Superconductivity Research Developments

SUPERCONDUCTIVITY RESEARCH DEVELOPMENTS SUPERCONDUCTIVITY RESEARCH DEVELOPMENTS JAMES R. TOBIN Editor Nova Science ...

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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


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


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].


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


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)


(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].


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.


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


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


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


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.


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


(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


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


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.


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)

22


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.


23

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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

28

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


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:


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


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.


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,


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


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]:


⎛ 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


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?


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
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


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.


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


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.


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 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


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 ) 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


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


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


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-


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 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


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


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]).


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.


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].


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


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.


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


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


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 ) exp [ ] 1.2 λ − μ* (1 + 0.62 λ) − λsp

(16)

where λsp represents contribution from spin fluctuations (paramagnons). 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


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.


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].

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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)

94

Svetlana G. Titova and John T. S. Irvine

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

Superconductivity Research Developments

Superconductivity Research Developments

New Developments in High Temperature Superconductivity

New developments in high temperature superconductivity

New Developments in High Temperature Superconductivity

New Topics in Superconductivity Research