Jc. Jc’s are fairly similar in the range of 0.98 ≤ x ≤ 1.0. At 20K, the discrepancy in Jc became less significant and samples 0.98 ≤ x ≤ 1.0 exhibited comparable Jc.

6K 5

10

MgB2 Mg0.98B2

-2

Jc (Acm )

Mg0.99B2 Mg0.95B2

4

10

20K

5

10

4

10

3

10

2

10

0

1

2

3

4

5

Applied field (T) Figure 13. Jc versus field for samples 0.95 ≤ x ≤ 1.0 (MgxB2) at 6K and 20K.

Similarly, Jc decreased with excess Mg with sample Mg1.5B2 showing the lowest Jc as shown in Figure 14. Variation of Jc(H) with nominal Mg non stoichiometry has been reported [39, 40] and ascribed to there being more grain boundary pinning in Mg deficient samples [39]. From Table 8, the obtained Δρ values on our samples show large differences across the range. While grain sizes are larger for sample Mg1.5B2, the other samples are all very similar sizes (Figure 7) indicating grain boundary is not the dominant factor in setting Jc. At low field

18

S. K. Chen and J. L. MacManus-Driscoll

(20K, < 2T) the sample with a grain size of ∼ 500 nm had a comparable Jc to the sample with a grain size of ∼ 300 nm, although Jc of the former dropped more rapidly at higher field [27]. Moreover, lattice distortion has a significant influence on the Jc behaviour [27, 83].

6K 5

10

MgB2

4

Mg1.1B2 Mg1.2B2

-2

Jc (Acm )

10

Mg1.5B2

3

10

20K

5

10

4

10

3

10

2

10

0

1

2

3

4

5

Applied field (T) Figure 14. Jc versus field for samples 1.0 ≤ x ≤ 1.5 (MgxB2) at 6K and 20K.

As shown in Figure 15, the form of Jc dependence on applied field at 20K did not change much in the composition range of x = 0.95 – 1.0 but there was a slight change for Mg1.1B2 and Mg1.2B2. Sample Mg1.5B2 showed a significantly stronger field dependence. It has previously been suggested that phase separation of MgB2 occurs into Mg vacancyrich and Mg vacancy-poor regions [35]. The relative volume of these phases changes with the overall Mg vacancy concentration [35, 36]. In fact, Rietveld refinement on the high resolution powder neutron diffraction data established clear evidence for the existence of temperatureindependent phase inhomogeneity with varying Mg stoichiometry in undoped samples, i.e. Mg1-δB2 where δ ≥ 0 [31]. Added to this, a Mg deficient layer has also been observed on the subsurface region by x-ray photoemission spectroscopy [32]. For the region where x < 1.1, Mg deficient MgxB2 exists. The amount of MgB4 in this region decreases with increasing Mg content as indicated by the intensity ratio of MgB4 to MgB2 (Table 7). The insignificant change of intensity ratio of Raman peaks, I730/I600 (Figure

Optimization of Critical Current Density in MgB2

19

11) and the RRR (Table 8) implies the degree of disorder in this region is fairly constant as also shown by the comparable normalised Jc(H) behaviour (Figure 15) and similar ρ(T) behaviour (Figure 8).

10

0

20K 10

-1

Normalised Jc

Mg0.95B2 10

Mg0.98B2

-2

Mg0.99B2 MgB2 10

0

20K 10

-1

MgB2 10

Mg1.1B2

-2

Mg1.2B2 Mg1.5B2 1

2

3

4

5

Applied field (T) Figure 15. Normalised Jc versus field at 20K. Jc is normalised at 1.2T.

The high volatility of Mg and hence the final Mg content in reacted MgB2 is strongly dependent on preparation conditions such as starting powders (purity, particle size and form) and reaction conditions. Due to high reactivity of the boron precursor and fine Mg powder used in our study [18] in addition to the way samples are reacted (i.e. Mg is always lost to the reaction atmosphere), Mg vacancies exist at x = 1.0 and even at higher x [30, 31]. It has been reported that the “clean” sample with no indication of the presence of excess Mg, MgO or MgB4 was shown by electron probe microanalysis to have ratio Mg:B of 0.95 : 2 [30]. Since the intensity ratio of Raman peaks and RRR values for x > 1.1 are still changing as shown in Figure 11 and Table 8, samples 1.1 ≤ x ≤ 1.5 are still within the Mg deficient region even though MgB4 is not present in this region. By increasing the Mg content beyond x > 1.5, it is

20

S. K. Chen and J. L. MacManus-Driscoll

possible that a near stoichiometric MgB2 phase can be obtained. In the region where unreacted Mg is observed, presumably MgB2 with minimal Mg deficiency should exist. As indicated in Table 8, the RRR of sample Mg1.5B2 is comparable to some nominally stoichiometric polycrystalline MgB2 (∼ 2 – 3, Table 2) samples which are believed to be Mgdeficient while it is generally accepted that “clean” samples have larger RRR (> 10, as shown in Table 3). Nevertheless, the increase of RRR (Table 8) indicates that the samples are moving towards stoichiometric Mg1B2 with increasing Mg content. The decreasing intensity ratio of Raman peaks (Figure 11) suggests that the Mg vacancy induced disorder decreases with increasing nominal Mg content. In addition, Jc of Mg1.5B2 dropped more rapidly with applied field (Figure 15) and its Hirr and Hc2 are lower than sample x = 1.0 (Figure 9). Although Mg1.5B2 has a slightly larger grain size (∼ 100 nm in x = 1.0 compared to ∼ 200 -300 nm x = 1.5) implying reduced grain boundary pinning, this can not solely account for either the steep drop of Jc with field (Figure 15) or the measured lower Hirr(T) and Hc2(T) (Figure 9). Varying Mg content with no change in Tc as observed in our samples is thought to predominantly distort the Mg site [84, 85] leading to π band but not σ band scattering. Thus, the stronger π band scattering in sample x = 1.0 should lead to higher Hc2 compared to x = 1.5 [6]. It should be pointed out that a Mg vacancy ordering superlattice structure was observed by Sharma et al. in electron diffraction patterns with the incident electron beam parallel to the (001) zone axis but not in (100) direction supporting vacancy ordering only from Mg sublattice [35]. Interestingly, dislocations and stacking faults were observed in the (001) basal plane [43] suggesting that they could be induced by Mg vacancies. Finally, it should be reflected that the variation of Mg content affects Hirr and Hc2 considerably without degrading Tc, agreeing with reference [6]. Therefore, this suggests that significant improvements can be made to the magnetic properties of undoped MgB2 without recourse to second phase doping which generally lowers Tc. The issue of Mg nonstoichiometry also provokes concern about some doping studies and whether the main cause of Hirr and Hc2 enhancement is due to the role of pinning centres (precipitates produced from reaction between dopants and parent elements of Mg and B), Mg vacancy induced disorder, Mg induced disorder as a result of doping or a combination of these three. For example, Mg vacancies may exist if a dopant D does not react with Mg or B in Mg1-xDxB2 or the “added” dopant reacts with Mg in nominally prepared MgB2.

5. Chemical Modification: Second Phase Additions To attempt liquid phase sintering, GaN (1, 3 and 5 at. %) was added into Mg + 2B during in situ reaction [86]. The amount of secondary phases increases with GaN additions. The reaction products are MgO, Mg5Ga2 and Mg-B-N phases showing that some of the N2 evolved from the decomposition of GaN reacted with Mg and B. Neither XRD data nor SEM shows any preferential alignment of the grain structure. However, the pure and GaN added sample shows strikingly different microstructure with the latter show larger and more platey grains. TEM image shows the presence of amorphous region at the grain boundary. These regions are believed to be the quenched liquid phase and they cover about 20% of the fracture surface. The additions of GaN even up to 5 at.% does not degrade Tc while the room temperature resistivity increases with amount of additive showing that the secondary phases of the reaction product may dirty the grain boundary or increase the intragrain scattering.

Optimization of Critical Current Density in MgB2

21

There is also a systematic increment in density though the increase in density is not as great as one can achieve through hot pressing [51]. Additions of Dy2O3 up to 5 wt. % does not decrease the Tc to any measurable extent [87]. Samples with 5 wt.% additions show a decrease in Tc for only 0.5 K with a broader transition curve. A relatively small change in Tc indicates there is no alloying effect, i.e. no doping of the Dy into the MgB2 lattice structure. The apparent decrease in the room temperature resistivity is believed to be due to the reaction product of Mg as there is deficiency in B due to the formation of DyB4 [87]. This effect overcomes the impurity scattering (increase in resistivity) from DyB4 and MgO. The decrease in room temperature resistivity suggests that the excess Mg resides in the grain boundary regions. 6

10

6K 5

10

Dy2O3

10

GaN

0.5 wt.% 1 wt.% 5 wt.% Pure

-2

Jc (Acm )

4

1 at.% 3 at.% 5 at.%

20K

5

10

4

10

3

10

2

10

0

1

2

3

4

5

Applied field (T) Figure 16. Field dependence of Jc for GaN and Dy2O3 added MgB2.

Figure 16 shows the Jc(H) at 6K and 20K for the GaN and Dy2O3 added samples. Jc’s of the pure samples from reference [86] are also included for comparison. It is obvious from Figure 16 that the Dy2O3 added samples show higher Jc than those added with GaN. However, they both exhibit the same Jc(H) behaviour. For liquid phase sintering, Jc (6K, 1T)

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S. K. Chen and J. L. MacManus-Driscoll

of the 3 at. % GaN added samples (highest Jc among the GaN added samples) is approximately one-half of the value of the HIPed samples [51]. In contrast to HIPed samples in which the enhancement of Jc is through densification, grain connectivity and dislocations [48], the enhancement in Jc for the GaN added sample was accompanied by the marginal increment in sample density (by around 8% for the 3 at.% GaN added sample). The presence of amorphous regions [86] is expected to obstruct the current path in the sample as also indicated by the increase of room temperature resistivity with GaN additions. Therefore, the overall improvement in Jc must originate from the enhanced intragrain Jc. Additions of Dy2O3 (starting particle size is 1-3 μm) led to much enhanced pinning. Jc (6K, 1T) of the 0.5 wt.% added samples is ∼ 6.5 × 105 Acm-2, equivalent to HIPed pure samples [51] or nano-Si added MgB2 [51] at ≤ 1T. TEM images show even distribution of nano scale precipitate of DyB4 and MgO within the MgB2 matrix.

6. Summary We have shown that the critical current density of MgB2 is strongly dependent on the starting boron precursor powders and the control of nominal magnesium content. Further, we also demonstrated that the critical current density can be enhanced by using simple and cost effective routes via chemical additions. The form and purity of boron precursors has a significant influence on the critical current density of MgB2. The lower Jc in samples prepared from crystalline boron compared to amorphous boron can be explained by the reduction of the current carrying cross section area by several oxide impurities. Excess oxide phases also degrade the field dependence of Jc presumably because grain connectivity is degraded. Therefore, the difference in Jc among the samples made from crystalline and amorphous boron was simply the presence of the amount of oxide phases. The differences between samples prepared from amorphous boron is less clear as they show large differences in Jc(H) despite showing very similar phase purity. In addition to starting boron precursor powders, magnesium content must also be well controlled in order to optimise Jc. We showed that nominal Mg non-stoichiometry affects Jc(H) considerably while leaving Tc relatively unchanged. Magnesium deficient samples show lower RRR values and more disorder as observed from Raman spectroscopy as well as higher Hc2(T) and Hirr(T) values. Finally, it is possible to enhance the Jc by second phase additions of GaN and Dy2O3 without affecting Tc appreciably. Dy2O3 additions of ∼ 0.5 wt.% results in a Jc ∼ of 6.5 × 105 Acm-2 at 6K, 1T. TEM imaging showed the presence of nano precipitates of DyB4 and MgO in the grains which may account for the enhanced pinning.

Acknowledgements We would like to express our sincere thanks to Dr. Karen A. Yates for her comments on the manuscript and her help with Raman spectroscopy, Dr. Xueyan Song and Dr. Ming Wei on TEM imaging, Mary Vickers on XRD analysis, Douglas Guthrie and Dr. John Cooper for the resistivity measurement, and Dr Adriana Serquis for high field transport measurement. Funding from EPSRC and Universiti Putra Malaysia is also gratefully acknowledged.

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In: Superconductivity Research Developments Editor: James R. Tobin, pp. 27-62

ISBN: 978-1-60021-848-4 © 2008 Nova Science Publishers, Inc.

Chapter 2

MAGNETIC PROPERTIES OF ARTIFICIALLY PREPARED HIGHLY ORDERED TWO-DIMENSIONAL SHUNTED AND UNSHUNTED NB–ALOX–NB JOSEPHSON JUNCTIONS ARRAYS Fernando M. Araújo-Moreira Department of Physics and Physical Engineering - UFSCar, Laboratory of Materials and Devices, Multidisciplinary Center for the Development of Ceramic Materials, Caixa Postal 676, São Carlos/SP 13565-905, BRAZIL

Sergei Sergeenkov Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russian Federation

Abstract Josephson junction arrays (JJA) have been actively studied for decades. However, they continue to contribute to a wide variety of intriguing and peculiar phenomena. To name just a few recent examples, it suffice to mention the so-called paramagnetic Meissner effect and related reentrant temperature behavior of AC susceptibility, observed both in artificially prepared JJA and granular superconductors. Employing mutual-inductance measurements and using a high-sensitive home-made bridge, we have thoroughly investigated the temperature and magnetic field dependence of complex AC susceptibility of artificially prepared highly ordered (periodic) two-dimensional Josephson junction arrays (2D-JJA) of both shunted and unshunted Nb–AlOx–Nb tunnel junctions In this Chapter, we report on three phenomena related to the magnetic properties of 2DJJA: (a) the influence of non-uniform critical current density profile on magnetic field behavior of AC susceptibility; (b) the origin of dynamic reentrance and the role of the Stewart-McCumber parameter, βC, in the observability of this phenomenon, and (c) the manifestation of novel geometric effects in temperature behavior of AC magnetic response. Firstly, we present evidences for the existence of local type non-uniformity in the periodic (globally uniform) unshunted 2D-JJA. Specifically, we found that in the mixed state region

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Fernando M. Araújo-Moreira and Sergei Sergeenkov AC susceptibility χ(T, hAC) can be rather well fitted by a single-plaquette approximation of the overdamped 2D-JJA model assuming a non-uniform distribution of the critical current density within a single junction. According to the current paradigm, paramagnetic Meissner effect (PME) can be related to the presence of π-junctions, either resulting from the presence of magnetic impurities in the junction or from unconventional pairing symmetry. Other possible explanations of this phenomenon are based on flux trapping and flux compression effects including also an important role of the surface of the sample. Besides, in the experiments with unshunted 2D-JJA, we have previously reported that PME manifests itself through a dynamic reentrance (DR) of the AC magnetic susceptibility as a function of temperature. Using an analytical expression we successfully fit our experimental data and demonstrate that the dynamic reentrance of AC susceptibility is directly linked to the value of βC. By simultaneously varying the parameter βL, a phase diagram βC-βL is plotted which demarcates the border between the reentrant and non-reentrant behavior. We show that only arrays with sufficiently large value of βC will exhibit the dynamic reentrance behavior and hence PME. The last topic reviewed in this Chapter is related to the step-like structure observed when the resolution of home-made mutual-inductance bridge is improved. That structure (with the number of steps n = 4 for all AC fields) has been observed in the temperature dependence of AC susceptibility in unshunted 2D-JJA with βL(4.2K) = 30. We were able to successfully fit our data assuming that steps are related to the geometric properties of the plaquette. The number of steps n corresponds to the number of flux quanta that can be screened by the maximum critical current of the junctions. The steps are predicted to manifest themselves in arrays with the inductance related parameter βL(T) matching a "quantization" condition βL(0)=2π(n+1).

I. Introduction Artificially prepared two-dimensional Josephson junctions arrays (2D-JJA) consist of highly ordered superconducting islands arranged on a symmetrical lattice coupled by Josephson junctions (Fig. 1), where it is possible to introduce a controlled degree of disorder. In this case, a 2D-JJA can be considered as the limiting case of an extreme inhomogeneous type-II superconductor, allowing its study in samples where the disorder is nearly exactly known. Since 2D-JJA are artificial, they can be very well characterized. Their discrete nature, together with the very well-known physics of the Josephson junctions, allows the numerical simulation of their behavior (see very interesting reviews by Newrock et al.[1] and by Martinoli et al.[2] on the physical properties of 2D-JJA).

Nb-island

Tunnel junction

Shunt resistor

(I)

(II)

Figure 1. Photograph of unshunted (I) and shunted (II) Josephson junction arrays.

Magnetic Properties of Artificially Prepared Highly Ordered Two-Dimensional…

29

Many authors have used a parallelism between the magnetic properties of 2D-JJA and granular high-temperature superconductors (HTS) to study some controversial features of HTS. It has been shown that granular superconductors can be considered as a collection of superconducting grains embedded in a weakly superconducting - or even normal - matrix. For this reason, granularity is a term specially related to HTS, where magnetic and transport properties of these materials are usually manifested by a two-component response. In this scenario, the first component represents the intragranular contribution, associated to the grains exhibiting ordinary superconducting properties, and the second one, which is originated from intergranular material, is associated to the weak-link structure, thus, to the Josephson junctions network [3-6]. For single-crystals and other nearly-perfect structures, granularity is a more subtle feature that can be envisaged as the result of a symmetry breaking. Thus, one might have granularity on the nanometric scale, generated by localized defects like impurities, oxygen deficiency, vacancies, atomic substitutions and the genuinely intrinsic granularity associated with the layered structure of perovskites. On the micrometric scale, granularity results from the existence of extended defects, such as grain and twin boundaries. From this picture, granularity could have many contributions, each one with a different volume fraction [7-10]. The small coherence length of HTS implies that any imperfection may contribute to both the weak-link properties and the flux pinning. This leads to many interesting peculiarities and anomalies, many of which have been tentatively explained over the years in terms of the granular character of HTS materials. One of the controversial features of HTS elucidated by studying the magnetic properties of 2D-JJA is the so-called Paramagnetic Meissner Effect (PME), also known as Wohlleben Effect. In this case, one considers first the magnetic response of a granular superconductor submitted to either an AC or DC field of small magnitude. This field should be weak enough to guarantee that the critical current of the intergranular material is not exceeded at low temperatures. After a zero-field cooling (ZFC) process which consists in cooling the sample from above its critical temperature (TC) with no applied magnetic field, the magnetic response to the application of a magnetic field is that of a perfect diamagnet. In this case, the intragranular screening currents prevent the magnetic field from entering the grains, whereas intergranular currents flow across the sample to ensure a null magnetic flux throughout the whole specimen. This temperature dependence of the magnetic response gives rise to the well-known double-plateau behavior of the DC susceptibility and the corresponding doubledrop/double-peak of the complex AC magnetic susceptibility [7-11]. On the other hand, by cooling the sample in the presence of a magnetic field, by following a field-cooling (FC) process, the screening currents are restricted to the intragranular contribution (a situation that remains until the temperature reaches a specific value below which the critical current associated to the intragrain component is no longer equal to zero). It has been experimentally confirmed that intergranular currents may contribute to a magnetic behavior that can be either paramagnetic or diamagnetic. Specifically, where the intergranular magnetic behavior is paramagnetic, the resulting magnetic susceptibility shows a striking reentrant behavior. All these possibilities about the signal and magnitude of the magnetic susceptibility have been extensively reported in the literature, involving both LTS and HTS materials [12-15]. The reentrant behavior mentioned before is one of the typical signatures of PME. We have reported its occurrence as a reentrance in the temperature behavior of the AC magnetic susceptibility of 2D-JJA [16,17]. Thus, by studying 2D-JJA, we were able to demonstrate that the appearance of PME is simply related to trapped flux and has nothing to do with

30

Fernando M. Araújo-Moreira and Sergei Sergeenkov

manifestation of any sophisticated mechanisms, like the presence of pi-junctions or unconventional pairing symmetry. In this Chapter we report on three phenomena related to the magnetic properties of 2DJJA: (a) the influence of non-uniform critical current density profile on magnetic field behavior of AC susceptibility; (b) the observability of dynamic reentrance and the role of the Stewart-McCumber parameter, βC, in this phenomenon, and (c) the manifestation of novel geometric effects in temperature behavior of AC magnetic response. To perform this work, we have used numerical simulations and both the mutual-inductance and the scanning SQUID microscope experimental techniques. The paper is organized as follows. In Sec. II we outline the main concepts related to the mutual-inductance technique (along with the physical meaning of the measured output voltage) as well as the scanning SQUID microscope experimental technique. In Sec. III we review the numerical simulations based on a unit cell containing four Josephson junctions. In Sec. IV we describe the influence of non-uniform critical current density profile on magnetic field behavior of AC susceptibility and discuss the obtained results. In Sec. V we study the origin of dynamic reentrance and discuss the role of the Stewart-McCumber parameter in the observability of this phenomenon. In Sec. VI we present the manifestation of completely novel geometric effects recently observed in the temperature behavior of AC magnetic response. And finally, in Sec. VII we summarize the main results of the present work.

II. The Mutual-Inductance Technique Complex AC magnetic susceptibility is a powerful low-field technique to determine the magnetic response of many systems, like granular superconductors and Josephson junction arrays. It has been successfully used to measure several parameters such as critical temperature, critical current density and penetration depth in superconductors. To measure samples in the shape of thin films, the so-called screening method has been developed. It involves the use of primary and secondary coils, with diameters smaller than the dimension of the sample. When these coils are located near the surface of the film, the response, i.e., the complex output voltage V, does not depend on the radius of the film or its properties near the edges. In the reflection technique [18], an excitation coil (primary) coaxially surrounds a pair of counter-wound pick up coils (secondaries). When there is no sample in the system, the net output from these secondary coils is close to zero since the pick up coils are close to identical in shape but are wound in opposite directions. The sample is positioned as close as possible to the set of coils, to maximize the induced signal on the pick up coils (Figure 2). An alternate current sufficient to create a magnetic field of amplitude hAC and frequency f is applied to the primary coil. The output voltage of the secondary coils, V, is a function of the complex susceptibility, χ AC = χ´+iχ´´, and is measured through the usual lock-in technique. If we take the current on the primary as a reference, V can be expressed by two orthogonal components. The first one is the inductive component, VL (in phase with the time-derivative of the reference current) and the second one the quadrature resistive component, VR (in phase with the reference current). This means that VL and VR are correlated with the average magnetic moment and the energy losses of the sample, respectively.

Magnetic Properties of Artificially Prepared Highly Ordered Two-Dimensional… Primary coil

31

Secondary coils

ip

δV

JJA SAMPLE Figure 2. Screening method in the reflection technique, where an excitation coil (primary) coaxially surrounds a pair of counter-wound pick up coils (secondaries).

We used the screening method in the reflection configuration to measure χAC(T) of Josephson junction arrays. Measurements were performed as a function of the temperature T (1.5K < T < 15K), the amplitude of the excitation field hAC (1 mOe < hAC < 10 Oe), and the external magnetic field HDC (0 < HDC < 100 Oe) parallel with the plane of the sample (Figure 3). The frequency in the experiments reported here was fixed at f = 1.0 kHz. The typical dimensions of the coils and samples are depicted in Fig. 4 The susceptometer was positioned inside a double wall μ-metal shield, screening the sample region from Earth's magnetic field.

hac

HDC Figure 3. Sketch of the experimental setup, where the excitation field field

H dc

h ac

and the external magnetic

are respectively perpendicular and parallel to the plane of the sample.

For a complete description of this technique, let us study now the relation between the measured complex voltage, V= VL + iVR, and the components of the AC magnetic susceptibility, χ’ and χ”. We assume that the current in the drive coil (primary) is given by

I D e iωt , which creates at the sample an average magnetic field H D e iωt . Considering the section of the sample as a simple loop, we model its response as an impedance ZS in series

32

Fernando M. Araújo-Moreira and Sergei Sergeenkov

with a geometrical inductance, L g . The impedance depends on the material parameters as well as the size of the loop. For a normal metal sample, ZS = 2πρ( rt ) Δr , with ρ the resistivity of the material, r the radius of the loop, t the thickness of the sample, and Δr the width of the loop.

2 mm

pick-up coils

primary coil sample

6 mm Figure 4. Typical dimensions of the coils and samples.

We can obtain equivalent equations for the specific case of a superconducting material. The equation relating the drive field to the current response IS of the loop is given by:

−

∂Φ ext = −iωμ 0 H D e iωt A = IS ( ZS + iωL g ) ∂t

(II.1)

where A is the area of the loop. Taking ZS = X = iY , Eq. (II.1) reduces to:

IS =

− iAωμ 0 H D e iωt X + i ( Y + ωL g )

(II.2)

The induced voltage in the pick-up coil is given by:

− M SP (iωIS ) = VP

(II.3)

where MSP is the mutual inductance between the sample and the pickup coil. Combining Eqs. (II.2) and (II.3), we obtain:

VP = −

ω2 AMSP μ 0 H D e iωt X + i(Y + ωL g )

(II.4)

To obtain the magnetic susceptibility, we first find a relationship between the effective magnetization <M> of the loop and IS. Since B = μ 0 ( H + M ) , we may write:

Magnetic Properties of Artificially Prepared Highly Ordered Two-Dimensional…

μ 0 [< H > + < M >]A =< B > A = Φ

33 (II.5)

From this, we identify the magnetic flux due to the current in the sample as being proportional to the average magnetization:

μ 0 < M > A = L g IS

(II.6)

Combining Eqs. (II.2) and (II.6), gives:

−

iωL g H D e iωt X + i(Y + ωL g )

=< M >= (χ'−iχ" )H D e iωt

(II.7)

where we have neglected higher harmonics considering the response of the loop given by the average magnetization:

< M >= (χ'−iχ" )H D e iωt

(II.8)

On the other hand, since the pickup coil is counter wound, it only responds to dM dt , so that:

VP ∝ −

∂M ∝ (−ωχ"−iωχ' )H D e iωt ∂t

(II.9)

From Eqs. (II.2) and (II.6)- (II.8), we obtain:

μ 0 M SP Aω (− χ"−iχ')H D eiωt = VP = VP '+iVP " Lg

(II.10)

which agrees with Eq.(II.9). From Eq. (II.7)we can write:

χ' =

χ" =

ωL g Y + ω2 L2g X 2 + ( Y + ωL g ) 2

ωL g X X 2 + (Y + ωL g ) 2

(II.11a)

(II.11b)

To get the complete response of a real sample, these equations should be integrated over the whole specimen. For the special case of a superconducting loop far below TC, where we can neglect the normal channel in a two-fluid model, the induced EMF in a magnetic field

34

Fernando M. Araújo-Moreira and Sergei Sergeenkov

H D eiωt is still given by ε = −iωAμ 0 H D e iωt . The loop has now a kinetic inductance LK as well

as

a

− iωAμ 0 H D e

geometrical i ωt

inductance

Lg

(

so

that

= iω(L K + L g )IS , or IS = − Aμ 0 H D e

i ωt

the

) (L

current K

is

given

by

+ L g ). Eq. (II.6) implies

that the magnetization is:

< M >= −

Lg LK + Lg

H D e i ωt

(II.12)

or, alternatively, that:

χ' = −

Lg L K + Lg

(II.13)

χ" = 0 which agrees with Eqs. (II.11) setting X = 0 and Y = ωL K . Therefore, we have:

χ'∝ VL

(II.14a)

χ"∝ VR

(II.14b)

This means that by measuring the output voltage from the secondary coils, we can obtain the components of the complex AC magnetic susceptibility, χ , as we stated in the beginning.

III. Numerical Simulations We have found that all the experimental results obtained from the magnetic properties of 2DJJA can be qualitatively explained by analyzing the dynamics of a single unit cell in the array [16, 17].

Figure 5. Unit cell of the array, containing a loop with four identical junctions.

Magnetic Properties of Artificially Prepared Highly Ordered Two-Dimensional…

35

In our experiments, the unit cell is a loop containing four junctions (Fig. 5) and the measurements correspond to ZFC AC magnetic susceptibility. We model a single unit cell as having four identical junctions, each with capacitance CJ, quasi-particle resistance RJ and critical current IC. We apply an external field of the form:

H ext = h AC cos(ωt )

(III.1)

The total magnetic flux, Φ TOT , threading the four-junction superconducting loop is given by:

Φ TOT = Φ EXT + LI

(III.2)

where Φ EXT = μ 0 a H EXT with μ 0 being the vacuum permeability, I is the circulating 2

current in the loop, L is the inductance of the loop and Φ EXT is the flux related to the applied magnetic field. Therefore the total current is given by:

Φ 0 dγ i C J Φ 0 d 2 γ i I = I Csinγ i + + 2πR J dt 2π dt 2

(III.3)

Here, γ i is the superconducting phase difference across the ith junction and IC is the critical current of each junction. In the case of our model with four junctions, the fluxoid quantization condition, which relates each γ i to the external flux, is:

γi =

π π Φ TOT n− 2 2 Φ0

(III.4)

where n is an integer and, by symmetry, we assume:

γ1 = γ 2 = γ 3 = γ 4 = γ i

(III.5)

In the case of an oscillatory external magnetic field of the form of Eq. (III.1), the magnetization is given by:

M=

LI μ 0a 2

It may be expanded as a Fourier series in the form:

(III.6)

36

Fernando M. Araújo-Moreira and Sergei Sergeenkov ∞

M( t ) = h AC ∑ [χ 'n cos(nωt ) + χ"n sin (nωt )]

(III.7)

n =0

We calculated χ' and χ" through this equation. Both Euler and fourth-order RungeKutta integration methods provided the same numerical results. In our model we do not include other effects (such as thermal activation) beyond the above equations. In this case, the temperature-dependent parameter is the critical current of the junctions, given to good approximation by [19]:

I C (T) = I C (0) 1 −

⎡ T T ⎤ T tanh ⎢1.54 C 1 − ⎥ TC ⎦ T TC ⎣

(III.8)

We calculated χ1 as a function of T. χ1 depends on the parameter β L , which is proportional to the number of flux quanta that can be screened by the maximum critical current in the junctions, and the parameter βC , which is proportional to the capacitance of the junction:

β L (T ) =

β C (T ) =

2πLIC (T) Φ0 2πI C C J R 2J Φ0

(III.9)

(III.10)

IV. Influence of Non-uniform Critical Current Density Profile on Magnetic Field Behavior of AC Susceptibility Despite the fact that Josephson junction arrays (2D-JJA) have been actively studied for decades, they continue to contribute to the variety of intriguing and peculiar phenomena. To name just a few recent examples, it suffice to mention the so-called paramagnetic Meissner effect and related reentrant temperature behavior of AC susceptibility, observed both in artificially prepared 2D-JJA and granular superconductors (for recent reviews on the subject matter, see Refs. [20–24] and further references therein). So far, most of the investigations have been done assuming an ideal (uniform) type of array. However, it is quite clear that, depending on the particular technology used for preparation of the array, any real array will inevitably possess some kind of non-uniformity, either global (related to a random distribution of junctions within array) or local (related to inhomogeneous distribution of critical current densities within junctions). For instance, recently a comparative study of the magnetic remanence exhibited by disordered (globally non-uniform) 3D-JJA in response to an excitation with an AC magnetic field was presented [25]. The observed temperature behavior of the remanence curves for arrays fabricated from three different materials (Nb,

Magnetic Properties of Artificially Prepared Highly Ordered Two-Dimensional…

37

YBa2Cu3O7 and La1.85Sr0.15CuO4) was found to follow the same universal law regardless of the origin of the superconducting electrodes of the junctions which form the array. In the section, through an experimental study of complex AC magnetic susceptibility χ(T,hac) of the periodic (globally uniform) 2D-JJA of unshunted Nb–AlOx–Nb junctions, we present evidence for existence of the local type non-uniformity in our arrays. Here, hAC corresponds to the amplitude of excitation field. Specifically, we found that in the mixed state region χ(T,hac) can be rather well fitted by a single-plaquette approximation of the over-damped 2DJJA model assuming a non-uniform (Lorentz-like) distribution of the critical current density within a single junction. Our samples consisted of 100 × 150 unshunted tunnel junctions. The unit cell had square geometry with lattice spacing a = 46 μm and a junction area of 5 × 5 μm2. The critical current density for the junctions forming the arrays was about 600 A/cm2 at 4.2 K, giving thus IC = 150 μA for each junction. We used the screening method [26] in the reflection configuration to measure the complex AC susceptibility χ = χ'+iχ" of our 2D-JJA (for more details on the experimental technique and set-ups see [27–29]). Fig. 6 shows the obtained experimental data for the complex AC susceptibility χ(T, h ac ) as a function of hac for a fixed temperature below TC. As is seen, below 50 mOe (which corresponds to a Meissner-like regime with no regular flux present in the array) the susceptibility, as expected, practically does not depend on the applied magnetic field, while in the mixed state (above 50 mOe) both χ' (T, h ac ) and χ" (T, h ac ) follow a quasi-exponential field behavior of the single junction Josephson supercurrent (see below). To understand the observed behavior of the AC susceptibility, in principle one would need to analyze the flux dynamics in our over-damped, unshunted 2D-JJA. However, given a well-defined (globally uniform) periodic structure of the array, to achieve our goal it is sufficient to study just a single unit cell (plaquette) of the array. (It is worth noting that the single-plaquette approximation proved successful in treating the temperature reentrance phenomena of AC susceptibility in ordered 2D-JJA [24,27,28] as well as magnetic remanence in disordered 3D-JJA [25]). The unit cell is a loop containing four identical Josephson junctions. Since the inductance of each loop is L = μ 0 a = 64 pH and the critical current of each junction is IC = 150 μA, for the mixed-state region (above 50 mOe) we can safely neglect the self-field effects because in this region the inductance related flux Φ L ( t ) = LI( t ) (here I(t) is the total current circulating in a single loop [29]) is always smaller than the external field induced flux Φ ext ( t ) = B ac ( t ) ⋅ S (here S ≈ a is the projected area of a single 2

loop, and B ac ( t ) = μ 0 h ac cos(ωt ) is an applied AC magnetic field). Besides, since the length L and the width w of each junction in our array is smaller than the Josephson penetration depth, then:

λj =

Φ0 2πμ 0 djc 0

(where jc0 is the critical current density of the junction, Φ 0 is the magnetic flux quantum, and

d = 2λ L + ξ is the size of the contact area with λ L (T ) being the London penetration depth

38

Fernando M. Araújo-Moreira and Sergei Sergeenkov

of the junction and ξ an insulator thickness), namely L ≈ w ≈ 5 μm and λ j ≈ 20 μm (using jc0 = 600 A/cm2 and λ L = 39 nm for Nb at T = 4.2 K), we can adopt the small junction approximation [29] for the gauge-invariant superconducting phase difference across the ith junction (by symmetry we assume that [27,28] φ1 = φ 2 = φ 3 = φ 4 = φ i ), then:

2πB ac ( t )d ⋅x φ0

φi (x, t ) = φ0 +

(IV.1)

where φ 0 is the initial phase difference. The net magnetization of the plaquette is

M( t ) = SI S ( t ) , where the maximum upper current (corresponding to φ 0 = π 2 ) through an inhomogeneous Josephson contact reads: L

w

0

0

I S ( t ) = ∫ dx ∫ dyjc ( x , y) cos φ i ( x, t )

(IV.2)

For the explicit temperature dependence of the Josephson critical current density:

⎡ Δ(T ) ⎤ ⎡ Δ(T ) ⎤ tanh ⎢ jc 0 (T ) = jc 0 (0) ⎢ ⎥ ⎥ ⎣ Δ(0) ⎦ ⎣ 2 k BT ⎦

(IV.3)

we used the well-known [30] analytical approximation for the BCS gap parameter (valid for all temperatures):

⎛ T −T ⎞ ⎟ Δ(T) = Δ(0) tanh⎜⎜ 2.2 C ⎟ T ⎝ ⎠ where Δ (0) = 1.76k B TC . In general, the values of χ' (T, h AC ) and χ" (T, h AC ) of the complex harmonic susceptibility are defined via the time dependent magnetization of the plaquette as follows:

χ' (T, h ac ) =

1 πh AC

χ" (T, h AC ) =

1 πh AC

∫

2π

0

∫

d(ωt ) cos(ωt )M( t )

2π

0

d (ωt )sin (ωt )M( t )

(IV.4)

(IV.5)

Using Eqs. (IV.1)–(IV.5) to simulate the magnetic field behavior of the observed AC susceptibility of the array, we found that the best fit through all the data points and for all temperatures is produced assuming the following non-uniform distribution of the critical current density within a single junction [29]:

Magnetic Properties of Artificially Prepared Highly Ordered Two-Dimensional…

⎛ L2 ⎞⎛ w 2 ⎞ ⎟ ⎟⎜ jc ( x , y) = jc 0 (T)⎜⎜ 2 2 ⎟⎜ 2 2 ⎟ ⎝ x + L ⎠⎝ y + w ⎠

39

(IV.6)

It is worthwhile to mention that in view of Eq. (IV.2), in the mixed-state region the above distribution leads to approximately exponential field dependence of the maximum supercurrent I S (T , h AC ) ≈ I S (T ,0) exp(−h AC / h0 ) which is often used to describe critical-state behavior in type-II superconductors [31]. Given the temperature dependencies of the London penetration depth λ L (T ) and the Josephson critical current density jc 0 (T ) , we find that: 14

Φ0

⎛T −T ⎞ ⎟⎟ ≈ h0 (0) ⋅ ⎜⎜ C h0 (T ) = 2πμ 0λ j (T ) L ⎝ TC ⎠

(IV.7)

for the temperature dependence of the characteristic field near TC. This explains the improvement of our fits (shown by solid lines in Fig. 6) for high temperatures because with increasing the temperature the total flux distribution within a single junction becomes more regular which in turn validates the use of the small-junction approximation.

χAC (SI)

χAC (SI)

-0.54

-0.63

0.0

10

100

10

hAC (mOe)

-0.50

100

hAC (mOe)

(a)

χ'

χ" 0.050

χAC(SI)

χAC(SI)

-0.55

-0.60

-0.65

0.025

0.000

-0.025

-0.70 1

10

hAC (mOe)

100

1

(b) Figure 6. Continued on next page

10

hAC (mOe)

100

40

Fernando M. Araújo-Moreira and Sergei Sergeenkov

χ'

χ"

-0.55

0.06

χAC(SI)

χAC(SI)

-0.60

0.04

-0.65

0.02

-0.70

0.00

10

10

100

100

hAC (mOe)

hAC (mOe)

(c) 0.0

χ'

-0.1

0.2

χ"

-0.2 0.1

χAC(SI)

χAC(SI)

-0.3 -0.4 -0.5

0.0

-0.6 -0.7 -0.1

-0.8 0.1

1

10

hAC (mOe)

100

0.1

(d)

1

10

100

hAC (mOe)

Figure 6. The dependence of both components of the complex AC magnetic susceptibilities, on AC magnetic field amplitude hAC for different temperatures: (a)T= 4.2 K, (b), T = 6 K, (c) T = 7.5 K, and (d) T = 8 K. Solid lines correspond to the fitting of the 2D-JJA model with non-uniform critical current profile for a single junction (see the text).

V. On the Origin of Dynamic Reentrance and the Role of the Stewart-McCumber Parameter According to the current paradigm, paramagnetic Meissner effect (PME) [32-37], can be related to the presence of π -junctions [38], either resulting from the presence of magnetic impurities in the junction [39,40] or from unconventional pairing symmetry [41]. Other possible explanations of this phenomenon are based on flux trapping [42] and flux compression effects [43] including also an important role of the surface of the sample [34]. Besides, in the experiments with unshunted 2D-JJA, we have previously reported [44] that PME manifests itself through a dynamic reentrance (DR) of the AC magnetic susceptibility as a function of temperature. These results have been further corroborated by Nielsen et al. [45] and De Leo et al. [46] who argued that PME can be simply related to magnetic screening in multiply connected superconductors. So, the main question is: which parameters are directly responsible for the presence (or absence) of DR in artificially prepared arrays?

Magnetic Properties of Artificially Prepared Highly Ordered Two-Dimensional…

41

Previously (also within the single plaquette approximation), Barbara et al. [44] have briefly discussed the effects of varying β L on the observed dynamic reentrance with the main emphasis on the behavior of 2D-JJA samples with high (and fixed) values of β C . However, to our knowledge, up to date no systematic study (either experimental or theoretical) has been done on how the β C value itself affects the reentrance behavior. In the present section of this review, by a comparative study of the magnetic properties of shunted and unshunted 2D-JJA, we propose an answer to this open question. Namely, by using experimental and theoretical results, we will demonstrate that only arrays with sufficiently large value of the StewartMcCumber parameter β C will exhibit the dynamic reentrance behavior (and hence PME). To measure the complex AC susceptibility in our arrays we used a high-sensitive homemade susceptometer based on the so-called screening method in the reflection configuration [47-49], as shown in previous sections. The experimental system was calibrated by using a high-quality niobium thin film. To experimentally investigate the origin of the reentrance, we have measured χ' (T) for three sets of shunted and unshunted samples obtained from different makers (Westinghouse and Hypress) under the same conditions of the amplitude of the excitation field h ac (1 mOe <

h ac <10 Oe), external magnetic field H dc (0 < H dc < 500 Oe) parallel to the plane of the sample, and frequency of AC field ω = 2πf (fixed at f = 20 kHz). Unshunted 2D-JJAs are formed by loops of niobium islands linked through Nb-AlOx-Nb Josephson junctions while shunted 2D-JJAs have a molybdenum shunt resistor (with R sh ≈ 2.2Ω ) short-circuiting each junction (see Fig. 1). Both shunted and unshunted samples have rectangular geometry and consist of 100 × 150 tunnel junctions. The unit cell for both types of arrays has square geometry with lattice spacing a ≈ 46μm and a single junction area of 5 × 5μm . The 2

critical current density for the junctions forming the arrays is about 600A/cm2 at 4.2 K. Besides, for the unshunted samples β C (4.2K ) ≈ 30 and β L ( 4.2K ) ≈ 30 , while for shunted samples β C (4.2K ) ≈ 1 and β L ( 4.2K ) ≈ 30 where β L and β C are given by expressions (III.9) and (III.10), respectively [50]. There, C j ≈ 0.58pF is the capacitance,

R j ≈ 10.4Ω the quasi-particle resistance (of unshunted array), and I C (4.2K ) ≈ 150μA the critical current of the Josephson junction. Φ 0 is the quantum of magnetic flux. The parameter β L is proportional to the number of flux quanta that can be screened by the maximum critical current in the junctions, while the Stewart-McCumber parameter β C basically reflects the quality of the junctions in arrays. It is well established that both magnetic and transport properties of any superconducting material can be described via a two-component response [51], the intragranular (associated with the grains exhibiting bulk superconducting properties) and intergranular (associated with weak-link structure) contributions [52,53]. Likewise, artificially prepared JJAs (consisting of superconducting islands, arranged in a symmetrical periodic lattice and coupled by Josephson junctions) will produce a similar response [54].

42

Fernando M. Araújo-Moreira and Sergei Sergeenkov

0.0 (a) unshunted

χ'(SI)

-0.2

hac=100 mOe

-0.4

-0.6 hac=10 mOe

-0.8 0.2

0.4

0.6 T/TC

0.8

1.0

0.00 (b)

shunted

-0.03

χ'(SI)

-0.05 -0.08

hac=200 mOe

hac=25 mOe

-0.10 -0.13 -0.15 0.4

Figure 7. Experimental results for mOe; (b) shunted 2D-JJA for

hac=10 mOe

0.6 0.8 T/TC

χ' (T, h ac , H dc ) :

1.0

(a) unshunted 2D-JJA for

h ac =

10 and 100

h ac = 10, 25, and 200 mOe. In all these experiments H dc = 0 . Solid

lines are the best fits (see text).

Since our shunted and unshunted samples have the same value of β L and different values of β C , it is possible to verify the dependence of the reentrance effect on the value of the

Magnetic Properties of Artificially Prepared Highly Ordered Two-Dimensional…

43

Stewart-McCumber parameter. For the unshunted 2D-JJA (Fig. 7a) we have found that for an AC field lower than 50 mOe (when the array is in the Meissner-like state) the behavior of χ' (T) is quite similar to homogeneous superconducting samples, while for h ac > 50 mOe (when the array is in the mixed-like state with practically homogeneous flux distribution) these samples exhibit a clear reentrant behavior of susceptibility [44]. At the same time, the identical experiments performed on the shunted samples produced no evidence of any reentrance for all values of h ac (see Fig. 7b). It is important to point out that the analysis of the experimentally obtained imaginary component of susceptibility χ" (T) shows that for the highest AC magnetic field amplitudes (of about 200 mOe) dissipation remains small. Namely, for typical values of the AC amplitude, h ac = 100 mOe (which corresponds to about 10 vortices per unit cell) the imaginary component is about 15 times smaller than its real counterpart. Hence contribution from the dissipation of vortices to the observed phenomena can be safely neglected.

0.1 Hdc=30.5 Oe

0.0 -0.1

χ'(SI)

-0.2 -0.3 -0.4 -0.5 -0.6

Hdc=26 Oe

-0.7

Hdc=19.5 Oe

-0.8

Figure 8. Experimental results for

Hdc=13 Oe

0.4

0.6 0.8 T/TC

χ' (T, h ac , H dc )

and 30.5 Oe. In all these experiments

Hdc=0 Oe

1.0

for unshunted 2D-JJA for

H dc =

0, 13, 19.5, 26,

h ac = 100 mOe. Solid lines are the best fits (see text).

To further study this unexpected behavior we have also performed experiments where we measure χ' (T ) for different values of H dc keeping the value of h ac constant. The influence of DC fields on reentrance in unshunted samples is shown in Fig. 8. On the other hand, the shunted samples still show no signs of reentrance, following a familiar pattern of fieldinduced gradual diminishing of superconducting phase (very similar to a zero DC field flatlike behavior seen in Fig.7b).

44

Fernando M. Araújo-Moreira and Sergei Sergeenkov

To understand the influence of DC field on reentrance observed in unshunted arrays, it is important to emphasize that for our sample geometry this parallel field suppresses the critical current I C of each junction without introducing any detectable flux into the plaquettes of the array. Thus, a parallel DC magnetic field allows us to vary I C independently from temperature and/or applied perpendicular AC field. The measurements show (see Fig. 8) that the position of the reentrance is tuned by H dc . We also observe that the value of temperature Tmin (at which χ' (T) has a minimum) first shifts towards lower temperatures as we raise H dc (for small DC fields) and then bounces back (for higher values of H dc ). This non-monotonic behavior is consistent with the weakening of I C and corresponds to Fraunhofer-like dependence of the Josephson junction critical current on DC magnetic field applied in the plane of the junction. We measured I C from transport current-voltage characteristics, at different values of H dc at T = 4.2 K and found that χ' (T = 4.2K ) , obtained from the isotherm T = 4.2 K (similar to that given in Fig. 8), shows the same Fraunhofer-like dependence on H dc as the critical current I C (H dc ) of the junctions forming the array (see Fig. 9). This gives further proof that only the junction critical current is varied in this experiment. This also indicates that the screening currents at low temperature (i.e., in the reentrant region) are proportional to the critical currents of the junctions. In addition, this shows an alternative way to obtain I C (H dc ) dependence in big arrays. And finally, a sharp Fraunhofer-like pattern observed in both arrays clearly reflects a rather strong coherence (with negligible distribution of critical currents and sizes of the individual junctions) which is based on highly correlated response of all single junctions forming the arrays, thus proving their high quality. Such a unique behavior of Josephson junctions in our samples provides a necessary justification for suggested theoretical interpretation of the obtained experimental results. Namely, based on the above-mentioned properties of our arrays, we have found that practically all the experimental results can be explained by analyzing the dynamics of just a single unit cell in the array. To understand the different behavior of the AC susceptibility observed in shunted and unshunted 2D-JJAs, in principle one would need to analyze in detail the flux dynamics in these arrays. However, as we have previously reported [44], because of the well-defined periodic structure of our arrays (with no visible distribution of junction sizes and critical currents), it is reasonable to expect that the experimental results obtained from the magnetic properties of our 2D-JJAs can be quite satisfactory explained by analyzing the dynamics of a single unit cell (plaquette) of the array. An excellent agreement between a single-loop approximation and the observed behavior (seen through the data fits) justifies a posteriori our assumption. It is important to mention that the idea to use a single unit cell to qualitatively understand PME was first suggested by Auletta et al. [55]. They simulated the field-cooled DC magnetic susceptibility of a single-junction loop and found a paramagnetic signal at low values of external magnetic field.

Magnetic Properties of Artificially Prepared Highly Ordered Two-Dimensional…

45

χ'

(solid

Figure 9. The critical current

IC

triangles) as a function of DC field

(open squares) and the real part of AC susceptibility

H dc

for T=4.2K (from Ref.44).

To understand the different behavior of the AC susceptibility observed in shunted and unshunted 2D-JJAs, in principle one would need to analyze in detail the flux dynamics in these arrays. However, as we have previously reported [44], because of the well-defined periodic structure of our arrays (with no visible distribution of junction sizes and critical currents), it is reasonable to expect that the experimental results obtained from the magnetic properties of our 2D-JJAs can be quite satisfactory explained by analyzing the dynamics of a single unit cell (plaquette) of the array. An excellent agreement between a single-loop approximation and the observed behavior (seen through the data fits) justifies a posteriori our assumption. It is important to mention that the idea to use a single unit cell to qualitatively understand PME was first suggested by Auletta et al. [55]. They simulated the field-cooled DC magnetic susceptibility of a single-junction loop and found a paramagnetic signal at low values of external magnetic field. In our calculations and numerical simulations, the unit cell is a loop containing four identical Josephson junctions and the measurements correspond to the zero-field cooling (ZFC) AC magnetic susceptibility. We consider the junctions of the single unit cell as having capacitance C j , quasi-particle resistance R j and critical current I C . As shown in previous sections, here we have also used this simple four-junctions model to study the magnetic behavior of our 2D-JJA by calculating the AC complex magnetic susceptibility χ = χ'+iχ" as a function of T, β L and β C . Specifically, shunted samples are identified through low values of the McCumber parameter β C ≈ 1 while high values β C >> 1 indicate an unshunted 2D-JJA. If we apply an AC external field B ac ( t ) = μ 0 h ac cos(ωt ) normally to the 2D-JJA and a DC field Bdc = μ 0 H dc parallel to the array, then the total magnetic flux Φ ( t ) threading the four-junction superconducting loop is given by Φ ( t ) = Φ ext ( t ) + LI( t ) where L is the loop

46

Fernando M. Araújo-Moreira and Sergei Sergeenkov

inductance, Φ ext ( t ) = SB ac ( t ) + (ld)B dc is the flux related to the applied magnetic field (with l × d being the size of the single junction area, and S ≈ a being the projected area of the loop), and the circulating current in the loop reads: 2

I (t ) = I C (T ) sin φi (t ) +

Φ 0 dφi C j Φ 0 d 2φi + 2πR j dt 2π dt 2

(V.1)

Here φi ( t ) is the gauge-invariant superconducting phase difference across the ith junction, and Φ 0 is the magnetic flux quantum. Since the inductance of each loop is L = μ 0 a ≈ 64 pH, and the critical current of each junction is I C ≈ 150μA , for the mixed-state region (above 50 mOe) we can safely neglect the self-field effects because in this region LI( t ) is always smaller than Φ ext ( t ) . Besides, since the length l and the width w of each junction in our array is smaller than the Josephson penetration depth λ j =

Φ 0 2πμ 0 djc 0 (where jc 0 is the critical current density of the

junction, and d = 2λ L + ξ is the size of the contact area with λ L (T ) being the London penetration depth of the junction and ξ an insulator thickness), namely l ≈ w ≈ 5μm and

λ j ≈ 20μm (using jc 0 ≈ 600A / cm 2 and λ L ≈ 39nm for Nb at T = 4.2 K), we can adopt the small-junction approximation [50] for the gauge-invariant superconducting phase difference across the ith junction (for simplicity we assume as usual [44] that φ1 = φ 2 = φ3 = φ 4 ≡ φi ):

φi ( t ) = φ 0 (H dc ) +

2πBac ( t )S Φ0

(V.2)

where φ 0 (H dc ) = φ 0 (0) + 2πμ 0 H dc ld Φ 0 with φ 0 (0) being the initial phase difference. To properly treat the magnetic properties of the system, let us introduce the following Hamiltonian: 4

H( t ) = J ∑ [1 − cos φi ( t )] + i =1

1 LI( t ) 2 2

(V.3)

which describes the tunneling (first term) and inductive (second term) contributions to the total energy of a single plaquette. Here, J (T ) = (Φ 0 2π) I C (T ) is the Josephson coupling energy.

Magnetic Properties of Artificially Prepared Highly Ordered Two-Dimensional…

47

The real part of the complex AC susceptibility is defined as:

∂M ∂h ac

(V.4)

1 ∂H V ∂h ac

(V.5)

χ' (T, h ac , H dc ) = where:

M (T, h ac , H dc ) = −

is the net magnetization of the plaquette. Here V is the sample's volume, and <...> denotes the time averaging over the period 2π ω , namely: 2π

A =

1 d (ωt )A( t ) 2π ∫0

(V.6)

Taking into account the well-known [56] analytical approximation of the BCS gap parameter (valid for all temperatures), Δ (T ) = Δ (0) tanh(2.2 (TC − T ) T ) for the explicit temperature dependence of the Josephson critical current:

⎡ Δ (T ) ⎤ ⎡ Δ (T ) ⎤ tanh ⎢ I C ( T ) = I C ( 0) ⎢ ⎥ ⎥ ⎣ Δ (0) ⎦ ⎣ 2k B T ⎦

(V.7)

we successfully fitted all our data using the following set of parameters: φ 0 (0) = π 2 (which corresponds to the maximum Josephson current within a plaquette), β L (0) = β C (0) = 32 (for unshunted array) and β C (0) = 1.2 (for shunted array). The corresponding fits are shown by solid lines in Figs.7 and 8 for the experimental values of AC and DC field amplitudes. In the mixed-state region and for low enough frequencies (this assumption is wellsatisfied because in our case ω << ωLR and ω << ωLC where ωLR = R L and

ωLC = 1

LC are the two characteristic frequencies of the problem) from Eqs.(V.3)-(V.6)

we obtain the following approximate analytical expression for the susceptibility of the plaquette:

⎡

⎤ ⎛ 2 H dc ⎞ ⎛H ⎞ ⎟⎟ + f 2 (b) sin ⎜⎜ dc ⎟⎟ − β C−1 (T )⎥ ⎝ H0 ⎠ ⎝ H0 ⎠ ⎦

χ ' (T , hac , H dc ) ≈ − χ 0 (T ) ⎢ β L (T ) f1 (b) cos⎜⎜ ⎣

(V.8)

48

Fernando M. Araújo-Moreira and Sergei Sergeenkov

where χ 0 (T ) = πS I C (T ) VΦ 0 , H 0 = Φ 0 2

(2πμ0 dl) ≈ 10 Oe, f1 (b) = J 0 (2b) − J 2 (2b) ,

f 2 (b) = J 0 (b) − bJ1 (b) − 3J 2 (b) + bJ 3 (b) with b = 2πSμ 0 h ac Φ 0 and J n ( x ) being the Bessel function of the nth order.

Figure 10. Numerical simulation results for different values of

β L (T = 4.2K )

h ac = 70 mOe, H dc = 0 , β C (T = 4.2K ) = 1

and for

based on Eqs.(V.4)-(V.7).

Notice also that the analysis of Eq.(V.8) reproduces the observed Fraunhofer-like behavior of the susceptibility in applied DC field (see Fig.9) and the above-mentioned fine tuning of the reentrance effect (see also Ref. 44). Indeed, according to Eq.(V.8) (and in agreement with the observations), for small DC fields the minimum temperature Tmin (indicating the beginning of the reentrant transition) varies with H dc as follows,

(TC − Tmin ) TC

≈ H dc H 0 .

To further test our interpretation and verify the influence of the parameter β C on the reentrance, we have also performed extensive numerical simulations of the four-junction model previously described but without a simplifying assumption about the explicit form of the phase difference based on Eq.(V.2). More precisely, we obtained the temperature behavior of the susceptibility by solving the set of equations responsible for the flux dynamics within a single plaquette and based on Eq.(V.1) for the total current I( t ) , the equation for the total flux Φ ( t ) = Φ ext ( t ) + LI( t ) and the flux quantization condition for four junctions, namely

φi ( t ) = (π 2)[n + (Φ Φ 0 )] where n is an integer. Both Euler and fourth-order Runge-Kutta

Magnetic Properties of Artificially Prepared Highly Ordered Two-Dimensional…

49

integration methods provided the same numerical results. In Fig. 10 we show the real component of the simulated susceptibility χ(T) corresponding to the fixed value of

β C (T = 4.2K ) = 1 (shunted samples) and different values of β L (T = 4.2K ) = 1, 10, 15, 20, 30, 40, 50, 60, 90, 150 and 200. As expected, for this low value of β C reentrance is not observed for any values of β L . On the other hand, Fig. 11 shows the real component of the simulated χ(T ) but now using fixed value of β L (T = 4.2K ) = 30 and different values of

β C (T = 4.2K ) = 1, 2, 5, 10, 20, 30 and 100. This figure clearly shows that reentrance appears for values of β C > 20 . In both cases we used hac=70 mOe. We have also simulated the curve for shunted ( β L = 30 , β C = 1 ) and unshunted ( β L = 30 , β C = 30 ) samples for different values of hac (see Fig. 12). In this case the values of the parameters β L and β C were chosen from our real 2D-JJA samples. Again, our simulations confirm that dynamic reentrance does not occur for low values of β C , independently of the values of β L and hac. The following comment is in order regarding some irregularities ("jumps" and "steps") visibly seen in Figs.(10)-(12) around the transition regions from non-reentrant to reentrant behavior. It is important to emphasize that the above irregularities are just artifacts of the numerical simulations due to the conventional slow-converging real-time reiteration procedure [44]. They neither correspond to any experimentally observed behavior (within the accuracy of the measurements technique and data acquisition), nor they reflect any irregular features of the considered here theoretical model (which predicts a smooth temperature dependence seen through the data fits). As usual, to avoid this kind of artificial (non-physical) discontinuity, more powerful computers are needed. 0.10 βC=50

0.05

χ' (S.I.)

0.00

βC=30

-0.05

βC=20 βC=100

-0.10 -0.15

βC=1,2,5,10

-0.20 4

5

6

7

8

Temperature (K) Figure 11. Numerical simulation results for for different values of

β C (T = 4.2K )

h ac = 70

mOe,

9 Figure 4a

H dc = 0 , β L (T = 4.2K ) = 30

based on Eqs.(V.4)-(V.7).

and

50

Fernando M. Araújo-Moreira and Sergei Sergeenkov 0.2 0.1

70 mOe 60 mOe

χ' (S.I.)

0.0 -0.1 20 mOe

-0.2 50 mOe

40 mOe 30 mOe

-0.3

10 mOe

-0.4 -0.5 4

5

6

7

8

9

10

Temperature (K) (a) 0.2 0.1

χ'(S.I.)

0.0 -0.1 -0.2 70 mOe

-0.3

30 mOe 40 mOe

10 mOe

-0.4

20 mOe

-0.5 4

5

6

7

8

9

Temperature (K) (b) Figure 12. Curves of the simulated susceptibility ( H dc corresponding to (a) unshunted 2D-JJA with shunted 2D-JJA with

β L (T = 4.2K ) = 30

=0

and for different values of

β L (T = 4.2K ) = 30 and

and

h ac )

β C (T = 4.2K ) = 30 ; (b)

β C (T = 4.2K ) = 1 .

Based on the above extensive numerical simulations, a resulting phase diagram β C - β L (taken for T=1K, hac=70 mOe, and Hdc=0) is depicted in Fig. 13 which clearly demarcates the

Magnetic Properties of Artificially Prepared Highly Ordered Two-Dimensional…

51

border between the reentrant (white area) and non-reentrant (shaded area) behavior in the arrays for different values of β L (T ) and β C (T ) parameters at given temperature. In other words, if β L and β C parameters of any realistic array have the values inside the white area, this array will exhibit a reentrant behavior.

Figure 13. Numerically obtained phase diagram (taken for

T = 1K , h ac =

70 mOe, and

H dc = 0 )

which shows the border between the reentrant (white area) and non-reentrant (shaded area) behavior in the arrays for different values of

βL

and

βC

parameters.

Figure 14. A qualitative behavior of the envelope of the phase diagram (shown in previous figure) with DC magnetic field

H dc

(for

T = 1K

and

h ac =

70mOe) obtained from Eq.(V.8).

52

Fernando M. Araújo-Moreira and Sergei Sergeenkov It is instructive to mention that a hyperbolic-like character of β L vs. β C law (seen in

Fig. 13) is virtually present in the approximate analytical expression for the susceptibility of the plaquette given by Eq.(V.8) (notice however that this expression can not be used to produce any quantitative prediction because the neglected in Eq.(V.8) frequency-related terms depend on β L and β C parameters as well). A qualitative behavior of the envelope of the phase diagram (depicted in Fig. 13) with DC magnetic field Hdc (for T=1 K and hac=70 mOe), obtained using Eq.(V.8), is shown in Fig. 14.

10

PARA 5

Φtotal/Φ

st

1 branch

DIA 0

-5 nd

2 branch rd

-10 -10

3 branch

-8

-6

-4

-2

0

Φext/Φ0

2

4

6

Figure 15. Numerical simulation results, based on Eqs.(V.4)-(V.7), showing shunted 2D-JJA with

β L (T = 4.2K ) = 30

and

8

Φ tot

10

vs.

Φ ext

for

β C (T = 4.2K ) = 1 .

And finally, to understand how small values of β C parameter affect the flux dynamics in shunted arrays, we have analyzed the Φ tot − Φ ext diagram. Similarly to those results previously obtained from unshunted samples [44], for a shunted sample at fixed temperature this curve is also very hysteretic (see Fig. 15). In both cases, Φ tot vs. Φ ext shows multiple branches intersecting the line Φ tot = 0 which corresponds to diamagnetic states. For all the other branches, the intersection with the line Φ tot = Φ ext corresponds to the boundary between diamagnetic states (negative values of χ' ) and paramagnetic states (positive values of χ' ). As we have reported before [44], for unshunted 2D-JJA at temperatures below 7.6 K the appearance of the first and third branches adds a paramagnetic contribution to the average value of χ' . When β C is small (shunted arrays), the analysis of these curves shows that there is no reentrance at low temperatures because in this case the second branch appears to be

Magnetic Properties of Artificially Prepared Highly Ordered Two-Dimensional…

53

energetically stable, giving an extra diamagnetic contribution which overwhelms the paramagnetic contribution from subsequent branches. In other words, for low enough values of β C (when the samples are ZFC and then measured at small values of the magnetic field), most of the loops will be in the diamagnetic states, and no paramagnetic response is registered. As a result, the flux quanta cannot get trapped into the loops even by the following field-cooling process in small values of the magnetic field. In this case the superconducting phases and the junctions will have the same diamagnetic response and the resulting measured value of the magnetic susceptibility will be negative (i.e., diamagnetic) as well. On the other hand, when β C is large enough (unshunted arrays), the second branch becomes energetically unstable, and the average response of the sample at low temperatures is paramagnetic (Cf. Fig. 7 from Ref. 44). In summary, in this section we have shown that our experimental and theoretical results demonstrate that the reentrance phenomenon (and concomitant PME) in artificially prepared Josephson Junction Arrays is related to the damping effects associated with the StewartMcCumber parameter β C . Namely, reentrant behavior of AC susceptibility takes place in the underdamped (unshunted) array (with large enough value of β C ) and totally disappears in overdamped (shunted) arrays.

VI. Manifestation of Novel Geometric Effects in Temperature Behavior of AC Magnetic Response Many unusual and still not completely understood magnetic properties of 2D-JJAs continue to attract attention of both theoreticians and experimentalists alike (for recent reviews on the subject see, e.g. Refs. [57-61] and further references therein). In particular, among the numerous spectacular phenomena recently discussed and observed in 2D-JJAs we would like to mention the dynamic temperature reentrance of AC susceptibility [57,58] (closely related to paramagnetic Meissner effect [59]) and avalanche-like magnetic field behavior of magnetization [60,61] (closely related to self-organized criticality (SOC) [62,63]). More specifically, using highly sensitive SQUID magnetometer, magnetic field jumps in the magnetization curves associated with the entry and exit of avalanches of tens and hundreds of fluxons were clearly seen in SIS-type arrays [61]. Besides, it was shown that the probability distribution of these processes is in good agreement with the SOC theory [63]. An avalanche character of flux motion was observed at temperatures at which the size of the fluxons did not exceed the size of the cell, that is, for discrete vortices. On the other hand, using a similar technique, magnetic flux avalanches were not observed in SNS-type proximity arrays [64] despite a sufficiently high value of the inductance L related critical parameter β L = 2πLIC Φ 0 needed to satisfy the observability conditions of SOC. Instead, the observed quasi-hydrodynamic flux motion in the array was explained by the considerable viscosity characterizing the vortex motion through the Josephson junctions. In this section of the present review article, we show experimental evidence for manifestation of novel geometric effects in magnetic response of high-quality ordered 2DJJA. By improving resolution of home-made mutual-inductance measurements technique

54

Fernando M. Araújo-Moreira and Sergei Sergeenkov

described in the beginning of this article, a pronounced step-like structure (with the number of steps n = 4 for all AC fields) has been observed in the temperature dependence of AC susceptibility in artificially prepared two-dimensional Josephson Junction Arrays (2D-JJA) of unshunted Nb-AlOx-Nb junctions with β L (4.2K ) = 30 . Using a single-plaquette approximation of the overdamped 2D-JJA model, we were able to successfully fit our data assuming that steps are related to the geometric properties of the plaquette. The number of steps n corresponds to the number of flux quanta that can be screened by the maximum critical current of the junctions. The steps are predicted to manifest themselves in arrays with the inductance related parameter β L matching a "quantization" condition β L (0) = 2π(n + 1) . To measure the complex AC susceptibility in our arrays with high precision, we used a home-made susceptometer based on the so-called screening method in the reflection configuration as described in the previous sections [65-67]. Measurements were performed as a function of the temperature T (for 1.5 K < T < 15 K), and the amplitude of the excitation field hac (for 1 mOe
Figure 16. Experimental results for temperature dependence of the real part of AC susceptibility

χ' (T, h ac )

for different AC field amplitudes

h ac =

41.0, 59.6, 67.0, 78.2 and 96.7 mOe.

Magnetic Properties of Artificially Prepared Highly Ordered Two-Dimensional…

55

It is important to recall that the magnetic field behavior of the critical current of the array (taken at T=4.2 K) on DC magnetic field Hdc (parallel to the plane of the sample) exhibited a sharp Fraunhofer-like pattern characteristic of a single-junction response, thus proving a rather strong coherence within arrays (with negligible distribution of critical currents and sizes of the individual junctions) and hence the high quality of our sample. The observed temperature dependence of the real part of AC susceptibility for different AC fields is shown in Fig. 16. A pronounced step-like structure is clearly seen at higher temperatures. The number of steps n does not depend on AC field amplitude and is equal to n = 4. As expected [58,67,68], for hac > 40 mOe (when the array is in the mixed-like state with practically homogeneous flux distribution) the steps are accompanied by the previously observed reentrant behavior with χ' (T, h ac ) starting to increase at low temperatures. To understand the step-like behavior of the AC susceptibility observed in unshunted 2DJJAs, in principle one would need to analyze in detail the flux dynamics in these arrays. However, as we have previously reported [58,67,68], because of the well-defined periodic structure of our arrays with no visible distribution of junction sizes and critical currents, it is quite reasonable to assume that the experimental results obtained from the magnetic properties of our 2D-JJAs could be understood by analyzing the dynamics of just a single unit cell (plaquette) of the array. As we shall see, theoretical interpretation of the presented here experimental results based on single-loop approximation, is in excellent agreement with the observed behavior. In our analytical calculations, the unit cell is the loop containing four identical Josephson junctions described in previous sections, and the measurements correspond to the zero-field cooling AC magnetic susceptibility. If we apply an AC external field H ac ( t ) = h ac cos ωt normally to the 2D-JJA, then the total magnetic flux Φ ( t ) threading the four-junction superconducting loop is given again by Φ ( t ) = Φ ext ( t ) + LI( t ) where L is the loop inductance, Φ ext ( t ) = SH ac ( t ) is the flux related to the applied magnetic field (with S ≈ a being the projected area of the loop), and the circulating current 2

in the loop reads I (t ) = I C (T ) sin φ (t ) . Here φ( t ) is the gauge-invariant superconducting phase difference across the ith junction. As is well-known, in the case of four junctions, the flux quantization condition reads [67,69]

φ=

π⎛ Φ ⎞ ⎜⎜ n + ⎟ Φ 0 ⎟⎠ 2⎝

(VI.1)

where n is an integer, and, for simplicity, we assume as usual that [58,67] φ1 = φ 2 = φ3 = φ 4 ≡ φ . To properly treat the magnetic properties of the system, let us introduce the following Hamiltonian

H (t ) = J (T )[1 − cos φ (t )] +

1 2 LI (t ) 2

(VI.2)

56

Fernando M. Araújo-Moreira and Sergei Sergeenkov

which describes the tunneling (first term) and inductive (second term) contributions to the total energy of a single plaquette. Here, J (T ) = (Φ 0 2π)I C (T ) is the Josephson coupling energy. Since the origin of reentrant behavior in our unshunted arrays has been discussed in much detail earlier [58,67,68] (see also the previous section of this Chapter), in what follows we concentrate only on interpretation of the observed here step-like structure of χ' (T, h ac ) . First of all, we notice that the number of observed steps n (in our case n = 4) clearly hints at a possible connection between the observed here phenomenon and flux quantization condition within a single four-junction plaquette. Indeed, the circulating in the loop current I( t ) = I C (T)sinφ( t ) passes through its maximum value whenever φ( t ) reaches the value of

(π 2)(2n + 1)

with n = 0,1,2... As a result, the maximum number of fluxons threading a

single plaquette (see Eq. (VI.1)) over the period

2π / ω

becomes equal to

< Φ ( t ) >= (n + 1)Φ 0 . In turn, the latter equation is equivalent to the following condition β L (T ) = 2π(n + 1) . Since this formula is valid for any temperature, we can rewrite it as a geometrical "quantization" condition β L (0) = 2π(n + 1) . Recall that in the present experiment, our array has β L (0) = 31.6 (extrapolated from its experimental value

β L (4.2K ) = 30 ) which is a perfect match for the above "quantization" condition predicting n = 4 for the number of steps in a single plaquette, in excellent agreement with the observations. Based on the above discussion, we conclude that in order to reproduce the observed temperature steps in the behavior of AC susceptibility, we need a particular solution to Eq.(VI.1) for the phase difference in the form of φ n ( t ) = (π 2 )(2n + 1) + δφ( t ) assuming

δφ( t ) << 1 .

After

substituting

this

Ansatz

into

Eq.(VI.1),

we

find

that

φ n ( t ) ≈ (π 2 )n + (1 4 )β L (T) + (1 4 )f cos(ωt ) where f = 2πSh ac Φ 0 is the AC field related frustration parameter. Using this effective phase difference, we can calculate the AC response of a single plaquette. Namely, the real part of susceptibility reads π

χ' (T, h ac ) =

1 d (ωt ) cos(ωt )χ n ( t ) π ∫0

(VI.3)

where

χ n (t) = −

Here V is the sample's volume.

1 ⎡ ∂ 2H ⎤ ⎢ 2 ⎥ V ⎣ ∂h ac ⎦ φ =φ

(VI.4) n (t)

Magnetic Properties of Artificially Prepared Highly Ordered Two-Dimensional…

57

Figure 17. Theoretically predicted dependence of the normalized susceptibility on reduced temperature according to Eqs.(VI.3)-(VI.5) for f=0.5 and for "quantized" values of

β L (0) = 2π (n + 1)

(from top

to bottom): n=0, 3 and 5.

For the explicit temperature dependence of

β L (T ) = 2πLI C (T ) Φ 0 we use again the

well-known [70,71] analytical approximation of the BCS gap parameter (valid for all temperatures),

Δ(T) = Δ(0) tanh(2.2 (TC − T ) T ) which governs the temperature

dependence of the Josephson critical current:

⎡ Δ (T ) ⎤ ⎡ Δ (T ) ⎤ tanh ⎢ I C ( T ) = I C ( 0) ⎢ ⎥ ⎥ ⎣ Δ (0) ⎦ ⎣ 2k B T ⎦

(VI.5)

Fig. 17 depicts the predicted by Eqs.(VI.3)-(VI.5) dependence of the AC susceptibility on reduced temperature for f=0.5 and for different "quantized" values of β L (0) = 2π(n + 1) . Notice the clear appearance of three and five steps for n = 3 and n = 5, respectively (as expected, n = 0 corresponds to a smooth temperature behavior without steps). In Fig. 18 we present fits (shown by solid lines) of the observed temperature dependence of the normalized susceptibility χ' (T, h ac ) χ 0 for different magnetic fields h ac according to Eqs.(VI.3)-(VI.5) using β L (0) = 10π . As is seen, our simplified model based on a singleplaquette approximation demonstrates an excellent agreement with the observations. In summary, in this section we have shown a step-like structure (accompanied by previously seen low-temperature reentrance phenomenon) which has been observed for the first time in the temperature dependence of AC susceptibility in artificially prepared twodimensional Josephson Junction Arrays of unshunted Nb-AlOx-Nb junctions. The steps are shown to occur in arrays with the inductance related parameter β L (T) matching the "quantization" condition β L (0) = 2π( n + 1) where n is the number of steps.

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Fernando M. Araújo-Moreira and Sergei Sergeenkov

Figure 18. Fits (solid lines) of the experimental data for according to Eqs.(VI.3)-(VI.5) with

h ac =

41.0, 59.6, 67.0, 78.2, and 96.7 mOe

β L (0) = 10π .

VII. Summary In summary, in this review article we report on three phenomena related to the magnetic properties of 2D-JJA: (a) the influence of non-uniform critical current density profile on magnetic field behavior of AC susceptibility; (b) the origin of dynamic reentrance and the role of the Stewart-McCumber parameter, βC, in observability of this phenomenon, and (c) the manifestation of novel geometric effects in temperature behavior of AC magnetic response. We have found clear experimental evidence for the influence of the junction nonuniformity on magnetic field penetration into the periodic 2D array of unshunted Josephson junctions. By using the well-known AC magnetic susceptibility technique, we have shown that in the mixed-state regime the AC field behavior of the artificially prepared array is reasonably well fitted by the single-plaquette approximation of the over-damped model of 2D-JJA assuming inhomogeneous (Lorentz-like) critical current distribution within a single junction. On the other hand, our experimental and theoretical results have demonstrated that the reentrance of AC susceptibility (and concomitant PME) in artificially prepared Josephson Junction Arrays takes place in the underdamped (unshunted) array (with large enough value

Magnetic Properties of Artificially Prepared Highly Ordered Two-Dimensional…

59

of the Stewart-McCumber parameter βC) and totally disappears in over-damped (shunted) arrays. Finally, we have shown a step-like structure (accompanied by previously seen lowtemperature reentrance phenomenon) which has been observed for the first time in the temperature dependence of AC susceptibility in artificially prepared two-dimensional Josephson Junction Arrays of unshunted Nb-AlOx-Nb junctions. The steps are shown to occur in arrays with the inductance related parameter β L (T ) matching the "quantization" condition

β L (0) = 2π(n + 1) where n is the number of steps.

Acknowledgements We thank P. Barbara, C.J. Lobb, A. Sanchez and R.S. Newrock for useful discussions. We thank W. Maluf for his help in running some of the experiments. S.S. and F.M.A.M. gratefully acknowledge financial support from Brazilian Agency FAPESP under grant 2003/00296-5.

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[45] A.P. Nielsen, A.B. Cawthorne, P. Barbara, F.C. Wellstood, and C.J. Lobb, Phys. Rev. B 62 (2000) 14380. [46] De Leo, G. Rotoli, P. Barbara, A.P. Nielsen, and C.J. Lobb, Phys. Rev. B 64 (2001) 14518. [47] Magnetic Susceptibility of Superconductors and Other Spin Systems, edited by R.A. Hein, T.L. Francavilla, and D.H. Liebenberg, (Plenum Press, New York, 1992). [48] J.L. Jeanneret, G.A. Gavilano, A. Racine, Ch. Leemann, and P. Martinoli, Appl. Phys. Lett. 55 (1989) 2336. [49] F.M. Araujo-Moreira, P. Barbara, A.B. Cawthorne, and C.J. Lobb, in Studies of High temperature Superconductors, edited by A.V. Narlikar (Nova Science Publishers, New York, 2002), vol. 43, pp. 227-254. [50] T.P. Orlando and K.A. Delin, Foundations of Applied Superconductivity (AddisonWesley Publishing Company, New York, 1991), p. 451. [51] F.M. Araujo-Moreira, O.F. de Lima, and W.A. Ortiz, Physica C 240-245 (1994) 3205; F.M. Araujo-Moreira O.F. de Lima, and W.A. Ortiz, J. Appl. Phys. 80, 6 (1996) 3390; F.M. Araujo-Moreira O.F. de Lima, and W.A. Ortiz, Physica C 311 (1999) 98. [52] W.A.C. Passos, P.N. Lisboa-Filho, R. Caparroz, C.C. de Faria, P.C. Venturini, F.M. Araujo-Moreira, S. Sergeenkov, and W.A. Ortiz, Physica C 354 (2001) 189. [53] R.S. Newrock, C.J. Lobb, U. Geigenmuller, and M. Octavio, Solid State Physics 54 (2000) 263. [54] P. Martinoli and C. Leeman, J. Low Temp. Phys. 118 (2000) 699. [55] Auletta, P. Caputo, G. Costabile, R. de Luca, S. Pase, A. Saggese, Physica C 235-240, 3315 (1994); C. Auletta, G. Raiconi, R. de Luca, S. Pase, Phys. Rev. B 51, 12844 (1995). [56] R. Meservey and B.B. Schwartz, in Superconductivity, edited by R.D. Parks (M. Dekker, New York, 1969), vol. 1, p.117; S. Sergeenkov, JETP Lett. 76 (2002) 170. [57] R.S. Newrock, C.J. Lobb, U. Geigenmuller, and M. Octavio, Solid State Physics 54, 263 (2000); Mesoscopic and Strongly Correlated Electron Systems-II, Ed. by M.V. Feigel'man, V.V. Ryazanov and V.B. Timofeev, Phys. Usp. (Suppl.) 44 (10) (2001); S. Sergeenkov, in Studies of High temperature Superconductors, Ed. by A.V. Narlikar (Nova Science Publishers, New York, 2001), vol. 39, p. 117. [58] F.M. Araujo-Moreira, P. Barbara, A.B. Cawthorne, and C.J. Lobb, in Studies of High temperature Superconductors, Ed. by A.V. Narlikar (Nova Science Publishers, New York, 2002), vol. 43, p. 227. [59] M.S. Li, Phys. Rep. 376 (2003) 133. [60] Altshuler and T.H. Johansen, Rev. Mod. Phys. 76 (2004) 471. [61] S.M. Ishikaev, E.V. Matizen, V.V. Ryazanov, V.A. Oboznov, and A.V. Veretennikov, JETP Lett. 72 (2000) 26. [62] H.J. Jensen, Self Organized Criticality: Emergent Complex Behavior in Physical and Biological Systems (Cambridge University Press, Cambridge, 1998). [63] S.L. Ginzburg and N.E. Savitskaya, JETP Lett. 73 (2001) 145. [64] S.M. Ishikaev, E.V. Matizen, V.V. Ryazanov, and V.A. Oboznov, JETP Lett. 76 (2002) 160. [65] Jeanneret, J.L. Gavilano, G.A. Racine, Ch. Leemann, and P. Martinoli, Appl. Phys. Lett. 55 (1989) 2336. [66] P. Martinoli and C. Leeman, J. Low Temp. Phys. 118 (2000) 699.

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[67] F.M. Araujo-Moreira, P. Barbara, A.B. Cawthorne, and C.J. Lobb, Phys. Rev. Lett. 78 (1997) 4625; P. Barbara, F.M. Araujo-Moreira, A.B. Cawthorne, and C.J. Lobb, Phys. Rev. B 60 (1999) 7489. [68] F.M. Araujo-Moreira, W. Maluf, and S. Sergeenkov; to be published in European Journal of Physics B (2005). [69] Barone and G. Paterno, Physics and Applications of the Josephson Effect (A WileyInterscience Publisher, New York, 1982). [70] R. Meservey and B.B. Schwartz, in Superconductivity, vol.1, ed. by R.D. Parks (M. Dekker, New York, 1969), p.117. [71] S. Sergeenkov, JETP Lett. 76 (2002) 170.

In: Superconductivity Research Developments Editor: James R. Tobin, pp. 63-92

ISBN: 978-1-60021-848-4 © 2008 Nova Science Publishers, Inc.

Chapter 3

SUPERCONDUCTING NOBLE METAL DIBORIDE A. K. M. A. Islam and F. Parvin Department of Physics, Rajshahi University, Rajshahi-6205, Bangladesh

Abstract The discovery of superconductivity in MgB2 (Tc~39K) revived interest in non-oxides and initiated a search for superconductivity in related materials. Currently about 100 binary compounds with an AlB2-type structure is known to exist. The noble metal diborides, AgB2 and AuB2 (quasi 2D structure with AlB2-type, space group P6/mmm), which correspond to effectively hole doped systems, have been predicted to be potential candidates for high-Tc superconductors. This is due to their larger density of B2p σ-like states near EF and electronphonon coupling constant λ than MgB2 and hence higher Tc. Despite difficulties successful synthesis of silver boride thin films was made with nominal composition AgB2 by a Japanese team. Experimental observation confirmed the superconductivity with Tc significantly lower than the theoretically predicted value. The observed value of Tc is comparable with those for some d-metal diborides: ZrB2 (5.5K), TaB2 (9.5K) and NbB2 (5.2K). We attempt to explain the discrepancy between the predicted and the observed Tc of AgB2 by invoking the possible role of spin-fluctuations in the system. A study of the mechanical and electronic properties of the noble metal diborides, in comparison with lighter metal (Mg) diboride, has been made using self-consistent density functional theory (DFT). The electronic band structure has also been analysed in order to shed further insight into the differences between the two groups of diborides. The study also includes a review of the existing literatures and indications for future direction of research.

1. Intermetalic Diborides The discovery of superconductivity in MgB2 (Tc~39 K) by Nagamatsu et al. [1] revived the interest in non-oxides and initiated a search for superconductivity in related materials. The research activities during the first few weeks after the presentation of the results by

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Akimitsu [2] showed how well the scientific communities were prepared for the study of new superconducting materials. Currently about 100 binary compounds with an AlB2-type structure is known to exist. MgB2 is only a member of a rich family of diborides. So also the other members of this family (Li, Be, Al, Ti, Zr, Hf, V, Nb, Ta, Cr, Mo) became the subject of intensive studies of different groups [3-14]. On the other hand Kwon et al. [15] suggest that noble metal diborides, AgB2 and AuB2, which correspond to effectively hole doped systems, are potential candidates for high-Tc BCS superconductors. The calculated critical temperature of AgB2 using McMillan’s empirical formula is 59 K [15], almost 1.5 times larger than that of MgB2. Immediately after the discovery of the medium-Tc MgB2 the prediction of the noble metal diboride AgB2 as a potential higher-Tc superconductor thus attracted attention [16-18]. About 30 years prior to the discovery of superconducting MgB2, Cooper et al. [19] discovered NbB2 with critical temperature equal to Tc = 3.87 K and in Zr0.13Mo0.87B2 with Tc ~ 11 K. Systematic study of diborides was conducted by Leyarovska et al. [20]. They looked for superconductivity in these compounds at temperatures down to 0.42 K and showed that only NbB2 was superconducting at Tc = 0.62 K. Other observations in boron-rich NbB2.5 (Tc = 6.48 K [19]), stoichiometric NbB2 (0.62 ≥ Tc ≥ 5.2 K [19]) and hole-doped Nb1-xB2 (Tc up to 9.2 K under pressure [5]) were reported. But there are some conflicting results among different groups about the claim of superconductivity of some compounds. For example Kaczorowski et al. [6] found superconducting transition at Tc= 9.5 K for TaB2 and no superconductivity for TiB2, HfB2, VB2, NbB2 or ZrB2. Although Felner [7] stated that BeB2 is not superconducting, according to Young et al. [8] BeB2.75 is superconducting with Tc= 0.7 K. Gasparov et al. [9] found ZrB2 superconducting with Tc = 5.5 K and simultaneously they did not confirm superconductivity for TaB2 and NbB2. Superconductivity in TaB2 was discovered in old, well aged material. Since the discovery of superconductivity in MgB2 [1] there have been several theoretical studies to search for the potential high Tc binary and ternary borides in isoelectronic systems such as BeB2, CaB2, transition metal (TM) diborides TMB2, hole doped systems Mg1-xLixB2, Mg1-xNaxB2, Mg1-xCuxB2. and related compounds [3] Noble metals such as silver, gold, platinum etc. are resistant to corrosion by all but the most powerful acids. The diborides of the noble metals such as AgB2 and AuB2 also came under study as possible superconductors. A list of several diborides with their critical temperature is given in Table 1.

Figure 1. Schematic representation of the unit cell of MB2 and some neighbour atoms. The grey and yellow spheres represent M and boron atoms, respectively.

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65

Structure

Noble metal diboride MB2 (M=Ag, Au) is a quasi 2D structure with AlB2-type, space group

D61h -P6/mmm. It consists of hexagonal M layers and plane graphite-like boron networks stacked in the order ...MBMB.... Figure 1 shows the schematic representation of the unit cell of MB2 and some neighbour atoms. The grey and yellow spheres represent M and boron atoms, respectively. Figure 2 shows different views of the MB2 unit cell represented in figure 1. Table 1. List of few diborides, their critical temperature and structure type. Compound

MgB2 NbB2

NbB2.5 Nb0.95Y0.5B2.5 Nb0.9Th0.1B2.5 MoB2 MoB2.5 Mo0.9Sc0.1B2.5 Mo0.95Y0.05B2.5 Mo0.85Zr0.15B2.5 Mo0.9Hf0.1B2.5 Mo0.85Nb0.15B2.5 TaB2 BeB2 BeB2.75 ZrB2 Zr0.13Mo0.87B2 ReB1.8-2 TiB2 HfB2 VB2 CrB2 OsB2 AgB2 AuB2

Tc (K) 39 0.62 5.2 3.87 6.4 9.3 7 8.1 9 8.6 11.2 8.7 8.5 9.5 0.7 5.5 ~11 4.5-6.3 6.7 (thin film) -

Structure AlB2 AlB2

AlB2 AlB2 AlB2 AlB2 AlB2 AlB2 AlB2 AlB2 AlB2 AlB2 AlB2 AlB2 not AlB2 AlB2

AlB2

Orthorhombic AlB2 AlB2

Ref. 1 9,19 20 2§ 19 19 19 19 19,20 19 19 19 19 19 19 9,20 6 7 8 9 19 10 6,20 6,20 6,20 20 14 21 15*

§ Akimitsu et al, as cited in Yamamoto et al, arXiv cond-mat-mat/0208331. * Noble metal diboride phases CuB2 and AuB2 (potential superconductor according to theoretical calc. [15]) have not yet been produced experimentally.

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Figure 2. Different views of the MB2 unit cell represented in figure 1.

2. Synthesis History of AgB2 and AuB2 2.1. Early Attempts AgB2 is a compound which is difficult to synthesize. The primary attempts to synthesize AgB2 led to inconsistent results. It is a phase, which can be synthesized, but its stability when exposed to high temperature for long time is unclear. The first synthesis of AgB2 phase was reported [22, 23] quite some time ago.

2.2. Recent Attempts a. Experiments Due to Sinder-Pelleg Sinder and Pelleg [24] used different experimental routes to obtain AgB2 samples as a bulk and thin films. The bulk pellets were obtained from the powders of the constituents and the films were produced by co-sputtering and sequential sputtering. The specimens were annealed and subjected for X-ray and Auger analysis. Preliminary results seem to indicate that AgB2 is an unstable phase. They used three experimental approaches to produce silver boride: a) powder approach, b) co-sputtering of Ag and B and c) sequential sputtering of Ag and B. •

Powder approach Powders of Ag and B were mixed in a ratio Ag : B = 1 : 2 and then pressed into pellets. Each pellet was sealed in a quartz ampoule under vacuum and backfilled with a small amount of Ar. Annealing at 1173 K was done in a box furnace for different durations of 2, 5, 9 and 24 hours. Ampoules were quenched in water to room temperature. A Rigaku diffractometer was used to obtain x-ray diffraction (XRD) pattern of the specimen. Figure 3 shows the XRD spectrum of the specimen for 9 hours. In addition to the Ag phase two AgB2 peaks of the (001) and (002) reflections are observed. The intensity of peaks is seen to increase with increasing annealing time, but the peaks disappear when a prolonged annealing of 24 hour is used. There is good explanation for this observation except the possibility that the phase is not stable at this high temperature for extended times. The specimen annealed for 9 hours is seen to reproduce the AgB2 peaks when

Superconducting Noble Metal Diboride

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tested after a year. The results indicate stability of the AgB2 phase at room temperature, but become unstable when exposed to high temperature for long time. The c parameter is found to be 3.18 Å, from the peaks which is very close to earlier report [22]. •

Co-sputtering of Ag and B Magnetron co-sputtering from Ag and B targets on Si (100) substrate in Ar atmosphere was used for the deposition. The base pressure prior to film deposition was ~ 10-5 Pa. RF power was applied on both targets. The film was annealed in ampoules of quartz at 933 K and then analyzed by XRD and Auger techniques [24]. An XRD spectrum of a specimen obtained by co-sputtering and annealed at 933 K for 30 min is shown in figure 4. Only the (100) AgB2 reflection was observed thus confirming the existence of the AgB2 phase in the co-sputtered film also. An Auger spectrum of a specimen co-sputtered under the same conditions as the specimen illustrated in figure 4 but annealed at 933 K for 4 min is shown in figure 5. The spectrum does not show the expected ratio between Ag and B in the AgB2 phase. The AgB2 peak disappeared in one of the thin film specimens after several days at room temperature.

Figure 3. XRD spectrum of a bulk specimen annealed at 1173 K for 9 hours. (001) and (002) peaks of AgB2 are indicated in addition to the Ag peaks. Adopted from [24] with permission.

Figure 4. XRD spectrum of a co-sputtered film annealed at 933 K for 30 min. Only the (001) AgB2 peak is observed in the general pattern of Ag peaks. Adopted from [24] with permission.

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Figure 5. Auger illustration of a specimen, which was prepared under the same conditions as the specimen shown in figure 4, but annealed only for 4 min. Adopted from [24] with permission.

Figure 6. XRD spectrum of a sequentially sputtered film annealed at 933 K for 1 h. Only the (001) AgB2 peak is observed in the general pattern of Ag peaks. Adopted from [24] with permission.

•

Sequential sputtering of Ag and B B and Ag were successively deposited by magnetron sputtering from the respective targets on Si (100) substrate respectively. The sputtering conditions were the same as those for the co-sputtering. The resulting film was annealed at 933 K in quartz ampoules and then analyzed by XRD technique. Figure 6 is the XRD spectrum of a film obtained by sequential sputtering and annealing at 933 K for 1 hour. Again only one AgB2 reflection from the (001) plane was observed. This specimen was retested after several days and the (001) peak could not be detected. A common observation of some of the films obtained by sputtering was that the AgB2 peaks disappear after a few days. In bulk specimens the AgB2 peaks disappeared only when exposed to high temperature for long times. Sinder and Pelleg [24] do not have yet a satisfactory explanation of these observations. But it may be speculated to be attributable to the relative stability of AgB2. Sinder and Pelleg [24] were unable to provide satisfactory explanation as yet of the instability as a function of time at high temperature or low temperature. AgB2 bulk specimens seem to be an unstable phase when exposed to high temperature for long times. The instability of thin films is observed at room temperature when exposed for a few days. These authors [24] are still continuing their efforts to understand this and to investigate the superconductive behaviour of the stable bulk specimens [24].

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b. Ag and Au Borides by Fresh Magnetron Sputtering J. Pelleg et al. [25] made fresh attempt to synthesize Ag and Au boride thin films by cosputtering from the elemental targets and characterize them. Specimens of AgB2 and AuB2 are prepared by magnetron co-sputtering from two separate targets of B and Ag or Au on Si (100) in Ar ambient in the case of Ag and in a mixture of 70% Ar + 30% H2 in the case of Au. First the Si specimens have been chemically cleaned in a solution of HF:H2O=1:10. It is then dry etched by RF after loading into the sputter chamber. The base pressure before sputtering was ~10-5 Pa. Co-sputtering is preceded by pre-sputtering of the targets for 20 min. The resulting films, with thickness ~0.1 μ (AgB2) and 0.6 μ (AuB2), are characterized. The methods used are XRD, Auger electron spectroscopy (AES), X-ray photo spectroscopy (XPS), optical microscopy (OM), scanning electron microscopy (SEM). Resistivity measurements of AgB2 have also been made in the temperature range ~ 10-200 K. The observed XRD peaks cannot be definitely identified with the AgB2 or AuB2 phase.

Figure 7. Schematics of cross sections of (a) type I sample and (b) type II sample fabricated by PLD technique. (Adopted from [21], ©2004 Physical Society of Japan).

c. Thin-Film Pulsed Laser Deposition Method The first successful synthesis of silver boride thin films with nominal composition AgB2 was performed by a pulsed laser deposition method [21]. The samples were prepared by a thinfilm pulsed laser deposition (PLD) technique. It was in fact developed for the synthesis of MgB2 thin films [26]. The annealing process in a high vacuum was as described below [21]. First, a Ag/B composite film was deposited on a MgO(100) single-crystal substrate at room temperature by irradiating an excimer laser on a rotating heterotarget of Ag/B. The background pressure of the PLD chamber was 10-8 Torr. The laser was a KrF excimer laser with a wavelength of 248 nm, an energy of 200–250 mJ and a repetition rate of 10 Hz. The Ag/B target composition had nearly the stoichiometric composition of AgB2. Next, a B film was in situ-deposited using a B target. The B layer acts as a cap for the Ag/B film since the annealing temperature is much less than the B melting point. The cross section of the fabricated sample (type I sample) is schematically shown in figure 7(a). The composite film was in situ-annealed at 820 oC for 4 hours using a halogen lamp heater in a high vacuum of 10-6 Torr or less in the PLD chamber. There is no possibility of the involvement of impurities in this process. The sample film was 2 mm long, 1.5 mm wide and about 130 nm thick. Tomita et al. [21] also fabricated a different type of sample (figure 7(b)). It was fabricated first by synthesizing a MgB2 film by the method given by Uchiyama et al. [26] and then depositing a Ag film and a B film at room temperature by the PLD method (type II sample). The MgB2 film exhibited superconductivity with a Tc of typically 25 K before Ag and B deposition. The composite film was again annealed at 820oC for 1 hour in a high vacuum of 10-6 Torr. MgB2 easily dissociated into Mg and B at this high temperature.

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A. K. M. A. Islam and F. Parvin

d. Powder Approach of Lal and Coworkers Lal et al. [27] synthesized MgB2, AlB2 and AgB2 compounds using high quality Mg, Al, Ag and B powder, by mixing them in stoichiometeric ratio. The mixed and ground powder, are further palletized. The pellets are then put in closed end soft iron (SS) tubes. The pellets containing SS tubes were then sealed inside a quartz tube at high vacuum of 10-5 Torr (figure 8). The encapsulated raw pellets are then heated at 750oC with a hold time of 3 hours and finally quenched in liquid nitrogen. XRD patterns were obtained at room temperature using CuKα radiation. MgB2 Capsule

Figure 8. Photograph of SS tube encapsulated raw MgB2 compound at 10-5 Torr. Adopted from [27] with permission.

2.3. Resistivity Studies of the Prepared Samples The resistivity of the prepared samples of diborides of Mg, Al and Ag were made by Lal et al. [27] in the temperature range of 12 - 300 K using a four-point-probe technique on a Close Cycle refrigerator.

Figure 9. ρ (T) of (a) MgB2 (b) AlB2 and (c) AgB2 compounds (source: Ref. [27]).

Figures 9(a-c) show the resistivity of MgB2, AlB2 and AgB2 up to 300 K. While there is superconducting phase transition in MgB2 at Tc= 38 K, the low temperature behaviour of the resistivity of AlB2 and AgB2 show an effect of localization. The main interaction in the four diborides studied (see [27]) is the electron-phonon interaction. Further the metallic nature of resistivity for large T suggests a weak localization. Thus the resistivity as a function of T can be written as [27]:

Superconducting Noble Metal Diboride ΘD

ρ (T ) = ρo + A ΘT

4 4 D

T

∫

z5 ( e z −1)(1− e − z

71

dz − B ln T

(1)

o

The first term gives the temperature independent contribution to resistivity due to the impurities present in the system. The second term arises due to the contribution of the electron-phonon interaction within the Bloch-Grüneisen theory [28]. In this term A, providing the contribution of the electron-phonon interaction, is a constant independent of both temperature T and Debye temperature θD. In the third term, the constant B is the contribution due to the weak localization [29]. Table 2. Values of the parameters ρo, A, θD and B for the systems MgB2, AlB2 and AgB2 [27]*.

System

MgB2 AlB2 AgB2

ρo (μΩcm) 10.3 26.8 1.18

A (μΩcm) 0.4 0.8 0.082

θD (K)

700 670 480

B (μΩcm) 0.0 0.02 0.001

* With permission.

The fitting of the resistivity data of MgB2, AlB2 and AgB2 with Eq. (1) yield the values of the parameters ρo, A, θD and B which are given in Table 2 [27]. The set of values ρo, A, θD and B for a given sample is considered to be unique. The reason is that ρo, B and (A, θD) correspond to qualitatively much different functional dependence of ρ(T), and (A, θD) govern the low-T curvature and high-T linear variation of ρ(T). As far as the separate uniqueness of A and θD is concerned, it may be noted that while the low-T curvature of ρ(T) [excluding the upturn] depends upon A/θD4, the high-T linear part of ρ (T) depends on A only. This means the value of θD will also become unique. In this way the set of values ρo, A, θD and B, given in Table 2, is a unique set, and no other set of values of these parameters can provide an equally well fit of the experimental data with Eq. (1). It appears from the values of ρo and A that the effect of impurities and electron-phonon interaction is maximum in the AlB2 sample, while it is least in the AgB2 sample. Values of B indicate that there is no weak localization in MgB2. The effect of weak localization is about 20 times less in AgB2 compared to AlB2. This is consistent with the values of ρo for these two samples, because for AlB2 ρo is about 22 times larger than ρo of AgB2. The Debye temperature is found to decrease in the sequence MgB2 > AlB2 > AgB2, which is expected because of dependence of θD on the mass of the constituent atoms. Lal et al. [27] found that the combined effect of the impurity scattering, electron-phonon interaction (Bloch-Grüneisen theory) and weak localization provides a reasonable explanation of the resistivity data with various parameter values given in Table 2. Figure 10 shows the resistance versus temperature curve for the thin-film sample due to Tomita et al. [21]. A sharp superconducting transition, with an onset at 7.4 K and zero resistance at 6.7 K, was observed. The transition was seen reversible for both decreasing and increasing temperature. The measured current was 1 mA; hence, the critical current was at

72

A. K. M. A. Islam and F. Parvin

least greater than 1 mA, suggesting the existence of bulk superconductivity. The film resistivity was estimated to be ~ 0.05 μΩcm at 10 K. The MgB2 decomposition rate approaches one monolayer/sec at 650oC and reaches about 100 monolayers/sec at 800oC [30]. Hence, it is very likely that all MgB2 decomposed during the annealing process and that the dissociated B atoms reacted with Ag atoms to form the silver boron intermetallic compound. From the T-Pvap curve, the melting point of Ag is 680oC in a vacuum of 10-6 Torr. Tomita et al. [21], in fact, observed at ~710oC an abrupt increase in residual film resistance ratio R(280K)/R(10K) for the Ag/B film, suggesting the occurrence of some reaction above 710oC. Figure 10 shows a sharp transition at 6.4 K for type II material. The transition width was 1 K, slightly broader than that of the type I sample (0.7 K). The measured current was again 1 mA. Tomita et al. [21] also fabricated the samples at 670oC and 720oC. Interestingly, the sample annealed at 670oC exhibited Tc = 25 K, corresponding to the superconductivity of MgB2, while that annealed at 720oC again exhibited Tc = 6.5 K. The results are quite consistent with the T-Pvap curve and the observed MgB2 decomposition rate given above.

Figure 10. R(T) for (a) type I sample and (b) type II sample. The film resistance (for both the samples) decreased nearly linearly and exhibited a sharp transition at 7.4 K (Tco = 6.7 K). The inset shows the expanded version of R(T) near the transition. (Adopted from [21], ©2004 Physical Society of Japan).

Since the observed Tc was almost the same for the intermetallic compound fabricated by two different methods and only the existence of one intermetallic compound between Ag and B (AgB2) has been reported, [22, 31] Tomita et al. [21] conjecture that the films are AgB2. The XRD measurements, however, did not show any new peaks corresponding to AgB2. This is the common feature of the samples fabricated by the cap method. In fact, the MgB2 thin films fabricated using the same method showed very small peaks or did not show any peaks in the X-ray diffraction measurements, although the films exhibited bulk superconductivity with high critical current density and high Tc above 25 K [26]. The absence of the diboride peak is attributed to the fact that the diboride films thus fabricated had poor crystallinity.

Superconducting Noble Metal Diboride

73

3. Mechanical Behaviour, Electronic and Bonding Properties and Critical Temperature 3.1. Computational Details We report here some results (where specifically mentioned) using SCF Hartree-Fock linear combination of atomic orbital computer programme CRYSTAL98 [32], which contains a density functional theory (DFT) option that permits one to solve the Kohn-Sham (KS) equation self-consistently. The choice of basis sets for a crystalline compound is very important for accurate description of the crystalline orbital of the system. In our earlier calculations for AgB2 we used valence electron basis sets HAYWSC-2111d31 [33] and DURAND-21d1 [34] for Ag and B, respectively. The basis sets used in the present study for AuB2 are INPUT-3111221d41 [35,36] and DURAND-21d1[34] for Au and B, respectively. The correctness of the basis set for B has already been seen in the case of the widely studied MgB2 and other related compounds. 15 atomic shells with 49 atomic orbitals (AO) have been used in the calculations for AuB2 using CRYSTAL98 [32]. Diffuse sp and d shells have been used in order to provide additional variational freedom accounting for the tails of the atomic wave function. The exponents of these most diffuse shells for each atom have been optimized by searching for the minimum crystalline energy. The exchange correlation potential proposed by PWGGA [37] is expanded in an auxiliary basis set of symmetrized atom-centred Gaussian-type functions. The quality of the calculation depends on the density of points with which the Brillouin zone (BZ) is sampled. The integrations over the BZ were performed using the Monkhorst-Pack scheme [38]. To ensure convergence for the BZ integration with accuracy very tight tolerances were utilized in the evaluation of the infinite Coulomb and exchange series. A dense Gilat net [39] was defined with a total of 1200 k-points in the reciprocal space, corresponding to a shrinkage factor of 28. Most of the results presented here have been calculated utilizing a full ab initio package implementing electronic structure and energy calculations, linear response methods (to calculate phonon dispersion curves, dielectric constants and Born effective charges) and thirdorder anharmonic perturbation theory [40]. We used it to perform total energy calculations, within the generalized gradient approximation. The electronic wave function and eigenvalue ε are calculated using the ab initio pseudopotential formalism. The electron-ion is represented by soft separable pseudopotential and the single particle wave functions are expanded in a plane-wave basis set. The package contains a set of programs for electronic structure calculations within Density Functional Theory (DFT) and Density-Functional Perturbation Theory (DFPT), using a plane-wave basis set and pseudopotentials. We employed plane waves with an energy cutoff (ecut) of 60 ry, as a basis to expand the electronic wavefunctions. The kinetic energy cutoff controls number of plane waves at given k. This is the single parameter which can have an enormous effect on the quality of the calculation, basically the large ecut is, the better converged the calculation is. We used the ultrasoft pseudopotentials for Ag and B. The pseudopotentials used in AgB2 are [41]: Ag.pbe-drrkjus.UPF and B.pbe-n-van.UPF. These are ultrasoft pseudopotentials (developed by Vanderbilt). As the name suggests, ultrasoft pseudopotentials attain much smoother (softer) pseudo-wavefunctions so use considerably fewer plane-waves for calculations of the same accuracy. This is achieved by relaxing the norm-conservation constraint, which offers greater

74

A. K. M. A. Islam and F. Parvin

flexibility in the construction of the pseudo-wavefunctions. UPF is a unified pseudopotential format that the package [40] uses. The exchange-correlation potential is due to Perdew Burke Ernzerhof (PBE). Linear response theory is used to calculate phonon frequencies and polarization vectors. The second-order change in the total energy, and hence the dynamical matrix, depends only on the first-order change in the electronic density to atomic displacements. The electronphonon matrix element, gqν, are computed from the first-order change in the self-consistent potential. The doubly constrained Fermi surface sums in equation (11) are performed using dense meshes of 1200 k points in the irreducible Brillouin zones (IBZ). The δ-functions in energy are replaced by Gaussians of width 0.04 Ry. Because of the large number of k points sampled, the results are not very sensitive to the Gaussian width. Phonon wave vectors are sampled on coarser meshes in the IBZ. At first we made a self-consistent calculation for AgB2 using a dense grid of k-points. We used 28 × 28 × 28 Monkhorst-Pack grid. This dense grid must contain all k and k+q grid points used in the subsequent electron-phonon (e-ph) calculation and must be dense enough to produce accurate e-ph coefficients (in particular the double delta integral at EF is very critical). For Brillouin zone integrations 18 × 18 × 12 grid of k-points are used. The choice of k-points should be such that it is suitable for good selfconsistency and phonon calculation. Both for AgB2 (and MgB2) the calculation of dynamical matrices were carried out using a 4 × 4 × 4 grid resulting in 12 irreducible q-points. The output contains the results for the e-ph coefficient at each q-point and the double delta integral at 10 different values of the Gaussian broadening. These are useful for convergence testing. The final result is summed over all q-vectors. A converged calculation can be obtained only with very dense grids.

3.2. Mechanical Behaviour Including Pressure Effects The results of a first principle calculations, for the mechanical behaviour of AgB2 and AuB2 at equilibrium and under pressure, are presented in this section. These hole-doped systems, potential candidates for superconductor, have been studied as a function of pressure in comparison to the medium-Tc MgB2 superconductor [17,18,42-44], where available. 0.03 (a) AgB2

0.004

0.000 28

(b) AuB2

ΔE (Hartree)

ΔE (Hartree)

0.008

0.02

0.01

0.00 30

32

V

(Å3)

34

36

28

30

32

34

36

3

V (Å )

Figure 11. Energy-volume curve for (a) AgB2 and (b) AuB2. ΔE = energy difference from equilibrium.

The total energy E of AgB2 and AuB2 has been calculated at different primitive cell volume (V). The results have been shown in figure 11 as a function of the normalized volume

Superconducting Noble Metal Diboride

75

V. The energy was minimized as a function of the c/a ratio for selected values of volume. Figure 12 shows normalized cell volume as a function of pressure for both AgB2 and AuB2. The structural parameters of hexagonal MX2 phase of AgB2 and AuB2 are shown in Table 3 along with those for superconducting MgB2. 1.00 (a) AgB2

(b) AuB2

Normalized volume

Normalized volume

1.00

0.96

0.92

0.95

0.90

0.85

0.88 0

5

10

15

20

25

0

10

P (GPa)

20

30

40

50

P (GPa)

Figure 12. Normalized cell volume (Vn) as a function of pressure (P) for (a) AgB2 and (b) AuB2.

Table 3. Structural parameters of hexagonal MX2 phases. a (Å)

c (Å)

X-X (Å)

M-X (Å)

V (Å3)

Ref.

MgB2

3.0640

3.4930

1.7690

2.4859

28.40

[42]

AgB2

3.0233 3.000 3.024 2.98 3.000 2.9572 3.0240 3.1340 2.98

4.0799 3.240 4.085 3.92 3.02 3.7854 4.1202 3.5130 4.05

1.7465 1.732 1.7073 1.7458 1.809 -

2.6869 2.372 2.5490 2.7006 2.522 -

32.3 25.25 32.36 30.15 25.25 28.66 32.63 29.88 31.15

[43] see [16] [45] [15] [23] [18] [44] [16] [15]

Phase

AuB2

The zero pressure bulk modulus B0 and its pressure dependence, B0′ (=dB0/dP) were determined by fitting the Murnaghan equation of state [46]: ' ⎡V 1 Vn1−B0 ⎤ n ΔE(V) = E − E0 = B0V0 ⎢ ' + + ' ' ⎥ ' ⎢⎣ B0 1− B0 B0(B0 −1) ⎥⎦

(2)

where E0 is the equilibrium energy. The pressure (P) versus the primitive-cell volume is obtained through the thermodynamic relationship: P=−

dE B0 = dV B0'

⎡V − B0' − 1⎤ ⎢⎣ n ⎥⎦

(3)

76

A. K. M. A. Islam and F. Parvin

The pressure dependence of normalized primitive-cell volume and lattice parameters of AgB2 and AuB2 are shown in figure 13. The linear bulk modulus at P=0 along the crystallographic axes a and c (Ba0 and Bc0) and their pressure derivatives are then obtained by fitting Eq. (3) to points in figure 13. The obtained results are shown in Table 4. (a) AgB2

1.04 1.00 a/ao 0.96

(b) AuB2

1.04

a/ao , c/co

a/ao , c/co

1.08

c/co

1.00

0.96

a/ao c/co

0.92 -10

0

10

0.92 -20

20

0

P (GPa)

20

40

P (GPa)

Figure 13. Normalized lattice parameters of (a) AgB2 and (b) AuB2.

Table 4. Bulk modulus, pressure derivative of bulk modulus and their in-and out-of-plane linear values for MgB2, AgB2 and AuB2.

Ba 0

B0 (GPa)

B0'

MgB2

122

3. 4

653

15

AgB2

142 142 167.4

4.6 4 4.9

596 564.9

15 15.8

Phase

AuB2

(GPa)

Ba' 0

Bc 0

Bc' 0

Ref.

397

5.5

[42]

274 411

10.6 12.8

[43] [45] [44]

(GPa)

The normalized lattice parameters as a function of pressure shown in figure 13 clearly shows the anisotropy in bonding of AgB2. As the pressure increases from 0 to 10 GPa, the c/a ratio decreases by 1.4% (0.9% for MgB2). Compression along the c-axis is larger than along the a-axis, consistent with the comparatively weaker Ag-B bonds that determine the c-axis length. A similar but smaller anisotropy has been seen for TiB2 [47]. The layered cuprates show much larger (~a factor of 2) compression anisotropy [48] than in MgB2. The fitted values for B a 0 , B a' 0 , Bc 0 , B c' 0 shown in the table clearly reveal the diversity in bonding interactions present. AgB2 is less compressible in the basal plane, in which the covalent B-B bonds lie. The interlayer linear compressibility, dlnc/dP = -0.0028 GPa-1 (-0.0019 for AuB2) is more than the in-plane value, dlna/dP = -0.0014 GPa-1 (-0.0014 for AuB2). It is worth noting that the structurally related alkali-metal intercalated graphite is strongly anisotropic (see [49]) with interlayer compressibility about ten times larger than the corresponding value in MgB2. MgB2 is an incompressible solid with volume compressibility, dlnV/dP = -0.0082 GPa-1 [42] compared to K3C60 which is a fairly soft material with weak intermolecular interactions with dlnV/dP =-0.036 GPa-1 [50]. On the other hand AgB2 is found to be a much more tightly-

Superconducting Noble Metal Diboride

77

packed incompressible material with dlnV/dP =-0.0056 GPa-1. On the other hand AuB2 is a solid with volume compressibility, dlnV/dP = -0.0046 (-0.0082 for MgB2) GPa-1 [42]. Another important parameter which implies the sensitivity of superconducting properties to the interatomic distances is the volume coefficient of Tc, dlnTc/dV. We utilize here McMillan’s Tc formula [51], as modified by Allen and Dynes [52] to get this quantity. The Tc equation is

Tc =

ω

⎡ ⎤ 1+ λ exp⎢− 1.04 *⎥ 1.2 λ - (1+ 0.62λ)μ ⎦ ⎣

(4)

where <ω> is the average phonon frequency, λ is the electron-phonon coupling constant. The Coulomb pseudopotential μ*= 0.1 describes the repulsive interaction between electrons. μ* does not depend on pressure (in principle, lattice hardening should lead to some increase of μ*, but this is a relatively weak effect [53]). Following Loya and Syassen [54] we differentiate Tc to get d ln Tc d ln N ( E F ) ⎤ 1 ⎡ α B0 ≡β =− + (1 − 2α )γ G ⎥ dV V 0 ⎢⎣ dP ⎦

[

where α = 1.04λ (1 + 0.38 μ * ) λ - (1 + 0.62 λ ) μ *

]

−2

P =0

(5)

.

In determining β we need dlnN(EF)/dP, λ and γG. First we calculate the density of states as a function of pressure (see section 3.4) from which we obtain dN(EF)/dP. For solids with isotropic compression, the mode Grüneisen parameter γG of zone-centre phonons are ~1. Loya and Syassen [54] assumed this to be 1 while estimating dlnTc/dP using McMillan’s formula for Tc. Recently Goncharov et al. [53] determined γG to be 2.9±0.3. On the other hand Islam et al. [42] find γG to be substantially larger than 1 which is expected for phonon in a compound with covalent bonding like graphite. For iron with partial metallic bonding γG =1.7 [55]. Thus for MgB2 and AgB2 we should not expect γG to be ~1 as has been assumed in ref. [54]. It is to be noted that larger γG is usually associated with enhanced anharmonicity of the particular normal mode of vibration, and is broadly consistent with theoretical prediction for the E2g inplane B stretching mode [53]. In order to see the variation of the volume derivative of Tc, β for AgB2 for any combination of the parameters λ and γG we plot β in figure 14 as a function of both λ and γG in the relevant range and γG = 2 - 3. Because γG is relatively uncertain, three sets of values of γG are used. The value of β for MgB2 [56, 57] is about the same order as that for AgB2 and is larger than ~ 0.07 Å-3 which is found in fulleride superconductors. It is thus evident from figure 14 that MgB2 in comparison to AgB2 shows nearly equal sensitivity of superconducting properties to the interatomic distances. Both MgB2 and AgB2 are characterized by moderately large anisotropy of compressibility. This is smaller than those of cuprates [48] but larger than other related diborides. Strong bonding, dominant phonon frequency and reasonable density of state at the Fermi level is believed [58, 59] to lead to the observed Tc of MgB2.

78

A. K. M. A. Islam and F. Parvin 0.3 0.5

d lnTc/dV (Å-3)

-3

dlnTc/dV (Å )

(a) AgB2

0.2

γG = 3.0 γG = 2.5 γG = 2.0

0.1 0.9

1.0

1.1

1.2

1.3

λ

0.4

(b) MgB2 γG = 3.0 γG = 2.5

0.3

γG = 2.0

0.2 0.1 0.6

0.7

0.8

0.9

1.0

λ

Figure 14. The variation of β as a function of λ for different values of γG for (a) AgB2 and (b) MgB2. The shaded region shows the most likely ranges of λ values in each case (see text).

3.3. Electronic Band Structure The calculations of the electronic band structure help one to understand the shape of the Fermi surface. Kortus et al. [59] made the first ab initio calculations for MgB2 and found that the electronic states near the Fermi level are mainly B in character. Since then the surface is understood to be comprised of four sheets (see figure 15). The band structure shows four bands crossing the Fermi energy leading to four topologically disconnected Fermi surface sheets shown in the figure. Two of these nearly cylindrical hole sheets about the Γ-A line arise from quasi-2D px,y B bands. The other two bands are derived from Boron pz orbitals. They form the so-called π bands seen as the red (electronlike) and blue (holelike) tubular networks in figure 15. The two bands which derive from Boron px and pyorbitals form the socalled σ bands, seen as the green and blue cylindrical Fermi surfaces centered around the Γ point (both holelike). These possess mainly 2D character. Interestingly, it is seen that all these bands are dominated by B p orbitals and contributions from Mg orbitals are very small at the Fermi level [59, 60]. Using the first-principles methods described earlier the energy bands were calculated. The full BZ is spanned in such a way that Γ-M-K-Γ-A-L-H-A directions are covered. The ΓM-K-Γ lines are in the basal plane, while A-L-H-A lines are on the top of the plane at kZ. The results of the band structure calculations of AgB2 and AuB2 are shown in figures 16 (a-c) and 17 (a-d), respectively. The band structures of MgB2 (not shown) previously calculated by us [57] are taken into consideration for comparison. In this superconductor, there are two distinct types of bands, both of which are contributed by boron. We observe the σ(2px,y) band along Γ−Α to be double degenerate, quasi two-dimensional and to make a considerable contribution to DOS at EF for MgB2 [57, 59]. The existence of degenerate px,y-states above EF at the Γ point in BZ has been shown to be crucial for superconductivity in MgB2. The weaker ppπinteractions result from B 2pz-bands. These 3D-like bands possess maximum dispersion along the Γ-A direction. It has been recognized that only the branch of the in-plane E2g phonon of B ions exhibits a large electron-phonon coupling. In our case for AgB2 and AuB2, B 2p states near EF, corresponding to the in-plane B-B px,y σ bands, are relatively dispersionless along the Γ−A−L line and yield hole Fermi surfaces. Like in MgB2 we can also take AgB2 phonon with

Superconducting Noble Metal Diboride

79

a frequency ω close to that of the E2g mode as responsible for the superconductivity. The B pσ bands are flatter in AgB2 and AuB2 than in MgB2, yielding higher DOS’s at EF. The band structures of AgB2 and AuB2, due to Kwon et al. [15] and Shein et al. [16] along high symmetry lines in the BZ are shown in figures 18 and 19, respectively. These results are compared with those from the present calculations. It is seen that the overall features mentioned above agree reasonably well. But when examined closely we see some differences even near the Fermi energy (see figures 18 and 19). It is likely to be due to the differences in parameters, basis sets and the method of calculations used. The bandwidths of B pσ calculated by Kwon et al. [15] are less than 5 eV in AgB2 and AuB2 due to band repulsion between Ag 4d (Au 5d) and B 2p, as is evident along the A−L line. The estimated bandwidths are as much as 9 eV in MgB2. The band repulsion drives the B 2pσ states to pile up at EF. Therefore, one can see that the Ag 4d (Au 5d) bands play a role in enhancing the DOS of B 2pσ states near EF [15].

Figure 15. The Fermi surfaces of MgB2. (Adopted from [59], ©2001 The American Physical Society). AgB 2

AgB 2

8.98 P = - 14 GPa

1.08

Energy (eV)

Energy (eV)

11.08

EF

-8.92

P=0 EF

-1.02

-11.02 Γ

M

K

Γ

A

L

H

Γ

A

M

Γ

K

k-space

A

L

H

A

k-space

AgB2

Energy (eV)

7.33 P = 17 GPa EF -2.67

-12.67

Γ

M

K

Γ

A

L

H

A

k-space

Figure 16(a-c). AgB2 band structures along high symmetry lines for (a) P=-14 GPa, (b) P=0 and (c) P=17 GPa. Along Γ-A-L lines B 2px,y σ hole bands are formed near EF, similar to MgB2.

80

A. K. M. A. Islam and F. Parvin

The calculated band structures away from the equilibrium are also included in figures 16 and 17. It is seen that the character of σ band is unchanged even after application of pressure. The σ band along Γ−Α shows weak dispersion that reflects its quasi-two-dimensionality. The dispersion increases slightly with increase of pressure. Neaton et al. [61] observed that the σ bands of MgB2 are nearly free electron-like: their dispersion is parabolic near the Γ point, and their overall bandwidth is comparable to the free electron value (~15.5 eV).

P = -15 GPa 0.82

EF

-7.35 Γ

M

Γ

K

A

L

H

Energy (eV)

Energy (eV)

8.44

AuB2

8.98

AuB 2 P=0

0.27

EF

-6.26

Γ

A

M

K

Γ

A

L

H

A

k-space

k-space

11.43 4.08

P = 17 GPa

AuB 2

EF -4.08

Energy (eV)

Energy (eV)

AuB2

[

P = 70 GPa

3.27

EF -4.90

-13.06

-12.25 Γ

M

K

Γ

A

k-space

L

H

A

Γ

M

K

Γ

A

L

H

A

k-space

Figure 17(a-d). AuB2 band structures along high symmetry lines for (a) P=-15 GPa, (b) P=0, (c) P=17 GPa and (d) P=70 GPa. Along Γ-A-L lines B 2px,y σ hole bands are formed near EF, similar to MgB2.

AgB2

AuB2

Figure 18(a,b). Band structures along high symmetry lines for (a) AgB2 and (b) AuB2 at equilibrium. (Kwon et al. [15]).

Superconducting Noble Metal Diboride

81

Figure 19 (a,b). Band structures of (a) AgB2 and (b) AuB2 along high symmetry lines at equilibrium due to Shein et al. [16]. The features are similar to those of MgB2.

3.4. Density of States The total and partial electronic density of states of AgB2 and AuB2 near the Fermi level are shown in figure 20. The shape and locations of the bands shown in figures 16 and 17 are reflected in the density of states N(EF). AgB2 has a larger DOS at EF. In fact there is a 27% increase in the DOS compared to MgB2. Figure 21 shows DOS of MgB2 and AgB2 as a function of pressure. The values at equilibrium (P=0) are 0.84 and 0.88 states/eV for AgB2 and AuB2, respectively. The value for AgB2 slightly reduces to 0.8 states/eV at P = 17 GPa. That the density of states decreases as the pressure increases is, contrary to expectation for a nearly-free electron metal, in agreement with other results [54, 61, 62]. This observation, via BCS equation, shows that Tc should decrease with the increase of pressure, a result in agreement with experiment for MgB2 [63]. Neaton et al. [61] noted that the dependence of DOS on pressure is nearly due to the changes in the width and position of a considerable van Hove peak. This originates from a saddle point in the highest occupied σ band at the M point. Further the decrease in bandwidth with decreasing pressure reduces the separation between the peak and the Fermi level, enhancing DOS. The singularity is further enhanced by an increase in two dimensionality. 4

(a) AgB2 Total EF

4

Ag(1d) 2

(b) AuB2

DOS (states/eV)

DOS (states/eV)

6

B(2p)

Total

3

EF 2

Au B

1 0

0 -10

-5

0

E-EF (eV)

5

10

-10

-5

0

5

10

E-EF (eV)

Figure 20. Total DOS of (a) AgB2 and (b) AuB2 at equilibrium. Ag and B-contributions are also indicated.

A. K. M. A. Islam and F. Parvin

N(EF) (states/eV)

82

0.9

AgB2 0.8 0.7

MgB2

0.6 -10

0

10

P (GPa) Figure 21. Total DOS as a function of pressure for AgB2 and MgB2.

3.5. Phonon Spectra and Electron-Phonon Coupling Strength Let us first of all discuss the density-functional calculations of the phonon modes and the electron-phonon interaction strength of MgB2 which can be found in refs. [64, 65, 66]. The highest phonon density of states is found in the energy range around 30 meV [64]. However, these phonons only couple weakly to the electrons at the Fermi level and thus do not contribute very much to superconductivity. This can be nicely recognized in figure 1 of ref. [64], where the interaction strength of the phonons is shown as the area of the black circles in the figure. In fact, the phonons that couple most strongly to the electrons at the Fermi level are found in the energy range around 70 meV. These phonon modes evolve from the E2g mode at the Γ point and correspond to a Boron-Boron bond-stretching vibration of the Boron sublattice. A comparison with the phonon modes in the isostructural but nonsuperconducting compound AlB2 in ref. [65] shows that these E2g phonon modes are strongly softened in MgB2 consistent with their strong coupling. Correspondingly, the socalled Eliashberg function α2F(ω), which weights the phonon density of states with the coupling strengths and appropriately describes the pairing interaction due to phonons, possesses a strong peak around 70 meV and significantly differs from the phonon density of states in contrast to conventional strong-coupling superconductors. The dimensionless electron-phonon coupling constant was found to lie between λ / 0.6-0.9 [64-66] from these first-principles calculations. Boron is a light element and the E2g phonon modes in question only involve vibrations of the Boron sub lattice yielding characteristic phonon frequency ωc to be large. Further such high frequency phonon possesses a strong coupling to the electrons at the Fermi level. We can look at the following BCS Tc (very crude approximation) to show qualitatively how these factors help to achieve high Tc: Tc ~ ωc e-1/v N(0)

(6)

The interaction strength V and the density of states N(0) at the Fermi level, with λ ~ V N(0) shows that in MgB2 we have a favourable coincidence to yield high Tc. [60].

Superconducting Noble Metal Diboride

83

We know that the normal mode frequencies, ω and displacement patterns, U

α I

for

Cartesian component α of atom I, at atomic position RI, are determined by the secular equation:

∑β (C αβ − M IJ

Iω

2

)

δ IJ δ αβ U Jβ = 0,

J,

(7)

αβ is the matrix of second derivatives of the energy with where MI is the mass matrix and CIJ

respect to atomic positions, i.e. inter-atomic force constants:

∂ 2 E ({R}) ∂RIα ∂RJβ

C IJαβ ≡

In crystals, normal modes are classified by a wave-vector q. Phonon frequencies, ω(q) and displacement patterns, Usα (q) , are determined by the secular equation:

∑β (C αβ (q) − M ~

st

sω

2

)

(q)δ stδ αβ U tβ (q) = 0

(8)

t,

Atomic perturbation u is introduced to atomic positions

R I = Rl + τ s

as

RI [us (q)] = Rl + τ s + us (q) e iq.Rl

where Rl = lattice vector, τ s = equilibrium position of the sth atom in the unit cell. This induces a response having the same wave vector q (at linear order). Fourier transform of force constants at q are second derivatives of the energy with respect to such monochromatic perturbations: ~ C stαβ (q) ≡

∑

e −i q . R C stαβ (R) =

R

∂2E 1 N ∂u *s α (q) ∂u tβ (q)

(9)

This can be calculated from the knowledge of the linear response ∂ n ( r ) ∂ u αs ( q ) and diagonalized to get phonon modes at q. The matrix element for scattering of an electron from one state to another state by a phonon with frequency ωqν is given by [40, 67]: ⎛ h g qυ ( k , i , j ) = ⎜ ⎜ 2 M ω qυ ⎝

⎞ ⎟ ⎟ ⎠

1

2

ψ i,k

∂Vscf ∂U (υ ) ( q )

ψ j,k + q

(10)

84

A. K. M. A. Islam and F. Parvin

where the gradient of Vscf is the self-consistent change in the potential due to atomic displacements. U(ν) is a displacement along phonon ν. This quantity can be easily calculated using Density Functional Perturbation Theory (DFPT). The scattering gives a finite line-width γqν. It is given by γ qv = 2πωqv ∑ ∫ ij

2 d 3k g qv ( k , i, j ) δ (ε k ,i − ε F )δ (ε k + q ,i − ε F ), Ω BZ

(11)

The dimensionless electron-phonon mass enhancement parameter (electron-phonon coupling strength):

λ = ∑ λ qυ = ∑ qυ

qυ

γ qυ πhN ( E F )ωq2υ

(12)

where N(EF) is the DOS at the Fermi level. We calculate the strength of electron-phonon coupling (e-ph) λ and phonon frequency ω utilizing computational calculations as in section 3.1. The calculation yields λ = 0.96 and ω(E2g mode) = 282.7 cm-1 = 406.6 K for AgB2. The corresponding values for MgB2 are 0.73 and 770 K [65], respectively. Lal et al. [27] have synthesized sample of MgB2, AlB2 and AgB2 and have measured their resistivity up to room temperature. The combined effect of the impurity scattering, electron-phonon interaction (Bloch-Grüneisen theory) and weak localization provides a reasonable explanation of the resistivity data with various parameters in a unique way. The unique value of Debye temperature is found to be 700 K, 670 K and 480 K for MgB2, AlB2 and AgB2, respectively. The Debye temperature would suggest a ω =340 K for AgB2.

3.6. Charge Density and Chemical Bonding The nature of bonding may be understood from the charge density (CD) plots. Figures 22 (ad) and 23 (a-d) show the total charge density profiles for AgB2, AuB2 and MgB2 at different pressures for (110) plane. Note that Ag, Au or Mg nuclei are located at the corners of the charge density map and B nuclei are at the (1/3, 1/2) and (2/3,1/2) positions. The density profile (figure 23d) for MgB2 [57] shows features similar to that in ref. [54]. The electron accumulation between Ag or Mg and B is found to be low. Further the electron population in the Mg site is lower than that for a neutral Mg atom. These indicate an ionic bonding between Mg and B. There may have been a smaller electron density transfer from (Ag, Au) to B compared to the Mg – B transfer in MgB2. The same can be said about bonding between Ag (or Au) and B but with a decreasing strength. Maximum charge density with a aspherical character is seen at the bond middle. This indicates a covalent bonding between B-B atoms in both compounds. Further a certain degree of metallic bonding between the Ag (Au or Mg) atoms is indicated by a somewhat homogeneous charge distribution between them.

Superconducting Noble Metal Diboride

Ag

Ag

Ag

Ag

Ag

Ag

85

Ag

Ag

B

B

B

B

B

B

B

B

Ag

Ag

(a) P = -14 GPa

Ag

Ag

Ag

Ag

(c) P = 18 GPa

(b) P = 0

Ag

Ag

(d) P = 23 GPa

Figure 22(a-d). Total electron charge density map on the (110) plane through Ag and B atoms for AgB2 at (a) P=-14 GPa, (b) P=0, (c) P=18 GPa, (d) P=23 GPa.

Au

Au

Au

Au

Au

Au

Au

B

B

B

B

B

B

Au

(a) P = -15 GPa

Au

Au

(b) P = 0

Au

Au

(c) P = 17 GPa

(d) P = 0

Figure 23(a-d). Total electron charge density map on the (110) plane through Au or Mg and B atoms for AuB2 (a) P=-15 GPa, (b) P=0, (c) P=17 GPa, and for MgB2 at (d) P=0.

Figure 24. Inter-plane charge-density difference maps for AgB2 [45]. Solid and dashed lines indicate an increase and a decrease of the electron density.

Shein et al. [45] studied the metastable nature of AgB2. They observed very weak interlayer Ag-B and intra-layer Ag-Ag interaction. They further examined the bonding nature of AgB2 using difference electron densities (Δρ) (figure 24). It is the difference between the crystalline charge ρcryst and the neutral atomic charge ρat densities. Negative Δρ around Ag and positive ones around B indicate a charge transfer from Ag to B. The Δρ map shows a strongly covalent B-B bonding in the hexagonal boron sheets. On the contrary, the partial ionic type of the inter-plane Ag-B bonding occurs. The in-plane Ag-Ag bonds are not significant: (i) the near-spherical symmetry of silver Δρ contours confirm the absence of Ag-

86

A. K. M. A. Islam and F. Parvin

Ag covalency; (ii) the ionic interaction between silver ions is repulsive and (iii) the Ag-Ag distance in diboride (3.004 Å) is about 4% higher than in fcc Ag (2.889 Å) i.e. the decrease in metallic-like bonding also occurs.

3.7. Superconducting Tc and Role of Paramagnon The calculated critical temperature of AgB2 [15] is significantly higher than the observed Tc of 6.7 K [21] for the fabricated thin films of AgB2. In comparison the MgB2 thin films fabricated by the same method also yielded Tc~ 25 K, 64% of the bulk Tc of 39 K. These would indicate that the synthesis conditions for AgB2 and MgB2 thin films were not properly optimized. This is evident in AgB2 thin films. In fact some inhomogeneity has been seen which indicates the coexistence of the AgB2 phase along with some remainders despite the use of a stoichiometric target. This is probably due to the difference in the laser ablation rate and stacking factor to the substrate between Ag and B atoms. Hence it is expected that Tc would increase once the synthesis conditions are improved. At this point we note that the observed Tcs are comparable with those for some d-metal diborides: ZrB2 (5.5K), TaB2 (9.5K) and NbB2 (0.6-5.2K) [68]. Theoretical calculation of λ and ω by Singh [69] predicted Tc for NbB2 as 3 K, which do not vary much from the corresponding experimental value. But for AgB2 the situation looks quite different. In MgB2 we find λ = 0.6-0.9 and ω = 61-70 meV [70, 71] and the calculated Tc ~38 K which agrees well with the observation. The large discrepancy between the theoretical estimate of Tc and the observed value for AgB2 may be due to several reasons. First reason is apparent from the crude approximation (i.e. ωMgB2 = ωAgB2) made by Kwon et al. [15] for the phonon frequency of AgB2. There may also be inaccuracy in the calculated e-ph coupling constant. Further an underestimation or omission of nonphononic mechanisms cannot be ruled out. The occurrence of paramagnons is related with the lining up of electron spins. Since superconductivity has its origin in the formation of electron pairs with opposite spins, paramagnons are expected to counteract superconductivity. Indeed the enhanced spin fluctuations (‘paramagnons’) have been found to counteract superconductivity in several materials [72-75]. A typical example for paramagnon effects is Pd, where superconductivity has been shown to be completely suppressed by paramagnons [72, 73]. In case of Nb and V, ferromagnetic spin fluctuations substantially reduce Tc [76]. Another example is Li, in which including the effects of spin-fluctuation one finds, in agreement with experiment, no superconductivity at low pressure [77]. But it is somewhat unexpected to find spinfluctuations to be of importance in Li, since one usually associates this phenomenon with high DOS transition metal [76]. The enhanced density of states in the case of AgB2 compared to MgB2 it is conjectured that the phenomenon of spin-fluctuation is likely to be associated with AgB2 [43]. We would discuss below the effects of paramagnons on the critical temperature of AgB2. Paramagnons We will now study the role of enhanced spin fluctuations on the critical temperature of AgB2. A virtual exchange of paramagnons leads to a contribution (λsp) to the electron mass enhancement m*/m [72, 76]. Thus one can write

Superconducting Noble Metal Diboride

m* /m =1 + λ ph + λ spin

where,

87

(13)

∞

∞

0

0

λ ph = 2∫ dω α 2 F (ω ) / ω and λ spin = 2∫ dω P( ω ) / ω

(14)

α2F(ω) is the Eliashberg function and P(ω) is paramagnon spectral weight function. P(ω) is related to particle-hole t-matrix t(q,ω) by [72, 76] P(ω ) =

3 N (E F ) 2π

∫

2k F 0

qdq Im t (q, ω ) 2k F2

(15)

Thus the Eliashberg function α2F(ω), the Coulomb repulsion μ* and the paramagnon spectral weight function P(ω) constitute the ingredients which determine Tc. The retarded nature of the spin fluctuation kernel P(ω) explains why Tc is more sensitive to λsp, than to μ* [77]. Moreover P(ω) is directly related to density of states (DOS) N(EF). The Modified McMillan expression of Tc including spin fluctuations is given by [51,78]:

Tc =

− 1.04 (1 + λ + λsp ) <ω> exp [ ] 1.2 λ − μ* (1 + 0.62 λ) − λsp

(16)

where λsp represents contribution from spin fluctuations (paramagnons). <ω> is the average phonon energy. Using Eq. (16), we calculate Tc with different values of λsp. The results are shown in figure 25. As can be seen the theoretically calculated Tc is 22.4 K for λsp= 0 which is significantly larger than the observed Tc of 6.4–6.7 K for thin film. We comment here that the MgB2 thin films fabricated by the same cap method as that used in AgB2 synthesis also exhibited Tc of only around 25 K, significantly lower than the bulk Tc of 39 K. These results indicate that the synthesis conditions for AgB2 and MgB2 thin films have not been optimized yet. In fact, the AgB2 films showed some inhomogeneous structure, indicating the coexistence of the AgB2 phase, and some remainders despite the use of a stoichiometric target, probably due to the differences in laser-ablation rate and sticking factor to the substrate between Ag and B atoms. Hence, it is expected that Tc may be increased by improving the synthesis conditions. In the absence of successful synthesis of AgB2 by the conventional powder or bulk material method, a rough estimate of Tcbulk of AgB2 may be inferred from a knowledge of experimental Tcbulk and Tcfilm of MgB2. Assuming a similar trend Tcbulk of AgB2 is roughly estimated to be ~ 10.5 K. From this we see that Tcbulk is nowhere near the theoretically estimated value. Both Tcbulk and Tcfilm have been shown in figure 25. From the plot we that λsp= 0.12 and 0.18 corresponding to bulk and film AgB2. The non-zero values of λsp indicate the effects of spin fluctuations which suppress the calculated Tc below the experimental limit. Since other effects may be present which may also contribute to small suppression of superconductivity the estimated λsp values may be considered the upper limit. If calculated λ and Tc are reliable, the spin fluctuations are primarily responsible for lowering of electron-

88

A. K. M. A. Islam and F. Parvin

phonon pairing and hence reducing Tc of AgB2. An average upper value of λsp = 0.15±0.03 would thus significantly suppress the predicted Tc of AgB2 phase below the experimental limit.

Tc (K)

20 15 10

Tcbulk Tcfilm

5 0 0.0

0.1

0.2

0.3

λsp

Figure 25. Tc vs λsp curve for AgB2. Tc bulk is as explained in the text.

Thus on the basis of the above discussion, it appears that the spin fluctuation plays a vital role in suppressing Tc in AgB2. In particular, since high electronic densities of states N(0) favour the occurrence of spin fluctuations, paramagnon effects are one, if not the limiting, factor for high superconducting transition temperatures. So like several other elements spin fluctuations do seem to play an important role in reducing the transition temperature.

4. Future Direction of Research The theoretical predictions of superconductivity in noble metal diborides have at least come true only in AgB2 for which it has been possible to obtain thin film with Tc=6.7 K much lower than the predicted value. Although attempts by Pelleg et al. [24, 25] to stabilize AuB2 and AgB2 have failed, they did not exclude the possibility of the formation of these compounds. They conjectured that the formation of a noble metal boride by a non-equilibrium process such as sputtering may be still possible. The formed boride phase must be kept in a condition, say high vacuum, where the possibility of water absorption is avoided. But according to Pelleg et al. [25] until further information is being provided on the material and material processing, the claim of forming an AgB2 film should be considered with suspicious caution. The mystery associated with the various reports regarding the existence of noble metal diborides and the lack of observing superconductivity or its Tc, in the range of investigation further observation requires additional work. Thus Pelleg et al. [25] suggested to address the following topics in future research work: a) the stability of the particles formed, their definite identification, the conditions of their disappearance; b) further evaluation of Tc , if superconductivity is observed, as indicated in ref. [79], and c) substantiate whether the theoretical predictions should be the guidelines in searching for superconductivity in noble metal-boride systems.

Superconducting Noble Metal Diboride

89

Theoretically higher Tc was expected to be obtainable in AgB2 and AuB2 compared to MgB2. It is thus desirable to find the right condition of synthesizing stoichiometric AuB2. The presence of Ag or Au as impurity in solid solution of MgB2 increases DOS [16] at the Fermi surface and hence can modify the superconducting properties. One should first of all try to synthesize hole doped compound like Mg1-xAuxB2 or grow it in film [15]. The calculations by Shein et al. [16] show that the band structure of the nonboride AlB2like phases, which involve sp elements making up graphite like networks, differs strongly from that of MgB2; therefore, a search for new MTSC phases among the former compounds (as well as among the d metal diborides) does not hold obvious promise. But the observation based on theoretical calculations was made before the discovery of AgB2 superconductors in thin film form. They also said that the main MTSC candidates among the AlB2-like structures are probably the diborides of Group I and II elements, their solid solutions, or superstructures. Shein et al. [16] further conjectured, from their theoretical study of Ag and Au diborides, that the presence of these elements, e.g., as impurities (or in atomic layers) in solid solutions (or superstructures), can modify the MTSC properties of MgB2 through an increase in the density of states at the Fermi level of the system.

Acknowledgements The authors would like to thank Prof. V.P.S. Awana [27], Prof. J. Pelleg [24, 25], Dr. B.J. Min [15]), Prof. I.R. Shein and coworkers [16, 5] and the Physical Society of Japan [21] for the permission to use their published and unpublished materials during preparation of the present work. The American Physical Society is also acknowledged for reproducing a figure in the present work from ref. [59].

References [1] Nagamatsu, J.; Nakagawa, N.; Muramaka, T.; Zenitani, Y.; Akimitsu, J. Nature (London) 2001, 410, 63. [2] Akimitsu, J. Symposium on Transition metal Oxides, Sendai, January 10, 2001. [3] Buzea, C.; and Yamashita, T. Supercond. Sci. Technol. 2001, 14(1), R115-R146. [4] Ravindran, P.; Vajeeston, P.; Vidya, R.; Kjekshus, A.; and Fjellvag, H. Phys. Rev. 2001, B 63, 224509. [5] Yamamoto, A.; Takao, C.; Masui, T.; Izumi, M.; Tajima, S. Physica C 2002, 383, 197. [6] Kaczorowski, D.; Zaleski, A. J.; Zogal, O. J.; Klamut, J. arXiv: cond-mat/0103571 (2001); Kaczorowski, D.; Klamut, J.; Zaleski, A. J. arXiv:cond-mat/0104479 25 April 2001, v1. [7] Felner, I. Physica C 2001, 353, 11. [8] Young, D. P.; Adams, P. W.; Chan, J. Y.; Fronczek, F. R. arXiv: cond-mat/0104063 (2001). [9] Gasparov, V. A.; Sidorov, N. S.; Izver’kova, I.; Kulakov, M. P. arXiv: condmat/0104323 (2001).

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[10] Strukova, G. K.; Degtyareva, V. F.; Shovkun, D. V.; Zverev, V. N.; Kiiko, V. M.; Inov, A. M.; Chaika, A. N. 2001 Superconductivity in the Re-B system, arXiv: condmat/0105293. [11] Heid, R.; Renker, B.; Schober, H.; Adelmann, P.; Earnst D.; and Bohnen, K. -P. arXiv: cond-mat/0302411 20 Feb 2003, v1. [12] Islam, F. N.; and Islam, A. K. M. A. Physica C 2005, 426-431, 464. [13] Islam, A. K. M. A.; Sikder, A. S.; and Islam, F. N. Physics Letters A 2006, 350, 288. [14] Hou, Z. F. arXiv:cond-mat/0601216 26 Feb 2006, v2. [15] Kwon, S. K.; Youn, S. J.; Kim, K. S.; and Min, B. J. arXiv: cond-mat/0106483 (2001); Kwon, S. K.; Min, B. I.; Youn, S. J.; and Kim, K. S. Journal of the Korean Physical Society 2005, 46(6), L1295. [16] Shein, I. R.; Medvedeva, N. I.; and Ivanovskii, A. L. Physics of the Solid State 2001, 43, 2213. [17] Parvin, F.; Islam, A. K. M. A.; Islam, F. N.; Wahed, A. F. M. A.; and Haque, M. E. Physica C 2003, 390, 16. [18] Parvin, F.; Islam, A. K. M. A.; Islam, F. N. Solid State Communications 2004, 130, 567. [19] Cooper, A. S.; Corenzerst, E.; Longinotti, L. D.; Matthias, B. T.; Zachariasen, W. H. Proc.Natl.Acad.Sci. 1970, 67, 313. [20] Leyarovska, L.; Leyarovski, E. J. Less-Common Metals 1979, 67, 249. [21] Tomita, R.; Koga, H.; Uchiyama, T.; Iguchi, I. J. Phys. Soc. Jpn. October, 2004, 73(10), 2639. [22] Obrowski, W. Naturwissenschaften 1961, 48, 428. [23] Gurin, V. N.; Korsukova M. M. in Boron and Refractory Borides; Matkovich, V. I.; Ed.; Springer-Verlag Berlin Heidelberg, NY, 1977, p. 301. [24] Sinder, M.; and Pelleg, J. (2002) arXiv: cond-mat/0212632. [25] Pelleg, J.; and Sinder, M.; Rotman, M. - paper under preparation – Personal communication with J. Pelleg (February 2007). [26] Uchiyama, T.; Koga, H.; and Iguchi, I.: Jpn. J. Appl. Phys. 2004, 43, 121. [27] Lal, R.; Awana, V. P. S.; Singh, K. P.; Kishan, H.; and Narlikar Ziman, A. V. arXiv: cond-mat/0509674; accepted Mod. Phys. Lett. B. [28] Ziman, J. M. Principles of The Theory of Solids, Second ed., Cambridge University Press, Cambridge, UK, 1972. [29] Lee, P. A.; Ramakrishnan, T. V. Rev. Mod. Phys. 1985, 57, 287. [30] Fan, Z. Y.; Hinks, D. G.; Newman, N.; and Rowell, J. M.: Appl. Phys. Lett. 2001, 79, 87. [31] Okamoto, H.: Desk Handbook: Phase Diagrams for Binary Alloys (ASM International, Materials Park, 2000) p. 3. [32] Saunders, V. R.; Dovesi, R.; Roetti, C.; Causa’, M.; Harrison, N. M.; Orlando, R.; and Zicovich-Wilson, C. M. CRYSTAL98 User’s Manual, 1998, University of Torino, Torino and the references cited therein. [33] Apra, E. et al. Chem. Phys. Lett. 1991, 186, 329. [34] Causa, M.; Dovesi, R.; and Roetti, C. Phys. Rev. B 1991, 43, 11939. [35] Doll, K.; Pyykkoe, P.; Stoll, H. J. Chem. Phys. 1998, 109, 2339. [36] Andrae, D.; Haeussermann, U.; dolg, M.; Stoll, H.; Preuss, H. Theor. Chim. Acta 1990, 77, 123.

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[37] Perdew, J. P.; and Zunger, A. Phys. Rev. B 1981, 23, 5048; Perdew, J. P.; and Wang, Y. Phys. Rev. B 1986, 33, 5048; Perdew, J. P.; and Wang, Y. Phys. Rev. B 1989, 40, 3399. [38] Monkhorst, H. J.; Pack, J. D. Phys. Rev. B 1976, 13, 5188. [39] Gilat, G.; and Raubenheimer, J. L. Phys. Rev. B 1966, 144, 390. [40] Baroni, S.; Dal Corso, A.; de Gironcoli, S.; Giannozzi, P.; Cavazzoni, C.; Ballabio, G.; Scandolo, S.; Chiarotti, G.; Focher, P.; Pasquarello, A.; Laasonen, K.; Trave, A.; Car, R.; Marzari, N.; Kokalji, A.; http://www.pwscf.org/.cc; Dal Corso, A. Lectures at the Summer School on Electronic Structures, Methods and Applications, and Workshop on Computational Materials Theory, JANCASR (Bangalore, India), 10-22 July 2006. [41] www.pwscf.org/pseudo/1.3/UPF/Ag.pbc-d-rrkjus.UPF; www.pwscf.org/pseudo/1.3/UPF/B.pbc-n-van.UPF. [42] Islam, A. K. M. A.; Islam, F. N.; and Kabir, S. J. Phys.: Cond. Matter. 2001, 13, L641. [43] Islam, A. K. M. A.; Parvin, F.; Islam, F. N.; Islam, M. N.; Islam, A. T. M. N.; Tanaka, I. Physica C 2007, 466, 76. [44] Parvin, F.; and Islam, A. K. M. A. – Paper to be submitted (2007). [45] Shein, I. R.; Medvedeva, N. I.; and Ivanovskii, A. L. 28 Dec 2004 arXiv: condmat/0412426, v2. [46] Murnaghan, F. D. Proc. Natl Acad. Sci. USA 1944, 30, 244. [47] Perottoni, C. A.; Pereira, A. S.; and da Jornada, J. A. J. Phys.: Cond. Matter 2000, 12, 7205. [48] Jorgensen, J. D.; Pei, S.; Lightfoot, P.; Hinks, D. G.; Veal, B. W.; Dabrowski, B.; Paulikas, A. P.; and Kleb, R.; Physica C 1990, 171, 93. [49] Prassides, K.; Iwasa, Y.; Ito, T.; Chi, D. H.; Uehara, K.; Nishibori, E.; Takata, M.; Sakata, S.; Ohishi, Y.; Shimomura, O.; Muranaka, Y.; and Akimitsu, J. 2001 arXiv: cond-mat/0102507. [50] Zhou, O.; Vaughan, G. B. M.; Zhu, Q.; Ficher, J. E.; Heiney, P. A.; Coustel, N.; McCauley, J. P.; and Smith, A. B.; Science 1992, 255, 835. [51] McMillan, W. L. Phys. Rev. 1968, 167, 331. [52] Allen, P. B.; and Dynes, R. C. Phys. Rev. B 1975, 12, 905. [53] Goncharov, A. F.; Struzhkin, V. V.; Gregoryanz, E.; Hu, J.; Hemley, R. J.; Mao, H.; Lepertot, G.; Bud’ko, S. L.; Canfield, P. C. Phys. Rev. B 2001, 64, 100509. [54] Loya, I.; and Syassen, K. Solid State Commun. 2001, 118, 279. [55] Markel, S.; Goncharov, A. F.; Mao, H. K.; Gillet, P.; and Hemley, R. J. Science 2000, 288, 1626. [56] Islam, A. K. M. A.; Islam, F. N.; and Islam, M. N. Phys. Lett. A 2001, 286, 357. [57] Islam, F. N.; Islam, A. K. M. A.; and Islam, M. N. J. Phys.: Cond. matter. 2001, 13, 11661. [58] Vogt, T.; Schneider, G.; Hriljic, J. A.; Yang, G.; and Abell, J. S. Phys. Rev. B 2001, 63, 220505. [59] Kortus, J.; Mazin, I. I.; Belashchenko, K. D.; Antropov, V. P.; and Boyer, L. L. Phys. Rev. Lett. 2001, 86, 4656. [60] Dahm, T. 2004 arXiv: cond-mat/0410158 v1.; Malisa, A. Ph.D. Thesis, Chalmers University of Technology, © A Malisa 2005, ISBN: 91-7291-565-X. [61] Neaton, J. B.; and Perali, A. arXiv: cond-mat/0104098. [62] Wan, X.; Dong, J.; Weng, H.; and Xing, D. Y.; Phys. Rev. B 2002, 65, 12502.

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[63] Tissen, V. G.; Nefedova, M. V.; Kolesnikov, N. N.; and Kulakov, M. P. Physica C 2001, 363, 194. [64] Kong, Y.; Dolgov, O. V.; Jepsen, O.; and Andersen, O. K. Phys. Rev. B 2001, 64, 020501 (R). [65] Bohnen, K. -P.; Heid, R.; and Renker, B. Phys. Rev. Lett. 2001, 86, 5771. [66] Liu, A. Y.; Mazin, I. I.; and Kortus, J. Phys. Rev. Lett. 2001, 87, 087005. [67] Liu, A. Y.; Quong, A. A. Phys. Rev. B 1996-II, 53(12), R7575; Liu, A. Y.; Quong, A. A.; Freericks, J. K.; Nicol, E. J.; and Jones, E. C. Phys. Rev. B 1999-II, 59(6), 4028. [68] Ivanovskii, A. L. Phys. Solid State 2003, 45, 1829. [69] Singh, Probhakar P. Solid State Commun. 2003, 125(6), 323. [70] Islam, A. K. M. A.; and Islam, F. N. Int. J. Modern Physics B 2003, 17, 3785. [71] Choi, H. J.; Roundy, D.; Sun, H.; Cohen, M. L.; and Louie, S. G. Phys. Rev. B 2002, 66, 020513. [72] Gladstone, G.; Jensen, M. A.; and Schrieffer, J. R. in Superconductivity, Parks, R. D.; Ed.; Marcel Dekker, New York, 1969, Vol. 2, p.665. [73] Berk, N. F.; and Schrieffer, J. R. Phys. Rev. Lett. 1966, 17, 433. [74] Schrieffer, J. R. J. Appl. Phys. 1968, 39, 642. [75] Bennemann, K. H.; and Garland, J. W. in superconductivity in d- and f-band Metals, Douglass, D. H.; Ed.; Plenum, New York, 1971. [76] Rietschel, H.; and Winter, H. Phys. Rev. Lett. 1979, 43, 1256. [77] Jarlborg, T. Phys. Scr. 1988, 37, 795. [78] Christensen, N. E.; and Novikov, D. L. Phys. Rev. Letters 2001, 86(9), 1861. [79] Wald, F.; and Stormont, R. W. J. Less-Common Met. 1965, 9, 423.

In: Superconductivity Research Developments Editor: James R. Tobin, pp. 93-123

ISBN: 978-1-60021-848-4 © 2008 Nova Science Publishers, Inc.

Chapter 4

PERSPECTIVES OF SUPERCONDUCTING TEMPERATURE INCREASE IN HTSC COPPER OXIDES Svetlana G. Titova1 and John T. S. Irvine2* 1

Institute of Metallurgy, Urals Division of Russian Academy of Sciences 2 School of Chemistry, St Andrews University

Abstract At the review part the influence of various structure parameters on the temperature of superconducting transition Tc are considered. The original results devoted to this topic are presented in the next chapter. Third part presents low temperature X-ray and neutron powder diffraction study for Bi- and Hg-based HTSC cuprates, where three different structure anomalies at temperatures T0~Tc+15 K, T1~160 K and T2~260 K are established and their origin is discussed. It is shown that the structural anomaly at T0, in vicinity of Tc, is connected with “quasi-ferroelectric” distortion of CuO2-planes and is a sign of presence of corresponding soft phonon mode. It is shown that T0 is linear function of Tc when optimally doped compounds for different systems are compared. This fact means that the mechanism of superconductivity of HTSC cuprates must involve the electron-phonon interaction. Systematic analysis of crystal structure features as function of temperature shows an enhancement of thermal atomic vibration amplitudes and compression of apical bond in the temperature interval T1-T2. The whole complex of observed data is interpreted as result of localization of part of charge carriers at participation of lattice deformation in temperature interval ~160-260 K. Independence of this interval from charge carrier concentration and even chemical composition of HTSC compound confirms this interpretation. Low temperature border of this interval, connected with delocalization of charge carriers, determines the maximal possible Tc value for HTSC cuprates.

Introduction All HTSC copper oxides have strong dependence of properties on concentration of charge carriers which is determined by a degree of chemical doping and it is achieved by means of * †

E-mail address: [email protected]. Current address: Institute of Metallurgy UrD RAS, Ekaterinburg 620016, Russia. (Corresponding author)

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non isovalent cation substitution or oxygen non-stoichiometry. For example, partial substitution of a trivalent La in La2CuO4 by bivalent Sr, this material, being an antiferromagnetic insulator without Sr, becomes superconducting. The generalized scheme of the phase diagram in coordinates temperature-doping degree is resulted in Fig. 1.

Figure 1. Scheme of phase diagram of HTSC cuprate compounds p-T, p- concentration of holes, T – temperature, after [1].

The common and conventional now for all HTSC cuprates is presence of temperature of antiferromagnetic ordering TN which sharply decreases at increase of doping state, doping dependence for critical temperature Tc in form of a symmetric parabola with branches downwards and curve Т*, so-called "pseudo-gap" or “spin gap”. The maximum of Tc corresponds so-called “optimum” level of doping. Feature Т* is visible on various physical properties, such as tunnel spectroscopy, ARPES, nuclear magnetic resonance, transport properties, thermal capacity, Raman scattering, magnetic neutron diffraction, etc. [3]. In all cases for materials at Т<Т* presence of antiferromagnetic correlations is characteristic. However, in opinion of some authors (for example, [1]) these correlations appear at more higher temperature Tmax (see Fig. 1). Experimentally observable below Т* suppression of density of spin states according to nuclear magnetic resonance and inelastic neutron scattering [2-4] can be interpreted differently: as presence of bosons without reach of superconductivity because of absence phase coherence or their too small concentration, or as effect of "phase separation” - occurrence of specific state with fractions with non-uniform concentration of charge carriers [3-4]. The question on origin of effect of "pseudo-gap" is extremely important for understanding of the mechanism of superconductivity of HTSC materials. In BCS model of superconductivity the approach of "an average field” is used and the superconducting gap 2D at T<
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between Tc and Т*, however such correlation can be connected with their common character of dependence on charge carrier concentration. It is known, that Т* is sensitive to isotope substitution. Most brightly this effect was shown for HoBa2Cu4O8 and Er2Ba4Cu7 in underdoped state at replacement of 16O on 18O [5]. The pseudo-gap opens at 170 K and 220 K for samples with 16O on 18O, respectively, that corresponds to isotope shift ΔT*>>50 K. Such a significant isotope effect demonstrates the doubtless contribution of phonon subsystem in formation of a pseudo-gap state, as, by the way, in all other properties. Generally speaking, concepts of formation of the pseudo-gap can be divided on two big categories: 1) the pseudo-gap state is a harbinger of superconductivity, but macroscopical phase coherence is destroyed by thermal fluctuations; 2) the pseudo-gap is not related to superconductivity and is an attribute of presence of any other type of ordering (for example, spin or charge). The basic complexity of experimental studying of a pseudo-gap state may be explained by difficulty to divide a superconducting gap and a pseudo-crack at low temperatures. For suppression of superconductivity it is possible to use high magnetic fields, but it creates technical difficulties at spectral researches. Alternative may be impurity, for example, Zn or Ni, replacing copper in CuO2-layers. The optical conductivity along c-axis was measured for (Sm,Nd)Ba2{Cu1-y(Ni,Zn)y}3O7-δ single crystals (analog of YBCO-123) [6]. The content of not magnetic Zn and magnetic Ni impurity up to 9% and 17%, accordingly, is enough to destroy superconductivity even in optimally doped by oxygen samples. It was revealed, that while impurity suppress superconductivity, they absolutely differently influence a pseudo-gap: the increase of Zn concentration leads to smooth reduction of a pseudo-gap whereas additive Ni leads to its sharp increase. Besides Cu/Ni substitution leads to existence of a pseudo-gap even in optimally doped and overdoped samples where it earlier was absent. Thus, clearly, the pseudo-gap and superconductivity are not connected, and connection of a pseudo-gap state and impurity heterogeneity becomes obvious. To the same conclusion the authors of work [7] have come; by means of the complex analysis of optical and kinetic properties it is shown, that the pseudo-gap behavior is common both for hole and electronic HTSC cuprates and is connected with antiferromagnetic spin correlations. Up to now the origin of pseudo-gap is under investigation. As there is no conventional interpretation of the phase diagram of HTSC materials so there is no also a uniform theoretical model of the phenomenon of their superconductivity, though, certainly, there are plural attempts to create such theory. Among the most perspective we should allocate various updatings of BSC model, the BE-models with interpretation of charge carriers as polarons or bipolarons (Alexandrov A.S. [8]), the model of spin fluctuations, RVB, marginal Fermiliquid, SO5 new order parameter, stripe mechanism, various variants of QCP - quantum critical points, etc. Analysis and comparison of these theories exceeds the topic of this chapter, these questions are considered in a number of reviews, see for example, [2, 9-13]. At the same time, as it was already mentioned, it is necessary to note, that strong electronphonon interaction has influence on behavior of system in any case because HTSC cuprates are systems with strongly connected electron, spin and phonon subsystems. The doubtless importance of phonon contribution was shown at research of influence of doping state on lattice softening for La1-xSrxCuO4 [14]. Softening is more pronounced for high energy longitudinal optical phonon (LO), concerning to Cu-O bond in a CuO2-plane. In figure 3 the experimental dispersion curves obtained by inelastic scattering of neutron and

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synchrotron radiation for single crystal La1-xSrxCuO4 are shown. In right part of the same figure, the concentration dependence of resulting phonon softening calculated as a difference in energy between the maximal and minimal value on the left part of the figure is shown. It is visible, that this curve has a maximum on a linearly increasing background, conterminous with optimum doped state. Such a direct link of phonon mode softening and Tc value allows assumption about necessity to consider the phonon contribution in the mechanism of superconductivity and so, to consider the structure parameters important for superconducting properties, for HTSC cuprates.

Figure 2. Dispersion of LO phonon for Cu-O bond for various Sr content in La1-xSrxCuO4 obtained by inelastic scattering (left) and resulting concentration dependence of LO phonon softening as function of doping state. After J. Mizuki, T. Fukuda and K. Yamada [14].

1. Crystal Structure Parameters Determining Tc (Review) The crystal structure for HTSC compounds may be interpreted as an alternation of layers with NaCl and perovskite structure. Examples of structures of the some HTSC phases are shown in Fig. 3. As usual for perovskites, HTSC cuprates can have high concentration of defects, in particular, vacancies on oxygen sub-lattice, shifts of atoms from their positions etc. Other prominent feature is the fact, that these materials even being high quality single crystals are not "homogeneous" though traditional methods cannot find out this inhomogeneity. The information on inhomogeneous state can be obtained, for example, using high resolution STM spectroscopy, see Fig. 4. The relation between defects due to chemical inhomogeneity because of oxygen non-stoichiometry or non-isovalent substitution and microscopic charge inhomogeneity is well established [15,16]. As all HTSC cuprates have relatively similar crystal structure, a question, which features are responsible for a high Tc value in a superconducting state, has appeared at once after discovering of these materials. As parameters - structural characteristics for cuprates - the degree of anisotropy, transition from orthorhombic to tetragonal phase, defect state of

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structural fragments, in particular the distortion of perovskite block of the compressionstretching type and others were chosen. In particular, studies of real structure for cuprates show, that CuO2-planes are not "flat", and represent double anion-cation block (naturally, this effect does not exist in materials with single CuO2-layer into bi-pyramid CuO6). The reason of this splitting or «buckling», according to S. Shilshtein [17], is the balance of an electric charge for cations with variable valency into MO-layer and copper into CuO2-layers. Also for various HTSC cuprates S.Sh. Shilshtein established that growth of Tc correlates with growth of splitting of Ba-O layer, which in turn, is almost in inverse proportion to buckling of CuO2layers. Thus, at comparison of different materials in optimally doped state it seems established, that the increase of buckling leads to Tc downturn.

Figure 3. Schematic crystal structure for HTSC compounds: Pb(1+x)/2Cu(1-x)/2Sr2(Y1-xCax)Cu2Oy (Pb,Cu-1212, left), HgBa2CuO4+δ (Hg-1201, straight) and Hg0.8Tl0.2Ba2Ca2Cu3O8+δ (Hg,Tl-1223, right). The following atoms are shown as color circles: (Y,Ca)–green, (Ba,Sr) – yellow, oxygen – red, (Hg,Tl,Pb,Cu) – blue. Copper-oxygen structural fragments are shown as red polyhedrons; apical oxygen is placed in the tops of the pyramids.

O. Chmaissem, J.D. Jorgensen et al. [18] investigated interrelation between Tc and buckling in the same compound (La1-xCax)(Ba1.75-xLa0.25+x)Cu3Oy varying the doping state. At 0<x<0.4 this material is single phase and supposes change of oxygen content in very wide range, down to y=7.3. Tc value for samples with various x and y varies from 0 up to 80 T K, and even at very low Tc~10 K the width of superconducting transition makes only few degrees that testifies to high quality of samples and their single-phase condition. The result has appeared opposite. It has been found, that at constant x the Tc values and measure of buckling state (α, an angle for O-Cu-O bond) have the same parabolic shape as functions of oxygen content y determining the charge carrier concentration and reach a maximum at the same y=7.15 (at x=0.1 αmax=6.3° and Tcmax=50 K; at x=0.4 αmax=5° and Tcmax=80 K). As a possible explanation of the found out correlation between Tc and buckling authors [18] assume, that structure distortion defines the shape of peak of density of electronic states near to Fermi level: in process of approach of optimal degree of doping the density of states at

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Fermi level grows, therefore Tc increases; deformation of CuO2-layers thus varies so that to lower and/or widen peak of density of states and to lower that full free energy of system. Thus, unlike low temperature superconductors of A15 type, the superconducting state appears energetically more favourably structural instability and consequently transition to dielectric phase does not occur. Authors note, that if thus it would be possible to decrease the buckling independently of doping, the maximal possible temperature of transition to a superconducting state could be essentially higher.

Figure 4. 56×56 nm2 map in real space of conducting (light spots) and insulating (dark spots) fractions for BSCCO-2212 obtained using STM from a surface of single crystals below 30 K. Upper – underdoped sample with Tc≤30 K, below - almost optimally doped sample with Tc=84 K. Reprinted by permission from Macmillan Publishers Ltd: Nature [15], copyright 2002.

Openers of high-temperature superconductivity I.G. Bednorz and K.A. Muller wrote [19], that leading idea in creation of the general concept of search of materials with high Tc values was superconductivity in system with polarons, to say exactly, Jahn-Teller (JT) polarons, theoretically developed by B.K. Chakraverty [20]. The Jahn-Teller theorem says that the nonlinear molecule or the molecular complex, having degeneration of electronic levels, will be spontaneously distorted, aspiring to take off this degeneration. This effect is shown in the complexes containing specific ions of transition metals with certain valency. At enough high energy of JT distortion the tendency to electron localization grows, at EJT equal on size to width of a band of localized states, there is formation of JT-polarons. (Ion Cu2+ is JT center, therefore CuO-octahedrons in systems Bi-2201 and Hg-1201 are distorted, extended in cdirection. In materials, where Cu-O fragments look like pyramids (see Fig. 4) it is impossible to say about JT effect as planar and apical Cu0O bond lengths are different because of the symmetry. In this case sometimes speak about "pseudo JT effect" connected with distortion in

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CuO2-plane. Really, E.V. Antipov et al [21] wrote, that lengthening of apical bond length, i.e. height of CuO5-pyramid correlates with Tc growth. Apical distance determines charge transfer from so-called “reservoir” layer to “superconducting” CuO2-layer, and its growth should reduce such a transfer. However at the same time at increase of apical distances the degree of hybridization between d x 2 − y 2 and d z 2 − r 2 copper orbitals decreases, that leads to the best development for important for charge transfer into CuO2-plane Cu d x 2 − y 2 -Opx,y hybridized states. Thus values of unit cell parameters, i.e. and cell volume as a whole, as a rule, decrease at increase of oxygen content and, accordingly, concentration of holes, see Figs 5-7. Reduction of cell volume at oxygen doping leads to sharp growth of conductivity [25]. Thus optical, kinetic properties [25] and concentration dependence of the attitude n/m* [26] where m* - effective mass of a hole, show, that the most adequate interpretation requests at least two-band model, i.e. assumption of presence of two types of holes: "heavy" and "normal". It is possible to assume, that distortion of structure in form of volume compression is caused with "heavy" holes, which concentration increases at doping. Other possible mechanism is change of a degree of oxidation for both cations and oxygen.

Figure 5. The unit cell parameters of YBCO-123 as function of oxygen content, according to R.J. Cava et al. [22]. Sharp reduction of c-parameter leads to reduction of cell volume, despite of small growth of b-parameter. Reprinted from [22], with permission from Elsevier.

Figure 6. Tc vs cell c-parameter (left), vs oxygen content (straight) and c-parameter as function of oxygen content (right) in Bi2Sr2CaCu2Oδ. After [23].

Oxygen content, (x) Figure 7. Unit cell a-parameter as function of oxygen content (left) and Tc as function of a-parameter for Hg-based materials HgBa2Can-1CunO2n+2+δ, n=3 and 4. Reprinted from [24], with permission from Elsevier.

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The analysis of properties shows (see for example, the review [27]), that HTSC cuprates cannot be considered as two-dimensional, i.e. the account of interaction between "superconducting" and other structural fragments, and also interaction of "superconducting" fragments among themselves is necessary. So there is a question on influence of number of CuO2-planes per unit cell on Tc. This question has been in detail examined in refs. [24, 28] for Hg- and Bi-based compounds. The greatest Tc values may be obtained materials with three CuO2-planes per unit cell. It is necessary to note, that Bi-based HTSC cuprates demonstrate incommensurate structural modulations because of some mismatch between BiO- and CuO2-blocks (see, for example, [29-30]). Conditions of synthesis determine type of structure modulation, moreover, it is possible to choose temperature area, heat treatment in which suppresses modulations. It seems there is not a link between Tc and structural modulations, some correlation is possible because both these parameters are functions of the temperature of synthesis. As the modulations are not characteristic for HTSC cuprates as class of objects, we shall not consider them as the structural factor determining Tc. Summarizing the aforesaid, it is possible to conclude, that a material possessing high temperature of transition in a superconducting state, should have three CuO2-planes per unit cell, these planes should be practically flat (minimization of buckling), long apical Cu-O distance and to contain heavy metals with variable valency (Bi, Tl, Pb, Hg).

2. Comparison of Crystal Structure Parameters for Isostructural Materials of 1212 Type In previous part, we summarized the structural criteria for high Tc value; let us now examine isostructural HTSC materials with purpose to establish those features of structure parameters, which at preservation of the common structure motive provide higher Tc. We will consider so called 1212 compounds having general formula M1Me2Y1-xCaxCu2O7-δ, where M=Hg, Pb, Cd, Cu or their mixes, Me=Sr or Ba. These materials have three various mechanisms of doping: varying of oxygen content, non-isovalent substitution for M-site, and Y/Ca replacement. Mentioned above Pb,Cu-1212 are the representative of this family, also widely known material YBCO-123 here concerns. In 1212 structure each copper atom in CuO2planes coordinated by five oxygen atoms, forming CuO5-pyramids. Distinction between YBCO-123 and the others 1212 structures consists of construction of MO-layer: oxygen takes of (0, ½, 0) site in YBCO-123 and (½, ½, 0) site in the others 1212. Despite of crystallographic similarity, 1212-materials have very various Tc vaues, see Table 1 and Fig. 8. We have analyzed the following systems (see Table 1): Hg-1212 - HgBa2CaCu2O6+δ; Tl,Pb1212 - (Tl0.5Pb0.5)Sr2Y1-xCaxCu2O7; YBCO-123 - CuBa2YCu2O7; Pb,Cu-1212 - Pb(1+x)/2Cu(1x)/2Sr2Y1-xCaxCu2O7+δ. The results are summarized in Fig. 9. As we already mentioned, decrease of buckilng of CuO2- planes, growth of degree of splitting of MeO-layer and apical distance should result in growth of Tc. However the materials with Tc=128 K and 40 K, Hg-1212 and Pb,Cu-1212 have relatively close parameters, having highest and lowest Tc values among considered systems. Apical distance for Pb,Cu-1212 is even higher than for YBCO-123 and Tl,Pb-1212. To understand this

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behavior we will study the evolution of structure parameters at cooling for these two materials.

Figure 8. Tc values for different 1212 materials. Reprinted from [31], with permission from Elsevier.

Table 1. Characteristics of studied 1212 compounds

System (notation)

Tc, K

Unit cell parameters, Å a, b

c

dMeO, Å dCuO, Å

dapical, Å

HgBa2CaCu2O6+δ (Hg-1212) (Tl0.5Pb0.5)Sr2Y1-xCaxCu2O7 (Tl,Pb-1212) CuBa2YCu2O7 (YBCO-123)

128 107

3.860 3.804

12.67 12.05

0.74 0.50

0.06 0.24

2.76 2.20

92

11.68

0.30

0.26

2.30

Pb(1+x)/2Cu(1-x)/2Sr2Y1-xCaxCu2O7+δ (Pb,Cu-1212)

40

3.823 3.887 3.816

11.86

0.53

0.15

2.34

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Figure 9. Structure parameters for 1212 type HTSC materials in optimally doped state as functions of superconducting transition temperature. Notations: 1 - splitting of MeO-layer dMeO=c⋅⎜zMe-zO⎜, where O – oxygen corresponding to MeO-block; 2 – apical distance (right axis); 3 splitting of CuO2-layer dCuO= c⋅⎜zCu-zO⎜, where O – oxygen corresponding to CuO2-plane; 4 – unit cell a-parameter; 5 – unit cell cparameter (the right axis). All dimensions are in angstroms.

Pb,Cu-1212 and Tl,Pb-1212 compounds were prepared using the procedure described in literature [32,33]. The characteristics of Pb,Cu-1212 samples with different doping state are listed in the Table 2. The crystal structure was studied using high-resolution neutron diffraction for Pb,Cu-1212, temperature range 300-4.5 K, ISIS, RAL, HRPD [34], time-of– flight mode, 30÷120 msec range, backscattering detector; and X-ray diffraction for Tl,Pb1212, temperature range 100-300 K, STOE diffractometer, Cu Kα1-radiation, graphite monochromator, 2ϑ range 20÷90°, reflection mode, the liquid nitrogen cryostat with He-filled sample chamber. The measurements were performed on cooling, for neutron diffraction the exposure time was ~15-20 min, while at room temperature and few temperature points we used higher exposure time up to 2 hours for possibility to get a good quality structure start model. An average exposure time for X-ray diffraction was ~70 min. Crystal structure refinement was carried out using GSAS [35] program. First order Chebyshev polynomial with 8-12 parameters was used for background approximation. Atomic form factor was taken as f = xT ' ' ( f 0 + f '+ f ' ' ) for X-ray diffraction and f = xT ' ' (b0 + ib' ) for neutron diffraction, where x- atomic site fraction, fo and bo are atomic scattering amplitudes, f ' , f ' ' 2 ⎡ ⎤ and b' -corrections for anomalous scattering, T ' ' = exp ⎢ − 8π 2U iso ⎛⎜ sin ϑ ⎞⎟ ⎥ , where Uiso – ⎝ λ ⎠ ⎥⎦ ⎢⎣ isotropic temperature factor. As we treated the powder data for compounds with several different atoms per unit cell, the thermal parameters Uiso were calculated in isotropic form. The profile pseudo-Voight function was used for diffraction line shape approximation. For neutron diffraction with exposure time 15 min the weight profile discrepancy factors Rwp~7-8 % were obtained, when exposure time was 2 hours Rwp becomes ~4.5 %. Unit cell parameters were determined within accuracy range Δa = 1.2*10-5 Å, Δc =1.7*10-5 Å. X-ray diffraction data are characterized by Rwp~6-8 %, Δa=1*10-3 Å, и Δc=2*10-3 Å. The values of calculation

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quality χ2 were about 1.5-3.5 for all cases. The Pb,Cu-1212 samples contained less than 3% impurity phase SrCuO2. As an initial, the following crystal structure [32] was taken, space group P4/mmm, atomic positions are the following: Pb/Cu1 – (0.055 0 0); Y/Ca – (½, ½, ½); Sr – (½,½, 0.2087); Cu2 – (0,0,0.3613); O1 – (0, 0, 0.1642); O2 – (0, ½, 0.3740); O3 – (0.3323, ½, 0). For Tl,Pb-1212 compounds the following structure model was used: space group P4/mmm, Tl/Pb – (0 0 ½); Y/Ca – (½ ½ 0); Sr – (½ ½ 0.284); Cu2 – (0 0 0.1255); O1 – (0 0 0.342); O2 – (½ 0 0.121); O3 – (½ ½ 0.5). The examples of experimental, calculated and difference diffraction patterns for studied samples are shown in Fig. 10. Table 2. Characteristics of Pb,Cu-1212 samples Sample ISIS2 ISIS3 ISIS4

Doping state Underdoped Underdoped Optimally doped

Ca content, x 0.2 0.3 0.4

Oxygen content, y 7 7 7

Tc, K 0 19 37

Figure 10. The experimental, calculated and difference neutron diffraction patterns for Pb,Cu-1212 ISIS3 sample at T= 80 K, the exposure time 15 min. (left), X-ray diffraction at room temperature for optimally doped Tl,Pb-1212 sample (right).

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Temperature evolution of structure parameters will be considered in next part, now we can say that most prominent difference in temperature evolution of structure parameters between studied materials is the behavior of apical oxygen O1. In Fig 11 we plot the calculated values of splitting of SrO-layer (Me-O). At room temperature this parameter for Tl,Pb-1212 material with higher Tc is less than for Pb,Cu-1212. The value of dSrO remains the same for Pb,Cu-1212 and this parameter is not function neither doping state nor a temperature. While for Tl,Pb-1212 material with higher Tc below T1~160÷180 K there is significant reduction of dSrO. Naturally, change of this parameter correlates with change of apical bond length. I.e., as follows from these data, apical bond should be not only long, but also enough "soft" – for compounds with high Tc it should have high compressibility both under cooling and pressure increase. 0,60 0,55 0,50

1

dSrO, (A)

0,45 0,40 0,35 0,30

2

0,25 0,20 0,15 0

50

100

150

200

250

300

Temperature (K) Figure 11. Splitting of SrO-layer for Pb,Cu-1212 (1) and Tl,Pb-1212 (2). There are the samples with different doping state: without doping (light rhombuses), underdoped (filled rhombuses) and near optimally doped (filled circles).

Close connection between high compressibility for apical Cu-O bond and influence of external pressure on the Tc value is noted as well in the literature [36]. Application of external hydrostatic pressure is known, that, leading to growth of Tc, leads also to sharp reduction of the relation unit cell parameters c/a [37]. To compare, how much increases c-parameter at doping for different HTSC phases, we have cited corresponding data in Fig. 12. From data of Fig. 12 it is visible, that growth of Tc is practically linearly connected with difference for cparameter in optimally doped (copt) and undoped (c0) state. And for all these phases, as we already mentioned, the increase in of doping state leads to reduction of c-parameter. Thus, really, high values of temperatures of transition in a superconducting state are connected with lattice softness in direction of c-axis, and especially with high compressibility for apical copper-oxygen chemical bond.

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(c0- copt), A

0,13

4

0,12 0,11

3

0,10 0,09 0,08

1

2

90

100

0,07

110

120

130

140

Tc , K Figure 12. Decrease of unit cell parameter c at doping, from undoped p=0 to optimally doped state p=popt as function of Tc for different systems: 1 - YBCO-123 [22]; 2 - Hg-1201 [38-39]; 3 - Hg-1212 [40-41]; 4 - Hg-1223 [36].

3. Crystal Structure Feature for HTSC Cuprates in Temperature Range 100-300 K Among the first works devoted HTSC were studies of different properties (adiabatic calorimetry, ultrasonic, dilatometric and structure researches, see [42-55] in temperature range between 300 K and Tc. Mainly these results correspond to pseudo-gap feature, but except them there were the data about structural instability both near T0~Tc +15 K [48-50,5355], near T1~140÷180 K and T2~220÷260 K [42, 44, 46-48, 50, 52]. The structural anomaly at T0~140 K which has not a shift while Tc varies was studied by A.I. Akimov et al. [48] using X-ray diffraction for Tl-based 2212 samples, see Fig. 13. Except T0-anomaly they found weak increase of thermal expansion coefficient in temperature range T1÷T2, these features disappear for non-superconducting samples. As one of the first it is possible to note the work of M. François with co-authors [44] when authors studied temperature evolution of unit cell parameters in atomic coordinates for ceramic sample YBa2Cu3O6.90 using neutron diffraction in temperature interval 5-320 K and have found out small anomaly of temperature dependence of all structure parameters close 90 K and 240 K. Below 90 K temperature dependence for unit cell parameters becomes very weak, same it is possible to tell about isotropic thermal vibration amplitudes Uiso. Small Uiso values as well as small TEC [48] below T0~Tc+15 K mean high whole lattice hardness. Bond lengths weekly depend on temperature, except for Ba-O4 (О4- apical oxygen), that correlates with stronger dependence c(T), than a(T) or b(T). Authors explained the weak structure anomalies at 90 and 240 K as possible ordering of oxygen atomic displacements in positions into zigzag Cu-O chains. A number of anomalies for different HTSC compounds were observed using ultrasound attenuation [42,43,46,51-52] in the temperature range Tc-300 K. These anomalies at

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temperatures T1~160÷180 K and T2~ 230÷260 K had a weak frequency dependence what means they are not accompanied by activation process [56] being rather connected with order-disorder second order phase transition [57]. The attempts to study this phase transition using high resolution X-ray and neutron diffraction were not successful up to now. Recently we found that HTSC compounds Bi-2212 (Bi2Sr2CaCu2O8.10) and Hg-1223 (Hg0.8Tl0.2Ba2Ca2Cu3O8.15) being near optimally doped (OD) state demonstrate a minimum of temperature dependence of unit cell constants at temperature T2 and negative thermal expansion occurs in temperature range between T1 and T2 [58-59]. The T1 and T2 anomalies were detected for practically all HTSC cuprates and even for non-superconducting antiferromagnetic relative compounds CuO and Y2BaCuO5 containing Cu2+-O clusters [42,56] in spite of significantly special behavior with negative thermal expansion was observed only for OD Hg-1223 and Bi-2212 up to now. Therefore, it was a motivation to search the common features of temperature evolution of crystal structure for a wide number of HTSC cuprates using high resolution X-ray and neutron diffraction in temperature range between Tc and room temperature.

(b)

Figure 13. Linear thermal expansion coefficients (TEC) along a- and c-axis for ceramic sample Tl2Ba2CaCu2O8+δ; the change 1→4 corresponds to decrease of Tc from 125 to 0 K due to thermal treatment. After [48].

The first group of studied materials, Pb(1+x)/2Cu(1-x)/2Sr2(Y1-xCax)Cu2Oy, belongs to so called 1212 type compounds considered in previous part. Maximal possible temperature of superconducting phase transition Tc for OD-compounds of Pb,Cu-1212 system is ~40 K. The materials are chemically stable in concentration range 0≤x≤0.5, increase of oxygen content above 7.0 makes the materials overdoped (OvD) by charge carriers and suppress superconductivity [32]. The second considered group, Hg-based compounds HgBa2CuO4+δ (Hg-1201) and Hg0.8Tl0.2Ba2Ca2Cu3O8+δ (Hg,Tl-1223) attracts a significant interest due to a

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record high values of Tc~134 K for OD Hg,Tl-1223 at ambient pressure. To obtain the oxygen content corresponding to OD state we annealed the samples at 300°C in oxygen during 5 hours. Pb,Cu-1212, Hg-1201 and Hg,Tl-1223 compounds were prepared using the procedure described in literature [32,60,33]. The characteristics of Pb,Cu-1212 samples were already described in previous part. The Hg-1201 sample contained impurity phases Ba2Cu3O5+x, BaCO3 and HgO, summary amount less than 5%. Tc values for as prepared (UD) and OD samples were 85 K and 97 K, respectively. Oxygen content for UD and OD sample was estimated comparing lattice constants with reference data [61] as 4.03 and 4.08, respectively. As an initial, the following crystal structure was taken: space group P4/mmm, atomic positions are the following: Hg – (½ ½ 0), Ba - (0 0 z), Cu - (½ ½ ½), O1 - (0 ½ ½), O2 - (½ ½ z), O3 - (0 0 0). The Hg,Tl-1223 compounds contained less than 5 % impurity phase BaCuO2. The values of Tc before and after annealing in oxygen atmosphere were 127 K and 133 K for UD and OD samples, respectively. The oxygen content was estimated as 8.10 and 8.33 for UD and OD samples, respectively. The following crystal structure model was applied as an initial for the refinement, space group P4/mmm, atomic positions are the following: Hg/Tl - (0, 0, 0), Ca - (½ ½ z), Cu1 - (0 0 ½), Cu2 - (0 0 z), Ba - (½ ½ z), O1 - (0 ½ ½), O2 (0 ½ z), O3 - (0 0 z), O4 - (½ ½ 0). Methods of high-resolution neutron and X-ray diffraction are described in part 2. Crystal structure refinement was carried out using GSAS [35] program. Experimental, calculated and difference diffraction patterns for studied samples are shown in Fig. 14, except Pb,Cu-1212 sample which is shown in Fig. 10.

a Figure 14. Continued on the next page.

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c Figure 14. The experimental, calculated and difference X-ray diffraction pattern at room temperature for UD Hg-1201 (a), neutron diffraction (b) and X-ray diffraction pattern (c) for OD Hg,Tl-1223, the impurity phase Ba0.98CuO2.07 [62] is taken into account, content ~9%, space group Im 3 m, unit cell parameter a=18.287 Å).

Unit cell parameters for Pb,Cu-1212 samples as function of the temperature are shown in Fig. 15. Calculated linear thermal expansion coefficient α c = dc c dT as a function of temperature is shown in Fig. 16. One can see that in spite of unit cell parameters are smooth monotonic functions of temperature, αc demonstrates a wide maximum between T1~140 and T2 ~250 K for all samples, what is very similar with data in Fig. 13 for Tl-2212 compounds. Temperature dependence of the neutron diffraction line width has an anomalous growth in temperature interval between T1 and T2. Increased line width for NMR data were also reported for the same temperature interval for the same Pb,Cu-1212 samples [63], see Fig. 17.

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a (A)

isis2

3,814 3,812 isis3 isis4

3,810 3,808 0

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300

Temperature, К

Temperature, K

Figure 15. Unit cell parameters a and c as function of temperature for Pb,Cu-1212 samples obtained from high resolution neutron diffraction.

αc⋅105, K-1

No structural anomalies were observed also by analysis of atomic coordinates and thermal vibration amplitudes for all atoms in the structure, including the temperature interval in vicinity of superconducting transition. Increased thermal expansion coefficient and linewidth for diffraction and NMR data may be a sign of defect state in defined temperature interval, the same for different state of doping, T1-T2. Into the error bars limit the atomic coordinates have not a temperature dependence, except Sr and apical oxygen O2, strongly connected with Sr, for OD sample ISIS4, see Fig. 18. Thermal vibration amplitudes slightly almost linearly decrease by temperature.

Temperature, K Figure 16. Linear thermal expansion coefficient αc as a function of temperature for Pb,Cu-1212. The lines are approximation by 5-th degree polynomial. The notations are the following: ISIS2 - filled triangles, ISIS3- open triangles, ISIS4 – circles.

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Figure 17. Left - linewidth for neutron diffraction (full width at half maximum FWHM, ISIS4 sample, open squares correspond to cooling, filled circles – heating). Right - the 89Y NMR linewidths [63] for ISIS3 and ISIS4 samples at right as functions of temperature.

One can see that shifts of Sr and O2 atoms (Fig. 18) are well correlated with each other, but these shifts are so small that do not affect the apical bond length, which decreases with temperature in accordance with c(T) behavior. We already discussed in previous part the hardness of apical bond as possible reason of relatively low Tc value for this particular material. The X-ray diffraction study was carried out for ISIS2 sample and in spite of less data precision the non-monotonic behavior for unit cell parameters was observed, Fig. 19. 0,1646

z (Sr, O2)

0,2090

0,1644

0,2088

0,1642

0,2086

0,1640 0,1638

0,2084 0,1636 0,2082

0,1634

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0,1632 0

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Temperature, K Figure 18. Atomic coordinates for Sr (filled circles, left axis) and apical oxygen О2 (open circles, right axis) for the ISIS4 sample as functions of temperature.

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11,805

с, А

11,800 11,795 11,790 11,785 11,780 11,775 100

150

200

250

300

Temperature, К Figure 19. Unit cell parameter c for ISIS2 sample from neutron diffraction (open symbols) and X-ray diffraction data.

Unit cell parameters for OD Hg-1201 as functions of temperature are shown in Fig. 20, the data for UD sample look very similar.

Figure 20. Unit cell parameters a (filled symbols) and c (open) for OD Hg-1201 as a function of temperature.

We see non-monotonic behavior for both unit cell constants what is in contrary with good quality data obtained by P. Bordet et al. [64] using neutron diffraction. From the X-ray data we can see a small minimum for both unit cell constants at T0~117 K, while neutron diffraction does not show such a feature, see Fig. 21, being very similar with smooth behavior for neutron diffraction data for Pb,Cu-1212 (Fig. 15).

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Т (К) Figure 21. Temperature dependence of Cu-O bond length into ab-plane obtained using neutron diffraction (squares) and EXAFS (circles). Reprinted from [64], with permission from Elsevier.

Calculated atomic coordinates for Hg-1201 well coincides with literature data [61]. Atomics coordinates for OD sample have not temperature dependence within accuracy in studied temperature range 100-300 K. Because of low resolution for oxygen in presence of heavy Hg and Ba atoms we were not able to determine the position of apical oxygen enough precisely to analyze it temperature dependence. Thermal parameters for Hg-atoms practically have not a temperature dependence and are equal to Uiso(Hg) = 0.010(5) Å and 0.040(5) Å for UD and OD samples, respectively. Possibly, higher temperature factor for the sample with higher oxygen content in HgOδ-plane is caused by static displacement of Hg from its position. Coordinate of Ba for OD sample is almost a constant zBa~0.2928(5) when temperature varies, while for UD sample this value decreases below T1≈170 K and has a complicate behavior between T1 and T2, as it is shown in Fig. 22. We can suppose that apical oxygen O2, strongly connected with Ba also demonstrates similar behavior with a minimum, similarly with behavior of Sr and apical oxygen for Pb,Cu-1212 compound (Fig. 18), Ba and apical oxygen for YBCO-123 [44]. Unit cell parameters for OD Hg,Tl-1223 sample are shown in Fig. 23. We see that at room temperature the parameters from both neutron and X-ray diffraction coincide well, but on cooling we observe quite different behavior; smooth monotonic shape for neutron diffraction and much more complicate shape for X-ray data: two maximums of unit cell constants are visible at T1~160 K and T2~260 K and a narrow minimum at T0~145 K. Calculated linear thermal expansion coefficients from neutron diffraction data plotted in Fig. 24 demonstrate a local maximum in temperature range T1-T2 similarly with data for PB,Cu1212 and Tl-2212 samples.

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2,04

c*(zCu-zBa), A

2,02 2,00 1,98 1,96 1,94 1,92 1,90 1,88 100

150

200

250

300

Temperature, К

Figure 22. Temperature dependence of the projection on c-direction of Cu-Ba distance for UD sample Hg-1201. a, A

c, A 15,92

3,870

15,90

3,865

15,88 3,860

15,86

3,855

15,84

3,850

15,82 15,80

3,845

15,78 100

Temperature, K

150

200

250

300

Temperature, K

Figure 23. Unit cell parameters a (filled symbols) and c (open symbols) for OD Hg,Tl-1223 sample from neutron (left) and X-ray diffraction (right) as functions of temperature.

Atomic coordinates and thermal parameters demonstrate very weak temperature dependence for neutron diffraction data and bright anomalous behavior for X-ray data, reported in [65] in studied temperature range. For neutron diffraction, the shift of Ba and apical oxygen towards CuO2-plane seems to be most visible in temperature range T1-T2 similarly with data for Sr and apical oxygen for Pb,Cu-1212, see Fig. 18. This effect is very small for neutron diffraction data and becomes much more pronounced from X-ray data; for comparison we plotted the apical distance dapical=c⋅(zO3-zCu2) obtained for the same sample with neutron and X-ray diffraction in Fig. 25. One can see that in spite of significant numerical difference, the shape of the apical distance change is similar; it has a local minimum in temperature range T1-T2. Similar behavior for Ba atoms and apical distance with minimum was reported for HgBa2CaCu2Ox [66], HgBa2CuO4-δ [64] and YBa2Cu3O7-δ [44]. So we can conclude, such a shift of Sr/Ba and apical oxygen between T2 and T2 is a common feature for HTSC cuprates. We should note that this feature might not be a sign of charge exchange between structural fragments because it has a reversible character by temperature.

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αc*10-5, K-1

2

1

0

0

50

100

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Temperature, K Figure 24. Linear thermal expansion coefficient αc as a function of temperature for OD Hg,Tl-1223 obtained from neutron diffraction data. 2,80

3,5

2,78

dapical, A

dapical, A

2,79

2,77

3,0

2,76 2,5

2,75 2,74 0

50

100

150

200

Temperature, K

250

300

100

150

200

250

300

Temperature, K

Figure 25. Temperature dependence of the apical distance for UD Hg,Tl-1223 for neutron (left) and Xray (right) diffraction data.

Diffraction line width demonstrates a growth in the same temperature range T1-T2 for Hg,Tl-1223 samples as well as for Hg-1201 and Pb,Cu-1212 materials from both neutron and X-ray diffraction data, see Fig. 26. It correlates with increase of thermal vibration amplitudes for all atoms in the structure and especially for apical oxygen, as it reported earlier [67]. These facts may be explained by disordered or/and defect state, which occurs in T1-T2 temperature interval. Most intriguing seems the behavior of Cu2 and O2 atoms belonging to “superconducting” planes in vicinity of superconducting transition. The minimum of unit cell constants at T0 visible with X-ray diffraction only is accompanied by shift of oxygen atoms from the plane and plane splitting, see Fig. 27. The direction of the shift is different for UD and OD sample, but the effect occurs at the same temperature range, as it was for Tl-2212 (Fig. 13). The weak structural anomaly in vicinity of Tc was reported in several papers where authors used either EXAFS or X-ray radiation but not neutron diffraction. The difference between X-ray data

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(synchrotron source) and neutron diffraction is well visible in Fig. 21, data of P. Bordet et al. [64] where smooth behavior for unit cell parameter for Hg-1201 was observed with neutron diffraction and a significant anomalous behavior with EXAFS data near 140 K. This difference P. Bordet et al. explain by local origin of data obtained using EXAFS. This explanation seems doubtful because the experiment time for EXAFS exceeds al least half an hour and the method is not a microprobe, so it will give the information averaged both by space and time. Actually, if at 140 K the local lattice displacement occurs and the average bond length separates into short and long values, it means that using neutron diffraction still the average bond length will be determined and using EXAFS both of them will be determined, not only high one as it was shown by P. Bordet et al.

FWHM, rel. units

1.0

0.8

0.6

0

50

100

150

200

250

300

Temperature, K Figure 26. Neutron diffraction data for Hg,Tl-1223 sample; linewidth (left) and isotropic thermal parameters Uiso for Cu2, O1 and O2 atoms (right, reprinted with kind permission from Springer Science and Business Media, after [65]) as functions of temperature.

Figure 27. Atomic coordinates for Cu and O atoms belonging to CuO2 “superconducting” planes for UD (left) and OD (right) Hg,Tl-1223 samples calculated using high resolution X-ray diffraction. The data after [65].

We explain the difference between X-ray and neutron diffraction data as photodeformation effect [68], because this anomaly as well as behavior at higher temperature up to 300 K is sensitive for radiation intensity for HTSC cuprates. The photo-deformation effect is

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known to be a property for ferroelectrics [69]. Possibly, the fact that HTSC cuprates demonstrate microscopic phase separation (see Fig. 4) and so they contain an insulating fraction may be a reason for influence of radiation of lattice distortion. The T0-anomaly as an appearance of “buckling” of CuO2-planes may be shifted by temperature by applying a higher intensity [68]. There is an important circumstance that the quasi-ferroelectric distortion vanishes when temperature drops towards Tc, that may be explained as suppression of ferroelectric order by superconducting fluctuations. Corresponding to enhancement of lattice deformation and accompanying polarization into CuO2-plane may be existing of soft phonon mode, as it occurs in ferroelectrics close to Curie temperature. This soft phonon mode may be important for superconducting state at following cooling. For Ginzburg-Landau theory the fluctuations of superconducting state may occur less than 1 K above Tc. But as the main properties of HTSC cuprates are determined by strong electron correlations, so it is possible to suppose the pairing mechanism with non zero and even high impulse, considered by Yu. Kopaev et al. [70]. In this case superconducting fluctuations may appear at T0~Tc+15 K. Of course, the pairing mechanism for cuprates is still under discussion and another explanation for superconducting fluctuations at relatively high temperature may exist. Moreover, A. Bianconi et al. using EXAFS for Bi2Sr2CaCu2O8+δ, LaCuO4.1 and YBa2Cu3Oy show differing data from ours for lattice deformation, which continues to increase below Tc [71]. Possibly, such a difference is caused by very high used intensity of synchrotron radiation ~1011 photon/second. Let us summarize the observed structure features in temperature range Tc-300 K. We found the temperature interval T1-T2 where thermal vibration amplitudes and thermal expansion coefficient are increased. The last fact means strong unharmonism of thermal vibrations. Moreover, T1-T2 temperature range is characterized by increased linewidth for diffraction, NMR and, possibly, other spectral properties. It means disordered and/or defect state in this specific temperature range. Increased thermal vibration amplitudes may also be a signs of a defect state. All considered in this work HTSC cuprates demonstrate shift of apical oxygen and related Sr/Ba atoms towards CuO2-plane also in this T1-T2 temperature range. In some cases such a shift is accompanied by negative thermal expansion [58-59]. Another feature established is structural anomaly close to Tc with temperature T0~Tc+15 K connected to shift of oxygen from its position into CuO2-plane and rising of buckling, which suppresses when temperature comes close to Tc. As an interpretation of observed features we assume maximal degree of localization for charge carriers localized with participation of lattice distortion in form of volume compression in temperature range Т1-Т2. In favor of this explanation the minimum of unit cell for optimally doped Hg,Tl- and Bi-based compounds says [58,59], together with reversible shift of Ba/Sr and apical oxygen in temperature range T1-T2. Defect state of the lattice leading to growth of linewidth for diffraction data may be a sequence of lattice softness caused by enhanced localization of charge carriers. There are few factors, which may lead to lattice softness at localization, for example, increase of electronic density of states if localized states are located in vicinity of Fermi level. High density of states will lead to screening of Coloumn interatomic repulsion and so, to lattice softness. There are other factors possible, as renormalization of characteristic frequencies for corresponding chemical bonds etc., all they may be very small but all they lead to lattice softness. If so, some growth of both thermal vibration amplitudes and thermal expansion coefficient may also be a sign of lattice softness. As there is local process – localization with participation of local lattice deformation – the

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temperature of this process is not a function of charge carrier concentration and even of type of the material, containing specific structural fragments. We should note that the character of deformation at localization is not only shortening of the apical bond. Except apical bond a little bit smaller deformation is visible for all chemical bonds in the lattice, so we can say about anisotropic volume compression, though this apical bond demonstrates most visible compression being most soft in the lattice. We already concluded that softness of apical bond is specific property of HTSC cuprates with high Tc values. As the data about anomaly in kinetic properties (electrical conductivity, thermo EDC etc.) with temperature, which is constant for varying doping state are absent in literature, it is possible to conclude that charge carriers undergoing localization with participation of lattice deformation do not form a majority. As Т0, Т1 and Т2 anomalies correlate with superconducting properties of HTSC cuprates and their origin is different, we can suppose, without even knowledge about their origin, that to being superconducting below Tc, the sample on cooling must perform the sequence of processes: Т2, than Т1 and Т0. Hence, Тс
Figure 28. Dependence of temperatures of structural anomalies T0~Tc+15 K (stars), Т1~160 K (grey circles) and T2~240 K (black circles) from temperature of transition in superconducting state for optimally doped samples: 1 - PbCu-1212, 2 - YBCO-124, 3 - YBCО(Ca)-124, 4 - BSCCO-2212, 5 YBCO-123, 6 - BSCCO-2223, 7 - BSCCO-2223, 8 - Hg1212, 9 - Hg,Tl-1223. Data from [72] and this work.

Analysis of the results shows that Т0(Тc) significantly differs from Тc+15 K expression. For materials with Тс>100 K the difference between Тc and Т0 decreases and becomes 5 K for compound with record high Тc (Hg,Tl-1223). In error bar limit, shown into the brackets, T0=49.9(7)+0.7(7)⋅Tc. There is linear function for the temperature of this structural anomaly, what might be a support of electron-phonon interaction in mechanism of pairing for HTSC cuprates. The T1=180(1)-0.1(9)Tc anomaly is limiting process for Tc, according to request Тс
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lines will determine maximal possible value Тс max=169(5). As the Т1-line is plotted using only four experimental points, this value is not precise. Nevertheless, as Т1 weekly changes as function of type of the material, it is possible to conclude that Тс max is not differs significantly from our estimation. From these data, one can conclude that next increase of the superconducting transition temperature Тс, if it in general is possible, requests a search of materials principally differ from HTSC cuprates. The mean of limit Тс200 K, when authors note that the results are irreproducible and unstable, are in accordance with that interpretation. The reason of unstable high temperature of superconductivity may be a specific thermal treatment for the material, for example, the rapid quench from high temperature, which allows to safe delocalized state of charge carrier. Evidently, crossing of the temperature interval Т1-Т2 will lead to localization of charge carriers and to vanish of “too high” Тс values.

Summary We review the data considering an influence of structure parameters (defect state in form of buckling of CuO2-planes, the length of the apical Cu-O bond, number of CuO2-planes per unit cell) on the temperature of superconducting transition Tc. It is possible to conclude, that highest Tc values may be obtained for some average parameters, except apical bond, which must be long to obtain high Tc value. Than we considered isostructural 1212 materials to find more specific structure features determining Tc and found that softness of apical bond is important parameter; the materials with apical bond demonstrating high compressibility under external pressure, doping and temperature change have high Tc values. We present low temperature X-ray and neutron powder diffraction study for number of HTSC cuprates, where three different structure anomalies at temperatures T0~Tc+15 K, T1~160 K and T2~260 K are established and their origin is discussed. It is shown that the structural anomaly at T0, in vicinity of Tc, is connected with “quasi-ferroelectric” distortion of CuO2-planes. Possibly, presence of corresponding soft phonon mode is important for superconductivity. It is shown that T0 is linear function of Tc when optimally doped compounds for different systems are compared. This fact means that the mechanism of superconductivity of HTSC cuprates must involve the electron-phonon interaction. Systematic analysis of crystal structure features as function of temperature shows an enhancement of thermal atomic vibration amplitudes and compression of apical bond in the temperature interval T1-T2. In spite of bond length compression, weak lattice softness is observed according to growth of amplitudes of atomic thermal vibration. The whole complex of observed data is interpreted as result of localization of part of charge carriers at participation of lattice deformation in temperature interval ~160260 K. Independence of this interval from charge carrier concentration and even chemical composition of HTSC compound confirms this interpretation. Low temperature border of this interval, connected with delocalization of charge carriers, determines the maximal possible Tc value for HTSC cuprates. Sporadic appearance in literature of data about HTSC cuprates with unusually high values of Tc>200 K, while authors note unstable and irreproducible character of this “super-high” Tc values is in agreement with our interpretation. The reason of such

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unstable and high Tc values may be specific heat treatment of the material, for example, a rapid quench from high temperature, which prevents charge carrier localization at participation of lattice distortion. When sample undergoes cooling in temperature range T1T2~ 160-260 K, charge carriers become localized and “super-high temperature superconductivity” disappears. From these data we conclude that next increase of the superconducting transition temperature Тс, if it in general is possible, requests a search of materials principally differ from HTSC cuprates.

Acknowledgements We would like to express our sincere gratitude to Prof. Ingrid Bryntse, Stockholm University and Prof. S.N. Putilin and Dr. V.A. Aleshin (MSU) for good quality ceramic samples of Hg,Tl-1223 and Hg-1201, respectively. In addition, we are very grateful to Dr. Kevin Knight (ISIS) for help with neutron diffraction. Funding from RFBR (Russia), Royal Society and Nuffield Foundation (GB) is also gratefully acknowledged.

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In: Superconductivity Research Developments Editor: James R. Tobin, pp. 125-148

ISBN: 978-1-60021-848-4 © 2008 Nova Science Publishers, Inc.

Chapter 5

EFFECT OF APICAL OXYGEN ORDERING ON TC OF CUPRATE SUPERCONDUCTORS H. Yang, Q. Q. Liu, F. Y. Li, C. Q. Jin and R. C. Yu* Key Laboratory of Extreme Conditions Physics, Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, P. O. Box 603, Beijing 100080, P. R. China

Abstract Chemical disorder introduced into the charge reservoir blocks by doping has been shown to be one of the parameters influencing the superconducting transition temperature (Tc) of high-Tc cuprate superconductors (HTSs). We address the question of whether the Tc is susceptible to the ordering of dopant atoms. The answer is found by studying the Sr2CuO3+δ superconducting system with K2NiF4-type structure in which oxygen atoms only partially occupy the apical sites next to the CuO2 planes and act as hole dopants. The numerous Tcs appearing in this system are revealed to arise from different modulated phases that are formed just by the ordering of apical oxygen, and each of the superconducting modulated phases is associated with a distinct type of the ordering. The superconductivity differences for the modulated phases are revealed to result, mainly, from the ordering of apical oxygen.

1. Introduction Crystallographically hole-doped high-Tc cuprate superconductor (HTS) is built up from alternating stackings of charge reservoir blocks and the CuO2 conducting planes. Chemical substitution and tuning the oxygen content in the charge reservoir blocks are the fundamental chemical doping mechanics of HTS. In most cases, chemical disorder is created in the charge reservoir blocks as doping is made. Besides the doping level [1] and the number of CuO2 planes (n) in a unit cell [2], the disorder induced by doping has been shown to be another important parameter influencing the superconducting transition temperature of HTS [3]. On the other hand, the oxygen atoms above or below the CuO2 plane(s), usually called “apical *

Corresponding author: [email protected]

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oxygen”, form the nearest-neighbor charge reservoir block [2,4,5], and the apical oxygen serves as the connection between the charge reservoir blocks and the CuO2 conducting planes in terms of electron exchange interaction [6]. Therefore, the apical oxygen doping (i.e., tuning the oxygen content in the nearest-neighbor charge reservoir block) and its ordering is expected to have an appreciable effect on the electronic structures of the CuO2 planes and hence on high-Tc superconductivity. However, for almost all HTSs, apical oxygen sites are fully occupied, leading to the apical oxygen amount being unchangeable, not less to modulate the ordering state. Study of a compound with apical sites partially occupied by oxygen is thus of fundamental physical interest for a deeper understanding of the doping/ordering effect on high-Tc superconductivity. The Sr2CuO3+δ superconducting system synthesized under high pressure is a unique one crystallizing into a highly apical oxygen-deficient K2NiF4-type tetragonal structure, and the apical oxygen acts as a hole dopant. Recently, by systematically studying the Sr2CuO3+δ superconducting system, we found that numerous Tcs appear in this system and arise from different modulated phases formed by the ordering of apical oxygen. This case shows that the ordering of apical oxygen dopants has dramatic effects on Tc. At ambient pressure, the Sr2CuO3 composition forms as an orthorhombic structure with Cu-O chains along the a-axis [7–9]. Introducing extra O leads to the formation of a K2NiF4type tetragonal structure of Sr2CuO3+δ, and at the same time holes are doped in this cuprate due to the introduction of extra O; however, superconductivity is displayed only in the samples synthesized using high-pressure techniques [10–16]. Although the high-pressure technique is a very effective method in searching for new HTS materials [17–21], preparation of a single-phase sample is generally difficult, and sometimes it is hard to identify the superconducting phase in the sample. Using KClO4 as an oxidizer, Hiroi et al. [10] first succeeded in fabricating the tetragonal Sr2CuO3+δ showing superconductivity at Tc = 70 K. They suggested that the main phase in this material is a highly apical oxygen-deficient K2NiF4-type tetragonal structure and the superconductivity results from this tetragonal phase. Later, several groups also repeatedly synthesized the Sr2CuO3+δ superconductors using highpressure technique [11–13], and Tc was enhanced up to 94 K by post annealing [13]. In all those high-pressure samples both as-prepared and post-annealed, however, modulated structures that have the K2NiF4-type tetragonal structure of Sr2CuO3+δ as their basic substructure were observed. For example, Hiroi et al. [10] and Laffez et al. [11] observed, separately, a 4 2 ap×4 2 ap×cp and a 5 2 ap×5 2 ap×cp modulated structure in their asprepared samples, and the latter one was also found by Wang and Zhang et al. [22,23] in their as-prepared and annealed samples. Contrary to the idea suggested in Refs. [10,11,13] that the non-modulated tetragonal form of Sr2CuO3+δ is the superconducting phase, Wang and Zhang et al. [22,23] suggested that the 5 2 ap×5 2 ap×cp modulated phase is actually responsible for the superconductivity. However, no convincing experimental results were given in all the previous works since the previous synthesized samples were multiphase mixtures such as multilayer phase or Cl containing phase. In addition, the tetragonal form of Sr2CuO3+δ synthesized at ambient pressure has been shown to be isostructural with that obtained at high pressure on the basis of powder x-ray diffraction (XRD) analysis, but no superconductivity was observed in the ambient-pressure samples [14,15]. The above works seem not to support the predication that the tetragonal form of Sr2CuO3+δ is responsible for the superconductivity. At the same time, Shimakawa et al. [12] studied the compounds of Sr2CuO3+δ synthesized at high pressure and ambient pressure separately by means of neutron diffraction, and suggested

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that the compounds prepared at different pressures both have an oxygen-deficient K2NiF4type tetragonal structure with oxygen vacancies located in the Cu-O planes instead of the SrO layers, which indicates that the main phase in the samples is possibly non-superconducting in terms of our current understanding of superconductivity in the cuprates, which relies on oxygen vacancy-free in the Cu-O plane. Furthermore, Kawashima et al. [24] reported that the superconducting phase with Tc = 70 K prepared from the nominal starting powders of Sr2CuO3 with KClO4 oxidizer was not of the 0201-type but of 0212-type Sr-Cu-O compound. Up to now, despite extensive studies on the Sr2CuO3+δ system, it is still not clear what phase in the high-pressure material of this system can lead to the superconductivity, and what causes the Tc to increase and further to disappear with increasing annealing temperature. In order to find the answer, preparation of high-quality samples and the characterization of their detailed structures by means of transmission electron microscopy (TEM) are needed. To make the phase as pure as possible, recently we synthesized the Sr2CuO3+δ samples under high pressure using SrO2 as an oxidizer [25]. A series of superconducting samples of Sr2CuO3+δ for various values of δ, commonly displaying different values of Tc, were prepared by changing the initial amount of SrO2. A single-phase superconducting sample (Tc = 75 K) detected to be a K2NiF4-type tetragonal structure from XRD data was obtained when nominal δ = 0.4. When this sample was annealed in N2 atmosphere, we found that with increasing the annealing temperature the Tc was increased from 75 K (as-prepared), in turn, to 89 K (heat treatment at 150 oC), 95 K (heat treatment at 250 oC), and then disappeared when further increasing the annealing temperature above 300 oC [25]. This is consistent with the observation in Ref. [13]. The XRD data show that the sample maintains the same tetragonal single phase as that of as-prepared up to 250 oC. For this superconducting system of Sr2CuO3+δ, the dependence of Tc on post annealing treatment, as well as the superconductivity differences observed in the samples for various values of δ, can not be explained in terms of the diffraction-averaged structures observed and the mere doping level. However, careful studies of these superconducting samples by means of TEM revealed that the numerous Tcs arise from different modulated phases formed by the ordering of apical oxygen. Our findings, which will be shown in detail in the following, clearly indicate that the ordering of apical oxygen dopants has dramatic effects on high-Tc superconductivity.

2. Experimental To make the phase as pure as possible, we synthesized the Sr2CuO3+δ samples under high pressure using SrO2 as an oxidizer. The starting material, Sr2CuO3, was prepared from high purity SrCO3 and CuO raw materials mixed at a molar ratio of 2:1. The powder mixture was calcined at 950 oC for 24 h with several intermediate grindings. Then, Sr2CuO3, SrO2 and CuO were mixed to yield the nominal composition Sr2CuO3+δ at different molar ratios and subjected to high-pressure synthesis (6 GPa and 1100 oC for 1 h) using a cubic-anvil-type apparatus. The role of SrO2 peroxide is to create an oxygen atmosphere during the highpressure synthesis as Jin et al. [20] previously used in the related Sr-Ca-Cu-O-Cl system. The oxygen pressure was controlled by the amount of SrO2 in the starting materials. The main reason for using SrO2 oxidizer instead of KClO4 is to avoid the formation of the superconducting phase containing Cl as found in Ref. [26] or other unwanted pollution from the third element, which will make the identification of the real superconducting phase easier.

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A series of superconducting samples of Sr2CuO3+δ with different nominal δ were prepared by changing the initial amount of SrO2. The sample structures detected from XRD using Cu Kα radiation gradually transform to tetragonal form from orthorhombic structure with increasing nominal δ, and a single-phase tetragonal form is formed when nominal δ = 0.4. Superconductivity of these samples was examined by dc magnetic susceptibility measurements using a SQUID magnetometer in an external magnetic field of 20 Oe, and the results reveal that the samples for nominal δ = 0.1 and 0.2 display a 60 K superconductivity, while the samples for nominal δ = 0.3 and 0.4 exhibit, respectively, a 68 K and a 75 K superconductivity. The 75 K single-phase superconducting sample (nominal δ = 0.4) was then annealed under 1 atm N2 atmosphere in a tube furnace at different temperatures for 12 h. We noticed a clear dependence of Tc on post annealing treatment as revealed in Fig. 1, which shows the magnetic susceptibility measured for as-prepared and annealed samples. It can be seen that with increasing the annealing temperature the Tc was first increased from 75 K (asprepared) to 89 K (heat treatment at 150 oC), and then to 95 K (heat treatment at 250 oC), which is the highest Tc observed in the Sr2CuO3+δ system, and further heating above 250 oC caused the sample to lose superconductivity. However, no obvious phase difference was detected from XRD between the as-prepared and the annealed samples up to 250 oC. In order to further identify the superconducting phase definitely and reveal the mechanics resulting in the superconductivity differences, in the following work we focus on the studies of these samples both as-prepared and post annealed by means of TEM and electron energy-loss spectroscopy (EELS).

Figure 1. Temperature dependence of the dc magnetic susceptibility in the field-cooling mode for asprepared Sr2CuO3+δ (nominal δ = 0.4) and those after annealed at different temperatures in N2 atmosphere.

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TEM thin foils were prepared by crushing the samples in an agate mortar filled with alcohol, and then dispersing the fine fragments suspended in alcohol on a microgrid. A Tecnai F20 electron microscope with a field emission gun, operated at an acceleration voltage of 200 keV, was used for TEM observations and EELS measurements. Our previous TEM work on the “apical oxygen doped” Sr2CuO2+δCl2–y superconductor has suggested that an electron beam with high intensity could cause the superconductor to lose oxygen, and consequently result in the formation of modulated superstructure [27]. Therefore, during the TEM experiments, we tried our best not to expose the Sr2CuO3+δ samples to the intense electron beam, so as to avoid the electron-irradiation effects on the samples. For example, selected area electron diffraction (SAED) instead of convergent beam electron diffraction (CBED) has been used to search for and tilt grains. In fact, the Sr2CuO3+δ samples appeared to be stable under illumination with a weak electron beam. Even when the samples were exposed to the electron beam with high intensity, a short irradiation duration was not found to cause their structure change, either.

3. Results and Discussion 3.1. TEM and EELS of As-Prepared Samples of Sr2CuO3+δ (Nominal δ=0.1, 0.2, 0.3, 0.4) A. Basic Structure Identification Figure 2 shows the XRD patterns of Sr2CuO3+δ samples for different nominal values of δ, revealing that with increasing δ the sample structures gradually transform from an orthorhombic structure to a tetragonal form, and a single-phase tetragonal structure is formed when nominal δ = 0.4. In the samples for nominal δ ≤ 0.3, the orthorhombic structure coexists with the tetragonal form. The orthorhombic structure in these high-pressure samples has a low-oxidized form of Sr2CuO3+y (where the “y” represents the real oxygen content doped in the orthorhombic structure), which can be suggested from oxygen 1s EELS spectra (shown in the following). Previous work [11] has also suggested that the low-oxidized form of Sr2CuO3+y crystallizes mainly as an orthorhombic structure nearly same as that of the ambient-pressure orthorhombic phase of Sr2CuO3. For the tetragonal form of Sr2CuO3+x in these high-pressure samples (where the “x” represents the real oxygen content doped in the tetragonal form), a complex picture was observed. SAED and high-resolution transmission electron microscopy (HRTEM) investigations showed that almost all the grains of the tetragonal form in the samples for nominal δ = 0.3 and 0.4 exhibit modulated structures, and most grains of the tetragonal form in the samples for nominal δ = 0.1 and 0.2 show up modulated structures. In the following studies by oxygen 1s EELS spectra, we can see that these modulated phases have obviously higher oxygen content than the non-modulated tetragonal structure. In addition, we should point out that no any other layered copper oxide such as copper-rich phases including the infinite-layer phase SrCuO2 (n = ∞) and the double Cu-O planes phase Sr3Cu2O5 (n = 2) was detected by ED and HRTEM in all the samples.

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Figure 2. X-ray powder diffraction patterns of the Sr2CuO3+δ samples for different nominal values of δ, where δ = 0.0 denotes the normal sample Sr2CuO3 synthesized at ambient pressure.

B. Superstructure Modulation of the Sr2CuO3+x Tetragonal Form The K2NiF4-type tetragonal form of Sr2CuO3+x was revealed by TEM to prefer to form modulated superstructures. Systemic tilting experiments suggested that the modulation plane for all the modulated structures lies in the apbp-plane, where the subscript “p” stands for the basic K2NiF4-type tetragonal structure. In the sample for nominal δ = 0.4, two types of superstructure modulation of the tetragonal form of Sr2CuO3+x were found. One exhibits a superlattice with a constant of ~5 2 ap×5 2 ap×cp same as reported in Ref. [11]. Fig. 3 presents a typical [001]p zone-axis ED pattern of this modulated structure. Based on the consideration that the superstructure spots do not construct an exact tetragonal array but an orthorhombic one, and according to the extinction rule (hkl are all odd or all even), this modulated phase is identified to be a facecentered orthorhombic superstructure. Its space group can be approximately described as Fmmm in consideration of the sub-structure of the modulated phase being the K2NiF4-type tetragonal structure with the space group I4/mmm. The lattice parameters are a ≈ b = 5 2 ap and c = cp. Fig. 4 shows the [001]p zone-axis ED pattern of the other modulated structure. By systemic tilting experiments this superstructure modulation is determined to be onedimensional and the superstructure peaks are characterized by a unique modulation wave

Effect of Apical Oxygen Ordering on Tc of Cuprate Superconductors 2 , vector q = (2π/ap)( 25

3 25

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, 0). It is clear that the superlattice spots construct a base-centered

monoclinic space array and the conditions for reflection are: hkl: h + k = 2n h0l: h = 2n 0k0: k = 2n

Since its sub-structure is the K2NiF4-type tetragonal structure with the space group I4/mmm, this monoclinic modulated phase has approximately a space group C2/m. The unit-cell parameters are determined to be: a = 5 2 ap, b = cp, c = 26 2 /2ap and β = 101.3o. We noted that the C2/m modulated phase exists only in the sample for nominal δ = 0.4, while the Fmmm one exists in all the samples with different nominal values of δ.

Figure 3. [001]p zone-axis ED pattern of the Fmmm modulated phase observed in the as-prepared sample of Sr2CuO3+δ for nominal δ = 0.4.

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Figure 4. [001]p zone-axis ED pattern of the C2/m modulated phase observed in the as-prepared sample of Sr2CuO3+δ for nominal δ = 0.4.

For the Sr2CuO3+x tetragonal form in the sample with nominal δ = 0.3, besides the Fmmm superstructure modulation as described above, we found another new superstructure modulation. Fig. 5(a) shows the ED pattern along [001]p zone-axis of this modulated structure. The schematic representation of the diffraction pattern along this direction is shown in Fig. 5(b). The superlattice spots were determined by systemic tilting experiments to construct base-centered orthorhombic space arrays and the reflection conditions are: hkl: h + k = 2n hk0: h + k = 2n h00: h = 2n 0k0: k = 2n

Therefore, this modulated structure has a Cmmm symmetry with the unit-cell parameters a = cp, b = 3 2 ap and c = 4 2 ap. The Cmmm modulated phase was found to exist only in the sample with nominal δ = 0.3.

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Figure 5. (a) ED pattern along the [001]p zone-axis of the Cmmm modulated phase observed in the asprepared sample of Sr2CuO3+δ for nominal δ = 0.3. (b) Schematic representation of the diffraction pattern along this zone-axis direction.

The Sr2CuO3+x tetragonal form in the samples for both nominal δ = 0.1 and 0.2 exhibits also two types of superstructure modulation. One is the Fmmm superstructure modulation, but the other is a new one. Fig. 6(a) and (b) displays, respectively, the [001]p zone-axis ED pattern and its schematic representation of this new modulated structure. Unlike those modulated structures described above, this one bears no extinction conditions, indicating that it belongs to primary (P) space group. So this modulated phase has reasonably a space group of Pmmm, and the unit-cell parameters are obtained to be: a = 5 2 ap, b =

2 ap and c = cp.

Figure 6. (a) A typical [001]p zone-axis ED pattern of the Pmmm modulated phase in the as-prepared samples of Sr2CuO3+δ for nominal δ = 0.1 and 0.2. (b) Schematic representation of the diffraction pattern along this zone-axis direction.

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Table 1. Properties of the Sr2CuO3+x tetragonal form in the high-pressure samples of Sr2CuO3+δ for different nominal values of δ. The primitive-cell volume of each modulated phase is obtained from reference to that of the Fmmm modulated phase assumed to be Vo.

Table 1 summarizes the properties of the Sr2CuO3+x tetragonal form in the high-pressure samples for different nominal values of δ together with the sample superconductivity. The primitive-cell volume of each modulated phase, obtained from reference to that of the Fmmm modulated phase assumed to be Vo, is also listed in the table. The primitive-cell volume of modulated phase directly reflects the periodicity of superlattice, and may be related to the total amount of oxygen in the structure as suggested in Ref. [11].

C. Oxygen 1s Absorption Edge For the hole doped cuprate compounds, the oxygen 1s edge fine structure can provide important information about the charge carriers and their crystallographic confinement [28, 29], and the presence of pre-peak in the low-energy part (usually E < 531 eV) of the O 1s edge is related to the unoccupied O 2p states. The intensity of the pre-peak is proportional to the number of holes introduced by doping and directly related to metallic behavior for cuprates [30]. Therefore, it is interesting to see how the pre-peak changes for different phases in the samples. In our EELS experiments, image mode was used to acquire the O 1s absorption edge in order to avoid the anisotropy on the hole-related pre-peak intensity.

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Figure 7. O 1s absorption edges obtained from (a) the ambient-pressure orthorhombic phase, (b) the high-pressure orthorhombic phase in the sample for nominal δ = 0.1, (c) the non-modulated tetragonal phase in the sample for nominal δ = 0.1, (d)–(f) the Pmmm, Cmmm and C2/m modulated phases, respectively, in the samples for nominal δ = 0.1, 0.3 and 0.4, (g) the Fmmm modulated phase in the sample for nominal δ = 0.4.

First we noted that a defined phase has a similar oxygen 1s absorption edge albeit it exists in different samples. Taking the Fmmm modulated phase for example, no obvious difference was observed between the O 1s edges acquired from that in different samples. This phenomenon is different from that reported previously [22]. Fig. 7(a) and (b) shows O 1s absorption edges obtained, respectively, from the ambient-pressure orthorhombic phase and the high-pressure orthorhombic phase in the sample for nominal δ = 0.1. A hole-related prepeak at ~ 528 eV is clearly seen for the latter orthorhombic phase, while the pre-peak disappeared for the former one, indicating that the high-pressure orthorhombic phase has obviously a low-oxidized form of Sr2CuO3+y. However, the pre-peak for the low-oxidized orthorhombic phase is much lower than that for the tetragonal form of Sr2CuO3+x as shown in Figs. 7(c)–(g). Fig. 7(c) corresponds to O 1s edge from the non-modulated Sr2CuO3+x tetragonal structure (in the sample for nominal δ = 0.1), while Fig. 7(d)–(g) corresponds to O 1s edge, respectively, from the Pmmm modulated phase (in the sample for nominal δ = 0.1),

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the Cmmm modulated phase (in the sample for nominal δ = 0.3), the C2/m modulated phase (in the sample for nominal δ = 0.4), and the Fmmm modulated phase (in the sample for nominal δ = 0.4). It can be seen that for the Sr2CuO3+x tetragonal form the non-modulated phase has a lower hole-related pre-peak than those modulated phases; while between those modulated phases a similar pre-peak is observed, indicating a close hole density doped in them. Since the primitive-cells of these modulated phases are very similar in volume (see Table 1), the total amount of oxygen doped in their structures, which directly determine the hole intensity, would be very close. The slight differences of the doped oxygen content for the modulated phases cannot be clearly detected by the hole-related pre-peak in O 1s absorption edge.

D. Discussion The sample structures of Sr2CuO3+δ were revealed by XRD to transform gradually from an orthorhombic structure to a tetragonal form with increasing nominal δ, and a single-phase tetragonal structure was formed when nominal δ = 0.4 (see Fig. 2). Although the orthorhombic phase in these high-pressure samples has a low-oxidized form of Sr2CuO3+y and a number of holes were doped in its structure, this phase can be said to be nonsuperconducting because the CuO2 planes, which is one of the requirements for occurrence of superconductivity, are missing in the orthorhombic phase, instead Cu-O chains are formed in its structure. The absence of superconductivity in the low-oxidized orthorhombic phase of Sr2CuO3+y formed under high pressure was also reported by Laffez et al. [11]. In addition, no trace of other layered copper oxide, such as SrCuO2 and Sr3Cu2O5 copper-rich phases, was detected in the samples by careful TEM investigations. Therefore, the observed bulk superconductivity in the high-pressure samples must result from the tetragonal form of Sr2CuO3+x. Preferring to form modulated superstructures is one of the main characteristics of the tetragonal form. TEM investigations reveal that of the tetragonal form almost all the grains in the samples for nominal δ = 0.3 and 0.4, and most grains in the samples for nominal δ = 0.1 and 0.2 exhibit modulated structures, which strongly suggests that the superconductivity is associated with the modulated phases. It should be noted that the Fmmm modulated phase always exists as major phase in all the samples. This means that, if the Fmmm modulated phase were superconducting, nearly a same Tc could have been observed for all the samples. Therefore, the Fmmm modulated phase is not superconducting. This will be confirmed by further studies performed on the sample with nominal δ = 0.4 annealed at different temperatures. Thus the Tcs can be definitely correlated with the modulated phases based on the data shown in Table 1, i.e., the Pmmm modulated phase is responsible for Tc at 60 K, the Cmmm modulated phase for Tc at 68 K, and the C2/m modulated phase for Tc at 75 K. Now one of the questions that arise is why the Fmmm modulated phase, distinct from the Pmmm, Cmmm and C2/m modulated phases, is not superconducting? A possible reason is that the main mechanism resulting in the formation of the Fmmm modulated phase is different from that for the latter three. Previous neutron powder diffraction suggested that the main phase in Sr2CuO3+δ samples formed as an oxygen-deficient K2NiF4-type tetragonal structure with oxygen vacancies located in the Cu-O planes, not in the Sr-O layers [12]. Zhang et al. [23] proposed an atomic model of the average commensurate lattice to explain the Fmmm modulated structure. They suggested that the modulated structure results from metal ion shear displacements with half sine and half cosine waves along <110>p directions. According to this

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model, the Cu ions have a half cosine wave displacement while Sr ions have a half sine wave displacement, indicating that oxygen atoms are lost in the Cu-O planes instead of in the Sr-O layers. Therefore, the Fmmm modulated phase, revealed to be the major phase in all our samples, results from the ordered oxygen deficiency in the Cu-O planes, and thus it is reasonable for it to be non-superconducting. For the Pmmm, Cmmm and C2/m superconducting modulated phases, the oxygen vacancies can only be located in the Sr-O layers (i.e., in the apical sites) due to the requirement of oxygen vacancy-free CuO2 planes for superconductivity. Thus the formation of the superconducting modulated phases is naturally explained to be the ordering of apical oxygen vacancies. Then another question that arises is why the Pmmm, Cmmm and C2/m modulated phases exhibit different Tcs. From Table 1, we can see that the primitive-cells of the three modulated phases are slightly different in volume from one another, indicating that the numbers of oxygen introduced in these modulated phases could be different. The correlation between the primitive-cell volume of the modulated phase and the total amount of doped oxygen has also been discussed by Laffez et al. [11]. This means that the three modulated phases could have different hole doping levels. However, the effect on Tc by the doping level would be slight, and can be neglected to some extent, because the difference between the doping levels of the modulated phases is very slight and cannot be detected clearly by the hole-related pre-peak in O 1s absorption edge. One the other hand, these superconducting modulated phases are induced by the ordering of apical oxygen, and exhibit clearly different symmetries from one another, in other words, each of the modulated phases has a distinct type of the apical oxygen ordering. It is consequently inferable that the apical oxygen ordering is the key factor influencing the superconducting transition temperature. The oxygen ordering has also been established in YBa2Cu3O6+δ [31], but as it takes place at the second nearest neighbor charge reservoir block, the ordering has little effect on the electronic structures of the CuO2 planes, and hence has little effect on the superconductivity.

3.2. TEM and EELS of the Annealed Sample of Sr2CuO3+δ with Nominal δ=0.4 A. Heat Treatment at 150 oC The 75 K superconducting sample of Sr2CuO3+δ with nominal δ = 0.4, which was revealed to be a single-phase K2NiF4-type tetragonal form by XRD data, was annealed in 1 atm N2 atmosphere at different temperatures. The XRD pattern (Fig. 8) shows that the sample postannealed up to 250 °C maintains the same tetragonal single phase as that of the as-prepared sample (the sample structure begins to undergo the tetragonal-orthorhombic transformation only when the annealing temperature is raised above 250 °C). However, the presence of modulated phase transformation induced by annealing was revealed from TEM observation.

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Figure 8. X-ray diffraction patterns of the Sr2CuO3+δ sample (nominal δ = 0.4) for both as-prepared and post annealed at different temperatures, showing no obvious phase difference in the XRD data between the post annealed up to 250 °C and the as-prepared materials.

The tetragonal form of the as-prepared sample has been revealed above to exhibit two types of superstructure modulation. One is the Fmmm modulated structure with the unit-cell parameters: a ≈ b = 5 2 ap and c = cp, and the other is the C2/m modulated structure with the unit-cell parameters: a = 5 2 ap, b = cp, c = 26 2 /2ap and β = 101.3o. An attempt has been made to evaluate the relative content of the modulated phases, although the TEM results usually suffer from poor statistics due to the observation of a limited number of individual grains. In our evaluation, we conclude, by observing 50 grains (in the following observation statistics we also used 50 grains), that about 20% of the grains have the C2/m modulated structure, and the rest exhibit the Fmmm modulated structure. After heat treatment at 150 oC, a new modulated structure was found in the sample. Careful study of diffraction patterns along various orientations indicates that the structure modulation is two-dimensional, and the modulation plane lies also in the apbp-plane. Fig. 9(a) shows the [001]p zone-axis ED pattern of this modulated structure. The schematic representation of the diffraction pattern along this zone-axis direction is shown in Fig. 9(b).

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The systematic extinction conditions associated with this modulated structure are observed to be: hkl: h + k = 2n hk0: h + k = 2n h00: h = 2n 0k0: k = 2n

From the reflection condition, this structure modulation of the K2NiF4-type tetragonal form of Sr2CuO3+x is determined to have a Cmmm symmetry with the unit-cell parameters a = cp, b = 5 2 ap and c = 5 2 ap. In order to make a knowledge from the Cmmm modulated phase described above, we term this one as Cmmm* modulated phase. According to the statistic observation, we conclude that at least 15% of the grains have the Cmmm* modulated structure, while the content of the C2/m modulated structure decreased to less than 5%. The rest of the grains (about 80%) remain with the Fmmm structure. This result suggests that the heat treatment at 150 oC converted most of the C2/m modulated phase to the Cmmm* one. The coexistence of the remnant C2/m and the Cmmm* modulated phases, as shown by the HRTEM image given in Fig. 10, further confirms the phase transformation from the former modulated phase to the latter one. The domains A and B in the image of Fig. 10 correspond, respectively, to the C2/m and the Cmmm* modulated phases, and the two modulated phases share one K2NiF4-type tetragonal substructure of Sr2CuO3+x.

Figure 9. (a) [001]p zone-axis ED pattern of the Cmmm* modulated phase in the Sr2CuO3+0.4 sample post annealed at 150 oC. (b) Schematic representation of the diffraction pattern along this zone-axis direction.

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Figure 10. HRTEM image along the [001]p direction showing the coexistence of the C2/m modulated phase (domain A) and the Cmmm* modulated phase (domain B).

B. Heat Treatment at 250 oC When the annealing temperature was raised to 250 oC, all the Cmmm* modulated phase and the remnant C2/m one were found to be converted to another kind of modulated phase, while the Fmmm modulated phase did not suffer change and continued to occupy 80% volume of the sample. Fig. 11(a) and (b) display, respectively, the [001]p zone-axis ED pattern and its schematic representation of this new modulated structure. No any extinction condition is observed in this modulated phase by analyses of the diffraction patterns along various orientations, indicating that this modulated phase belongs to primary space group. Its basic sub-structure is revealed by ED and HRTEM to remain the K2NiF4-type tetragonal structure. So the space group of the modulated phase is reasonably thought to be Pmmm. The unit-cell parameters are: a ≈ b = 4 2 ap and c = cp. In order to make a knowledge from the Pmmm modulated phase described above, we term this one as Pmmm* modulated phase.

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Figure 11. (a) [001]p zone-axis ED pattern of the Pmmm* modulated phase in the Sr2CuO3+0.4 sample post annealed at 250 oC. (b) Schematic representation of the diffraction pattern along this zone-axis direction.

C. Heat Treatment at 350 oC When the annealing temperature was further increased to 350 oC, it was found that all the Pmmm* modulated phase and a small part of the Fmmm modulated phase were converted to an orthorhombic structure with the low-oxidized form of Sr2CuO3+y. Statistic observation suggests that about 30% of the grains have the orthorhombic structure, and the rest remain as the Fmmm modulated phase. The transformation from the K2NiF4-type tetragonal form to the orthorhombic structure has also been suggested by the XRD patterns presented in Fig. 8, in which we can see that the orthorhombic structure peaks begin to appear when the annealing temperature is increased to 350 oC. In addition, it should be pointed out that after annealed at 500 oC, the sample was revealed by both TEM observation and XRD data to be completely converted to the orthorhombic structure, and no modulated structure was found in the sample. Table 2 summarizes the phase characteristics as well as the Tc of the Sr2CuO3+δ (nominal δ = 0.4) both as-prepared and post annealed at different temperatures. The primitive-cell volume of each modulated phase is obtained from reference to that of the Fmmm modulated phase assumed to be Vo. The relative phase content is approximately deduced from the statistic TEM observation of 50 grains.

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Table 2. Phase characteristics and Tc of Sr2CuO3+δ (nominal δ = 0.4) both as-prepared and post annealed at different temperatures. The relative phase content is approximately deduced from the statistic TEM observation of 50 grains.

D. Oxygen 1s Absorption Edge The modulated phases found in the Sr2CuO3+δ (nominal δ = 0.4) sample both as-prepared and post-annealed at different temperatures were also studied by the oxygen 1s electron energyloss spectroscopy, and the results are shown in Fig. 12(a) by the curves A~D. The spectra A and B were obtained, respectively, from the C2/m and the Fmmm modulated phases in the asprepared sample, and those marked C and D were obtained, respectively, from the Cmmm* and the Pmmm* modulated phases in the post-annealed samples. The oxygen 1s edge from the orthorhombic phase in the sample annealed at 350 oC is also shown in Fig. 12(a) by curve E, for comparison. It can be seen that the hole-related pre-peak at ~ 528 eV is visible in all the spectra, but the intensity of that for the orthorhombic phase is much weaker than that for the modulated phases. In addition, we can see that the pre-peak for the modulated phases appears to exhibit not the same intensity. In order to get a better impression of the intensity differences of the pre-peak for the modulated phases, we show in Fig. 12(b) their difference spectra with reference to the edge of the orthorhombic phase. To obtain the difference spectra we normalized all the spectra in Fig. 12(a) to have the same integrated area between 535 and 545 eV and then subtracted the edge of the orthorhombic phase from those of the modulated

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phases. The difference spectra were then smoothed by splines. From the difference spectra, the pre-peaks at ~ 528 eV for these modulated phases are seen to show slight differences in intensity, except that the Fmmm and the C2/m modulated phases exhibit a similar pre-peak. The intensity of the pre-peak can be correlated with the primitive-cell volume of the modulated phases on the base of such a relationship, i.e., the larger the primitive-cell volume is then the weaker the intensity of the pre-peak. In addition, it should be pointed out that the Fmmm modulated phase exhibits almost the same O 1s absorption edge between as-prepared and post-annealed, suggesting that the annealing caused no obvious changes of the hole density for this modulated phase. This phenomenon is different from the previous results showing a systematic decrease of the hole density caused by annealing for this modulated phase [22].

Figure 12. (a) Oxygen 1s absorption edges obtained from the modulated phases and the orthorhombic structure in the sample for nominal δ = 0.4. The spectra A and B were obtained, respectively, from the C2/m and the Fmmm modulated phases in the as-prepared sample, and the spectra C and D, respectively, from the Cmmm* and the Pmmm* modulated phases in the post-annealed samples. The spectrum E was obtained from the orthorhombic phase in the sample annealed at 350 oC. (b) The smoothed difference spectra of the modulated phases with the reference to the O 1s edge of the orthorhombic phase.

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E. Discussion The XRD pattern (Fig. 8) shows that the post-annealed sample maintains the same tetragonal single phase as that of the as-prepared sample up to 250 °C, and then begins to convert to an orthorhombic structure as the annealing temperature was increased to 350 oC, where the superconductivity is destroyed. This means that the orthorhombic phase, which is also observed in the as-prepared samples for nominal δ = 0.1~0.3, is not superconducting, although this orthorhombic phase is revealed by the oxygen 1s EELS spectra to be doped with a number of holes. TEM investigations show that almost all the grains in samples both asprepared and post-annealed in the temperature range ≤ 250 °C exhibit modulated structures, which strongly suggests that the observed superconductivity is related to those modulated phases. The as-prepared sample with Tc = 75 K contains two types of modulated phases, i.e., the Fmmm and the C2/m modulated phases, and the former one is revealed to be the major phase occupying about 80% volume of the sample. After heat treatment at 150 oC, most of the C2/m modulated phase was converted to the Cmmm* modulated phase. At the same time, Tc increased from 75 K to 89 K. As the annealing temperature was raised to 250 oC, the Cmmm* modulated phase as well as the remnant C2/m one was converted to the Pmmm* modulated phase. Correspondingly, the Tc further increased to 95 K, which is the highest Tc observed in the Sr2CuO3+δ superconducting system. Further heating to 350 oC led to the Pmmm* modulated phase being converted to the low-oxidized orthorhombic structure, and simultaneously caused the sample to lose superconductivity. Under the annealing, however, the Fmmm modulated phase appeared to be stable, and only when the annealing temperature was raised to 350 oC, this modulated phase begin to convert to the low-oxidized orthorhombic structure. The results of our experiments suggest that the Fmmm modulated phase is nonsuperconducting, for if it were superconducting, nearly a same Tc could have been observed for the superconducting samples both as-prepared and post-annealed since the Fmmm modulated phase always exists as a major phase in these superconducting samples. The absence of superconductivity for the Fmmm modulated phase has also been verified above by the studies of the as-prepared samples of Sr2CuO3+δ for different nominal compositions. Thus the superconductivity (or Tc) can be definitely correlated with the modulated phase, i.e., the C2/m modulated phase is responsible for Tc at 75 K, the Cmmm* modulated phase for Tc at 89 K, and the Pmmm* modulated phase for Tc at 95 K. The TEM observations clearly suggest that the transformation induced by annealing from the C2/m modulated phase to the Cmmm* modulated phase, and then to the Pmmm* modulated phase is just the reason leading to the evolution of Tc, correspondingly, from 75 K to 89 K, and then to 95 K; but why do these modulated phases exhibit clearly different superconducting transition temperatures from one another? The doping level seems to be one of the possible reasons leading to the superconductivity differences, as the hole densities for these modulated phases appear to be slightly different, as revealed from the O 1s EELS spectra in Fig. 12. In this respect, the Pmmm* modulated phase with the highest Tc = 95 K seemly has an optimum-doped hole density. However, the doping level would be a minor or secondary factor influencing the Tc, because a thermogravimetry (TG) analysis on the Sr2CuO3+δ superconductor found little weight change below 300 oC as shown in Fig. 13, indicating a negligible change of oxygen content and hence doping level after annealing, although slight differences of the hole intensity are detected by O 1s absorption edge for these modulated phases. Since the superconducting modulated phases are induced by the ordering

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of apical oxygen, and each of them is related to a distinct type of the ordering, we suggest that the apical oxygen reordering of the modulated structure induced by annealing is the key factor resulting in the change of Tc.

Figure 13. The thermogravimetry (TG) analysis on the Sr2CuO3+δ superconductor, showing almost no weight loss below 300 oC.

3.3. Effect of Apical Oxygen Ordering on Tc The origin of superconductivity in the high-pressure phases of the Sr2CuO3+δ system, which has long been mysterious, has been clarified through the systematical TEM investigations. The numerous superconducting transition temperatures appearing in this system are revealed to arise from different modulated phases, and those superconducting modulated phases and their Tc values are presented as follows: Pmmm modulated phase → 60 K Cmmm modulated phase → 68 K C2/m modulated phase → 75 K Cmmm* modulated phase → 89 K Pmmm* modulated phase → 95 K

These superconducting modulated phases are formed by the ordering of apical oxygen. Besides the slight effect of the doping level, the reordering of apical oxygen was found to be the key factor resulting in the superconductivity differences for the modulated phases. Why does the apical oxygen ordering has such a significant effect on Tc? The localdensity approximation (LDA) band-structure calculation illustrates that the distance da of apical oxygen with respect to the CuO2 plane has a substantial effect on the second-nearest neighbor hopping integral t′ between nearest Cu-Cu atoms in the CuO2 plane [6] and t′ has a

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correlation with the maximum Tc of each cuprate material. The LDA band calculation [6] and also the evaluation of the Madelung potential difference between apical and planar O sites [32] find a general trend that Tc becomes higher as ‫׀‬t′‫ ׀‬becomes larger or as the apical oxygen distance da becomes longer. In this respect, the CuO2 plane without apical oxygen atoms such as that in T′-Nd2CuO4 would sustain highest Tc if it could be doped with holes. Normally, it is hard to dope holes into such CuO2 planes due to relatively electropositive circumstances around the CuO2 plane, but an exceptional situation is realized in the multilayered cuprates with the consecutive CuO2 plane number n larger than 3 (inclusive). In the multilayer systems, the inner CuO2 planes have no apical oxygen and thus fewer hole density compared with the two outer layers [33]. It might be that the outer layers supply a sufficient density of holes, while the inner layers provide a place for strong pairing correlation, both working cooperatively to enhance Tc. In analogy with this we suppose that a similar situation may be realized in Sr2CuO3+δ system in which the CuO4 plaquettes with apical oxygen and those without apical oxygen form some ordered structure within the same CuO2 plane. Further, the ordering of apical oxygen atoms would minimize the disorder effect, and probably enhance Tc in this monolayer cuprate. In this respect, the Pmmm* modulated phase with the highest Tc = 95 K has an optimal ordering of apical oxygen. Interestingly, this optimally superconducting modulated phase can be obtained only by post annealing. The high-pressure synthesis and the subsequent quenching to ambient pressure would leave some residual strains in the lattice. These strains are expected to be relaxed by both the apical oxygen relocation and slightly tuning the oxygen content under the annealing at low temperature. The release of strain was suggested by Attfield et al. [34] to account for the disorder effects on Tc in the cationsubstituted La2CuO4 type superconductor. Normally, the chemical doping introduces disorder into the charge reservoir blocks owing to random distribution of dopant atoms. It has been suggested that the disorder might be responsible for the observed electronic inhomogeneity on the nanometer scale in the CuO2 plane [35–37]. It has also been demonstrated that in Bi2212 and Bi2201, intentionally introduced cation disorder in the nearest neighbor (SrO) block containing apical oxygen sites gives rise to an appreciable effect on Tc [35–37]. The present system is an example showing that the ordering of dopant atoms influences Tc. Application of this method to other cuprates will lead to further enhancement of Tc of cuprate superconductors.

4. Conclusion The Sr2CuO3+δ superconducting system with highly apical oxygen-deficient K2NiF4-type tetragonal structure supplies us a good example to study the effect of apical oxygen ordering on the superconducting transition temperature of HTS. By systematic TEM investigations, we find that the numerous Tcs observed in this system arise from different modulated phases, and these superconducting modulated phases are formed just by the ordering of apical oxygen. The results suggest that the enhancement of Tc from 60 K to 95 K in this system is mainly associated with the reordering of apical oxygen. In this respect, the 95 K superconducting phase, i.e., the Pmmm* modulated phase, has an optimal ordering of apical oxygen. Our findings point a new route toward the further enhancement of Tc of cuprate superconductors.

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Acknowledgement We thank S. Uchida for valuable discussions.

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[24] Kawashima, T.; Takayama-Muromachi, E. Physica C 1996, 267, 106-112. [25] Liu, Q. Q.; Yang, H.; Qin, X. M.; Yu, Y.; Yang, L. X.; Li, F. Y.; Yu, R. C.; Jin, C. Q.; Uchida, S. Phys. Rev. B 2006, 74, 100506(R). [26] Scott, B. A. et al. Nature 1997, 389, 164-167. [27] Yang, H.; Liu, Q. Q.; Li, F. Y.; Jin, C. Q.; Yu, R. C. Supercond. Sci. Technol. 2005, 18, 1360-1364. [28] Yang, H.; Liu, Q. Q.; Jin, C. Q.; Li, F. Y.; Yu, R. C. Appl. Phys. Lett. 2006, 88, 082502. [29] Fink, J. Unoccupied Electronics States (Springer, Berlin, 1992) p. 203-242. [30] Chen, C. T.; Sette, F.; Ma, Y.; Hybertsen, M. S.; Stechel, E. B.; Foulkes, W. M. C.; Schulter, M.; Cheong, S. -W.; Cooper, A. S.; Rupp, L. W.; Batlogg, Jr. B.; Soo, Y. L.; Ming, Z. H.; Krol, A.; Kao, Y. H. Phys. Rev. Lett. 1991, 66, 104-107. [31] Shaked, H. et al. Phys. Rev. B 1995, 51, 547-552. [32] Ohta, Y.; Tohyama, T.; Maekawa, S. Phys. Rev. B 1991, 43, 2968-2982. [33] (a) Tokunaga, Y. et al. Phys. Rev. B 2000, 61, 9707-9710. (b) Kotegawa, H. et al. ibid. 2001, 64, 064515. [34] Attfield, J. P. et al. Nature (London) 1998, 394, 157-159. [35] Eisaki, H.; Kaneko, N.; Feng, D. L. Phys. Rev. B 2004, 69, 064512. [36] Fujita, K.; Noda, T.; Kojima, K. M.; Eisaki, H.; Uchida, S. Phys. Rev. Lett. 2005, 95, 097006. [37] McElroy, K.; Lee, J.; Slezak, J. A.; Lee, D. -H.; Eisaki, H.; Uchida, S.; Davis, J. C. Science 2005, 309, 1048-1052.

In: Superconductivity Research Developments Editor: James R. Tobin, pp. 149-166

ISBN: 978-1-60021-848-4 © 2008 Nova Science Publishers, Inc.

Chapter 6

PAIRING CORRELATIONS IN COPPER OXIDE SUPERCONDUCTORS Rongchao Ma Department of Physics, University of Alberta, Edmonton, Alberta, Canada

Yuefei Ma Xiamen Tungsten CO., Ltd, Haicang, Xiamen 361026, Fujian, P. R. China

Abstract The copper oxide superconductors, or high-Tc superconductors, possess a number of unusual properties due to their complicated interplay of electronic, spin, and lattice degrees of freedom. The mechanism of high-Tc superconductivity is one of the most enduring and important problems in physics, and has never been solved explicitly in theories or clarified thoroughly in experiments, because the multi-layered crystal structures of the materials make the theoretical modelling extremely difficult and the search for the mechanism of high-Tc superconductivity is not successful so far. The main problem is how pairs arise in these materials at such higher temperatures. Lattice vibration (phonon) has long been implicated in conventional low-temperature superconductivity under the BCS theory, but in some sense, has been ignored in high-Tc superconductivity. This article provides a short review on the recent progress in high-Tc superconductivity research - the paring mechanisms which are supported by the recent experimental evidences. Here we underline the phonons again based on the recent experimental results that they could also have a supporting role in the high-Tc superconductivity. The explanations to some of the physical phenomena are also given.

1. Introduction Since the first high-Tc (high temperature) superconductor Ba-La-Cu-O was discovered in 1986 by Bednorz and Muller [1], an unprecedented wave of scientific activity has been caused in the study of superconductivity. Experimental researchers are now actively looking for new compounds of high-Tc superconductors and theoretical researchers are eagerly searching

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for the better theoretical models to explain the unusual properties observed in these novel compounds. There is hope that these activities will lead to new discoveries or creation of the high-Tc superconductors with properties appropriate for technical applications. High-Tc superconductors are usually referred to the copper-oxide superconductors. These compounds possess a number of unusual properties due to their complicated interplays of electrons, spins, and lattice degrees of freedom. To study these properties, researchers have to make use of various experimental methods, such as ‘magnetic field measurement’, ‘electron transport property measurement’, and ‘scanning tunneling spectroscopy’, ‘optical spectroscopy’, ‘neutron scattering’ and so on. A large number of papers are devoted to the general problem of designing high-Tc superconductors and their properties have been well studied by physicists as well as chemists and material scientist. Despite the powerful modern methods of statistical physics, the study of various microscopic models has not so far resulted in an unambiguous interpretation of all the physical phenomena and the mechanism for the formation of the superconducting state in these copper oxides. It has emerged that many other classes of unconventional superconductors are also discovered, such as heavy-fermion superconductors [2, 3, 4, 5, 6], superconducting ferromagnets [7, 8], organic superconductors [9, 10, 11], Sr2RuO4 [12], and borocarbide superconductors [13]. A couple of theoretical models were also proposed to explain the exotic mechanisms in these classes of materials. The theoretical studies in these superconductors are also unsuccessful. Up to now, none of the theoretical models can account for all the phenomena observed in these unconventional superconductors. Usually, a theoretical model can account for some of the phenomena, but fails in the application to the others. In this article, we will focus on the copper oxide superconductors, which are considered as the most promising ones for the upcoming engineering applications.

2. Properties of High-Tc Superconductors The ceramic high-Tc superconductors possess a number of novel properties, of which many are in common with the conventional superconductors, but some of them are beyond the understanding presently, for example, 1. High transition temperature Tc, which cannot be explained by electron-phonon paring mechanism alone [1, 14, 15, 16]. 2. Very short coherence length ξ [17]. 3. Strong anisotropy. Such as critical magnetic field Hc1 and Hc2, critical current density j, penetration depth λ , coherence length ξ , energy gap Δ , and so on all show anisotropy. 4. The Linear dependence of resistivity on temperature with a wide range from Tc to 1000K [18, 19, 20], which is in contradiction with the usual metallic behavior and requires a novel electron scattering mechanism. 5. The anomalous temperature dependence of Hall coefficient [21] 6. Unusual thermoelectric power [22] 7. The pseudogap that happens at the temperatures above the Tc. It indicates that the electron pairing starts in normal state [23, 24, 24].

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3. What Should a Theoretical Model Do? In developing a theory of high-Tc superconductivity, it is necessary to solve at least the following problems, which are foremost important and definitely interrelated: 1. Describe the non-Femi liquid conductivity behaviors. A successful theoretical model should fully explain the general properties of different copper oxides, such as: thermoelectric coefficient (positive under-doped, negative over-doped), Hall coefficient (temperature dependence), and why we cannot detect the de Haas-van Alphen magnetic oscillatory signals in the over-doped region. 2. The Origin of pseudo-energy gap. Since the pseudo-energy happens at the temperatures above Tc, what is the relationship between pseudo-energy gap and the superconducting parameter, or pairing mechanism? 3. Explain why the higher pressure can strongly enhance the Tc and Hall coefficient. The higher pressure can enhance the Tc of Hg-1223 almost 30K. Apparently, this phenomenon cannot be explained by the compression of lattice only. 4. Explain the unusual properties in normal state, such as the linear temperature dependence of resistivity. 5. Predict the limit of Tc. Explain why the maximum Tc is different in different compounds. 6. Account for some of the properties of vortex phase, such as explain why the flux pinning energy is affected by dimension and doping.

4. Pairing Mechanisms In conventional superconductors, the electron-phonon interaction (electron-lattice interaction) [26, 27] plays a central role in the mechanism of superconductivity. The crystal lattice vibrations —phonons — provide an attractive mechanism that pairs up electrons possessing opposite spins and moments. An electron moving through a conductor will attract nearby positive charges in the lattice. This deformation of lattice causes another electron, with opposite spins and moments, to move into the region of higher positive charge density. The two electrons are held together with a certain binding energy and then form the so-called Cooper pairs. A finite amount of energy, usually called the energy gap, is needed to break these apart into two independent electrons. This energy gap is the highest at low temperatures but vanishes at the transition temperature Tc where superconductivity ceases to exist. Based on the above ideas, Bardeen, Cooper, and Schrieffer created the so-called BCStheory [28], which is currently the most elegant and successful theory in conventional superconductors. The BCS theory succeeds in the following aspects, (1) The prediction of the variation of energy gap with temperature. It also gives an expression that shows how the gap grows with the strength of the attractive interaction and the single particle density of states of normal phase at the Fermi surface. (2) The description of how the DOS (density of states) is changed on entering the superconducting state, where there are no electronic states any more on the Fermi surface. (3) The prediction of the Meissner effect, i.e. the expulsion of a magnetic field from the superconductor and the variation of the penetration depth with temperature.

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(4) An approximation for the quantum-mechanical states of the system of (attractively interacting) electrons inside the metal. This state is now known as the “BCS state”. In the normal metal, the electrons move independently, whereas, in the BCS state they are bound into “Cooper pairs” by the attractive interaction. (5) The important theoretical prediction of the properties that are independent of the details of interaction, which have been confirmed in numerous experiments (the quantitative predictions mentioned below hold only for sufficiently weak attraction between the electrons, which is however fulfilled for many low temperature superconductors - the socalled “weak-coupling case”). After the first high-Tc superconductor was discovered in 1986, many physicists also tried to explain the properties of these novel materials by using the electron-phonon interaction mechanism. But the first theoretical calculation of Tc based on the theory of electron-phonon coupling was not successful due to the fact that it contradicts some experiments. On the other hand, the discovery of strong Coulomb correlations led to the development of non-phononic models of high-Tc superconductivity, so it looks like that the electron-phonon pairing mechanism does not work in the oxide high-Tc superconductors. It is now widely accepted that other effects are at plays, which are not yet fully understood. It is possible that these unknown effects also control the superconductivity at lower temperatures for some materials. The formation of electron pairs is essential to superconductivity. The pairing mechanism in high-Tc superconductors is certainly one of the top unsolved problems in modern physics. Generally speaking, the carriers in the cuprates with typical layered structure are strongly correlated and the superconductivity sensitively depends on the holes doped in the CuO2 plane. As a result, many usual properties are observed, non Femi-liquid behavior, rich phasediagram, antiferromagnetism and so on. It is very much important that many experiments have indicated the singlet Cooper-pairing and the d-wave symmetry in high-Tc superconductors [29]. The main question is how pairs arise in these materials at such higher temperatures. Theorists have put forward many theoretical models to explain the pairing mechanism in the materials that contain the copper oxide layers. However, In view of the complicated character of the interplay, any theoretical model of oxide superconductors encounters a number of difficulties. The theoretical modeling is extremely difficult because the materials in question are generally very complicated multi-layered crystals (for example, BSCCO). The search for the mechanism of high-Tc superconductivity is not successful so far presently. But with the rapid rate of new important discoveries in the field, many researchers are optimistic that a complete understanding of the process is possible within the next decades or so. In the following paragraphs, we will focus on the most foundational problem in superconductivity - pairing correlation.

4.1. Electron-Phonon Mechanism As mentioned before, the formation of electron pairs in conventional superconductors is due to the attraction originated from the electron-phonon interaction. The interaction between electrons, resulting from the exchange of phonons, is attractive when the energy difference between the electrons states involved is less than the phonon energy. It is favorable to form a

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superconducting phase when this attractive interaction dominates the repulsive screened Coulomb interaction. The normal phase is generally described with the Bloch individualparticle model, and the superconducting ground state is described by a linear combination of normal state configurations. In the ground state, electrons are virtually in pairs with opposite spins and moments, so the energy of ground state is lower than that of normal state by amount proportional to an average. Fröhlich [26] suggested that the effective electron-electron interaction may be wrote as

H eff =

1 Vkq c k++ q ,σ c k+′− q ,σ ′ c k ′σ ′ c kσ ∑ 2 k ′,k ,q

(1)

σ ′,σ

2

Vkq =

where

Eq. (2) shows that Vkq < 0 if

2 D q hω q

(2)

(ε k − ε k + q ) 2 − (hω q ) 2

ε k − ε k + q < hω q , which means it is possible that there is

an attractive interaction between two electrons under certain conditions. Nakajima [27] later derived the Eq.(1) directly from the Hamiltonian of an electron-phonon system. In the history of superconductivity, Cooper [30] made the crucial step to the successful formulation of the microscopic theory of superconductivity in 1956, his calculation leads to minimum energy of two bound electron system

ε = −2hω D e − (2 / N ( 0 )V )

(3)

The negative sign in Eq. (3) showed that a bound state could be obtained regardless of how weak the attractive interaction is. To describe the superconductivity, we need the so-called BCS wavefunction [28], which was first conceived by Schrieffer

ΨBCS = ∏ (u k + v k c k+↑ c −+k ↓ ) Vac

(4)

k

where the product extends over all plane wave states, and

uk

2

+ vk

2

ΨBCS ΨBCS = 1 ,

= 1.

In conventional superconductivity, the calculation of Cooper pair is based on the ‘isotropic’ wave, but the electron-phonon interactions in high-Tc superconductors are essentially ‘anisotropic’ and the superconductivity order parameter is a function of (θ , ϕ ) .

( ) ( ) (

)

Thus, there are four choices for spin pairing: ↑↑ , ↓↓ , ↑↓ ± ↓↑ / 2 . To formulate a description for the high-Tc superconductors, we need to generalize the conventional ‘isotropic Cooper pair’ to the general ‘anisotropic Cooper pair’.

154

Rongchao Ma and Yuefei Ma Let us consider two electrons with spins of arbitrary direction (r1 , σ 1 ) and (r2 , σ 2 )

outside a filled Fermi-Surface. The wave equation of the two electrons can be then written as ψ (r1 , σ 1 ; r2 , σ 2 ) , we now have the following Schrödinger equation for the pair

⎧ h2 ∇12 + ∇ 22 + V ( r1 − r2 ⎨− m 2 ⎩

(

)

)⎫⎬ψ (r , σ ; r , σ ) = Eψ (r , σ ; r , σ ) ⎭

1

1

2

2

1

1

2

2

(5)

The solutions to Eq. (5) can be regarded as a generalization to the wave function of Cooper pairs. But new problems rise when applying to the layered copper oxides; it may be that we ignored some important effects that play roles in the pairing. The detailed calculations and the extension for BCS theory can be found in other literatures [31, 32, 33] and the calculation for more general case of pairing of Fermions with arbitrary spin is also available in literature [34]. Recently, more and more experimental evidences come out and seem to support the electron-phonon pairing mechanism in high-Tc superconductivity.

4.1.1. Isotope Effect Historically, the importance of the electron-lattice interaction in explaining the superconductivity in conventional superconductors was first confirmed experimentally by the −1 / 2

discovery of isotope effect, i.e., the proportionality of Tc and Hc to M for the isotopes of same element [35, 36]. However, in high- Tc superconductors, the oxygen-isotope exchange has been found to have a very small effect on Tc, particularly for optimally doped samples, which has the highest transition temperature. It was therefore surprising when Gweon et al. [37] recently reported a large isotope effect on the electronic structure in optimally doped Bi2 Sr2 CaCu 2 O8+δ using angle-resolved photoemission spectroscopy (ARPES). They provided a detailed comparison of the electron dynamics in Bi-2212 samples that contain different oxygen isotopes. This isotope effect was seen mainly at a broad, high-energy hump feature and was unusually large (10–40 meV shift), by contrast with the expected isotopic phonon shifts of the order of 3 meV. The authors argued that this is the experimental evidence that shows that the coupling between lattice and electrons plays a particularly significant role. Gweon and colleagues’ data shows definite and strong isotope effects, which mainly appear in broad high-energy humps and are commonly referred as ‘incoherent peaks’. The magnitude of the isotope effect is a function of temperature and electron momentum, and is closely correlates with the superconducting gap — the pair binding energy. Gweon et. al. also argue that their results can be explained in a dynamic spin-Peierls picture [38], where the “singlet pairing of electrons” and the “electron–lattice coupling” mutually enhance each other. It indicates that the phonons play a special importance role in high-Tc superconductors.

4.1.2. Optical Spectroscopy Optical spectroscopy, the scattering of light by materials, is a vital tool for revealing the fundamental information about superconductivity. The BCS theory born partially because of

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the existence of a gap in the excitation-energy spectrum of electrons observed in optical studies. By using the angle-resolved photoemission spectroscopy, Lanzara et. al. [39] recently investigated the electron dynamics — velocity and scattering rate — for three different families of hole-doped copper oxide superconductors: Bi2Sr2CaCu2O8 (Bi2212) and Pb-doped Bi2212 (Pb-Bi2212), Pb-doped Bi2Sr2CuO6 (Pb-Bi2201) and La2-xSrxCuO4 (LSCO). They found that in all of these materials an abrupt change of electron velocity happens between 50– 80 meV, which cannot be explained by any known process other than to invoke the coupling with the phonons associated with the movement of oxygen atoms. Lanzara et. al. then claimed that electron–phonon coupling strongly influences the electron dynamics in the high-Tc superconductors, and must therefore be included in any microscopic theory of superconductivity.

4.1.3. Inelastic Scattering Measurements Reznik et. al. [40] recently shown a strong anomaly in the Cu–O bond-stretching phonon in the copper oxide superconductors La2-xSrxCuO4 (with x = 0.07, 0.15) by the inelastic scattering measurements. The authors also claimed that their results suggest a giant electron– phonon anomaly associated with charge inhomogeneity. This giant electron–phonon anomaly is absent in undoped and over-doped non-superconductors. So it indicates that the electron– phonon coupling may be important in high-Tc superconductivity, although its contribution is likely to be indirect. Reznik and colleagues’ data present further evidence in support of quantum stripes [68, 41]. By exploiting the motions of the ions that form the copper oxide lattice, they show that the collective vibrations of the atomic lattice of certain superconducting copper oxides behave in a manner which indicates that the motion characteristics of the quantum stripes are – to shake the ion lattice. These motions give rise to quantized lattice vibrations, known as phonons, which can be easily observed by the inelastic neutron scattering. The authors observed that the electronic stripes undergo coherent vibrations by checking the spectrum of the phonons. The experiments also show strong bond-stretching phonon anomalies that are common to the stripe-ordered superconducting copper oxides. The disappearance of the phonon anomaly at non-superconducting extremes of doping suggests that electron–phonon coupling may be important to the mechanism of high-Tc superconductivity.

4.1.4. Scanning Tunneling Spectroscopy The tunneling spectroscopy has shown that the electron pairing in conventional superconductors is mediated by phonons: a peak in the second derivative of tunnel current d2I/dV2 corresponds to each phonon mode [42]. However, no such phonon mediating electron pairing has been identified in high-Tc superconductors. It was explained that the electron pair formation and the related electron–phonon interactions are heterogeneous at atomic scale in high-Tc superconductors, so they are difficult to be characterized. Recently, Lee et. al. [43] reported the d2I/dV2 imaging studies in Bi2Sr2CaCu2O8+x superconductor. The authors used the technique of scanning tunneling spectroscopy, in which the voltage measurements from a conducting probe provide a precise picture of the density of energy states on an atomic scale. They have performed the ultimate test to confirm the phononic origin of the effect by growing their Bi2Sr2CaCu2O8+x superconductors in an environment that is rich in the heavy oxygen isotope 18O, rather than the normal 16O. The

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intense disorder of electron–boson interaction energy at the nanometer scale was found along with the expected modulations in d2I/dV2. It was also found that the changing of the density of holes has minimal effects on both average mode energies and modulations, which indicates that the bosonic modes are unrelated to electronic or magnetic structure. Instead, the modes appear to be local lattice vibrations, as the substitution of 18O for 16O throughout the material reduces the average mode energy by approximately 6 per cent — the expected effect of this isotope substitution on lattice vibration frequencies. It was found that the mode energies are always spatially anticorrelated with the superconducting pairing-gap energies, which suggests the interplay between these lattice vibration modes and the superconductivity. The results also indicates that, because of the inhomogeneity of superconductivity, the ‘technical’ average of phonon energies has concealed them as the enabling force behind high-Tc superconductivity, and then make it very much difficult to detect the ‘truth’ of superconductivity.

4.2. Magnetic Excitation 4.2.1. Theoretical Description In conventional superconductors, the interaction that pairs the electrons to form the superconducting state is mediated by lattice vibrations (phonons). However, in high-Tc copper oxides, the superconductivity develops near the antiferromagnetic phases. So the magnetic excitations [44, 45] probably contribute to the superconducting pairing mechanism and play a fundamental role in the superconducting mechanism of high-Tc copper oxides because in these materials the superconductivity occurs when mobile ‘electrons’ or ‘holes’ are doped into the antiferromagnetic parent compounds. The most obvious feature in the magnetic excitations of high-Tc superconductors such as YBa2Cu3O6+x is the so-called ‘resonance’ [67, 46], which is believed strongly couple to the superconductivity. Indeed, the sharp magnetic excitations has been observed by neutron scattering in a number of hole-doped high-Tc superconductors [47, 48, 49]. As already mentioned in section 4.1.3, let us now talk more about the concept ‘stripes’ in the Cu-O plane. The fundamental building block of the copper oxide superconductors is the Cu-O planes, which are usually distorted to form a rectangular lattice. The valence electrons in these planes are localized — one per copper site — by the strong Coulomb repulsion between the atoms. The spins of localized electrons alternate between up and down, and form an antiferromagnetism background, where superconductivity appears when mobile ‘holes’ are doped into this insulating state, and it coexists with antiferromagnetic fluctuations. One theory predicted that the holes self-organize into ‘stripes’ [50, 51], which alternate with antiferromagnetic (insulating) regions within copper oxide planes. The ‘stripes’ should align along one of the axes of the Cu-O plane. This theory received strong support from experiments that indicated a one-dimensional character for the magnetic excitations in the high-Tc material YBa2Cu3O6.6, but the measurements of magnetic excitations in superconducting YBa2Cu3O6+x near optimum doping are incompatible with the naive expectations for a material with stripes.

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The resonance is intimately related to high-Tc superconductivity and its interaction with charged quasi-particles were observed by photoemission, optical conductivity, and tunneling. It is suggested that the resonance might play a part similar to that of the phonons in conventional superconductors. On the other hand, it is known that the phonons in conventional superconductivity are predicted to produce an ‘isotropic’ spherical pattern, while the quantum-mechanical wave function that describes a superconducting state in highTc superconductors is widely accepted as a d-wave function, which resembles a four-leaf clover when traced in momentum (‘k’) space, but how well such predictions work when the electron–phonon interaction is strong and the phonons are very unevenly distributed around the crystal is unclear. Most physicists tend to believe that magnetic excitation may play a main role in the high-Tc superconductivity. Recently, there have emerged that a number of strong experimental evidences which support the magnetic excitation pairing mechanism in high-Tc superconductivity as will be disscussed in following paragraphs. To investigate the magnetic excitation pairing mechanism, physists usually start with the Hamilitonian of Hubbard model [52],

H = ∑∑ t ij ci+σ c jσ + i, j

σ

U 2

nσnσ ∑∑ σ i

i

(6)

i

+

where niσ = ciσ ciσ . From Eq. (6), many theoretical model are derived based on different physical approxmation condsideration or constraints [53, 54, 55, 56, 57, 58, 59, 60]. Most of the theoretical model are assumed the so-called d x 2 − y 2 pairing symmetry [45, 61, 62].

4.2.2. Optical Spectroscopy Hwang et. al. [63] recently reported the systematically investigation on the optical self-energy of Bi2Sr2CaCu2O8+ (Bi-2212) sample. They shown that the sharp ‘kink’ [64, 65, 66], which is believed to be related to magnetic resonance [67, 68, 69] and also phonons [39], can be separated from a broad background. The kink is also found weakening with doping before disappearing completely at a critical doping level of 0.23 holes per copper atom, at which superconductivity is still strong in terms of the transition temperature at this doping (Tc 55 K). The authors found that, for temperatures above the superconducting transition temperature, the so-called self-energy has a rather broad and featureless distribution, up to the very high electron excitation energies. Such behavior is inconsistent with that expected if the moving electrons are interacting with the phonon-vibrations in the atomic lattice – as occurs in low-temperature superconductors. They concluded that the self-energy distribution is due to the interactions between electrons, which strongly supports the magnetic-resonance interpretation of self-energy peak. The authors have tracked the optical self-energy peak at various temperatures over a wide range of chemical doping (vary the amount of mobile charge-carriers added to the sample and then affect the performance of a superconductor) and find that the peak is directly correlated with magnetic resonance. They also find that, as doping increases, the peak eventually disappears at a doping level for which the superconductivity is still strong — electrons are still joined into Cooper pairs. So they suggest that the peak itself cannot represent their binding glue.

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Hwang et al. have again demonstrated the power of optics to reveal the fundamental information on the underlying interactions in high-Tc superconductivity. They have provided strong evidences which show that the origin of high-Tc superconductivity in copper oxides is connected with magnetic resonances.

4.2.3. Inelastic Neutron Scattering Recently, Motoyama et. al. [70] reported inelastic magnetic neutron-scattering measurements that indicate the genuine long-range antiferromagnetism does not coexist with superconductivity. They data revealed a magnetic quantum critical point where superconductivity first appears, consistent with an exotic quantum phase transition between the two phases. They also demonstrated that the pseudogap phenomenon in the electrondoped materials which is associated with pronounced charge anomalies arises from a build-up of spin correlations. This is in agreement with recent theoretical proposals. Hinkov et. al. [71] reported the data on the so-called ‘untwined’ YBa2Cu3O6+x crystals, in which the orientation of rectangular lattice is maintained throughout the entire volume. They identified that the geometry of magnetic fluctuations is two-dimensional. Rigid stripe arrays therefore appear to be ruled out over a wide range of doping levels in YBa2Cu3O6+x, but the data may be consistent with liquid-crystalline stripe order. So, there is possibility that these data may prove that, the theories that predict the coexistence of high-Tc superconductivity with a rigid array of stripes are incorrect. Hayden et. al. [72] claimed that they detected the possible mediating excitations at higher energies in YBa2Cu3O6.6 by using inelastic neutron scattering and observed a square-shaped continuum of excitations peaked at incommensurate positions. These excitations have energies greater than the superconducting pairing energy that present at Tc, and have spectral weight far exceeding that of the ‘resonance’. The discovery of the similar excitations in La2– xBaxCuO4 superconductors suggests that the excitations is a general property of the copper oxides, and then is a good candidate for mediating the electron pairing. Tranquada et. al. [73] reported the measurements on stripe-ordered in La1.875Ba0.125CuO4, and showed that the excitations are quite similar to those in YBa2Cu3O6+x that predicted the spectrum of magnetic excitations is wrong. They find instead that the observed spectrum can be understood within a stripe model by taking account of quantum excitations. The authors argued that the ladder model, within the stripe picture, provides a more compelling explanation to the results. The results provide supports for the concept that the charge inhomogeneity, possibly dynamic in nature, is essential to achieve the superconductivity with a high transition temperature in copper oxides In the last decades, the relevance of the resonance to high-Tc superconductivity has been in doubt because the magnetic excitations related resonance has not been found in the La2x(Ba,Sr)xCuO4 family and is not universally present in Bi2Sr2CaCu2O8+x, in other words, the ‘resonance’ are mostly been observed in hole-doped materials. However, Wilson et. al. [74] recently reported the discovery of the resonance in electron-doped superconducting Pr0.88LaCe0.12CuO4-x (Tc = 24 K). They find that the resonance energy (Er) is proportional to Tc via E r ≈ 5.3kBTc for all high-Tc superconductors and is irrespective to electron- or holedoping. This may demonstrated that the resonance is a fundamental property of superconducting copper oxides and therefore must be essential in the mechanism of high-Tc

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superconductivity. Several important conclusions can be sum up from Wilson and colleague’s experimental results as following, (1) The magnetic resonance in electron-doped PLCCO with E r ≈ 5.3kBTc suggests that the resonance is a common feature for high-Tc superconductors irrespective of electron- or hole-doping. (2) The commensurate spin fluctuations below resonance imply that the intimate connection between the incommensurate spin fluctuations and the resonance in hole-doped materials is not a universal feature. (3) The magnetic excitations ( 0.5 ≤ hω ≤ 16meV ) in the electron-doped PLCCO are gapless, decrease monotonically with increasing energy, and are virtually temperatureindependent between 2 K T 30 K except for the appearance of the resonance below Tc. Such behavior differs from those observed in the optimally hole-doped YBCO and La2xSrxCuO4, but the temperature-independent low-energy ( 0.5meV ≤ h ≤ 4.5meV ) magnetic scattering is remarkably similar to the quantum critical scattering in heavy fermions UCu5-xPdx. (4) The magnetic resonance in electron-doped PLCCO allows a systematic comparison of their properties in the same bulk sample, which has not hitherto been possible for any other high-Tc superconductors.

5. The Explanations for the Universality Law in HTSC Since the discovery of the first high-Tc superconductor [1], a considerable number of physicists are actively looking for the universal trends and correlations amongst physical quantities of these novel materials. One of the earliest patterns was the so called Uemura relation [75], which predicted a linear scaling of the superfluid density ρ s with the superconducting transition temperature Tc. This relation works reasonably well for the underdoped materials, but fails in describing optimally doped or overdoped materials [76]. Similarly, an attempt to scale the superfluid density with the D.C. conductivity ( σ dc ) was only partially successful [77]. Recently, Homes et. al. [78] reported a simple scaling relation,

ρ s = (125 ± 25)σ dcTc This relation holds for all tested high-Tc materials (with

(7)

σ dc measured at approximately

Tc). As shown in Fig. 1, Eq. (7) holds regardless of doping level, nature of dopant (electrons versus holes), crystal structure and type of disorder, and direction (parallel or perpendicular to the copper–oxygen planes).

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Rongchao Ma and Yuefei Ma

Figure 1. (a) Plot of the superfluid density ( ρ s ) versus the product of the d.c. conductivity ( σ dc ) and the superconducting transition temperature (Tc) for a variety of copper oxides and some simple metals in a-b plane. (b) Plot of the

ρs

versus the product of the

σ dc

and Tc for copper oxides only, and

including data for the poorly conducting c axis. (The figure is from ref. 78)

Homes et. al. first demonstrate the scaling for the a–b plane properties of single- and double-layer copper oxide materials, as well as for conventional metals Nb and Pb. The results for scaling relation are shown on a log–log plot, as shown in Fig. 1 (a). It shows that within the error all of these points fall onto a single line with a slope of unity. The optimal and overdoped materials, which fall well off of the Uemura plot, now scale with the underdoped materials onto a single line. Then Homes and colleagues further searched for the scaling relations along the poorly conducting c axis, where the charge transport is thought to be incoherent. As shown in Figure 1 (b), the results demonstrate that the c-axis data for all of the single and double-layer materials are again well described by a line with slope of unity. The a–b-plane and c-axis results may all be described by the same universal line, even though the two results correspond to very different ranges of ρ s . The scaling relation for the a–b planes is interpreted by assuming that all of the spectral weight associated with the free-carriers of the normal state (nn) collapses into the superconducting condensate ( n s = n n ) below Tc, and the low-frequency conductivity at

T ≈ Tc can be described by the simple Drude theory for a metal, σ 1 (ω ) = σ dc /(1 + ω 2τ 2 ) (where ω is the frequency), which has the shape of a Lorentzian centered at zero frequency with a width at half-maximum given by the scattering rate 1 / τ , the area under this curve may be approximated simply as σ dc / τ . The transport measurements for copper oxides suggest that 1 / τ near the transition scales linearly with Tc, so the strength of the condensate is just as Eq. (7). But the properties along c axis are interpreted in a way which is a little bit different [79, 80].

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It is well known that in copper oxide superconductors, Tc is a function of the number of CuO2 layers, n, in the unit cell of a crystal. In a given family of these superconductors, Tc rises with the number of layers, reaches a peak at n=3, and then declines with the number of layers. The Tc-n curve is of bell-shape. Chakravarty et. al. [81] present an explanation for the universality of transition temperatures in families of copper oxide superconductors. They shown that the quantum tunneling of Cooper pairs between the layers simply and naturally explains the experimental results, when combined with the recently quantified charge imbalance of the layers and the latest notion of a competing order nucleated by this charge imbalance that suppresses superconductivity. They calculate the bell-shaped curve and show that Tc can be raised further if materials can be engineered so as to minimize the charge imbalance as n increases,. The authors argued that the tunneling between layers must be regarded as a mechanism by which pairing is enhanced and should not be construed as the sole reason for high Tc, especially for single-layer materials in which the effect of tunneling is negligible. To take advantage of tunneling between the close pairs of Cu-O planes within a unit cell, it is necessary that a low-energy electron in the normal state is forbidden to tunnel coherently perpendicular to the planes. In the superconducting state this kinetic energy is recovered, resulting in enhanced pairing. The frustrated kinetic energy in the normal state may either be due to a non-Fermi-liquid nature of this state, in which an electron breaks up into more fundamental constituents, or due to the pseudogap that is present over much of the phase diagram. From a simple renormalization group argument, in the pseudogap state, the singleparticle tunneling is irrelevant at low energies below the gap, and it should simply drop out of all macroscopic considerations. In contrast, the pair tunneling, in which a Cooper pair of electrons tunnels together, is a coherent zero-energy process and must have macroscopic consequence. For high-Tc superconductors, the tunneling matrix element in the momentum space is peaked where the superconducting gap is large, and therefore the existence of nodes in the gap cannot invalidate this argument. By referring to the mean-field free energy functional of complex order parameters ψ , Chakravarty and colleagues concluded that the free energy of the competing order parameter is ultimately responsible for the downturn of Tc with n. Although phase fluctuations do play a role in determining Tc, it is the competition of two order parameters that plays the dominant role in determining the superconducting dome. According to this theory, to increase Tc further, it would be necessary to dope the system in such a way that the layers do not develop a charge imbalance and nucleate competing order. Another consequence of this theory would be that site-specific NMR relaxation rates should show a pseudogap in the inner layers, but not in the outer layers, owing to the charge imbalance, which is in agreement with recent experiments [82, 83]. This is because the suppression of the superconducting order parameter in the inner layers will enhance the pseudo gap. The authors also predict that the maximum superconducting gap measured in ARPES will be a bell-shaped curve as a function of n with a maximum at n = 3. A more elaborate picture is the development of what makes some materials superconductive at relatively high temperatures. With it come hints for how to design materials with still higher transition temperatures. In high-Tc superconductors, the under-doped materials with lower transition temperatures actually develop a larger gap than optimally doped materials, even at higher temperatures where they have not yet become superconducting. It is produced by a second order parameter that competes with

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superconductivity. The authors have melded the ideas of tunneling, charge transfer and order into a simple Landau–Ginzburg expression for the total energy of the system. Looking beyond the order connected with correlated electrons, the basic idea of their theory is that, for more than three layers of copper oxide, a second, competing order parameter nucleates in the interior under-doped layers and lowers the superconducting transition temperature. By fitting the properties of a single-layer material and using the measured doping profile, they are then able to compute how the superconducting transition temperature evolves with the number of layers, obtaining good qualitative agreement with experimental data. In short, Chakravarty and colleagues’ theory brings together three important effects into the physics of high-Tc superconductivity: (1) Electron tunneling between the superconducting layers. Inside a superconductor, electrons are bound into pairs. When two superconducting layers are brought close together, the pairs can ‘tunnel’ or ‘jump’ from one layer to the others. This tunneling effect is called a ‘Josephson coupling’ after its discoverer. Chakravarty and colleagues argue that, although it is not the main engine of superconductivity in the layers, such coupling is responsible for the rise in transition temperature when going from one-layer to three-layer materials. (2) Interlayer charge distribution. The authors also take account of how charge is distributed between the layers of a superconductor. A remarkable aspect of high-Tc superconductors is that, in many ways, these materials are closer to insulators than to metals. The mother compounds of high-Tc superconductors are insulating, and develop superconductivity only when extra charge carriers (electrons or holes) are introduced into the materials by chemical doping. (3) Order parameter. Many experimental results indicate that the unique properties of high-Tc superconductors result from the competition between more than one types of order parameter. Chakravarty and colleagues skillfully hand the problem by using the orderparameter approach to analyze the interaction between the superconductivity in each layer of the material though the electron correlations that responsible for high-Tc superconductivity are still a mystery presently.

6. Conclusion The pairing mechanism of high-Tc superconductors has been extensively studied by theoretical and experimental physicsts in last decade, but the high-Tc superconductivity still remain in a mystery presently. The idea that vibrations of a solid crystal lattice might be also crucial in high-Tc superconductivity has been shot down from time to time, and raised again and again under new experimental supports. Current experimental data and theoretical calculations show that both electron-phonon interaction and magnetic excitation may play a role in the pairing mechanism of high-Tc superconductivity. A new theoretical model of superconductivity for copper oxides is urgently needed. There is hope that the mystery of high-Tc superconductivity can be clarified in the near future.

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[80] Dordevic, S. V.; Singley, E. J.; Basov, D. N.; Komiya, Seiki; Ando, Yoichi; Bucher, E.; Homes, C. C.; Strongin, M. Phys. Rev. B 2002, 65, 134511 [81] Chakravarty, Sudip; Kee, Hae-Young and Völker, Klaus. Nature 2004, 428, 53 [82] Tokunaga, Y.; Ishida, K.; Kitaoka, Y.; Asayama, K.; Tokiwa, K.; Iyo, A.; Ihara, H. Phys. Rev. B 2000, 61, 9707 [83] Kotegawa, H.; Tokunaga, Y.; Ishida, K.; Zheng, G.-q.; Kitaoka, Y.; Kito, H.; Iyo, A.; Tokiwa, K.; Watanabe, T.; Ihara, H. Phys. Rev. B 2001, 64, 064515

In: Superconductivity Research Developments Editor: James R. Tobin, pp. 167-185

ISBN 978-1-60021-848-4 c 2008 Nova Science Publishers, Inc.

Chapter 7

S INGLE I NTRINSIC J OSEPHSON J UNCTION FABRICATED FROM Bi2 Sr2CaCu2 O8+x S INGLE C RYSTALS L. X. You1,2,∗, A. Yurgens2 , D. Winkler2 and P. H. Wu3 1 State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, PR China 2 Quantum Device Physics Laboratory, Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-412 96 G¨oteborg, Sweden 3 Research Institute of Superconductor Electronics, Department of Electronic Science and Engineering, Nanjing University, Nanjing 210093, PR China

Abstract Due to the short superconducting coherent length of high temperature cuprate superconductor (HTS), intrinsic Josephson effect is an exclusive tunneling effect which can be observed with HTS till now. With conventional photolithography and precise control of Ar-ion etching, we have first successfully isolated a single intrinsic Josephson junction in two geometries: a U-shaped mesa on top of- and a zigzag structure inside a Bi2Sr2 CaCu2 O8+x single crystal. The refined fabrication methods are introduced and compared. The single intrinsic Josephson junction (SIJJ) in both structures shows typical single junction behavior, however, with some different characteristics at a high current bias. Both two methods are quite controllable and reproducible. Subgap structures are observed in SIJJ with U-shaped mesa structure. However, the heating effect is still evident. In the SIJJ/IJJs with double-sided zigzag structure, the heating effect is much less. We can observe a few upturn (peak) structures in I-V (dI/dV-V) curves, which may originate from the in-plane superconducting transition and/or the energy gap. The SIJJ/IJJs with double-sided zigzag structure are important for both fundamental research like macroscopic quantum tunneling and applications like HTS SQUID. ∗

E-mail address: [email protected]

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Introduction

High temperature cuprate superconductors (HTS) are known to be of quasi-twodimensional (2D) nature. In Bi2Sr2 CaCu2O8+x (BSCCO) with Tc up to over 90 K, conducting/superconducting CuO layers alternate with insulating BiO and SrO layers along the c axis thus forming a natural stack of tunnel junctions. This intrinsic tunneling (Josephson) effect was first observed in BSCCO single crystals, and later in many other HTS materials [1–3]. The intrinsic Josephson effect has turned out to be very useful in probing electronic properties of HTS “from inside” [4, 5]. The stacks of intrinsic Josephson junctions (IJJs) appear to be promising for high frequency applications like heterodyne receivers and quantum voltage standards [6, 7]. Besides, recent experiments of macroscopic quantum tunneling observed in IJJs implies possible applications for quantum information processing [8, 9]. On the other hand, intrinsic Josephson effect is the one and only tunneling effect which can be observed in HTS materials till now due to their short superconducting coherent length. With its good single-crystal quality and high anisotropy, BSCCO is the best material for studies on IJJs. There are three popular structures of IJJs made of single crystals: (I) the simple- or (II) U-shaped mesa structures and (III) the double-sided zigzag structure. Because one intrinsic junction is exceedingly thin (1.5 nm), it is hard to isolate and study small number of these junctions and most of the research results are therefore obtained on IJJs’ stacks enclosing many junctions. Recently, with the refined fabrication process, we achieved precise control of the junction number in the stacks [10]. Any small number of junctions could be fabricated, including the most interesting case of a single intrinsic Josephson junction (SIJJ) [11–13]. A SIJJ-sample can make it possible to study intrinsic tunneling with less or without the negative effects of self-heating and sample inhomogeneities. Indeed, the self-heating effect in a stack with many junctions cannot be neglected [14, 26] due to a rather low thermal conductivity of the material. The more junctions in the stack, the more severe is the heating while in a SIJJ this side effect is reduced to some extent. Next, many results obtained from IJJs like the tunneling spectroscopy of the superconducting energy gap are based on the assumption that all the junctions are equal in properties. Difference in critical currents or superconducting gaps can result in smearing of the tunneling peaks or appearance of extra features. It is clear that to deal with this problem one has to either make sure that all the junctions in the stack are identical or reduce the junction number down to one. In this chapter, we present the refined fabrication techniques and results obtained on the SIJJ-samples. The fabrication process and junction properties in different structures are discussed in detail. Measurement results and physical properties of the SIJJ are also analyzed. The content is subdivided in several sections in accordance with different fabrication processes and final sample shapes.

2.

Simple Mesa Structure

A mesa formed on the surface of a single crystal is the simplest and easiest-to-make structure of IJJs (see Fig. 1(a)). Other geometries of IJJs/SIJJ are derived from this structure. The fabrication process is as follows. First, we deposit a metal (gold or silver) layer on

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a freshly-cleaved BSCCO single crystal. Both thermal/E-beam evaporation and ion beam aided deposition work quite well. However, experience shows that the latter provides the best metallic contact with smallest contact resistance and dc/ac sputtering is not recommended for this process. Then, we fix it in the center of a silicon or MgO substrate using polyimide as the glue. With conventional photolithography and Ar-ion milling, mesas covered with Au are formed on the top surface of the BSCCO single crystal. Several different mesas can be made during this step, with lateral sizes from a few up to several tens of microns. The mesas are then surrounded by an insulating layer of CaF 2 or SiO which is evaporated to isolate and protect the mesas. After that, we make thin-film metallic electrodes reaching from contact pads on the substrate to the top of the mesas, which is realized by the process of gold deposition, photolithography and Ar-ion etching. The height of the mesas (the number of the junctions) is controlled by the timing of the first Ar-ion etching. The typical beam energy of 230 eV and beam intensity of 7 × 1014 s−1 cm−2 result in the etching rate of ∼1.5 nm/min for BSCCO, i.e. corresponding to about one junction per minute. More details of the fabrication process can be found elsewhere [10]. Final sample structure may be slightly different from Fig. 1(a) if the two base electrodes are formed onto other isolated larger mesas instead of the base crystal. However, it will not give evident difference to the measurement of the target mesa. Fig. 1(b) shows typical current-voltage (I-V) curves of IJJs in the simple mesa structure. All the branches are traced out by sweeping the bias current technique. Seven quasi-particle branches tell us that there are seven junctions in this stack. We note that there is one junction with a critical current (∼ 7 µA) about 30 times smaller than that (∼ 210 µA) of other six junctions with rather uniform critical currents. This is a surface junction just below the metal layer on the top of the mesa. Superconductivity is suppressed in this surface junction because it is in contact to the top metal layer. The property of the surface junction is different from that of the inner IJJs and detailed analysis has been discussed earlier [15,16]. In the simple mesa structure, we can precisely control the junction number in the stack even down to one by controlling the etching time. However, the surface junction always exists, which means that the obtained single junction in this geometry is the surface junction with different property from the inner intrinsic junctions [17]. In other words, it is impossible to obtain a genuine SIJJ in a simple mesa structure.

3.

U-Shaped Mesa Structure

The U-shaped mesa structure is developed from the simple mesa structure. After a simple mesa is fabricated, a slit can be etched from the top in the middle of the mesa by conventional photolithography and Ar-ion etching. This slit divides the top gold layer and a few junctions beneath into two parts, thus forming a U-shaped mesa with two top electrodes to the junctions below the slit (see Fig. 2). It is clear that only the junctions below the slit in the mesa will be ‘seen’ in the four-probe measurements as all the junctions in the arms of the U-shaped mesa including the surface junctions can be considered as parts of the electrodes. The number of effective junctions below the slit can be reduced down to one by controllable etching of the slit [11]. It is noted here that, because of the existence of an insulating

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Figure 1. (a) Schematic view of IJJs with a simple mesa structure. For simplicity, the isolating layer around the mesa is not drawn; (b) I-V curves of a IJJs’ stack 3 µm × 4 µm large and enclosing 7 junctions at 4.2 K. The inset shows the I-V curves at a low bias current indicating the existence of a degraded surface junction.

Figure 2. Schematic view of a SIJJ with a U-shaped mesa structure.

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BSCCO layer (∼ 3 nm thick) caused by Ar-ion etching, the upper surface of the one effective junction under the slit is not open to the ambient [10]. As the result, no evident degradation was observed in this ‘inner’ intrinsic junction. The detailed process and property of a U-shaped SIJJ have been discussed before [11, 12, 18]. Fig. 3(a) shows an optical image of a sample integrated with four identical U-shaped mesas, all enclosing just one effective junction. There are two extra large mesas on both sides which work as the base electrodes. The four-probe I-V characteristics of the four different mesas are very much alike as are seen in Fig. 3(b). This shows the good reproducibility and controllability of the U-shaped mesa structure. At lower voltages, subgap structures can be observed in the quasiparticle branches. Fig. 4 shows the I-V and the differential conductance (dI/dV-V) curves of the quasiparticle branch. Marked with solid arrows, the subgap structures at voltages of 6.2, 8.1, and 11.6 mV (less clear) are observed. It is consistent with earlier results from multi-junction stacks [19]. These features can be explained by considering coupling of the Josephson oscillations to the optical c-axis phonons within a modified RSJ model [20]. Although there can just be one effective junction in a U-mesa, it is still difficult to avoid the self-heating problems. Indeed, there are always several extra junctions under the electrodes formed after the etching of the slit. These junctions are smaller in size and therefore readily switch to the resistive state even at relatively small bias current. Despite they are not seen in the four-probe measurements, the heat dissipated in them affects the SIJJ and is usually manifested in back-bending of the quasiparticle branch of the SIJJ’s I-V curve. Fig. 5 shows an I-V curve of a U-shaped SIJJ with a high current bias. To illustrate the level of heating, the I-V curves of the mesa in the three-probe measurements are also given in the inset. From the last quasi-particle branch in the three-probe I-V curves, we may calculate the overall dissipated power of the whole stack at different bias currents. This dissipated power is the same for both four-probe and three-probe measurements because of the same current bias. The corresponding dissipation powers are shown in the four-probe IV curves as well as the inset of the three-probe I-V curves. Assuming the thermal resistance of ∼ 50 K/mW [21], we can estimate the temperature of the sample: 12, 17, 36, 54, and 73 K at the bias current of 0.7, 1, 2, 3, and 4 mA, respectively. The data shows a severe temperature increase at the bias corresponding to the back-bending region. At the bath temperature of 4.2K, the temperature of the mesa can in fact reach over 100 K at the highest bias current (6 mA). This is consistent with the previous report on [14]. Despite all that, U-shaped mesas with a SIJJ were very useful in measurements of the in-plane critical current of a single CuO plane [22]. The main idea is to let current distribute itself over the top electrode of the U-shaped mesa while injecting the current non-uniformly from one side of the mesa. Then, a part of the total current will flow along the ab-plane. There are some distinct features of the I-V characteristics (break) at the moment when this part of the current exceeds the critical value for the single plane [22]. Although the in-plane critical current density is three orders of magnitude higher than in the c-axis direction, jab  jc , the overall critical current of a single CuO plane iCuO is quite small and is comparable to the c-axis critical current of a micron-sized SIJJ. For example, for a 5 µm wide CuO plane, iab = 0.15 mA assuming jab = 2 MA/cm2 , which is slightly smaller than the critical current of a mesa 5×5 µm2 large for which ic = 0.25 mA assuming

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Figure 3. (a) Optical image of a sample integrating four SIJJ’s with the same size and geometry on one single crystal. The bright ones are the gold electrodes. (b) I-V curves of 4 SIJJ’s in the sample shown in (a) at 4.2 K. Each panel corresponds to the measurement of a different mesa.

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Figure 4. I-V and dI/dV-V curves of the quasiparticle branch of one mesa in low voltage range. The subgap structures are marked by solid arrows.

Figure 5. I-V curve of a SIJJ with U-shaped mesa structure at a high bias current. The dissipation power at different bias currents 0.7, 1, 2, 3, and 4 mA are marked on the curve. The inset shows the corresponding three-probe I-V curves with the dissipation power indicated as well.

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jc = 1 KA/cm2 . So when the bias current in a U-shaped SIJJ/IJJs stack is close to ic which is above iCuO , the single CuO plane may switch into its resistive state due to the relatively high current flow. In the U-shaped SIJJ system, we can observe a sharp upturn structure in the quasi-particle branch [22]; and in the U-shaped IJJs system, we may observe some extra quasi-particle branches besides the normal branches [23]. Those distinct features in I-V characteristics give us a way to measure the in-plane critical current of a single CuO plane and estimate the in-plane critical current density of a bulk material. The self-heating at such currents is relatively low and can only result in temperature increase inside the mesa by about 5-10 K which is acceptable.

4.

Double-Sided Zigzag Structure

Compared with the two structures described above, double-sided zigzag structure is more complicated and difficult to realize [6]. First, we make a simple mesa ∼100 nm thick using photolithography and Ar-ion etching. Looking from above, the mesa has the shape of a bow-tie antenna with a micro-bridge in the middle. Then, a slit ∼50 nm deep (plus the thickness of the gold layer) is etched down in the middle of the bridge (see Fig. 6(a)). The whole sample is then turned upside down and glued onto another substrate thus forming a ‘sandwich’ with the single crystal in between. Separating the two substrates cleaves the single crystal into two pieces, one with the flipped mesa glued onto the second substrate. We subsequently remove all the single crystals but the mesa by iteratively cleaving it using Scotch tape and inspecting the sample in an optical microscope. Four gold electrodes are then made on the banks of the bridge. Finally, we deposit protecting thin CaF 2 layer in which an open window across the bridge is left for subsequent etching with Ar ions through it (see Fig. 6(b)). During this final slow etching, the second slit is formed in the bridge. When the sum depth of the two slits becomes larger than the thickness of the bridge, the bottoms of the two slits overlap along the c axis. There will be a stack of junctions formed between the two slits and the final sample acquires a zigzag structure. Fig. 7(a) shows the schematic side view of the double-sided zigzag structure enclosing a SIJJ. Optical images of a sample are also shown in Fig. 7 (b) and (c). There are two slits in each image. the left one corresponds to the slit open on the top in (a); and the right one is the one on the bottom in (a). The bridge area between the two slits is the junction area with the sizes of 5.5 µm × 2.0 µm. We found that the long-term Ar-ion etching, even if using a low-energy- and current beam, can result in a degradation of superconductivity in BSCCO. Possible heating of the crystal by the ion beam during the etching in vacuum can easily be ruled out because the total energy of the beam that is dissipated (230 V × 0.1 mA/cm2 ∼ 25 mW/cm2 ) is very low. Indeed, assuming the thickness of polyimide to be 0.1 mm and its thermal conductivity of about 1 mW/cm·K, we get the overall temperature rise: (25 mW)/(100 mW/K) = 0.25 K. Another possible source for heating is a quite bright tungsten wire of the ion neutralizer. However, the similar heat-balance estimations show that the radiation from this wire can only result in a temperature increase not higher than 10 K. Moreover, switching the neutralizer off during the etching did not help anything regarding the degradation. It seems that other reasons like the Ar-ion implantation and/or the change of the oxygen content might explain the degradation. Although the reason remains to be proven exper-

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Figure 6. Schematic view of the fabrication process. (a) the sample just before flipping; (b) the sample ready for the consecutive etching-measurements cycles

Figure 7. (a) Schematic Side view of the sample when a stack of a SIJJ forms by continuous etching. The gold and CaF2 layers are marked by yellow and blue colors, respectively. The substrates are not shown for simplicity. The red area between the two slits indicates the effective junction; (b) and (c) Optical images of a real sample illuminated from the top and bottom respectively.

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Figure 8. R-T curves of a micro-bridge made of BSCCO single crystals. Three curves (from top down) are measured after 30 minutes etching without annealing, after 30 minutes etching with annealing, and the original micro-bridge without etching of the slit. imentally or at least by simulations, we note that just 5-10 min of annealing at 100 ◦C in air is enough to restore the superconductivity of samples. The critical temperature of the sample can therefore be retained practically unchanged if every five-minutes of etching is followed by five-minutes hot-plate annealing. Fig. 8 shows the effect of the annealing. The bottom green curve is the R-T curve of a sample before etching. After 30 minutes etching without annealing, the degradation is evident with a lower critical temperature (see the top square curve in the Fig. 8). However etching with annealing result in no degradation of the critical temperature at all (see the middle solid diamond curve in the figure). In the case of U-shaped mesas, the number of effective junctions decreases with the increase of the etching time. For the double-sided zigzag structure, the situation is opposite. After the two slits overlapped, the number of junctions increases one by one with further etching. For making a SIJJ in this geometry it is therefore crucial to determine the exact moment when the two slits overlap during the etching. There are two possible ways of detecting the appearance of the overlap. One way is to measure the resistance of the bridge during the etching. Before the overlap is reached, the resistance R increases with the etching time t because the bridge gets thinner in the region of the slit. At the same time, R can be considered as largely determined by the in-plane (ab) resistivity which shows a metallic behavior when the temperature decreases. After the overlap has been reached, the much higher c-axis resistivity starts contributing to the overall R and the slope dR/dt increases, i.e. the overlap is marked by a break in the R(t)-curve, especially prominent at the temperature close to the transition temperature. However, it

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Figure 9. Schematic view of the sample close to the overlap. n= 2, 1, 0, and -1 is the total number of those spacings that is still left between the bottoms of the slits in the c-axis direction. The scales of w, s and d are indicated. is difficult to precisely detect the moment of overlap by this method because the etching should be continued for some time after the moment of overlap in order to clearly reveal the break in R(t). Moreover, when the bias current is non-uniform due to the high anisotropy of material, both the ab- and c- axis resistivities contribute to R at all times making the break in R(t) less pronounced [24]. A more accurate way is to measure the I-V curves of the bridge at low temperature after each etching session, to track the changes of the critical current and I-V curves. Indeed, before the two slits overlap, the critical current of the sample is mainly determined by jab and the thickness of the bridge between the two slits because of jab  jc . When the slits are close to the overlap, the critical current of the bridge can be roughly approximated by Ic ≈ ws(n + 1)jab + wd jc

(1)

where w is the width of the bridge, d is the distance between the closest walls of the slits, s = 1.5 nm is the interlayer spacing, and n = −1, 0, 1, 2 . . . is the total number of those spacings that is still left between the bottoms of the slits in the c-axis direction before the overlap. Fig. 9 shows the schematic view of the sample close to the overlap with current flow indicated. From the Eq. 1 it is seen that i) the critical current of the bridge decreases in quite large steps on approaching the overlap and ii) stays constant after the overlap is reached. Fig. 10(a) and (b) shows the R-T and I-V characteristics after consecutively increasing the etching time. R(T ) in Fig. 10(a) changes from a typical metallic one (35 minutes of etching) to a slightly semiconductor-like (40 minutes of etching) and finally turns to be of dominantly semiconductor type (after 42 minutes of etching). The change indicates the transition from in-plane dominant resistivity to c-axis dominant resistivity.

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Figure 10. (a) R(T ) with the etching time as a parameter. The resistance R is normalized to the resistance at 273 K; (b) The I-V curves of the sample before and after the overlap appears. The thin (blue) curve with a higher critical current corresponds to the state before-, while the thick (red) curve shows the I-V curve of a SIJJ just after the overlap happens.

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Figure 11. I-V curves of the sample at a low bias current after further etching. The number of quasi-particle branches (and correspondingly, the number of junctions) increase from two to five. At t = 40 min, the bottom of the two slits are close to overlap, the critical current estimated from the thin (blue) I-V curve in Fig. 10(b) is about 1.3 mA. After two more minutes of etching, the critical current decreases abruptly to ≈ 50µA. The I-V curve (the thick red one in Fig. 10(b)) with strong hysteresis is seen, i.e. shows a typical behavior for an intrinsic Josephson junction. From the value of the critical current and lateral sizes of this SIJJ (5.5 × 2 µm2 ) we calculate the c-axis critical current density jc ≈ 500 A/cm2 which is typical for our single crystals. After the overlap is registered, we can revisit the earlier data and assign certain values of n for different etching times. Assuming that Eq. 1 holds, we can even estimate the inplane critical current density from the step-like decrease of the overall critical current of the bridge on approaching the overlap. The obtained jab (10 − 30 MA/cm2 ) was reproducibly found for a few samples. However, this value of jab is several times larger than the results reported earlier [22]. The reason for this discrepancy is not clear, but could be due to our oversimplified model (Eq. 1) and a more complicated current flow before the overlap. The number of junction increases with further etching. We followed the details of the emerging junctions (up to five) by tracing out the I-V characteristics after each etching step. The I-V curves with a low bias current are shown in Fig. 11. All the curves show standard multi-branch structures and consistent critical currents, which shows the perfect controllability of the fabrication process. To reach the superconducting energy gap of SIJJ/IJJs, we increase the bias range up to ±2 mA (see Fig. 12). The differential conductances σ(V ) = dI/dV (V ) are then computed from the I-V data. Fig. 13(a) and (b) show σ(V ) for several different etching times (number

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Figure 12. I-V curves of the same sample with different number of junctions (1-5) in a bias current range of ±2 mA. The number of junction has increased after irretrievable etching. of junctions). The voltage is normalized to the junction number in the first figure. Before discussing the results on the energy gap, we shall estimate if the self-heating effect in the stacks of SIJJ/IJJs in the zigzag structure is important or not. Since all the ‘leads’ to the stack are made of BSCCO, i.e. they are superconducting, the heating originates only from the stack itself. No back-bending quasiparticle branches were observed in the I-V curves with sample sizes in the range of 0.5 − 11 µm2 and the number of junctions less than five. This implies that there is not much of self-heating in stacks with so few IJJ’s. Latest experiments show that the thermal resistance is about 120 - 160 K/mW for the zigzag structure IJJs [25]. Some results showed even higher value (about 300 K/mW) [26]. However, they are all based on the sample of the junction number  5, and the backbending structure is evident at a high bias current. So it is reasonable to believe that the real thermal resistance in our samples is much smaller. In the ideal I-V curve of an SIS tunneling junction, there is one steep upturn at the energy-gap voltage. On the contrary, the I-V curves of IJJs/SIJJ show several smeared upturns both at low and high bias current. At a low bias current, this energy-gap-like upturn structure can be also observed when junction number increases up to five. However, it turns to be less evident with increasing the junction number (see Fig. 10(b) and Fig. 11). The upturns in I(V ) transform into peaks in σ(V ) dependence. To be able to compare samples with different number of junctions N , the voltage should be normalized by N (see Fig. 13(a)). Usually, N is taken as the number of distinct quasiparticle branches seen in the overall I-V curve, see Fig. 11. The figure shows that the voltage of the first conductance peak (Vp ≈ 32 mV ) is independent of N . The measurements of four other samples give

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Figure 13. σ(V ) = dI/dV (V ). The I(V ) raw data is from Fig. 12 while the voltage in (a) is normalized to the number of the junction number.

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similar results Vp ≈ 28 − 36 mV . No clear correlation between Vp and Tc could firmly be established, except for the sample with the smallest Vp = 28 meV that had the lowest Tc = 60 K. The peaks persist in a wide temperature range (T ≤ 0.8Tc) and for all low N ≤ 5, although becoming weaker with T and N . If the peak originates from a superconducting energy gap, it corresponds to the superconducting order parameter ∆ = Vp/2 ≈ 14 − 18 meV. Although being significantly smaller than in the majority of STM data (30 − 50 meV) [27–30], the suggested ∆ is not totally odd. In fact, similar values were repeatedly measured in break-junction experiments on overdoped samples and in point-contact measurements of the gap anisotropy [31–33]. In spite of the reproducibility of the results, one may argue that the peaks are due to, e.g. the current-driven transition of a surface CuO plane in the bottom of the slits [22]. When the current flowing in/out of the junction stack is larger than the in-plane critical current in its top/bottom superconducting CuO plane, the CuO plane may enter its resistive state with a high resistance. Then a part of the current will flow in c-axis and an extra junction will be biased into its quasiparticle state. When the critical current of the CuO plane is less than the critical current of the c-axis junction (long junction), we will observe an extra quasiparticle branch (see Fig. 2(a) in Reference [23]). When the critical current of the CuO plane is larger than the critical current of the c-axis junction (wide junction), the extra junction will show as a upturn in I-V’s and correspondingly as a peak in dI/dV-V. Indeed, taking the sheet critical current density of 0.3 − 0.7 A/cm for a CuO plane and the width of the bridge (5.5 µm), we get a range of the critical currents (0.16 − 0.39 mA) which obviously includes the current at which the first upturn is seen in the I-V curves. In the I-V curves of the sample with two and four junctions 1, there are two features (marked by the arrows in Fig. 12) which are evidently caused by such transitions. When the bias current is further increased, the two curves go closer to the one corresponding to the stacks with three and five junctions, i.e. the stacks can have one more effective IJJ that is only revealed at a high bias current. When the bias current exceeds 1 mA, all the I-V curves enter the state of approximately constant resistance, which probably means that it is the last upturn in the I-V curves (the last conductance peak in the dI/dV-V curves) that should be taken as the energy gap. Now we can assume that all but the last conductance peaks are merely due to ‘hidden’ IJJs that are reveal themselves one-by-one when the current-induced in-plane superconducting transitions take place (see Fig. 14). These IJJs lie under or above the actual stack and are normally short-circuited by the superconducting planes and Josephson current between them. The number of effective junctions should then be recalculated in order to obtain the correct value of the energy gap ‘per junction’. The result is shown in Fig. 13(b) where the positions of major conductance peaks are assumed to be integer multiples of some characteristic voltage (= 2∆/e). In most cases the integer used here is the number of branches seen at low bias (‘visible’ IJJs) plus the number of extra peaks at high bias (‘hidden’ IJJs). For example, in the dI/dV-V curves of the two quasi-particle branches’ IJJs, we have two extra peaks besides the last one indicating the energy gap. So the effective junction num1

The ‘4 IJJs-v2’ correponds to one more minute etching after ‘4 IJJs-v1’. The I-V curves of them are same at a low bias current with four quasi-particle branches. However; they are different at a high bias current; as shown in Fig. 12.

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Figure 14. Schematic view shows the possible ‘hidden’ junctions at high bias current. ‘Hidden’ junctions are marked by net under and above the IJJ stack and current flows are indicated by arrows.

ber for computing the energy gap should be four. Based on this assumption, the effective junction number in the 6 curves at high bias current are 2, 4, 4, 5, 6 and 6 respectively. This assumption gives us an estimate of the energy gap parameter ∆ ≈ 23 − 25 meV, which is consistent for different junctions number. Although this procedure and results are quite reasonable, we admit that it is in fact quite a rough estimation based on several assumptions. To make this issue clear, the inplane transition should somehow be avoided. Junction size should be reduced down to a sub-micron scale to make the bias current needed to reach the energy gap feature much lower than the current for the in-plane transition.

5.

Conclusion

With the improvement in the fabrication process, we can realize a SIJJ with both U-shaped mesa structure and double-sided zigzag structure. Both two methods are quite controllable and reproducible. Subgap structures are observed in SIJJ with U-shaped mesa structure. However, the heating effect is still evident. In the SIJJ/IJJs with double-sided zigzag structure, the heating effect is less evident. We can observe a few upturn (peak) structures in I-V (dI/dV-V) curves, which may originate from the in-plane superconducting transition and/or the energy gap. The SIJJ/IJJs with double-sided zigzag structure are important for both fundamental research like macroscopic quantum tunneling and applications like HTS SQUID.

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References [1] Kleiner R.; Steinmeyer F.; Kunkel G.; M¨uller P. Phys. Rev. Lett. 1992, 68, 2394-2397. [2] Oya G.; Aoyama N.; Irie A.; Kishida S.; Tokutaka H. Jpn. J. Appl. Phys. Part 2 1992, 31, L829-831. [3] M¨uller P. EURESCO conference on Future Perspectives of Superconducting Josephson Devices 2002, Pommersfelden, Germany. [4] Krasnov V. M. ; Kovalev A. E.; Yurgens A.; Winkler D. Phys. Rev. Lett. 2001, 86, 2657-2660. [5] Suzuki M.; Watanabe T. Phys. Rev. Lett. 2000, 85(22), 4787-4790. [6] Wang H. B.; Wu P. H.; Yamashita T. Phys. Rev. Lett. 2001, 87, 107002. [7] Wang H. B.; Wu P. H.; Chen J.; Maeda K.; Yamashita T.; Appl. Phys. Lett. 2002, 80, 4060-4062. [8] Inomata K.; Sato S.; Nakajima K.; Tanaka A.; Takano Y.; Wang H. B.; Nagao M.; Hatano H.; Kawabata S. Phys. Rev. Lett. 2005, 95, 107005. [9] Jin X. Y.; Lisenfeld J.; Koval Y.; Lukashenko A.; Ustinov A. V.; M¨uller P. Phys. Rev. Lett. 2006, 96, 177003. [10] You L. X. ; Wu P. H.; Ji Z. M.; Fan S. X.; Xu W. W.; Kang L.; Lin C. T.; Liang B.; Supercond. Sci. Technol. 2003, 16, 1361-1364 . [11] You L. X.; Wu P. H.; Xu W. W.; Ji Z. M.; Kang L. Jpn. J. Appl. Phys. Part 1 2004, 43(7A), 4163-4165. [12] Wu P. H.; You L. X.; Chen J.; Ji Z. M.; Xu W. W.; Kang L.; Lin C. T.; Liang B.; Physica C 2004, 405, 65-69. [13] You L. X.; Torstensson M.; Yurgens A.; Winkler D.; Lin C. T.; Liang B. Appl. Phys. Lett. 2006, 88, 222501. [14] Krasnov V. M.; Sandberg M.; Zogaj I. Phys. Rev. Lett. 2005, 94(7), 077003. [15] Kim N.; Doh Y.-J.; Chang H.-S.; Lee H.-J. Phys. Rev. B 1999, 59, 14639-14643. [16] Zhu X. B.; Zhao S. P.; Chen G. H.; Tao H. J.; Lin C. T.; Xie S. S.; Yang Q. S. Physica C 2004, 403, 52-56. [17] Odagawa A.; Sakai M.; Adachi H.; Setsune K. IEEE Tran. on Appl. Supercond. 1999, 9(2), 3012-3015. [18] You L. X.; Wu P. H.; Chen J.; Xu W. W.; Kajiki K.; Watauchi S.; Tanaka I. Supercond. Sci. Technol. 2004, 17, 1160-1164. [19] Schlenga K.; Kleiner R.; Hechtfischer G. et al. Phys. Rev. B 1998, 57, 14518-14536.

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[20] Helm Ch.; Preis Ch.; Forsthofer F.; Keller J.; Schlenga K.; Kleiner R.; M¨uller P. Phys. Rev. Lett. 1997, 79, 737-740. [21] Yurgens A.; Winkler D.; Claeson T.; Ono S.; Ando Y. Phys. Rev. Lett. 2004, 92, 259702. [22] You L. X.; Yurgens A.; Winkler D. Phys. Rev. B 2005, 71, 224501. [23] You L. X.; Yurgens A.; Winkler D.; Torstensson M.; Watauchi S.; Tanaka I. Supercond. Sci. Technol. 2006, 19; S209-S212. [24] Busch R.; Ries G.; Werthner H. et al. Phys. Rev. Lett 1992, 69, 522-525. [25] Verreetz B.; Sergeantz N.; Negretez D. M.; Torstensson M.; Winkler D.; Yurgens A. Supercond. Sci. Technol. 2007, 20; S48-S53. [26] Wang H. B.; Hanato T.; Yamashita T.; Wu P. H.; M¨uller P. Appl. Phys. Lett. 2005, 86, 023504. [27] Renner C.; Fischer Ø. Phys. Rev. B 1995, 51, 9208-9219. [28] Fang A. C.; Capriotti L.; Scalapino D. J.; Kivelson S. A.; Kaneko N.; Greven M.; Kapitulnik A. Phys. Rev. Lett. 2006, 96, 017007. [29] Lang K. M.; Madhavan V.; Hoffman J. E.; Hudson E. W.; Eisaki H.; Uchida S.; Davis J. C. Nature 2002, 415, 412-416. [30] McElroy K.; Lee J.; Slezak J. A.; Lee D.-H.; Eisaki H.; Uchida S.; Davis J. C. Science 2005, 309, 1048-1052. [31] Zasadzinski J. F. et al. Phys. Rev. Lett. 2001, 87, 067005. [32] Ozyuzer L.; Zasadzinski J. F.; Kendziora C.; Gray K. E.; Phys. Rev. B 2000, 61, 3629-3641. [33] Kane J.; Ng K. -W.; Phys. Rev. B 1996, 53, 2819-2826.

In: Superconductivity Research Developments Editor: James R. Tobin, pp. 187-222

ISBN: 978-1-60021-848-4 © 2008 Nova Science Publishers, Inc.

Chapter 8

RARE EARTH MODIFIED (Bi,Pb)-2212 SUPERCONDUCTORS A. Biju and U. Syamaprasad National Institute for Interdisciplinary Science and Technology (CSIR), Trivandrum-695019, India

Abstract This chapter deals with a new class of (Bi,Pb)-2212 based superconductors with highly enhanced superconducting properties by modifying the system with the addition of Rare earths (RE) such as La, Nd, Gd, Dy and Yb. The effect of stoichiometric addition of the REs on the structural, superconducting and flux pinning properties of the bulk superconductor was studied and presented in detail. The samples were prepared by solid state synthesis in polycrystalline form. The RE content in the samples were varied from 0.0 to 0.5 on a general stoichiometry of Bi1.7Pb0.4Sr2.0Ca1.1Cu2.1RExOy (where x = 0.0, 0.1, 0.2, 0.3, 0.4 and 0.5. RE = La, Nd, Gd, Dy and Yb). The samples were characterized using Differential thermal analysis (DTA), X-ray diffraction analysis (XRD), Scanning electron microscopy (SEM), Energy dispersive X-ray analysis (EDX) density measurements and R-T measurements. Superconducting parameters such as critical temperature (TC), critical current density (JC) in self field and applied field, at a comparatively higher temperature of 64 K, of the samples were also measured. It was found that, when RE ions are added to (Bi,Pb)-2212 system, they enter into the crystal structure replacing Sr and/or Ca with significant changes in the lattice parameters, microstructure, normal state resistivity, hole concentration and flux pinning strength of the system. Consequently the TC, JC and the field dependence of JC (JC-B characteristics) of the system enhance considerably for an optimum doping level. At higher doping levels these properties decrease from the maximum values. The enhancement in these properties are explained to be due to the substitution of RE3+ ions in place of Sr2+/Ca2+ ions with consequent change in charge carrier concentration (holes) in the Cu-O2 planes. The decrease in the number of charge carriers in (Bi,Pb)-2212 change the system from ‘overdoped’ to ‘optimally-doped’ condition. The substituted RE3+ ions also act as pinning centers as point like defects and improve the field dependence of JC and hence the flux pinning properties. There is a possibility of formation of nano-size secondary precipitates, which may also act as flux pinning centers. At higher levels of addition, the system again changes from ‘optimally-doped’ condition to ‘under-doped condition’. Further the chemical inhomogenity and secondary phase fraction increases at higher levels of RE in the system, which in turn brings down the superconducting properties.

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1. Introduction New electronic materials have been created by doping their inert parent compounds. Doping generally involves, introduction of impurities or charge carriers, which can be holes or electrons into the inert materials. The parent compounds of cuprate superconductors are Mott insulators. Superconductivity arises when they are doped away from stoichiometry. Temperature-charge carrier concentration phase diagram of these cuprate superconductors ( Fig.1) shows that the variation of transition temperature (TC) with carrier concentration is parabolic. TC becomes maximum at an optimum doping level, because it decides the charge carrier density. TC decreases when the system is 'overdoped' or 'underdoped'. Thus the superconducting properties such as TC and also critical current density (JC) of cuprate superconductors can be improved by appropriate doping. The structural and flux pinning properties of the system can also be tailored by doping, to make the material suitable for application at higher temperatures and higher fields. Doping can also be used as a powerful tool to explore the mechanism of high temperature superconductivity (HTS). For example, substitution of impurity atoms for Cu strongly perturbs the surrounding electronic environment and can therefore be used to probe HTS at atomic scale. This has provided the motivation for several experimental and theoretical studies [1-4]. Scanning tunneling microscopy (STM) is an ideal technique for the study of such effect at the atomic scale.

PG AFM

Temperature

NFL

TC SC

Under doped

M

Over doped

Hole doping Abbreviations are Superconductor (SC), Antiferromagnetic insulator (AFM), Pseudogap (PG), non-Fermi liquid (NFL) and metallic (M).

Figure 1. Temperature-Hole doping phase diagram.

Rare Earth Modified (Bi,Pb)-2212 Superconductors

189

Among the efforts that have been made to improve the superconducting properties of Bi2212, many research groups have investigated the substitution of Bi by Pb [6-13] and Ca or Sr by Rare earths (RE) [14-45]. When Pb is substituted for Bi, it occupies the Bi site due to the reason that Pb and Bi cations have rather similar radii and can occupy octahedral sites of Bi-2212. The replacement of trivalent Bi3+ by divalent Pb2+ increases the hole concentration in the Cu-O2 planes [7-9]. But there are a few reports [12, 13] which show that Pb incorporation does not affect the carrier concentration significantly. X-ray powder diffraction (XRD) patterns from Pb-substituted Bi-2212 [(Bi,Pb)-2212] samples show same reflections with similar intensities indicating that the fundamental structure is unchanged by Pb substitution and Pb-atoms occupy the Bi-site. Reports suggest that the maximum solubility limit of Pb in Bi-2212 corresponds to the formula Bi1.6Pb0.4Sr2CaCu2O8+δ [46]. Several investigations have pointed out that Pb substitution reduces structural modulation by the removal of oxygen atoms in the Bi-O layers [47-49]. As a result the anisotropy of the crystal is reduced. Cationic substitutions of RE in Bi-2212 confirm the existence of solid solutions over a wide range and the doped structure does not change fundamentally [14-16]. RE doping studies are carried out in many forms, such as bulk polycrystalline forms prepared by solid state synthesis and melt texturing, thin films or single crystalline samples. Majority of RE substitution studies are in Ca site and a few in Sr site. These studies have concluded that even though RE substitution improves the structural stability and induces a change in TC, at higher levels of substitution a metal -insulator transition occurs [20,25,34-40]. The substitution studies also discuss properties such as thermoelectric power (TEP), Hall effect, thermal conductivity etc [15-23]. Bi-2212 exhibit poor flux pinning properties due to its anisotropic nature and absence of effective pinning sites [11]. One of the effective ways to improve the flux pinning properties of this material is the creation of pinning centers by doping. The impurities chosen for flux pinning should not deteriorate the superconducting properties. There are few reports that RE substitution at Ca site improve the flux pinning properties [23, 43-45]. Almost all these previous RE substitution studies are on Pb free Bi-2212. Our recent studies on RE addition in Pb doped Bi-2212 show highly enhanced superconducting and flux pinning properties comparing with Bi-2212 doped with either Pb or RE [50-61]. This chapter reviews the effect of stoichiometric addition of RE such as La, Nd, Gd, Dy, and Yb on (Bi,Pb)-2212 superconductor. These REs are chosen from left, middle and right end of the periodic table with different ionic size including magnetic and nonmagnetic rare earths. Ce additions have shown entirely different behavior [56] in comparison with the other RE additions and hence it is not included in this chapter.

2. Experimental RE-added (Bi,Pb)-2212 superconductors were prepared by the conventional solid state synthesis method, with an initial stoichiometry of Bi1.7Pb0.4Sr2Ca1.1Cu2.1RExO8+δ (x = 0.0, 0.1, 0.2, 0.3, 0.4 and 0.5). This is the optimized stoichiometry based on our initial trials. Stoichiometric quantities of high purity chemicals such as Bi2O3, PbO, SrCO3, CaCO3, rare earth oxides and CuO (Aldrich> 99.9%) were weighed using an electronic balance (Mettler AE 240) and mixed using a planetary ball mill (Frisch-Pulverisette 6), with agate bowl and balls in an acetone medium for 1 h. The dried powders were subjected to a three stage

190

A. Biju and U. Syamaprasad

calcination process (800 0C/15 h + ~825 0C/~30 h + 830~840 0C/40 h) with intermediate wet grinding in acetone medium. All calcinations were done in air with a heating rate of 3 0C/min. The average particle size of the precursor powders was estimated to be 4-6 μm using a particle size analyser (Micromeritics sedigraph 5100). The samples were pelletized using a cylindrical die of 12 mm diameter under a force of 60 KN. Heat treatments of the pellets were done in two stages [845~850 0C /(60 + 60) h] with one intermediate repeated pressing.

Figure 2. Schematic diagram of JC and JC-B measurement set up.

Rare Earth Modified (Bi,Pb)-2212 Superconductors

191

The phase analysis and crystallographic study of the samples were done using XRD (Philips X’pert Pro) employing ‘X’celerator’ and a monochromater on diffracted beam side. The phase identification was performed using ‘X’pert High Score software’ supported with ICDD-PDF 2 database. Densities of the samples were determined by measuring the mass and dimensions of the pellets before and after the heat treatments. Thermal analysis was done using DTA (SERTAM setsys 16/18). Microstructural examination and elemental analysis of the samples were done using an SEM (JOEL JSM 5600LV) with EDX (PHOENIX) attachment. The R-T vatiations and critical temperature measurements were conducted by using a homemade set up and a bath cryostat in the temperature range of 64 to 300 K using the four-probe method using a nano voltmeter (KEITHLEY 181), and a constant current source (KEITHLEY 220). A temperature controller (LAKESHORE 340) was used to accurately monitor and control the temperature. Transport critical current measurements were conducted at 64 K with the standard criterion of 1 μV cm-1. The critical current dependence with magnetic field (JC-B) was studied using a home made set up with an electromagnet (Fig.2). Bar shaped samples of dimensions about 12mm x 3mm x 1mm cut from round pellets were used for the JC-B measurements. Electrical contacts were made using highly conducting silver paste applied to the pellet and subsequently cured at about 600oC for one hour. In some cases the electrical contacts were made by placing silver strips at the surface of the pellets during pressing. This type of contact reduces the contact resistance to a large extent. The measuring instruments were interfaced to a PC using GPIB-PC and data collection and analysis were performed using appropriate software. We label RE0, RE1, RE2, RE3, RE4, RE5 [where RE = La, Nd, Gd, Dy and Yb] for the samples with RE content in Bi1.7Pb0.4Sr2Ca1.1Cu2.1RExO8+δ with x = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5.

3. Results and Discussion 3.1. Differential Thermal Analysis (DTA) of the Samples Fig. 3 shows the DTA curves of the Nd and Gd added samples after calcinations taken in air at a heating rate of 10 oC/min. Two endothermic peaks are observed above 800 oC for all the samples. The first endothermic peak which is smaller corresponds to the formation of (Bi,Pb)2212 phase and the second endotherm corresponds to the complete melting of the system. For pure sample the second endothermic peak which appears at 851 oC is sharp. But for the RE added samples the endotherm broadens and shifts to higher temperatures with RE concentration. This shows that the added RE oxides not only causes an increase in the melting point of the system but also broadens it. This increase in the melting point of the RE added samples can be related to the melting temperature RE2O3 which is much higher (2270 oC for Nd2O3 and 2350 oC for Gd2O3) as compared to that of (Bi,Pb)-2212. Similar type of behavior occurs in the case of other RE additions also. It is also found that the Pb doping at Bi site can reduce the melting point of pure and RE added Bi-2212 [61]. The DTA results together with our preliminary data on (Bi,Pb)-2212 phase formation with respect to temperature enabled us to choose a temperature in the range 845-850 oC for the preparation of the pure sample.

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A. Biju and U. Syamaprasad

650

700

750

800

850

650

GD4

Nd4

GD3

Nd3

GD2

Nd2

GD1

700

750

800

850

Nd5

(b)

Endo/Exo

Endo/Exo

GD5

900

900

(a)

Nd1

GD0

Nd0

650

700

750

800

o

850

900

Temperature ( C)

650

700

750

800 o

Temperature ( C)

850

900

Figure 3. DTA curves of (a) Nd and (b) Gd added samples.

3.2. X-ray Diffraction (XRD) Analysis 3.2.1. Phase Analysis of Calcined Powders The XRD patterns of the samples after the second stage calcination at 825 oC are shown in Fig. 4. With every batch of doped samples, a pair of pure sample is also prepared for measurements to avoid the effect of slight variations in the processing conditions. Irrespective of the kind of rare earth added peaks of (Bi,Pb)-2212, Bi-2201, Ca2PbO4 and CuO are detected for all samples at this stage, among which (Bi,Pb)-2212 is the major phase. This result shows that for all RE additions the intermediate phases formed are Bi-2201 and Ca2PbO4. The CuO is not fully reacted at this stage. Some additional peaks of very small intensity are also observed in the Gd and Yb additions. These phases were analysed using X’pert Highscore software and identified as intermediate phases and unreacted RE oxides. For example, in the case of Gd addition peaks of Gd2O3 [88-2165] is present and in the case of Yb addition, patterns of Yb4 and Yb5 peaks of SrPbO3 [25-0898] and Yb2O3 [88-2161] are present and very small fraction of Cu0.73Pb2.03Sr3O7.7 [42-0146] is also present in all the samples.

Rare Earth Modified (Bi,Pb)-2212 Superconductors

0-Bi-2201 1-Bi-2212 @-Ca2PbO4 $-CuO

0

1

0

1 1

@ 0 1

1

1

$ 11

@

$

0

$

1

@

1 La5

Intensity in arb.units

@

193

La4

La3

La2

La1

La0

15

20

25

30

35 2 theta

40

o

45

50

55

Figure 4(a). XRD patterns of the La added samples after second stage calcination. 0-Bi-2201 1-(bi,Pb)-2212 @-Ca2PbO4 $-CuO

0 1

1

@

1 0 @

0

1

1

1 1

@

$

$ $

@

0

1 1

Intensity in arb. units

Nd5

Nd4

Nd3

Nd2

Nd1

Nd0

15

20

25

30

35 2 theta

40

45

50

55

o

Figure 4(b). XRD patterns of the Nd added samples after second stage calcination.

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A. Biju and U. Syamaprasad 1

1 1

@

1 * @ *

0

1

1

*-Gd2O3 0-Bi-2201 1-Bi-2212 @-Ca2PbO4 1 Bi Ca O #-Sr 8.5 6 2.5 22

1

0 #

1

@

@

#

# 0

#1

Gd5

Intensity in arb. units

Gd4

Gd3

Gd2

Gd1

Gd0 15

20

25

30

35

40

45

50

55

0

2 theta

Figure 4 (c). XRD patterns of the Gd added samples after second stage calcination. 1

1

1

1 @ 1

1

1

1 $

1-(Bi,Pb)-2212 0-Bi-2201 @-Ca2PbO3 $-CaO 1

$

Intensity in arb units.

Dy5

Dy4

Dy3

Dy2 0

0 Dy1

0 Dy0

15

20

25

30 o

35 2 theta

40

45

50

Figure 4 (d). XRD patterns of the Dy added samples after second stage calcination .

55

Rare Earth Modified (Bi,Pb)-2212 Superconductors

1 $

$ 1& 0 #

1

@

Intensity (arb. unit)

1

1

1 - (Bi,Pb)-2212 @-Ca2PbO4 # - SrPbO3, + - CuO & - Yb2O3 0 - Bi-2201 $ - Sr3Pb2.03Cu0.73O7.7

1

1

1

@

# 0 &

0

195

@

+ + +

#

1

1 Yb5

Yb4

Yb3

Yb2

Yb1

Yb0

15

20

25

30 35 2 theta (degree)

40

45

50

55

Figure 4(e). XRD patterns of the Yb added samples after second stage calcination.

The volume percentage of different phases was calculated from these XRD patterns using the relation:

Fx =

ΣI x X 100 ΣI

(1)

where Fx is the volume percentage of the phase x, ΣIx is the sum of the integrated peak intensities of x, and ΣI is the sum of the integrated peak intensities of all the phases present. The values calculated using the relation are tabulated in Table 1. From the Table it is clear that larger RE ions such as La and Nd inhibit the formation of (Bi,Pb)-2212. The volume percentage of (Bi,Pb)-2212 decreases from 72% to 55.2%, when La is added to a stoichiometric level x = 0.5. At the same time volume percentage of Bi-2201increases from 11.6% to 34.9%, but the amount of Ca2PbO4 decreases with increase in La content. In the case of Nd addition also volume percentage changes from 85.8% for Nd0 to 75.2% for Nd4. Dy and Yb additions show an increase of 9.1 % and 16.3% increase in the formation of (Bi,Pb)-2212 comparing with undoped samples and there is a corresponding decrease in the volume percentage of Bi-2201. These results clearly show that smaller RE ions such as Dy, and Yb favours the formation of (Bi,Pb)-2212. But in the case of Gd addition there is no significant variation in the formation of (Bi,Pb)-2212 with increase in Gd content.

196

A. Biju and U. Syamaprasad Table 1. Volume percentage of different phases in the samples after second stage calcinations RE

La

Nd

Gd

Dy

Yb

Sample

Bi-2212

Bi-2201

Ca2PbO4

CuO

Other*

La0

72.1

11.6

13.9

2.4

0

La1

71.9

16.6

9.5

2.0

0

La2

67.6

21.7

8.2

2.5

0

La3

64.8

22.3

7.6

5.3

0

La4

64.2

27.4

5.7

2.7

0

La5

55.2

34.9

5.1

4.8

0

Nd0

85.8

10.3

2.8

1.1

0

Nd1

84.0

10.9

3.4

1.7

0

Nd2

78.0

13.7

4.8

3.5

0

Nd3

76.6

14.1

5.1

4.2

0

Nd4

75.2

15.4

5.1

4.3

0

Nd5

82.6

10.6

4.2

2.6

0

Gd0

89.4

6.5

3.1

1.0

0

Gd1

87.0

8.9

1.6

2.2

0.5

Gd2

90.3

5.7

1.7

2.2

1.1

Gd3

84.8

9.0

2.2

2.6

1.4

Gd4

89.5

4.4

2.0

2.7

1.6

Gd5

86.2

6.2

2.4

3.3

1.9

Dy0

76.2

12.8

9.8

1.2

0

Dy1

87.0

5.3

6.2

1.5

0

Dy2

82.5

4.6

10.0

2.9

0

Dy3

85.6

4.7

7.5

2.3

0

Dy4

84.9

3.4

8.5

3.2

0

Dy5

85.3

1.7

9.9

3.1

0

Yb0

70.0

15.3

11.2

3.5

0

Yb1

75.4

10.5

10.8

1.6

1.7

Yb2

84.2

6.7

6.5

1.4

1.2

Yb3

86.3

3.7

6.7

1.6

1.7

Yb4

85.6

1.6

8.5

1.1

3.2

Yb5

84.5

1.3

6.7

1.1

6.4

* Other phases are mainly Gd2O3 in Gd added samples and SrPbO3 and Cu0.73Pb2.03Sr3O7.7 in Yb added samples.

(200/020)

(220)

(0210)

(117)

197

All peaks are of (Bi,Pb-2212) (0012)

(113)

(0010)

(006)

(008)

(115)

Rare Earth Modified (Bi,Pb)-2212 Superconductors

Intensity in arb.units

La 5 La 4 La 3 La 2 La 1 La 0 15

20

25

30

35 2Theta

o

40

45

50

55

(220)

(0210)

All peaks are of (Bi,Pb-2212) (0012)

(200/020)

(0010)

(006)

(008)

(113)

(115)

(117)

Figure 5 (a). XRD patterns of the La added samples after last stage heat treatment.

Intensity in arb. units

Nd5

Nd4

Nd3

Nd2

Nd1

Nd0

15

20

25

30

35 2 theta

o

40

45

50

55

Figure 5 (b). XRD patterns of the Nd added samples after last stage heat treatment.

(220)

(0012)

(0210)

All peaks are of (Bi,Pb-2212)

(0010)

(113)

(008)

(006)

(200/020)

(117)

A. Biju and U. Syamaprasad

(115)

198

Intensity in arb.units

Gd5

Gd4

Gd3

Gd2

Gd1

Gd0 15

20

25

30 0

35

2 theta

40

45

50

55

(200/020)

(0210)

(220)

All peaks are of (Bi,Pb-2212) (0012)

(117)

(115)

(113)

(0010)

(006)

(008)

Figure 5 (c). XRD patterns of the Gd added samples after last stage heat treatment.

Intensity in arb units.

Dy5

Dy4

Dy3

Dy2

Dy1

Dy0

15

20

25

30o

35 2 theta

40

45

50

55

Figure 5 (d). XRD patterns of the Dy added samples after last stage heat treatment.

(117)

(200/020)

1 #

1

1

#

1

1

@

#

Intensity in arb. units

@

(220)

(0210)

(006)

1

1

199

1 - (Bi,Pb)-2212) # - SrPbO3 @ - Cu0.73Pb2.03Sr3O7.7

(0012)

1

1

1

1 (0010)

(113)

(008)

(115)

Rare Earth Modified (Bi,Pb)-2212 Superconductors

Yb 5

Yb 4

Yb 3

Yb 2

Yb 1

Yb 0

15

20

25

30

35 2 theta

40

0

45

50

Figure 5 (e). XRD patterns of the Yb added samples after last stage heat treatment.

30.85 30.80

o

c- Axis Length ( A)

30.75 30.70 30.65 30.60

La Nd Gd Dy Yb

30.55 30.50 30.45 0.0

0.1

0.2

0.3

0.4

RE-content

Figure 6. Variations of c-axis length with RE concentration.

0.5

55

200

A. Biju and U. Syamaprasad

3.2.2. Phase Analysis of the Sintered Pellets Fig. 5 shows the XRD patterns of the sintered pellets after the last stage heat treatment. In all the rare earth additions except Yb, the only phase detected is (Bi,Pb)-2212. The (hkl) values of each peaks are marked on their top. In Yb4 and Yb5 samples very small fractions of SrPbO3 [70-1682] and Cu0.73Pb2.03Sr3O7.7 [42-0146] were also detected, when the XRD patterns were analysed using the X’pert high score software. It is interesting to note that even though the REs are added upto a stoichiometric level of x = 0.5, no secondary phases containing RE ions or other cations are observed in the doped system except for Yb addition. In Yb4 and Yb5 samples though secondary phases are detected these do not contain Yb. These results demonstrate that the added RE enter in the lattice site of (Bi,Pb)-2212.

3.2.3. Lattice Parameter Variations The lattice parameters a, b, c of different samples were calculated from the final stage XRD patterns [Fig.5] of the samples using the following relation assuming orthorhombic symmetry for (Bi,Pb)-2212 [44]: 2

2

1 ⎛h⎞ ⎛k ⎞ ⎛l ⎞ =⎜ ⎟ +⎜ ⎟ +⎜ ⎟ d2 ⎝ a ⎠ ⎝ b ⎠ ⎝ c ⎠

2

(2)

where ‘d’ is the inter-planar spacing and (hkl) are the Miller indices. The variations of c-axis length with RE concentration of different samples are plotted in Fig. 6. It is observed that in all the cases of RE additions except for Nd, the c-axis length decreases with increase in RE content, while in the case of Nd addition an increasing tendency is observed after Nd3 sample. The reported results on RE substitution at Ca site also show decrease of c-axis length with increase in RE content [16, 18, 20,24-27]. The reasons for c-axis contraction are explained by many authors [20,24,26]. High-TC copper oxides exhibit intergrowth structure consisting of superconducting layers of a fixed oxygen concentration and interactive layers like Bi-O layers of variable oxygen concentration which impart anisotropy, internal electric fields and mismatch in the bond lengths. Whenever a divalent cation such as Ca2+ or Sr2+ is being replaced by a trivalent rare earth ion the charge neutrality is established by incorporating additional oxygen ions in Bi-O planes of the crystal. As a result the net positive charge in the planes reduces. So the repulsion between them is reduced. This induces a contraction of Bi-O layers and causes an increase in the covalancy of Bi-O bonds. This results in a reduction of c-axis length and thereby improves the stability of Bi-2212 phase. It is interesting to note that the reduction in the c-axis length is least for La addition and gradually increases with decrease in ionic size and c-axis reduction is maximum for Yb, which has the smallest ionic radius among the REs chosen for the present study. This result shows that ionic radius also affects c-axis contraction. The average of a-and b-axis lengths increase with increase in RE content [Fig.7]. The elongation of a/b axis length is generally associated with the increase in the Cu-O bond length in the Cu-O2 planes, which controls the dimensions in the basal planes. A decrease in the difference in lengths of a- axis and b-axis as observed in Fig.7 shows that the orthorhombic nature of the crystals changes with an increase in concentration of RE. In the case of La addition a and b axis lengths become equal from x = 0.3 onwards and hence the crystal

Rare Earth Modified (Bi,Pb)-2212 Superconductors

201

structure change from orthorhombic to tetragonal. When Yb is added in Bi-2212 the a/b axis change is not considerable comparing with the other RE additions. 5.44 5.420

(a)

(b)

5.415

(a/b) axis length (A )

5.42

5.410

o

o

a/b axis length (A )

5.43

5.41

5.40

5.39

a-axis b-axis

5.38

5.405 5.400 5.395 5.390 5.385

a-axis b-axis

5.380

5.37

5.375 0.0

0.1

0.2

0.3

0.4

0.5

0.0

0.1

5.420

0.3

0.4

0.5

5.43

(d)

(c) 5.42

5.410

o

a/b-axis Length ( A)

0

a/b axis length ( A )

5.415

0.2

Nd stoichiometry

La-stoichiometry

5.405

5.400

5.395

5.41

5.40

5.39

5.38

5.390

a-axis b-axis

a-axis b-axis

5.37

5.385 0.0

0.1

0.2

0.3

0.4

Dy-Stoichiometry

0.0

0.1

0.2

0.3

0.4

0.5

Gd stoichiometry

(e)

0

a/b axis length ( A )

5.400

0.5

5.395

5.390

5.385

5.380

a-axis b-axis 5.375 0.0

0.1

0.2

0.3

0.4

0.5

Yb-Stoichiometry

Figure 7. Variations of a/b-axis with RE-stoichiomery.

3.3. Density Measurements Fig. 8 shows the variations of average density of the samples after different stages of sintering as a function of RE stoichiometry. The density values as percentages of the theoretical density of (Bi,Pb)-2212 (6.6 g/cm3) are shown on the right axis of the figures. From the graph, it is clear that the sintered densities of all the samples are less than the corresponding density prior to heat treatment (green density). This reduction in the sintered density occurs because of the layered growth mechanism of BSCCO wherein additional Ca-O and Cu-O layers are intercalated into Bi-2201 to form Bi-2212 during the sintering process. It is usually referred to as ‘retrograde densification’ [62]. Repeated intermediate pressing can reduce the retrograde

202

A. Biju and U. Syamaprasad

87.9

5.6

84.8

5.4

81.8

5.2

78.8

5.0

75.8

4.8

72.7

0.0

0.1

0.2

0.3

0.4

0.5

5.0

75.8

5.4

81.8

5.2

78.8

5.0

75.8

4.8

72.7

4.6

69.7

0.3

Gd stoichiometry

0.4

0.5

0.2

0.3

Nd stoichiometry

0.4

87.9

84.8

5.4

81.8

5.2

78.8

5.0

75.8

(d) 0.0

0.1

0.2

0.3

0.4

84.8

5.4

81.8

5.2

78.8

3

5.6

Theoretical Density(%)

87.9

Green Density Sintered Density Green Density after Repelletisation Sintered Density after Repelletisation

(e) 0.0

0.1

0.2

0.3

0.4

0.5

0.5

Dy stoichiometry

5.8

5.0

0.5

5.6

4.8 0.2

0.1

3

84.8

Density (g/cm )

5.6

0.1

72.7

(b)

75.8

Yb content

Figure 8. Variation of density of the samples at different stages of heat treatment.

72.7

Theoretical Density(%)

3

78.8

0.0

Theoretical Density(%)

Density(g/cm )

5.2

5.8

(c)

Density (g/cm )

81.8

66.7

La stoichiometry

0.0

5.4

4.8

(a) 4.4

84.8

3

Density (g/cm )

3

69.7

4.6

5.6

Theoretical Density(%)

5.8

Theoretical Density(%)

Density (g/cm )

densification and in the present case one such intermediate pressing has reduced the retrograde densification to a great extent as seen from the Fig. 8. For example the sintered density attains ~80% of theoretical density from 73~75 % after one stage repeated pressing. The decrease in density of RE added samples is primarily related to the microstructure of the samples which will be discussed in detail under microstructural analysis. As the RE content increases the porosity also increases as a result of formation of secondary grains of a different morphology and this reduces the apparent density of the samples. Both the green density and sintered density of RE added samples show a significant amount of reduction as the RE content increases. But at higher level of Yb addition density increases despite the increase in porosity. Yb5 sample shows an increase of density comparing with other Yb added samples at all stages. This is due to the higher atomic weight of Yb.

Rare Earth Modified (Bi,Pb)-2212 Superconductors

203

3.4. Microstructural Analysis

Figure 9(a). SEM images of La added samples.

Figure 9(b). SEM images of Nd added samples.

Fig.9. show the SEM micrographs of fractured surfaces of the samples taken in the back scattered mode. Clean and flaky grains typical of Bi-based superconductors are observed for pure samples. From RE1 onwards the characteristic flaky grain structures evolve into spherical or cuboidal shape of smaller size for all the RE added samples. A secondary phase having dark contrast is found distributed in the main matrix. At higher levels of doping secondary phase colonies are distinctly seen. SEM images of these samples are taken where such colonies are visible. However in most cases such a secondary phase could not be detected in the XRD analysis probably due to overlapping of peaks with (Bi,Pb)-2212 or below the detection level of XRD. Peaks with very low intensity, less than the sensitivity

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merge into the noise background of the pattern. However the EDX results give the composition of the phases which will be discussed under the section EDX analysis. SEM of Gd, and Yb added samples show the colony formation only at x = 0.5 concentration. While in the case of La addition La4 and La5 sample contain secondary phase colonies. Welldeveloped secondary phase colonies are visible from Dy3 sample onwards in the case of Dy addition. It is also clear from the microstructure that the porosity of the samples increases with increase in RE content. This may be the reason for the decrease in density of the RE added samples. From the SEM patterns it can also be seen that the grain orientation is destructed due to RE addition. This is also verified by finding the Lotgering index (£) of the samples.

Figure 9(c). SEM images of Gd added samples.

Figure 9(d). SEM images of Dy added samples.

Rare Earth Modified (Bi,Pb)-2212 Superconductors

(0012)

(0210) (220)

(117) (200/020)

(113) (115)

(006)

Intensity in arb. units

(0010)

(008)

Figure 9(e). SEM images of Yb added samples.

Nd5 Nd4 Nd3 Nd2

Nd1 Nd0

15

20

25

30 35 o Angle ( 2θ)

40

45

50

Figure 10. XRD patterns of the sintered pellets of Nd added sample as typical example.

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3.5. Lotgering Index (£) A measure of texture or grain orientation known as Lotgering Index (£) was calculated from the XRD peaks. Fig. 10 shows the XRD patterns of the repressed and sintered pellets of Nd added samples as a typical example. Here the reflections from the planes other than (00l) are highly diminished due to the high intensity from the oriented (00l) planes. Due to repeated pressing of the pellets the grains are highly oriented in the [00l] direction. From the XRD peaks of the powder (Fig.4) and that of the pellets (Fig.10) the Lotgering index (£) is calculated using the relation £=

( Pa − P0) (1 − P 0)

(3)

Where Pa = (ΣI00l) /(ΣIhkl) of the pellets surface and Po refers to that of the randomized powder [63]. Fig.11 shows the variation of £ of the samples with RE content. As the RE stoichiometry increases £ decreases. That is, the orientation or texturing of (Bi,Pb)-2212 grains decreases monotonically upto x = 0.4 and then levels off. 0.9

Lotgering Index

0.8

0.7

La Nd Gd Dy Yb

0.6

0.5

0.0

0.1

0.2

0.3

0.4

0.5

RE stoichiometry

Figure 11. Variation of Lotgering index with RE stoichiometry

3.6. EDX Analysis Atomic compositions of the phases of the RE5 samples were analysed using EDX. EDX spectra of relatively large area of the (Bi,Pb)-2212 grains (Main Matrix) and those of the secondary phases are shown in Fig. 12 and 13 respectively. The spectra of main phase contain the added RE, but the secondary phase is free from RE even for the samples with x = 0.5, except for La. This shows that the added RE completely enter into the (Bi,Pb)-2212 structure. But in the La5 sample a minor fraction of La is present in the secondary phase. The quantitative values of the main phase are given in Table 2. These values clearly show that there is distinct reduction of cation stochiometry with respect to Sr, Ca and Bi compared to

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the initial stoichiometry as a result of RE addition. The EDX spectra of the secondary phases (Fig.13) show that all these are of Sr rich phase. This indicates that the added RE entering into the crystal structure of (Bi,Pb)-2212 replace mainly Sr and a few Ca and Bi atoms which react with Pb and Cu forming the Sr rich oxide phase .

La5

Nd5

Gd5

Dy5

Yb5

Figure 12. EDX spectra of relatively large area of (Bi,Pb)-2212 grains of RE5 samples.

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La5

Nd5

Gd5

Dy5

Yb5

Figure 13. EDX spectra of the secondary phases found in RE5 samples.

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Table 2. Cationic compositions of the RE added (Bi,Pb)-2212 grains of RE5 samples. Cationic compositions Sample

Bi

Pb

Sr

Ca

Cu

RE

La5

1.53

0.44

1.88

1.00

2.43

0.5

Nd5

1.7

0.42

1.66

1.04

2.65

0.5

Gd5

1.46

0.38

1.55

1.04

2.26

0.5

Dy5

1.60

0.48

1.96

0.97

2.33

0.5

Yb5

1.67

0.40

2.05

0.92

2.35

0.5

3.7. Resistance – Temperature (R-T) Plots Fig. 14 shows the variation of normalized resistivity as a function of temperature (R-T plots) for the (Bi, Pb)-2212 samples with different RE contents. All the samples show a metallic behaviour above the TC-onset. It is noticeable that a pseudo transition that precedes the actual transition to zero resistivity is clearly observed for the pure samples. This pseudo transition

La1

La0 1.0

1.0

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1.0

La 2

0.5

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La4

100

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50

100

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250

300

La5

0.5

0.0

0.0

50 1.0

0.5

0.5

50

0.0 50

300

0.0 100

150

200

250

300

50

100

Figure 14 (a). Normalised R-T Plots of La added samples.

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A. Biju and U. Syamaprasad

Nd0

1.0

Nd1

1.0

1.0

0.8

0.8

0.8

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0.6

0.4

0.4

0.4

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0.0

0.0

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Nd3

1.0

50

300

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Nd4

1.0

50

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0.8

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300

50

100

150

200

250

Nd2

300

100

150

200

250

300

Nd5

50

100

150

200

250

300

Figure 14 (b). Normalised R-T Plots of Nd added samples.

1.0

Gd0

Gd1

1.0

1.0

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

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Gd3

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Gd4

50

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Gd2

50

100

150

200

250

300

50

100

150

200

250

300

150

200

250

300

Gd5

100

Figure 14 (c). Normalised R-T Plots of Gd added samples.

Rare Earth Modified (Bi,Pb)-2212 Superconductors

Dy0

1.0

1.0

0.5

Dy1

Dy2

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

211

0.0

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Dy4

1.0

1.0

100

150

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Dy5

50

1.0

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0.0

0.0

0.0

50

100

150

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250

300

50

100

150

200

250

100

150

200

250

300

100

150

200

250

300

Dy6

300

50

Figure 14(d). Normalised R-T Plots of Dy added samples. 1.2

1.2

1.2

Yb1

Yb0 1.0

Yb2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

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0.2

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0.0

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0.0 50

100

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-0.2 350 50

100

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1.2

1.2

Yb3

-0.2 350 50

Yb4 1.0

1.0

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0.0

0.0

0.0

100

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350

-0.2 50

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350

150

200

250

300

350

Yb5

1.0

-0.2 50

100

1.2

100

150

200

250

300

350-0.250

100

Figure 14(e). Normalised R-T Plots of Yb added samples.

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occurs at a temperature of ~ 110 K, which indicates the presence of the high TC phase (Bi2223). In the RE added samples this behaviour is not observed. Earlier reports [64, 65] confirmed the intergrowth of the TC phase in Bi-2212 by ac magnetic susceptibility and Miessner effect measurements. The co-existance of the high TC phase, along with Bi-2212 phase could not be detected from XRD in the previous studies also and this is believed to be due to its small quantity available in the sample (less than a few percent by volume). It is inferred from the large resistivity drop and the small volume fraction that the 110 K phase is most likely to be distributed in high aspect ratio morphology ie thin plates, needles or thin layers at interface such as grain boundaries, phase boundaries etc [64].

3.8. Normal State Resistivity The normal state resistivities at 300 K (ρ300) and the temperature coefficient of resistance (α) of the samples are calculated from R-T plots (Fig.14). Fig.15 shows the variation of normal state resistivity (at 300 K) as a function of RE content for different RE added samples. In all RE additions the normal state resistivity increases significantly with increase in RE content and at higher concentrations the increment is rapid. The temperature coefficient of resistance (α) values in the normal state regions are shown in Table 3. The α values are positive and show a decrease with increase in RE content. Comparing with the other RE added (Bi,Pb)2212, La5 sample shows much higher normal state resistivity. In all other cases of RE additions the variations of ρ300 with RE concentration are almost similar. When RE3+ ions enter into the system these may be substituted to any of the three cationic sites such as Sr2+, Ca2+and Bi3+. When they replace the divalent ions such as Sr2+ / Ca2+ the charge carrier (holes) concentration decreases. Each substitution of RE3+ at Sr2+ / Ca2+ site fills one hole and a decrease in hole concentration occurs on the Cu-O2 planes. This decrease in hole concentration increases the resistivity of the sample. The presence of secondary phase may also increase the normal state resistivity. 160 140

La Nd Gd Dy Yb

Resistivity (μΩm)

120 100 80 60 40 20 0 0.0

0.1

0.2

0.3

0.4

0.5

RE content

Figure 15. Variations of Normal state resistivity (at 300 K) as a function of RE content.

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Table 3. Temperature Coefficient of resistance (α) of the samples in K-1 Sample La Nd Dy Gd Yb

0 0.0430 0.0400 0.0450 0.0304 0.0243

0.1 0.0150 0.0138 0.0050 0.0088 0.0061

0.2 0.0092 0.0052 0.0037 0.0056 0.0057

0.3 0.0064 0.0048 0.0032 0.0044 0.0045

0.4 0.0045 0.0029 0.0029 0.0026 0.0037

0.5 0.0032 0.0015 0.0023 0.0013 0.0033

3.9. Critical Temperature (TC) 98 96 94

Temperature (K)

92 90 88 86

La Nd Gd Dy Yb

84 82 80 78 76 74 0.0

0.1

0.2

0.3

0.4

0.5

RE-Content

Figure 16. Variation of TC-onset as a function of RE content.

The critical temperature ( TC-onset and TC-zero) of the samples were determined from the R-T plots. The variation of TC-onset as a function of RE stoichiometry of different RE added samples are shown in Fig. 16. A remarkable feature is that all the RE added samples show much higher TC values than the pure sample. Depending on the processing conditions TC-onset of the pure sample show slight variations from batch to batch. All RE additions show parabolic variation of TC-onset with RE content. ie. TC-onset reaches a maximum at an optimum concentration of RE and thereafter it slightly decreases. The optimum concentration for maximum TC-onset changes with RE. Addition of La shows maximum TC-onset at the concentration of x = 0.3, but Gd and Yb additions show maximum TC-onset at x = 0.2. But in the case of Dy and Nd additions Dy4 and Nd4 samples show maximum TC-onset. Along with TC-onset, TC-zero values also increase to a maximum and then decrease (Table 4). The transition width ΔTC (TC-onset- TC-zero) also changes with the REs and their concentrations. With increase in RE content ΔTC, which is a measure of or inhomogeneities of the samples, also increases. Our results show that the inhomogenity of the RE added Bi-2212 can be reduced by the doping of Pb in Bi site [58]. The temperature-hole concentration phase diagram (Fig.1) shows clear that for high TC systems, TC becomes maximum at an optimum carrier concentration.

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Pure Bi-2212 is in the ‘over doped regime’ when the charge carrier density is considered [66]. Due to the replacement of Sr2+/Ca2+ ions by RE3+ ions the carrier concentration (holes) decreases and the system changes from the ‘over-doped’ to an ‘optimally doped’ regime. This increases the TC of the system. At higher levels of doping the system again shifts from ‘optimally doped’ regime to ‘under doped’ regime and causes a decreases in TC. The change in charge carrier concentration causes chemical as well as electronic inhomogenity in the system and thus the transition width (ΔTC) increases with RE addition. The impurity phases present in the samples also increases ΔTC. When the secondary phase volume increases in the sample, the superconducting properties decrease. Buzea and Yamishitra [67] give a theoretical explanation that the TC variation with doping concentration (x) depends on the ratio of the superconducting volume and total volume and becomes maximum at a particular value of x and then decreases. Table 4. TC-zero values of the samples in K X 0.0 0.1 0.2 0.3 0.4 0.5

La 67.5 78.2 83.5 85.1 83.6 74.5

Nd 68.3 78.9 80.8 80.8 80.9 76.3

Gd 76.7 78.3 82.1 84.0 81.9 66.7

Dy 66.5 75.8 78.9 79.5 79.7 68.8

Yb 67.5 72.0 81.5 85.0 83.0 80.0

3.10. Critical Current Density (JC) 800

2

Critical current density (A/cm )

700 600 500 400 300

La Nd Gd Dy Yb

200 100 0 0.0

0.1

0.2

RE-Content

0.3

0.4

Figure 17. JC variations of the samples with RE content

0.5

Rare Earth Modified (Bi,Pb)-2212 Superconductors

215

The self field transport JC variations of the samples with RE content at 64 K is shown in Fig.17. The JC of polycrystalline bulk Bi-2212 are less comparing with tapes and thin films. The variation of JC with RE content is similar but the values vary with RE. In the case of all REs, x = 0.2 samples show the maximum JC, thereafter it decreases and the decreasing rate is comparatively higher for rare earths having higher ionic radius. The significant improvement of JC can not be attributed to any improvement in microstructure such as grain growth or orientation, because SEM and Lotgering index results show that the pure samples have better microstructure, grain growth and grain orientation than doped samples. The improvement in JC is due to the decrease in hole concentration in the Cu-O2 plane, which brings the hole density to it the optimum value for the best superconducting properties. The replacement of cations by RE can also lead to electronic or chemical inhomogeneities of the charge reservoir layers (Bi-O/Si-O) adjacent to the Cu-O2 layers through which the actual supercurrent is believed to flow. The decrease in hole concentration leads to an increase in the TC and thereby to JC upto the optimum value. Comparing with the decrease of TC, decrease of JC is rapid. At higher levels of doping the porosity and the fraction of secondary phase increases. These will act as weak links inhibiting the flow of supercurrent. This can be the reason for the reduction of transport JC of the samples with higher RE content and higher TC.

3.11. The JC-B Characteristics The variation of JC with respect to external magnetic fields (JC-B characteristics) of the different RE added samples are plotted in Fig.18. The JC-B behavior improves significantly with RE addition. The best JC-B behavior is observed for RE content from x = 0.2 to 0.3. Even though x = 0.4 sample shows higher self field JC than pure sample, their Jc-B characteristics are not better than pure sample in the case of La, Gd and Dy addition. But Nd4 and Yb4 samples show better Jc-B characteristics comparing with pure sample. These characteristics show that addition of RE elements La, Nd, Gd, Dy and Yb in optimum concentration (x = 0.2 to 0.3) in (Bi,Pb)-2212 improves the JC-B characteristics of the superconductor along with significant enhancement in TC and self field JC. This improvement of JC in applied field is a clear indication of improvement of flux pinning properties of the system. Among these additions the best JC-B characteristics are shown by the rare earth, Yb. JC-B characteristics of Gd and Dy added samples are not so good as La, Nd and Yb. These results clearly show that the field dependence of JC varies with RE and the flux pinning strength of RE added samples has considerably enhanced compared to the pure sample. More interestingly, we have recently shown that for improved pinning and high transport JCs at high temperatures the Bi-2212 must be co-doped with RE and Pb [60]. The single doping with Pb or RE only cannot give the sufficient pinning and high JCs at the high temperatures (64 –77 K). The Pb doping improves the coupling between the vortices and the strongly coupled vortices are effectively pinned by the crystal defects at Sr or Ca site created by RE doping [60].

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A. Biju and U. Syamaprasad Table 5 FPmax values of the samples FPmax ( x I03 N/m3)

RE Stoichiometry(x)

La

Nd

Gd

Dy

Yb

0.0 0.1 0.2 0.3 0.4 0.5

12.99 68.05 269.94 110.01 19.97 0.59

13.0 46.9 240.8 111.3 70.3 8.16

12.99 60.19 176.26 86.70 7.97 2.59

13.0 40.00 176.10 127.5 13.37 0.21

12.99 24.10 330.31 289.98 84.93 56.5

In high TC superconductors there are different pinning mechanisms describing the interaction of a flux core with a pinning site: δl pinning, due to the scatter of electron free path caused by the non-superconducting particles size of the order of coherence length (ξ) embedded in the superconducting matrix and δT pinning, due to a spatial scatter of superconducting TC, through out the sample caused by other superconducting phases causing fluctuation in TC. Kim et al. [46] observed that the pinning mechanism was changed from δl pinning to δTC pinning at low fields and from the catastrophic instability to a combination of the catastrophic instability and the avalanche instability at high fields with temperature. The flux pinning properties of superconductor can be investigated by calculating the volume pinning force density FP = JC x B [68-71]. Table 5 shows the maximum value of Fp (FPmax) of the samples for different RE concentrations. In all the rare earths the x = 0.2 stoichiometry shows higher values for FPmax. Fig.19 shows the normalized volume pinning force density FP/FPmax as a function of the applied field. For RE added samples the peak positions of FP/FPmax get shifted towards higher fields. When the concentration of the added rare earth is in the optimum level (x = 0.2 to 0.3) the shift is maximum. Above this optimum level the peak positions of FP/FPmax shift towards lower fields. For example, in the case of pure sample, FPmax appears at 0.08 T whereas for La3 it appears at 0.48 T, the maximum shifting observed towards higher fields while La4, FPmax appears at 0.08 T only. This implies that the irreversibility line (IL) of the RE added samples shift towards higher fields and higher temperatures if the RE content is in the range 0.2-0.3. In other words flux pinning strength of RE added samples improves considerably compared to the pure sample when RE content is in the optimum level and the flux pinning strength reduces if the RE content increases from this range. Comparing the flux pinning force of the rare earths added samples Yb addition shows the maximum flux pinning force (Table 5). When RE atoms are added to (Bi,Pb)-2212, the RE3+ ions are likely to replace the 2+ Sr /Ca2+ ions and there is a consequent change in the carrier concentration in the Cu-O2 planes. The replacement of divalent cations by trivalent ions will cause changes in the hole concentration and can lead to chemical as well as electrical inhomogenities. By the addition of the rare earths the inhomeogenity introduced may be random distributions of charges of Sr2+ and / or Ca2+ and RE 3+ and local lattice distortion caused by the difference between the ionic radii for the two ions. The substituted RE3+ ions can also act as pinning centers as point like defects. The enhancement of JC is due to the change in the charge carrier (hole) concentration and also due to the changes in the flux pinning properties. When the RE content increases beyond a critical level the number of defects increase which are likely to form

Rare Earth Modified (Bi,Pb)-2212 Superconductors

217

larger defects larger than the coherence length of the system. These larger defects inhibit the transport of supercurrent and thus reduce critical current density and flux pinning properties. There is the possibility of formation of nanosize precipitate, which can enhance the flux pinning strength of the system [72, 73]. Presence of secondary phases are observed in the SEM of all RE added samples and these phases are analysed using EDX (Fig. 13). At higher 900

1000

(a)

(b) 100

2

JC(A/cm )

100

2

JC (A/cm )

10 10

Lao La1 La2 La3 La4 La5

1

Nd0 Nd1 Nd2 Nd3 Nd4 Nd5

1

0.1

0.1

0.01 0.0

0.1

0.2

0.3

0.4

0.5

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0.7

0.0

0.1

0.2

Field (T)

0.3

0.4

0.5

0.6

Field (T)

800 1000

Gd0 Gd1 Gd2 Gd3 Gd4 Gd5

(c)

Dy0 Dy1 Dy2 Dy3 Dy4 Dy5

100

2

JC (A/cm )

2

JC(A/m )

100

(d)

10

10

1

1 0.1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Field (T)

-0.1

0.0

0.1

0.2

1000

(e) 100

10

1 -0.1

0.3

Field (T)

2

0.0

JC(A/m )

-0.1

Yb0 Yb1 Yb2 Yb3 Yb4 Yb5 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Field (T)

Figure 18. JC-B Plots of the samples at 64 K.

0.4

0.5

0.6

0.7

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level of doping the secondary phase formation is higher and this will also reduce the flux pinning properties even though JC and TC are improved compared to the pure sample. Further investigations are needed to get more information about the mechanism that changes the flux pinning strength due to rare earth additions in (Bi, Pb)-2212.

1.0

(a)

0.8

0.6

0.6

Fp/Fpmax

FP/FPmax

0.8

0.4

0.4

La0 La1 La2 La3 La4 La5

0.2

0.0 -0.1

0.0

(b)

1.0

0.1

0.2

Nd0 Nd1 Nd2 Nd3 Nd4 Nd5

0.2

0.0 0.3

0.4

0.5

0.6

0.0

0.7

0.1

0.2

1.0

(c)

1.0

0.4

0.5

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0.8

0.6

0.6

Dy0 Dy1 Dy2 Dy3 Dy4 Dy5

(d)

Fp/Fpmax

Fp/Fpmax

0.8

0.4

0.4

Gd0 Gd1 Gd2 Gd3 Gd4 Gd5

0.2

0.0 0.0

0.1

0.2

0.3

0.4

0.2

0.0 0.5

0.6

0.7

-0.1

0.0

0.1

0.2

Field (T) 1.0

0.3

Field (T)

(e)

0.8

Fp/Fpmax

-0.1

0.3

Field(T)

Field (T)

0.6

0.4 Yb0 Yb1 Yb2 Yb3 Yb4 Yb5

0.2

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Field (T)

Figure 19. Normalised Fp Plots of the samples (at 64 K).

0.4

0.5

0.6

0.7

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4. Conclusion Some of the interesting features with respect to the structural and superconducting properties of rare earth modified (Bi,Pb)-2212 superconductors are presented here based on the original and published works of the authors. RE addition in (Bi,Pb)-2212 leads to significant changes in the formation temperature, reaction rate, microstructure and enhancements in the superconducting and flux pinning properties of the system. When RE ions are doped they mainly enter into the cationic sites such as Ca/Sr and replace them. The superconducting properties such as TC, JC and the flux pinning properties are highly enhanced by RE addition. Best superconducting properties are found when RE content in (Bi,Pb)-2212 is in the range x = 0.2 to 0.3. The enhancement of these properties are attributed to a decrease in the hole concentration and disorder of the lattice due to the replacement of Sr2+ and/or Ca2+ by RE3+ ions in the system. The decrease in the hole concentration brings (Bi,Pb)-2212 from ‘overdoped’ to ‘optimally doped’ condition and thus TC and JC of the system increase. At higher levels of addition the number of holes still decreases and the system changes to ‘underdoped’ condition and thus superconducting properties deteriorate. The substituted RE3+ ions can act as pinning centers as point like defects and improve the field dependence of JC and hence the flux pinning properties. There is a possibility of formation of nano-scale secondary precipitates, which may also act as flux pinning centers. The improved flux pinning properties also improves the JC. At higher level of RE addition, chemical inhomogenity and secondary phase formation increases which brings down the flux pinning properties along with superconducting properties of the system. Further work is necessary to find out more details about the various aspects of RE addition in (Bi,Pb)-2212. The results are highly relevant to the development of RE modified (Bi,Pb)-2212 conductors in bulk and tape forms for practical applications.

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In: Superconductivity Research Developments Editor: James R. Tobin, pp. 223-239

ISBN: 978-1-60021-848-4 © 2008 Nova Science Publishers, Inc.

Chapter 9

NOVEL APPROACHES TO DESCRIBE STABILITY AND QUENCH OF HTS DEVICES V. S. Vysotsky1, A. L. Rakhmanov2 and Yu. A. Ilyin3 1

Russian Scientific R&D Cable Institute, 5, shosse Entuziastov, 111024, Moscow, Russia 2 Institute Theoretical and Applied Electrodynamics, 13/19 Izhorskaya str. 125412, Moscow, Russia 3 Twente University, P.O. Box 217, 7500 AE, Enschede, the Netherlands

Abstract In R&D of HTS devices most researchers and designers still use the traditional approach for their stability and quench development analysis based on normal zone determination, and consideration of its appearance and propagation. On the other hand most peculiarities of HTS and their relatively high operating temperature make this traditional approach quite impractical and inconvenient. The novel approaches were developed that consider the HTS device as a cooled medium with non-linear parameters with no mentioning of “superconductivity” in the analysis. The approaches showed their effectiveness and convenience to analyze the stability and quench development in HTS devices. In this review we present these approaches being well confirmed and verified by the experiments as well as their development for long HTS objects like HTS cables where "blow-up" regimes may happen. The difference of HTS (1-st and 2-nd generations) from LTS is discussed that lead to the difference of their stability and quench development. We consider these approaches as very useful for any researchers and designers of modern HTS devices from both first and second generation HTS.

I. Introduction The goal of this edited collection was determined as calling attention to the new research trends in superconductivity. We consider this goal as very important and very timely started. That is why we would like to present our modest contribution to the item.

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Modern superconductivity, especially its applied part, is under the Zodiac of high temperature superconductors (HTS) that are used for real and important devices more and more widely. As we are in the large scale applications R&D business we would like to discuss what should be considered in this area as something new or different from very well developed science for low temperature superconducting (LTS) devices. We will try to convince the readers that it is really necessary to look for some new approaches in describing such important items as stability and quench of HTS devices which could differ from those for LTS devices. We will present the novel approaches developed that consider the HTS device as a cooled medium with non-linear parameters with no mentioning of “superconductivity” in the analysis. The approaches showed their effectiveness and convenience to analyze the stability and quench development in HTS devices. They are well confirmed and verified by the experiments. They also could be used for long HTS objects like HTS cables where "blow-up" regimes may happen. The difference of HTS (1-st and 2-nd generations) from LTS is discussed that lead to the difference of their stability and quench development. We consider these approaches as very useful for any researchers and designers of modern large scale HTS devices from both first and second generation HTS.

II. Why the New Approaches Are Necessary – Two Important Notes The euphoria after discovery of high temperature superconductors delivered their fruits. Many large scale devices were developed and tested. HTS magnets, windings of HTS motors and generators, HTS transformers, HTS cables become usual items. Their sizes are rising; power cables developed have lengths up to 200-650 m; HTS windings of different types reach meters in sizes. Like their predecessors, made from LTS, HTS devices have their operation limits, namely critical current, field and temperature. It means that at certain conditions HTS device may lose their stable superconducting state, heating may start with the temperature rise. In the worst case it could lead to full destroying of a device. For LTS devices such transition usually is called as a quench. The efforts to avoid a quench were called as stabilization of superconductors. It was necessary to learn how to reach the stability of superconducting devices. To describe the stability of LTS devices, usually the energy is analyzed that is necessary to initiate a normal zone in a superconductor while it carries current below its critical level. If due to some disturbance the normal zone appears in LTS, the normal zone propagation is studying to describe the quench development and heating of a LTS device. Thus, the LTS stability and quench description is based on the determination of the normal zone and analysis of the normal zone propagation. This approach was very fruitful and permitted to solve most stability and quench problems for LTS devices. It is not surprising that when HTS devices came to the scene, the same approaches were used for their stability and quench analysis [1, 2]. In principle, this is fair because from the general or formal point of view there is no difference between HTS and LTS superconductivity except operating temperature. But just high operating temperature and, as a consequence, the sufficient difference in some material parameters make old fashioned

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description of HTS devices quite inconvenient. First of all it applies to the basic determination of the “normal zone” in HTS devices. Important note 1: It is very difficult to determine properly where is “normal” and where is “superconducting” part of an HTS device.

It is very important to mention that the most prospective applications of HTS are power electro-technical devices: power cables, transformers, generators, etc. All these devices must have one general feature – they must withstand fault currents dozens times more than their operating currents (if one not considers special current limiting devices). It is the standard for electric power grids. This situation is absolutely different from the quench of LTS devices, where the transport current is below or about the critical current during quench. In HTS power devices, the overload current forcibly becomes much more than the operating/critical current of a device. In this case the usual approaches to analyze quench and heating in superconducting devices are not valid. There is no normal zone and its propagation in the usual sense used for LTS devices. Important note 2: In HTS power devices sufficient overload currents are possible when no normal zone and its propagation in the usual sense used for LTS devices could be considered.

These two important notes makes HTS devices, especially energy devices, different from LTS ones. It looks reasonable to develop some new approaches for more physically clear and simple description of the stability and quench of HTS devices, especially at overload conditions. Below we present such approaches developed and validated.

III. Comparison of HTS with LTS What is Common? In principle, the old approach to analyze HTS devices is fair, because for both LTS and HTS it is based on the same standard equation. For example the one-dimensional, general, simplified, differential equation that governs the quenching process in any superconductor is given by [1-4]:

C (T )

∂ ∂T ∂T = ( k (T ) ) + Q (T ) − W (T ) ∂t ∂x ∂x

(1)

where C (T ) is the volumetrically averaged heat capacity, the first term on the right-hand side represents thermal conduction along the superconductors, k (T ) is the volumetrically averaged thermal conductivity and Q(T) represents the heat generation, particularly due to voltage-current characteristics (VCC). The last term represents the cooling, that is usually linear in temperature [1, 4], while sometimes it also could be non-linear [5].

226

V. S. Vysotsky, A. L. Rakhmanov and Yu. A. Ilyin The traditional presentation of VCC of superconductors E(I,T) is:

⎛ I ⎞ ⎟⎟ E ( I , T ) = E 0 ⎜⎜ ⎝ I 0 (T ) ⎠

n

(2)

Here, n - is the parameter called index, I0(T) - is a current corresponding to the electric field level E0 that is defined usually as 1 μV/cm or 0.1 μV/cm. The current I0(T) is what we usually call "critical current". In this case heat release term in (1) will look as:

⎛ I ⎞ ⎟⎟ Q(T , n) = IE = I 0 E0 ⎜⎜ ⎝ I 0 (T ) ⎠

n +1

(3)

Equation (1) is a basic heat balance equation to evaluate hot spot temperature in superconductors at a quench or loss of superconductivity. This equation is the same for LTS and HTS superconductors. Generally, this equation should be solved numerically, because of non-linearity and complexity of all terms included. And exact results (if all parameters are known properly) could be obtained. Of course, this equation can be enhanced for two or three dimensional cases (see for example [4] for LTS). But for practical purposes some simplified models were developed permitting welljustified analysis of the quench development in superconducting devices, for example normal zone appearance and propagation analysis for quench development and minimum propagating energy analysis for stability. We would like to offer another approach we consider as more convenient for HTS devices.

What Makes Difference? The major difference between HTS and LTS superconducting devices is the parameters’ magnitude. Table 1 lists a comparison between the major material parameters for LTS and two types of HTS superconductors. Strong differences in parameters of low-Tc and high-Tc superconductors and in their temperature dependencies do exist. Let us consider what the sequences from these differences are. Low n index or smooth voltage current characteristics (VCC) – is one of the fundamental peculiarities of HTS superconductors, at least for 1-G HTS. Very smooth voltage rise with current rises the question when and how to determine “normal” or “superconducting” parts of HTS. As it was shown by several experiments [6, 7] HTS devices at some conditions can safely and stably work at currents more than those called as “critical current” determined by, say, 1 μV/cm criteria.

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Table 1. Comparison of LTS and HTS characteristic parameters LTS High index value n≥30, n~50 – is a common value Specific heat C is low C~ 103 J/m3K Matrix resistivity ρ – const with temperature up to ~3035 K

HTS 1-st generation Low index value n<30, n~10 or less – is a common value Specific heat C is high C ~ 2·106 J/m3K Matrix resistivity ρ – nonlinearly rises with temperature

Difference between critical temperature operating temperature Tc-T < 1-10 K Critical current criteria I c (1μV / cm) ~ 1.05 I c (0.1μV / cm)

Difference between the critical temperature and operating temperature Tc-T >> 10 K Critical current criteria I c (1μV / cm) ~ 1.25 I c (0.1μV / cm)

the and

HTS 2-d generation Average index value n~2030 – is a common value Specific heat C is high C ~ 2·106 J/m3K Matrix may be absent except small stabilizing layer. Substrate resistivity const with the temperature. If matrix is present (Cu coating) – the resistivity ρ – nonlinearly rises with temperature Difference between the critical temperature and operating temperature Tc-T >> 10 K Critical current criteria

I c (1μV / cm) ~ 1.08 − 1.1 I c (0.1μV / cm)

In LTS devices very sharp VCC lead to the quick loss of stability as soon as voltage became close to 1 μV/cm or even before. The sharpness of voltage current characteristics permits to determine easily normal and superconducting parts of superconductors. While generally speaking the presence, let sharp, VCC should be taken into account to explain some properties of LTS superconductors and devices as it was done in [8, 9]. Another impact of low value of index n is the determination of the critical current (see Table 1). For LTS only 5% difference for two critical current criteria (of 1 μV/cm and 0.1 μV/cm) permits more or less precise determination of Ic. For HTS critical current is a very relative or conditional parameter. For n~10 there is no sense to be very serious with Ic determination. Superconducting devices can work at currents even two times more than the critical ones [6]. To underscore this we use in our equations the term I0 instead of usual Ic to show that it is just conditional parameter used in formulas. We again have to make two notes. •

•

Novel second generation high temperature superconductors have much sharper VCC with indexes close to those in LTS. It means that their critical current is better determined and it is fairer for them to determine normal zone appearance using critical current criteria. Some low Tc superconductors, like large cable-in-conduit conductors used for example for ITER magnets have low indexes n due to several reasons [10]. We believe that for such LTS some ideas developed in our approaches could be fruitfully used. The major question is: low index sometimes leads to lower “critical current”

228

V. S. Vysotsky, A. L. Rakhmanov and Yu. A. Ilyin determined by 1 μV/cm voltage criteria. But low “critical current” does not mean worse stability! Quite the contrary, stability could become better [11]!

Temperature margin – is another parameter that makes distinguishing of superconducting and normal parts difficult. For HTS superconductors this margin is huge even at operating currents. The rise of a voltage which often determine as the normal zone appearance could not be reasonable, as it is very relative or conditional parameters as we determined above. Even at rather high currents, smooth voltage current characteristics makes differentiation of normal and superconducting zones much undetermined. Even for 2-G HTS where VCC are more sharp ones, large temperature margin still make questionable the distinguishing of normal and superconducting parts.

The dependencies on temperature of matrix resistances also make their impact. Their non-linearity along with non linearity of VCC may lead to specific regimes called as “blowup” regimes considered in [12]. One of the major parameter that makes difference between HTS and LTS is the high specific heat at usual operating temperatures for HTS. It leads to the very strong difference of characteristic times of the heat processes development th=CA/Ph. Here A – is a cross-section of a superconductor, P –its cooling perimeter and h – heat removal coefficient. Specific heat is about 2000 times more for HTS. Heat removal coefficient h is about 15 times (LN cooling) more for HTS, but much less for indirect cooling by cryocoolers. In any case, the specific heat development time τh is at least 200 times and more long for HTS. The transition processes in HTS develop much more slowly than in LTS. Due to high n-values and low C(T), fast thermal instability happens in LTS with the appearance of clearly determined and propagating normal zones. The model with the appearance of a normal zone and its propagation is well justified to analyze quench in LTS devices [1,3,8]. To summary this chapter we should to conclude the follows. In high temperature superconductors and devices it is difficult to distinguish clearly normal and superconducting parts, transition processes develop very slowly in comparison with LTS. Due to smooth VCC and high specific heat HTS devices are much more stable than LTS devises. Thus, the transition processes in HTS devices are not really “quench” with normal zone propagation, but relatively slowly developing heating of a device. We need new approaches to describe this process, different from normal zone propagation model. Below we describe them.

IV. Novel Approaches to Describe Stability and Heating Development in HTS Devices The Analytical Model The analytical model has been developed for stability and heating. This model was developed for uniform or quasi-uniform heating of a HTS device.

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229

Temperature

I > Iq

Tq Tq-Tf

I < Iq

tq Time

Figure 1. Temperature traces for two general modes – unstable (when thermal runaway happens) (I

> I q ) and stabile at I < I q .

The specific “thermal quench current” (TQC) Iq or thermal runaway current (TRC) [13] has been introduced resembling the critical current for LTS devices. It was shown [14] that analytical expressions could be found for two cases. If IIq, the temperature rises with strong acceleration after the time tq [7, 14-19]. Two major regimes do exist: stable and unstable shown in Fig.1 The following expressions have been found [15]. Time evolution of temperature and the electric field in a HTS device:

T (t ) − Tq Tf

=

E (t ) − Eq Ef

= tan

t − tq tf

, I > Iq ;

(4)

for the unstable regime;

T (t ) − Tq Tf

=

E (t ) − E q Ef

=

1 + g exp(2t / t f ) 1 − g exp(2t / t f )

, g=

Tq − T0 + T f Tq − T0 − T f

, I < Iq

(5)

for the stable regime. Threshold thermal quench – thermal runaway current: n ⎡ hP(Tc − T0 ) ⎤ I q = I c (T0 ) ⎢ ⎥ n + 1 ⎣ nE 0 I c (T0 ) ⎦

1/( n +1)

(6)

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V. S. Vysotsky, A. L. Rakhmanov and Yu. A. Ilyin Characteristic time of the quench development: ⎛ t q = th ⎜ ⎜ ⎝

⎞ ⎛ ⎞ 2Iq Iq ⎟ ⎟ arctan ⎜ ⎜ 2 ⋅ I − I ( n + 1) ⎟ I − I q ( n + 1) ⎟ q ⎠ ⎝ ⎠

(7)

Characteristic temperatures and voltages:

Tq = T0 +

Tc − T0 , n +1

T f = (Tc − T0 )

2 I − Iq (n + 1) I q

Eq =

hPTc , I c(TO )n

E f = nE q

2 I − Iq (n + 1) I q

(8)

Characteristic time: t f = th

2I q

.

(9)

I − I q (n + 1)

Here T0 and Tc are the ambient temperature and critical temperature of a superconductor, Tq is a characteristic temperature at which fast temperature rise starts at the time tq, and tf is a time necessary to heat a sample to the equilibrium temperature Tq-Tf at IIq are universal and permit the scaling for the widest variety of superconducting devices. The parameters, describing the heat development of superconducting devices could be made dimensionless by dividing on the proper scaling factor and in this case one can obtain the universal dependence for different devices. The example of such dependencies for dimensionless temperatures and voltages on dimensionless time is shown in Fig.2 for different superconducting objects. It was shown that the theory well coincides with the experimental data for quite different devices [14]. It was shown also, that heating development time tq may be well scaled too [14, 17]. Equations (4) are quite universal and valid for any medium where heat release is sufficiently non-linear. One of the examples of such a behavior may be well known LTS superconductors. In Fig. 1 in [19], we showed the experimental results for the electric field trace measured during the quench of LTS superconductor. The sample is a typical multifilamentary NbTi/CuNi/Cu superconducting wire tested at the liquid helium. It was obvious that, the electric field rise in this wire obeys the universal curve calculated by Eq. (4). It means that heating process in the LTS wire has the same nature as for HTS superconductors. The only difference is the initial normal zone propagation in LTS wire (~50 m/s), which we never observed in the experiments with HTS objects. After normal zone filled entire LTS sample, the heating process for the LTS wire is similar to the heating for HTS devices. This possibility to use for LTS and HTS in describing of heating development is just due to use the standard equation that is valid for any heated media with cooling. The difference is for the instability development time as we mentioned above.

Novel Approaches to Describe Stability and Quench of HTS Devices

231

YBCO thin film, 77.4K, 1770 mA, voltage YBCO thin film, 77.4K, 362 mA, temperature Bi2212 coil, 13 K, 160 A, middle Bi2212 coil, 30K, 100A, bottom Bi2223, textolite bobbin, 20K,120A,TCC Bi2223, textolite bobbin, 40K, 80A,TCD Bi2223, copper bobbin, 40K,120A, TCB Bi2223, copper bobbin,40 K,120A, TCA Bi2223, copper bobbin, 60K,70A, TCC Pancake coil, 20K, 142 A, TC3 Pancake coil, 20K, 145 A, TC3, heat drains Pancake coil, 40K, 102A, TC3, heat drains Pancake coil, 60K, 58 A, TC3 Pancake coil, 62K, 55 A, TC3, heat drains Pancake coil, 80K, 26 A, TC3 θ=tan(τ)

10

q

θ=(T-T )/T

f

5

0

-5

-1

0

1

τ=(t-tq)/tf

Figure 2. Example of scaling [14]. Dimensionless temperature θ versus dimensionless time τ for experiments with different HTS objects.

Inevitable Consequences from the Theory The major consequence from the implementation of this theory is the necessity of changing design criteria for HTS devices. Usually, design criteria for any superconducting device is including: maximum allowable operating current Ioper and the time tprot when protection devices should be fully activated if a quench happens. The time mentioned is determined by the maximum allowed heating temperature Tmax. LTS devices work if operating current is less then the critical current Ioper
232

V. S. Vysotsky, A. L. Rakhmanov and Yu. A. Ilyin

Relative Thermal Quench Current

2.0

1.5

Cryo

Cry

1.0

coo ler 6 0K

LN , 2 77.8 K

oco oler 40 K Cryo coo ler 2 0K

Coil sizes rise

3

10

4

10

5

10

Total conductor length/ Cooling surface, m

-1

Figure 3. Relative thermal quench current (TQC divided by the critical current) versus inverse effective cooling perimeter [17]. Symbols are data from different HTS coils from literature [7, 23-26], solid lines – calculations by (6) for n=12.

The reasonable criteria for the heating temperature should be changed also. From the maximum heating temperature Tmax used for LTS devices [1,4] to the temperature Tq, at which the slope of T(t) is drastically changing for HTS devices (see Fig.1). The change of the protection time criteria is: from the time necessary to heat a device up to Tmax (LTS devices) to the time tq, when the slope of T(t) curve is changing (for HTS devices), i.e. tprot must be less than tq. Both these criteria for tprot and Tmax should be used because beyond Tq and tq the temperature rise is so fast that it could barely be controlled. It is necessary to note that the low index n while reducing the “critical current” or increasing “temperature of current sharing” may lead to better stability if good enough cooling is providing. It may be important for large CICC cables where the reduction of n was observed [10]. The theory developed for HTS superconducting objects with the low index n may be used for large CICC cables also. In our opinion, the analytical model for quasi – uniform heating provides better, more convenient and more adequate understanding of HTS quench – heating development. We successfully verified it by many experiments, both our and from literature. This model is working and for the overloading cases, when current in an HTS device is much more than the “critical current” and any talks about “normal zone” and its “propagation” have no sense. Next step – is analysis of non-uniform heating for long or large objects.

Novel Approaches to Describe Stability and Quench of HTS Devices

233

V. Non-uniform Cases and HTS Device at Overloading Conditions Non-uniform cases are important if the characteristic heat length lh =

Ak / Ph is less than

the characteristic size of HTS – devices. Here k is heat conductivity. Once the characteristic heat length is large enough (good heat conductivity and low cooling, that are most cryocooler cases), one may consider a winding as quasi-uniform and use the formulas (4) – (9) above. For windings, the heat length may be presented [14] as lh = Vol ⋅ k ⋅ G −1 , where Vol is the total volume of the winding and G is the effective cooling per surface unit. So, the heat length depends on a winding's volume and cooling. In [17] the typical values of the characteristic heat length were analyzed in dependence on characteristic winding size (cubic root of the winding volume) for different temperatures at cryocooler cooling and liquid nitrogen (LN) cooling. It was shown that lh are rather large (~1 m and more) in the case of cryocooler cooling and even large windings may be considered as quasi-uniform with all parameters averaged along the winding [7, 15]. In case of LN cooling, heat length lh becomes low (1-10 cm) and strong non-uniformity may appear. We observed the non-uniformity at LN cooling in the four-pancake magnet described in [7, 16, 17]. As we mentioned above, HTS cables being installed to a power grid may undergo overloading regimes – when current forcibly becomes much more than its operating value and critical current. In this case there is no way to talk about normal zone and its propagation! HTS cables are obviously “long HTS objects” cooled by liquid nitrogen and, therefore, they are rather vulnerable to the non-uniformity appearance. 100 100 A/mm

2

70 A/mm

95

2

60 A/mm

2

90 2

50 A/mm 85

2

40 A/mm 80

75 -6 10

10

-4

10

-2

10

0

10

2

10

4

Time, s

Figure 4. Instability developments in the adiabatic cases. Slow decay eventually changes to the fast rise of the temperature [21].

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V. S. Vysotsky, A. L. Rakhmanov and Yu. A. Ilyin

Due to strong non-linearity with temperature of the heat release and material parameters of HTS devices some specific phenomena could happen: blow- up regimes with the heat localization [12, 20-22]. 100

100 ΔT=4K

Temperature, K

95

ΔT=0.5K

95 180 A/mm

2

167 A/mm 90

161 A/mm2

2

90 165 A/mm

85

85

80

80 2

75 0.01

160 A/mm 100 A/mm

2

150A/mm

160 A/mm

0.1

1

10

2

75 0.01

100

0.1

1

10

2

2

100

Time, s

Time, s

Figure 5. The examples of the instability developments with time in the cases with cooling. Different initial disturbances levels are shown. Small change of the transport current leads to the change of regimes. 2

103 A/mm Unstable regime 100 Heat Localization

80.0

2

100 A/mm Stable regime No Localization

0.1744 s 79.5 0.17439 s 90 0.174 s

85

Temperature, K

Temperature, K

95

1s 79.0 0.02 s 78.5

80

0s

-0.50 -0.25

0

0.25

Distance, m

0s 0.50

78.0 -0.50 -0.25

0

0.25

0.50

Distance, m

Figure 6. The examples of the heating developments in the presence of the initial disturbance in the cases with cooling. Different initial disturbances levels are shown. Left - in the unstable mode the fast, actually a catastrophic temperature rise happens with the heat localization [21], no normal zone propagation observed. Right - in the stable mode the initial disturbances disappear and temperature stabilizes at the certain level.

Novel Approaches to Describe Stability and Quench of HTS Devices

235

The study of these phenomena (beside the experiments) could be done by numerical analysis only. We performed such study with several computer experiments [12, 20-22]. The numerical solution of the standard equation (1) was performed with parameters as much as possible close to the reality. Depend on parameters combinations (current density, cooling, initial disturbance, etc.) different heating modes may appear [21, 26]. In adiabatic cases – eventually, a fast temperature rise happens with heat localization (while may be after very long time). This is illustrated in Fig. 4. In the presence of cooling, two modes do exist of the heat development, absolutely similar to the analytical model – stable and unstable. This is illustrated in Fig.5, 6. In the stable mode the initial disturbances disappear. In the unstable mode the very fast, actually catastrophic temperature rise happens with the heat localization (see Fig.6). The time till the temperature runaway starts in the unstable regime becomes rather short with a current density rise. Like in the analytical model this time can be considered as the safety parameter. It is necessary to note that switch from stable to unstable mode happens of transport current changes very little. Say, in Fig.5 left, one can see that current density 160 A/mm2 corresponds to the stable mode; while at 161 A/mm2 the instability happens. The difference in current densities less than 1% led to the change of heating regime. Similar sharp switch between regimes we observed in experiments with uniform cooling also [14].

Relative quench current, A

4 Peak nucleate boiling cooling for LN2

n=10, Ic=40 A

3 n=10, Ic=100 A

2

n=15, Ic=40 A n=15, Ic=100 A

1 2 10

10

3

10

4

10

5

2

Heat removal coefficient, W/m K

Figure 7. Relative thermal runaway currents (transport current divided by the critical current) versus the heat removal coefficient. Solid lines – analytical calculations by theory [15, 17]. The upgraded model calculations [21] (short dashed lines), the experimental data [22] (symbols) and the data from calculations by the model [21] (long dashed lines) are shown also.

In Fig.7 the relative thermal runaway currents are shown in dependence on cooling for two HTS tapes with different critical currents. In Fig. 7 solid lines are calculations by the analytical model [14, 17] (zero disturbance) and dashed lines are numerical calculations by the upgraded model [22] and by the model [21] with the temperature disturbances in the

236

V. S. Vysotsky, A. L. Rakhmanov and Yu. A. Ilyin

center of the sample (ΔT~0.5-4 K). In the model [21] the heat release was approximated for the simplicity by the power function, while in the upgraded model [22] we used the real heat release function described by Eq.(3). One can see the practical coincidence of calculations with different methods. Symbols shown in Fig. 7 are the experimental data. It was shown in [15] that the thermal runaway current Iq is proportional to the heat removal coefficient h like Iq~h1/n+1 [15]. That means very weak dependence on cooling at the high index n. It permits also more or less precise evaluation of the thermal runaway current for different HTS devices.

VI. Closing Remarks for Quasi-uniform and Non-uniform Cases and 2-G HTS One can see that the behavior of the long HTS devices with non-uniform heating is quite close to the behavior of HTS devices with uniform (or quasi-uniform) heating. The models for both cases are working and provide close results that are well coinciding with the experiments when determining limiting operating current or heating development time. In both models we did not use any “superconducting” words: like “normal zone”, “normal zone propagation” or “minimum propagating energy”. Anyway, reasonable predictions of HTS devices behavior could be done and models are quite clear from the physical point of view. For both the uniform and non-uniform heating, sharp switching between stable and nonstable mode does exist. Less than 1% of the transport current change may lead from the stability to the fast temperature rise. The time of the temperature rise reduces drastically with the current rise, inversely proportional to the square of the current density [21]. Thus, the little excess of the thermal runaway current Iq may lead to the catastrophic sequences for an HTS device for both, uniform or non-uniform heating. The numerical experiments for the non-uniform heating of HTS at overload conditions showed the possibility of some dangerous phenomena like heat localization with local overheating. Important condition for appearance of the heat localization is some initial disturbance in temperature. This initial disturbance always could be expected in long HTS objects like power cables, for example due to local change of the critical currents of basic superconductors. The heat localization due to local disturbances still should be confirmed by appropriate experiments. But experience of burning of many HTS (and LTS) devices showed that it is always local. In any cases, the opportunity of the heat localization in long HTS objects should be kept in minds of cables’ designers. New 2-G HTS usually have higher indexes n then 1-G HTS, while magnetic field is low. It leads to better determining of critical current for 2-G superconductors and weaker dependence of thermal runaway current Iq on heating. Thermal runaway current becomes more close to the critical current. Nevertheless, specific thermal time remains long and heating development time will be also as long as for 1-G HTS devices. All conclusions about stability and heating development made for1-G HTS based devices are valid for devices from 2-G HTS.

Novel Approaches to Describe Stability and Quench of HTS Devices

237

VII. Conclusions We tried to sound our opinion about the necessity of some new approaches in describing the stability and quench or heating development in HTS devices and presented such approaches for cases of uniform and non-uniform heating. The models consider the instability development in HTS devices, while HTS device is considered not as a superconductor but the medium with non-linear material parameter. The analytical model developed is scalable and it is good to use for the HTS devices with sizes less than characteristic heat length l h =

Ak / Ph . This model may be used rather

universally, even for LTS - devices with low index n. The numerical model has been developed for long HTS objects (for example, power cables at overloading conditions). It was shown that blow-up regimes with heat localization can appear in long HTS objects. Instability development time can be rather short (due to heat localization) in comparison with usual time of the heating development in HTS devices. The threshold current Iq weakly depends on cooling characterizes the stability of HTS devices. This current separates stable and unstable modes. The increase of the transport current in an HTS device by less than 1% may lead to the switching from the stable to unstable mode. Important safety parameter is the heating development time that quickly decays with the transport current rise inversely proportional to the square of the current. We believe that these new approaches developed are useful to analyze HTS – devices stability/quench/heating behavior, without using “superconducting” terms, like a normal zone and its propagation.

Acknowledgments These are the works done by many researchers of course. It has been started in the late 1990-s in Kyushu University in Japan with experiments made by Vitaly Vysotsky. Then Yuri Ilyin joined to the experimental works. The first theoretical basis of these works was developed mostly by Alexander Rakhmanov during his visit to Kyushu University in 1998. In 2000 Yuri Ilyin defended his PhD Thesis in Kyushu University based on some of these works. So, these three researchers should be considered as initial fathers of these approaches. We have to mention also the contribution of Prof. T. Kiss from Kyushu University, Prof. N.V. Zmitrenko from Moscow Institute of Mathematical Modeling and Prof. V.E. Sytnikov from Russian Cable Institute for their invaluable support for further development of the first ideas. Many Master students, both from Japan and Russia contributed to this work as well as other our colleagues whose list of names would be very large to put here. In any case their names can be found in the list of references. Authors are happy to express his great gratitude to everyone who helps us to start, to move and to continue this work.

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References [1] Y. Iwasa , Case Studies in Superconducting Magnets, Plenum Press, N.Y. (1994). [2] R.H Bellis, Y. Iwasa, Quench propagation in high Tc superconductors, Cryogenics 34 (1994) 129-134. [3] M.N. Wilson, Stabilization, protection and current density: some general observations and speculations, Cryogenics 31 (1991) 499-505. [4] M.N. Wilson , Superconducting Magnets, Clarendon Press, Oxford, UK (1983). [5] V.S. Vysotsky, S.S. Fetisov and V.E. Sytnikov, Peculiarities on voltage – current characteristics of HTS tapes at overloading conditions cooled by liquid nitrogen, the paper to be presented at EUCAS – 2007, Brussels, Belgium, September 2007 [6] V.S.Vysotsky, T.Kiss, Yu.A.Ilyin, M.Takeo, H. Saho, K.Funaki, T.Hasegawa, Heat and quench propagation in HTSC coil with bobbins from different materials (Part1, Experiment) Proceedings of ICEC-17, Institute of Physics Publishing, Bristol, 1998, 583-586 [7] Yu. A. Ilyin, V. S. Vysotsky, T. Kiss, M. Takeo, H. Okamoto, F. Irie, Stability and quench development study in small HTSC magnet, Cryogenics 41 N9 (2001) 665-674. [8] A.V. Gurevich, R.G. Mints, A.L. Rakhmanov, The physics of composite superconductors, Beggel House Inc. N.Y. (1997). [9] Dorofejef G.L., Klimenko E.Yu., Imenitov A.V. , Cryogenics, v.20, pp.307-311, 1980 [10] N. N. Martovetsky, P-L. Bruzzone, B. Stepanov, R. Wesche, Chen-Yu Gung, J.V. Minervini, et al, Effect of conduit material on CICC performance under high cycling load, IEEE Trans. Appl. Supercond. 15 No.2 (2005) 1367-1370. [11] V.S. Vysotsky, A.L. Rakhmanov, Yu. Ilyin Influences of voltage-current characteristic difference on quench development in low-Tc and high-Tc superconducting devices. (Review), Physica C, v.401, N1, pp.57-65, 2004 (Proceedings ICMC-Topical 2003). [12] A.L. Rakhmanov, V.S. Vysotsky, N.V. Zmitrenko, Quench development in long HTS objects – the possibility of “blow-up” regimes and a heat localization, IEEE Trans. Appl Supercon. (2003) 13 1942-1945. [13] Ishiyama , H. Asai, A stability criterion for cryocooler-cooled HTS coils, IEEE Trans. Appl. Supercond. 10 No. 1 (2000) 1834-1837 [14] Yu.A. Ilyin, Quench development in high temperature superconducting devices, PhD Thesis, Kyushu University, Fukuoka, Japan, (2000) [15] A.L. Rakhmanov, V.S.Vysotsky, Yu.A.Ilyin, T.Kiss, M.Takeo, Universal scaling law for quench development in HTSC devices,. Cryogenics 40 N1 (2000) 19-27. [16] V.S. Vysotsky, Yu.A. Ilyin, T. Kiss, M. Inoue, M. Takeo, F. Irie et al. Thermal quench study in HTSC pancake coil, Cryogenics 40 N1 (2000) 9-17. [17] V.S. Vysotsky, Yu.A. Ilyin, A.L. Rakhmanov, Stability and quench development in HTS magnets: influence of cooling and material parameters, Adv. in Cryog. Eng. 47 (2002) 481-488. [18] V.S. Vysotsky, Yu.A. Ilyin, A.L. Rakhmanov, M. Takeo, Quench development analysis in HTSC coils by use of the universal scaling law, IEEE Trans. Appl. Supercond. 11 N1 (2001) 1824 –1827. [19] V.S. Vysotsky, V.E. Sytnikov, A.L. Rakhmanov, Yu.A. Ilyin, Analysis of stability and quench in HTS devices - new approaches, Fusion Engineering and Design 81 (2006)

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[20]

[21]

[22]

[23]

[24] [25] [26]

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2417–2424 (Paper presented at International ITC-15 conference, Toki, Japan, December, 2005) V.S. Vysotsky, N.V. Zmitrenko, A.L. Rakhmanov, Quench development analysis in long HTS objects, IOP Conference Series 181, (Proceedings of EUCAS-2003) (2004) 580-584. V. S. Vysotsky, V. E. Sytnikov, V. V. Repnikov, E. A. Lobanov, N. V. Zmitrenko, A. L. Rakhmanov, Heating Development Analysis in Long HTS Objects with Cooling, IEEE Trans. Appl Supercon. (2005) 15 1655-1658. V.S.Vysotsky, V.V.Repnikov, E.A.Lobanov, G.H. Karapetyan and V.E.Sytnikov Heating Development Analysis in Long HTS Objects – Updated Results, J. Phys.: Conf. Ser. 43 877-880, 2006 (Proceedings EUCAS – 2005) S. Torii S., S. Akita, K. Ueda, Transport current properties of double-pancake coils wound by Ag-sheated Bi-2223 tapes, IEEE Trans. on Appl. Supercond. 9 N.2 (1999) 944-947. Kumakura H., Kitaguchi H., Togano K., Hasegawa T. Cryogenics 38 pp. 163-167 (1998). H. Kumakura , H. Kitaguchi, K. Togano, H. Wada, Stability of a Bi-2223 refrigerator cooled magnet, Cryogenics 38 (1998) 639-842. Godeke, O. Shevchenko, H.J.G. Krooshoop, B. ten Haken, H.H.J ten Kate, G. Rutten, et al, An optimized BSCCO/Ag resonator coil for utility use, IEEE Trans. Appl. Supercond. 10 No. 1 (2000) 849-852.

INDEX A absolute zero, vii AC, viii, 27, 29, 30, 31, 34, 35, 36, 37, 38, 40, 41, 43, 44, 45, 47, 53, 54, 55, 56, 57, 58, 59 accounting, 73 accuracy, 49, 73, 103, 113 acetone, 189 activation, 107 AFM, 188 alcohol, 129 alloys, vii alternative, 44 aluminium, vii amorphous phases, 4, 7 amplitude, 30, 31, 37, 40, 41, 43, 54, 55 anion, 97 anisotropy, 76, 77, 96, 134, 150, 168, 177, 182, 189, 200 annealing, 3, 7, 66, 68, 69, 72, 108, 126, 127, 128, 137, 140, 141, 143, 144, 146, 176 antiferromagnetic, 188 argument, 161 aspect ratio, 212 atomic force, 83 atomic orbitals, 73 atomic positions, 83, 104, 108 atoms, ix, 64, 65, 71, 72, 84, 85, 86, 87, 96, 97, 101, 103, 110, 111, 113, 114, 115, 116, 117, 125, 137, 145, 146, 155, 156, 188, 189, 207, 216 attachment, 191 attention, 53, 64, 223 Auger electron spectroscopy, 69 averaging, 47

B backscattering, 103

bandwidth, 80, 81 Bangladesh, 63 banks, 174 basis set, 73, 79 BCS theory, x, 149, 151, 154 BED, 129 behavior, viii, x, 3, 27, 28, 29, 30, 36, 37, 38, 39, 41, 43, 44, 45, 48, 49, 51, 52, 53, 55, 56, 57, 58, 95, 102, 105, 107, 111, 112, 113, 114, 115, 116, 134, 150, 152, 157, 159, 167, 176, 179, 189, 191, 215, 230, 236, 237 Beijing, 125 Belgium, 238 bending, 171, 180 bias, x, 167, 169, 170, 171, 173, 174, 177, 179, 180, 182, 183 binding, 151, 154, 157 blocks, ix, 101, 125, 146 bonding, 76, 77, 84, 85 bonds, 76, 85, 118, 200 boron, vii, 1, 2, 6, 7, 9, 10, 19, 22, 64, 65, 72, 78, 85 boson(s), 94, 156 burning, 236

C Ca2+, xi, 187, 200, 212, 214, 216, 219 cables, xi, 223, 224, 225, 232, 233, 236, 237 calorimetry, 106 Canada, 149 candidates, ix, 63, 64, 74, 89 capacitance, 35, 36, 41, 45 carrier, ix, xi, 93, 94, 97, 118, 119, 187, 188, 189, 212, 213, 216 cation, 94, 97, 146, 200, 206 cell, 30, 34, 35, 37, 41, 43, 44, 45, 53, 55, 64, 65, 66, 74, 75, 76, 83, 99, 100, 101, 102, 103, 105, 106, 107, 109, 110, 111, 112, 113, 114, 115, 117, 119,

242

Index

125, 131, 132, 133, 134, 137, 138, 139, 140, 141, 143, 161 ceramic, vii, 106, 107, 120, 150 Chalmers, 91, 167 charge density, 84, 151 chemical bonds, 117 chemical composition, ix, 93, 119 China, 125, 149, 167 Chinese, 167 classes, 150 clusters, 107 coherence, 29, 44, 55, 94, 95, 150, 216, 217 combined effect, 71, 84 communication, 90 competition, 161, 162 complexity, 95, 226 components, 30, 31, 34, 40 composition, ix, 3, 11, 18, 63, 69, 126, 127, 204 compounds, ix, 63, 64, 70, 73, 84, 88, 89, 93, 94, 96, 97, 101, 102, 103, 105, 106, 107, 108, 109, 117, 119, 126, 134, 149, 150, 151, 156, 162, 188 compressibility, 76, 77, 105, 119 computers, 49 computing, 183 concentration, ix, xi, 2, 18, 93, 94, 95, 96, 97, 99, 107, 118, 119, 187, 188, 189, 191, 199, 200, 204, 212, 213, 215, 216, 219 conduction, 225 conductivity, vii, 95, 99, 151, 157, 159, 160, 168, 174, 189, 225, 233 conductor, vii, 151, 231 configuration, 31, 37, 41, 54 confinement, 134 conjecture, 72 connectivity, 4, 5, 6, 10, 13, 22 conservation, 73 constraints, 157 construction, 74, 101 control, x, 22, 152, 167, 168, 169, 191 convergence, 73, 74 cooling, 2, 11, 29, 45, 53, 55, 102, 103, 105, 111, 113, 117, 118, 120, 128, 225, 228, 230, 231, 232, 233, 234, 235, 237, 238 cooling process, 53 copper oxide, vii, x, 93, 95, 97, 99, 101, 105, 129, 136, 149, 150, 151, 152, 154, 155, 156, 157, 158, 159, 160, 161, 162, 200 correlation(s), 3, 73, 74, 94, 95, 97, 101, 117, 137, 146, 152, 158, 159, 162 corrosion, 64 costs, 2 Coulomb interaction, 153 coupling, 82, 86, 152, 154, 155, 162, 171, 215

covalency, 86 covalent bond(ing), 77, 84 critical current density (Jc,), viii, x, 22, 28, 187, 188, 191, 214, 215 critical temperature (TC), vii, x, 29, 33, 37, 39, 54, 94, 187, 188, 189, 200, 209, 212, 213, 214, 215, 216, 218, 219, 230 critical value, 171 crystal structure(s), ix, x, xi, 93, 96, 97, 103, 107, 108, 119, 121, 149, 159, 187, 201, 207 crystalline, vii, 1, 2, 4, 7, 10, 11, 15, 22, 73, 85, 158, 189 crystallinity, viii, 1, 72 crystals, 3, 29, 83, 152, 158, 200 cuprates, ix, 76, 77, 93, 94, 95, 96, 101, 107, 114, 116, 117, 118, 119, 127, 134, 146, 152 current limit, 7, 225 cycles, 175 cycling, 238

D damping, 53 data collection, 191 database, 191 decay, 233 decomposition, 11, 20, 72 defects, vii, xi, 4, 29, 96, 187, 215, 216, 219 deficiency, 3, 11, 13, 17, 20, 21, 29, 137 deformation, ix, 93, 98, 116, 117, 119, 151 degenerate, 78 degradation, 171, 174, 176 density, vii, viii, ix, x, 1, 4, 11, 21, 22, 27, 30, 37, 38, 39, 41, 46, 58, 63, 72, 73, 74, 77, 81, 82, 84, 85, 86, 87, 89, 94, 97, 117, 136, 143, 144, 145, 150, 151, 155, 159, 160, 171, 174, 179, 182, 187, 188, 201, 202, 204, 214, 215, 216, 217, 235, 236, 238 density functional theory, ix, 63, 73 density values, 201 deposition, 67, 69, 70, 169 derivatives, 76, 83 designers, xi, 223, 224, 236 detection, 203 DFT, ix, 63, 73 diboride(s), ix, 63, 64, 65, 70, 72, 77, 86, 88, 89 dielectric constant, 73 Differential Thermal Analysis (DTA), 191 differentiation, 228 diffraction, ix, 8, 18, 93, 94, 103, 104, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 119, 120, 126, 127, 130, 132, 133, 136, 138, 139, 140, 141, 189 dimensionality, 80, 81

Index discontinuity, 49 disorder, viii, ix, 1, 13, 16, 19, 20, 22, 28, 107, 125, 146, 156, 159, 219 dispersion, 73, 78, 80, 95 displacement, 83, 84, 113, 116, 137 distribution, viii, 6, 7, 22, 28, 36, 38, 39, 43, 44, 45, 55, 58, 84, 146, 157, 162 diversity, 76 dopants, ix, 20, 125, 126, 127 doping, ix, xi, 2, 6, 20, 21, 93, 94, 95, 96, 97, 99, 101, 103, 105, 106, 110, 118, 119, 125, 127, 134, 137, 144, 145, 146, 151, 155, 156, 157, 158, 159, 162, 187, 188, 189, 191, 203, 213, 215, 218 DTA curve, 191, 192 duration, 129

E earth, 189, 192, 200, 215, 216, 218, 219 eigenvalue, 73 electric charge, 97 electric field, 200, 226, 229, 230 electric power, 225 electrical conductivity, 118 electrical resistance, vii electrodes, 37, 169, 171, 172, 174 electron(s), ix, x, 4, 9, 19, 20, 63, 70, 71, 73, 74, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 93, 95, 98, 117, 118, 119, 126, 128, 129, 142, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 161, 162, 187, 188, 216 electron charge, 85 electron density, 84, 85 electron diffraction, 9, 20, 129 electron microscopy, x, 187 electron pairs, 86, 152 electronic materials, 188 electronic structure, 73, 126, 137, 154 electron-phonon coupling, ix, 63, 77, 78, 82, 84, 152 electron-phonon interaction(s), ix, 70, 71, 82, 93, 95, 118, 119, 151, 152, 155, 162 elongation, 200 emission, 129 endothermic, 191 energy, x, 4, 30, 46, 56, 69, 73, 74, 75, 78, 82, 83, 87, 95, 98, 128, 134, 142, 150, 151, 152, 153, 154, 155, 157, 158, 159, 161, 167, 168, 169, 174, 179, 180, 182, 183, 224, 225, 226, 236 environment, 156, 188 equilibrium, 74, 75, 80, 81, 83, 88, 230 estimating, 77 etching, x, 167, 169, 171, 174, 175, 176, 177, 178, 179, 180, 182

243

Euler, 36, 48 euphoria, 224 evaporation, 11, 169 evidence, 18, 37, 43, 53, 58, 94, 154, 155 evolution, 102, 105, 106, 107, 144, 229 EXAFS, 113, 115, 117 excitation, 30, 31, 36, 41, 54, 155, 157, 162 exclusion, vii exposure, 103, 104 expulsion, 151 external magnetic fields, 215 extinction, 130, 133, 139, 140

F fabrication, x, 167, 168, 175, 179, 183 family, vii, 64, 101, 158, 161 Fermi level, 77, 78, 81, 82, 84, 89, 97, 117 Fermi surface, 74, 78, 79, 89, 151 fermions, 159 ferroelectrics, 117 ferromagnets, 150 filament, 5 film(s), 5, 6, 30, 41, 65, 66, 67, 68, 69, 72, 86, 87, 88, 89, 169 financial support, 59 fine tuning, 48 flexibility, 74 flight, 103 fluctuations, ix, 63, 86, 87, 88, 95, 117, 156, 158, 159, 161 fluid, 33 flux pinning, x, 4, 29, 151, 187, 188, 189, 215, 216, 218, 219 foils, 129 force constants, 83 Fourier, 35, 83 France, 121, 122 free energy, 98, 161 freedom, x, 73, 149, 150 Friedmann, 163 fruits, 224 frustration, 56

G gases, 3 Gaussian, 73, 74 generalization, 154 generation, 225, 227 Germany, 184 gold, vii, 64, 168, 169, 172, 174, 175

244

Index

grain boundaries, 1, 2, 4, 13, 212 grains, 2, 4, 7, 11, 20, 22, 29, 41, 129, 136, 138, 139, 141, 142, 144, 202, 203, 206, 207, 209 graph, 201 graphite, 65, 76, 77, 89, 103 grids, 74, 225 groups, ix, 63, 64, 126, 189 growth, 3, 94, 97, 99, 101, 105, 109, 115, 117, 119, 201, 215 growth mechanism, 201 guidelines, 88

H halogen, 69 Hamiltonian, 46, 55, 153 hardness, 106, 111 heat, 101, 127, 128, 138, 139, 144, 171, 174, 191, 197, 198, 199, 200, 201, 202, 225, 226, 227, 228, 230, 231, 232, 233, 234, 235, 236, 237, 238 heat capacity, 225 heat conductivity, 233 heat release, 226, 230, 231, 234, 236 heat removal, 228, 231, 235, 236 heating, x, 3, 11, 111, 128, 144, 167, 168, 171, 174, 180, 183, 190, 191, 224, 225, 228, 230, 231, 232, 234, 235, 236, 237 heating rate, 3, 190, 191 heavy metals, 101 height, 99, 169 helium, 2, 230 heterogeneity, 95 hot pressing, 4, 5, 21 HRTEM, 129, 139, 140 Hubbard model, 157 hybridization, 99 hydrogen, 2 hysteresis, 179

indication, 19, 215 indices, 200 inelastic, 94, 95, 96, 155, 158 infinite, 73, 129 inhomogeneity, 10, 18, 86, 96, 146, 155, 156, 158 insight, ix, 63 instability, 68, 98, 106, 216, 228, 230, 234, 235, 237 instruments, 191 insulators, 162, 188 integration, 36, 49, 73 intensity, 11, 12, 15, 18, 19, 20, 66, 116, 129, 134, 136, 142, 144, 169, 192, 203, 206 interaction(s), 70, 71, 76, 77, 78, 82, 84, 85, 93, 95, 101, 118, 119, 126, 151, 152, 153, 154, 156, 157, 158, 162, 216 interface, 212 intermolecular interactions, 76 interpretation, ix, 44, 48, 55, 56, 93, 95, 99, 117, 119, 150, 157 interval, ix, 93, 106, 109, 110, 115, 117, 119 ion implantation, 174 ions, xi, 78, 86, 98, 137, 155, 174, 187, 195, 200, 212, 214, 216, 219 iron, 70, 77 irradiation, 14, 129 Islam, 63, 77, 90, 91, 92 isotope(s), 95, 154, 156

J Japan, 69, 72, 89, 237, 238, 239 Josephson coupling, 46, 56, 162 Josephson junction, viii, x, 4, 27, 28, 29, 30, 31, 36, 37, 41, 44, 45, 53, 54, 55, 58, 167, 168, 179 justification, 44

K kernel, 87

I idealization, vii identification, 88, 127, 191 illumination, 129 images, 8, 9, 12, 22, 174, 175, 203, 204, 205 imaging, xiii, 7, 22, 155 implementation, 231 impurities, vii, viii, 2, 7, 22, 28, 29, 40, 69, 71, 89, 188, 189 in situ, viii, 1, 4, 5, 11, 20, 69 in transition, 162 India, 91, 187

L Landau theory, 117 laser, 69, 86, 87 laser ablation, 86 lattice parameters, xi, 3, 76, 130, 187, 200 linear function, ix, 93, 118, 119 links, 215 liquid nitrogen, 70, 103, 233, 238 liquid phase, 20, 21 literature, 29, 103, 105, 108, 113, 118, 119, 154, 232

Index localization, ix, 70, 71, 84, 93, 98, 117, 119, 234, 235, 236, 237, 238 London, 37, 39, 46, 89, 147, 148, 164 low temperatures, vii, 29, 52, 55, 95, 151

M magnesium, 2, 3, 22 magnet(s), 2, 224, 227, 233, 238, 239 magnetic field, vii, viii, 10, 27, 29, 30, 31, 33, 35, 36, 37, 38, 40, 41, 43, 44, 45, 46, 51, 52, 53, 55, 57, 58, 95, 128, 150, 151, 191, 236 magnetic moment, 30 magnetic properties, viii, 20, 27, 29, 30, 34, 41, 44, 45, 46, 53, 55, 58 magnetic resonance imaging, 2, 94, 157, 158, 159 magnetic structure, 156 magnetization, 32, 33, 34, 35, 38, 47, 53 magnetoresistance, 13 Magnetron Sputtering, 69 Malaysia, xiii, 1, 22 matrix, 22, 29, 74, 83, 87, 161, 203, 216, 227, 228 measurement, xiii, 15, 22, 169, 172, 190 media, 230 Meissner effect, vii, viii, 27, 28, 36, 40, 53, 151 melting temperature, 3, 69, 72, 191 metals, vii, 64, 160, 162 microscope, 30, 129, 174 microscopy, 188 microstructure(s), xi, 11, 20, 187, 202, 204, 215, 219 mixing, 11, 70 modeling, 152 models, 95, 150, 152, 226, 236, 237 modulus, 75, 76 molar ratios, 127 molybdenum, 41 momentum, 154, 157, 161 monolayer, 72, 146 morphology, 202, 212 Moscow, 27, 223, 237 motion, 53, 155 motivation, 107, 188 movement, 155 MRI, 2 multiples, 182

N NaCl, 96 nanometer scale, 146, 156 Nd, x, 95, 187, 189, 191, 192, 193, 195, 196, 197, 200, 203, 205, 206, 210, 213, 214, 215, 216

245

needles, 212 neglect, 33, 37, 46 neon, 2 Netherlands, 223 network, 29 New York, 23, 25, 59, 60, 61, 62, 92 niobium, 1, 41, 54 NMR, 109, 110, 111, 117, 161 noble metals, vii, 64 nodes, 161 noise, 16, 204 novel materials, 152, 159 nuclear magnetic resonance, 94 nuclei, 84 numerical analysis, 235

O observations, 4, 48, 56, 57, 64, 68, 129, 144, 238 observed behavior, 37, 44, 45, 49, 55 omission, 86 optical microscopy, 69 optics, 158 orientation, 158, 204, 206, 215 overload, 225, 236 oxidation, 2, 3, 99 oxides, ix, 63, 155, 156, 160, 189, 191, 192 oxygen, ix, 2, 3, 4, 29, 94, 95, 96, 97, 99, 100, 101, 103, 104, 105, 106, 107, 108, 110, 111, 113, 114, 115, 117, 125, 126, 127, 129, 134, 135, 136, 137, 142, 143, 144, 145, 146, 154, 155, 156, 159, 174, 189, 200

P pairing, viii, 28, 30, 40, 82, 88, 117, 118, 146, 150, 151, 152, 153, 154, 155, 156, 157, 158, 161, 162 parallelism, 29 parameter, viii, 27, 30, 36, 38, 41, 43, 45, 47, 48, 52, 53, 54, 56, 57, 58, 59, 67, 71, 73, 77, 84, 95, 99, 100, 103, 105, 106, 109, 112, 116, 119, 125, 151, 153, 161, 162, 178, 182, 183, 226, 227, 228, 231, 235, 237 particles, 88, 157, 216 performance, 2, 157, 238 periodicity, 134 permeability, 35 permit, 230 perovskite, vii, 96, 97 perovskites, 29, 96 peroxide, 127 pH, 37, 46

246

Index

phase boundaries, 212 phase diagram, viii, 28, 50, 51, 52, 94, 95, 161, 188, 213 phase transformation, 137, 139 phonons, x, 77, 82, 149, 151, 152, 154, 155, 156, 157, 171 photoemission, 18, 154, 155, 157 photolithography, x, 167, 169, 174 physical properties, 28, 94, 168 physics, vii, x, 28, 149, 150, 152, 162, 238 plane waves, 73 platinum, 64 polarization, 74, 117 pollution, 127 polyimide, 169, 174 poor, 3, 6, 18, 72, 138, 189 population, 84 porosity, 4, 6, 11, 202, 204, 215 power, vii, 13, 67, 150, 158, 171, 173, 189, 224, 225, 233, 236, 237 prediction, 52, 64, 77, 151, 152 pressure, 3, 4, 5, 64, 67, 69, 74, 75, 76, 77, 80, 81, 82, 86, 105, 108, 119, 126, 127, 129, 130, 134, 135, 136, 145, 146, 151 probability distribution, 53 probe, 19, 70, 155, 169, 171, 173, 188, 191 production, 3 program, 103, 108, 121 propagation, xi, 223, 224, 225, 226, 228, 230, 232, 233, 234, 236, 237, 238 proportionality, 154

Q quanta, viii, 28, 36, 41, 53, 54 quantization, viii, 28, 35, 48, 54, 55, 56, 57, 59 quartz, 66, 67, 68, 70

R race, 230 radiation, 70, 96, 103, 115, 116, 128, 174 radius, 30, 32, 200, 215 Raman spectra, viii, xiii, 1, 15, 18, 19, 20, 22, 94 Raman spectroscopy, viii, xiii, 1, 22 range, 2, 3, 11, 14, 16, 17, 18, 69, 70, 77, 82, 88, 94, 97, 103, 106, 107, 113, 114, 115, 117, 120, 144, 150, 157, 158, 173, 179, 180, 189, 191, 216, 219 raw materials, 127 reaction rate, 2, 7, 219 reactivity, 7, 19 reality, 235

recall, 13, 14, 55 reduction, 2, 6, 22, 95, 99, 105, 200, 201, 206, 215, 232 reflection, 30, 31, 37, 41, 54, 67, 68, 103, 131, 132, 139 relationship, 32, 75, 143, 151 relaxation rate, 161 relevance, 158 renormalization, 117, 161 resistance, vii, 35, 41, 45, 72, 169, 176, 178, 182, 191, 212, 213 resolution, viii, 18, 28, 53, 96, 103, 107, 108, 110, 113, 116, 129 resonator, 239 RF, 67, 69 room temperature, 3, 9, 10, 11, 13, 20, 21, 22, 66, 67, 68, 69, 70, 84, 103, 104, 105, 107, 109, 113 Royal Society, 120 Russia, 93, 120, 223, 237

S safety, 235, 237 scaling, 159, 160, 230, 231, 238 scaling law, 238 scaling relations, 160 scanning electron microscopy, 69 scattering, 11, 13, 14, 20, 21, 71, 83, 84, 94, 95, 96, 103, 150, 154, 155, 156, 158, 159, 160 Schrödinger equation, 154 science, 1, 224 search(ing), ix, x, 63, 64, 73, 88, 89, 98, 107, 119, 120, 126, 129, 149, 152 second generation, xi, 223, 224, 227 selected area electron diffraction, 129 self-consistency, 74 SEM micrographs, 203 semiconductor(s), vii, 123, 177 sensitivity, 77, 203 separation, 3, 18, 81, 94, 117 series, vii, 1, 11, 14, 15, 31, 35, 73, 127, 128 shape, 3, 30, 78, 81, 97, 103, 113, 114, 160, 161, 174, 203 sharing, 232 shear, 136 shoulders, 16 sign(s), ix, 43, 93, 110, 114, 117, 153 signals, 151 silicon, 169 silver, vii, ix, 63, 64, 66, 69, 72, 85, 168, 191 similarity, 101 simulation, 28, 48, 49, 52 sine wave, 137

Index single crystals, 6, 95, 96, 98, 168, 174, 176, 179 sintering, 4, 5, 6, 20, 21, 201 sites, ix, 125, 126, 137, 146, 189, 212, 219 SNS, 53 software, 11, 191, 192, 200 solid solutions, 89, 189 solid state, x, 187, 189 solubility, 189 specific heat, 120, 228 spectroscopy, 4, 18, 69, 94, 96, 128, 142, 154, 155, 168 spectrum, 16, 66, 67, 68, 143, 155, 158 spin, ix, x, 63, 86, 87, 88, 94, 95, 149, 153, 154, 158, 159 sputtering, 66, 67, 68, 69, 88, 169 SQUID, x, 10, 30, 53, 128, 167, 183 stability, xi, 66, 67, 68, 88, 189, 200, 223, 224, 225, 226, 227, 228, 232, 236, 237, 238 stabilization, 224 stages, 190, 201, 202 standards, 168 stars, 118 statistics, 138 STM, 96, 98, 188 stoichiometry, vii, x, 1, 2, 3, 11, 17, 18, 20, 22, 94, 96, 187, 188, 189, 201, 206, 207, 213, 216 strain, 3, 14, 146 strength, xi, 82, 84, 151, 160, 187, 215, 216, 217, 218 stretching, 77, 82, 97, 155 structural characteristics, 96 students, 237 substitution, xi, 94, 95, 96, 101, 125, 156, 187, 188, 189, 200, 212 substrates, 174, 175 Sun, 24, 92, 122, 220, 221 superconducting gap, 94, 95, 154, 161, 168 superconducting materials, 64 superconductivity, vii, ix, x, xi, 63, 64, 69, 72, 78, 82, 86, 87, 88, 92, 93, 94, 95, 96, 98, 107, 119, 125, 126, 127, 128, 134, 136, 137, 144, 145, 149, 151, 152, 153, 154, 155, 156, 157, 158, 161, 162, 174, 176, 188, 223, 224, 226 superconductor(s), vii, viii, ix, x, 1, 2, 27, 28, 29, 30, 36, 39, 40, 63, 64, 65, 74, 77, 78, 82, 89, 98, 125, 126, 129, 144, 145, 146, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 161, 162, 168, 187, 188, 189, 203, 215, 216, 219, 224, 225, 226, 227, 228, 230, 236, 237, 238 superfluid, 159, 160 superlattice, 20, 130, 132, 134 supply, 146 suppression, 87, 94, 95, 117, 161

247

susceptibility, viii, 10, 16, 27, 29, 30, 31, 32, 34, 35, 36, 37, 38, 40, 41, 43, 44, 45, 47, 48, 50, 52, 53, 54, 55, 56, 57, 58, 59, 128, 212 Sweden, 167 switching, 174, 236, 237 symbols, 112, 114, 235 symmetry, viii, 28, 29, 30, 35, 38, 40, 79, 80, 81, 85, 98, 132, 139, 152, 157, 200 synthesis, ix, x, 3, 63, 66, 69, 86, 87, 101, 127, 146, 187, 189 systems, ix, 30, 63, 64, 71, 74, 88, 93, 95, 98, 101, 106, 119, 146, 213

T tantalum, 3 targets, 67, 68, 69 technology, 36 TEM, xiii, 7, 8, 9, 20, 22, 127, 128, 129, 130, 136, 137, 138, 141, 142, 144, 145, 146 temperature dependence, viii, 9, 12, 28, 29, 38, 39, 47, 49, 54, 55, 57, 59, 106, 107, 110, 113, 114, 150, 151 Texas, 121 theory, vii, 53, 71, 73, 74, 84, 95, 151, 152, 153, 155, 156, 160, 161, 162, 230, 231, 232, 235 thermal activation, 36 thermal analysis, x, 187 thermal expansion, 106, 107, 109, 110, 113, 115, 117 thermal resistance, 171, 180 thermal treatment, 107, 119 thermogravimetry, 144, 145 thin films, ix, 30, 63, 66, 68, 69, 72, 86, 87, 189, 215 threshold, 237 time, 2, 30, 38, 43, 47, 49, 57, 59, 66, 68, 70, 95, 99, 103, 104, 116, 126, 144, 162, 169, 176, 177, 178, 195, 228, 229, 230, 231, 232, 234, 235, 236, 237 total energy, 46, 56, 73, 74, 162, 174 transformation, 137, 141, 144 transition(s), vii, ix, x, 1, 10, 14, 17, 21, 48, 49, 64, 70, 72, 86, 88, 93, 96, 97, 98, 101, 103, 105, 107, 110, 115, 118, 119, 125, 137, 144, 145, 146, 150, 151, 154, 157, 158, 159, 160, 161, 167, 176, 177, 182, 183, 188, 189, 209, 213, 224, 228 transition metal, 64, 86, 98 transition temperature, vii, ix, 1, 10, 88, 103, 119, 120, 125, 137, 144, 145, 146, 150, 151, 154, 157, 158, 159, 160, 161, 176, 188 transmission electron microscopy, 127, 129 transport, xiii, 4, 14, 22, 29, 41, 44, 94, 150, 160, 215, 217, 225, 234, 235, 236, 237 trend, 87, 146 tungsten, 174

248

Index

tunneling effect, x, 46, 56, 150, 155, 157, 161, 162, 167, 168, 180, 183, 188

U UK, 90, 238 ultrasound, 106 unconventional superconductors, 150 uniform, viii, 14, 27, 30, 36, 37, 38, 40, 58, 94, 95, 169, 177, 228, 232, 233, 235, 236, 237 universality, 161

V vacancies, 4, 19, 20, 29, 96, 127, 136 vacuum, 3, 35, 66, 69, 70, 72, 88, 174 valence, 73, 156 values, xi, 10, 11, 12, 13, 14, 17, 19, 22, 38, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 57, 71, 74, 75, 76, 77, 78, 81, 84, 87, 97, 98, 101, 102, 103, 105, 106, 108, 116, 118, 119, 127, 129, 130, 131, 134, 145, 179, 187, 195, 200, 206, 212, 213, 214, 215, 216, 228, 233 vapor, 3, 6 variable, 97, 101, 200 variation, vii, 1, 10, 13, 20, 71, 77, 78, 151, 188, 195, 206, 209, 212, 213, 215 vector, 83, 131 velocity, 155 vibration, ix, x, 77, 82, 93, 106, 110, 115, 117, 119, 149, 156 viscosity, 53 volatility, 3, 4, 19

W water absorption, 88 wave vector, 74, 83, 131 weight loss, 145 wetting, 7 Wisconsin, 9

X XPS, 69 x-ray, viii, 1, 11, 14, 18, 66, 126 x-ray analysis, x, 187 x-ray diffraction (XRD), viii, x, xiii, 1, 6, 7, 10, 11, 12, 20, 22, 66, 67, 68, 69, 70, 72, 103, 104, 106, 108, 109, 111, 112, 113, 114, 115, 116, 126, 127, 128, 129, 136, 137, 138, 141, 144, 187, 189, 191, 192, 193, 194, 195, 197, 198, 199, 200, 203, 205, 206, 212 x-ray diffraction data, 103, 112, 115

Y yield, 71, 78, 82, 127

Z zirconium, 3