The Oxford Handbook of Small Superconductors 9780198738169, 0198738161

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THE OXFORD HANDBOOK OF SMALL SUPERCONDUCTORS

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The Oxford Handbook of Small Superconductors Edited by A.V. Narlikar

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1 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Oxford University Press 2017 The moral rights of the authors‌have been asserted First Edition published in 2017 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2016952433 ISBN 978–​0–​19–​873816–​9 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Oxford University Press makes no representation, express or implied, that the drug dosages in this book are correct. Readers must therefore always check the product information and clinical procedures with the most up-​to-​date published product information and data sheets provided by the manufacturers and the most recent codes of conduct and safety regulations. The authors and the publishers do not accept responsibility or legal liability for any errors in the text or for the misuse or misapplication of material in this work. Except where otherwise stated, drug dosages and recommendations are for the non-​pregnant adult who is not breast-​feeding Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

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Contents

Plan of the Book and Acknowledgments List of Contributors

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Part I  Introduction and Basic Studies 1

Small Superconductors—​Introduction A.V. Narlikar

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1.1 Two characteristic length scales of superconductors

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1.2 Two size effects in superconductors

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1.3 QSE, quantum fluctuations, Anderson limit, parity and shell effects, etc.

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1.4 Factors influencing small size effects

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1.5 Behavior of nanowires and ultra-​thin films

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1.6 Vortex states of small superconductors

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1.7 Proximity effect behaviors

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1.8 Synthesis of small superconductors

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1.9 Summary and outlook

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2 Local-​Scale Spectroscopic Studies of Vortex Organization in Mesoscopic Superconductors D. Roditchev, T. Cren, C. Brun, and M. Miloševic

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2.1 Basic properties of quantum vortices in superconductors

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2.2 Experimental requirements for studying vortex confinement phenomena

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2.3 Observation of confinement effects on vortices

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2.4 Conclusions and outlook

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3 Multi-​Vortex States in Mesoscopic Superconductors N. Kokubo, S. Okayasu, and K. Kadowaki

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3.1 Introduction

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3.2 Magnetic imaging of superconducting vortices

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3.3 Observation of multi-​vortex states in mesoscopic superconductors 3.4 Summary and outlook 4 Proximity Effect: A New Insight from In Situ Fabricated Hybrid Nanostructures J.C. Cuevas, D. Roditchev, T. Cren, and C. Brun

87 103 108

4.1 An introduction to proximity effect

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4.2 In situ fabricated hybrid nanostructures and tunneling spectroscopy

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4.3 Proximity effect in a correlated 2D disordered metal

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vi  Contents 4.4 Proximity effect in diffusive SNS junctions

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4.5 Proximity Josephson vortices

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4.6 Proximity effect between two different superconductors

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4.7 Conclusions and outlook

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5 Andreev Reflection and Related Studies in Low-​Dimensional Superconducting Systems D. Daghero, G.A. Ummarino, and R.S. Gonnelli

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5.1 Basics of point-​contact Andreev-​reflection spectroscopy

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5.2 Andreev reflection in a nutshell

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5.3 Length scales in mesoscopic systems

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5.4 Examples of PCARS in superconductors with reduced dimensionality

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5.5 Summary and outlook

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6 Topological Superconductors and Majorana Fermions Y.Y. Li and J.F. Jia

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6.1 Introduction

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6.2 TI/​SC heterostructures

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6.3 Nanowire/​SC junctions

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6.4 FM atomic chain on SCs

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6.5 Summary and outlook

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7 Surface and Interface Superconductivity S. Gariglio, M.S. Scheurer, J. Schmalian, A.M.R.V.L. Monteiro, S. Goswami, and A.D. Caviglia

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7.1 Introduction

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7.2 Superconductivity in two dimensions

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7.3 Superconductivity in ultra-​thin metals on Si(111)

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7.4 Superconductivity at the LaAlO3/​SrTiO3 interface

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7.5 Summary and outlook

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Part II  Materials Aspects 8 Mesoscopic Effects in Superconductor–​Ferromagnet Hybrids G. Karapetrov, S.A. Moore, and M. Iavarone

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8.1 Theories underpinning S/​F hybrid structures

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8.2 Domain wall and reverse domain superconductivity

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8.3 Vortex behavior in planar S/​F hybrids

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8.4 Conclusions and outlook

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9 Theoretical Study of THz Emission from HTS Cuprate H. Asai

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9.1 Intrinsic Josephson junction (IJJ) in HTS cuprate

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9.2 THz emitter utilizing IJJs

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Contents vii 9.3 Temperature inhomogeneity in IJJ-​based THz emitter

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9.4 THz emission from IJJs with temperature inhomogeneity

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9.5 Summary

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10 Micromagnetic Measurements on Electrochemically Grown Mesoscopic Superconductors A. Müller, S.E.C. Dale, and M.A. Engbarth

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10.1 Introduction

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10.2 Electrochemical preparation of β-​tin samples

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10.3 Measurement techniques and sample preparation

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10.4 Summary and outlook

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11 Growth and Characterization of HTSc Nanowires and Nanoribbons M.R. Koblischka

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11.1 HTSc nanowires prepared by the template method

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11.2 HTSc nanowires prepared by electrospinning

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11.3 Use of HTSc nanowires as building blocks

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11.4 Summary and outlook

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12 Mesoscopic Structures and Their Effects on High-​Tc Superconductivity H. Zhang

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12.1 Introduction and motivation

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12.2 Model

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12.3 Calculating results and discussion

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12.4 Strain between two blocks and its effect on superconductivity

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12.5 Carrier-​compensated system and mesoscopic structures

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12.6 Existence of fixed triangle (local mesoscopic structure) by x-​ray diffraction

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12.7 The existence of the fixed triangle (local mesoscopic structure) demonstrated by Raman spectroscopy

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12.8 Low wave number evidence about mesoscopic structure

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12.9 Discussions

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12.10  Summary and outlook 13 Magnetic Flux Avalanches in Superconducting Films with Mesoscopic Artificial Patterns M. Motta, A.V. Silhanek, and W.A. Ortiz

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13.1 Avalanches in superconductors

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13.2 Artificial pinning centers in superconducting films

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13.3 Effects of the antidot geometry and lattice symmetry in flux avalanches

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13.4 Summary and outlook

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viii  Contents Part III  Device Technology 14 Superconducting Spintronics and Devices M.G. Blamire and J.W.A. Robinson

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14.1 Conventional spintronics

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14.2 The rationale for superconducting spintronics

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14.3 S/​F proximity effects and Josephson junctions

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14.4 Spin transport in the superconducting state

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14.5 Superconducting spintronic memory

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14.6 Superconducting spintronic logic

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14.7 Superconductor/​ferromagnet thermoelectric devices

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14.8 Materials and device structures

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14.9 Summary and outlook

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15 Barriers in Josephson Junctions: An Overview M.P. Weides

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15.1 Josephson effect

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15.2 Tunnel barriers

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15.3 Metallic barriers

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15.4 Semiconducting barriers

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15.5 Magnetic barriers

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15.6 Summary and outlook

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16 Hybrid Superconducting Devices Based on Quantum Wires K. Grove-​Rasmussen, T.S. Jespersen, A. Jellinggaard, and J. Nygård

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16.1 Introduction

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16.2 Experimental aspects of hybrid devices

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16.3 Superconducting junctions with normal quantum dots

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16.4 Superconductivity-​enhanced spectroscopy of quantum dots

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16.5 Sub-​gap states in hybrid quantum dots

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16.6 Non-​local signals in hybrid double quantum dots

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16.7 Epitaxial superconducting contacts to nanowires

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16.8 Summary and outlook

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17 Superconducting Nanodevices J. Gallop and L. Hao

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17.1 The drive to the nanoscale

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17.2 Types of Josephson junction

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Contents ix 17.3 NanoSQUIDs imply improved energy sensitivity

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17.4 Applications

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17.5 Future developments

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17.6 Summary and outlook

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18 Superconducting Quantum Bits of Information—​Coherence and Design Improvements J. Bylander 18.1 Introduction: superconducting qubits

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18.2 Single-​qubit Hamiltonians and reference frames

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18.3 Decoherence. Characterization and mitigation of noise

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18.4 Superconducting qubits

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18.5 Circuit quantum electrodynamics (c-​QED)

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18.6 Second-​generation superconducting qubits

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18.7 Summary and outlook

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19 NanoSQUIDs Applied to the Investigation of Small Magnetic Systems M. J. Martínez-​Pérez, R. Kleiner, and D. Koelle

Index

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19.1 SQUID basics

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19.2 NanoSQUIDs

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19.3 Measurement techniques using nanoSQUIDs

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19.4 Particle positioning

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19.5 Applications

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19.6 Summary and outlook

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Plan of the Book and Acknowledgments

Plan of the book This Oxford Handbook is about the remarkable materials that are now commonly referred to as mesoscopic superconductors. The word “meso” has a Greek root, and refers to the “middle”; in this case, the meso level operates between the bulk and atomic levels. In fact, mesoscopic superconductors are, for all practical purposes, tiny: often fabricated at dimensions that range from about a nanometer to about a few micrometers. When the size of the superconductor is reduced to this level, its properties are dramatically changed. For simplicity, we have called them small superconductors. The subject matter would be of interest to graduates and final-​year undergraduates reading physics, materials science, and engineering with a focus on nanoscience and nanotechnology. It is also likely to be directly relevant to specialists and trained electronic engineers interested in the possible new generation of ultra-​small high-​sensitivity superconducting probes. The theoretical and experimental research studies in mesoscopic superconductivity have been groundbreaking. Further, in a world obsessed with miniaturization of electronic device technology, mesoscopic superconductors are acquiring even greater relevance and timeliness. But here, miniaturization is not meant to be just a choice of fashion; it is, in fact, an inescapable commitment, if one is to exploit the quantum effects that manifest themselves only at the nano levels. This Handbook is meant to provide a lens into what might emerge as a giga world of nano superconductors. Chapters contributed by a host of eminent frontrunners in the field investigate the novel and intriguing features and theoretical underpinnings of the phenomenon of mesoscopic superconductivity, offer accounts of the latest fabrication methods and characterization tools, and discuss the opportunities and challenges associated with technological advances. In its totality, the book addresses the current status and great promise of small superconductors in the theoretical, experimental, and technological spheres. This Handbook comprises 19 chapters describing cutting-​edge developments in research and applications of small superconductors. The chapters are grouped and presented in three parts. The distribution is kept near-​even with 7 chapters in Part I, 6 in Part II, and 6 in Part III. The first part carries an extended introduction and relates to the developments in basic research of small superconductors. Part II is materials-​specific, while Part III reviews the current progress in their device technology. However, perhaps it is worth pointing out that the progress

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xii  Plan of the Book and Acknowledgments described in most of the topics presented in the three parts is of recent origin. Consequently, the boundaries separating some of them are as yet not sufficiently sharp, and therefore the pertinent chapters could arguably have been grouped differently in the three categories from how we have presented them below.

PART I Introduction and Basic Studies Part I starts with an introductory chapter (Narlikar) whose prime motivation is to make the interested reader, new to this important area, familiar with some of the basic definitions, prominent characteristics, and important effects manifested by small superconductors. We have therefore briefly introduced some of the mainstay topics of the field which include various size effects, surface effects, electron-​ mean-​free-​path effects, different types of phase slips, unusual vortex states, a variety of proximity effects, and some of the experimental techniques commonly used in the synthesis of small superconductors. This should allow the reader to understand the specialized and more comprehensive chapters covering various novel facets of this evolving field which this book is all about. The next two chapters discuss the intriguing vortex matter of small superconductors with numerous vortex states that do not exist in bulk superconductors. Here (Chapter  2), Roditchev and his collaborators have studied, using STM/​STS techniques, the organization of vortex cores at different levels of confinement. They show that the sample size and shape govern the vortex distribution and pinning, leading to ultra-​dense configurations unachievable in bulk superconductors. These authors further present the peculiar features of vortices in atomically thin superconductors which exhibit mixed Abrikosov–​Josephson vortices. In Chapter 3, Kokubo, Okayasu, and Kadowaki present a state-​of-​the-​art overview of their recent work on the multi-​vortex state carried out on mesoscopic superconducting dots of different geometrical shapes using a scanning SQUID microscope. They discuss the formation of multiple shell structures, their sequential filling, and the commensurability effect, which hold immediate relevance to future developments of magnetic flux-​based superconducting mesoscopic instrumentation and quantum computers. Cuevas et  al. in Chapter  4 describe the proximity effect on small length and energy scales in novel low-​dimensional systems, studied with the combination of in situ fabricated superconducting nanostructures and STM/​STS techniques. Gonnelli and his collaborators, in Chapter  5, have emphasized the potential of the point contact Andreev reflection spectroscopy (PCARS) technique for measuring the symmetry of the energy gap and other key parameters of various 0-​, 1-​, and 2-​dimensional systems. Topological superconductors and Majorana fermions form the exciting theme of Chapter 6 by Li and Jia. The topological superconductor has led to new insights about mesoscopic superconductivity and has revealed that some superconductors can support Majorana fermions

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Plan of the Book and Acknowledgments xiii and non-​Abelian statistics. The closing chapter of Part I by Gariglio and his co-​ authors focuses on surface and interface superconductivity, another pivotal area of mesoscopic superconductivity. The current experimental status and theoretical understanding of the field have been reviewed and the perspective on unconventional phenomena occurring at the surfaces and interfaces is discussed.

PART II Materials Aspects The contents of Part II focus on mesoscopic materials, their synthesis, and their properties, although as mentioned earlier some of the chapters included in this part could just as well have been in the previous one. It starts with Chapter  8 where Karapetrov and his collaborators present the mesoscopic effects in superconducting (SC)/​ferromagnetic (FM) hybrids, an area which has matured during the last 10  years. The authors have discussed interesting situations that lead to spontaneous formation of flux vortices pinned to the magnetic domain structure. Formation of vortex–​antivortex molecules, vortex chains, disordered vortex matter, etc., which until recently were only theoretically predicted, have now been experimentally corroborated through local scanning probe microscopy (SPM) techniques. In Chapter  9, the focus is on HTS cuprates. Asai has theoretically considered the intrinsic Josephson junction (IJJ) model to discuss the THz emission from these materials in the mesoscopic state. The intense emission is ascribed to the strong excitation of transverse Josephson plasma waves in IJJs under a DC bias. The chapter further discusses the recent theoretical and experimental studies aimed towards realizing practical THz sources based on mesoscopic HTS cuprates. In the next chapter, André Müller and his coauthors present their work on micromagnetic measurements on electrochemically grown mesoscopic superconductors, namely lead and tin. The technique allows fabricating samples in a variety of shapes and sizes in the mesoscopic regime. Its particularly remarkable feature is that one can use the technique to grow composite core–​shell structures of the two superconductors and subsequently study their mutual proximity interaction. Using the micro-​Hall probes and by looking at only one sample at a time, the authors have been able to observe individual flux lines entering and leaving the sample and discuss their size-​dependent behavior near Tc. Growth, structure, and properties of nanowires of high-​temperature cuprate superconductors form the mainstay of Chapter 11 by Koblischka. He has described two methods, namely the templating technique and electrospinning method, for preparing nanowires and nanobelts of HTS cuprates such as YBCO, NdCO, LSCO, and Bi2212. The methods yield continuous long lengths, albeit of somewhat larger thickness of 100–​250  nm, suitable for applications. In Chapter  12, Han Zhang has related mesoscopic structural features of the HTS crystal structures to understand their characteristic high Tc and anisotropy by developing model calculations. As the

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xiv  Plan of the Book and Acknowledgments closing chapter of Part II, Ortiz and his collaborators, in Chapter 13, look into the practical problem of thermally driven high-​speed flux avalanches occurring in superconducting thin films. The thin films are synthesized with artificial pins in the form of sub-​micrometric antidots, which though enhancing the critical current give rise to large-​scale flux avalanches, adversely affecting the superconducting device performance. The authors have discussed the vortex dynamics and avalanche morphologies in relation to antidot geometries and the current crowding effects. These considerations provide them with some useful insights to meet and overcome the above challenge.

PART III Device Technology In this last part of the Handbook we have six contributions describing the progress in the technological development of small superconductors. As stated at the beginning, in technology, the prime motivation in making superconductors small in size has been to meet the challenge posed by the ever-​increasing demand of device miniaturization and to improve their capabilities in terms of resolution and sensitivity. By using a combination of a superconductor and a ferromagnet one can have Cooper pairs in the triplet state in which the electron spins are aligned like a ferromagnet. The superconducting spin currents generated in such a situation are sensitive to the magnetization direction of a ferromagnet. These two features together constitute the basis of the exciting new field of superconducting spintronics, covered by Blamire and Robinson in Chapter 14. This way, as the authors discuss, within a single circuit, in principle, one can integrate the data storage capabilities of magnetics with the low-​energy dissipation of superconducting electronics. The chapter reviews the current status of this new field, drawing attention to the critical issues and developments needed for its application to low-​power quantum computing. In Chapter 15, Weides focuses on Josephson junction (JJ) barriers where the mesoscopic superconductivity resides and which are the most intriguing components of JJ-​devices. The author has critically discussed a host of different barriers made from insulators, metals, semiconductors, magnets, and nanowires and new developments in device technology are identified. In Chapter 16, Nygård and his co-​authors present an overview of their studies of hybrid superconducting devices based on quantum wires, in the form of semiconductor nanowires or carbon nanotubes, which are coupled to superconducting electrodes. These coupled structures serve as highly tunable mesoscopic systems whose characteristics are sensitively dependent on the strength of coupling between the two. Weak coupling results in quantum dots while strong coupling gives rise to quantized supercurrents in a one-​dimensional nanowire. The authors discuss a host of exciting possibilities such as topological superconductivity,

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Plan of the Book and Acknowledgments xv superconductivity-​enhanced quantum dot spectroscopy, and non-​local signals in Cooper pair splitter devices that emerge primarily due to changes in the coupling strength between the superconductor and quantum wire. In the next chapter Gallop and Hao review the recent progress made in superconducting nanodevices by presenting details of the fabrication methods developed for superconducting nanowires and nanoscale JJs based on different barrier materials that also include the recent microbridges and weak link methods. Future potential of these nanodevices is assessed in the light of improvements in nanoscale fabrication and manipulation techniques and their likely impacts on future quantum technology and measurement are evaluated. In Chapter 18, Bylander presents the very important topic of superconducting quantumbits. During the last few years there has been extraordinary progress in this area, although the construction of a universal quantum computer still remains a big challenge. In this chapter, Bylander starts with the basics of modern superconducting qubit devices and their architectures, and reviews major improvements in circuit designs, materials, and experimental tools. This way we learn about the experimental state of the art, including the latest research directions pursued in this highly competitive area of small superconductors. Finally, this brings us to the last chapter of Part III, Chapter 19, by María José Martínez-​ Pérez et al., where the authors describe the nanoSQUID for investigating small magnetic systems. The authors discuss how the already very high sensitivity of conventional SQUIDs gets further enhanced by reducing the size of the SQUID loop down to the nanoscale. At the same time they draw attention to the practical constraints and challenges encountered in using the nanoSQUID technology to study small spin systems ranging from magnetic nanoparticles to molecule magnets. To achieve a proper alignment of the magnetic nanoentity with the nanoSQUID as the measuring device remains a non-​trivial problem with the new technology. The contents of the Handbook, as mentioned at the beginning, should interest advanced-​level science and engineering students and further reach out to the nanotechnology experts from electronic industries, interested to know the current status of the theory, manufacture, and future of mesoscopic superconductors. Various reviews and overviews might also answer the queries and curiosities of non-​specialists interested in nanoscale superconductivity. In doing so, the present volume will offer to all types of the mentioned readership the opportunity to engage with the cutting edge of research in arguably the most exciting nanolevel discipline of Physics, Materials Science, and Engineering of today and tomorrow.

ACKNOWLEDGMENTS At the outset, I remain grateful to all the contributors of this Handbook, from more than 15 countries, for their enthusiastic participation and sustained

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xvi  Plan of the Book and Acknowledgments cooperation all along without which this project could not have been feasible. I am thankful to the Indian National Science Academy, New Delhi and its Science Promotion program for supporting my superconductor research and the preparation of the present book. Thanks are extended to the Director of UDCSR, for making available various facilities and resources of the Consortium. I  am especially grateful to my colleague Dr.  U.P. Deshpande for his help and suggestions. Helpful assistance provided by Arjun Sanap is acknowledged. I  am thankful also to my former colleagues at NPL, New Delhi, in particular Drs. Hari Kishan, Anurag Gupta, and V.P.S. Awana, for many useful discussions and for drawing my attention to some of the recent publications. A substantial part of the work of preparing the present book was carried out during my various extended visits to Hamburg, Germany where I  am grateful to Professor Dr.  Kurt Scharnberg of the Theoretical Physics Group of the University of Hamburg for many stimulating discussions and very useful suggestions in preparation of this book. I am further thankful to him and to Professor Dr.  Roland Wiesendanger of the Nanoscience Center and the Physics Department of the University of Hamburg, for their courtesy in providing the office space and the basic infrastructure facilities. Several useful suggestions and enthusiastic support that came from Ms. Julia Kramer and Mr. Olaf Kruithoff are heartily acknowledged. I remain thankful to Professor Dr. B.A. Glowacki, Head of the Applied Superconductivity and Cryoscience Group of the Department of Materials Science and Metallurgy at Cambridge University, for many years of interaction and his vast experience on nanostructured superconductors which was invaluable in the present project. Very special thanks are due to my wife Dr.  Aruna Narlikar for her invaluable help, patience, and constant support. Her valued suggestions and unmatched cooperation throughout during the preparation of the book were most admirable. It is worth mentioning that the original suggestion of the present handbook devoted to nanoscale superconductors came from our daughter, Professor Dr. Amrita Narlikar, who had an active role also in my drafting the OUP proposal for the present book. I am thankful to her for this and for her sustained enthusiasm in helping me in numerous ways, including in getting me the references and publications that I often needed rather urgently. At OUP, Oxford, the superb cooperation and the efficient help provided by Ms. Ania Wronski were instrumental in overcoming the unexpected hurdles and in meeting the time targets. Along with her I would like to record my personal thanks to Dr. Sonke Adlung at OUP for his experienced advice and timely suggestions at all stages of the project. It has been a pleasure working with him, as always. June 2016

A.V. Narlikar

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List of Contributors

H. Asai, Nanoelectronics Research Institute, National Institute of Advanced Industrial Science & Technology (AIST), Umezono 1-​1-​1, Tsukuba, Ibaraki 305-​8568, Japan. hd-​[email protected]

J.C. Cuevas, Department of Theoretical Condensed Matter Physics and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid (UAM), E-28049 Madrid, Spain. [email protected]

M.G. Blamire, Device Materials Group, Department of Materials Science, University of Cambridge, 27 Charles Babbage Road, Cambridge, CB3 0FS, UK. [email protected]

D. Daghero, Department of Applied Science and Technology, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy. [email protected]

C. Brun, Institut des NanoSciences de Paris, INSP-​UMR7588, CNRS and UPMC Sorbonne Universités, 4 place Jussieu, 75252 Paris, France. [email protected] J. Bylander, Chalmers University of Technology, Department of Microtechnology and Nanoscience, Kemivägen 9, SE-​41296 Gothenburg, Sweden. [email protected] A.D. Caviglia, Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands. [email protected] T. Cren, Institut des NanoSciences de Paris, INSP-​UMR7588, CNRS and UPMC Sorbonne Universités, 4 place Jussieu, 75252 Paris, France. [email protected]

S.E.C. Dale, Department of Physics, University of Bath, Bath, BA2 7AY, UK. S.Dale@ Bath.ac.uk M.A. Engbarth, Department of Physics, University of Bath, Bath, BA2 7A, UK. [email protected] J. Gallop, National Physical Laboratory, Hampton Road, Teddington TW11 0LW, UK. [email protected] S. Gariglio, Department of Quantum Matter Physics, University of Geneva, 24 Quai E.-Ansermet, CH-1211 Geneva, Switzerland R.S. Gonnelli, Department of Applied Science and Technology, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy. renato. [email protected]

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xviii  List of Contributors S. Goswami, Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands. [email protected] K. Grove-​Rasmussen, Niels Bohr Institute, Center for Quantum Devices and Nano-​ Science Center, University of Copenhagen, DK-​2100 Copenhagen, Denmark. k_​grove@ fys.ku.dk L. Hao, National Physical Laboratory, Hampton Road, Teddington TW11 0LW, UK. [email protected] M. Iavarone, Department of Physics, Temple University, Philadelphia, PA 19122, USA. [email protected] A. Jellinggard, Niels Bohr Institute, Center for Quantum Devices and Nano-​ Science Center, University of Copenhagen, DK-​2100 Copenhagen, Denmark. anders@ jellinggaard.dk T.S. Jespersen, Niels Bohr Institute, Center for Quantum Devices and Nano-​ Science Center, University of Copenhagen, DK-​2100 Copenhagen, Denmark. tsand@nbi. ku.dk J.F. Jia, Department of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dongchuan Rd, Minhang District, Shanghai 200240, China. [email protected]

M.J. Martínez-​Pérez, Physikalisches Institut—​ Experimentalphysik II and Center for Collective Quantum Phenomena in LISA+, Universität Tübingen, Auf der Morgenstelle 14, D-​72076 Tübingen, Germany. mariajose.martinez@ uni-​tuebingen.de K. Kadowaki, Institute of Materials Science, University of Tsukuba, 1-​1-​1, Ten-​nodai, Tsukuba, Ibaraki 305-​8573, Japan. kadowaki@ims. tsukuba.ac.jp G. Karapetrov, Department of Physics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104-​2875, USA. [email protected] R. Kleiner, Physikalisches Institut—​ Experimentalphysik II and Center for Collective Quantum Phenomena in LISA+, Universität Tübingen, Auf der Morgenstelle 14, D-​72076 Tübingen, Germany. kleiner@uni-​tuebingen.de M.R. Koblischka, Institute of Experimental Physics, Saarland University, Bldg. C6.3-​3rd Floor, D-​66041 Saarbrucken, Germany. [email protected]​saarland.de D. Koelle, Physikalisches Institut—​ Experimentalphysik II and Center for Collective Quantum Phenomena in LISA+, Universität Tübingen, Auf der Morgenstelle 14, D-​72076 Tübingen, Germany. koelle@uni-​tuebingen.de

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List of Contributors xix N. Kokubo, Department of Engineering Science, Graduate School of Informatics Engineering, The University of Electro-​Communications, 1-​5-​1, Chofugaoka, Chofu, Tokyo 182-​8585, Japan. kokubo@ uec.ac.jp Y.Y. Li, Department of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dongchuan Rd, Minhang District, Shanghai 200240, China. [email protected] M.V. Milošević, Department of Physics, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerpen, Belgium. [email protected] A.M.R.V.L. Monteiro, Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft,The Netherlands. [email protected] S.A. Moore, Department of Physics, Temple University, Philadelphia, PA 19122, USA. [email protected] M. Motta, Departmento de Física, Universidade Federal de São Carlos, Rod. Washington Luis, km. 235, 13565-​905 São Carlos, SP, Brazil. [email protected] A. Müller, Department of Physics, University of Bath, BA2 7AY Bath. a.mueller@ bath.edu

A.V. Narlikar, UDCSR, University Campus, Khandwa Road, Indore 452001, India. [email protected] J. Nygård, Niels Bohr Institute, Center for Quantum Devices and Nano-​ Science Center, University of Copenhagen, DK-​2100 Copenhagen, Denmark. nygard@ nbi.ku.dk S. Okayasu, Advanced Science Research Center, Japan Atomic Energy Agency, 2-​4 Shirakata Shirane, Tokai-​mura, Naka-​gun, Ibaraki 319-​1195, Japan. [email protected] W.A. Ortiz, Departmento de Física, Universidade Federal de São Carlos, Rod. Washington Luis, km. 235, 13565-​905 São Carlos, SP, Brazil. [email protected], wilsonortiz@ uol.com.br J.W.A. Robinson, Device Materials Group, Department of Materials Science, University of Cambridge, 27 Charles Babbage Road, Cambridge, CB3 0FS, UK. [email protected] D. Roditchev, ESPCI-​ParisTech PSL, Research Director at CNRS, LPEM-​ESPCI-​ ParisTech, INSP-​UMR7588, CNRS and UPMC Sorbonne Universités, 10 rue Vauquelin, 75005 Paris, France. dimitri. [email protected]

xx

xx  List of Contributors M.S. Scheurer, Institute for Theory of Condensed Matter & Institute for Solid State Physics, Karlsruhe Institute of Technology, Wolfgang-​Gaede-​Str. 1, 76128 Karlsruhe, Germany. [email protected] J. Schmalian, Institute for Theory of Condensed Matter and Institute for Solid State Physics, Karlsruhe Institute of Technology, Wolfgang-​Gaede-​Str. 1, 76128 Karlsruhe, Germany. [email protected] A.V. Silhanek, Département de Physique, Université de Liège, Quartier Agora, Allée du 6 Août, 19, Bât.B5a, B-​4000 Sart Tilman, Belgium. asilhanek@ ulg.ac.be

G.A. Ummarino, Department of Applied Science and Technology, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy. giovanni. [email protected] M.P. Weides, Institute of Physics, Karlsruhe Institute of Technology, WolfgangGaede-Str. 1, 76128 Karlsruhe, Germany and Materials Science in Mainz, Johannes GutenbergUniversitaet Mainz, Staudinger Weg 9, 55128 Mainz, Germany [email protected] H. Zhang, State Key Laboratory for Mesoscopic Physics, Department of Physics, Peking University, Beijing 100871, China. [email protected]

  1

Part I Introduction and Basic Studies

2

  3

Small Superconductors—​ Introduction A.V. Narlikar UGC-​DAE Consortium for Scientific Research, University Campus, Khandwa Road, Indore-​452001, India

1 1.1 Two characteristic length scales of superconductors 

1.3 QSE, quantum fluctuations, Anderson limit, parity and shell effects, etc. 

By small superconductors we mean superconducting objects whose one, two, or all three dimensions are shorter than some small characteristic length. As depicted in Fig. 1.1, this respectively makes a (a)  three-​dimensional (3D) bulk material (b) quasi-​two dimensional (2D), (c) quasi-​one dimensional (1D), or (d) quasi-​zero dimensional (0D) in the form of a thin film, a narrow wire, and a fine particle. Although studies of small superconductors began in the 1960s, the field received a significant impetus only 15–​20  years ago when technological progress made various new experimental tools of fabrication and characterization of nanosized samples more abundantly available. Studies of small superconductors are important for many reasons. One of the prime motivations is to assess if there is any miniaturization limit for superconducting nanodevices. How do various superconducting parameters such as the critical temperature Tc, critical magnetic field Hc, critical current density Jc, etc. respond to sample size reduction and is there any limiting size just below which superconductivity completely disappears or is destabilized? Further, above all, there exists a perpetual curiosity to know if some entirely new properties or phenomena emerge when the sample size is significantly reduced down to the nanoscale. Finally, in a world obsessed with miniaturization of technology, mesoscopic superconductors are acquiring even greater relevance and timeliness for revolutionary innovations in novel devices and applications.

3

1.2 Two size effects in superconductors  8

9

1.4 Factors influencing small size effects  12 1.5 Behavior of nanowires and ultra-​thin films 

16

1.6 Vortex states of small superconductors  21 1.7 Proximity effect behaviors 

25

1.8 Synthesis of small superconductors  28 1.9 Summary and outlook 

33

References (Chapter-1) 

34

1.1 Two characteristic length scales of superconductors A superconductor possesses two fundamental length scales, namely the magnetic penetration depth λ and coherence length ξ. The former, according to F. and H.  London (1935), arises when a superconductor is subjected to an external

A.V. Narlikar, ‘Small Superconductors: Introduction’ in The Oxford Handbook of Small Superconductors. First Edition. Edited by A.V. Narlikar. © Oxford University Press 2017. Published in 2017 by Oxford University Press. DOI 10.1093/acprof:oso/9780198738169.003.0001

4

4  Small Superconductors—Introduction (a)

L

BULK

(b)

THIN FILM

L

L

Fig.  1.1 (a)  Bulk (3D), (b)  thin film (2D), (c)  fine (nano) wire (1D), (d)  fine particle/​nanodot (0D).

L (c)

NANO WIRE

2 σ=ξ

ξ

3 V=ξ (d)

NANO DOT

L MACROSCOPIC LENGTH ξ COHERENCE LENGTH

magnetic field, smaller than the critical field Hc. Due to one of its fundamental properties, namely, the Meissner effect, when a bulk material turns superconducting it expels the magnetic field from its interior. This, however, leaves a thin surface layer, called the penetration depth (or the London penetration depth) λ over which the magnetic field has exponentially decayed and there exist supercurrents that flow to keep the external magnetic field excluded from the bulk of the sample. The second characteristic length ξ, after Pippard (1950) and Ginzburg and Landau (1950), represents the spatial range over which superconductivity gets affected by any local perturbation like thermal fluctuation/​heating and magnetic field. For type-​I, or positive surface energy superconductors, that mostly include superconducting elements, such as Al, In, Pb, Sn, Zn, etc., ξ > λ (Fig.  1.2a), while for type-​II, or negative surface energy superconductors, like alloys and compounds that include Chevrel phases, A-​15 superconductors, high-​temperature superconducting (HTS) cuprates, and Fe-​based superconductors (FBS), ξ < λ (Fig. 1.2b). In terms of microscopic theory, the coherence length measures the separation of the two charge carriers (both negative or positive), which are generally of opposite spins, forming a Cooper pair which makes the pairs function as extended entities. Table 1.1 lists ξ and λ values along with the critical temperature for some representative superconductors. The materials with one or more dimensions smaller than ξ or λ are considered as small superconductors where the two length scales lie between a few nanometers and several hundred nanometers. Small superconductors thus clearly belong to the nanoscale, but more broadly they are also referred to as mesoscopic. The Greek word meso means “in between” and the smallness lies in between the macro, or the bulk, obeying the classical laws of physics, and the micro, or molecular size, where quantum laws dominate. On practical considerations, the nanometer is taken as the lower limit of the length

  5

Two characteristic length scales of superconductors 5 (a)

ξ>λ S.C. ORDER PARAMETER

ξ λ MAGNETIC FIELD

SUPERCONDUCTING (b)

NORMAL

ξ ξ. (From Narlikar, A.V. (2014), Courtesy Oxford University Press, Oxford, UK.).

6

6  Small Superconductors—Introduction Table 1.1  Critical temperature, coherence length and penetration depth of different superconducting materials. (Data compiled from Narlikar (2014) and Narlikar and Ekbote (1983). Superconductor

Critical Temperature (K)

Coherence Length ξ(0Κ) (nm)

Penetration Depth λ(0Κ) (nm)

Al

1.2

1600

16

Cd

0.51

760

110

In

3.4

364

64

Nb

9.30

38

39

Pb

7.19

87

39

Sn

3.7

230

34

V

5.38

40

100

MgB2

39

6.0

185

Nb3Sn

18.3

3.1

100

NbN

16.1

4.5

200

PbMo6S8

13

2.5

240

BKBO

30

4.0

250

MgCNi3

7.0

4.6

225

NaxCoO2

4.5

5.7

790

KOs2O6

9.6

3.7

243

YBa2Cu3O7

91

0.3–​2.0

500–​150

SmFeAsO0.8F0.2

55

4.0

200

Rb3C60

29.5

2.0

440

k-​(ET)2Cu(NCS)2

9.0

3.0

510

field μ0Hc. This contention was shortly afterwards corroborated by Pontius (1937) who found the critical field of a lead wire to be markedly increased when its wire diameter was reduced. Bean et  al. (1962) forced soft type-​I superconducting metals like Hg, In, Pb, etc. into the interconnected 3–​5-​nm-​ sized pores of a Vycor glass which resulted in networks of very fine filaments (Fig. 1.3).  The critical fields of the filamentary structures were found to be around 5 T, i.e. more than 100 times their bulk values. Further, at 4 K, the filaments were found to sustain supercurrent densities of 108A/​m2, about 1000 times larger than the bulk values. A similar large increase of the critical field has more recently been reported for Pb and In nanoparticles (Li et al. 2003, 2005),

  7

Two characteristic length scales of superconductors 7 primarily resulting from their incomplete Meissner effect. In type-​II superconductors, the Meissner effect is observed only until the lower critical field Hc1 < Hc and here again the results of Blinov et al. (1993) and Fleischer et al. (1996) exhibit an order of magnitude increase in Hc1 of YBCO fine powders of diameter  100 K and a large gap coefficient in the range 4 to 12, the σc is predicted to be < unit cell length, and therefore the Anderson limit may, in fact, be unattainable (Ivanov et al. 2003). Interestingly, many of the mentioned effects in fact were long ago theoretically predicted (Blatt and Thompson 1963, Hwang et al. 2000, Jankó et al. 1994, Kresin and Ovchinnikov 2006, Wei and Chou 2002, Yu and Strongin 1976) and a good many of them have also since been experimentally corroborated (Bao et al. 2005, Bose et al. 2009, 2010, Czoschke et al. 2003, García-​García et al. 2008, Orr et  al. 1984, Ralph et  al. 1995, 1996a, 1996b, Romero-​Bermudez and García-​García 2014, Tinkham 2000, Upton et  al. 2004, von Delft and Ralph 2001). Table 1.3  Anderson limit for nanoparticles of some elemental superconductors. The values mentioned are from Bose and Ayuub (2014) and references therein, and Yang et al. (2011). Superconductor

Al

In

Nb

Pb

Sn

V

Anderson limit σc (nanoparticle diameter) (nm)

6.2

5.5

7.0

3.5

4.3

3

12

12  Small Superconductors—Introduction

1.4 Factors influencing small size effects The superconducting behavior in the regime of small size effects, i.e. in the 100–​20  nm range, is commonly linked to two main factors arising from size reduction. (a) The disruption in the periodicity at the sample surface causes the latter to possess vastly altered properties. At the nanolevel, the surface to volume ratio becomes significantly prominent to cause the surface properties to override the bulk properties (surface effects). (b) A decrease in the electron mean free path due to spatial confinement (electron mean free path effects) which, as we will see, has interesting consequences. Besides (a) and (b) there are fluctuations (thermal and quantum) in the form of phase slips that become important as the sample size diminishes. Their effect (Sec. 1.5) is primarily to broaden the superconducting transition and adversely affect superconductivity.

1.4.1 Surface effects With samples of reduced size, their surface properties grow more relevant than bulk properties and this influences many of their important parameters like the critical temperature, the critical magnetic field, and the critical current. The atoms located on the sample surface possess a lower number of bonds than those located in the bulk interior, which makes the surface phonon modes much softer than the bulk and this thereby lowers its average phonon frequency. The dimension-​less parameter λe, which measures the electron–​phonon interaction strength in conventional superconductors to determine their Tc, is related to the electron DOS N(0) at EF, ionic mass M, the electronic matrix element ⟨I2⟩, and mean square phonon frequency ⟨ω2⟩ through the relation,



λe =

N ( 0) I 2 M * ω 2 (1.1)

Decrease in ⟨ω2⟩ in the above situation favors an increase in λe and consequently also the critical temperature. In this way, fine pressed powders (quasi-​ 0D category) of elemental platinum, which is otherwise non-​superconducting, exhibit superconductivity at 0.02 K (Schindler et al. 2002). The softened surface phonon modes in the case of small superconductors make a positive contribution to their critical temperature. There is also a negative contribution to Tc coming from structural distortion occurring as a result of the sample size reduction. This causes a decrease in the electronic DOS N(0) at EF which suppresses Tc. As we have seen earlier, discretization of energy levels occurs in small particles due to quantum confinement effects that subsequently get broadened as the particle size is reduced. This again lowers the DOS which makes a negative contribution to Tc. The overall effect of these competing factors is found to be different in different materials. The variation of Tc with particle size results in three types of behavior

  13

Factors influencing small size effects 13 which are schematically shown in Fig. 1.6. A pronounced peak-​like behavior such as shown by curve II is manifested by weakly superconducting metal. For instance, in the case of In, which is weakly superconducting, the Tc goes through a sharp maximum at about 30 nm and thereafter, in the ultra-​small regime, it shows a fast decrease until the Anderson limit is reached, below which the phenomenon disappears (Li et al. 2005). For intermediate electron–​phonon coupling superconductors, the behavior is as shown by (III), i.e. the curve is initially flat, and then there is a decrease in Tc with particle size which becomes faster as the Anderson limit is approached. This kind of behavior is manifested by Nb which possesses intermediate electron–​phonon coupling and its superconductivity is lost when the particle size is smaller than around 7 nm (Bose et al. 2009). In general, the negative contributions in small superconductors dominate over the aforesaid positive contribution when the sample dimensions are lowered below around 20 nm and the overall effect is therefore to suppress Tc. The behavior represented by (I) in Fig. 1.6 is for strongly coupled superconductors such as Pb. From bulk down to below 10 nm there is no change in Tc which Bose et al. (2009) attribute to the almost exact compensation of the positive phonon softening effect by the negative quantum size effect until the Anderson limit is approached. Interestingly, the behavior (Fig. 1.7) of MgB2 (Li and Dou 2006) seems very similar to that of Pb and may also be explained similarly. Interestingly, in the case of weakly coupled superconductors like In, Al, and Sn, the gap coefficient 2Δ(0)/​kBTc increases on size reduction; it remains invariant for Nb whose gap coefficient is of an intermediate value between weak and strongly coupled superconductors; while for Pb which is strongly coupled, its gap coefficient exhibits an increase (Bose and Ayyub 2014) on size reduction. Surface effects are important also in influencing the critical magnetic field of small superconductors. Saint-​ James and de Gennes (1963) had showed that the solution of Ginzburg–​Landau equations for a type-​II superconductor subjected to a magnetic field was quite different at its surface than for the bulk. Even when the magnetic field had exceeded the upper critical field Hc2 to quench the bulk superconductivity, a thin surface layer of thickness ξ, parallel to the magnetic field, continued to remain superconducting up to a much

40

Tc (K)

30 MgB2

20 10 0

10

20

30

40

50

GRAIN SIZE (nm)

60

70

(III)

(I)

Tc (K) Arb. Units

(II) 0 20 40 60 80 100

BULK

PARTICLE DIAMETER (nm)

Fig. 1.6  Three types of behavior observed for the size effect of critical temperature of superconducting nanoparticles.

Fig.  1.7 Until about 12  nm grain size the critical temperature of MgB2 remains the same as of the bulk sample. Tc subsequently is fast suppressed on further lowering the grain size. No superconductivity is observed below 2.5  nm, i.e. the Anderson limit for MgB2 (based on the reported measurements of Li and Dou 2006).

14

14  Small Superconductors—Introduction higher magnetic field, known as the sheath critical field Hc3 ≅ 1.695Hc2. For thin discs the numerical factor exceeds even 2.5 (Schweigert and Peeters 1999). The surface phonon modes, mentioned earlier, are believed to be responsible for the origin of sheath superconductivity. For type-​II superconductors of low Ginzburg–​Landau parameter κ (see Narlikar 2014), the existence of a surface sheath promotes magnetic irreversibility and a higher critical current density jc. However, for large-​κ materials, which are used for winding high-​field superconducting magnets, different types of pinning entities in the form of small normal (non-​superconducting) particles and structural defects are essential for strong flux pinning and enhanced jc. Surfaces of small precipitates of a second phase, various structural defects, and of artificially introduced fine particles (Table 1.2) embedded in wires, tapes, or thin films of superconductors serve as effective pinning centers for Abrikosov’s flux vortices. Their presence vastly enhances the critical current density of the superconducting matrix in which they are embedded. In general, for optimum pinning, the number density of the pinning entities should be large, their volume fraction small, and the surface area of each entity or the pinning center should be optimum. Ideally, since the normal core of the flux vortex has a radius equal to the range of coherence ξ of the matrix, the size of the pinning centers should be in the nano range, for optimum pinning. This makes the pinning entities mesoscopic. Since the magnetization of the pinning entity, in general, differs from that of the superconducting matrix, there is always a circulating supercurrent around the pinning center at its interface with the matrix. This surface current interacts with the circulating supercurrent of the moving flux vortex and creates an irreversible surface barrier responsible for flux pinning (Bean and Livingston 1964, Campbell et al. 1968). This approach of introducing nanoparticles of different materials and metallurgical phases, such as Gd-​211 in HTS cuprates (Muralidhar et al. 2004), or nano-​diamond (Cheng et al. 2003, Vajpayee et al. 2007), nano-​SiC (Dou et al. 2002), nano-​C (Yeoh et  al. 2006), nano-​SiO2 (Rui et  al. 2004), carbon nanotubes (Yeoh et al. 2006a), etc. as nanosized artificial pinning centers (APCs), has been effectively used to enhance the jc of superconducting MgB2. In fact, this has emerged as a successful route to increase the current carrying capacity of several high-​field superconductors for various applications.

1.4.2 Electron mean free path effects When a metallic particle or a grain is reduced in size, a wire is made narrower, or a film is made thinner, all lead to a decrease in their electron mean free path. The electron mean free path stands out as an important factor in superconductors which in different ways influences several of their important properties and parameters. These prominently include the two characteristic lengths λ and ξ, the normal stat resistivity ρn, the Ginzburg–​Landau parameter κ, and the three critical fields Hc1, Hc2, and Hc3.

  15

Factors influencing small size effects 15 1.4.2.1  λ, ξ, and κ When the sample size is reduced, the electron mean free path lF decreases, which has mutually opposite effects on the coherence length ξ and the penetration depth λ. The former decreases while the latter increases, following the relations from the Ginzburg–​Landau theory (1950):

ξ(T ) = 0.60( ξ ol F )1/ 2 (1 − t )1/ 2 (1.2)



λ(T ) = 0.62λ o{ξ o 0 l F (1 − t )}1/ 2 (1.3)

where ξο is the coherence length for the pure material at 0 K.  In the case of thin superconducting discs (thickness t) the effective penetration depth, Pearl’s penetration depth (Pearl 1964), is given by Λ = 2λ2/​t. The Ginzburg–​Landau parameter κ, which varies as λ/​ξ, therefore increases as the sample thickness is reduced. Interestingly, this makes a type-​I superconductor like Pb behave like a type-​II when it is formed as a nanoparticle of around 15 nm diameter (Bose 2007). As a result very thin type-​I superconductors show an interesting vortex structure, normally seen only in type-​II, but, as we will see later, they also carry many other surprising features (Deo et al. 1997, Schweigert and Peeters 1998, 1999). 1.4.2.2 Critical fields Hc1 and Hc2 The lower and upper critical fields Hc1 and Hc2 are respectively related to λ and ξ,

µ o H c 2 (0) = ϕ o 2πξ 2 (1.4a)



µ o H c1 (0) = ϕ o 4πλ 2 (1.4b)

where φo (= h/​2e = 2×10−​15Wb) is a quantum of magnetic flux. Consequently, with size reduction the lower critical field would decrease while the upper critical field would increase. However, if the sample size is smaller than λ, due to an incomplete Meissner effect, as discussed in 1.1, the Hc1 would increase. Thus, in the case of Hc1 there is a competition occurring between the electron mean free path effect and the consequence of the incomplete Meissner effect, and the latter effect should dominate as the sample size is reduced. As mentioned earlier, the lower critical field does indeed go up with decreasing particle size (Blinov et al. 1993). There is a similar competition also with Hc2. Decrease in the electron mean free path enhances the normal state resistivity of the small superconductor, which is related to the upper critical field through the relation (see Narlikar 2014),

µ o H c 2 (0) = 3.09 × � γρnTc (T ) (1.5)

16

16  Small Superconductors—Introduction The coefficient of the electronic specific heat in the normal state, γ, and the critical temperature Tc do not normally change significantly by electron mean free path effects and consequently an increase of ρn as per the above equation would make a positive contribution to the upper critical field. However, there is a competing negative contribution that arises from discretization of energy levels due to the quantum confinement effect which, as discussed earlier, causes a decrease in the DOS at EF and thereby suppresses Tc. The latter effect would surely dominate when the sample size is reduced down to the ultra-​small regime and the upper critical field would finally vanish when the Anderson limit is reached. The reported observations and their explanation of findings by Bose et al. (2006) of such a non-​monotonic behavior of the upper critical field of nano-​sized grains of Nb are in accord with the above reasoning. Nanostructured thin films of some superconductors are noted for their exceptionally high upper critical fields in comparison with their bulk values, which makes them attractive candidates for high magnetic field applications. For instance, for Chevrel phase superconductors the upper critical field for nanostructured films is reported to increase from 50 T to 100 T while for MgB2 from 16 T to about 75 T (Narlikar 2014).

1.5 Behavior of nanowires and ultra-​thin films 1.5.1 Nanowires and fluctuation effects: thermally activated phase slips and quantum phase slips Fluctuation effects, in general, become relevant for low-​dimensional systems and all the more so in 1D superconducting nanowires. Quantum fluctuations were briefly mentioned in Sec. 1.3. In a strictly 1D system, superconducting long-​ range order and a zero-​resistance state are, in fact, forbidden by the Mermin–​ Wagner (1966) theorem. Langer and Ambegaokar (1967) and McCumber and Halperin (1970) in their LAMH theory had pointed out that the process responsible for transient breaking down of superconductivity just below Tc was the so-​ called phase slips, which originated from thermal fluctuations. In addition to these thermally activated phase slips (TAPS), the experimental evidence (Giordano 1994) has indicated that for ultra-​thin wires, at much lower temperatures < Tc, when thermal excitation is not relevant, quantum phase slips (QPS) can occur due to quantum fluctuations via quantum tunneling. Superconducting order is characterized by a wave function ψ = |ψ|exp.(iϕ) where ϕ is its spatially coherent phase that is, so to speak, locked everywhere in the superconducting state. Locally, near Tc, the coherence can, however, get disturbed by thermal fluctuations or the occurrence of phase slips which results in the local transient loss of superconductivity (t ≈10−​12 s) over a region of size ξ where the order parameter ψ locally fluctuates to zero and the phase ϕ becomes

  17

Behavior of nanowires and ultra-thin films  17 indeterminate. The minimum energy cost for this is a function of the condensation energy density ½μ0Hc2 per unit volume, and for a nanowire of cross-​sectional area A becoming normal over a segment ξ, the corresponding energy needed, i.e. the barrier height, is (Langer and Ambegaokar 1967) ΔF = K.μ0Hc2Aξ, where Hc is the thermodynamic critical field and K is a constant. Subsequently, as ψ builds back up to its original value, the phase undergoes a slip and rotates by ±2π from its value just prior to the phase slip event. This change of phase corresponds to a voltage pulse, with more slips giving rise to more resistive voltages. The process is represented in Fig. 1.8a by a spiral of constant radius where with each phase slip event the spiral loses one complete loop corresponding to a phase of 2π. As shown in Fig. 1.8b, the process (TAPS) can take place by thermal activation over the barrier of height ΔF, just below Tc or, if the wire is ultra-​thin, it (QPS) can

(a) BEFORE PHASE SLIP REAL Ψ SINGLE PHASE SLIP AFTER PHASE SLIP

IMAG. Ψ

DISTANCE ALONG THE WIRE (b)

∆φ

Thermal Activation Thermally Activated Phase Slip (TAPS) ∆φ+2π

Tunneling Quantum Phase Slip (QPS)

Fig. 1.8  Phase slips in 1D superconductors. (a) The two axes shown represent the real and imaginary parts of the order parameter ψ, and the third axis is along the nanowire length. The spiral of constant radius represents a constant supercurrent and prior to phase slip, the spiral remains intact (the top spiral). A phase slip event (10–​12s) causes the order parameter |ψ| to become locally zero and the spiral loses one loop (middle spiral). During the phase slip, the phase at some point in the nanowire becomes indeterminate. During the recovery it is changed by ±2π, resulting in a finite voltage with loss of superconductivity, and as shown in (b) the superconducting system passes from one state to another across a potential barrier.The energy to cross the barrier is provided by thermal fluctuations, leading to thermally activated phase slips (TAPS), or by quantum fluctuations via the tunneling process resulting in quantum phase slips (QPS).

18

18  Small Superconductors—Introduction NANOWIRES

TAPS

R (Arb Units) QPS 20 nm 40 nm

0

60 nm

80 nm >100 nm T(K)

Fig. 1.9  Typical Resistance–​Temperature behavior of superconducting nanowires, displaying thermally activated phase slip (TAPS) and quantum phase slip (QPS).

occur quite far below Tc, through the above barrier due to quantum fluctuations via tunneling. In both situations the system moves from one local potential minimum into the neighboring one separated by ±2π in the phase space. In 2D and 3D samples the supercurrent transport is, however, little affected by the creation of the normal (non-​superconducting) region that can be readily bypassed by moving Cooper pairs without interruption. But, in nanowires of radius smaller than ξ, the normal region formed blocks the entire cross-​section of the nanowire, and therefore it can no longer be bypassed by electron pairs. The wire therefore ceases to be superconducting. The experimental results show that the normal state resistance RN of the nanowire has to be smaller than the quantum resistance RQ  =  h/​(2e)2 ≈ 6.5 kOhm to follow the predictions of the LAMH theory. When RN > RQ, the nanowire fails to exhibit superconductivity (Bezryadin 2008). Because of the two characteristic phase slips, TAPS and QPS, mentioned earlier, the resistance (R)–​temperature (T) behavior of nanowires differs from that of the bulk wires (thickness/​diameter of around 100 nm or more) in two ways. Instead of the very sharp transition (width < 0.01 K) of bulk wires the TAPS results in the rounding and broadening of the transition point at Tc-​onset, while the QPS results in an elongated tail at T < Tc, which frequently does not lead to a R = 0 state even at 0 K. The behavior is schematically shown in Fig. 1.9. This happens when the wires are sufficiently small (≤ 30  nm) in diameter or thickness. The behavior has been studied in a number of elemental superconductors like Sn, Zn, Al, Pb, and In (Zgirski et  al. 2005, Sharifi et  al. 1993, Giordano 1994, Bezryadin et al. 2000, Lau et al. 2001, Tian et al. 2005, Wang et al. 2005). However, the transition from TAPS to QPS in superconducting nanowires still remains to be systematically studied. As with 0D superconductors, discussed earlier, 1D ultra-​thin nanowires do exhibit the quantization of electron motion and discretization of energy levels, but only along the directions transverse to the length. These have been experimentally corroborated by observations of shape resonances with large oscillations in Tc of ultra-​thin nanowires of Al, Sn, and Pb (Guo et al. 2004, Shanenko et al. 2006) and also theoretically explained (Shanenko and Croitoru 2006).

1.5.2 Ultra-​thin films The 2D superconducting system holds some similarity with the behavior of 1D nanowires described above. When the normal state resistance RN of a thin film exceeds the quantum resistance RQ = 6.5 kOhm, superconductivity is not observed (Haviland et al. 1989, Hebard and Paalnen 1990, Bollinger et al. 2011). Ultra-​thin superconducting films with thickness < 20 nm, if they are not of a continuous form, comprise small islands and exhibit granular behavior, characteristic of 0D superconductors discussed earlier. The films are formed on substrates which can influence their intrinsic behavior through the proximity effect. Similar to what we previously discussed for 0D superconductors, fluctuations too play

  19

Behavior of nanowires and ultra-thin films  19 16

Tc (K)

14 NbN ULTRA-THIN FILMS

12 10

Kang et al. (2011) Marsili et al. (2008)

8 6

0

6

12

18

100

FILM THICKNESS (nm)

a significant role in 2D systems. The Berezinskii–​Kosterlitz–​Thouless (BKT) transition originating from phase fluctuations adversely affects superconductivity. Interestingly, despite this, superconductivity does occur in ultra-​thin films of thicknesses much smaller than ξ and, in fact, even down to the level of a single monolayer of the superconducting elements (Zhang et al. 2010) and compounds (Lee et al. 2014, Xi et al. 2015), albeit with reduced Tc and broadened transition as compared to the bulk samples. Fig. 1.10 depicts the results of Tc depression in ultra-​thin films of NbN reported by Marsili et al. (2008) and Kang et al. (2011) when the film thickness varied from 100 nm to about 2.5 nm. Clearly, the two sets of observations are in close mutual agreement. Down to 100 nm the critical temperature remains practically invariant from the bulk value. From 100 to about 10  nm thickness the films display a small decrease of Tc, which, however, in the ultra-​thin regime below 10 nm depicts a much faster drop. As also discussed previously, a decrease in the film thickness can give rise to different competing factors influencing Tc. Firstly, due to the previously discussed small size effects (SSE), the structural disorder in the film increases with decreasing thickness, causing a reduction in the DOS at EF which lowers the Tc. On the other hand, as the film gets thinner, surface phonons become more dominant to increase the electron–​ phonon interaction which promotes Tc. As with 0D and 1D superconductors, in the ultra-​thin regime of 2D films, the electron wave function is quantized along the direction normal to the film surface, which results in discretization of energy levels as part of the quantum size effects (QSE). This contributes to a decrease in the DOS at EF and lowers Tc. The observed behavior of Tc in Fig. 1.10 is the overall impact of the above factors in which the QSE finally dominate and depress Tc through decrease in the DOS (Kang et al. 2011) at EF. Similar to 0D and 1D superconductors, discretization of energy levels in ultra-​thin films gives rise to large oscillations (Bao et al. 2005, Shanenko et al. 2007) in various superconducting parameters in the form of shape resonances as we discussed earlier.

Fig.  1.10 Critical temperature of epitaxial ultra-​ thin NbN films. The filled circles present the results of Kang et  al. (2011). The open circles represent the data points of Marsili et al. (2008). In both cases the films were deposited on MgO substrates by magnetron sputtering.

20

20  Small Superconductors—Introduction (a)

1.0 0.75 R(T)/R(300K) 0.5 0.25 0

YBCO ULTRATHIN FILMS

8 nm 2 nm 12 nm

5 nm 50

100 150 200 TEMPERATURE (K)

250

(b) MgB2

102

RESISTIVITY µΩ cm

101

Fig. 1.11  Resistance–​Temperature curves for ultra-​thin films of superconducting (a)  YBCO (after Gao et  al. 1999)  and (b) magnesium diboride (after Wang et al. 2009), both showing Tc decrease and broadening of the transition with decrease of film thickness.

7.5 nm

ULTRATHIN FILMS

10 nm 20 nm 40 nm

100

10–1 30

34

38

42

46

50

TEMPERATURE (K)

Fig.  1.11a presents the results of R–​T measurements of Gao et  al. (1999) on ultra-​thin films of YBCO while Fig. 1.11b depicts the results on MgB2 films (Wang et al. 2009). In both cases the transition gets rounded and broadened and Tc goes down. In the case of YBCO, the 2.5 nm film does not seem to indicate any possibility of reaching the zero-​resistance state even at 0 K. Interestingly, the MgB2 films of thickness ≤ 20 nm exhibit a localization effect by displaying the characteristic negative dR/​dT at low temperatures, possibly due to the presence of structural disorder occurring at the ultra-​small thickness. However, from the R–​T curves of Fig. 1.11a and Fig. 1.11b one cannot unambiguously distinguish TAPS and QPS regimes.

1.6 Vortex states of small superconductors We have already seen above how size effects influence Tc and other related properties of superconductors. In this section we take a quick look at the influence of

  21

Vortex states of small superconductors 21 confinement on the formation of magnetic field induced vortex states in them. Normally, the vortex structure is a characteristic feature only of type-​II superconductors. However, type-​I superconductors, as we have seen earlier, if they are made sufficiently small, start behaving akin to type-​II ones and thus may have flux vortices under an applied magnetic field. When the field exceeds the lower critical field Hc1 the macroscopic Meissner state gives way to a mixed state, comprising a triangular lattice-​like arrangement of Abrikosov’s vortex lines (Fig. 1.12a) threading the bulk of the superconductor (Abrikosov 1957). The triangular vortex lattice, as imaged by Nishio using scanning Hall microscopy, is depicted in Fig.  1.12b. Each vortex line has a normal cylindrical core of radius ξ carrying a single quantum of magnetic flux ϕo (= h/​2e  =  2  ×  10−15Wb), which is surrounded by vortices of supercurrents spread over a radius of λ (Fig. 1.12c). The smallest subdivision of magnetic flux is a consequence of the negative surface energy existing between the normal core and its superconducting surrounding, the perennial characteristic of type-​II superconductivity. A single rotational turn around the vortex changes the phase of the order parameter by 2π. In general, for a bulk superconductor, it is energetically impermissible for a flux line, whose self-​energy varies as ϕo2, to contain n ≥ 2 flux quanta. Such a vortex line, known as the giant vortex, will normally be unstable and split into n vortices, each with a

(a)

(c)

SUPERCONDUCTING

NORMAL

φ0

SUPER CURRENT VORTEX

NORMAL CORE

λ

2ξ

(b)

Ψ S

N MAGNETIC FIELD Ψ=0 20 nm

Fig.  1.12 (a)  Schematic of triangular vortex lattice, (b) vortex lattice in a superconductor imaged using scanning Hall microscopy (courtesy Nishio), (c)  structure of a vortex line (Narlikar 2014).

22

22  Small Superconductors—Introduction single flux quantum. However, a giant vortex state (GVS), as we will see, is one of the novel and stable vortex states of small superconductors. The triangular vortex lattice in bulk superconductors is the outcome of mutual repulsion between the macroscopic number of vortex lines. In small or confined superconductors of size 14, the inner shell grows, leading to the state (5,11) at L = 16. The third shell appears at L = 17, with the added vortex again occupying the shell center, and the state is represented by three digits, i.e. (1,5,11). The next three vortices, leading to L = 20, are all added to the outermost shell, i.e. (1,5,14) and thereafter, up to L = 32, all the shells grow intermittently to (5,11,16). The fourth shell is formed at L = 33, again in the form of a single vortex at the center of the existing three shells, and the vortex configuration is represented by four numbers, i.e. (1,5,11,16). The numbers 6, 17, and 33, at which a new shell is formed, are referred to as magic numbers. These magic numbers and the described pattern of shell formations and filling are both theoretically and experimentally in mutual agreement. It would be interesting to ask how the vortex configuration might change in small noncircular superconducting samples possessing a discrete symmetry, such as squares and equilateral triangles having pointed edges of the size ≈ 5ξ to 10ξ? In this situation the ensuing vortex configuration grows more complex since the vorticity has to adjust with the corresponding boundary shapes respectively of C4 and C3 symmetries (Chibotaru et al. 2001, 2005, Misko et al. 2003, Teniers et al. 2003, also see Chapter 3 of this book). L = 1 and L = 4 in a square-​shaped

24

24  Small Superconductors—Introduction Table 1.4  Filling up of magnetic flux vortices/​flux quantum at the center of mesoscopic superconducting triangles and squares with increasing vorticity. Vorticity L

0

1

2

3

4

5

6

7

8

9

10

11

Mag.flux (ϕo) at triangle center

0

+1

–​1 AV

0

+1

–​1 AV

0

+1

–​1 AV

0

+ 1

–​1 AV

Mag.flux (ϕo) at square center

0

+1

+2 GV

–​1 AV

0

+1

+2 GV

–​1 AV

0

+1

+2 GV

–​1 AV

GV: giant vortex state; AV: antivortex state. In the case of a triangle the repetitive pattern is 0:+1:–​1:0:+1: … …, while for a square: 0:+1:+2:–​1:0:+1: … ….

sample present no problem as one and four vortices, each with a single flux quantum, can respectively be accommodated at the square center and at its four corners or diagonal points without disturbing the fourfold symmetry. Similarly L = 2 can be readily realized by replacing the single quantum vortex of L = 1 by a giant vortex with L = 2 at the square center. The vorticity can be increased to L = 5 by adding a single quantum vortex at the center of the four corner points of the L = 4 configuration and further raised to L = 6 by replacing the central vortex of L = 5 by a giant vortex with L = 2. Both these configurations represent two concentric shells. Vorticities of L = 3 and L = 7 are, however, not consistent with the square symmetry of the sample. Here an additional vortex–​antivortex pair is formed to geometrically adjust the vortex distribution to meet the fourfold square symmetry of the sample; here antivortex represents an oppositely directed vortex with a flux quantum –​ϕo. Thus, for L = 3, we have four vortices at the square corners and one antivortex at its center (i.e. 3 vortices + 1 vortex–​ antivortex pair). The vortex–​antivortex pair is also called a vortex–​antivortex molecule. Similarly, the configuration for L = 7 is formed with four vortices on each of its two diagonals with an antivortex at the square center (i.e. 7 vortices + 1 vortex–​antivortex pair/​molecule) constituting a three-​shell arrangement (4ϕo.+ 4ϕo.–​1ϕo.). A similar situation exists for a triangle-​shaped sample, for example, for L = 2 which is not compatible with the triangular shape. This can be realized by three ϕo. vortices, one each at the three apex points, and a single –​ϕo. (antivortex) at the center of the triangle. Interestingly the filling up of the triangle and square centers with magnetic flux quanta, as the vorticity rises, follows distinct repetitive patterns (Table 1.4). The former is 0:+1:–​1:0:+1:–​1:0:+1:  … while for a square it is 0:+1:+2:–​1:0:+1:+2:–​1:0  … This holds relevance for superconducting technology applications, which can use the triangular and square nanodots as components of supercomputers in quantum computing. The antivortex formation in square and triangle-​shaped samples, described above, is however symmetry induced which to date remains to be experimentally corroborated. One of the reasons for this is that vortex–​antivortex pair formation is sensitive to the presence of defects in the sample and moreover their mutual separation is too small, i.e.  0. The order parameter obeys the equation:   2  αψ (r ) + β ψ (r ) ψ (r ) = 0 (2.2)



that minimizes the free energy F of the system. Eq. 2.2 has a solution S α  2  ψ (r ) = − = ∆ 2 , ϕ (r ) = const , meaning that in the absence of currents and fields β the order parameter is homogeneous inside a superconductor.

 When a bulk superconductor is put in a uniform magnetic field µ 0 H 0,1 the Cooper pairs start moving, supporting a dissipation-​less current flow. In a low field these Meissner supercurrents screen the external magnetic field which cannot penetrate deep inside the superconductor, but rapidly decays instead. For the macroscopic superconducting cylinder represented in Fig.  2.2a,    R −r , where the origin of the cylindrical coordinate Binside (r ) = B (r = R ) exp  −  λ L  r is taken at the cylinder axis, λ L is the London penetration depth, λ L =

m* 4µ 0nS e 2

(typically 50–​1000nm), R (D) is the cylinder radius (diameter), m * and e are  2 respectively the electron effective mass and charge, and nS = ψ (r ) —​the superfluid density (London 1935). The Meissner currents are responsible for the ideal diamagnetism of bulk superconductors of size D  λ L (Meissner and Ochsenfeld 1933). Indeed, in a low applied field µ 0 H 0 < BC1,2 near the edges where ( R − r ) ~ λ L ,  H 0 is the externally applied field that would exist without the superconductor. 2 B is commonly called the first critical field. C1 1µ

0

  43

(a)

(b)

(c) D >> λ

D = 2R >> λ

D >> λ

µ0H0 r ~λ

(d)

µ0H0 < BC1 D B ∼ C1 D ∆ ,

(2.16)

for |E| ≤ ∆

10 When the STM tip is made of a normal metal, the resolution is limited by the width of the Fermi–​Dirac distribution, ≈ 3.5kBT . However, if the STM tip is made of a superconductor, the spectral resolution may be yet improved owing to singular quasiparticle peaks in the excitation spectrum of a superconductor. The disadvantage of superconducting STM tips is that one needs to process the tunneling data to de-​convolve the a priori unknown density of states of the studied material from that of the superconducting tip.

  51

Experimental requirements for studying vortex confinement phenomena 51 where N N ( E F ) = const is the DOS of the material in the normal state.11 The spectrum has a gap within the spectral window eV < ∆ followed by two singularities at eV = ± ∆ (see for example the spectrum B = 0 mT in Fig. 2.5i). In the tunneling conductance curves the excitation spectrum is smeared by ≈ 3.5 kBT due to the ∂f ( E − eV ) convolution with − in the integral (Eq. 2.15). The local quasiparticle ∂E excitation spectrum is very sensitive to spatial variations of the superconducting order. For instance, in the vortex cores the gap vanishes, and the excitation spectrum has a finite value at zero energy. Hence, by mapping the zero-​bias tunneling  dI (V = 0, r ) conductance of a superconductor subject to a magnetic field, one dV reveals the vortex cores in the superconducting medium (see Fig. 2.1a, Fig. 2.3a). Since the pioneering works by Hess and collaborators (Hess et al. 1989), scanning tunneling spectroscopy has been widely applied to explore the vortex phase diagram in bulk superconductors. The vortices were visualized and their arrangements and cores studied in several conventional (Renner et al. 1991, Nishimori et al. 2004, Guillamon et  al. 2008)  and high-​TC superconducting materials (Maggio-​ Aprile et al. 1995, Pan et al. 2000, see Hoogenboom et al. 2002 for a review, Shan et al. 2011). Different regimes of vortex organization and motion were revealed in bulk superconductors near critical temperature (Troyanovski et al. 1999), close to critical field (Guillamón et al. 2009) or in the presence of a disorder potential (van Baarle et al. 2003, Guillamón et al. 2014); several works focused on vortex pinning phenomena at grain boundaries (Maggio-​Aprile et al. 1997, Song et al. 2012). In general, STM/​STS studies have been strongly contributing to a deeper insight into the microscopic behavior of the vortex matter in bulk superconductors.

2.2 Experimental requirements for studying vortex confinement phenomena 2.2.1 Size effects: basic considerations The first deviation from a “bulky” behavior is expected when the sample size becomes comparable to or smaller than the magnetic penetration depth, D ≤ λ (or D ≤ Λ in thin samples with thickness t  λ ). As we already saw in Sec. 2.1, inside such samples the Meissner screening is not complete, and the externally applied magnetic field penetrates everywhere (see Fig. 2.2d–​f, Sec. 2.1.1, and Sec. 2.1.2). Spatially inhomogeneous supercurrents circulate deep inside the sample, even in the Meissner phase (Fig. 2.2d). This influences the field at which vortices penetrate into the sample, as well as the vortex arrangements inside. In this limit ( D ≤ λ ) magnetic probes such as Scanning Hall probe, Bitter decoration, or scanning SQUID microscopy are pertinent as the magnetic response remains measurable. Moreover, 11 In general, the DOS of a conductor depends on energy, N ( E ) ≠ N ( E ); S S F here we consider the simplest case of a constant DOS around the Fermi energy.

52

52  Local Studies of Vortex Organization as the magnetic methods are not sensitive to the very surface, the samples can be fabricated by the usual lithography methods. As a result there is a significant body of literature concerning studies of confinement phenomena on the scale of Λ in ensembles of micron-​sized particles (Buisson et al. 1990) or on individual particles (Geim et al. 1997, Hata et al. 2003, Grigorieva et al. 2007, Nishio et al. 2008a). When the lateral size of a sample becomes D  λ (or D  Λ in thin samples with t  λ ), the diamagnetic field generated by the screening currents is very  weak, M j (r )  H 0 (see Sec. 2.1.2). In  this case the magnetic field inside and  outside the sample is simply B (r )  µ 0 H 0 . The vanishing of the magnetic response makes the magnetic field-​sensitive probes much more difficult to apply. STM spectroscopy naturally appears as the most appropriate method to study locally vortex matter in small superconductors of size D  λ.

2.2.2 Elaboration of mesoscopic superconducting systems for STM spectroscopy In most superconducting materials the characteristic length ξ lies in the range 1–​500 nm. This range is achievable by modern means of nano-​elaboration such as electron or optical lithography. These techniques are commonly applied to fabricate mesoscopic superconducting samples for various types of measurements including electron or thermal transport. However, lithographied samples are generally not suitable for STM studies. The main drawback of lithography resides in the alteration of the surface during sample processing and its exposure to air between the elaboration and the introduction into the STM chamber. Both effects cause surface pollution leading to an unavoidable modification of the structural and electronic properties of the topmost atomic layers, which are probed by STM. That is why most reliable STM studies of vortices in bulk superconductors are performed on samples cleaved in an ultrahigh vacuum (UHV) (Renner et al. 1998, Pan et al. 2000). The study by STS of “as-​grown” samples or samples cleaved in non-​UHV conditions led to nice successes only in a few inert systems (Hess et al. 1989, Maggio-​Aprile et al. 1995). Recent success of STM/​ STS on ex situ lithographied but in situ cleaved samples in UHV (Stolyarov et al. 2014)  could provide a new way of preparing superconducting nanostructures and devices for STS experiments with the required surface quality. In order to satisfy the stringent surface quality requirements, several groups started elaborating samples directly inside the UHV environment shared with low-​temperature STM. The samples are usually produced by evaporating a controlled flux of atoms of the superconducting material onto an insulating or semiconducting substrate maintained at the desired temperature. The advantage of this approach is double: (a) STS experiments are provided in situ on very clean surfaces, and (b) the samples can be characterized, if required, by powerful and well-​established surface analysis tools, such as low-​energy electron diffraction, Auger-​spectroscopy, photo-​emission spectroscopy, STM, etc. Let us consider a specific case of elaboration of a collection of mesoscopic superconducting islands by deposition at room temperature of a few atomic layers of Pb onto the (111)-​oriented surface of Si (Fig. 2.4). The choice of Pb and

  53

Experimental requirements for studying vortex confinement phenomena 53

Monoatomic steps separating atomically flat terraces Pb nanocrystals (3–15 ML)

Underlying Si(111)

Amorphous Pb layer

Atomic order in Pb(111)

100 nm

Si (111) + Pb wetting layer (1ML)

Si was made by several groups, mainly for two reasons: (a) Pb and Si are not miscible in the bulk, which favors the on-​top growth of Pb, and (b) there is a significant lattice mismatch between Pb(111) and Si(111) atomic planes, favoring the so-​called Stranski–​Krastanow growth (Stranski and Krastanow 1938). First of all, Si(111) substrate is cleaned in UHV to achieve atomically smooth terraces, several hundreds of nanometers wide, separated by monoatomic steps. At this stage the Si-​surface presents the well-​known 7 × 7 atomic reconstruction.12 Then, Pb-​ atoms are evaporated. First, an ultrathin, 1–​2 atom thick, disordered wetting layer (WL) of Pb grows on a naked Si(111). After the WL is completed, (111)-​oriented fcc Pb nanocrystals start growing (Jalochowski and Bauer 1988, Weitering et al. 1992, Altfeder et al. 1997) (Fig. 2.4). Various lateral sizes (typically 10–​500 nm) and thicknesses (1–​10 nm) can be achieved, depending on numerous deposition parameters such as growth and annealing temperature, deposition rate, source and substrate cleanness, base pressure, etc. (Serrier-​Garcia et  al. 2015). Using lower growth temperature, it is possible to achieve a layer-​by-​layer growth to form continuous films instead of islands (Jalochowski and Bauer 1988). Notably, these (111)-​oriented ultrathin Pb-​films have peculiar electronic properties due to quantum size effects (Jalochowski and Bauer 1988, Chiang 2000). The Fermi length of Pb matches the thickness of four monolayers of Pb in the (111) direction, resulting in standing wave conditions realized for selected film heights. This quantum electron confinement in the growth direction discretizes the electronic spectrum, and the films behave as two-​dimensional (2D) metallic quantum wells (Jalochowski and Bauer 1988, Chiang 2000). This quantum 12 To remove the native silicon oxide from the surface the substrate is heated up to 1200°C, usually by electron bombardment or direct current heating. Wide monoatomic terraces with high-​quality 7   ×   7 reconstruction are achieved by annealing the substrate between 900°C and 500°C.

Fig.  2.4 Pb growth on Si(111). In the center: a 1,000 nm × 1,000 nm STM image of a Pb/​ Si sample after deposition of 6 nominal atomic monolayers of Pb onto the clean 7 × 7-​reconstructed (111)-​ oriented surface of Si. Insets are atomic-​scale STM images showing: (left top)—​7 × 7 reconstruction at the surface of the Si(111) substrate before deposition, (left bottom)—​atomic order on top of the (111)-​oriented Pb islands, and (right)—​ nanometer-​scale amorphous character of the Pb wetting layer covering the entire Si(111) surface after Pb deposition.

54

54  Local Studies of Vortex Organization size effect has direct consequences on all physical properties of the islands that become modulated with thickness, including the superconducting gap and critical temperature (Guo et  al. 2004, Özer et  al. 2006). Generally, the superconducting gap, critical temperature, and mean-​free path are lower in thinner islands (Pfennigstorf et al. 2002, Guo et al. 2004, Özer et al. 2006, Brun et al. 2009). By adjusting the growth parameters one can produce Pb-​crystals of lateral size D comparable with the effective superconducting coherence length in these systems, ξ eff = (20 − 40) nm. This procedure delivers well-​ defined mesoscopic samples perfectly suitable for studies of confinement effects on vortices. The clear advantage of this approach relies on the atomic cleanness of the elaborated nano-​ systems, yielding high-​quality tunneling conditions in the regime D, t  λ , Λ. The counterpart is the difficulty of reproducing identical samples of controlled size and shape in distinct growth sequences, even when nominally the same external growth parameters are used (Serrier-​Garcia et al. 2015).

2.3 Observation of confinement effects on vortices In this section we focus on recent experimental scanning tunneling microscopy/​spectroscopy (STM/​STS) studies of vortex cores in superconductors whose dimensions D are comparable to ξ. Following a “bottom-​up” order, we first present the situation in the smallest islands (D ~ ξ) where the strongest confinement occurs, and then consider larger islands (up to D ~10ξ) with weaker confinements. We start with the description of the ultimate situation of an island so small that in its superconducting state it can only accept zero vortices or one individual vortex inside (Sec. 2.3.1). By considering slightly larger superconductors, accepting only a few vortices at maximum, we show how strong lateral confinement makes the vortices merge to form a single giant vortex with an unusual core (Sec. 2.3.2). In Sec. 2.3.3 we describe how in yet larger mesoscopic samples a weaker confinement still strongly influences the organization of the vortex lattice and leads to novel ultra-​dense vortex configurations, impossible in bulk superconductors. The chapter ends with the description of vortices in single atomic monolayer superconductors. In such ultimately thin films mixed Abrikosov–​Josephson vortices were recently observed (Sec. 2.3.4).

2.3.1 Ultimate vortex confinement When the lateral size D of the sample is significantly smaller than ξ eff , a vortex cannot exist in the superconducting state. Indeed, the creation of a vortex core (of typical size ξ eff ) inside the specimen is equivalent to the destruction of the order parameter everywhere in the superconductor, which will automatically drive the specimen to its normal state. Thus, for a strong enough magnetic field, the superconductor should transit directly from the Meissner phase to the normal state.13 On the other hand, we 13 Notice that the spatial distribution of the order parameter in such ultra-​small superconductors was not yet studied experimentally.

  55

Observation of confinement effects on vortices  55 know that in macroscopically large superconductors many vortices exist. Thus, there should exist a size threshold between vortex-​less superconductivity in ultra-​small superconductors, and superconductivity with vortex phases. While the existence of such a threshold was anticipated theoretically many years ago, only recently have experimental STM studies addressed this issue. Nishio and collaborators (Nishio et al. 2008) studied single nanocrystals of Pb/​Si(111) with D = 50 − 150 nm and t = 1 − 5 nm. The samples were grown in situ, as described above. In larger islands, the authors visualized individual vortices, analyzed their profile, and estimated the effective coherence length to be ξ eff ≈ 26 nm. Then, they studied smaller samples and experimentally demonstrated that the crystals with lateral sizes smaller than (2.5 − 2.8) ξeff do not accept vortices at all (Nishio et al. 2008).

Slightly larger samples D = (3.5 − 5) ξ eff were studied independently by Nishio and collaborators (Nishio et al. 2008) and by Cren and collaborators (Cren et al. 2009). These nano-​superconductors were revealed to host a single vortex in their vortex phase. In order to understand the phenomena revealed in these observations we introduce now some relevant concepts linked to the effects produced in superconductors by superconducting currents, both in the Meissner and single-​ vortex states. 2.3.1.1 Description of the Meissner and single-​vortex states Let us consider a small, thin superconducting disc of size ξ eff < D  λ eff  Λ, subject to a perpendicular external magnetic field (upper part of Fig. 2.5). To compare the theory with the experiment, a series of STS maps of a superconducting island of a near-​disc shape is presented in the lower part of Fig. 2.5. These maps were created by acquiring local tunneling spectra dI / dV (V ) at T = 300 mK at each location of the sample. The bottom curve in Fig. 2.5i presents a typical spectrum acquired at zero field. The shape of the spectrum is in agreement with theoretical expectations for a conventional BCS superconductor (Eqs. 2.15, 2.16); it is characterized by the superconducting gap ∆ ≈ 1.2 meV, close to ∆ Pb = 1.35 meV of bulk Pb.14 The color palette in STS maps represents the strength of the local superconducting gap through the energy-​integrated conductance area (dashed area in Fig. 2.5i). Locations in red correspond to a fully open superconducting gap while the blue color encodes the absence of a gap (normal state). At zero field (Fig. 2.5e), the superconducting gap is observed everywhere in the island, as demonstrated by a spatially uniform red map. It corresponds well to the expected spatially homogeneous superconductivity in our disc at zero field (Fig. 2.5a). When the field is applied, the Meissner currents start flowing. Since the field fully penetrates, and taking into account the circular symmetry the sample,  of  it  is useful to express the vector-​potential in the central gauge A (r ) = µ 0 2 H 0 × r  , with r = 0 at the disc center. In this gauge the phase of the order parameter is

14 The energy of the superconducting gap gives an estimate for the critical temperature of the superconducting transition, TC ≈ 6.2 K.  Thus, the experimental conditions (T = 300 mK) correspond to T ≈ TC / 20 , i.e. T  TC . The Pb-​wetting layer around the islands is not superconducting; the corresponding tunneling spectra show no gap (for more information, see Chapter 4).

56

56  Local Studies of Vortex Organization (b) N

(c)

→ B

(d)

S

(e)

(f) N 0T

S

0.4 T

(g) 0.6 T

(h) 1.8 T

(i)

8

Normalized Conductance (a.u.)

(a)

1.8T

7

1.4T

6

1.0T

5

0.8T

4

0.6T

3

0.4T

2

0.2T

1 0 –4 –2

B=0T 0

2

4

Bias Voltage (mV)

Fig. 2.5  Top row: Schematic view of the superfluid density and circulation of supercurrents in a mesoscopic disc of size D ≈ 3.5 − 5 ξeff under a magnetic field. (a) In the ground state B = 0 and superconductivity is homogeneous. (b) In a weak field the Meissner currents circulate (white arrows) owing to the vector-​potential; their intensity rises with radius. As a consequence of current-​induced depairing effects, superconductivity is gradually weakened with a stronger effect near the edges. (c) The situation with one vortex in the center; the apparent vortex size gives an estimate for ξeff ~ 20 − 30 nm. The vortex entry provokes the suppression of the superconductivity in its core (white); around the core the vortex currents circulate in the direction opposite to the Meissner ones (white arrows represent the resulting total currents). There exists always a loop along which the current is zero (dashed line). (d) At the critical field the depairing effect of Meissner currents drives the whole system to the normal state. Bottom row: (e–​h) A series of experimental STM/​STS spectroscopic maps measured on a small Pb-​island of lateral size D ≈ 4ξeff at magnetic fields 0 T, 0.4 T, 0.6 T, and 1.8 T, respectively. These maps illustrate the phase diagram presented in the top row (a–​d). In each map is represented the local dI / dV (V ) quantity integrated in energy in the range shown by dashed areas in spectra Fig. 2.5i. (i) Local tunneling conductance spectra acquired close to the island edge in increasing field. The spectra show the progressive weakening of superconductivity: smaller energy gaps and broader coherence peaks until superconductivity is completely destroyed.   constant in the Meissner state, ∇ϕ (r ) = 0, and the currents circulate owing to the vector-​potential (Eq. 2.5):

   2µ e 2  j (r ) = − 0* nS H 0 × r (2.17) m

The currents induce a depairing effect, weakening superconductivity (Anthore et al. 2003, Kohen et al. 2006). Thus with increasing magnetic field, the superconductivity in the disc should become more and more suppressed. Moreover,    as j (r ) increases with r , the depairing should depend on position: the larger r is, the greater are the depairing effects, as shown in Fig. 2.5b. This prediction is confirmed experimentally in Fig. 2.5f which displays the color-​coded STM map of the island at low field: 0.4 T. The red color at the island center gradually changes to yellow at the island periphery. The color thus encodes the reduction of the superconducting gap in the LDOS and the broadening of the superconducting  coherence peaks with increasing r .

  57

Observation of confinement effects on vortices  57 At low field the total kinetic energy of the currents circulating in the disc is:

2  E j ( H ext , D ) ∝ ∫∫ j (r )d 2S ∝ H 02 ⋅ D 4 (2.18) S

Hence the total free energy in the Meissner phase has a parabolic dependence with the magnetic field. This simple expression is in agreement with theoretical calculations known since the 1960s (Saint-​James 1965, Fink and Presson 1966). In Fig. 2.6 we reproduce the result of calculations done within GL theory by Fink and Presson (Fink and Presson 1966). It represents the Gibbs free energy of a small superconducting disc as a function of magnetic field. The lowest theoretical curve in Fig. 2.6 corresponds to the Meissner state we consider here. This curve has a nearly ideal parabolic shape, in agreement with our Eq. 2.18. Fink and Presson predicted that if in a rising field a superconductor would remain in the Meissner state, its free energy would rapidly rise and reach the normal state at a rather low critical field hs (Fig. 2.6). The vortex entry (Fig.  2.5c, g) allows the superconductivity to survive until a higher field. From one side, the entry of a vortex provokes the suppression of superconductivity in its core (white color in theoretical Fig. 2.5c, and dark blue in the experimental map Fig. 2.5g). The corresponding energy cost is equal to the conden 1 2 1 2 sation energy density ∆ 2 N ( E F ) times the core volume, E core ≈ ∆ N ( E F )· πξ t , with 2 2 ξ eff (t ) defined by Eq. 2.12 (ξ eff is about 28 nm, as justified in the next section). From the other side, the vortex currents partially compensate the Meissner ones, as seen in Fig. 2.5c, thus substantially reducing the total kinetic energy of Cooper pairs in the disc (see the corresponding branch b = 1 in Fig. 2.6). The detailed energy balance between these two contributions decides at which field the vortex enters and how many vortices at maximum can be present before the disc transits to the normal state. For instance, the calculations reproduced in Fig. 2.6 were done for a mesoscopic disc D = 6ξ eff which accepts only a limited number (up to five) of vortices. The calculations showed that the Gibbs energy as a function of magnetic field indeed has several branches, each corresponding to a fixed vorticity L (noted “b” in Fig. 2.6) inside the sample. The shape of each branch reflects the parabolic dependence of the total kinetic energy of the magnetic field (Eq. 2.18). The branches (vorticities) with the minimum Gibbs free energy exceeding the normal state energy are not realized, as the sample transits to the normal state. In the considered island it occurs at 1.8 T (Fig. 2.5h); the STS spectra show no gap, and the map appears in blue.15 Looking at Fig. 2.5c one notices that there should exist a zero-​current line—​a closed loop where the Meissner and vortex currents perfectly compensate each other. Using Eq. 2.7 one finds that the magnetic flux inside such a zero-​current  loop is simply Φ S = 2πL = L Φ 0, where L is the vorticity, that is, the number of 2e 2 vortex cores inside the loop. At the same time Φ S = BSloop = BπRloop . Therefore

15 The slight depletion in the DOS at zero bias is related to the Altshuler–​ Aronov (Altshuler and Aronov 1985) effect due to electron–​electron correlations in the disordered wetting Pb layer. It has no connection to superconductivity.

58

58  Local Studies of Vortex Organization hs

hℓ (b = 1) 0

hu (b = 1) 5

4

3

hC3 b=6

∆g

2 –0.1

1 b=0

hC1

κ=1 χ=3

hm (b = 1)

–0.2

–1

0

+1

+2

+3

h0

Fig. 2.6  Calculated Gibbs free energy (Eq. 2.3) of a long superconducting disc of radius 3 ξ as a function of external perpendicular magnetic field H 0 (h0 = H 0 / HC ) (after Fink and Presson 1966. Copyright © 1966 by the American Physical Society).The diagram contains several branches each corresponding to different vorticity L (noted as “b” on the original plot). In zero field the superconductor has the lowest energy state.When the field is applied, the Meissner currents start to circulate.The Gibbs free energy is enhanced by the kinetic energy of Cooper pairs,16 in agreement with Eq. 2.3, resulting in parabolic branches, each corresponding to a given vorticity L . In this calculation a cylindrical geometry of the system is assumed for all L . The maximum vorticity is L = 6. the radius Rloop of the loop is directly related to the external field and to the numL Φ0 ber of vortices inside the disc, Rloop = . For a fixed multi-​vortex configuraπB tion characterized by L = cte, the zero-​current loop shrinks in a rising field as Rloop ~ 1 / B , thus pushing the vortex cores towards the island center. This can be interpreted as an effective “pressure” exerted by Meissner currents on vortices; the effect will be illustrated in Sec. 2.3.3.1. To summarize this subsection, the series of experimental STM spectroscopy maps presented in Fig. 2.5(lower row) demonstrated that the Meissner and vortex phases discussed above can indeed be realized, and that the STM/​STS method catches all the significant features of the superconducting phase diagram. 2.3.1.2 Critical field enhancement in ultra-​small superconductors Interestingly, in the experiments made on Pb/​Si nano-​islands the critical value of the magnetic field was found to be up to 20 times higher than the thermodynamical 16 To be precise, in the studied case of a long (thick) cylinder the magnetic energy also    comes into play, ∇ × B = µ 0 j . However, both kinetic and magnetic terms scale as ~H 02 (see Eq. 3), resulting in parabolic branches.

  59

Observation of confinement effects on vortices  59 critical field of bulk Pb (0.09 T). For instance, in Fig. 2.5 the upper critical field evaluated from STS data was 1.6–​1.7 T.  This strong enhancement is due to two distinct effects. The first one is related to the renormalization of the coherence length, ξ becoming ξ eff (t ) < ξbulk , due to the increased electron diffusion in ultrathin films (Eq. 2.12). This results in a very strong thickness-​dependent Φ0 enhancement of the second critical field BC 2 = . 2πξ 2eff (t ) The second effect is related to the reduced energy of Meissner currents in smaller samples, which scales as H 02 D 4, as Eq. 2.18 demonstrates. As the total condensation energy is proportional to the sample volume and scales as D 2 with the disc diameter, this implies that the critical field of vortex-​less nano-​superconductors should diverge as BC ∝1 / D when reducing the sample size.17 2.3.1.3 Interplay between Meissner and vortex currents We now present more subtle effects, resulting from an interplay between Meissner and vortex currents, occurring in ultimate mesoscopic superconductors hosting only zero vortices or one single vortex in their phase diagram. Cren et al. (2009) measured the local quasiparticle tunneling spectra in an ultrathin hexagonal single nanocrystal of Pb having the lateral size ≈ 3ξ eff , Fig. 2.7a. The authors observed a spatially homogeneous superconducting ground state at zero field, the Meissner current depairing effect at the edges at low field, one vortex entry at an intermediate field, and finally a transition to the normal state at a higher field, in agreement with theoretical expectations presented in Fig. 2.5. The dI / dV (V ) spectra were acquired at the center of the island and at its edge. With increasing magnetic field the Meissner phase lasts until µ 0 H 0 < 0.24T. Superconductivity is progressively weakened (the gap reduces and closes), as witnessed by the rise of the tunneling conductance at zero bias (zero bias conductance, ZBC) in Fig. 2.7b. As shown in Sec. 2.3.1.1, this phenomenon is due to the depairing effect of Meissner currents. The depairing effect is weaker at the island center (curve C) where currents are smaller than at the periphery (curve E), in agreement with Fig. 2.5b. Importantly,   there is no perfect match between the measured local ZBC and the local j (r ) current distribution in the island, as could be naively expected. In fact, the local  superconducting properties measured at r are affected by any perturbation (currents, defects, etc.) situated within the spatial extent of the superconducting wave packet ~ ξ eff . Thus, even in the island center where j (0) = 0, there   exists a measurable depairing effect due to non-​zero currents in the vicinity, j (r ≠ 0) ≠ 0 (Cren et al. 2009). At µ 0 H 0 = 0.25T the vortex sets in the center of the island; it is witnessed by a sudden jump of the ZBC to its normal state value (curve C), as expected for a normal vortex core. A remarkable effect of reentrance of superconductivity is revealed at the island edge (curve E). Right after the vortex entry in the center, 17 This effect still requires experimental confirmation. Notice also that the latter dependence should be only valid for fields below the paramagnetic limit Bc < Bpar, Bpar ≈ ∆/gμB (typically 10T for a superconductor with ∆ =1 meV).

60

60  Local Studies of Vortex Organization (a)

(c)

(d) L=0 H λ). Thus, as given in Eq. 3.4, the mutual repulsive interaction between the vortices is effective in the low density of vortices [< Φ 0 / Λ 2 (~100 μT)] where our SQUID microscope works. In a weak pinning material, one should bear in mind that during scanning of samples with respect to the magnetic sensor chip, vortices may be dragged and/​or shifted in positions by the interaction between the vortex and the superconducting pickup loop (Plourde et al. 2002, Kokubo et al. 2010). This interaction is sensitive to the distance between the pickup loop and the sample surface, and is reduced by separating (lowering) the sample stage from the sensor chip. However, when the distance is larger than the penetration depth, the vortex image becomes blurred due to the spread of the magnetic flux above the sample surface (Fig. 3.1b). Thus, the height of the sample stage needs careful adjustment with respect to the sensor chip. The multi-​vortex states are also susceptible to the field history via the influence of a surface energy barrier for a flux entry formed at the sample boundary (Zeldov et al. 1994). To exclude this, we focus mainly on equilibrium vortex states prepared by the field-​cool procedure, where the magnetic field is applied in some temperatures (~10 K) above Tc, followed by cooling the samples in the magnetic field to temperatures (~3–​5 K) below Tc where scanning SQUID microscopy measurements are carried out. Since the magnetic flux is quantized (vortices are

  87

Observation of multi-vortex states in mesoscopic superconductors  87 nucleated) during the field cooling, no entry of vortices from the sample boundary via the surface energy barrier occurs. Thus, vortices are believed to be uniformly distributed over samples with their equilibrium configurations.

3.3 Observation of multi-​vortex states in mesoscopic superconductors We turn to the observation of multi-​vortex states in mesoscopic dots of weak pinning, amorphous MoGe thin films with our scanning SQUID microscope. In Sec. 3.3.1, we present the formation of vortex polygons and concentric vortex rings in mesoscopic disks (circle dots), and construct the packing sequence of vortices in concentric multiple shells. This is intended to serve as introduction to the following subsections about the formation of square (Sec. 3.3.2), triangle (Sec. 3.3.3), and other-​shaped multiple vortex shells (Sec. 3.3.4). Subsequently, manipulations of vortex states with (artificial) pinning sites and the progress made in a vortex-​ based, prospective application are presented in Sec. 3.3.5.

3.3.1 Disks To illustrate how vortices are distributed in a circle-​shaped mesoscopic dot (disk), selected images are shown in Fig. 3.2. After the Meissner state, a single vortex

(a)

(c)

(f)

(b)

(d)

(g)

(a’)

(e)

(h)

(c’)

(f’)

(b’)

(d’)

(g’)

(e’)

(h’)

Fig. 3.2  Magnetic images of vortices in an amorphous MoGe disk with 34 μm diameter for vorticities L = 1–​5 (after Kokubo et al. 2010). For clarity, the corresponding patterns of vortices for these images of (b–​h) are sketched in (b′–​h′), respectively. Reproduced by permission of the American Physical Society from [Kokubo et al., 2010], Copyright (c) 2010 by the American Physical Society.

88

88  Multi-Vortex States in Superconductors appears and sits at the disk center. This originates from a radial force exerted on the vortex from the disk boundary, which is balanced (or the net force on the vortex from the boundary disappears) when the vortex is situated at the disk center. When two vortices are induced in the disk, the geometrical confinement and its interplay with the mutual repulsion between the vortices determine the configuration of the vortices. This is intuitively reduced to a classical one body problem (or central force problem) in a circular well by considering the center of a vortex pair to be like a rigid body. As imaged in Fig. 3.2c–​e, two vortices are located symmetrically with respect to the disk center, although the pair orientation is rotatable. On further increasing vorticity, vortices are equally distributed with respect to the disk center by forming triangle (Fig.  3.2f), square (Fig.  3.2g), and pentagon clusters (Fig.  3.2h). These polygonal configurations are symmetric with respect to the disk center, and originate from the interplay. To know how far vortices are separated from the edge boundary and also from each other at a given field and temperature, it needs to be numerically worked out by using the Ginzburg–​Landau theory (Palacios 1998, 2000, Schweigert et  al. 1998, Schweigert and Peeters 1998, Baelus and Peeters 2002) and/​or the London theory (Baelus et al. 2004, Cabral et al. 2004, Misko et al. 2007). For large disks (on the scale of λ), it is useful to employ molecular dynamic simulations by treating vortices as classically interacting particles and imbedding them in a circular potential well. It turns out that the lowest-​energy configurations are formed by vortices distributed uniformly in simple regular polygons (as observed in Fig. 3.2), which are not sensitive to the disk diameter Dd as far as small vorticities (L ≤ 5)  are considered. The polygonal patterns can be orientated in all directions in the plane of a mesoscopic disk, since the disk is the most symmetric 2D structure and accompanies the rotational degrees of freedom, as seen in a series of vortex pair images at L = 2 (Fig. 3.2c–​e). The number of vortices induced in a mesoscopic disk varies with an applied magnetic field in a discrete manner. As seen from Fig.  3.3a, the vorticity exhibits a stepwise behavior with the magnetic field. Each jump corresponds to one vorticity and the width of plateaus indicates the field stability of corresponding vortex states. One thing we would like to emphasize in this plot is that the staircase behavior of L is well below the full penetration line given by L F  =  µ 0 HS / Φ 0 with the disk area S (the origin of the line is shifted by the amount of the ambient magnetic field around the sample space). Namely, the difference between L and L F clearly indicates the diamagnetic response of the disk. We note that the corresponding curve (the difference “L – LF” vs magnetic field) accompanies discontinuities when the vorticity changes. Such a behavior is qualitatively consistent with a magnetization M curve of the disk shown in Fig.  3.3b where M (= Φ d / µ 0S − H ) was evaluated from the magnetic images by using magnetic flux Φ d in the disk integrated over the surface area S of the disk (Nisho et al. 2004). For understanding of the geometrical confinement, it is important to know how the size of vortex polygons varies with the magnetization of the disk (or simply the magnetic field). Referring back to the series of vortex pair images taken in different

  89

Observation of multi-vortex states in mesoscopic superconductors  89 (a) 4

L

3 2 1 0 0

(b)

5

10

15

20

25

0 –1

M (arb.)

–2 –3 –4

Fig.  3.3  (a)  Vorticity vs magnetic field in a 34 μm disk. A solid line represents a condition of the full penetration of magnetic flux in the disk. (b)  A  magnetization curve of the disk evaluated from the magnetic images.

–5 –6

0

5

10

15 µ 0 H (µT)

20

25

magnetic fields (Fig. 3.2c–​e, c′–​e′), one may see some reduction in the vortex spacing upon increasing the magnetic field. The field-​induced shrinkage in the vortex spacing is also observed in vortex triangle (L = 3), square (L = 4), and pentagon clusters (L = 5). These features originate from the field-​dependent radial force on vortices which becomes strong as the shielding current flowing along the disk boundary is enhanced, and agree reasonably with the London model developed by Cabral et al. (2004). Following their argument, the diameter Dring of a vortex polygon formed with N vortices for a given magnetic field H can be written as



Dring = Dd

N −1 N −1 = Dd µ 0 HS / Φ 0 Φd / Φ0

(3.5)

As seen from the main panel of Fig. 3.4, the agreement between the experimental results and Eq. 3.5 is reasonable; however, some deviations seem to appear as the size of the vortex polygons grows by increasing N (and/​or lowering H).

90

90  Multi-Vortex States in Superconductors 1.0 0.8 Dring/Dd

Fig. 3.4  Normalized diameter of vortex polygons vs magnetic field (after Kokubo et  al. 2010). The dotted lines represent theoretical results for polygon sizes with different N  =  2–​5 as denoted. The inset illustrates how we determine the diameter Dring of a polygon ring. Reproduced by permission of the American Physical Society from [Kokubo et al., 2010], Copyright (c) 2010 by the American Physical Society.

N=5

0.6

N=2

D ring

0.2 0.0

N=4

N=3

0.4

0

2

4

6 Φd/Φ0

8

10

This stems from the attractive interaction with the disk boundary and requires a proper boundary condition(s), e.g. the image method (Buzdin and Brison 1994, Cabral et al. 2004). The formation of concentric vortex shells appears when more vortices are confined in disks (L > 5). As seen from Fig. 3.5a–​c , a double shell structure is formed for L = 6–​8, where one vortex appears near the disk center and the others form the outer shell around. On increasing L from 9 to 16, the “inner” vortices form pair, triangle (Fig.  3.5d), square, and pentagonal clusters as observed in small vorticities. A triple shell structure appears at L = 17. The observed shell structures can be characterized by a standard notation for multiple shells (N1,N2,N3,…,Ni) where Ni is the occupation number of vortices in the ith shell from the center (Juan et al. 1998). It turns out that the inner and outer shells are filled with vortices by increasing L as follows; … → (5) → (1,5) → (1,6) → (1,7) → (2,7) → (2,8) → (3,8) → (3,9) → (3,10) → (4,10) → (5,10) → (5,11) → (1,5,11). The resultant filling sequence can be simply understood by the fact that, when a vortex is added in one of the concentric shells, the adjacent shell(s) would become relatively sparse and have the space to insert a new vortex. It is worth pointing out that double shell configurations contain a hexagonal pattern of (1,6) where the vortex in the center is sixfold coordinated and is stably coupled with the outer shell vortices as seen from Fig. 3.5b’. Five-​and sevenfold coordinated patterns also appear as (1,5) (Fig. 3.5a′) and (1,7) (Fig. 3.5c′), respectively, but not four-​ or eightfold ones. These are similar to the defect structure in the vortex lattice (e.g. edge dislocation) in weak pinning, bulk superconductors (Guillamón et al. 2014), and would explain the reason why a (2,7) configuration appears, instead of a (1,8) configuration at L  =  9. Other sixfold coordinated stable patterns of (3,9) and (4,10) also appear at L  =  12 and 14, respectively. To

  91

Observation of multi-vortex states in mesoscopic superconductors  91 (a)

(b)

(c)

(a’)

(b’)

(c’)

(d)

(d’)

form these stable configurations with increasing L, vortices are filled in both shells as listed above. There are some limitations to filling vortices in concentric shells. The first shell or the innermost shell can contain up to 5 vortices, while up to 11 vortices fit in the second shell. Thus, configurations of (5) and (5,11) are “closed” in the sense that they need a new vortex shell for filling additional vortices. The nucleation of a new shell consisting of only one vortex (or single vortex core) emerges at the disk center, which results in the increase in the order of the existing shell(s), i.e. the first (second) shell becomes the second (third) one. Thus, the corresponding occupation of vortices in the shells varies discontinuously with vorticity when a new shell is nucleated. As seen from Fig. 3.6, after the occupation of 5 vortices in the first shell at L = 5, N1 drops to 1 at L = 6 due to the formation of a double shell structure. When a triple shell structure is formed at L = 17, N1 exhibits also a discontinuous drop from 5 to 1. Thus, the occupation of the first shell is ranged (or oscillates) from 1 to 5. Based on the resultant packing sequence of vortices in multiple concentric shells, together with the repeated (oscillating) occupation for each shell, we made a periodic table for vortex configurations in Table 3.1, where the first, second, and third rows summarize single, double, and triple shell configurations, respectively. The vortex configurations are also separated into subgroups on the basis of the occupation N1 of the first shell. The closed shell configurations are given in the

Fig.  3.5 Magnetic images of vortices in disks with different diameters for large vorticities (after Kokubo et al. 2010). Images of (a–​c) were observed in a disk with 56 μm diameter, while that of (d) was in a 106 μm disk.The corresponding patterns of vortices are traced in (a′–​d′). Reproduced by permission of the American Physical Society from [Kokubo et al., 2010], Copyright (c) 2010 by the American Physical Society.

92

92  Multi-Vortex States in Superconductors 1st shell 2nd shell 3rd shell

Occupation of shell

10

Fig. 3.6  Occupation of different shells vs vorticity in disks (after Kokubo et al. 2016). Reprinted from doi:10.1016/j. physc.2016.05.009, N. Kokubo et al., Vortex shells in mesoscopic triangles of amorphous superconducting thin films; copyright (2016), with permission from Elsevier.

5

0

5

0

10 Vorticity

15

rightmost column, while vortex configurations accompanying a new shell at the center are in the leftmost column. This is reminiscent of Mendeleev’s periodic table of elements. Unlike quantum atoms where new electrons are added in the outermost shell, vortices are filled in different shells with increasing vorticity.  This table can also be regarded as an inclusion structural hierarchy of vortices in multiple shells: the top level (single shell structure) reveals the most frequently observable vortex patterns, which are included as “cores” in the lower levels (double and triple shell structures). For example, as seen in the table, a double shell configuration of (5,11) has a pentagonal core of (5). More inclusion can be found in numerical simulation studies (Baelus et  al. 2004, Cabral et al. 2004, Misko et al. 2007). Table 3.1  Periodic table of vortex shells in mesoscopic superconducting disks (after Kokubo et al. 2010). Occupation in N1

1

2

(1)

3

(2)

4

(3)

5

(4)

(5)

(4,10)

(5,10)

Single shell (1,5)

(1,6)

(1,7)

(2,7)

(2,8)

(3,8)

(3,9)

Double shells (1,5,11*) Triple shells

*A vortex(vortices) trapped at the dot edge is ignored.

(3,10)

(5,11*)

  93

Observation of multi-vortex states in mesoscopic superconductors  93 The alternative packing in concentric shells with periodically oscillating occupation was observed in different physical systems of strongly charged μm-​sized dust particles (104 electrons/​particle) suspended in a cylindrical plasma trap (Juan et  al. 1998), and electrostatically charged sub-​mm-​sized stainless steel balls (109 electrons/​ball) confined in a circular frame (Jean et al. 2001). These are model systems to image the fundamental properties of 2D mesoscopic (or macroscopic) Wigner clusters, and share some general features including that (a) the occupations of first and second shells never exceed 5 and 11, respectively, (b) double and triple shell structures appear at characteristic packing numbers of 5 and 17, respectively. Numerical studies revealed that the packing sequence of concentric shells is not sensitive to inter-​electron interaction forms including Coulomb-​and Yukawa-​type interactions as far as a simple parabolic potential well is assumed. Readers may find more details in the literature on how concentric shell structures and corresponding packing sequences are influenced by both the interaction form and confinement potential (Partoens and Peeters 1997, Lai and I 1999).

3.3.2 Squares In geometry, a square has four discrete mirror lines of symmetry as illustrated in Fig. 3.7a′. Due to this discreteness, the square confinement leads to the formation of a piece of the square vortex lattice as seen in vortex images at the square numbers of

(a)

(b)

(e)

(c)

(f)

40 µm

(a’)

(d)

(g)

(b’)

(e’)

(c’)

(f’)

(d’)

(g’)

Fig. 3.7  (b–​g) Magnetic images of vortices for L  =  1–​4 observed in a square dot of 76 × 76 μm2 in size (after Kokubo et al. 2014). The corresponding patterns of the vortex images are sketched in (b′–​g′). The electron micrograph of the dot is shown in (a). Reproduced by permission of the Physical Society of Japan from [N. Kokubo et al., 2014], Copyright © 2014 by Physical Society of Japan.

94

94  Multi-Vortex States in Superconductors vorticities, i.e. L = 4 (Fig. 3.7e), 9 (Fig. 3.8c), and 16 (Fig. 3.8h). These configurations are commensurate with the dot geometry by sharing the discrete lines of symmetry, and appear in relatively wide field ranges (or accompany wide field plateaus of L when the vorticity is plotted against the applied magnetic field). Such symmetry is partly present at other vorticities. As seen from Fig. 3.7d′, g′, vortex triangle patterns at L = 3 are highly frustrated with respect to the confined geometry, but retain one of the discrete lines of symmetry. It is worth pointing out that the orientation of these triangle patterns is discrete due to the line symmetry. Vortex pair patterns at L = 2 have also symmetry-​induced discrete orientations as shown in Fig. 3.7c′, f′. More symmetry and orientations in other incommensurate vortex states can be found in traced patterns given in Fig. 3.8a′–​l′. The observed vortex patterns are explainable in terms of multiple shells. As seen from Fig. 3.8a, a′, a double shell structure is clearly formed with a configuration of (1,4) at L = 5. The occupations of vortices in the inner and outer shells increase with L as follows; … → (1,4) → (1,5) → (1,6) → (1,7) → (1,8) → (2,8) → (3,8) → (4,8) → (4,9) → (4,10) → (4,11) → (4,12) → …. Namely, the occupation N2 in the outer shell grows first from 4 to 8, while the occupation in the inner shell remains

(a)

Fig. 3.8  (a–​l) Magnetic images of vortices in square dots of 76 × 76 μm2 in size for L ≥ 5 (after Kokubo et al. 2014). The corresponding patterns of vortices are sketched in (a′–​l ′). Reproduced by permission of the Physical Society of Japan from [N. Kokubo et al., 2014], Copyright © 2014 by Physical Society of Japan.

40 µm

(b)

(c)

(a’)

(b’)

(c’)

(d)

(e)

(f)

(d’)

(e’)

(f’)

(g)

(h)

(i)

(g’)

(h’)

(i’)

(j)

(k)

(l)

(j’)

(k’)

(l’)

  95

Observation of multi-vortex states in mesoscopic superconductors  95 12

1st shell 2nd shell 3rd shell

Occupation of shell

10 8 6 4 2 0

0

5

10 Vorticity

15

20

unchanged, i.e. N1  =  1. After forming the (1,8) configuration (Fig.  3.8c, c′), the occupation N1 in the inner shell in turn increases from 1 to 4. Then, the outer shell grows again until the formation of the (4,12) configuration (Fig. 3.8h, h′). The corresponding changes in the packing sequence are marked by the occupations of multiple shells filled by four and/​or multiples of four vortices, i.e. the formation of square vortex shells stabilized by the dot geometry. The resultant filling rule of vortices in different shells is plotted against vorticity in Fig. 3.9. The occupation N1 of the first shell varies from 1 to 4 for L = 1–​4, and this repeats for L = 5–​16. N2 of the second shell changes from 4 to 12 with increasing vorticity. Thus, the maximum numbers of occupations in the first and second shells are 4 and 12, respectively. This indicates that vortex configurations of (4) and (4,12) are “closed” as shell structures and need a new shell for filling additional vortices. It is worth recalling that these closed shell patterns correspond to pieces of the square lattice at square numbers of vorticities, L = 4 and 16, and they are indeed stable from the viewpoint of the commensurability between the dot geometry and vorticity. Based on these findings, a periodic table (or an inclusion hierarchy of vortices in multiple shell structures) for square dots is formed in Table  3.2, where vortex patterns are again separated into subgroups by using the occupation N1 of the first shell. The closed shell configurations are given in the rightmost column, while the configurations with a new shell are in the leftmost column. Again, the vortex patterns in the top level (single shell structure) are included as cores in the lower levels (double and triple shell structures). What is unique in this table is that the packing sequence changes after the second shell is “half” filled with N2 = 8 (N2 varies from 4 to 12), which is reminiscent of one of Hund’s rules for filling electrons in atomic orbitals. The observed feature originates simply from the fact that the square vortex shell(s) with occupation of 4, 8, or 12 vortices is stable. The filling of vortices predominantly occurs in non-​stable shell(s), until the occupation with four or multiples of four vortices.

Fig. 3.9  Occupation of different shells vs vorticity in square dots (after Kokubo et al. 2016). The occupation of ground and metastable states are indicated by solid and open symbols, respectively. Reprinted from doi:10.1016/j.physc.2016.05.009, N. Kokubo et al., Vortex shells in mesoscopic triangles of amorphous superconducting thin films; copyright (2016), with permission from Elsevier

96

96  Multi-Vortex States in Superconductors Table 3.2  Periodic table of vortex shells in mesoscopic superconducting squares (after Kokubo et al. 2014, Copyright © by Physical Society of Japan). Occupation in N1

1 (1)

2

3

(2)

(3)

(2,8)

(3,8)

4 (4)

Single shell (1,4)

(1,5)

(1,6)

(1,7)

(1,8)

(4,8)

(4,9) (4,10)

(4,11) (4,12)

Double shells (1,4,11) (1,5,11) Triple shells

Meanwhile, at L = 5, another configuration of a pentagon cluster appears as a metastable state (Fig. 3.8k, k′), in addition to a double shell configuration of (1,4) (Fig. 3.8a, a′). A numerical simulation study revealed that the difference in free energy of the two configurations is very small and would be negligible in experiment (Zhao et al. 2008). The situation is similar to L = 17, where vortices form either a triple shell configuration of (1,4,12) as the ground state (Fig.  3.8i, i′) or a double shell configuration of (5,12) as a metastable state (Fig. 3.8l, l′). Such metastability stems from the incommensurability between the geometrical confinement and vorticity, and is one of the characteristic features of multi-​vortex states confined in mesoscopic square dots. Many missing metastable states were numerically observed as the sample size or temperature was varied. Metastable shell configurations with pair or triangle vortex cores [i.e. (2,10), (2,12), and (3,11)] can be formed at L  =  12, 13, and 14 (Misko et al. 2009). Moreover, as illustrated in Fig. 3.10, an antivortex (a vortex with opposite polarity with respect to the applied magnetic field) may appear in the dot center and coexist with a square vortex shell at L = 3 due to a remarkable effect of square symmetry (Chibotaru et al. 2000). This is called a vortex–​antivortex state, its configuration being characterized by (–​1,4). A similar argument leads also to a vortex–​antivortex state of (–​1,3) in a mesoscopic triangle dot at L = 2 (Chibotaru et al. 2001). The vortex–​antivortex states were

+1

+1

Fig.  3.10  Schematic illustrations of vortex–​antivortex states for triangle and square dots.

+1

–1 L = 3–1

+1 –1

+1

+1

+1 L = 4–1

  97

Observation of multi-vortex states in mesoscopic superconductors  97 numerically shown to be stable only in a narrow region close to the mean field line in very small dots (Mertelj and Kabanov 2003). The direct observation of vortex–​antivortex states remains an experimental challenge.

3.3.3 Triangles Obviously, in an equilateral triangle-​shaped mesoscopic dot, vortices form a piece of the triangular vortex lattice when the vorticity becomes a triangular number, i.e. L = 3 (Fig. 3.11b), 6 (Fig. 3.11e), 10 (Fig. 3.11i), 15, …, n(n + 1)/​2 with an integer n. The corresponding vortex arrangement is commensurate with the dot geometry by sharing three discrete mirror lines of symmetry of the triangle dot (e.g. Fig.  3.11e′). Such symmetry is also seen at L  =  4 (Fig.  3.11c, c′) where a vortex appears at the dot center and the other three vortices form a triangle around. For other vorticities, vortex patterns are not commensurate with the confined geometry, but maintain mirror symmetry with respect to triangle dots as shown in traced patterns in Fig. 3.11a′–​l′. Let us try to explain the observed vortex patterns in terms of multiple shells. As seen from Fig. 3.11c′, a double shell structure is formed with a configuration

(a)

40 µm

(b)

(c)

(a’)

(b’)

(c’)

(d)

(e)

(f)

(d’)

(e’)

(f’)

(g)

(h)

(i)

(g’)

(h’)

(i’)

(j)

(k)

(l)

(j’)

(k’)

(l’)

Fig.  3.11 Magnetic images of vortices for L  =  2–​ 11 in regular triangle dots in different magnetic fields (after Kokubo et  al. 2015). The corresponding patterns of vortices are sketched in (a′–​l  ′). Reproduced by permission of the Physical Society of Japan from [Kokubo et al., 2015]; Copyright © 2015 by Physical Society of Japan.

98

98  Multi-Vortex States in Superconductors of (1,3) at L = 4. The occupations of vortices in the inner and outer shells increase with L as follows: … → (3) → (1,3) → (5) → (6) → (1,6) or (7) → (1,7) → (1,8) → (1,9) → (1,10) or (2,9) → (2,10) or (1,11) → (2,11) or (3,10) → (3,11) or (2,12) → (3,12). Many configurations have the occupation of three and/​or multiples of three vortices in the shells. This indicates the formation of a triangular vortex shell(s) which is stabilized with respect to the confined geometry. Some vorticities accompany metastable states created by the incommensurability between the geometrical confinement and vorticity. These include a single shell configuration of (7) at L = 7 (Fig. 3.11l, l′) and double shell configurations of (2,9), (1,11), (3,10), and (2,12) for L  =  11–​14 (Kokubo et  al. 2016). It is worth pointing out that after the formation of the double shell configuration of (1,3) at L = 4, single shell configurations reappear at L = 5 and 6 (also L = 7 as a metastable state). Thus, the packing sequence of vortices in multiple shells is not monotonic with vorticity. A different definition for vortex shells was proposed by Zhao et al. (2008). In their model, three vortices near the corners of a triangle dot form always the outermost shell (open circles in traced images given in Fig. 3.11b′–​l′). As a result, the V-​shaped vortex pattern at L  =  5 can be regarded as a (2,3) configuration (Fig.  3.11d′). The triangle pattern at L  =  6 becomes a double shell configuration of (3,3) (Fig. 3.11e′). The corresponding packing sequence becomes rather monotonic with vorticity as follows: … → (3) → (1,3) → (2,3) → (3,3) → (4,3) → (5,3) → (1,5,3) → (1,6,3) → (2,6,3) or (1,7,3) → (2,7,3) or (1,8,3) → (2,8,3) or (3,7,3) → (3,8,3) or (2,9,3) → (3,9,3). Namely, after the formation of (1,3), double shell configurations appear up to L  =  8, above which triple shell configurations are formed. We would like to emphasize that inner parts of shell configurations (excluding three vortices in the outermost shell) resemble well ones observed in mesoscopic disks, i.e. (1) → (2) → (3) → (4) → (5) → (1,5) → (1,6) → (1,7) → …. Thus, the Zhao picture is a sort of combination of the three corner vortices and concentric shell configurations. When it comes to the reiteration of the occupations of multiple shells, neither model works well. The maximum shell occupation varies with the multiplicity of vortex shells, and therefore closed shell configurations are not reasonably defined. Thus, it is controversial whether the concept of multiple vortex shells is fully applicable to vortex patterns in mesoscopic triangle dots (Kokubo et al. 2016). Apart from the discrepancy of vortex shells in mesoscopic triangles, another idea of “vortex bricks” was proposed by Cabral and Aguiar (2009). In this model, a vortex arrangement is interpreted as either a commensurate vortex pattern(s), a linear vortex chain(s), or their combination. As indicated by a line in traced images of Fig. 3.11d′, the V-​shaped vortex arrangement at L = 5 can be separated into a commensurate triangle pattern (at L = 3) and a linear chain of two vortices. The vortex arrangement at L = 9 can also be regarded as a combination of a commensurate triangle pattern (at L = 6) and a linear chain of three vortices (Fig.  3.11h′). Such identification works well in other incommensurate vortex arrangements. As listed in Table 3.3, the notation of cN describes a commensurate pattern of N vortices and lM means a linear chain of M vortices. This

  99

Observation of multi-vortex states in mesoscopic superconductors  99 Table 3.3 Vortex configurations observed in equilateral and isosceles triangle dots (Reproduced by permission of the Physical Society of Japan from [Kokubo et al.], 2015, Copyright © by Physical Society of Japan). L

Configuration Isosceles Triangle A

Isosceles Triangle B

1

Equilateral Triangle c1

“c1”

“c1”

2

l2||​,l2

“l2”

“l2⊥ ”

3

c3

“c3”

“c3”

4

c4

“c4”

“c4”

5

c3 + l2

“c3” + l2

l2 + l3

6

c6

“c6”

“c6”

7

c4 + l3, c3 + l4

“c4 + l3”

“c3 + l4”

8

c4 + l4

“c3” + l2 + l3

“c4” + l4

9

c6 + l3

“c6” + l3

l2 + l3 + l4

10

c10

“c10”

“c10”

11

c6 + l5, c3 + l4 + l4

“c4” + l3 + l4

“c6” + l5

picture is straightforward for representing vortex configurations and is useful also to distinguish commensurate and incommensurate vortex states in mesoscopic triangle dots. The vortex-​brick model also works in vortex states in isosceles triangle dots (Kokubo et al. 2015). As seen from selected images in Fig. 3.12, the combination of a triangular vortex pattern and a linear vortex chain appears at “incommensurate” vorticities of L = 5 (Fig. 3.12d, d′), 7 (Fig. 3.12j, j′), 9, …, and so on. It is worth noting that the incommensurate vortex configurations vary sensitively with the isosceles geometry when its height is either expanded (isosceles triangle A) or reduced (isosceles triangle B) with respect to the regular triangle. At L = 5, for instance, a combination of a triangle pattern and a linear chain appears for the isosceles triangle A dot, while two linear chains appear for the isosceles triangle B dot (Fig. 3.12i, i′). More differences can be found in Table 3.3. Meanwhile, at commensurate vorticities, the vortex configurations are essentially unchanged, although the corresponding patterns are either expanded or compressed in accordance with the isosceles geometry (e.g. vortex triangle images of Fig. 3.12b, g). Thus, the commensurate configurations are stable with respect to the deformation of the dot geometry, whereas the incommensurate ones are not. These facts justify the identification of commensurate/​incommensurate vortex states in isosceles triangle dots from the stability point of view. There are many coincidences in vortex configurations between equilateral and isosceles triangle dots. At L  =  5, equilateral and isosceles A  triangle dots

100

100  Multi-Vortex States in Superconductors (a)

Fig.  3.12 Magnetic images of vortices for L = 2–​7 in two types of isosceles triangle dots in different magnetic fields (after Kokubo et al. 2015). The images of (a–​e) were observed in isosceles A  dots, and those of (f–​j) in isosceles B dots. The corresponding patterns of vortices are sketched in (a′–​j  ′). Reproduced by permission of the Physical Society of Japan from [Kokubo et al., 2015]; Copyright © 2015 by Physical Society of Japan.

40 µm

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(a’)

(b’)

(c’)

(d’)

(e’)

(f’)

(g’)

(h’)

(i’)

(j’)

share the c3 + l2 configuration, while at L = 8 the c4 + l4 configuration appears in both equilateral and isosceles B triangle dots. At L = 7, c4 + l3 and c3 + l4 configurations respectively appear in isosceles A and B triangle dots, while both configurations do in equilateral triangle dots. This would imply that the configurational “degeneracy” in equilateral triangular dots can be lifted by introducing deformation in the dot geometry, i.e. the transformation from regular to isosceles triangle dots. Thus, for this purpose, an investigation on mesoscopic dots of various shapes other than simple regular polygons would be useful.

3.3.4 Other geometries So far, we have shown that the concept of multiple vortex shells is reasonably applicable to vortex patterns observed in mesoscopic circle and square dots (partly in mesoscopic regular triangle dots). The shell formation is not limited to them, but relevant also to other geometries including regular pentagon (Huy et al. 2013), ellipse (Meyers and Daumens 2000), and rectangle-​shaped dots (Misko et al. 2009). In a regular pentagon dot, the packing sequence of vortices in multiple shells was partly observed. It turns out that a vortex pentagon at L = 5 is commensurate with the dot geometry, and seems to be closed as a single shell, since double shell configurations appear for L ≥ 6. More vortices can be involved

  101

Observation of multi-vortex states in mesoscopic superconductors  101 in the closed shell in ellipse dots. Our observation made on an ellipse dot with (first) flattening f = 17% reveals that a single elliptical shell is closed at L = 6 and a new shell is nucleated at L = 7. The occupation of vortices in the closed shell is expected to become even larger as the flattening of ellipse dots is enhanced. However, in largely flattened ellipses, vortices are likely aligned in a single linear chain along the major axis of the ellipse (Meyers and Daumens 2000) and the concept of shells seems not to be applicable. The formation of a single linear vortex chain also occurs in rectangular dots with large aspect ratios (Sardella et al. 1999). Thus, the formation of multiple shells is not stable when introducing large deformation in the dot geometry. To know how much deformation is allowed requires further understanding of the formation of multiple shell structures.

3.3.5 Mesoscopic superconductors with (artificial) pinning centers The multi-​ vortex states would be drastically changed when one strong pinning site [a material defect(s)] is involved in a small superconductor. This originates from the appearance of a firmly trapped vortex at the strong pinning site which hinders the formation of symmetric, regular vortex polygons. As seen in Fig. 3.13d, d′, in a disk with a (off-​centered) strong pinning site, vortices form a dice pattern of six at L = 6, in contrast to (1,5) in weak pinning disks (Fig. 3.5a). The rest of the images given in Fig. 3.13 reveal clearly, for other vorticities, how vortices are distributed in the presence of the pinning site as marked by the cross. It is worth noting that the pinning site induces the “symmetry” in vortex patterns characterized by the mirror reflection with respect to a line defined by the off-​centered pinning site and the disk center (a dotted line in the traced image of Fig. 3.13a′). In a rhombus-​shaped pattern at L = 4 (Fig. 3.13c, c′), for instance, two vortices lie on the line and the others are symmetrically located with respect to the line. A  physical reason for the symmetry is not simple since the vortex

(a)

(d)

40 µm

(b)

(e)

(c)

(f)

(a’)

(d’)

(b’)

(e’)

(c’)

(f’)

Fig.  3.13 (a–​ f) Magnetic images of vortices for different vorticities in a disk with a strong pinning site [a material defect(s)] indicated by a cross (after Kokubo et  al. 2010). The corresponding patterns of vortices are sketched in (a′–​f  ′). Reproduced by permission of the American Physical Society from [Kokubo et al., 2010], Copyright (c) 2010 by the American Physical Society.

102

102  Multi-Vortex States in Superconductors patterns are determined by the subtle balance between many competing interactions (Kokubo et al. 2010). Unusual multi-​vortex states can also be created at will by fabricating artificial pinning sites in a small superconductor. This can be simply achieved by digging a small defect(s) in a superconducting dot where a vortex(ices) is trapped due to the modulation of the thickness in the superconductor, since the line energy of a vortex trapped in the defect is less than one for penetrating the whole thickness of the dot. The situation is also similar to a small hole through the dot where the quantized magnetic flux, “fluxon,” penetrates (the supercurrent flows around the hole, but the normal core is not accompanied). Fig. 3.14 shows the penetration of fluxons in five small holes in a square dot. As seen in an optical micrograph shown in Fig. 3.14a, one hole is fabricated at the dot center and the other holes are in a square arrangement around. The penetration of three fluxons is frustrated by the hole arrangement and the resulting fluxon patterns are unusual. As seen from Fig. 3.14d, the three fluxons are arranged in a diagonal row. But, for samples with five dense holes, three fluxons are occupied in three corners of the square hole arrangement and the centered fluxon disappears. Thus, the fluxon state depends on the spacing between holes and can be controlled by the hole arrangement. Meanwhile, the diagonal state of two fluxons is “bistable” in the sense that two orientations of a fluxon pair are doubly degenerated (see Fig. 3.14b, c). This is analogous to configurations of two electrons localized in two of four (2 × 2) semiconductor quantum dots which allow setting binary logic states (i.e. “0” and “1”) as elements for performing quantum cellular automata (Lent et  al. 1993). The bistability of fluxons was numerically shown in half-​filled 2 × 2 antidot clusters in a superconducting mesoscopic square dot (Puig et al. 1997), and also later in an array of 2 × 2 blind holes (defects) in a superconducting single layer (Milošević et  al. 2007). The latter study also demonstrated the possibility of transmitting

(a)

Fig.  3.14 (a)  An optical micrograph of an arrangement of through holes in a square dot 68 × 68 μm2 in size. (b–​e) Magnetic images of fluxons observed in different magnetic fields.

40 µm

(b)

(c)

(d)

(e)

  103

Summary and outlook 103 two logic states by assuming a locally interconnected architecture of blind hole (defect) arrays, analogous to the principle of the cellular automata. The experimental demonstration of flux(vortex)-​based cellular automata has been recently made on a more feasible structure of “vortex cells” fabricated in a superconducting bilayer film (Kokubo et al. unpublished). Another structure was numerically proposed by incorporating the ratchet effect in nanostructured superconductors (Hastings et al. 2003). These would be fundamental components for the computations based on vortices/​fluxons manipulated in nanostructured superconductors. The vortex/​fluxon-​based computations would be made more unique by incorporating the non-​ trivial topology, e.g. a vortex–​ antivortex state or a fluxon–​antifluxon state of nanostructured superconductors. The latter state was numerically proposed to appear stably in the arrangements of five holes in square dots (see an optical micrograph in Fig. 3.14a). Similar to a vortex–​ antivortex state, an antifluxon penetrates in the center and four fluxons thread the other four holes with a square arrangement. This state can be prepared by using the properties of flux pinning and magnetic hysteresis as follows: after trapping a fluxon at the center by applying a magnetic field, the magnetic field is reversed. Because of the flux pinning, the “antifluxon” is stably trapped at the center. Then, the magnetic field is increased until the other holes are filled by fluxons, resulting in the fluxon–​antifluxon state. A  numerical simulation revealed that the fluxon–​antifluxon state appears stably in a large dot (Geurts et  al. 2009)  and should be distinct from the symmetry-​ induced vortex–​ antivortex state which is expected to appear only in a narrow temperature and field region close to the mean field line in small dots (Mertelj and Kabanov 2003). Thus, the fluxon–​antifluxon state is more feasible for experimental observation, but remains an experimental challenge.

3.4 Summary and outlook An overview has been given to the current state regarding our knowledge of multi-​ vortex states imaged in mesoscopic superconducting dots with different geometries by using the scanning SQUID microscope. The presented key features, including commensurability effect, multiple shell structures, repeated packing sequences, inclusion structural hierarchy, and pinning effect, are not only useful for the understanding of the fundamental properties of vortex matter confined in mesoscopic scales, but also will advance various vortex-​(fluxon)-​based prospective applications. Thanks to the continuous development of the scanning SQUID microscope, the spatial resolution has been improved down to nanometer scales and the observation of (multi-​)vortex states in nanostructured superconductors is within reach (Finkler et al. 2012, Granata and Vettoliere 2016). This has the potential to play a major role for unveiling properties of the elementary depinning process, (quantum) melting transition, and/​or possible entanglements of vortex clusters confined in nanostructured superconductors (Filinov et al. 2001, Embon

104

104  Multi-Vortex States in Superconductors et al. 2015), and also for future development in fluxon-​based (quantum) measurement instrumentation and quantum computations.

Acknowledgements N.K. acknowledges M. Kato, V.R. Misko, T. Nojima, S. Okuma, and B. Shinozaki for many helpful discussions. This work was supported by JSPS KAKENHI Grant Numbers 23540416, 26287075, and 26600011, Nanotechnology Network Project of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), and the Inter-​university Cooperative Research Program of the Institute for Materials Research, Tohoku University (Proposal No. 13K0029, 14K0004, 15K00059).

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106  Multi-Vortex States in Superconductors Kes, P.H. (1999) in Physics and materials science of vortex states, flux pinning and dynamics, edited by R. Kossowsky, S. Bose, V. Pan, and Z. Durusoy, Kluwer Academic, Dordrecht, The Netherlands, 109–​136. Kirtley, J.R. and Wikswo, J.P.J. (1999) Annu. Rev. Mater. Sci. 29, 117. Kokubo, N., Okayasu, S., Kanda, A. and Shinozaki, B. (2010) Phys. Rev. B 82, 014501. Kokubo, N., Okayasu, S., Nojima, T., Tamochi, H. and Shinozaki, B. (2014) J. Phys. Soc. Jpn. 83, 083704. Kokubo, N., Miyahara, H., Okayasu, S. and Nojima, T. (2015) J. Phys. Soc. Jpn. 84, 043704. Kokubo, N., Miyahara, H., Okayasu, S. and Nojima, T. (2016) Physica C, doi:10.1016/​j.physc.2016.05.009. Kokubo, N., Mochizuki, R., Okayasu, S., Nojima, T. and Sasaki T. unpublished. Lai, Y.J. and I, L. (1999) Phys. Rev. E 60, 4743. Leiderer, P., Ebner, W. and Shikin, V.B. (1982) Surface Science 113, 405. Lent, C.S., Tougaw, P.D., Porod, W. and Bernstein, G.H. (1993) Nanotechnology 4, 49. Madison, K.W., Chevy, F., Wohlleben, W. and Dalibard, J. (2000) Phys. Rev. Lett. 84, 806. Mangold, K., Birk, J., Leiderera, P. and Bechinger, C. (2004) Phys. Chem. Chem. Phys. 6, 1623. Mertelj, T. and Kabanov, V.V. (2003) Phys. Rev. B 67, 134527. Meyers, C. and Daumens, M. (2000) Phys. Rev. B 62, 9762. Milošević, M.V., Berdiyorov, G.R. and Peeters, F.M. (2007) Appl. Phys. Lett. 91, 212501. Misko, V.R., Fomin, V.M., Devreese, J.T. and Moshchalkov, V.V. (2003) Phys. Rev. Lett. 90, 147003. Misko, V.R., Xu, B. and Peeters, F.M. (2007) Phys. Rev. B 76, 024516. Misko, V.R., Zhao, H.J., Peeters, F.M., Oboznov, V., Dubonos, S.V. and Grigorieva, I.V. (2009) Supercond. Sci. Technol. 22, 034001. Nishio, T., Okayasu, S., Suzuki, J. and Kadowaki, K. (2004) Physica C 412–​414, 379. Nishio, T., An, T., Nomura, A., Miyachi, K., Eguchi, T., Sakata, H., Lin, S., Hayashi, N., Nakai, N., Machida, M. and Hasegawa,Y. (2008) Phys. Rev. Lett. 101, 167001. Nishio, T., Okayasu, S., Suzuki, J., Kokubo, N. and Kadowaki, K. (2008) Phys. Rev. B 77, 052503. Olson Reichhardt, C.J. and Reichhardt, C. (2010) Phys. Rev. B 81, 224516. Palacios, J.J. (1998) Phys. Rev. B 58, R5948. Palacios, J.J. (2000) Phys. Rev. Lett. 21, 1796. Partoens, B. and Peeters, F.M. (1997) J. Phys.: Condens. Matter 9, 5383. Pearl, J. (1964) Appl. Phys. Lett. 5, 65. Plourde, B.L.T., Van Harlingen, D.J., Saha, N., Besseling, R., Hesselberth, M.B.S. and Kes, P.H. (2002) Phys. Rev. B 66, 054529.

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References (Chapter-3) 107 Puig, T., Rosseel, E., Baert, M., Van Bael, M.J., Moshchalkov, V.V. and Bruynseraede, Y. (1997) Appl. Phys. Lett. 70, 3155. Sardella, E., Doria, M.M. and Netto, P.R.S. (1999) Phys. Rev. B 60, 13158. Schweigert, V.A., Peeters, F.M. and Deo, P.S. (1998) Phys. Rev. Lett. 81, 2783. Schweigert, V.A. and Peeters, F.M. (1998) Phys. Rev. B 57, 13817. Tarucha, S., Austing, D.G., Honda, T., van der Hage, R.J. and Kouwenhoven, L.P. (1999) Phys. Rev. Lett. 77, 3613. Thomson, J.J. (1904) Philos. Mag. 39, 236. Thuneberg, E.V. (1989) Cryogenics 29, 236. Tsuei, C.C. (1981) Treatise on Materials Science and Technology 20, 395. Uehara, S. and Nagata, K. (1981) Appl. Phys. Lett. 39, 992. Wei, J. and Yang, T. (1996) Jpn. J. Appl. Phys. 35, 5696. Williams, G.A. and Packard, R.E. (1980) J. Low Temp. Phys. 39, 553. Yarmchuk, E.J. and Packard, R.E. (1982) J. Low Temp. Phys. 46, 479. Zeldov, E., Larkin, A.I., Geshkenbein, V.B., Konczykowski, M., Majer, D., Khaykovich, B., Vinokur, V.M. and Shtrikman, H. (1994) Phys. Rev. Lett. 73, 1428. Zhao, H.J., Misko, V.R., Peeters, F.M., Oboznov, V.S., Dubonos, V. and Grigorieva, I.V. (2008) Phys. Rev. B 78, 104517. Zhao, H.J., Misko, V.R., Peeters, F.M., Dubonos, V., Oboznov, V.S. and Grigorieva, I.V. (2008) Euro Phys. Lett. 83, 17008.

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4 4.1 An introduction to proximity effect  109

Proximity Effect: A New Insight from In Situ Fabricated Hybrid Nanostructures

4.2 In situ fabricated hybrid nanostructures and tunneling spectroscopy  123

J.C. Cuevas1, D. Roditchev2,3, T. Cren2, and C. Brun2

4.3 Proximity effect in a correlated 2D disordered metal 

126

1Departamento

de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-​28049 Madrid, Spainand

4.4 Proximity effect in diffusive SNS junctions 

129

4.5 Proximity Josephson vortices 

131

4.6 Proximity effect between two different superconductors 

135

des Nanosciences de Paris, Université Pierre et Marie Curie (UPMC) and CNRS-​UMR 7588, 4 place Jussieu, 75252 Paris, France

4.7 Conclusions and outlook 

139

3Laboratoire

References (Chapter-​4) 

139

2Institut

de Physique et d’Etude des Matériaux, ESPCI-​ParisTech, CNRS and Université Pierre et Marie Curie (UPMC)-​ UMR 8213, 10 rue Vauquelin, 75005 Paris, France

When a normal metal (N)  and a superconductor (S)  are brought into contact their electronic properties near the interface are modified. In particular, the leakage of Cooper pairs inside the normal metal induces superconducting properties in this material such as a gap in the density of states (DOS) or the ability to sustain a dissipationless electrical current. These phenomena, collectively known as proximity effect, constitute one of the central topics of modern mesoscopic superconductivity. The first study of this effect was already reported in 1932 and it was extensively investigated, both experimentally and theoretically, in the 1960s. Since the early 1990s there has been a renewed interest in this subject due to the possibility to investigate proximity-​related phenomena at much smaller length and energy scales as well as in novel low-​dimensional materials and devices. Thus for instance, proximity superconductivity is being currently investigated in materials like graphene, semiconductor nanowires, and topological insulators, just to mention a few. Moreover, proximity effect lies at the heart of other hot topics such as the realization of triplet superconductivity in the context of superconductor–​ ferromagnet hybrid systems or the study of topological superconductors and the search for Majorana fermions.

J.C. Cuevas, D. Roditchev, T. Cren, C. Brun, ‘Proximity Effect: A New Insight from In-situ Fabricated Hybrid Nanostructures’ in The Oxford Handbook of Small Superconductors. First Edition. Edited by A.V. Narlikar. © Oxford University Press 2017. Published in 2017 by Oxford University Press. DOI 10.1093/acprof:oso/9780198738169.003.0004

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An introduction to proximity effect 109 Very recently, the topic of proximity effect has received yet another new impetus from the field of surface science. It has been shown that the combination of in situ fabricated superconducting nanostructures and scanning tunneling microscopy/​spectroscopy (STM/​STS) techniques enables one to investigate the proximity effect with unprecedented spatial resolution, thus providing a new insight into this fundamental quantum phenomenon. In this chapter, and after a general introduction to proximity effect, we shall review the recent progress on this topic made with the help of in situ fabricated superconducting hybrid nanostructures, focusing mainly on our local STM/​STS studies.

4.1 An introduction to proximity effect 4.1.1 Brief historical review and scope of this chapter The first study of the interaction between a superconductor and a non-​ superconducting metal was reported by Holm and Meissner in 1932 (Holm and Meissner 1932). These researchers studied the electrical resistance of the contact between two metals separated by thin oxide films of both elements. By varying temperature they found that the contact resistance reduces when one of the metals becomes superconducting, a clear signature of the induction of superconductivity in the normal metal. Surprisingly, there do not seem to be new entry points in the literature of proximity effect until the mid-​1950s, when Bedard and Meissner reported a more detailed experimental study of the temperature dependence of the contact resistance between normal and superconducting metals (Bedard and Meissner 1956), which basically confirmed the original observations by Holm and Meissner. Another significant contribution was made by Meissner in 1958 and 1960 (Meissner 1958, 1960). Meissner measured the current–​voltage (I–​V) characteristics of two crossed copper-​coated tin wires forming superconductor–​ normal metal–​superconductor (SNS) sandwiches and found that the resistance between the wires for low currents vanishes for sufficiently low thicknesses of copper. Although it was not recognized back then, this was probably the first observation of the dc Josephson effect, i.e. the observation of a supercurrent in a superconducting weak link, reported two years before Brian Josephson’s seminal paper (Josephson 1962). The 1960s are considered as the first golden era of superconductivity, and it was definitively so also for the topic of proximity effect. Not only were many new experiments reported investigating different aspects of this phenomenon, but the first rigorous theoretical treatments of the proximity effect were developed too (de Gennes 1964, McMillan 1968, Deutscher and de Gennes 1969, Clarke 1968). On the experimental side, there were mainly three different kinds

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110  Proximity Effect: In Situ Fabricated Hybrid Nanostructures of experiments that helped to establish some of the basic concepts. First, the superconducting transition temperature of SN bilayers was extensively investigated. It was shown that the transition temperature of a thin superconducting film deposited onto a normal metal can be significantly lower than that of the bulk superconductor. This demonstrated the suppression of superconductivity by proximity to a normal metal (referred to as inverse proximity effect). Second, the critical current in different SNS sandwiches was measured, showing that it is possible to pass a supercurrent through a normal conductor, provided it is not too thick. These experiments demonstrated that superconductivity can indeed penetrate into a normal metal. Finally, numerous tunneling experiments on SN bilayers were performed in which an oxide layer was grown onto one side of the bilayer and covered with a third metallic layer. The I–​V curves of those tunnel junctions enabled measuring the quasiparticle DOS of the bilayer and showed, in particular, that some degree of superconductivity extends well into the normal metal. These old experiments were reviewed by Clarke (1968). The first theoretical studies of the proximity effect in the early 1960s relied either on the phenomenological Ginzburg–​Landau (GL) equations or on the linearized Gorkov equations in the framework of the microscopic BCS (Bardeen–​ Cooper–​ Schrieffer) theory, which are both only valid close to the critical temperature of the hybrid structures. The early theory of proximity effect based on these approaches was mainly developed by de Gennes and co-​workers and it is reviewed in de Gennes (1964) and Deutscher and de Gennes (1969). As we explain below, this early theory provided the first qualitative picture of how a normal metal and a superconductor are mutually influenced and, in particular, it provided a qualitative explanation of most of the early experiments. In the 1970s the focus on proximity effect changed slightly and researchers concentrated on the study of its influence on the non-​equilibrium properties of hybrid normal–​superconductor structures:  I–​V curves, ac Josephson effect, microwave-​assisted transport, etc. (Likharev 1979). On the other hand, the theoretical framework known as quasiclassical theory of superconductivity came to its full maturity in the early 1970s. This approach, which will play a central role later in this chapter, was originally developed by Eilenberger to describe the equilibrium properties of non-​homogeneous superconductors (Eilenberger 1968). His theory is built upon a simplification of the full Gorkov equations based on the assumption that all relevant variations of superconductivity occur on length scales larger than the Fermi wavelength. Later on this theory was simplified by Usadel to describe disordered (or dirty) superconductors (Usadel 1970). Then, the quasiclassical theory was generalized by Larkin and Ovchinnikov to deal with arbitrary non-​equilibrium problems (Larkin and Ovchinnikov 1968, 1975, 1977), by Eliashberg (1971), and by Schmid and Schön (1975). The 1980s are often considered as a period in which there were no noticeable advancements in the field of proximity effect. However, this field greatly benefited from progress in related topics such as quantum transport in mesoscopic devices (Datta 1995, Imry 1995) and disordered systems (Altshuler and Aronov

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An introduction to proximity effect 111 1985, Lee and Ramakrishnan 1985). During this time, many basic concepts were finally elucidated such as the role of the phase coherence in mesoscopic systems, the importance of correlations between electrons in disordered systems, and the role played by the Thouless energy and the inelastic scattering length. On the other hand, the development of the celebrated BTK (Blonder–​Tinkham–​ Klapwijk) model (Blonder et al. 1982) helped to clarify the role of the Andreev reflection (Andreev 1964) in the physics of hybrid superconducting systems (see Sec. 4.1.2). Moreover, Zaitsev (1984) introduced appropriate boundary conditions to describe interfaces between different materials in the quasiclassical theory of superconductivity, which is a key ingredient in the theory of proximity effect. The 1990s witnessed the revival of proximity effect and since then this topic has become one of the central subfields of superconductivity. The advances in fabrication techniques in this decade helped to take up two basic experimental challenges:  (a)  to pattern hybrid structures at the submicron scale; and (b)  to control SN interfaces. Thanks to these advances, a large variety of SN hybrid structures, including phase-​sensitive devices, were studied, leading to a clarification of the basic proximity mechanism as well as to the discovery of many previously unforeseen effects (Pannetier and Courtois 2000, Klapwijk 2004). Subsequently, the advances in material science and nanofabrication techniques also enabled the exploration of proximity effect in other non-​superconducting material and in novel nanoscale devices. Thus for instance, since the late 1990s a lot of attention has been devoted to the study of the induction of superconductivity in ferromagnets and, indeed, this topic has evolved as a field of its own in the last 15 years (Bergeret et al. 2005, Buzdin 2005, Eschrig 2011, Linder and Robinson 2015). In recent years, proximity effect has also been studied in the context of hybrid quantum dots devices (De Franceschi et al. 2010), where the interplay between superconductivity and phenomena like the Coulomb blockade or Kondo physics can be studied. The discovery of novel carbon-​based structures such as carbon nanotubes or graphene has allowed studying the peculiarities of proximity effect in these materials (Beenakker 2008). On the other hand, the development of semiconductor nanowires and the discovery of two-​dimensional and three-​dimensional (2D and 3D) topological insulators are having a tremendous impact in the field of proximity effect mainly due to the possibility of creating and manipulating Majorana fermions in proximity structures (Beenakker 2013). The fact that there are still many open problems in the context of new hybrid systems along with the frequent discovery of new low-​dimensional materials will surely keep the topic of proximity at the forefront of research in superconductivity for many years. In this chapter we shall mainly focus on the proximity effect between conventional low-​TC superconductors and diffusive metals, including a discussion of the proximity effect between two dissimilar superconductors. We shall concentrate on the study of how the electronic structure of a metal is altered by proximity to a superconductor and we shall not address transport properties. Probably, the main quantity that reflects the proximity effect is the local

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112  Proximity Effect: In Situ Fabricated Hybrid Nanostructures DOS, and for this reason researchers have developed different techniques over the years to have access to this crucial information in hybrid superconducting structures. As mentioned above, such studies started in the 1960s with the help of planar tunnel junctions formed with thin oxide barriers. They accessed some energy features related to the proximity effect, but completely missed the spatial aspects of the phenomenon. In the 1990s Guéron et al. (1996) were able to incorporate tunneling probes in a normal wire connected to a superconductor and to measure the local DOS at different distances from the SN interface, which constituted the first spatially resolved experimental study of proximity effect. Since then, and in order to improve the spatial resolution of these tunneling experiments, different groups have employed STM/​STS (Vinet et al. 2001, Moussy et al. 2001, Escoffier et al. 2004, le Sueur et al. 2008, Wolz et al. 2011). However, there are intrinsic difficulties in applying STM/​STS techniques to ex situ fabricated mesoscopic systems, which are mainly due to surface contamination. This experimental hurdle has been recently overcome by borrowing the surface science methods to build atomically clean superconducting materials and proximity devices under ultrahigh vacuum and by integrating STM/​STS in the same ultrahigh vacuum environment. This has enabled probing the proximity effect in situ in STM or STS experiments with high spatial and energy resolution (Kim et  al. 2012, Serrier-​Garcia et  al. 2013, Cherkez et  al. 2014, Stepniak et  al. 2015, Roditchev et  al. 2015). In this chapter we review these recent experiments, which help to resolve long-​standing problems and discover new striking phenomena.

4.1.2 Proximity effect in diffusive metals: a qualitative description In this subsection we provide a qualitative and concise summary of the modern view of proximity effect in SN structures and introduce the relevant length and energy scales of this phenomenon. But before doing that, it is convenient to remind ourselves of the view that emerged in the 1960s (de Gennes 1964, Deutscher and de Gennes 1969). The degree of superconductivity in a normal metal is described   (r ) is a  (r )ψ  (r ) , where ψ by means of the condensation amplitude F (r ) = ψ ↑ ↑ ↓ spin-​up electron annihilation operator. The modulus square of this amplitude,   | F (r )|2 , is essentially the probability density of finding a Cooper pair at r and represents the superfluid density. In this notation, the pair potential, which plays the    role of a space-​dependent order parameter, is given by ∆(r ) = V (r )F (r ), where  V (r ) is the pairing interaction constant. This constant is positive inside a superconductor (attractive interaction between the electrons) and it vanishes inside a  normal metal. Thus, ∆(r ) reduces to the BCS order parameter in the bulk of a superconductor and it vanishes in a normal metal, although the superconducting  correlations described by F (r ) remain finite close to the interface with a superconductor due to the leakage of Cooper pairs.

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An introduction to proximity effect 113 The first quantitative description of the proximity effect was based on the linearized Gorkov equations, which are only valid close to the critical temperature of the proximity structure (de Gennes 1964). The qualitative results for the magnitude of the condensation amplitude for a SN and a SNS junction are summarized in Fig. 4.1. In the SN case (see Fig. 4.1a), one can see that the Cooper pair leakage through the SN interface leads to a partial suppression of the superconducting correlations inside the S electrode (inverse proximity effect), while it induces the appearance of superconducting correlations inside N over a certain distance. In the case of diffusive metals, it was found that the correlation amplitude inside N decays exponentially with a characteristic decay length given by LT = DN / (2πkBT ), where T is the temperature and DN is the diffusion constant in the normal metal, which is given by DN = (1 / 3)vF , N  N , where vF , N is the Fermi velocity and  N the elastic mean free path. The length LT was termed as coherence length, but nowadays is more correctly referred to as thermal length (see discussion below). In the case of a SNS junction (see Fig. 4.1b), if the length of the junction L is smaller than the thermal length, it was concluded that superconductivity can be induced throughout the normal wire and the junction can sustain a supercurrent that decays exponentially with the junction length, having LT as the characteristic decay length. This appealing picture of proximity effect is, however, incomplete for several reasons. First, it does not clarify the charge transfer mechanism across the SN interfaces. Second, it does not really identify the most fundamental length in this problem, namely the true coherence length (see below). Finally, it does not tell us how the proximity effect modifies the electronic structure of normal metals, a central issue in this chapter. In the modern view of proximity effect, which emerged in the 1990s (Pannetier and Courtois 2000, Klapwijk 2004), this phenomenon is intimately connected to the concept of Andreev reflection (Andreev 1964). The idea goes as follows. The induction of superconductivity in a normal metal connected to a superconductor requires:  (a)  the leakage of Cooper pairs through the SN interface, which occurs via the Andreev reflection; and (b)  the propagation of electron pairs in the normal metal. Moreover, it is (a)

(b)

|F(x)|2

S

N

ds

dN

|F (x)|2

S

N

x

S

x dN

Fig. 4.1  Qualitative spatial dependence of the magnitude of the condensation amplitude at temperatures close to the transition temperatures in a SN sandwich (a) and in a SNS junction (b).

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114  Proximity Effect: In Situ Fabricated Hybrid Nanostructures worth stressing that these two processes do not necessarily take place sequentially, but as usual in quantum coherent transport, they influence each other and often it is not possible to clearly separate them. As illustrated in Fig. 4.2a, Andreev reflection is a tunneling process in which an electron coming from N with an energy E (measured with respect to the Fermi energy) is converted into a reflected hole of energy –​E, thus transferring a Cooper pair of zero energy to the S electrode. Alternatively, the Andreev reflection can be seen as the process in which a Cooper pair tunnels into N and becomes a pair of time-​reversed electron states. This tunneling process governs the charge transfer in a SN junction for energies below the superconducting gap in S, which are indeed the relevant energies for proximity effect, as will become clear in this chapter. The two time-​reserved electron states involved in an Andreev reflection diffuse inside the normal metal and preserve their coherence over a certain distance (see Fig. 4.2a), which can be roughly determined as follows (Guéron 1997). Since the energy difference between the electron and hole states is 2E , the relative amplitude of these two states dephases in time as the pair propagates in the normal metal by a factor exp( −2iEt /  ), where t is the time elapsed after the pair has left the superconductor. A dephasing of order 1 is obtained for t ≈  / E . During this time, the electrons diffuse over a distance of the order of L E = DN t = DN / E , where DN is the diffusion constant in the normal metal. Thus, L E = DN / E is the coherence length, i.e. the typical energy-​dependent length at which pair correlations decay in the normal metal. Thus for instance, the coherent propagation in N of two states with the energy E = ∆ will extend over a characteristic length scale ξ = DN / ∆ . For states with a smaller energy the propagation of superconducting correlations extends further; it becomes theoretically infinite for the two states with zero energy. However, even for these states the maximum coherence length is limited by other dephasing processes such as spin-​flip scattering by magnetic impurities or inelastic interactions. In other words, the actual coherence length is limited by the phase-​breaking length Lφ of a single electron (see Fig. 4.2a).

(a)

(b) e

S mean free path ℓN coherence length LE =

Fig.  4.2 Relevant length and energy scales in (a)  a SN sandwich and (b)  a SNS junction, where N is a diffusive metal.

S

ℓN

N

S

ℓN

L (junction length) LE

LE

Thouless energy

h DN

ETh =

E

thermal length LT =

N

h

LT

LT

h DN

2πkBT phase-breaking length Lϕ

Lϕ

Lϕ

hDN 2

L

  115

An introduction to proximity effect 115 From the discussion above, there seems to be some kind of contradiction between the old and the new pictures of proximity effect. What is the actual relevant length scale, LT or L E ? The thermal length LT is indeed a characteristic of a thermal equilibrium electron distribution and in this sense, it appears in properties that are sensitive to the occupation of the electron states such as the supercurrent (Clarke 1969, Dubois et al. 2001). However, this length does not directly determine the extent of the penetration of the superconducting correlations inside the normal metal. That extent is determined by the coherence length L E , as will become clear later on. In particular, L E is the length scale that is revealed by tunneling experiments that probe the local DOS. To conclude this discussion, let us introduce another relevant energy scale in the proximity effect in diffusive metals, namely the Thouless energy, E Th . For this purpose, let us consider a SNS junction where the normal wire has a length L. From our discussion above, it is clear that at a given energy E, the superconducting correlations will extend throughout the wire as long as L < L E , or equivalently, as long as E < E Th = DN / L2. In other words, at a given distance L, only electrons with energies below the Thouless energy E Th are still correlated in pairs. Let us recall that the Thouless energy is also related to the inverse of the escape time: the time necessary for a quasiparticle to exit the N wire.

4.1.3 Quasiclassical theory of superconductivity The most complete theoretical framework for the description of proximity effect is provided by the quasiclassical theory of superconductivity. This approach simplifies the full microscopic Gorkov equations of the BCS theory by assuming that the spatial variations of superconductivity occur on a scale larger than the Fermi wavelength (Eilenberger 1968). Thus, this theory ignores atomic-​scale spatial changes of the electronic structure, but it is able to completely describe the electron–​hole coherence that is responsible for the spatial variations of the superconducting correlations. The great advantage of this theory, as compared to other approaches like the Bogoliubov–​de Gennes equations (de Gennes 1966), is the fact that it is able to deal with impurity scattering and with inelastic interactions such as electron–​electron and electron–​phonon interactions. In this subsection, and since we are interested in disordered systems, we shall briefly describe the Usadel equations (Usadel 1970) that summarize the quasiclassical theory of superconductivity in the diffusive (or dirty) limit, where the mean free path is smaller than the superconducting coherence length. This approach is reviewed in Larkin and Ovchinnikov (1986) and Belzig et al. (1999). In the Usadel approach, all equilibrium properties  are described in terms of a ˆ ( R , E ), which depends on posiG momentum-​ a veraged retarded Green function  tion R and energy E.This propagator is indeed a 2 × 2 matrix in electron–​hole space

116

116  Proximity Effect: In Situ Fabricated Hybrid Nanostructures g Gˆ =   f



f . g 

(4.1)

The general definitions of the different functions can be found in Serene and Rainer (1983). It is sufficient for our discussion here to know that while the diagonal elements of Gˆ describe the quasiparticle states, the off-​diagonal ones contain the information on the superconducting correlations. Thus for instance,  the local DOS,  normalized by its value in the normal state, is given by ρ( R , E ) = − Im g( R , E ) / π , while the imaginary part of f determines the condensation or, more precisely, the correlation amplitude, see discussion after Eq. 4.4 below. Our goal here is to describe the local DOS  in a hybrid superconducting structure. In such a structure, the propagator Gˆ ( R , E ) satisfies the stationary Usadel equations, which read

{



}

(

)

D ∇ Gˆ ∇Gˆ +  E τˆ 3 + ∆ˆ + Σˆ sf + Σˆ in ,Gˆ  = 0, (4.2) π

where τˆ3 is the Pauli matrix in electron–​hole space, Σˆ sf is a self-​energy describing spin-​flip scattering (due for example to magnetic impurities), Σˆ in is a self-​energy describing inelastic interactions, and ∆ˆ is given by

  0 ∆( R ) ∆ˆ =  ∗   , (4.3) 0   ∆ (R )

 where ∆( R ) is the space-​dependent order parameter that needs to be determined self-​consistently via the following equation:



  εC dE  E  ∆( R ) = λ ∫ Im f ( R , E ) tanh  (4.4) − εC 2π  2kBT 

{

}

Here, T is the temperature, εC is a cut-​off energy (basically the Debye energy), and λ is the electron–​phonon coupling constant. This latter constant is often written as λ = VN(0), where V is the pairing interaction constant and N(0) is the normal DOS at the Fermi energy. In the normal parts of the proximity structure λ is zero and the order parameter vanishes. Of course, the superconducting correla tions, which are described by the anomalous Green function f ( R , E), are different from zero in the normal parts. Let us remark that the quantity ∆( R ) / V is simply the correlation amplitude F ( R ) introduced above. The presence of a magnetic field in the structure can be described with  Eq. 4.2 ˆ − (ie /  )Aτˆ , where by replacing the gradient ∇ by the gauge-​ i nvariant gradient ∇ 1 3  A is the vector potential. On the other hand, Eq. 4.2 must be supplemented by the normalization condition Gˆ 2 = − π 2 . In order to solve numerically the Usadel equations, it is necessary to use a convenient parameterization that automatically

  117

An introduction to proximity effect 117 takes into account the normalization condition. Moreover, in order to describe interfaces between different materials within the quasiclassical theory, one needs to supplement the Usadel equations with appropriate boundary conditions for the Green functions. For a detailed discussion of these technical issues, we refer to Hammer et al. (2007) and Cherkez et al. (2014) and references therein.

4.1.4 Proximity density of states: some representative examples In this subsection we shall make use of the quasiclassical theory to illustrate how the proximity effect is reflected in the local DOS of a normal wire connected to one or several superconductors. In particular, we shall address situations that will be discussed in the following sections in the context of in situ fabricated structures. Since we are mainly interested in the proximity-​induced modification of the DOS in normal systems, we shall assume that the superconductors are infinite and behave as ideal reservoirs (no inverse proximity effect) and all of them will be identical and characterized by a constant energy gap ∆. Moreover, for simplicity, we shall ignore spin-​flip mechanisms and inelastic interactions. We shall see that these interactions do not need to be taken into account to explain the observed behavior in the experiments discussed below. In what follows, lengths and position will be normalized by ξ = DN / ∆ , which provides a natural energy-​independent length scale for the superconducting coherence inside a normal wire. In terms of this scale, if the length of a normal wire L is larger than ξ, it means that the Thouless energy E Th = DN / L2 is smaller than ∆. Conversely, if L < ξ , then E Th > ∆. 4.1.4.1 SN bilayer As a first example, let us consider the case of a normal layer (N)  of thickness L connected to a semi-​infinite superconductor (S). We assume here that the SN interface has a negligible resistance, i.e. it is perfectly transparent. This problem is effectively one-​dimensional and we denote the distance from a point inside the N layer to the SN interface as x. In Fig.  4.3a we show the spatially resolved DOS in the N layer as a function of energy for L = 4ξ. As one can see, the DOS evolves continuously from the standard BCS DOS at the SN interface to a smooth DOS inside the N layer that has no singularities. The width of the energy interval in which the DOS is modified at a given distance is roughly given by min ∆ / ( x / ξ )2 = DN / x 2 , ∆ , which is consistent with the idea that

{

}

L E = DN / E is the actual coherence length in the proximity effect. Moreover, notice that the local DOS exhibits a minigap ∆ g < ∆ that remains constant throughout the N layer. The magnitude of this minigap decreases as the layer thickness increases, as we show in Fig. 4.3b where we display the DOS at the end of the N layer for different layer thicknesses. In fact, within this simple model the minigap decays with thickness as ∆ g ≈ 0.765E Th in the long junction limit (L >> ξ ), i.e.

118

118  Proximity Effect: In Situ Fabricated Hybrid Nanostructures (a)

(b)

4

2 2 1 0 –2 –1.5

x/ξ

3

DOS (x = L)

4

Fig.  4.3  (a)  Local DOS inside N as a function of energy and position for a SN bilayer where N is a diffusive layer of thickness L = 4ξ . (b) Local DOS as a function of energy at the end of the normal layer in a SN bilayer for different values of layer thickness.

1 –1 –0.5

0 E/∆

0.5

1

1.5

2

L = 0.0 L = 1ξ L = 2ξ L = 3ξ L = 4ξ L = 5ξ

5

0

3 2 1 0 –1.5

–1

–0.5

0

0.5

1

1.5

E/∆

it is determined by the Thouless energy, which again confirms the basic picture described in Sec. 4.1.2. The existence of a minigap in a diffusive normal metal in contact with a superconductor was first predicted by McMillan (1968) within a tunneling model, where the normal region was a thin layer and the spatial variation of the superconducting correlations was ignored. In the 1990s, the minigap was extensively investigated in various hybrid diffusive SN and SNS structures (Golubov and Kupriyanov 1988, Belzig et al. 1996, Zhou et al. 1998). From the experimental point of view, the appearance of a minigap has been tested indirectly in several tunneling experiments, see for instance Scheer et al. (2001) and Rubio-​Bollinger et al. (2003) and references therein, and very directly in recent STM experiments of SN bilayers (Wolz et al. 2011). 4.1.4.2 SN junction (infinite N wire) We again consider a SN junction with a perfect interface, but this time both the S and N electrodes are semi-​infinite. We show the corresponding spatially resolved DOS inside N in Fig. 4.4 in two different ways. As one can see, there is a strong suppression of the local DOS below the superconducting gap and again, the DOS at a certain distance differs from that of the normal state when the magnitude of the energy is smaller than min ∆ / ( x / ξ )2 = DN / x 2 , ∆ . Notice that in this case there is no minigap and the DOS only vanishes at the Fermi energy. In this situation, any phase-​breaking mechanism or inelastic interaction would induce a finite zero-​energy DOS (Belzig et al. 1996). These predictions were experimentally verified by Guéron et al. (1996) by incorporating tunneling probes into a SN junction at different distances from the interface.

{

}

4.1.4.3 SNS junctions (no magnetic field) We consider now a SNS junction where N is a diffusive wire of length L. We display in Fig. 4.5a the spatially resolved DOS inside the N wire for L = 4ξ assuming perfect interfaces. The most notable feature is the appearance in the spectrum of a minigap ∆ g < ∆ in the spectrum that is constant throughout the wire. Again, this

  119

An introduction to proximity effect 119 (a)

(b) 2.5

10

10 9 8 7 6 5 x/ξ 4 3 2 1 0

2 1 0 –2 –1.5

–1 –0.5

0 0.5 E/∆

1

1.5

2

x/ξ

9 8 7

2

6

1.5

5 1

4 3

0.5

2 1 0 –1.5

–1

–0.5

0

0.5

1

1.5

0

E/∆

Fig.  4.4  (a)  Local DOS inside N as a function of energy and position for a SN junction where both S and N are semi-​ infinite. (b) The same as in (a), but in a different representation.

minigap becomes progressively smaller as the wire length increases, see Fig. 4.5b. In the long junction limit when L >> ξ (or E Th > 1. As we show in Fig. 4.5d for a junction with L = 4ξ, the minigap is quite sensitive to the interface quality and it diminishes progressively as the parameter r increases, i.e. as the interface becomes more opaque. For this particular example, the minigap decays roughly as 1 / r; see Hammer et al. (2007) for a more detailed discussion of this issue. 4.1.4.4 SNS junctions under a magnetic field Let us now discuss the influence of an applied magnetic field in the local DOS of a SNS junction. By following Cuevas and Bergeret (Cuevas and Bergeret 2007, Bergeret and Cuevas 2008) we assume that: (a) the diffusive SNS junction is sub ject to a magnetic field H applied in the z-​direction perpendicular to the normal wire, (b) the wire is thin enough so that the magnetic field completely penetrates inside, (c) the magnetic field does not penetrate inside the superconducting electrodes (i.e. we assume a superconductor having zero penetration depth), and (d)  we have ideal SN interfaces. The wire length (x-​direction) will be denoted by L, x ∈[0, L ], while W will denote its width (y-​direction), y ∈[0,W ]. For an arbitrary aspect ratio, this problem is quite complex because, contrary to all the previous cases discussed above, the superconducting correlations can also vary spatially along the transverse y-​direction (y-​coordinate) and thus, the problem becomes effectively 2D. The characteristic length scale for this transverse modulation is given by the magnetic length ξ H = Φ 0 / H , where Φ 0 is the flux quantum. Thus, if the field is sufficiently high that ξ H < W , then the DOS will also change along the y-​direction. In Fig. 4.6 we show the spatially resolved DOS at the Fermi energy inside a SNS junction for L = 4ξ and W = 4L and different values of the magnetic flux Φ = HLW enclosed in the N wire. The first thing to notice is that, as anticipated, the local DOS is strongly modulated by the magnetic field and the modulation becomes more pronounced as the field increases. In particular, there appear regions in the middle of the wire (x = L / 2) where the proximity superconductivity is completely suppressed. As demonstrated by Cuevas and Bergeret, this is the signature of the appearance of a line of Josephson vortices in the middle of the normal metal with true vortex cores in which superconducting

  121

An introduction to proximity effect 121

4 3 2 1 0

x/ξ 2

4

6

8

10

12

14

16

(c) Φ = 2.5Φ0

0

2

4

6

8

10

12

14

16

(e) Φ = 3.5Φ0

0

2

4

6

4 3 2 1 0 0

1 0.5

2

4

6

8

10

12

14

16

(d) Φ = 3.0Φ0

x/ξ

4 3 2 1 0

(b) Φ = 2.0Φ0

x/ξ

x/ξ

x/ξ

x/ξ

(a) Φ = 1.5Φ0

4 3 2 1 0 0

8 y/ξ

10

12

14

16

4 3 2 1 0 0 4 3 2 1 0

0

1 0.5

2

4

6

8

10

12

14

16

(f) Φ = 4.0Φ0

0

1 0.5

0

2

4

6

8

10

12

14

16

0

y/ξ

correlations are suppressed, in analogy with suppression of the superconducting order in Abrikosov vortex cores. Notice that those cores are separated by regions where the minigap shows up and resembles that in the absence of magnetic field. The effect can be better appreciated in the DOS spectra that we show in Fig. 4.7 where we display the energy dependence of the DOS along the y-​coordinate in the middle of the wire ( x = L / 2). In this case, the minigap appears as black spots centered about zero energy and separated by lines where the DOS becomes exactly that of the normal state. The structure of the Josephson vortices in N and corresponding modulation of the local DOS are easy to understand in the wide junction limit (W >> L ). In this case, one can show that the role of the magnetic field is to change the superconducting phase difference, φ , between the two S parts into the gauge-​invariant combination ϕ = φ − 2π( Φ / Φ 0 )( y / W − 1 / 2) with y ∈[0,W ]. More importantly, one can show that in this limit the pair correlations (or correlation amplitude) vanish at points given by:  x = L / 2 and ϕ = (2m + 1)π, where m = 0, ±1,... This means that there are vortex cores located exactly on the middle of the wire forming a regular linear array along the y-​direction. As usual, the number of vortices is determined by the number of flux quanta enclosed in the junction and the vortex cores are separated by a distance ( Φ 0 / Φ )W . On the other hand, the structure of the DOS is simply determined by the local value of the gauge-​invariant phase ϕ, which depends on the y-​coordinate. The vortex cores appear at those points at which ϕ is an odd multiple of π, where the minigap vanishes, as we showed in the previous example. Furthermore, the minigap is fully recovered when ϕ is zero or a multiple of 2π , in agreement with our discussion above of the phase dependence of the DOS in a SNS junction. These arguments nicely explain the structure observed in the local DOS in Fig. 4.6 and Fig. 4.7. Let us also say that, apart from the cores, these proximity Josephson vortices share other properties with standard

Fig.  4.6  (a–​f) Spatially resolved DOS at the Fermi energy (E  =  0) in a SNS junction where N is a diffusive wire of length L = 4ξ and width W = 4 L . The magnetic field is considered to penetrate only the N-​part of the junction; it is fully screened in S-​parts. The different panels correspond to different values of the magnetic flux enclosed in the junction. The white regions correspond to regions where the N wire becomes normal, indicating the appearance of vortex cores.

122

122  Proximity Effect: In Situ Fabricated Hybrid Nanostructures (a) Φ = 1.5Φ0

y/ξ

1

0.5

–1

–0.5

(c) Φ = 2.5Φ0

0 E/∆

0.5

1

0.5

–0.5

16 14 12 10 8 6 4 2 0 –1.5

0 E/∆

0.5

1

1 0.8 0.6 0.4 –0.5

0 E/∆

1.2 1 0.8 0.6 0.4 –1

–0.5

16 14 12 10 8 6 4 2 0 –1.5

0.5

1

16 14 12 10 8 6 4 2 0 –1.5

0 E/∆

0.5

1

0.2

1.5 1

0.5

–1

–0.5

(f) Φ = 4.0Φ0

1.2

–1

1.4

0 E/∆

0.5

1

1.2 1.1 1

y/ξ

–1

y/ξ

y/ξ

1

(e) Φ = 3.5Φ0

16 14 12 10 8 6 4 2 0 –1.5

(d) Φ = 3.0Φ0

1.5

y/ξ

Fig. 4.7  (a–​f) Local DOS as a function of energy and transversal coordinate y in the middle (x = L/​2) of a SNS junction where N is a diffusive wire of length L = 4 ξ and width W = 4 L . The different panels correspond to different values of the magnetic flux Φ = HLW enclosed in the junction. The black regions correspond to regions where a minigap is present in the N wire. Those regions are separated by places where the DOS is that of the normal state and they correspond to the vortex cores forming a line in the middle of the wire.

16 14 12 10 8 6 4 2 0 –1.5

(b) Φ = 2.0Φ0

1.5

y/ξ

16 14 12 10 8 6 4 2 0 –1.5

0.9 0.8 0.7 –1

–0.5

0 E/∆

0.5

1

Abrikosov vortices. Thus for instance, there are circulating current loops around the cores, and the phase of the pair correlation changes by 2π around the cores. However, they also have clear differences with Abrikosov vortices such as the size, the shape, or the arrangement. These issues will be further discussed in Sec. 4.5 when we present the first experimental observation of this novel type of vortices. It is worth stressing that, as shown by Cuevas and Bergeret, in the opposite limit of a narrow junction (W min  e ,  e ′ , but it is still much smaller than the diffusion length for inelastic  

Fig.  5.3 The regimes of conduction through a point contact between two different metals M and M′. (a)  Ballistic regime; (b) intermediate (diffusive) regime (c) Maxwell (thermal) regime.

150

150  Andreev Reflection and Related Studies scattering Λ =  i  e , that is, a 3–​4) and 1 (when K ≤ 0.02) and thus its correction to RM is important only when  e  ≥ d. Of course, in this situation the backflow of electrons preserving energy-​resolved information—and thus the intensity of the spectroscopic signal (∝ − d 2 I (V ) dV 2 )— are strongly reduced by elastic scattering in the contact region (see Fig. 5.3b). This means that, in this intermediate regime, spectroscopy is not forbidden but it is usually more difficult to extract good spectroscopic information from the PC curves. Finally, when a is so large that it widely exceeds the inelastic mean free path, that is, a >> min  i ,  i ′  , the contact enters the thermal (or Maxwell) regime in   which electrons can undergo inelastic scattering processes in the contact region (see Fig. 5.3c) and, as a consequence, the spectroscopic information is completely lost. Further details can be found in Naidyuk and Yanson (2004) and Daghero and Gonnelli (2010).

5.2 Andreev reflection in a nutshell Up to now we have discussed the conditions under which PC spectroscopy is possible in normal metals. When one makes a PC on a superconductor, other quantum physical processes occur at the interface, namely Andreev reflection, that can give quantitative (energy-​resolved) information on the superconducting energy gap, provided that such information is not lost due to inelastic scattering events in the contact. Therefore, the aforementioned spectroscopic conditions must still be fulfilled. In the following we will describe Andreev reflection and the models used to extract information about the gap from the I–​V characteristics of a PC.

5.2.1 The physics of Andreev reflection at a N/​S interface Let us consider a PC between a normal metal (N) and a superconductor (S), as depicted in Fig. 5.4a. Let us call x the axis normal to the interface, with the origin

  151

Andreev reflection in a nutshell 151 (a)

Normal metal

Superconductor ℓ, ξ

e–

e–

h+

h+

2e–

N S E

(b) e– eV2 > ∆ e– eV1

2e–

∆

h+ N (E)

U (x) = U 0δ (x) = 0

on the S/​N boundary, and model the potential barrier by a repulsive potential U0δ(x). Just for simplicity, let us make the following assumptions (that do not affect the general physical description): (a) quasiparticles orthogonally cross the interface, that is, the whole process is unidirectional; (b)  there is no potential barrier at the interface (i.e. U0 = 0 so there is a direct contact between N and S); (c) the temperature is zero; and (d) the Fermi velocities in N and S are equal, that is, vF , N = vF , S and τ = 1. Under these conditions, when a bias voltage is applied across the N/​S interface the electrons (quasiparticles) in the normal side would be driven by the electric field to cross the interface. However, in the S side the quasiparticles exist only at energies above the superconducting energy gap Δ, while below Δ only Cooper pairs exist with charge 2e and total spin equal to zero (for conventional singlet superconductors). So, quasiparticles can be transmitted in S through the interface (as happens in a PC between metals) only when they gain an energy eV > Δ. Let us call C(eV) the probability of this process, that is zero for V < ∆ e and rapidly increases at V > ∆ e , reaching 1 at V ≈ 4∆ e . For V < Δ/​e, however, the current through the interface is not zero (as it would be in a tunnel junction with large U0) thanks to Andreev reflection: an electron coming from N can be reflected as a hole in N, while a Cooper pair is transmitted in S (see Fig. 5.4a, b). In this process the total charge, momentum, and spin of the involved particles are conserved across the interface and thus (if the Cooper

Fig.  5.4 (a)  Scheme of a point contact (in the needle-​anvil configuration) between a normal tip (N)  and a superconductor (S), showing the phenomenon of Andreev reflection from the point of view of the quasiparticles’ trajectories. (b) The phenomena occurring at a direct N/​S interface depicted from the point of view of energy. In both (a) and (b) white arrows refer to electronlike quasiparticles, grey arrows to holelike quasiparticles, and black arrows to Cooper pairs.

152

152  Andreev Reflection and Related Studies pairs are singlets) the hole reflected in N has a spin opposite to that of the incoming electron and almost an opposite wavevector, because of the tiny difference in their energy, as shown in Fig. 5.4b (Heslinga et al. 1994). The hole traces back the incoming electron trajectory until an inelastic scattering event occurs in the N bank, as shown in Fig. 5.4a. The probability of this process is called A(eV) and, if U0 = 0 and T = 0, is equal to 1 for V ≤ ∆ e while it decreases to zero for V > ∆ e . The main consequence of the Andreev reflection process is that, when V ≤ ∆ e V, for each incoming electron of charge e a Cooper pair of charge 2e propagates in S, that is, the conductance of the junction is doubled with respect to the case where only conventional transmission occurs. It can be shown that the normalized conductance of the junction (dI NS / dV ) (dI NN / dV ) (where dI NN / dV is the conductance when both the electrodes are normal) is simply given by 1 + A(eV), thus varying from 2 (when V ≤ ∆ e ) to about 1 (when V ≈ 4∆ e ). In this ideal, simplified case the energy gap of the superconductor can be determined by simply detecting the bias voltage at which the normalized conductance begins to decrease below the value of 2 measured at low bias. In real experimental conditions most of the previous assumptions have to be relaxed, but the physics remains the same. For example, at finite temperature T ≠ 0 there is a smoothing and rounding of the gap edge in the quasiparticle density of states of the superconductor (the continuous curve in the right side of Fig. 5.4b) which, in turn, produces a smoothing of the normalized conductance. The result is that, even if U0 = 0, the normalized conductance is smaller than 2 at energies slightly below the gap voltage ∆ e, as shown in Fig. 5.5a. The effect of the removal of the other assumptions will be discussed in the next subsection.

5.2.2 The Blonder–​Tinkham–​Klapwijk (BTK) model and beyond When a potential barrier U0 ≠ 0 exists at the N/​S interface the physical situation becomes more complex. The usual way to describe it consists in introducing an adimensional parameter Z proportional to the repulsive potential, Z = U 0 vF , where vF is the Fermi velocity in the normal metal (supposed to be equal to the one in the superconductor in our present simplified analysis). This parameter easily accounts for the transparency of the interface in the normal state since the transmission coefficient is simply given by τ N = 1 1 + Z 2 (note that here we are still under the hypothesis of a unidirectional AR process and that this τ N has nothing to do with the previously defined τ which can arise from the mismatch of the Fermi velocities in N and S). For Z = 0 the interface is perfectly transparent in the normal state (τ N = 1) and this is the situation described in Sec. 5.2.1. The opposite limit is Z → ∞ and thus τ N → 0, which corresponds to a S/​I/​N tunnel junction. The intermediate situation, when 0 < Z < 1, is the most typical in PCARS experiments on clean or freshly cleaved S surfaces. In this case the

(

)

  153

Andreev reflection in a nutshell 153 electrical transport depends also on a term that describes the probability of normal specular reflection, usually called B(eV). In a seminal paper that appeared in 1982, G.E. Blonder, M. Tinkham, and T.M. Klapwijk were able to calculate for the first time (by solving the Bogoliubov–​de Gennes equations at an N/​S interface) the coefficients A(eV, Δ, Z), B(eV, Δ, Z), C(eV, Δ, Z), and an additional term D(eV, Δ, Z) that is always very small and corresponds to the probability, present only at eV > ∆ and Z > 0, of transmission of an electron with reflection of a hole in N at eV < −∆ (Blonder et al. 1982). In the same paper the authors calculated the current flowing through the interface and, thus, the conductance of the N/​S junction that turns out to be: dI NS (eV ,T ) d = 2N (0)evF Seff ⋅ dV dV

∫

+∞ −∞

 f ( E − eV ,T ) − f ( E ,T ) [1 + A( E ) − B( E )] dE (5.4)



where f ( E ,T ) is the Fermi function at finite T, N(0) and vF are the density of states at the Fermi level and the Fermi velocity in the normal metal, respectively, and Seff is an effective cross-​sectional area for the electron transport. For example, in the case of a PC modeled as an orifice of radius a, Seff = π a 2 4. In Eq. 5.4, for simplicity of notation, we dropped the explicit dependence on Δ and Z in A(E) and B(E). When the conductance of Eq. 5.4 is divided by the conductance of the junction with the S electrode in the normal state (obtained, for example, in a magnetic field higher than the critical one) we obtain the normalized conductance (dI NS / dV ) (dI NN / dV ) introduced in the previous subsection. The normal-​state conductance at T = 0 is simply given by dI NN / dV = 2N (0)e 2vF Seff τ N and has a nonzero value even when Z = 0, essentially being the Sharvin conductance of the PC defined in Sec. 5.2.1. As a consequence of what we said before, the theoretical normalized conductance of a PC made on a conventional, single-​band, s-​wave superconductor (singlet state) is a function of T, Z, and Δ. In the three panels of Fig. 5.5 this dependence is illustrated in detail. In Fig. 5.5a we show how the normalized AR conductance for Δ(0) = Δ(T = 0) = 2 meV and Z = 0 evolves at the increase of T. The gap value is here supposed to have a Bardeen–​Cooper–​Schrieffer (BCS) temperature dependence ΔBCS(T) with Tc = 13.2 K. It is clear that the progressive gap reduction and thermal broadening of the curves transform the original plateau at eV < ∆ into a broad zero-​bias peak of decreasing amplitude that finally disappears at Tc. Fig. 5.5b illustrates the effect of the Z parameter on the normalized conductance at T  =  0 for Δ(0)  =  2 meV. As previously mentioned, when Z ≠ 0 the conductance depends on both the AR probability A(E) and the normal reflection probability B(E). On increasing Z the plateau at eV < ∆ transforms into a pair of peaks at V ≈ ∆ e which become progressively more and more pronounced at the increase of Z. When Z ≥ 5 the normalized conductance is almost identical to the tunneling conductance of a N/​I/​S junction which, at T = 0 (as in Fig. 5.5b), corresponds to the BCS quasiparticle density of states of the superconductor. In most of the actual PCARS experiments, Z ranges between 0.1 and

154

154  Andreev Reflection and Related Studies

Normalized conductance

(a)

(b)

2.2

T (K) 0 3.0 6.0 9.0 11.0 12.0 12.5 13.0 13.2

∆(0) = 2 meV 2.0 Z=0 1.8 Tc =13.2 K 1.6 1.4 1.2 1.0

(c)

∆(0) = 2 meV T=0K

Z 0 0.1 0.2 0.3 0.4 0.5

T=2K Z = 0.1

∆ (meV) 2.0 1.5 1.0 0.5 0.2

0.8 0.6 0.4 –10

–5

0 5 Voltage (mV)

10 –10

–5

0 5 Voltage (mV)

10 –10

–5

0 5 Voltage (mV)

10

Fig.  5.5  Effects of different parameters on the shape of the PCARS spectra calculated within the BTK model. (a)  Effect of the temperature. A  zero-​temperature gap amplitude Δ(0)  =  2 meV has been used, and the barrier parameter Z has been set to zero. A  BCS temperature dependence of the gap amplitude has been assumed, i.e.

(

)

∆( T ) = ∆(0)tanh 1.74 (Tc / T) − 1 . (b) Effect of the barrier parameter Z. The temperature T has been set to zero. (c) Effect of the gap amplitude. Here finite values of temperature (T = 2 K) and barrier parameter (Z = 0.1) have been used. The spectra present a single maximum when Δ becomes comparable to the thermal broadening kBT ≅ 0.2 meV. In all the calculations, no broadening parameter has been used. Note, however, that if a broadening was present (Γ≠ 0) all the spectra would be smeared and the “merging” of the two conductance maxima depicted in panel (c) would occur at larger values of Δ.

0.5 and thus the conductance shows peaks at energies close to the gap that are usually much lower and more broadened, due to the thermal smearing and to extrinsic effects we will briefly discuss below. Finally, in Fig. 5.5c we show the effect of the gap amplitude Δ on the PCARS spectra at a finite temperature T, when a small potential barrier exists at the interface. Starting from reasonable values for a PCARS experiment in a low-​Tc superconductor (T = 2 K, Z = 0.1, and Δ = 2 meV) we see that when the gap is reduced below 0.5 meV the original broad two-​peak structure transforms into a single zero-​bias peak. Even if these zero-​bias conductance peaks are often a sign of a non-​conventional symmetry of the order parameter in the k space (d wave, p wave, …), here we simply show that similar peaks (but with a different shape) can also appear in s-​wave superconductors as a consequence of the combined effect of temperature, potential barrier, and smallness of the gap. These particular conditions could be present in experiments on mesoscopic superconductors or on thin-​film multilayer structures where quantum-​confinement or proximity effects can strongly reduce the “effective” gap value (see Sec. 5.4). In the past 30 years, many extensions and integrations of the original 1D BTK model have appeared in the literature. Their detailed discussion is far beyond the scope of the present chapter, but we would like to briefly mention some of the most important ones.

  155

Length scales in mesoscopic systems 155 The first improvement was necessary to account for the fact that the experimental PCARS spectra are often depressed in amplitude and spread in energy with respect to the BTK predictions. This broadening effect has intrinsic reasons (lifetime of quasiparticles) and far more substantial extrinsic ones (inelastic quasiparticle scattering processes occurring near the N/​S interface). It has been shown that it can be taken into account within the BTK model by introducing an imaginary part of the energy so that E + iΓ replaces E in all the equations (Plecenik et al. 1994). Γ is the so-​called lifetime-​broadening parameter, and its inclusion in the BTK model allows a very good fit of most of the high-​quality experimental conductance curves. In 1996 S.  Kashiwaya, Y.  Tanaka, and coworkers (Kashiwaya et  al. 1996) extended the BTK model to the two-​dimensional (2D) case, in order to make it suitable for the description of anisotropic superconductors with a k dependence of the gap such as the high-​Tc cuprates (that display the well-​known dx2−​y2 symmetry of the order parameter). In this 2D BTK model, electrons can approach the interface from any direction (in the ab-​plane) and the only condition that has to be fulfilled is the conservation of the component of the k vector parallel to the interface. A more complete description of this 2D BTK model can be found in Daghero and Gonnelli (2010). To cope with the advent of multiband superconductors with a complex 3D Fermi surface, in 2011 we proposed an extension of the BTK model to the 3D case (Daghero et al. 2011) in which electrons can approach the NS interface from any direction in the k space. The model requires the knowledge of the shape of the Fermi surface, but allows calculating the theoretical spectra for any k dependence of the energy gap. The model is much more complex than the original 1D version from the computational point of view, but the analysis of directional PCARS measurements in very high-​quality single crystals can provide information not only on the number, value, and k-​dependence of the gap(s) but also on the shape of the FS and its evolution with doping, pressure, etc. Further details can be found in Daghero et al. (2011) while some examples of application to anisotropic, multiband, and Fe-​based superconductors are presented in Daghero et al. (2013).

5.3 Length scales in mesoscopic systems 5.3.1 The typical length scales of  a “large” superconductor As well known, all superconducting materials can be divided into type-​I and type-​ II superconductors (Tinkham 2004) depending on the value of the ratio between the penetration length λL and the coherence length ξ = hvF / π∆, that can roughly be interpreted as the size of Cooper pairs. If ξ ≤ 2λ L the superconductor is of type-​II, otherwise it is of type-​I.

156

156  Andreev Reflection and Related Studies In general, the electronic mean free path 𝓁 does not play an important role unless it is smaller than the coherence length. In this situation (dirty limit) the coherence length is given by: 1 1 1 = + (5.5) ξ ξ0 



where ξ0 is the value of the coherence length in the clean limit, and the effective penetration depth turns out to be (De Gennes 1999)

λL = λL0

ξ0 ξ = λ L 0 1 + 0 (5.6)  ξ

This has interesting implications in small particles, where the electronic mean free path is limited by the size: the coherence length decreases with decreasing the particle size, while the effective penetration depth increases. Consequently, a “small” type-​I superconductor tends to behave like a type-​II superconductor.

5.3.2 The typical length scales of a “low-​dimensional” superconductor When at least one of the dimensions of a superconductor is dramatically limited, the size comes into play since neither the superconducting coherence length (ξ) nor the magnetic penetration depth (λ) define the critical length scale for the destabilization of the Cooper pairs (Bose and Ayyub 2014). As for quasi-​0D superconductors (i.e. nanoparticles), the Anderson criterion states that superconductivity should disappear when the particle size D is smaller than a critical value Dcrit (Anderson 1959) that, for most elemental superconductors, is of the order of a few nanometers. Indeed, the confinement of the electronic wavefunctions leads to a discretization of the energy levels, whose energy spacing is the so-​called Kubo gap δ ≈ EF /​N (N being the number of electrons in the particle (Kubo et  al. 1984)). When δ exceeds the superconducting gap at zero temperature, superconductivity cannot take place. Before vanishing at Dcrit, the superconducting properties evolve with reducing the particle size under the effect of two competing phenomena: surface effects (that tend to increase Tc) and the quantum size effect (QSE) (that instead is detrimental for Tc). Surface effects arise from the increase in the surface to volume ratio when the particle size is reduced, and consist in the transfer of spectral weight from high-​ frequency (bulk) to low-​frequency (surface) phonon modes (Dickey and Paskin 1968). As a consequence, the representative phonon frequency



Ω log

  α 2F (Ω)  ∫ log Ω Ω d Ω  = exp   (5.7) α 2F (Ω)  dΩ   ∫ Ω 

  157

Length scales in mesoscopic systems 157 α2F(Ω)

(where is the electron–​phonon spectral function) decreases on decreasing the particle size (phonon softening). As shown elsewhere (Ummarino 2005) the relationship between the electron–​phonon coupling constant λ and Ωlog is:

λ=



N ( 0) < I 2 > (5.8) M Ω2log

where M is the ion mass, N(0) is the normal density of states at the Fermi level, and < I2> is the average of the electron–​phonon interaction over the whole Fermi surface. The above equation tells us that λ increases when Ωlog decreases. This does not automatically mean that Tc increases; according to the McMillan formula (McMillan 1968)—​valid for λ < 2—​the critical temperature is  1.04(1 + λ )  Tc = Ω log exp  − (5.9) λ − µ * (1 + 0.62λ )  



and thus also depends on Ωlog. For Pb in the bulk form, Ωlog = 4.45 meV and λ = 1.55 (Carbotte 1990). The Coulomb pseudopotential μ* can be fixed to the value 0.0937 so as to obtain the experimental critical temperature (Tc = 7.22 K) by solving the Eliashberg equations. Assuming as a first approximation that N(0) does not depend on the size, it turns out that Tc actually depends only on Ωlog. Instead of using the McMillan formula, which is still an approximation, we numerically solved the Eliashberg equations to calculate how the critical temperature responds to variations in Ωlog. The result (Fig. 5.6) shows that surface effects enhance Tc as the size of the particles is reduced, not only in Pb but in all single-​band s-​wave superconductors. However, the relevance of surface effects on Tc is limited to very small particles (with diameter smaller than 5 nm) because the

8

10

6

λ

9

2

8 Tc (K)

4

0

7

2

Pb bulk

3

4

5

Ωlog (meV)

6

6 5 4 2.0

2.5

3.0

3.5

4.0 Ωlog (meV)

4.5

5.0

5.5

6.0

Fig. 5.6  Dependence of the critical temperature of Pb on the representative phonon energy Ωlog. The electron–​phonon spectral function of Pb is here simulated by a Dirac delta function peaked at the frequency Ω0 = Ωlog. The result is obtained by solving the Eliashberg equations and also accounts for the change in the electron–​ phonon coupling strength. Inset: dependence of the electron–​phonon coupling constant on Ωlog.

158

158  Andreev Reflection and Related Studies phonon softening occurs only at the free surface and is confined to the outermost atomic layers. In addition to surface effects, the critical temperature of quasi-​0D superconductors is also affected by the QSE that consists in the discretization of the electronic energy levels resulting in a decrease in the effective density of states at the Fermi level N(0). According to Eqs. 5.8 and 5.9, the reduction in N(0) leads to a reduction in Tc with decreasing size (Ummarino 2013). The interplay between these two opposite effects gives rise to a different behavior of Tc upon decreasing the particle size in different materials, before the complete suppression of superconductivity when the Anderson limit is achieved. As described in Bose and Ayyub (2014) and shown in Fig. 5.7, Tc increases monotonically on decreasing the size in weak-​coupling superconductors like Al, decreases in intermediate-​coupling superconductors like Nb, and shows almost no size dependence in strong-​coupling materials like Pb (Bose et al. 2009). Real quasi-​0D systems studied experimentally do not consist of isolated particles, but rather of nanoparticles in a matrix or in granular systems (such as the quench-​condensed thin films of elemental superconductors (Bose and Ayyub 2014)). In these cases, the surfaces of the nanograins are not “free” and so the above considerations should be corrected by taking into account the interaction with the matrix or with the neighboring particles. Moreover, these systems are really quasi-​0D superconductors only if the grains are decoupled, that is, if the superconducting wavefunction is confined within a single grain. This condition is controlled by the intergrain regions, that is, by the microstructure of the films, that also deeply affects the size dependence of the normal-​state properties and of the thermodynamic quantities (Bose and Ayyub 2014). Other physical aspects of

2.5 Pb (Reich 2003) Pb (Bose 2009) Nb (Bose 2005) Nb (Hazra 2009) Al (Abeles 1966) Sn (Li 2008)

Fig.  5.7 Size dependence of the normalized transition temperature Tc/​ (Tc)bulk, for Pb, Nb, Al, and Sn nanoparticles, i.e. 0D superconductors (from Bose and Ayyub 2014). Up triangles and down triangles are experimental data in Pb by Reich et al. (2003), and Bose et al. (2009), respectively. Squares and circles are data for Nb by Bose et al. (2005) and Hazra et  al. (2009). Stars are data for Al (Abeles et al. 1966) and diamonds for Sn (Li et al. 2008). Lines are only guides to the eye.

Tc / Tc bulk

2.0

1.5

1.0

0.5

0.0

0

5

10

15 20 25 Grain size (nm)

30

35

40

  159

Length scales in mesoscopic systems 159 the problem that are usually not taken into account in theoretical descriptions are the possible effects of the substrate, such as proximity and strain/​stress due to lattice mismatch. Recent theoretical calculations have also predicted a “shell effect” (Kresin and Ovchinnikov 2006), which leads to oscillations in Tc when the particle size is reduced (García-​García et  al. 2011). An experimental demonstration of this effect has been given in Bose et al. (2010) by means of scanning tunneling microscope measurements on individual Sn nanoparticles. Another important aspect that must be taken into account is the role of thermodynamic fluctuations. In quasi-​ 0D superconductors, thermodynamic fluctuations may lead to a finite superconducting order parameter above the bulk Tc (Skocpol and Tinkham 1975). Measurements of fluctuation diamagnetism and specific heat in nanostructured superconductors have provided results that have found a reasonably good explanation within the Ginzburg–​Landau theory (Buhrman and Halperin 1973). In quasi-​1D systems like nanowires (Altomare and Chang 2013)  the transverse dimension (let’s say, the diameter D) is small, but still larger than the Fermi wavelength λF, which is of the order of a few Angstrom in all metals. In conventional superconductors, the coherence length ξ is typically 10–​1000 times the Fermi wavelength λF so that λF < D < ξ. This means that from the point of view of the condensate the system is 1D, and the order parameter is uniform and position-​independent in the transverse directions. Instead, from the point of view of the quasiparticle excitations, the system is effectively 3D. In this limit, as pointed out in Altomare and Chang (2013), collective excitations with energies higher than the superconducting gap are no longer pair breaking, but create “phase slips,” topological defects in the ground-​state configuration, that result in a non-​zero resistance down to low temperature (Zaikin et al. 1997). For instance, in single-​crystal Sn nanowires (Tian et al. 2005), for which ξ = 200 nm, a transition from quasi-​bulk to quasi-​1D properties (with a residual resistive tail) was observed when D < 70 nm (i.e. 40 nm and 20 nm). In Al nanowires (Singh et al. 2009) for which ξ = 1600 nm, Tc was enhanced above the bulk value of 1.2 K (up to 1.9 K) in 30-​nm wires, as a manifestation of the “shape resonance” (Shanenko et al. 2006). A study of carbon nanotubes coated with a Mo–​Ge alloy (D ≤ 10  nm) evidenced that quantum tunneling of phase slips does not necessarily destroy superconductivity in 1D systems since it can be prohibited by strong damping when the normal-​state resistance of the wire is smaller than the quantum resistance for Cooper pairs (Bezryadin et al. 2000). As for the role of the mean free path 𝓁, it can be said that if the longitudinal dimension L is smaller than 𝓁 the conduction along the nanowire is ballistic; in general, however, 𝓁 is larger than the transverse dimension D. Another interesting fact is that, when D is sufficiently small, the spacing between electronic energy levels in the transverse direction can significantly exceed the superconducting gap. The possibility of singularities in the normal density of the electronic states associated with each transverse channel can cause an oscillatory behavior in the superconducting properties as a function of diameter.

160

160  Andreev Reflection and Related Studies From the above discussion, it is clear that the regime of interest is set by the condition that the transverse dimension (D) must be smaller than the superconducting coherence length. The experimental data show that the critical temperature remains the same or slightly changes when D is reduced below ξ, but the zero-​resistance state is gradually destroyed because of the formation and proliferation of phase slips (Bose and Ayyub 2014). Quasi-​2D systems are generally obtained at interfaces (e.g. by electrostatic confinement of charge carriers), in ultrathin films, and in other mesoscopic structures (Bose and Ayyub 2014). In the case of ultrathin films, the technological difficulty is to ensure continuity, so that the film does not behave as a granular system. In this case, the important dimension is of course the thickness t. As t is varied, in addition to the QSEs which can tune superconductivity by controlling the density of states around the Fermi energy, there may be a modification to the electron–​phonon coupling strength (as for example in Pb films, Moore et al. 2015). The superconducting properties of these systems are thus again determined by an unusual interplay between QSEs and electron–​phonon coupling modulation, which has been deeply studied (at least for Pb films and islands) both experimentally and theoretically (Moore et al. 2015, Li et al. 2003, García-​ García et al. 2008, Ovchinnikov and Kresin 2010, Kresin and Ovchinnikov 2006, Guo et al. 2004). The role of superconducting fluctuations is of course important also in 2D systems. 2D films exhibit a Berezinskii–​Kosterlitz–​Thouless transition that results in a weakening of superconducting correlations due to phase fluctuations (Tinkham 2004) but superconductivity can persist when t is reduced much below ξ (Moore et al. 2015) or even down to a single layer, as recently shown for Pb (Zhang et al. 2010), FeSe (Lee et al. 2014), and NbSe2 (Xi et al. 2015). Another relevant observation is the connection between the sheet resistance of thin films and the emergence of superconductivity. If the normal-​state sheet resistance exceeds the critical value h/​4e2  =  6.45 kΩ, the film does not show superconductivity (Hebard and Paalanen 1990), as has been observed in films of Bi (Haviland et al. 1989) but also of unconventional high-​temperature superconductors (Bollinger et al. 2011). Recently, it has been shown that the substrate may play a fundamental role: monolayers of FeSe are not superconducting if deposited on graphene, but show high-​temperature superconductivity when deposited on SrTiO3. A discussion of this interesting point can be found in Mazin (2015). A completely different situation is envisaged when a thin film of a normal metal is deposited on top of a superconducting substrate; in this case, the thickness of the normal layer determines whether or not it can display proximity-​ induced superconductivity (McMillan 1968). Although in this case there is no “small” superconductor involved, the interface is a 2D system whose properties can be measured by PCARS, as we will show in the discussion of experimental results in Sec. 5.4.

  161

Length scales in mesoscopic systems 161

5.3.3 Is PCARS suitable for spectroscopy of “small” superconductors? In this section we will describe the role of the length scales from the point of view of PCARS. When a PC is made on a 3D superconductor, as shown in Fig. 5.8a, the contact radius should be: (a) smaller than 𝓁 to ensure that the contact is ballistic; (b) smaller than ξ to prevent the weakening of superconductivity in S due to the proximity with N, and to ensure that superconductivity is not quenched in the vicinity of the contact when the current becomes overcritical—​which happens when the bias voltage is of the order of the gap (Deutscher 2005). Let us now briefly explain the reasons of these requirements. As for point (a), the requirement of ballistic transport through the contact, which is necessary for spectroscopy of the phonon modes in the normal state (see Sec. 5.1.2), remains essential in the superconducting state. Indeed, the measurement of the gap requires that the bias voltage coincides with the excess energy Eexc (divided by the elementary charge) of the electrons injected in the superconductor, measured with respect to the chemical potential in S. This condition can only be approximated in a real experiment (see Fig. 5.8) since the bias voltage is the sum of various contributions: V = V1 + VPC + V2 where V1 = R1I is the voltage drop in the normal counterelectrode, that can be made very small by using I+, V+

(a) N

V–

I–

S

(b)

Potential

(c)

R1

RPC

RS(RAR)

R2

RM

VAR

V

Fig.  5.8  (a)  The pseudo-​four-​probe configuration used in most PCARS experiments. (b)  Scheme of the series of resistances actually present between the voltage leads (i.e. V + and V–​): R1 is the resistance of the normal bank (i.e. the tip); R2 the resistance of the material under study, between the point contact and V–​(this is zero in the superconducting state); RPC is the resistance of the point contact that contains RAR (affected by Andreev reflection, and whose normal-​ state value is the Sharvin resistance RS) and RM (Maxwell resistance). (c) Qualitative plot of the potential along the circuit. In an ideal ballistic contact, the applied voltage V is equal to the potential drop at the interface, VAR.

162

162  Andreev Reflection and Related Studies a good conductor; VPC is the voltage drop due to the contact, discussed below; and V2 is the voltage drop in the superconducting bank, which is generally zero when the large superconductor is cooled below Tc—​apart from special cases as discussed in Daghero et al. (2006). According to Wexler’s formula (see Eq. 5.3), VPC contains two contributions, VAR and VM (Chen et  al. 2010). VAR is the voltage affected by Andreev reflection; its value in the normal state is equal to the Sharvin resistance RS times the current I and is precisely the voltage drop at the interface, that is, the maximum excess energy of a quasiparticle injected in the S side. The second contribution, VM , is associated with dissipation effects (Maxwell term) and, as shown in Eq. 5.3, contains the resistivity of the banks, that here will be called ρ1 and ρ2. Note that, as pointed out in Chen et al. (2010), half of this term arises from the contact region, and half from the banks. If the S bank is in the zero-​resistance state, ρ2 = 0 but dissipation can occur in N, possibly leading to considerable heating in the contact region (Daghero and Gonnelli 2010, Daghero et al. 2006) and eventually to quenching of superconductivity in the S side. As discussed in Sec. 5.1.2, the requirement of a ballistic contact means that there is no inelastic scattering in the contact area, and therefore the Maxwell term is negligible; hence, VPC ≅ VAR. This ensures that, as long as V1 is negligible and V2 = 0 the experimentally accessible quantities V and T correspond to the microscopic quantities in the contact. As we will see in the following, these conditions may not be true in low-​dimensional superconductors. As for the condition a < ξ, what is its origin? When one uses PCARS to measure the gap Δ, the voltage bias is swept from –​Vmax to +Vmax where Vmax must necessarily become bigger than Δ/​e. In these conditions the current density in the contact becomes overcritical (Deutscher 2005) but after the contact bottleneck the current spreads in all directions, and its density decreases rapidly below the critical value over a distance r from the contact center. As long as r < ξ superconductivity is not quenched at the interface. In the limit condition r = ξ, one finds 2

 ξ that the current density in the contact, j0, is given by j0 = 2   jc . This means  a that the bigger the ratio ξ/​a, the more the voltage drop at the interface can exceed Δ/​e without quenching superconductivity (Deutscher 2005). In other words, if a < ξ no disruption of superconductivity occurs for energies of the order of Δ. If instead r > ξ, the N/​S interface is pushed forward a short distance in the S bank so that Andreev reflection does not occur at the interface, but at a distance r from it—​where the order parameter does not vary sharply as assumed in the BTK model. In principle, this is not detrimental to the detection of retroreflected holes, provided that they can be collected by the contact; but this is true only as long as quasiparticles are not scattered inelastically within a distance r from the point contact (i.e. r must be smaller than the inelastic mean free path). Actually, due to the (tiny) difference in the wavevectors of the incoming electron and reflected hole (Heslinga et al. 1994), even multiple elastic scattering can reduce the probability of detection of the reflected holes, with a consequent loss of Andreev signal. The situation is similar to that of PC experiments in a thin metallic layer backed

  163

Length scales in mesoscopic systems 163 by a superconductor, that will be described in Sec. 5.4. The relationships between the mean free path, the contact radius, and the size of the normal-​state region in that configuration are discussed in Heslinga et al. (1994). Now, what happens in a low-​dimensional superconductor? The number of papers reporting PCARS measurements in these systems is very small, and a complete understanding of all the effects of spatial confinement on the mechanism of Andreev reflection is missing. From an experimentalist’s point of view, the strong limitations to the use of PCARS in low-​dimensional systems are related to the critical current and to the spreading resistance. In 2D, the condition to avoid quenching of superconductivity at the N/​S interface is no longer a < ξ, but a < ξ t where t is the thickness. This condition can be more demanding than in 3D when t < ξ, which is actually a realistic situation in 2D systems. If t is very small, it may be impossible to attain voltage values of the order of Δ/​e without quenching superconductivity in the vicinity of the contact. Once this happens, the radius of the normal region has a stronger dependence on V than in 3D systems so that the available voltage range for spectroscopy is considerably reduced. In other words, PCARS spectra in 2D systems are more likely to display spurious dips (Sheet et al. 2004) associated with the breakdown of superconductivity. Moreover, on increasing temperature the critical current decreases and superconductivity can thus be locally quenched in the S bank. The resistivity 𝜌2 in the Maxwell term thus starts to play a role, giving rise to heating, which eventually results in resistance that increases with voltage. The spreading resistance R2 due to the normal region of the film can be very large, for geometrical reasons, and this gives rise to a downward shift of the PCARS spectra accompanied by a stretching of the voltage scale (see Fig. 5.9). The whole effect prevents a determination of the gap up to the critical temperature, which instead is possible by scanning tunneling spectroscopy, where the resistance of the NIS junction is much bigger (Zhang et al. 2010). In 1D superconductors, like nanowires, PCARS is even more challenging. First of all, the range of diameters for which PCARS can be reasonably tried is limited by the progressive destruction of the zero-​resistance state by phase slips (Lau et al. 2001, Tian et al. 2005). Second, even in wires that are superconducting, the observed steps in the I–​V curves (Tian et al. 2005) make spectroscopy of the gap impossible unless within a very precise range of parameters (temperature, diameter, current). Even leaving aside the various complications due to transverse quantum confinement (such as the existence of various values of depairing velocity, see Croitoru et  al. 2009), the main problem is actually set by the current, which must not become overcritical. In a possible PCARS setup, the contact is made on one end of the nanowire (by embedding it into a normal electrode). In this case, the contact cannot be smaller than the nanowire cross-​section and the current density is the same through the contact and along the whole wire. This means that when V = Δ/​e the whole wire suddenly becomes normal. No spectroscopy of the gap is then possible. Indeed, the few reported examples of Andreev-​reflection spectroscopy experiments

164

164  Andreev Reflection and Related Studies (a) PCARS on BaFe2 (As1–x Px)2

0.08

Conductance (Ω–1)

T (K)

0.06

24.1 25.3 26.0 26.4 26.9 27.2 27.8

0.04

–50

–40

–30

–20

–10

0

10

20

30

40

50

Voltage (mV)

(b) 28

0.3

26 0.2 24

Transport Point contact (RN @ –30 meV)

22

0.1

Normalized resistance

RN @ –30 meV (Ω)

Fig. 5.9  (a) An example of PCARS spectra taken in a 50 nm thick film of the Fe-​ based superconductor BaFe2(As1−​xPx)2 with x = 0.2 (optimal doping). The shift of the conductance curves that starts at 24 K is due to the onset of a spreading resistance R2. Here the junction was biased with DC current in the range (–​3, +3) mA; note the stretching of the voltage scale due to the spreading resistance. (b)  Normalized resistance (R(T)/​ R(200 K)) of the BaFe2(As,P)2 films (line) and equivalent resistance of the tip–​contact–​sample series (R1 + RPC + R2), deduced from the PCARS spectra at V= –​30 meV (symbols). The comparison shows that the shift of the PCARS spectra, which is actually due to the onset of a spreading resistance in the S bank, occurs in the proximity of the superconducting transition.

4.3 13.1 18.1 22.1

20 0.0 20

22

24

26

28

30

32

34

Temperature (K)

involve a normal 1D system (for example a carbon nanotube) that connects 3D superconducting banks, thus acting as the tip in the needle-​anvil configuration. Nonetheless, zero-​bias spectral features associated with proximity-​induced or intrinsic superconductivity in these systems have been observed, as we will show in Sec. 5.4. As for quasi-​0D systems, PCARS is obviously impossible in isolated grains and even in nanoparticles embedded in an insulating matrix. Instead, granular systems (films or bulks) can be used provided that the superconducting wavefunction is confined within each grain and that the contact can be made with an individual grain (Bose 2007). Actually, a good electrical connection between

  165

Examples of PCARS in superconductors with reduced dimensionality 165 adjacent grains is mandatory. The reason is that the PCs must have a relatively small resistance to display Andreev reflection (normally less than 100 Ω) and the spreading resistance R2 must be much smaller than that, to allow energy-​ resolved spectroscopy. This is not trivial in granular systems that may display a non-​zero resistance R2 even below Tc when single grains are superconducting but intergrain regions are not. For this reason, PCARS is less versatile than tunnel spectroscopy, which can be used also in systems with poor connectivity because of the much larger resistance of the N/​I/​S junction (Barber et al. 1994). Indeed, all the PCARS measurements in quasi-​0D systems reported in the literature have been made in films with low normal-​state resistance and thus fairly good intergrain connectivity. Clearly, another experimental limitation is associated with the current density, that in this case is limited by the critical current of the weak links rather than that of the grains. In all cases (quasi-​2D, quasi-​1D, and quasi-​0D systems) a role of the scattering from surfaces can be expected, that can give rise to a reduction of the Andreev signal with respect to the 3D case, due in turn to the partial possible loss of retroreflected holes. However, the interplay between all the possible factors that can affect Andreev reflection is very complex and not completely understood.

5.4 Examples of PCARS in superconductors with reduced dimensionality 5.4.1 PCARS in quasi-​2D superconductors Quasi-​2D superconductors can belong to different classes: interfaces, ultrathin films, layered materials, and single-​layer materials. Depending on the class they belong to, they display a variety of phenomena whose description is well beyond the scope of this chapter. PC spectroscopy is rarely used to investigate these systems, for various reasons (in addition to those listed in the previous section): (a) the interfaces are, by definition, not directly accessible; (b) ultrathin films are usually very sensitive to oxidation and contamination and must be handled in situ after deposition (Zhang et al. 2010) or must be protected by a cap layer; and (c) single-​layer or few-​layer materials that show superconductivity are not very common. Graphene is not superconducting (unless it is decorated with Li ions); single-​layer FeSe on strontium titanate (STO) (Ge et al. 2015), and few-​layer MoS2—​that has been made superconducting by electrochemical (Ye et al. 2012) and electrostatic (Biscaras et al. 2015)  doping—​have not been studied yet by Andreev reflection spectroscopy. Cuprates, that can be considered to some extent 2D superconductors, have been widely studied by PCARS (Deutscher 2005, Tortello and Daghero 2015), but mainly in the bulk form—​though ultrathin films have been grown and used for field-​effect experiments.

166

166  Andreev Reflection and Related Studies An early interesting example of “classic” PCARS in a quasi-​2D system is shown in van Son et al. (1987) where the proximity effect at a Ag/​Pb interface is studied. The contact is made by touching the Ag layer (backed by the Pb film) with a sharp Au or Cu tip (Fig. 5.10a). The characteristic lengths of the problem are the contact radius, a, the mean free path in Ag, 𝓁, the thickness of the Ag layer, t, and the coherence length ξ (83 nm in Pb). The interplay of these length scales is intriguing. First of all, a ≪ 𝓁 so that the contact is ballistic. As long as t < (ξ, 𝓁) no scattering occurs in the Ag layer and the whole bilayer is superconducting. In these conditions, “classic” PCARS spectra are observed, apart from the fact that the gap is smaller than in bulk Pb because of the order parameter depression at the interface (see Fig.  5.10b). On increasing t, (a)  the scattering

(a)

Ag, Cu Ag

(d)

d t

t’

1.45 1.40

1.2 t = 200 nm

1.0

(c)

1.4 1.2 1.0

t = 400 nm

–2

1.25 1.20

exp

calc

1.15 1.10

exp

calc

1.00

t = 900 nm –3

1.30

1.05

t = 700 nm

–4

(e)

1.35

t = 20 nm Normalized conductance

Normalized conductance Normalized conductance

1.4

Si3N4

Nb

∆ (x)

(b)

1.6

d

p++ Si

Pb 1.8

Ag Nb

–1 0 1 Voltage (mV)

2

3

4

–4

–2

0 Voltage (mV)

2

4

Fig. 5.10  (a) The experimental setup for the study of proximity effect at the Nb/​Ag interface by van Son et al. (1987).The profile of Δ(x) at the interface (when t > ξ) is also shown. (b) Normalized PCARS spectra showing only the (reduced) gap of Pb, i.e. in the case of no detectable proximity effect. (c) Some spectra taken in the proximity regime (t > ξ) showing both the shoulders associated to ΔPb and the zero-​bias maximum associated to the proximity gap in Ag. (d) Experimental setup in Heslinga et al. (1994). Here the p-​doped Si membrane is 50 nm thick, the Nb film is 300 nm thick, and ξ = 38 nm. (e) Two examples of experimental normalized spectra (solid lines) showing features associated to the bulk Nb gap and to the proximized gap. Dashed lines: calculations within the model that accounts for the spatial variation of the gap at the interface, and which is qualitatively able to account for the observations.

  167

Examples of PCARS in superconductors with reduced dimensionality 167 in the Ag layer becomes important and (b)  proximity effect makes the Δ(x) profile at the interface become smoother. In these conditions, electrons with low energy are Andreev-​reflected at an effective distance t′ < t from the contact (see Fig.  5.10a), do not feel the effect of the potential barrier (i.e. feel a small Z), and the retro-​reflected holes can easily reach the contact; higher-​energy electrons instead are Andreev-​reflected close to the Ag/​Pb interface, feel the effect of the barrier, and the reflected holes can be scattered and laterally dispersed before reaching the PC. As a result, Z and the intensity of the attenuation of the signal become energy dependent—​which means that the system cannot be described by the BTK model. The spectra look like those of multiband superconductors, in which the proximity gap is so small as to appear as a single maximum (compare with Fig.  5.5c) and the other (corresponding to ΔPb) results in higher-​energy shoulders (see Fig. 5.10c). Some years later, point contacts were used to probe the proximity-​induced gap in a thin semiconductor (Nishino et al. 1990, Heslinga et al. 1994). In Nishino et al. (1990) a coplanar configuration was used, in which the Nb/​Si point contact and the Nb drain electrode were both made on the P-​doped surface of a Si single crystal. This complicated the interpretation of the data so that they were not conclusive. In Heslinga et al. (1994) an Ag tip was brought into contact with an ultrathin (t = 50 nm) membrane of metallic, heavily p-​doped Si backed by a Nb film of thickness 300 nm, much larger than ξ = 38 nm (see Fig. 5.10d). The elastic mean free path in the doped Si was very small, 𝓁e = 4.8 nm, so that the conditions for ballistic conduction through the PC were out of reach, and the diffusion regime at most could be realized, in which a <  e  i where  i ≈ 250 nm. As in van Son et al. (1987) the multiple elastic scattering within the doped Si slab is expected to reduce the probability to collect the backscattered holes and thus to suppress the Andreev signal by a factor exp( −2t / λ att ) where λatt is an attenuation length. However, the large size of the contact with respect to the thickness of the normal layer (a > t) practically compensates for the lateral dispersion. At the same time, the Fermi velocity mismatch at the Nb/​Si interface reduces the transparency of the PC (Heslinga et  al. 1994)  since, as discussed in Sec. 5.1.2, here τ < 1. Nevertheless, a very clear and unexpectedly high Andreev signal was detected. Many spectra showed multiple gap features associated with the bulk Nb gap, ΔNb, and with the proximity gap in Si, ΔSi (see Fig.  5.10e). The clear detectability of ΔSi (that shows up as a pair of conductance maxima unlike in van Son et al. 1987) was ascribed to the very localized jump in the gap value at the Si/​Nb interface (in turn due to the large Fermi velocity mismatch at the interface), since the Andreev reflection probability is peaked at a gap discontinuity. As in van Son et al. (1987), the parameter Z becomes energy dependent because electrons with energy E ≈ ∆ Si are Andreev-​reflected far from the interface and with Z ≈ 0, while electrons with higher energy are reflected close to the interface and feel the contribution of the Fermi velocity mismatch that suppresses the Andreev signal (Z > 0). The attenuation term is neglected and a model is made that overcomes

168

168  Andreev Reflection and Related Studies the limiting assumption of the BTK model, instead using an ansatz profile for the gap at the interface and containing as fitting parameters ΔSi, Z,|T |2 (i.e. the transmission coefficient of the PC) and an imaginary part of the gap as a broadening term. Fig. 5.10e shows a comparison between some experimental spectra (solid lines) and the results of the model (dashed lines). To the same class of experiments, aiming to probe proximity-​induced superconductivity, belongs a recent study in topological insulator/​cuprate interfaces (Zareapour et al. 2012). Here a large direct planar junction is made by transferring a thin BSCCO layer on top of a Bi2Se3 (or Bi2Te3) cleaved crystal. The I–​V curves of the junction display multiple features that are associated with the proximity-​induced gap in the topological insulator, with the depressed BSCCO gap at the interface, and with the bulk BSCCO gap. This interpretation is supported by the disappearance of these features in the case of low-​transparency barriers. An example of PCARS in a 2D electron gas (2DEG) has been recently reported in Gallagher et al. (2014). The 2DEG is created on the topmost surface of a STO single crystal by electrochemical doping, which makes STO become conductive and even superconducting below 300 mK. Source and drain contacts are made on the 2DEG and separated by a 2-​μm wide top gate (TG) strip electrode, isolated from the 2DEG by a thin alumina layer (see Fig. 5.11b). Tuning VTG allows tuning the I–​V characteristics of the device and the transparency of the “weak link” beneath the top gate. For some intervals of VTG , a normal channel is formed separating the two banks of the 2DEG (Fig. 5.11b). In zero magnetic field, the two banks are superconducting and SNS spectra would be expected; however, the differential conductance of the device actually shows the features of Andreev reflection at a single N/​S interface (see Fig.  5.11a). The reason is probably an asymmetry of the channel so that only one end of the normal “neck” is small enough to act as a PC. Interestingly, the same channel shows quantized conductance and thus behaves as a quantum point contact (QPC) when superconductivity is quenched by a magnetic field:  in particular, the differential conductance shows zero-bias plateaus at e 2 / h and 2e 2 / h . This result is puzzling for various reasons. Indeed, the first plateau occurs at one half of the quantum conductance; this indicates a broken spin degeneracy that is in turn ascribed to ferromagnetism. Then, a QPC with conductance e 2 / h is a half metal, which typically gives zero Andreev reflection probability at the interface with a s-​wave superconductor. The authors propose that spin-​flip Andreev reflection is actually occurring at the QPC (Gallagher et al. 2014). Fig. 5.11a reports some examples of spectra, taken at different values of VTG and divided by the “normal state” spectra measured at the same VTG by quenching superconductivity with a magnetic field. The shape of the spectra evolve rapidly in a narrow range of VTG between 1.92 V and 2.12 V; even though the original BTK model should not hold for a spin-​polarized QPC, these spectra can be fitted fairly well to such a model including a broadening parameter Γ, because the effect of spin polarization is qualitatively similar to the effect of the barrier Z. The fit shows that the gap amplitude oscillates with VTG (not shown in Fig. 5.11).

  169

Examples of PCARS in superconductors with reduced dimensionality 169 (a)

(c) T (mK) 14 45 86 140 195 229 261 300 330 356 380 421

2

1.5

VTG (V)

1.0

2.12 2.11

0.5

2.04 2.02

0.0 1.94

–0.5

1 dI/dV (e2/h)

Normalized conductance

2.0

0

–1 –2

1.92 –1.0 –400

0

–200

200

–3 –400

400

–200

VSD (µV)

(b)

200

400

(d)

QPC

80 S

STO

Ohmic contacts

S N

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Top gate (TG)

Al2O3

0 VSD (µV)

∆ Γ

60 40 20 0 0.0

0.1

0.2

0.3

0.4

Temperature (K)

The fit of the unnormalized spectra that show the Andreev reflection features on top of the 2e2/​h quantization plateau (of the normal-​state conductance) (Fig.  5.11c) allows the authors to obtain the Δ(T) curve, that (considering all the limitations of the models and the complexity of the system) resembles rather well a BCS-​like curve (see Fig. 5.11d). The reason why the authors do not use the curve above Tc to normalize the spectra, as is commonly done in other systems (provided that Tc is not too high), is that the normal-​state conductance of a QPC is not featureless but presents quantization plateaus that are temperature-​dependent.

5.4.2 PCARS in quasi-​1D superconductors The quasi-​1D superconductors studied so far include nanowires of low-​Tc superconducting metals, semiconducting nanowires or carbon nanotubes proximized by superconducting contacts, and bulk materials with quasi-​1D conduction properties (that will not be considered here). Presumably for the reasons explained in the previous section, no examples of PCARS in superconducting quasi-​1D systems have been reported in the literature. Andreev reflection plays instead a determinant role in the case of semiconducting nanowires proximized by superconducting contacts. In single-​wall, 300-​nm

Fig.  5.11 (a)  Some normalized spectra taken at different values of the top gate voltage VTG in the setup shown in panel (b)  (Gallagher et  al. 2014). The effect of VTG is essentially to modify the interface potential barrier. (b) Sketch of the experimental setup. The topmost surface of STO is electrochemically doped (the gate electrode that causes migrations of ions in the electrolyte is not shown). For suitable values of VTG a single QPC is established between the two superconducting regions. (c) Temperature dependence of the unnormalized spectrum at the second quantization plateau, 2e2/​h. Symbols:  BTK fit of the lowest-​temperature curve. Guides to the eye highlight the shrinking of the Andreev maxima. (d)  Temperature dependence of the gap and of the broadening parameter Γ. The line is the BCS prediction.

170

170  Andreev Reflection and Related Studies long carbon nanotubes (CNTs) whose diameter was less than 1.6 nm, superconducting proximity effect was observed already in 1999 (Morpurgo et al. 1999). The nanotube was used as the active channel of a field-​effect transistor, with superconducting (Nb) source and drain contacts on a SiO2-​capped highly-​doped Si substrate that could be used as a back gate (Fig. 5.12a). By changing the gate voltage VG , the shape of the I–​V curves (measured in the two-​probe configuration between source and drain) changed, suggesting that the transparency of the Nb/​nanotube interfaces was tuned. In the high-​transparency regime (attained for VG = –​40 V) the differential conductance measured across the nanotube showed a zero-​bias peak (that disappears at the Tc of Nb), ascribed to Andreev reflection at the two interfaces in series, superimposed on a broad nonlinear background similar to what was seen for normal-​metal contacts (Fig. 5.12b). Below 4.2 K, other features of unknown origin appeared. The observation of Andreev reflection features clearly indicates that the nanotube is not completely proximized and a pair of independent N/​S interfaces are present (indeed, neither subharmonic gap structures nor a supercurrent were observed). More recently, similar experiments were carried out on field-​effect transistors (Fig. 5.13a) made with individual single-​walled CNTs with different diameters and lengths, and with either superconducting (Nb) or metallic (Pd) contacts (Zhang et al. 2006). In the first case, a thin (< 5 nm) Pd layer was used anyway CNT

Nb

(a)

Catalyst SiO2

Doped Si

dI/dV (Ω–1)

(b)

Fig.  5.12 (a)  Sketch of the field-​effect devices based on a carbon nanotube as in Morpurgo et  al. (1999). (b)  Differential conductance (unnormalized) of the device at various temperatures, with a gate voltage VG = –​40 V. The zero-​bias peaks are assumed to arise from Andreev reflection at the two CNT/​Nb interfaces.

1.8 x 10–4

T (K)

2∆Nb

1.7 x 10–4

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  171

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SiO2 Doped Si

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28 26 24 22 20

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(e)

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Vsd (mV) 1.4

∆ Γ

1.2 1.0

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∆, Γ (meV)

30

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Normalized conductance

Differential conductance (kΩ–1)

32

1.10

GN

1.2

DOS 4.2 K 6.3 K 10 K

VG = 10 V

0.95 –6 –4 –2

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1.3

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2 4 6 8 10 Temperature (K)

Fig. 5.13  (a) Sketch of the experimental setup for transport measurements on isolated nanotubes with superconducting contacts as in Zhang et al. (2006),Tselev et al. (2009), and Yang et al. (2012). (b) Differential conductance dIsd/​dVsd as a function of Vsd at two different values of VG, recorded with the contacts in the superconducting state (GS) and in the normal state (GN), obtained by applying a magnetic field. Inset: fit of the normalized curve at VG = +10V with the model proposed in Tselev et al. (2009). (c) Sketch of the band structure and of the density of states of a semiconducting CNT. (d) Examples of the zero-​bias peaks observed in a device with Pd contacts when the gate voltage VG shifts the Fermi level to a van Hove singularity, and the fit of the normalized conductance with a suitable model derived from BTK (Yang et al. 2012). (e) The values of the gap in the CNT extracted from the fit, compared to the temperature dependence of the gap in bulk Nb.

172

172  Andreev Reflection and Related Studies between the Nb electrode and the nanotube to ensure high-​transparency contacts. In this configuration, if the length of the nanotube is larger than the phase coherence length, the two interfaces are independent, no supercurrent is observed, and the nanotube acts as a PC able to detect Andreev reflection at both the Pd/​Nb interfaces (Tselev et  al. 2009). An enhancement of the zero-​bias conductance measured across source and drain (dIsd/​dVsd at Vsd  =  0) is observed when the contacts are superconducting, independent of the gate voltage (provided that VG is sufficient to make the nanotube conducting). The normalized conductance GS/​GN (GN being the conductance when superconductivity in the contacts is quenched by a magnetic field of 3 T) always shows a peak if plotted as a function of Vsd (inset to Fig. 5.13b) that disappears at the critical temperature of Nb (Tselev et al. 2009). The half-​width of the Andreev structure is minimum when the gate voltage (VG ≈ 20 V) makes the CNT highly transparent (as in Morpurgo et al. 1999), and corresponds to twice the gap in Nb (2Δ ≅ 3 meV). The normalized conductance curves, GS/​GN (of which an example is shown in the inset to Fig. 5.13b), are fitted to a suitable model that accounts for the two interfaces in series, the quantized conductance of the nanotube, and the scattering by the CNT/​Pd barriers. For each 1D channel in the nanotube, Andreev reflection is introduced using the BTK model, so that the system is schematized as a S/​N/​ CNT/​N/​S sandwich. Interestingly, the same authors also observed a zero-​bias anomaly (i.e. a peak in the unnormalized differential conductance dIsd/​dVsd vs. the source-​drain voltage Vsd) for specific, large negative values of the gate voltage (depending on the device), that disappeared above the critical temperature of the superconducting electrodes and were observed even in devices with metallic (Pd) contacts. At the same values of gate voltage, the zero-​bias differential resistance dVsd/​dIsd displayed a low-​temperature downturn that resembled the onset of (incomplete) superconductivity. The authors claimed that the observed features are due to intrinsic superconductivity in the nanotubes, that appears only when VG is sufficient to shift the Fermi level to the van Hove singularities in the density of states (Fig. 5.13c) (whose position incidentally depends on the diameter of the nanotube). By modeling the nanotube and the two contacts as a single N/​CNT/​N or SN/​CNT/​NS system, including Shottky barriers at the nanotube ends and Andreev reflection at all the interfaces, the authors were able to reproduce the behavior of the zero-​bias anomaly as a function of temperature. In a following paper (Yang et al. 2012) they also provided a fit of the conductance curves (Fig. 5.13d), obtaining an estimation of the gap within the nanotube. The latter turned out to be smaller than the gap in Nb, though the relevant critical temperature was higher (up to 30 K) (Fig. 5.13e). By the way, also the critical field turns out to be very high (≫7 T). It is worth mentioning that superconductivity had previously been reported in various nanotube samples with critical temperatures between 0.5 K and 15 K, without gate voltage. An evaluation of the temperature-​dependent gap from the I–​V curves was also given by Tang et  al. (2001), and showed a similar non-​BCS behavior (attributed to the 1D nature of the nanotubes).

  173

Examples of PCARS in superconductors with reduced dimensionality 173

5.4.3 PCARS in quasi-​0D superconductors Quasi-​0D superconductors have been extensively studied by a number of experimental techniques (Bose and Ayyub 2014), but only in a few cases by PCARS. Experimental investigations have been mainly carried out on isolated grains or compacted powders, or on quench-​condensed disordered thin films of elemental superconductors such as Nb, Al, Pb, etc. Ensemble-​averaged (“bulk”) information is usually provided by magnetization, transport, and specific heat measurements. Also electron tunneling and point-​contact spectroscopy can be considered “bulk” techniques when the junction size is bigger than the size of individual grains, for example when planar junctions between a nanocrystalline/​granular film and a metal electrode are used (Giaever and Zeller 1968, Bose et al. 2009, Barber et al. 1994). The first PCARS measurements in “small” superconducting particles were reported in 1996 and were actually made by using a scanning tunneling microscope (STM) brought in contact with Pb nanoparticles, deposited by evaporation of pure Pb on glass in a He atmosphere (Poza et al. 1996). The diameter of the particles was not controlled and ranged from 100 nm to 10 nm, smaller than the coherence length of Pb (about 300 nm); the spectra, taken at 4.2 K, showed only a broad maximum and the signal was largely suppressed. Many years later, PCARS measurements were systematically carried out on 500-​nm thick nanostructured films of Nb (Bose et al. 2005) deposited by magnetron sputtering in Ar pressure, and consisting of a network of superconducting nanograins weakly connected through interfacial regions made up of an amorphous Nb–​O phase. Unlike in the case of quench-​condensed films, the thickness of the film did not affect either the superconducting properties or the resistivity, which only depended on the particle diameter D, controlled by the Ar pressure and ranging between 5 nm and 60 nm. In all cases, the films had a rather small residual resistivity (less than 200 μΩ cm for particles of average diameter down to 11 nm) and showed a well-​defined superconducting transition. For D = 8 nm a diamagnetic signal was observed below Tc = 4.7 K, but the resistivity showed no full superconducting transition, which prevented PCARS measurements even if the single grains could still be superconducting (Bose 2007). The resistance of the point contacts, made by using a sharp Pt–​Ir tip in the needle-​anvil configuration, was about 10 Ω, which is in the usual range for PCARS spectroscopy, and their radius a (evaluated by using Eq. 5.2) was of a few nanometers. This ensures that the contact is ballistic and that probing an individual grain is possible (at least in the films with larger D values). Hence, a < min(D, ξ) (where ξ = 38 nm), which is the requirement to prevent quenching of superconductivity due to critical current effects, as discussed in the previous section. The measurement of the gap by PCARS together with the Tc determined by more conventional techniques (i.e. magnetization and resistivity) aims to investigate the interplay between the QSE (that reduces the density of states and thus Tc) and the surface effects (that induce a phonon softening and tend to increase Tc). Experimentally, Tc remains approximately equal to the bulk value (≈ 9.4 K)

174

174  Andreev Reflection and Related Studies from D = 60 nm to D = 30 nm, then decreases to 4.7 K between D = 20 nm and D = 8 nm; for D < 8 nm no superconductivity is observed either by transport or by magnetization, consistent with the Anderson criterion (Anderson 1959). The decrease of Tc (by about 50%) seems to suggest that QSE is dominant over SE; but if this is the case, the electron–​phonon coupling should not change with the particle size. A direct check of this is possible only if direct measurements of the energy gap are carried out. When the PCs were made on more than one grain, multiple-​gap features were clearly observed, due to the distribution in the particle size (about 20%). Only the spectra featuring a single pair of conductance peaks can thus be reasonably assumed to pertain to a single grain, although the possibility that different grains with very similar gap amplitude are probed cannot be excluded. Indeed, this possibility is discussed in Bose (2007) as a possible origin of the broadening parameter Γ which is necessary to fit the normalized PCARS spectra with the BTK model (Blonder et al. 1982). Fig. 5.14a reports some examples of these spectra with the relevant fit. They show no anomalies at V > Δ/​e, which confirms the spectroscopic nature of the contacts; the energy gap was found to be reduced from Δ = 1.6 meV (i.e. the bulk value) for D = 60 nm down to Δ = 0.9 meV for D = 11 nm. The good correlation between Δ and the Tc measured by transport and magnetization means that: (a) the “bulk” Tc is not affected by the intergrain regions, but only by the intrinsic properties of the grains; (b) the gap ratio 2Δ/​kBTc remains constant (and equal to 3.6) and is not affected by the particle size (see Fig. 5.14b); and (c) Nb persists in the intermediate coupling down to the complete suppression of superconductivity. Indeed, the Δ(T) curve obtained by fitting the PCARS spectra at different temperatures shows no deviation from the BCS weak-​coupling curve, as shown in Fig. 5.14c (Bose et al. 2005, Bose 2007). This result indicates that in Nb the phonon softening does not play a significant role. However, SEs play a major role in weak-​and intermediate-​coupling type-​I superconductors like Al, In, and Sn (where Tc increases on decreasing D) and also in Pb (Li et al. 2003, Bose et al. 2009) in which Tc decreases much less than in Nb before the complete disappearance of superconductivity (Fig.  5.15b). PCARS measurements in 200-​nm thick nanocrystalline films of Pb were reported in Bose (2007). The average particle size ranged from 60  nm down to 5  nm with an uncertainty of about ±15%. Due to the softness of Pb, the needle-​anvil technique was not suitable and planar Al/​Al2O3/​Pb junctions were made with the actual intent to make tunnel measurements (Bose et al. 2009). In a subset of junctions, however, clear Andreev reflection spectra were recorded, probably because of the formation of small pinholes through the alumina barrier (Bose 2007). The normalized spectra were analyzed by using the BTK model including a broadening parameter Γ that, as in the case of Nb nanoparticles, accounts for the possibility that multiple grains with slightly different gaps are probed. The fulfillment of the correct conditions for Andreev-​reflection spectroscopy is witnessed by the consistency (within 5%) of the gap amplitude obtained by fitting the PCARS spectra with that resulting from the fit of tunnel-​like spectra taken at the same

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(b) D = 60 nm T = 2.6 K Tc = 9.4 K

∆ = 1.60 meV Z = 0.6 Γ = 0.327 meV

D = 18 nm T = 2.7 K Tc = 7.2 K

1.25 1.20 1.15

1.5

Energy gap ∆(0)

1.35 1.30 1.25 1.20 1.15 1.10 1.05 1.00

∆ = 0.95 meV Z = 0.576 Γ = 0.1 meV

1.10

(c)

1.05

2∆/kBTc = 3.6 1.0

0.5

0.0

1.00

0

2

6

4

8

10

Tc (K) 1.2 1.0

1.30 1.25 1.20 1.15 1.10 1.05 1.00

D = 11 nm T = 2.9 K Tc = 5.9 K

∆ = 0.82 meV Z = 0.56 Γ = 0.08 meV

∆(T) /∆(0)

Normalized conductance

(a)

0.8 0.6 0.4 0.2

–8

–6

–4

–2

0

2

Voltage (mV)

4

6

8

0.0 0.0

D = 11 nm, Tc = 5.9 K D = 60 nm, Tc = 9.4 K BCS

0.2

0.4

0.6

0.8

1.0

T/Tc

Fig. 5.14  (a) Examples of PCARS spectra in Nb nanoparticles of decreasing diameter D (from top to bottom) with the relevant BTK fit (lines).The dashed line is a guide to the eye, to highlight the decrease in the energy of the Andreev maxima. The values of the fitting parameters Δ, Z, and Γ are reported in the labels. (b) Correlation between the zero-​temperature energy gap Δ(0) and the critical temperature.The solid line corresponds to a gap ratio of 3.6. (c) Normalized gap amplitude Δ(T)/​Δ(0) as a function of the normalized temperature T/​Tc for particles of diameter D = 60 nm and Tc = 9.4 K (solid symbols) and for particles of diameter D = 11 nm and Tc = 5.9 K. The solid line is the BCS curve. All data are taken from Bose et al. (2005).

176

1.3 1.2

(b) D = 64 nm T = 4.3 K Tc = 7.25 K

∆ = 1.35 meV Z = 0.55 Γ = 0.06 meV

(c)

1.0 1.3

10 8 Pb Nb

6 4 2 0

D = 22 nm T = 4.3 K 1.2 T = 7.25 K c

∆ = 1.48 meV Z = 0.56 Γ = 0.25 meV

Gap ratio

6.5

1.1

6.0 5.5 5.0 4.5 4.0

1.0

(d)

1.5

D = 15 nm 1.4 T = 4.3 K Tc = 7.19 K 1.3

∆ = 1.38 meV Z = 0.35 Γ = 0.08 meV

1.2

∆(T) /∆(0)

Normalized conductance

1.1

8 7 6 5 4 3 2 1 0

Tc (K)

1.4

Tc (K)

(a)

–4

–2

0

Voltage (mV)

2

4

10

20 30 40 50 Particle diameter D (nm)

60

70

1.0

0.5

1.1 1.0

0

0.0

D = 64 nm D = 22 nm D = 11 nm BCS

0.4

0.6

0.8

1.0

T/Tc

Fig. 5.15  (a) Examples of PCARS spectra at 4.3 K in Pb nanoparticles of decreasing diameter D (from top to bottom) with the relevant BTK fit (solid lines). The values of the fitting parameters Δ, Z, and Γ are reported in the labels. The data are taken from Bose (2007). (b) Critical temperature (measured by bulk techniques) as a function of particle diameter for Pb and Nb (black and grey circles). (c) Gap ratio 2Δ(0)/​kBTc in Pb films, as a function of the particle diameter, as extracted from tunneling spectra (Bose et al. 2009). The PCARS results shown in (a) are in fairly good agreement with this trend. (d) Temperature dependence of the gaps in Pb nanoparticles as extracted from tunneling spectra (Bose et al. 2009). On decreasing the particle size, the stronger electron–​phonon coupling results in increasing deviations from the BCS curve.

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Summary and outlook 177 temperature (4.3 K). Note that the PCARS measurements were carried out only in well-​conductive samples with zero resistivity below Tc (i.e. D > 10 nm) (Fig. 5.15a). Thanks to the mechanical stability of planar junctions, the Tc of each contact was measured and turned out to coincide with that measured by transport and susceptibility. Since the base temperature of 4.3 K is not low enough to determine the gap amplitude Δ(T→0)—​especially because the temperature dependence cannot be safely extrapolated to low T, since Pb is not BCS—​the gap was measured in Bose et al. (2009) at 2.2 K, but only from the fit of tunneling spectra. The resulting gap was found to increase on decreasing the particle size, while Tc remains approximately constant (and equal to the bulk value) down to D  =  14  nm, and then drops by only 13% between 14  nm and 7  nm, just before the disappearance of superconductivity. The conclusion was that the gap ratio 2Δ/​kBTc increases monotonically from about 4.37 for D = 64 nm to 6.29 for D = 9 nm (Fig. 5.15c). This means that the electron–​phonon coupling increases and indeed the Δ(T) dependence deviates more and more from the BCS curve (Fig. 5.15d). This would demonstrate that surface effects are dominant over the QSE. According to the McMillan equation for strong-​coupling superconductors, an increase in the coupling strength λ should result in an increase in Tc which is not observed. To solve this puzzle, the authors propose that the phonon softening effects are almost exactly offset by the QSE that reduces the density of states and thus Tc (Bose et al. 2009).

5.5 Summary and outlook In the previous sections we have introduced the simple but rather powerful technique of point-​contact Andreev-​reflection spectroscopy (PCARS), which is widely used in the determination of the superconducting energy gap and its properties in 3D single crystals, bulk materials, and films. We have also summarized the few results obtained so far by using this technique in quasi-​2D, -​1D, or -​0D superconductors. It is quite clear from the discussion of Sec. 5.3.3 that several constraints appear when one seeks to apply PCARS to materials with reduced dimensionality and this is probably one of the reasons for the limited literature on the subject. In quasi-​1D samples, the conditions for obtaining an AR spectroscopic signal are so demanding that they have prevented up to now (and, probably, will also prevent in the future) a clear determination of the gap and its temperature and magnetic-​ field dependence. The few cases presented in Sec. 5.4.2 refer indeed to situations where the determination of a proximized (or gate-​induced) gap in a quasi-​1D normal material (a carbon nanotube) is somehow controversial. In our opinion there are more chances to successfully use this technique in quasi-​ 2D and quasi-​ 0D materials, where the geometrical and physical constraints can be more simply fulfilled—​at least in samples with specific

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178  Andreev Reflection and Related Studies characteristics or in a particular temperature range. In Pb nanoparticles, for example, the results discussed in Sec. 5.4.3 and summarized in Fig. 5.15 have already contributed to the determination of an anomalous and unexplained behavior in the physics of “small” superconductors, that is, the strong increase of the ratio 2∆ kBTc at the decrease of the particle diameter. These results suggest that PCARS could fruitfully be used to investigate other nanostructured materials, contributing to the future developments of this increasingly interesting field of research. In the case of quasi-​2D systems, which are among the hottest present subjects of research, what seems particularly promising is the possibility (only preliminarily explored up to now in the results shown in Fig. 5.11) to combine the tremendous charge-​doping capabilities of the electrochemical gating technique with the high-​resolution spectroscopic capabilities of PCARS. Here, the challenge is to find a technological way to create the N/​S junctions within a 2DEG and to control its transparency in a reproducible way. In materials where superconductivity is induced by field-​effect gating (as recently done in transition-​metal dichalcogenides) the possibility to measure the energy gap by exploiting Andreev reflection would open up completely new perspectives and allow a complete characterization of the superconducting state, presently not possible with other techniques.

Acknowledgements The authors wish to thank Sangita Bose and Pratap Raychaudhuri for help in retrieving information about point-​contact spectroscopy in quasi-​0D superconductors, and for enlightening discussions. One of the authors (GAU) was supported by the Competitiveness Programme of NRNU MEPhI. Special thanks to all the present and former members of our group, for their contributions in the achievement of important results in point-​contact spectroscopy in the past 15 years.

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Topological Superconductors and Majorana Fermions Y.Y. Li and J.F. Jia Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

6 6.1 Introduction 

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6.2 TI/​SC heterostructures 

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6.3 Nanowire/​SC junctions 

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6.4 FM atomic chain on SCs 

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6.5 Summary and outlook 

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References (Chapter-​6) 

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The search for new states that exhibit topological order is currently a very active and exciting area of research. Superconducting order can also exhibit topological order, which is different from that of a conventional superconductor (SC), and similar to that of a topological insulator (TI). These are so-​called “topological superconductors” (TSCs), which have a full pairing gap in the bulk and gapless surface states consisting of Majorana fermions (MFs). This chapter highlights a few of the recent developments in this area including the proposals for realizing TSCs by proximity effects through TI/​SC heterostructure, and the experimental efforts to fabricate artificial TSCs and explore MFs in various systems. TSCs have led to new insights about superconductivity, and revealed that some SCs can support MFs and non-​Abelian statistics. It might open a door to probing the novel physics of MFs.

6.1 Introduction Topological superconductivity is a novel quantum state of matter. However, by now, there is no unanimous definition for a TSC. TSCs have a full pairing gap in the bulk and gapless surface states consisting of MFs (Qi et al. 2011). They can be any SCs (3D, 2D, or 1D) with MFs at their surfaces or edges. Due to their scientific importance and potential applications in quantum computing, MFs have attracted lots of attention recently. Since TSCs are closely associated with MFs, much attention has also been paid to TSCs. The MF was first introduced as a fermion which is its own antiparticle by Ettore Majorana in 1937. It is believed that neutrinos are MFs, but this remains unproven (Majorana 1937). In the past decade, condensed matter physicists

Y.Y. Li, J.F. Jia, ‘Topological Superconductors and Majorana Fermions’ in The Oxford Handbook of Small Superconductors. First Edition. Edited by A.V. Narlikar. © Oxford University Press 2017. Published in 2017 by Oxford University Press. DOI 10.1093/acprof:oso/9780198738169.003.0006

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184  Topological Superconductors and Majoranas have become interested in the possibility of creating the solid-​state counterpart of MFs called Majorana bound states (MBSs). Unlike neutrinos, these are states that are localized in space and can be thought of as half a fermion: when two MBSs are brought closer to each other, their wave functions overlap, resulting in a conventional fermion state, which may be occupied or empty. Furthermore, the MBSs are expected to obey non-​abelian statistics, that is, in a collection of MBSs, the interchange of two of them does not merely change the sign of the wave function, but causes an evolution in the Hilbert space of the fermionic state. Such exotic objects have not been seen in nature, and represent a prime example of the notion of “emergence.” The tantalizing prospect of using MBSs as robust quantum memory and qubits for quantum computation adds further excitement. Being its own hole means that a MF must be an equal superposition of an electron and a hole state. It is natural to search for such excitations in superconducting systems, where the wavefunctions of Boguliubov quasiparticles have both an electron and a hole component (Leijnse and Flensberg 2012). In SCs, electrons form so-​called Cooper pairs. The Cooper pairs can form a dense “condensate,” which is responsible for superconductivity. As a consequence, electron number is in effect no longer conserved: two electrons (in a Cooper pair) can be added or subtracted from the condensate without substantially changing its properties. Crucially too, the SC screens electric and confines magnetic fields so that charge is no longer observable. Thus, the absolute distinction between electrons and holes is blurred. So, the most daunting barrier to producing Majorana-​like excitations—​the charge-​conjugation hurdle—​seems vulnerable in a SC. However, in most types of SCs, the electrons forming the Cooper pairs have opposite spin projections, which results in the electrons and holes in the Majorana-​like excitations also having opposite spin projections. So, MBSs do not occur in most SCs. MBSs can only occur in some exotic SCs, in which the two electrons in each Cooper pair have the same spin, that is, with triplet pairing symmetry. However, the existence of MBSs is a topological invariant (Hasan et al. 2010) (hence the name TSCs). As a result, they will exist in all superconducting systems with the same topological properties. A few years ago, the search for MFs took a big step forward when Fu et al. (2008) showed that px ± ipy-​wave-​like pairing may also occur for the surface states of a strong TI when brought into tunneling contact with an ordinary s-​ wave SC (giving rise to proximity-​induced superconductivity (de Gennes 1964, Doh et al. 2005, van Dam et al. 2006) in the TI). Theoretically, MBSs are predicted to emerge in exotic SCs in the vortex cores or at the end of one-​dimensional (1D) wires. Recently, there have been concrete proposals to create the requisite exotic SCs by the proximity effect of conventional SCs with TIs or materials with strong Rashba-​type spin–​orbit coupling. This chapter focuses on a few of the recent developments in this area including the proposals for realizing TSC by proximity effects through TI/​SC

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TI/SC heterostructures 185 heterostructure, and the experimental efforts to fabricate artificial TSCs and explore MFs in various systems.

6.2 TI/​SC heterostructures TI is a new quantum phase of matter that exhibits insulating bulk states and gapless spin-​polarized helical edge or surface states protected by time reversal symmetry (Hasan et al. 2010, Qi et al. 2011). Following the recent demonstration of topological Dirac states in several classes of TIs (Ando 2013), extensive research is now devoted to investigate exotic phenomena in quantum physics. The interplay between the symmetry-​protected surface states of the TI and broken-​symmetry states in magnets and SCs may lead to anomalous quantum Hall effects (Yu et al. 2010, Chang et al. 2013), time-​reversal invariant TSCs (Schnyder et al. 2008), MFs (Fu et al. 2008, Linder et al. 2010), and fault-​tolerant quantum computation (Nayak et al. 2008). One proposal to combine superconductivity with topological states is to use the superconducting proximity effect (Fu et al. 2008, Linder et al. 2010), either between a superconducting TI’s bulk and surface states or between an s-​wave SC and a TI’s surface state. Bulk superconducting states were recently observed in Cu-​intercalated Bi2Se3 (CuxBi2Se3) (Hor et al. 2010, Kriener et al. 2011, Sasaki et al. 2011, Levy et al. 2013). CuxBi2Se3 retains the Dirac surface state, but its superconducting volume fraction is low. Very recently, strontium (Sr)-​doped Bi2Se3 samples also showed bulk superconductivity with a transition temperature of Tc ~ 2.5 K (Liu et al. 2015, Maurya et al. 2015). The superconducting volume fraction is higher than 80% at a low temperature. The Sr-​doped Bi2Se3 is a promising candidate for TSC research (Han et al. 2015). Another way to realize the superconducting proximity effect between a TI and a SC is to grow TI/​SC heterostructures with an atomically sharp yet electronically transparent interface.

6.2.1 Bi2Se3/​NbSe2 heterostructures As theoretically proposed, preparation of TI/​SC heterostructures is a challenging task because of interface reaction, lattice mismatch between TI and available SC materials, and the poor thermostability of TIs. Nevertheless, the problem can be solved by van der Waals epitaxy by which a layered material is grown on a cleaved face of another layered material (Koma 1999). TI Bi2Se3 has a quintuple-​layered structure consisting of Se–​Bi–​Se–​Bi–​Se hexagonal sheets (Zhang et al. 2009). SC NbSe2 has a triple-​layered structure that consists of two hexagonal Se sheets with an intercalated Nb sheet (Se–​Nb–​Se) (Meerschaut et al. 2001). In both matters, the quintuple layers (QLs) or the triple layers are connected through van der Waals bonds. The TI Bi2Se3 thin films grown on SC NbSe2 by molecular beam epitaxy have been studied by low-​temperature scanning tunneling microscope and spectroscope (STM/​STS) (Wang et al. 2012). Fig. 6.1a shows a large-​scale

186

186  Topological Superconductors and Majoranas (a)

(b)

2nd QL

Height (nm)

3

NbSe2

2 1 0

Fig.  6.1 (a)  Morphology of Bi2Se3 thin films grown on NbSe2 substrate. (b)  Line profile along the line in (a). (c)  Atomically resolved STM image of the Bi2Se3 film. (Wang et  al. 2012, reprinted with permission from AAAS)

50

1st QL

3rd QL

100 150 200 250 Distance (nm)

(c)

250 nm

12 nm

STM image of the atomically flat Bi2Se3 film with a nominal thickness of 2 QLs. The majority of the surface is covered by 2 QL films, while there are small areas with a thickness of 1 QL or 3 QL. Fig. 6.1b shows the thickness of different layers. The atom-​resolved STM image (Fig. 6.1c) reveals the hexagonal atomic lattice of Se atoms with a spacing of 0.41 nm, implying that a well-​defined (111) surface of Bi2Se3 is formed. Local density of states (LDOS) can be obtained with STS by measuring differential conductance (dI/​dV) spectra. On Bi2Se3 films, superconducting gap-​ like spectra are observed where a pronounced dip is in the DOS at the Fermi level and there are peaks on both sides. Fig. 6.2 shows the spectra measured

4.0

3 QL (a)

6 QL @ 4.2 K

Fig.  6.2 Superconducting energy gap detected in Bi2Se3 thin films grown on NbSe2 substrate (Wang et  al. 2012, reprinted with permission from AAAS). dI/​dV spectra measured on 3 QL Bi2Se3 films at (a) 4.2 K and (b) 0.4 K. dI/​dV spectra measured on 6 QL Bi2Se3 films at (c) 4.2 K and (d) 0.4 K.

dI/dV (0.1 nA/V)

3.5

3

(c)

@ 4.2 K

(d)

@ 0.4 K

3.5

@ 0.4 K

(b)

1

0 –4

3

1

–2

0

0 2 4 –4 –2 Sample bias (mV)

0

2

4

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TI/SC heterostructures 187 on the Bi2Se3 films at a thickness of 3 QL and 6 QL. The STS data measured at 4.2 K show a depression in the DOS at the Fermi level, whereas at 400 mK sharp coherence peaks near ±1 meV are observed in both films. The results suggest that the Bi2Se3 films become superconducting due to the proximity effect of the NbSe2 substrate. The superconducting transition is further supported by magnetic field and temperature-​dependent measurements. As the magnetic field or temperature increases, the zero-​bias conductance (ZBC) increases and the coherence peaks on both sides of the gap diminish. Moreover, the superconducting gap can be observed on the Bi2Se3 films up to 7 QL at 0.4 K. The energy band dispersions of the Bi2Se3 thin films at different thicknesses are measured with angle-​resolved photoemission spectroscopy (ARPES) as shown in Fig. 6.3. There is an energy gap at the binding energy of 0.6 eV on the ARPES spectra when the film thickness is 3 QL. Compared with the electronic states of an intrinsic Bi2Se3 crystal, the Fermi level of the 3 QL sample is shifted upward as a result of possible charge transfer from the substrate. The charge transfer generates a large gradient of electric field that enhances the Rashba-​type spin–​orbit coupling. The energy band splitting resulting from spin–​orbital coupling is observed at a binding energy of ~0.15 eV (Fig. 6.3a). When the film thickness is increased to 6 QL, the gap disappears and the Dirac point emerges at ~0.45 eV below Fermi level, indicating decoupling of the interface and surface (Fig. 6.3b). The quantum-​well-​like bands within the Dirac cone do not show spin–​orbital splitting, which indicates that the electric field becomes weak on the surface of 6 QL films. In Bi2Se3/​NbSe2, the Dirac point is clearly visible on 6 QL films, implying that the interface is very sharp. Dirac points are also observed in the films with a thickness of 9 QL and 12 QL. The experimental data measured with ARPES reveal that the topologically ordered surface states persist despite the formation of the superconducting gap in the Bi2Se3 films. 3 QL

6 QL

(a)

9 QL

(b)

12 QL

(c)

(d)

Binding energy (eV)

EF

0.2

Rashba splitting

QW

SS 0.4

QW

SS

DP

QW

SS DP

DP

0.6

0.8 –0.1

0

0.1

–0.1

0

0.1 –0.1 Momentum k (Å–1)

0

0.1

–0.1

0

0.1

Fig.  6.3 Band structure of (a)  3 QL, (b)  6 QL, (c)  9 QL, (d)  12 QL Bi2Se3 thin films grown on NbSe2 substrate. (Wang et  al. 2012, reprinted with permission from AAAS)

188

188  Topological Superconductors and Majoranas

6.2.2 Bi2Te3/​NbSe2 heterostructures The coexistence of the superconductivity and topological surface states makes the Bi2Se3/​NbSe2 heterostructure a good platform for detecting Majorana mode; however, the topological surface states form only on films thicker than 6 QL (Zhang et al. 2010, Wang et al. 2012), where the superconducting gap induced by proximity effect is already very small. In contrast, the topological surface states form on 3 QL Bi2Te3 films (Li et al. 2010), so it’s more feasible to study superconducting proximity effect on the Bi2Te3 thin films grown on NbSe2 (Xu et al. 2014). The growth mode of Bi2Te3 on NbSe2 is layer-​by-​layer growth, the same as that of Bi2Se3 on NbSe2 (Fig. 6.4). Due to the hybridization between the top and bottom surface states of the film, the topological surface state of a Bi2Te3 thin film grown on Si(111) does not form a Dirac cone until the thickness reaches 2 QL to 4 QL (Li et al. 2010, Park et al. 2010, Liu et al. 2012). On the NbSe2 substrate, there is a similar evolution of the electronic states in STS data with the increase of Bi2Te3 thickness, as shown in Fig. 6.4. The dI/​dV spectra taken on Bi2Te3/​NbSe2 at a thickness of 3 QL have a deformed U-​shape segment in the energy range between the bulk valence band maximum (VBM) and the conduction band minimum (CBM), as indicated by the arrows in Fig. 6.4c, respectively. With the increase of Bi2Te3 thickness, the deformed U-​shape segment shifts to a higher binding energy while retaining its size in the energy scale. This indicates that the topological surface states on Bi2Te3/​NbSe2 come into existence at a thickness of 3 QL, which is a very similar behavior to that of Bi2Te3 films grown on Si(111) substrates (Li et al. 2010). The Fermi level

∆h =1.0 nm

3 QL

4 QL

dI/dV (a.U.)

(a)

(c)

0.4 0.2 0.0

–0.4 –0.3 –0.2 –0.1 Sample bias (V)

dI/dV (a.U.)

dI/dV (a.U.)

Fig. 6.4  (a) Morphology of Bi2Te3 thin films grown on NbSe2 substrate. dI/​ dV spectra measured at 4.2 K on (b)  2 QL, (c)  3 QL, (d)  5 QL Bi2Te3/​NbSe2. (Xu et al. 2014, copyright 2014 by the American Physical Society)

2 QL

(b)

0.4 0.0 –0.2 –0.1 0.0 0.1 0.2 Sample bias (V)

325 nm 3 QL

0.8

0.8

5 QL

(d)

0.4

0.0

–0.3 –0.2 –0.1 0.0 Sample bias (V)

0.1

  189

TI/SC heterostructures 189

11 QL

(b) 6 3 QL Bi2Te3

Normalized dI/dV

9 QL 8 QL 7 QL 6 QL 5 QL 4 QL 3 QL 2 QL

Normalized dI/dV

10 QL

(c)

STS curve BCS fitting

1.0 0.8

4 2 QL Bi2Te3

∆ (meV)

(a)

Normalized dI/dV

moves down with increasing film thickness, and it is almost in the bulk band gap in 5 QL films. On the Bi2Te3 thin films grown on NbSe2, the superconducting gap is visible up to 11 QL at 0.4 K as shown in Fig. 6.5a. Fig. 6.5b presents the dI/​dV spectra of bare NbSe2, 2 QL, and 3 QL Bi2Te3 films. Up to a thickness of 2 QL, the STS curve has a flat bottom that touches the zero value of the differential conductance at around zero bias. The STS curve can be well fitted by an s-​wave BCS-​type spectrum function (Fig.  6.5b, bottom and middle). The data up to 11 QL are fitted by thermally broadened s-​wave BCS-​like curves, and the results are summarized in Fig.  6.5c. Roughly speaking, the gap value decreases exponentially as the thickness increases, qualitatively consistent with the energy gap decay for proximity effect-​induced superconductivity. In contrast, the spectra of Bi2Te3 films of more than 2 QL have a non-​flat bottom that does not touch the zero of the differential conductance. Clearly, an s-​wave BCS-​type spectral function can no longer fit the spectrum of a 3 QL Bi2Te3 film (Fig. 6.5b, top). A simple s-​wave BCS tunneling spectrum cannot reproduce the spectrum near zero bias and the sharp coherence peak near the position of the energy gap. Such behavior is not observed on 1 QL or 2 QL films. As the film increases in thickness, an abrupt change is seen on 3 QL films, where the topological surface states start to form. The formation of TI surface states could play an important role in the observed deviation of the dI/​dV spectrum from s-​wave BCS behavior in the 3 QL structure (Black-​Schaffer et al. 2013, Tkachov 2013). To confirm this, the dI/​dV spectrum taken on 3 QL Bi2Se3/​NbSe2 at 0.4 K is also compared with a standard BCS tunneling spectrum (Fig. 6.5c, inset); the agreement is much better than on 3 QL Bi2Te3 film. Therefore, it can be concluded that the topological surface states become superconducting due to proximity effects and their contribution induced the observed deviation in the gap. In other words, the Bi2Te3/​NbSe2 heterostructure is an artificial TSC (Xu et al. 2014).

2 3 QL Bi Se

1 0

2

3

–3 –2 –1 0 1 2 3 Sample bias (mV)

0.6 0.4

2 NbSe2

1 QL

0.2

NbSe2 –3–2–1 0 1 2 3 Sample bias (mV)

0

–3 –2 –1 0 1 2 3 Sample bias (mV)

0.0

1 2 3 4 5 6 7 8 9 1011 Thickness (QL)

Fig.  6.5 Superconducting energy gap observed on Bi2Te3/​NbSe2 (Xu et  al. 2014, copyright 2014 by the American Physical Society). (a)  A  series of dI/​dV spectra taken on different thicknesses of Bi2Te3 thin films at 0.4 K. (b) dI/​dV spectra taken on pristine NbSe2, 2 QL, and 3 QL Bi2Te3/​NbSe2. (c) Thickness dependence of the superconducting energy gap. Inset: dI/​dV spectra measured at 0.4 K on 3 QL Bi2Se3/​NbSe2.

190

190  Topological Superconductors and Majoranas

EB (eV)

0.2

0.6 0.8

–0.2

0.0

hν = 18 eV

k2

3

EB (meV)

1K 2K 3K 4K 7K 12 K

2

4

8

1K 2K 4K 7K 12 K

4 0 2

0

–2

–4

6

4

2

(e)

0

–2

–4

EB (meV)

k1 topo. SSs SC

2

k1

(f)

k1 bulk SSs SC

2

1

0.2

1 K (symm.)

4

0.4 0.6

–0.3 –0.2 –0.1

12 –0.1 0 0.1 k (Å–1)

k2 bulk band SC Raw

EB (meV)

Intensity (a.u.)

EB (eV)

40

0.2

–1

(c)

T = 1K

0

20

6

k (Å )

0.8

4

1

(d) 0.0

Raw

k (Å–1)

0.1

0.2

0.3

0

4 2

1K 3K 7K

1K 3K 7K

2 0

0 0.0

1 1 K (symm.)

0

Intensity (a.u.)

Fig. 6.6  (a) Band structure of a 4 QL Bi2Se3/​NbSe2 measured at 12 K using an incident photon energy of 18 eV. Temperature dependence of ARPES spectra at (b) k1 and (c) k2 indicated in (a). (d) Band structure of a 7 QL Bi2Se3/​ NbSe2 measured at 12 K using an incident photon energy of 18 eV.Temperature dependence of ARPES spectra at (e)  k1 and (f)  k2 indicated in (d). (Xu et  al. 2014a, reprinted by permission from Macmillan Publishers Ltd)

0.4

Intensity (a.u.)

hν = 18 eV T = 12 K

0.0

k1 topo. SSs SC

(b)

k1 k2 k3 3 –k 2 –k 1 –k

(a)

6

4

2

0

–2

–4

6

4

EB (meV)

2

0

–2

–4

EB (meV)

Recently, Xu et al. (2014a) have directly observed the superconducting gap on the topological surface states of 7 QL Bi2Se3 thin films grown on NbSe2, using low-​temperature ultrahigh-​resolution ARPES. Fig.  6.6a shows the band dispersion of 4 QL Bi2Se3/​NbSe2. The ARPES spectra of the surface states at k1 (Fig.  6.6b) and the bulk states at k2 (Fig.  6.6c) are taken at different temperatures. Leading-​edge shifts and coherence peaks are observed at low temperatures. The superconducting signals disappear at higher temperatures such as 7 K and 12 K. For 7 QL Bi2Se3 films grown on NbSe2, the Dirac point is clearly visible and the surface states are topologically nontrivial (Fig. 6.6d). The superconductivity signals of the topological surface states at k1 (Fig. 6.6e) and the bulk states at k2 (Fig. 6.6f) are also observed. The data strongly support the view that TI/​NbSe2 is a most ideal platform for the superconducting proximity-​ induced novel physics.

6.2.3 Vortices of TI/​SC heterostructures Large atomically flat terraces of the Bi2Te3/​NbSe2 surface make it possible to image Abrikosov vortices with STS (Hess et al. 1989). Fig. 6.7 shows dI/​dV maps at zero bias, that is, the contour of ZBC, recorded on 3 QL Bi2Te3/​NbSe2 and bare NbSe2 surfaces under perpendicular magnetic fields (Xu et al. 2014). It is seen from Fig. 6.7b that the vortices exhibit a highly ordered hexagonal lattice, just like those observed on the clean NbSe2 surface shown in Fig. 6.7a. Because of the crystalline band structure and the interaction of the neighboring vortices in the hexagonal lattice, a sixfold symmetry is explicitly observed in the vortex images of the bare NbSe2 surface (Fig. 6.7c). The same symmetry is also present on the

  191

TI/SC heterostructures 191 (a)

NbSe2 0.75T

(b)

150 nm (c)

150 nm NbSe2

100 nm

3 QL 0.75T

(d)

5 QL

80 nm

ZBC contour of the 5 QL Bi2Te3 film (Fig. 6.7d). The growth of Bi2Te3 films on NbSe2 does not change the orientation of the vortex lattice; this is to avoid extra energy consumption for a magnetic flux penetrating the Bi2Te3/​NbSe2 samples. The spatial extension of the vortex for different Bi2Te3 thicknesses has been systematically investigated by Xu et  al. (2014). The ZBC line profile crossing through the center of the vortex can be very well fitted by the formula derived from the Ginzburg–​Landau (GL) expression for the superconducting order parameter (Eskildsen et al. 2002). The experimental data and fitted results for bare NbSe2 and 3 QL Bi2Te3/​NbSe2 are shown in Fig.  6.8a, giving ξNbSe2  =  16  nm and ξ5QL = 29 nm at 0.4 K and 0.1 T. Similar analyses were also performed on other samples, finding a monotonic increase of coherence length with increasing Bi2Te3 thickness, as shown in Fig. 6.8b. Because the in-​plane ξ of a NbSe2 single crystal varies from 7.2 nm to 28.2 nm in previous reports (Sonier et al. 1997, Miller et al. 2000), ξ as obtained above for the bare NbSe2 is a reasonable value. For epitaxial Bi2Te3 films, the superconducting coherence length should be estimated using its expression in the clean limit (Tkachov 2013), ξ0 = hvF/​(π2Δ), in which vF is the Fermi velocity, previously reported to be 3.32 × 105 m/​s. This yields a ξ0 value of 116 nm, much longer than that which has been seen experimentally. The discrepancy between the estimation and the above experimental results may be due to the difference between a proximity-​induced and a pure SC. As the thickness of the Bi2Te3 film increases, the influence from the NbSe2 substrate on the Bi2Te3 film becomes weaker, resulting in a longer coherence length. Variation of the magnetic field leads to changes of the vortex size and the coherence length. The coherence length’s dependence on the magnetic field is shown in Fig. 6.8c for 5 QL Bi2Te3/​ NbSe2. As the magnetic field increases, the coherence length decreases initially, and then saturates as the magnetic field reaches about 0.7 T. For a single-​band s-​wave SC, the vortex size or the coherence length is insensitive to the magnetic

Fig.  6.7  Large-​scale zero-​bias dI/​dV maps measured at 0.4 K and 0.75 T on (a)  NbSe2 and (b)  3 QL Bi2Te3/​NbSe2. Zero-​bias dI/​dV maps for a single vortex measured at 0.4 K and 0.1 T on (c)  NbSe2 and (d)  5 QL Bi2Te3/​NbSe2. (Xu et  al. 2014, copyright 2014 by the American Physical Society)

192

192  Topological Superconductors and Majoranas Fitting 3 QL NbSe2

1.5 1.0 2ξ

0.5 0.0

(b)

(c)

@ 0.4 K 0.1 T

40 35

25 20

0

20

Distance (nm)

40

5 QL @ 0.4 K

32 24 16

15 –40 –20

48 40

30

ξ (nm)

@ 0.4 K 0.1 T 2.0

ξ (nm)

Normalized dI/dV

(a)

0

1

2

3

4

5

6

Thickness (QL)

8

0.0 0.2 0.4 0.6 0.8 1.0 Magnetic field (T)

Fig. 6.8  (a) Normalized ZBC profiles crossing through the centers of vortices at 0.4 K and 0.1 T on NbSe2 and 3 QL Bi2Te3/​NbSe2. (b) Thickness dependence of the coherence length. (c)  The coherence length as a function of the magnetic field measured on 5 QL Bi2Te3/​NbSe2 (Xu et al. 2014, copyright 2014 by the American Physical Society). field at weak fields. The strong dependence of the vortex size on the field may be a characteristic feature of TSCs.

6.2.4 Majorana mode within a vortex As predicted, the Majorana mode should be observed in the vortex of the TSC Bi2Te3/​NbSe2 heterostructure (Fu et  al. 2008). However, it is extremely difficult to distinguish the zero-​energy Majorana mode in a vortex due to the tiny energy gap (~0.01 meV) separating it from the conventional quasiparticle states. Fortunately, the Majorana mode is not pinned at the central point of a vortex core, but extensively distributes around the core center (Chiu et al. 2011), which gives an opportunity to detect the MF by investigating the spatial distribution of the bound states in the vortex core. By applying a perpendicular magnetic field, a vortex can be observed on the top surface of Bi2Te3/​NbSe2. With STM and STS performed at 400 mK, the spatial distributions of the bound states in vortices of Bi2Te3/​NbSe2 heterostructures at different Bi2Te3 thicknesses were studied systematically by Xu et al. (2015). Fig. 6.9a shows a typical contour of ZBC taken on a 5 QL Bi2Te3 film in an external magnetic field of 0.1 T. An Abrikosov vortex is clearly seen, which exhibits higher ZBC values due to the suppression of superconductivity within the vortex. Along the dashed line directing to a nearest neighbor vortex, as indicated in Fig. 6.9a, the spatial variation of the dI/​dV spectra as a function of distance r away from the vortex center was measured. The results are given in Fig. 6.9b. Only one peak appears at zero bias in the dI/​dV spectra near the vortex center, and the peak splits into two at a finite distance r. The splitting energy increases linearly with r. For a better view, the dI/​dV spectra as functions of r and sample bias V in a fake color image are plotted in Fig. 6.10, where the positions of the dI/​dV peaks are indicated by crosses. Two dotted lines are drawn to illustrate the linear relation between the energy of the split peaks and the distance r. Extrapolating the

  193

TI/SC heterostructures 193 (a)

(b)

0

dI/dV (a.u.)

1.3

100 nm

0.6

–3 –2 –1 0 1 2 Sample bias (mV)

45 nm 3

lines, the cross point also gives out the splitting start point. The results for 1–​6 QL Bi2Te3 films are shown in Fig. 6.10a–​f. Although the splitting can be resolved almost at the same position ~20  nm from the center, the splitting start points (the cross points of the dotted lines) are obviously different for films with different thicknesses. For 1–​3 QL Bi2Te3 films, the peak splits right off the vortex center (zero-​distance splitting), similar to that in a conventional s-​wave SC, such as NbSe2 (Hess et al. 1989, Gygi et al. 1991). In contrast, for the thicker Bi2Te3 films (4–​6 QL), the splitting starts at a spatial point away from the vortex center (finite-​distance splitting), an apparent deviation from that in a conventional SC. The peak splitting start position as a function of the thickness of Bi2Te3 films has a transition at 4 QL. The finite-​distance splitting behavior of the bound states has not been reported before. This new feature is interpreted as related to the topological property of the local electronic structure. For the 4–​6 QL films, the Fermi level lies near the top of the Dirac bands, and also crosses the bottom of the bulk conduction bands (Xu et al. 2015). The LDOS of a vortex is contributed from both the bulk and the topological surface states. The bulk contribution is similar to that in a conventional SC, and the LDOS or the dI/​dV spectra contributed from the bulk has a maximum (peak) at a final energy value proportional to the spatial distance r, see Fig. 6.10a for instance. In what follows it’s expected that the MF mode of the 2D surface state may change the profile of the dI/​dV spectra. For simplicity, the LDOS contribution from the quasiparticle bound states in 2D is neglected, since their contribution is expected to be similar to that from the bulk. The Majorana mode in the vortex core has been studied theoretically (Chiu et al. 2011). They calculated the LDOS for a Nb/​Bi2Se3/​Nb sandwich structure, and showed that the MF mode has a spatial distribution of about 40 nm, with a sharp peak at zero bias in the dI/​dV spectrum near the vortex core. The Bi2Te3/​NbSe2 heterostructure has similar parameters, so the spatial extension of the Majorana mode should be similar, although the envelope function depends on the Fermi wavevector. The Majorana mode is then expected to enhance the zero-​bias LDOS within a range of spatial distance r ~ 40 nm away from the vortex core, hence to possibly shift the maximum of the LDOS from a finite energy to zero-​bias energy for small r.

Fig. 6.9  (a) A vortex mapped by zero-​ bias dI/​dV on 5 QL Bi2Te3/​NbSe2 at 0.1 T and 0.4 K. (b) Spatially resolved dI/​dV spectra taken along the dashed line in (a)  (Xu et  al. 2015, copyright 2015 by the American Physical Society).

194

194  Topological Superconductors and Majoranas

Distance (nm)

0

(a)

1 QL

1.6

0

1.5

0 10

20

20

20

30

30

30

0.3

40

0

(d)

4 QL

0.4

1.4

0

(e)

5 QL

1.5

0

10

10

20

20

20

30

30

30

0.6 –3 –2 –1 0 1 2 3 Sample bias (mV)

40

0.4 –3 –2 –1 0 1 2 3 Sample bias (mV)

3 QL

1.4

0.5 –3 –2 –1 0 1 2 3 Sample bias (mV)

10

40

(c)

40

–3 –2 –1 0 1 2 3 Sample bias (mV)

–3 –2 –1 0 1 2 3 Sample bias (mV)

Distance (nm)

2 QL

10

40

Fig.  6.10 Spatially resolved bound states within a vortex in (a) 1 QL, (b) 2 QL, (c) 3 QL, (d) 4 QL, (e) 5 QL, (f) 6 QL Bi2Te3/​NbSe2 heterostructures (Xu et al. 2015, copyright 2015 by the American Physical Society).

(b)

10

(f)

6 QL

40

1.2

0.7 –3 –2 –1 0 1 2 3 Sample bias (mV)

The large zero-​bias LDOS at small r of the MF mode should be the underlying physics for the deviation of the zero-​distance splitting behavior of the bound state as shown in Fig. 6.10. The STM measurement has an energy resolution of about 0.2 meV. The LDOS within this energy resolution is expected to be enhanced due to the MF mode. The maximum LDOS for a fixed r may shift towards lower energy and the energy shift is less for larger r. These factors may explain the basic feature in the observation of the finite-​distance splitting. Note that the effect of the MF mode on the change of the LDOS in the vortex core depends on the relative weight of the MF mode. A systematic study of the LDOS of a vortex with both the bulk and Dirac surface states will require further study. The explanation is also supported by the magnetic field dependence of the LDOS of a 5 QL system. The dI/​dV spectra taken at a vortex center of a 5 QL Bi2Te3 film in various magnetic fields are shown in Fig.  6.11a. The zero-​bias peak (ZBP) is very strong at a field less than 0.1 T. As the field reaches 0.18 T, the ZBP becomes much weaker. In a conventional s-​wave SC, the vortex density is proportional to the magnetic field below a critical field Hc2. A single vortex structure is not sensitive to the external field and the LDOS near a vortex core is essentially unchanged as the field increases. The bound states of a vortex in NbSe2 do not show the abrupt change when the magnetic field increases from 0.025 T to 1.25 T (Fig. 6.11b). The abrupt change should not be related to the proximity effect since it does not occur in the 2 QL Bi2Te3 film as shown in Fig.  6.11c. The dramatic change in the ZBP intensity in 5 QL Bi2Te3 film can be interpreted as the result of the coupling between adjacent vortices. At a small field, the distance between vortices is much larger than the vortex size, so the interaction between the vortices can be neglected. As the field increases to 0.18 T, the distance between two adjacent vortices is reduced to about 110 nm

  195

Nanowire/SC junctions 195 (c)

(d)

0.025 T

0.025 T

0.05 T

0.05 T

0.05 T

0.075 T 0.10 T 0.18 T 0.25 T

0.075 T 0.10 T 0.18 T 0.25 T

Normalized dI/dV

0.025 T

0.075 T 0.10 T 0.18 T 0.25 T

0.50 T

0.50 T

0.75 T

0.75 T

0.75 T

1.00 T

1.00 T

1.00 T

0.50 T

50 nm

(e)

0

Distance (nm)

(b)

Normalized dI/dV

Normalized dI/dV

(a)

10

1.2

20 30 40

0.6

–3 –2 –1 0 1 2 3

–3 –2 –1 0 1 2 3

–3 –2 –1 0 1 2 3

–3 –2 –1 0 1 2 3

Sample bias (mV)

Sample bias (mV)

Sample bias (mV)

Sample bias (mV)

(Fig. 6.11d). Moreover, the higher magnetic field suppresses the superconducting gap and the coherence length becomes longer, so the interaction between the vortices becomes strong enough to destroy the Majorana modes. In this case, the LDOS at vortex cores is governed again by conventional quasiparticle bound states and a zero-​distance splitting pattern recovers in the spatial variation of the dI/​dV spectra. This situation is shown in Fig. 6.11e, which is in contrast to the finite-​distance splitting at 0.1 T in Fig. 6.10e. By growing TI thin films on an s-​wave SC NbSe2 substrate, a sharp TI/​SC interface is obtained. The topological surface states are proved to be superconducting by proximity effects. Self-​consistent evidence is obtained indicating that the Majorana mode exists in the vortex of the Bi2Te3/​NbSe2 heterostructure. The observation of a finite-​distance split pattern is interpreted as a demonstration of the MF in the central area of the vortex. This conclusion is fully supported by a very recent theoretical investigation on the MBS in a vortex of proximity-​induced superconductivity on the surface of a TI by the Bogoliubov–​de Gennes approach (Kawakami and Hu 2015). Since the topological surface states are protected by the time reversal symmetry, the Majorana mode in the configuration is free from impurities and defects. It is also found that the Majorana mode can be tuned off by increasing the magnetic field. This might provide a route to controlling MF for quantum computation.

6.3 Nanowire/​SC junctions On the basis of earlier semiconductor-​based proposals, conditions to harbor the MFs can be achieved using more conventional ingredients (Sau et al. 2010, Alicea 2010, Lutchyn et al. 2010, Oreg et al. 2010). One of them is that engineering a nanowire (NW) device should accommodate pairs of Majoranas. The starting point is a 1D NW made of semiconducting material with strong spin–​orbit interaction. In the presence of a magnetic field B along the axis of the NW, a gap is

Fig.  6.11  Magnetic field-​dependent dI/​dV spectra at a vortex center of (a) 5 QL Bi2Te3/​NbSe2, (b)  pristine NbSe2, and (c)  2 QL Bi2Te3/​NbSe2. (d)  Vortex lattice measured at 0.18 T on the 5 QL Bi2Te3/​NbSe2. (e)  Spatially resolved bound states within a vortex shown in (d). The peak-​splitting start point is zero, in sharp contrast to that in Fig.  6.10e (Xu et  al. 2015, copyright 2015 by the American Physical Society).

196

196  Topological Superconductors and Majoranas opened at the crossing between the two spin–​orbit bands. If the Fermi energy is inside this gap, the degeneracy is twofold, whereas outside the gap it is fourfold. The next ingredient is to connect the semiconducting NW to an ordinary s-​wave SC. The proximity of the SC induces pairing in the NW between electron states of opposite momentum and opposite spins and induces a gap. Combining this twofold degeneracy with an induced gap creates a TSC. Majoranas arise as zero-​ energy bound states—​one at each end of the wire (Kitaev 2001, Lutchyn 2010, Oreg 2010, Alicea 2012, Beenakker 2013). InSb NWs are the most promising material systems for the formation of hybrid devices with an s-​wave SC, because they possess a large electron g factor (∼30–​70), a strong spin–​orbit interaction strength with a spin–​orbit interaction energy in the order of ∼0.3 meV, and a small electron effective mass (∼0.015me) (Nilsson et  al. 2009, 2010). These properties should allow one to generate a helical liquid in the InSb NW, by applying a relatively small magnetic field. The s-​wave SC will introduce superconductivity into the InSb NW by the proximity effect (Nilsson et  al. 2012)  and the external magnetic field will then drive the strongly spin–​orbit coupled NW system to a TSC phase through Zeeman splitting. The giant Landé g-​factor of InSb guarantees a significantly large Zeeman splitting at a magnetic field well below the critical magnetic field of the s-​wave SC. Therefore, it is experimentally feasible to fabricate a nontrivial TSC (Lutchyn et al. 2010, Oreg et al. 2010, Stanescu et al. 2011) which supports a pair of MFs, by coupling the InSb NW to an s-​wave SC.

6.3.1 N–​NW–​S device A typical sample is shown by Mourik et al. (2012) in Fig. 6.12a. InSb NWs of 80–​120 nm diameter are transferred onto the substrate containing gate patterns. Normal Ti/​Au contacts (20 nm Ti/​125 nm Au) are made to the NWs and to the gates. The superconducting contacts are NbTiN thin film with thickness of 75  nm and critical temperature Tc ~7 K.  The superconducting contacts only cover half of the top part of the NWs to avoid the complete screening of the underlying gates. The proximity effect-​induced SC energy gap of the InSb NWs is about 250 μeV. The corresponding peaks are indicated by the arrows shown in Fig. 6.12b. The magnetic field-​dependent spectroscopy in Fig. 6.12b–​c exhibits that a ZBP appears at finite B and sticks to zero bias over a range from 0.07 to 1 T. The measurements of gate voltage dependences in Fig. 6.12d display that when B is in the range from 0.07 to 1 T, the ZBP remains at zero bias while changing the gate voltage over large ranges. ZBPs due to the Kondo effect (Sasaki et  al. 2000)  or Andreev states bound to s-​wave SCs (Zareyan et  al. 2002) can occur at finite B; however, with changing B, these peaks then split and move to finite energy. As in the case of weak antilocalization, the resulting ZBP is maximal at B  =  0 and disappears when B is increased. Therefore the above options for a ZBP do not provide natural explanations for the experimental results.

  197

Nanowire/SC junctions 197 0.5

(a)

dI/dV (2e2/h)

3 2 1

1 µm

490 mT

4

0 mT

0.1

B

N

0.3

(b)

–400

0

–200

200

400

V (µV)

(c)

(2e2/h) 0.6

400

0

0.0

(d)

(2e2/h) 0.3

200 V (µV)

V (µV)

500

175 mT

0

0.2

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0.1

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0

Voltage on Gate 2 (V)

10

–400

–0.25

0

0.25

0.5

0.75

B (T)

6.3.2 S–​nanowire (NW) quantum dot (QD)–​S device NW–​SC hybrid devices with other configurations have also been designed to study the signatures of MFs. Fig. 6.13a shows a scanning electron microscope (SEM) image of the Nb–​InSb NW QD–​Nb junction device (Deng et al. 2012). Two Nb-​based SC contacts are deposited on an InSb NW. In a S–​NW–​S device, the presence of a strong supercurrent also induces ZBPs, which hinders the experimental identification of the transport signatures of MFs in the system. To avoid this problem, the separation between the two Nb-​based SC contacts is designed to be very small and the device is tuned to a low-​conductance region in which the quasiparticle transport shows the Coulomb blockade characteristics at a low temperature. In the Coulomb blockade regions, the Josephson supercurrent is strongly suppressed. Fig. 6.13b shows a clear quasiparticle Coulomb blockade diamond structure, indicating the formation of a quasiparticle quantum dot (QD) in the InSb NW junction region between the two SC contacts. In a quasiparticle Coulomb blockade region, magnetic field-​ dependent measurements of the differential conductance for the device are presented in Fig. 6.13c. The place corresponds to the white dashed line in Fig. 6.13b. A distinct peak at zero bias voltage is clearly visible at magnetic fields of B ~1–​2.5 T. In addition, along with the ZBP, there are two side peaks. The origin of the two side peaks is not known, but one possible cause for the two side peaks could be transport through the two normal fermion states created as a result of the

Fig. 6.12  (a) SEM image of a N–​InSb NW–​S device. (b) Magnetic field-​dependent spectra dI/​dV versus V at 70 mK taken at magnetic field B from 0 to 490 mT. (c)  Intensity plot of gate voltage-​ dependent spectra dI/​dV versus V and voltage on gate 2 at 175 mT and 60 mK. (d) Intensity plot of dI/ dV versus V and B (Mourik et  al. 2012, reprinted with permission from AAAS).

198

198  Topological Superconductors and Majoranas annihilation of the pair of MFs adjacent to the InSb NW QD. Moreover, the measurements of the charge stability diagram in the center of the quasiparticle Coulomb blockade region at B = 1.8 T also show a strong ZBP (see Fig. 6.13d). Clearly, the ZBP does not have any shift towards finite bias voltages as the back gate voltage changes. Mechanisms such as the Kondo effect, supercurrent, and high-​order multiple Andreev reflections should be excluded, even though they can also cause a ZBP. For the Kondo effect, the ZBPs have to split and move to finite bias voltage positions with an increasing magnetic field. However, in the measurements of Deng et al. (2012), the ZBP appears at much stronger magnetic fields ~1.2–​2.7 T and it does not show splitting over about a 1.5 T range of magnetic fields. Therefore, the ZBP observed in the experiment could not be a feature of the Kondo effect. Supercurrent and high-​order multiple Andreev reflections can also give a ZBP. However, in both cases, the ZBP will decrease with increasing magnetic field. This is clearly not consistent with the observations,

0.0 0.5 1.0 1.5 2.0

dIsd/dVsd (e 2/h)

(a)

–9

Ti/Nb/Ti

(b)

Vbg (V)

–10

InSb

–11 –12

500 nm

–6

0

2

4

Vsd (mV) 0.1

0.4

dIsd/dVsd (e 2/h)

(c)

Vbg = –11.1 V, T = 25 mK

4.0

3.5 T

3.5

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2.5

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2.0

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1.5

0.7

(d)

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B

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0.5 T

0.5 0.0 –0.2

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4.0 T

6

Vbg (V)

4.5

dIsd/dVsd (e2/h)

Fig. 6.13  (a) SEM image of a S–​InSb NW QD–​S device. (b)  Intensity plot of differential conductance versus source-​ drain bias voltage Vsd and back gate voltage Vbg at 0 T and 25 mK. The measurements show a quasiparticle Coulomb blockade diamond structure. (c)  Differential conductance as a function of Vsd and magnetic field B at Vbg = –​11.1 V and T = 25 mK. The measurements are taken in the region marked by the dashed square in (b). (d) Intensity plot of differential conductance as a function of Vsd and Vbg at B = 1.8 T and T = 25 mK (Deng et  al. 2012, copyright 2012 American Chemical Society).

–4 –2

0.0 T –0.1

0.0

Vsd (mV)

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0

40

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80

  199

FM atomic chain on SCs 199 and therefore, all these superconductivity effects cannot be considered as the physical origin of the ZBP.

6.4 FM atomic chain on SCs Not only 1D semiconductor NWs but also ferromagnetic (FM) atomic chains have been proposed to couple with a conventional SC to form a 1D TSC at the ends of which MFs are predicted to localize (Nadj-​Perge et al. 2013). Recently, the system of FM iron atomic chains on the surface of superconducting lead has been studied by Nadj-​Perge et al. (2014). A Pb(110) single crystal is used as a SC with strong spin–​orbit coupling. Following submonolayer evaporation of Fe on the Pb surface at room temperature and light annealing, Fe chains grow alone the anisotropic structure of the underlying surface (Fig. 6.14). Spin-​polarized STM studies demonstrate ferromagnetism on the Fe chains and strong spin–​orbit coupling on the Pb surface (Fig. 6.15). The spin-​polarized measurements on the Fe chains show hysteresis loops characteristic of tunneling between two ferromagnets (Bode et  al. 1998, Pietzsch et  al. 2004). Tunneling with the same tip on the Pb(110) surface far from the Fe chains also shows field-​ dependent conductance. In contrast to the asymmetric behavior observed on the

(a) STM tip Ferromagnet

Superconductor (b)

[001]

200 Å

[110]

10 Å

Fig.  6.14 (a)  Schematic diagram of a ferromagnetic atomic chain on the surface of strongly spin–​orbit-​coupled SC. (b)  STM image of atomic Fe chains on Pb(110) (Nadj-​ Perge et  al. 2014, reprinted with permission from AAAS).

200

200  Topological Superconductors and Majoranas 3.5

dI/dVchain (+30 meV) (a.u.)

H 3.4

Pb

3.20

3.15 3.2

(b) dI/dVchain–dI/dVsubstrate (a.u.)

Fe

3.3

3.1

Fig.  6.15  Spin-​polarized measurements of ferromagnetic atomic Fe chains on Pb(110) (Nadj-​ Perge et  al. 2014, reprinted with permission from AAAS). (a)  STM tunneling conductance as a function of out-​ of-​ plane applied magnetic field H on the atomic chain and the substrate measured with a spin-​ polarized bulk Cr/​Fe tip. (b)  Difference between conductance on and off the chain shown in (a).

3.25

Cr

on chain on substrate –1

–0.5

0

+0.5

+1

dI/dVsubstrate(+30 meV) (a.u.)

(a)

3.10

H (T ) 0.3

0.15

0 –1

–0.5

0

+0.5

+1

H (T )

chains, the field dependence on the substrate is symmetric with magnetic field. Similar tunneling magnetoresistance curves have previously been reported for tunneling from a ferromagnet into semiconductors and have been attributed to spin-​polarized tunneling in the presence of strong spin–​orbit interactions (Moser et al. 2007). Fig. 6.16 shows spatially resolved spectra and spectroscopic maps with a normal tip on both the substrate and the Fe chain. On the Pb surface, Pb’s bulk pairing gap measured with normal tips is 1.36 meV. However, the energy gap of the proximity-​induced superconductivity on the Fe chains measured with superconducting tips is estimated to be 0.25 meV, 20% of the Pb bulk pairing gap. In the middle of the chains, spatially resolved spectroscopic measurements show the formation of features in the Pb gap, with asymmetric peaks at roughly ±1 meV, which evolve to include a sharp ZBP that is prominently detected at about 10–​20 Å from the ends of the chain. Fig. 6.16b–​e show such ZBPs at two ends of the same chain. Spectroscopic maps in Fig. 6.16f exhibit the spatial structure of excitations at different energies and clearly show the localized nature of the ZBPs at one end of the chain and the delocalized nature

  201

FM atomic chain on SCs 201 (b)

600

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400

(d)

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25 150

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–5

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50

dI/dV (nS)

(f)

150

–1.44

–1.24

–0.76

–0.44

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+1.24

+1.44

0

Energy (meV)

of the excitations at higher energies throughout the chain. These maps also resolve the spatially modulated decay of the ZBP away from the chain ends. In the experiments of Nadj-​Perge et al. (2014), the ZBPs are not associated with the Kondo effect. When the magnetic field increases to 0.1 T, superconductivity in the Pb substrate is suppressed. However, all features associated with the gap in the middle of the chains and the ZBPs at their ends also disappear in the absence of superconductivity in Pb. If the ZBPs at the ends of the chains were due to the Kondo effect, the increasing DOS near EF in the normal state would only enhance the ZBP rather than suppress it. The ZBPs are also not induced by structural or potential defects, because nonmagnetic adsorbates or step edges cannot produce in-​gap states for conventional s-​wave SCs. Moreover, in very short Fe chains (~30–​40 Å), the ZBPs are strongly suppressed, which suggests that the ZBPs do not come from disorder effect.

Fig. 6.16 (a) dI/​dV spectra measured on the atomic chain indicated in the zoom-​ in topography of the upper (b) and lower (c) ends of the chain. The scale bars are 2.5  nm. (d)  and (e)  dI/​dV spectra measured at marked locations in (b) and (c). (f)  Spatial and energy-​resolved conductance maps of another Fe atomic chain close to its end. The scale bar is 1  nm. (Nadj-​Perge et  al. 2014, reprinted with permission from AAAS)

202

202  Topological Superconductors and Majoranas

6.5 Summary and outlook In this chapter, our focuses are on several very recent experiments on possible MBSs in 2D and 1D systems contacted by s-​wave SCs. For TI/​SC heterostructures, ZBPs are spatially resolved at vortex cores. The finite-​distance splitting of the ZBPs suggests that the bound quasiparticle states within a vortex could possibly contain the Majorana mode (Xu et al. 2015), which is fully supported by a very recent theoretical investigation on the MBS in the vortex (Kawakami and Hu 2015). So far, no theory or experiment has come out against the conclusion of Xu et al. (2015). For semiconductor NWs, ZBPs are detected in the charge transport measurements (Mourik et  al. 2012, Deng et  al. 2012). For chains of magnetic adatoms, ZBPs are observed at the ends of the chains in STM measurements (Nadj-​Perge et al. 2014). Physical mechanisms which lead to the anomalous ZBP, such as the Kondo effect (Lee et al. 2012), the Josephson supercurrent (van Dam et al. 2006, Deon et al. 2011), Andreev reflections (Lee et al. 2013), level crossing (Nilsson et al. 2009), defects (Sau et al. 2013), and disorder (Adagideli et  al. 2014), can be excluded from the reported experiments (Deng et  al. 2014). Nevertheless, there still exist other interpretations for the anomalous ZBP in the 1D systems (Sau et al. 2015, Ruby et al. 2015). In brief, the experiments on proposed setups have been carried out, but no consensus has been reached as to whether MBSs have been realized in the laboratory. It is now generally thought that a truly conclusive experiment showing Majorana particles will have to test one of their other unique properties in addition to the ZBP (Elliott et al. 2015). The effects that can be probed include the fractional Josephson effect (Kitaev 2001), quantized conductance in the ballistic regime (Law et al. 2009, Wimmer et al. 2011), various tests of nonlocality (Nilsson et al. 2008, Fu 2010, Burnell et al. 2013), and the non-​Abelian exchange statistics (Alicea et al. 2011). Further experiments will be necessary to obtain truly unambiguous evidence. Recently, Sun et al. (2016) have applied spin-​polarized STM/​STS to probe spin-​selective Andreev reflection of MBSs in the Bi2Te3/​NbSe2 heterostructure, which provides direct evidence of MBS and reveals its magnetic property in addition to the ZBP. The work will stimulate the research on the novel physical properties of MFs.

REFERENCES (CHAPTER-​6 ) Adagideli, I., Wimmer, M. and Teker, A. (2014) Phys. Rev. B 89, 144506. Alicea, J. (2010) Phys. Rev. B 81, 125318. Alicea, J., Oreg, Y., Refael, G., von Oppen, F. and Fisher, M.P.A. (2011) Nature Physics 7, 412.

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206  Topological Superconductors and Majoranas Zhang, H.J., Liu, C.X., Qi, X.L., Dai, X., Fang, Z. and Zhang, S.C. (2009) Nat. Phys. 5, 438. Zhang, Y., He, K., Chang, C.Z., Song, C.L., Wang, L.L., Chen, X., Jia, J.F., Fang, Z., Dai, X., Shan, W.Y., Shen, S.Q., Niu, Q., Qi, X.L., Zhang, S.C., Ma, X.C. and Xue, Q.K. (2010) Nat. Phys. 6, 584.

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Surface and Interface Superconductivity  S. Gariglio Department of Quantum Matter Physics, University of Geneva, 24 Quai E.-​Ansermet, CH-​1211 Geneva, Switzerland

M.S. Scheurer, J. Schmalian

7 7.1 Introduction 

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7.2 Superconductivity in two dimensions  208

Institute for Theory of Condensed Matter and Institute for Solid State Physics, Karlsruhe Institute of Technology, Wolfgang-​Gaede-​Str. 1, 76128 Karlsruhe, Germany

7.3 Superconductivity in ultra-​thin metals on Si(111) 

210

7.4 Superconductivity at the LaAlO3/​SrTiO3 interface 

215

A.M.R.V.L. Monteiro, S. Goswami, and A.D. Caviglia

7.5 Summary and outlook 

231

References (Chapter-7) 

232

Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands

7.1 Introduction During the last decade, superconductivity has been observed in a variety of two-​ dimensional (2D) systems ranging from one monolayer of Pb on Si to the conducting interface between LaAlO3 and SrTiO3. These discoveries have revived interest in: (a) superconductivity in the 2D limit; (b) pairing symmetry in systems with broken inversion symmetry and in the presence of Rashba spin–​orbit interaction; and (c) coupling of substrate phonon modes to layer electronic states to induce or enhance the superconducting condensate. In this chapter, we will illustrate these theoretical ideas with a particular focus on oxide materials. We will then review the experimental ongoing efforts to fabricate, characterize, and measure these systems. We will conclude discussing theoretical propositions aimed at realizing and testing novel superconducting states.

S. Gariglio, M. Scheurer, J. Schmalian, A.M.R.V.L. Monteiro, S. Goswami, and A. Caviglia, ‘Surface and Interface Superconductivity’ in The Oxford Handbook of Small Superconductors. First Edition. Edited by A.V. Narlikar. © Oxford University Press 2017. Published in 2017 by Oxford University Press. DOI 10.1093/acprof:oso/9780198738169.003.0007

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208  Surface and Interface Superconductivity

7.2 Superconductivity in two dimensions Can superconductivity persist when a material is made thinner and thinner, eventually reaching the atomic limit? This longstanding question has been the focus of intense research for many years, both theoretically and experimentally. Progress in thin film growth and in low-​temperature in situ measurement techniques has only recently enabled the realization and probing of monoatomic layers of several superconductors like lead on silicon or FeSe on perovskite SrTiO3. Before describing the current status of the research, let us briefly review the arguments of the discussion. Blatt and Thompson (1963), following the formulation of the Bardeen–​ Cooper–​Schrieffer (BCS) theory, predicted that the quantization effect of the electronic states due to finite layer thickness results in a modulation of the superconducting gap. Their model shows that when the layer thickness is increased, and hence the sub-​band energy splitting is reduced, the abrupt jumps occurring in the density of states (DOS) when each sub-​band gets close (within the cut-​off energy) to the Fermi level induce a rise of the superconducting (SC) energy gap. The oscillations of the energy gap are known as shape resonances and are illustrated in Fig. 7.1. According to their calculations, the SC energy gap can be pushed higher by a factor of two than for infinite thickness; the existence of a condensation energy, however, does not guarantee a superconducting state. The symmetry breaking that occurs at the phase transition between the normal and superconducting states is indeed described by a complex quantity, the order parameter Φ, which has an amplitude and a phase ϕ: Φ = |Φ|eiϕ. Φ represents the superconducting condensate and can have arbitrarily small phase excitations. In particular, in two dimensions, these excitations destroy the long-​range order and the average of the order parameter becomes zero at all temperatures.

E

Increasing thickness

∆

EF k

Thickness

Fig. 7.1  (Left) Schematics of a 2D band structure with different sub-​bands split by the quantum confinement. Increasing the thickness reduces the energy sub-​band splitting (dashed lines) and brings more states close to the Fermi energy within the cut-​off energy (colored region around EF). (Right) Superconducting energy gap ∆ vs thickness of the film. The dashed horizontal line indicates the gap value for infinite thickness. Adapted from Blatt and Thompson (1963).

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Superconductivity in two dimensions 209 This situation is general to 2D systems:  the Mermin–​Wagner (1966) theorem indeed states the absence of long-​range order in 2D systems with a continuous symmetry. Therefore, we could conclude that a superconducting state cannot occur in two dimensions! Berezinskii (1971a, 1971b) and Kosterlitz and Thouless (1972) (BKT) observed that such systems can nevertheless undergo a phase transition but into a quasi-​long-​range ordered state; this was shown later by Doniach and Huberman (1979) and Halperin and Nelson (1979) to be also valid for the superconducting case. This state corresponds to a phase where vortex–​antivortex pairs, thermal excitations of the homogeneous SC state, are bound by a logarithmic attractive interaction. Up to now, the experimental observation of a BKT transition is still debated (Benfatto et al. 2013). The possible existence of superconductivity in ultra-​thin layers has also been challenged by the theory of localization which predicts that 2D electronic systems are localized for any level of disorder. Experimental evidence supporting this prediction has been provided by the work of Strongin et  al. (1970) and Haviland et al. (1989) on superconducting Pb and Bi amorphous films: a quantum phase transition from a superconducting state into an insulating state was observed upon reduction of layer thickness. The last decade has been rich in discoveries that have taken the community by surprise. In 2007, the electron gas that forms at the interface between a LaAlO3 layer and a SrTiO3 single crystal (Ohtomo and Hwang 2004) was shown to be superconducting by Reyren et  al. (2007). The 2D character of the condensate was confirmed by analysis of the resistive transitions in magnetic fields applied parallel and perpendicular to the interface:  the estimated in-​plane coherence length (∼60 nm) is much larger than the vertical extension evaluated at 10 nm (Reyren et al. 2009). In 2010, Zhang et al. (2010) reported the observation of a superconducting state in a monoatomic layer of Pb (and of In) grown epitaxially onto the (111) surface of Si. Performing scanning tunneling spectroscopy, they could measure the superconducting gap as well as observe vortices in the magnetic field, proof that quasi-​long-​range order establishes in these layers. More recently, an atomic layer of FeSe epitaxially grown on SrTiO3 has revealed signatures of a superconducting critical temperature at ∼70 K, well above the Tc (9 K) of bulk FeSe (Wang et al. 2012). Also a new field-​effect technique which allows the doping of an insulating material with surface charges up to 1015 cm−2 (Misra et al. 2007) has been used to search for superconductivity in novel compounds. Beyond the proof-​of-​principle case reported by Ueno et al. (2008) for a material already known to superconduct once doped, SrTiO3, this approach has allowed the discovery of superconductivity in a dielectric material, KTaO3 (Ueno et al. 2011), albeit at extremely low temperature (Tc = 50 mK). These results have spurred the superconducting research community to look deeper into these phenomena, which raise questions such as: Is the pairing mechanism in a one-​atom-​thick layer the same as the one in bulk? Is the symmetry of the superconducting state modified by the confinement of the electrons in these

210

210  Surface and Interface Superconductivity structures? In the next sections, we will review the experimental results on superconductivity appearing at the interface or in ultra-​thin layers on the surface of crystalline substrates. We then discuss theoretical ideas on the role of the confinement potential with respect to the superconducting state. We will finally describe the fabrication and properties of nanostructures realized at the LaAlO3/​SrTiO3 interface to probe the nature of the superconducting order.

7.3 Superconductivity in ultra-​thin metals on Si(111) Superconductivity at the atomic limit has been observed in layers of Pb and In on Si(111) substrates (Zhang et al. 2010, Uchihashi et al. 2011). For a long time, researchers have investigated metals (Pb, In, Bi) and their superconducting behavior in thinner and thinner layers. In the amorphous form, transport measurements show that superconductivity is progressively lost when the layer thickness is decreased; concurrently, the sheet resistance increases until a localized state is reached (Strongin et al. 1970, Haviland et al. 1989). With technological advances enabling the fabrication of increasingly thin crystalline films and their in situ measurements at low temperatures, the search for the thickness limit for the superconducting state has opened a new chapter. Nowadays, clean Si(111) surfaces, usually showing a 7 × 7 reconstruction, are used as a substrate for the growth of ordered layers. For Pb and In, deposition occurs by thermal evaporation in a molecular beam epitaxy system at room temperature. A subsequent thermal anneal defines the layer phase: for a Pb monolayer annealed at ∼500 K for ∼30 s a striped incommensurate phase (SIC) is obtained; for a 2-​min annealing time, the Pb layer is in a 7 × 3 phase (Hupalo et al. 2003, Zhang et al. 2010). Other phases have been observed for thicker films, according to the thermal treatment (Ganz et al. 1991, Eom et al. 2006, Qin et al. 2009). The SIC phase where four Pb atoms cover three Si atoms is shown in Fig. 7.2a: this configuration results in covalent bonding between Si and three Pb atoms and metallic bonding between the Pb atoms (Pb is in a 2+ valence state). The 7 × 3 Pb phase has a smaller coverage, with 6 Pb atoms for 5 Si atoms: similarly, 5 Pb atoms covalently bond to the Si surface and 1 Pb atom remains without bonding to Si. High-​resolution scanning tunneling microscopy (STM) and low-​energy electron diffraction (LEED) reveal the surface structure of the Pb and In layers, allowing the different phases to be clearly identified. Fig. 7.2b displays a STM image of the SIC-​Pb phase: the Pb ordering is clearly visible. Angle resolved photoemission spectroscopy (ARPES) shows that the electronic structure of the Pb and In monolayers is made of metallic surface-​ state bands having a parabolic dispersion and centered at high symmetry points (Γ and K) (Rotenberg et al. 2003, Choi et al. 2007, Zhang et al. 2010). For both compounds, independent of the ordering phase, the Fermi contours measured by ARPES can be reproduced by a 2D free-​electron model.

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Superconductivity in ultra-thin metals on Si(111) 211 (a)

(b)

Pb Si (c)

(d)

2.00 K 1.82 K

1.2

1.78 K 1.70 K

1.50 K

α F (ω)

1.42 K 1.30 K

2

dI/dV (a.u.)

Bulk

λ = 1.41

1.0

1.58 K

1.15 K

0.8 0.6 0.4

1.05 K 0.82 K

SIC-Pb

λ = 0.72

0.2

0.65 K

0.0 0

0.42 K

–4 –3 –2 –1 0 1 2 3 4 Bias (mV)

2

4

6

8

10

12

14

Energy (meV)

Fig. 7.2  (a) Side view (top) and top view (bottom) projections of a Pb monolayer in the striped incommensurate phase (SIC) grown on a Si(111) 7 × 7 surface.Two different Pb–​ Pb bond lengths are present, the shorter one depicted by the blue lines. Image reprinted from Noffsinger and Cohen (2011) with permission from Elsevier. (b)  High-​resolution STM image of the SIC–​Pb phase. The scan size is 9.6 nm × 9.6 nm. Reprinted by permission from Macmillan Publishers Ltd:  Nature Physics (Zhang et  al. 2010). (c) The differential conductance of the Nb tip/​vacuum/​SIC–​Pb tunneling junction measured for different temperatures. Reprinted by permission from Macmillan Publishers Ltd:  Nature Physics (Zhang et  al. 2010). (d)  The total electron–​ phonon coupling λ = 2 ∫ (α 2 F (ω )/ω ) dω calculated for Pb from first-​principles decreases from 1.41 for the bulk (McMillan and Rowell 1965, Franck et al. 1969) to 0.72 for the monolayer. Image reprinted from Noffsinger and Cohen (2011) with permission from Elsevier.

7.3.1 The experimental evidence The superconducting state and the critical temperature of atomically flat Pb and In films with thicknesses down to a monolayer have been measured with different techniques and it was found that fluctuations do not destroy superconductivity. In one-​atom-​thick films, superconductivity was initially probed by scanning tunneling spectroscopy (STS) (Zhang et al. 2010): the differential conductance of the Pb monolayer/​vacuum/​Nb tip junctions, shown in Fig. 7.2c, reveals a superconducting gap with well-​defined coherence peaks. The analysis of the spectra

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212  Surface and Interface Superconductivity yields an energy gap ∆ of 0.35 meV and a superconducting critical temperature, Tc, of 1.8 K, resulting in a BCS ∆/​kBTc ratio of 4.44 (kB is the Boltzmann constant) for the SIC phase; for the 7 × 3 phase, Tc is lower, 1.5 K, as is the gap, 0.27 meV. In these monolayers, the Tc of lead is strongly suppressed with respect to the bulk value of 7.2 K. Monolayers of In can also be prepared in the 7 × 3 phase: their critical temperature, estimated from the tunneling spectra, is 3.2 K, closer to the bulk value of 3.4 K. STS imaging in a magnetic field reveals superconducting vortices with a core size of 50 nm, giving a first estimation of the coherence length. Transport measurements have been performed using an in situ van der Pauw configuration to observe the macroscopic supercurrent (Uchihashi et  al. 2011, Yamada et al. 2013). In contrast to amorphous layers that display an increase of the sheet resistance and a loss of the superconducting transition when the thickness is reduced, crystalline films exhibit a normal state sheet resistance RN at 10 K of a few hundreds of Ohms, a value much smaller than the critical sheet resistance h/​4e2 (= 6.45 kΩ) for the superconducting-​insulator transition and sharp superconducting transitions. For the 7 × 3 -​In phase, transport data reveal a Tc of 1.8 K and RN ∼400 Ω. I−V characteristics display sharp switching between superconducting and normal states. Analysis of the temperature dependence of the critical current indicates that the supercurrent has a behavior like the one observed for a Josephson junction. This effect stems from the presence of the step edges, which act as weak links/​Josephson junctions between the superconducting regions on the terraces. Careful analysis of tunneling spectra confirms the Josephson junction behavior of the condensate across a step edge: a sharp discontinuity is observed in the zero-​bias conductance when going from the upper to the lower terrace (Brun et al. 2014). These data point to a disruption of the superconducting order occurring across a step edge like in a junction. This is not surprising as the step edge height (∼0.33 nm) is of the order of the monolayer thickness. This configuration has a profound effect on the vortex physics. When a magnetic field is applied perpendicular to the layer, vortices appear in these type-​II superconductors: for a 2D system, these are Pearl vortices with a magnetic extension given by the Pearl length Λ = 2λ2/​d where λ is the London penetration depth and d the layer thickness (Pearl 1964). These Pearl vortices have been imaged by STS on the flat terraces of the ( 7 × 3 )-​In phase (Yoshizawa et al. 2014). More interesting is what happens at the step edges: here, even when the magnetic field is removed, vortices can get pinned. These latter can be better described as Josephson vortices: their shape is elongated along the step edge and the variation of the differential conductance (in particular at zero bias) across the vortex is less than the one observed for Pearl vortices (Blatter et al. 1994, Kogan et al. 2001). These are the characteristics of Josephson vortices:  the field penetrates in the barrier region and hence the superconducting state is less perturbed. This observation further confirms that the atomic steps work as Josephson junctions. STS mapping has also revealed Josephson vortices in a network of Pb superconducting

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Superconductivity in ultra-thin metals on Si(111) 213 islands connected by a metallic amorphous Pb layer (Roditchev et  al. 2015). These objects also attract interest for applications in devices for quantum computing (Devoret and Schoelkopf 2013). What is the effect of disorder on these layers? STS experiments have revealed that, despite the crystalline perfection over regions as large as the terraces, the height of the coherence peaks fluctuates (∼20%) over distances (2–​10 nm) much shorter than the coherence length (∼50  nm). Conductance maps acquired at zero bias do not show variations, indicating that no sub-​gap electronic states are generated by structural defects observed topographically in these regions. These are puzzling data: How can the amplitude of the coherence peaks fluctuate over such short distances? The Anderson theorem moreover predicts that disorder has no effect on the coherence peaks. Brun and coworkers suggest that virtual processes induced by the electron–​electron interaction could be at the origin of such behavior. According to their model, a broadening of the coherence peak should be induced, without the appearance of in-​gap states. The DOS computed in this scenario provides a good fit to the experimental spectra. These are marvelous results:  beyond many effects that could hamper the superconducting coupling and the establishment of an order parameter with long-​ range correlation, superconductivity is observed in a one-​atom-​thick Pb layer.

7.3.2 The behavior of Tc in Pb layers at  the atomic limit Previous work has shown that the critical temperature (Ozer et  al. 2007)  and the superconducting gap (Brun et al. 2009) decrease as a function of the inverse layer thickness. For two-​monolayer-​thick films, STS reveals a Tc ∼4.9 K for the 1×1 atomic structure and 3.65 K for the 3 × 3 phase (Qin et al. 2009). Considering the Pb layers in the strong-​coupling regime, we can estimate their superconducting transition temperature from the McMillan formula (Allen and Dynes 1975). According to this formula, modifications of the DOS at the Fermi level (EF) due to quantization effects or changes in the electron–​phonon coupling λ would directly affect Tc. Concerning the DOS, the decrease in thickness indeed induces a quantization effect, creating electronic sub-​bands whose energy separation increases as the thickness is reduced. This results in an oscillatory behavior of the DOS at EF and has been suggested to be at the origin of the oscillations of Tc observed in the 2–​15 monolayer range (Blatt and Thompson 1963, Guo et al. 2004, Eom et al. 2006). For the overall decrease of Tc with thickness, ab initio calculations indicate that the electron–​phonon coupling strength is suppressed with the inverse of the layer thickness (Brun et al. 2009, Noffsinger and Cohen 2010). In a phenomenological picture, the electron–​phonon coupling can be described as the averaged sum of a Pb/​Si interface term, a Pb bulk term, and a Pb surface term. This simple model fits the experimental data well. The decrease in the average λ is due to stiffer phonon modes at the surface and at the interface due to the different bondings of Pb. Remarkably, first-​principle density functional

214

214  Surface and Interface Superconductivity theory calculations (Noffsinger and Cohen 2011) for a single layer of Pb grown on Si(111) in the SIC phase provide a Tc = 1.5 K in extremely good agreement with the experimental value of 1.8 K.

7.3.3 Beyond the Pauli limit in single-​atom Pb layers A single-​atom layer is also an interesting system for the study of the effects of the broken inversion symmetry on the superconducting state. The asymmetry of the confining potential, together with the atomic spin–​orbit coupling present in a heavy element like Pb, is expected to lift the spin degeneracy of the electronic states, a phenomenon known as the Rashba effect (Rashba 1960). So far, experiments to probe this particular electronic configuration in the superconducting state have focused on the effect of a magnetic field applied parallel to the conducting layer. Applied perpendicularly, a magnetic field affects the orbital motion of the electrons in the 2D plane, while in a parallel configuration, the field is expected to couple only to the spin degree of freedom. In this orientation, the magnetic field will destroy superconductivity when the condensation energy (∆) is compensated by the reduction of the normal state energy due to the Pauli paramagnetic susceptibility ( g µ 0µ B H , with g the electron spin g-​factor): for this reason this limit is known as the Pauli paramagnetic limit (Chandrasekhar 1962, Clogston 1962). Magnetotransport measurements on one-​atom-​thick Pb layers (Tc0 = 0.9 K) deposited on GaAs(110) surfaces show that the superconducting transition has a very weak dependence on the parallel magnetic field: for a field of 14 T, well above the Pauli paramagnetic limit estimated at 1.7 T, a decrease of Tc of only 60 mK is observed (Sekihara et al. 2013). To account for such robustness of the superconducting state, different theories have been proposed that consider the presence of a Rashba spin splitting on the pairing interaction (Gor’kov and Rashba 2001, Barzykin and Gor’kov 2002). The data could be best fit in the framework of the long-​wavelength helical state proposed by Dimitrova and Feigel’man (2007). In this model, the Rashba field induces a non-​zero momentum on the Cooper pairs and thereby causes a spatial variation of the order parameter. The theory predicts that the critical parallel field scales as Tc with a proportionality factor related to the elastic scattering rate. The root square temperature behavior of H‖ is a common feature of 2D superconducting layers and is also observed in the data. Interestingly, the best fits to the experimental curves yield an estimate of the elastic scattering times in good agreement with those estimated from the normal state resistance. Indications of a strong Rashba term have been also revealed by scanning tunneling spectroscopy on 7 × 3Pb monolayers (Brun et al. 2014). The tunneling spectra show coherence peaks which are lower and broader than one predicts from BCS theory. They also display local variations on a scale that is shorter than

  215

Superconductivity at the LaAlO3/​SrTiO3 interface 215 the coherence length. The current interpretation of these results points to the presence of a triplet component in the superconducting phase. This momentum-​ dependent part of the order parameter would be extremely sensitive to local non-​ magnetic scattering centers in contrast to an s-​wave component, which, according to the Anderson theorem, is insensitive to non-​magnetic disorder: the coherence peaks would hence be reduced and broadened by the presence of non-​magnetic impurities.

7.4 Superconductivity at the LaAlO3/​SrTiO3 interface The emergence of a metallic 2D electron liquid near the interface of two insulators, LaAlO3 and SrTiO3, is a beautiful example for an engineered state of matter (Ohtomo and Hwang 2004). At low temperatures the 2D metallic state becomes superconducting (Reyren et al. 2007), a phase that can be tuned by the application of an external gate voltage (Caviglia et al. 2008). A unique aspect of such an interface is the broken inversion symmetry which inevitably leads to momentum-​ dependent Dresselhaus–​ Rashba spin–​ orbit interactions (Dresselhaus 1955, Rashba 1960). The effect of such an interaction was indeed seen in magnetotransport measurements (Ben Shalom et al. 2010, Caviglia et al. 2010). The associated splitting of the Fermi surface was observed via ARPES on the surface of SrTiO3 (Santander-​Syro et al. 2014, King et al. 2014). It was shown by Gor’kov and Rashba that this splitting leads to a mixed singlet–​triplet pairing state, regardless of the microscopic details of the pairing mechanism (Gor’kov and Rashba 2001). In what follows we give a brief summary of the properties and electronic structure of LaAlO3/​SrTiO3-​interfaces, discuss the allowed pairing states and the mechanisms proposed for the Cooper pairing, and demonstrate that these materials might even be candidates for topological superconductivity. Finally we will present nanodevices fabricated at the interface to probe the properties of the electron liquid.

7.4.1 The LaAlO3/​SrTiO3 interface SrTiO3 (STO) belongs to the family of perovskite oxides: at room temperature it has a cubic close-​packed lattice with Sr atoms at the corners, O atoms at the face center, and a Ti atom in the middle of the cube. The Ti4+ ion is in an octahedral coordination with the six surrounding O2− ions. STO possesses a variety of interesting electronic properties which make this material an excellent candidate for several applications. An impressive example is its large dielectric constant (300 at room temperature), which tends to diverge at low temperature (∼10,000 at 4 K) due to its quantum paraelectric (or incipient ferroelectric) state (Müller and Burkard 1979, Zhong and Vanderbilt 1996). STO can be electron-​doped by

216

216  Surface and Interface Superconductivity

LaAIO3 SrTiO3

Fig. 7.3  Side view of a LaAlO3/​SrTiO3 interface acquired using Scanning Transmission Electron Microscopy:  the [001] axis is along the vertical direction while the horizontal direction is [100]. On a TiO2-​ terminated (001) SrTiO3 substrate, a 10 unit cell thick LaAlO3 layer is grown by pulsed laser deposition. The superconducting gas sits in the first 20 SrTiO3 unit cells close to the interface. The image was acquired by G.  Tieri, A.  Gloter, and O.  Stephan at LPS, University of Paris-​Sud.

introducing oxygen vacancies, substituting Nb on Ti or La on Sr:  an insulator to metal transition occurs (Spinelli et al. 2010) when the electron concentration exceeds 1016 cm−3. For larger carrier densities (> 1018 cm−3), a superconducting state appears below a few hundred mK, and disappears for doping larger than 1021 cm−3 (Koonce et al. 1967). STO crystals are also the reference substrates for the growth of high-​Tc superconductors and other complex oxide perovskites due to their chemical and structural compatibility. For this application, an etching treatment allows one to achieve a single chemical termination (TiO2) (Kawasaki et al. 1994), enabling the growth of epitaxial films with higher crystalline quality. The deposition of an insulating LaAlO3 (LAO) layer onto such a terminated surface gives rise to the appearance of an electron liquid at the interface (Ohtomo and Hwang 2004). Fig. 7.3 shows the interface viewed with a scanning transmission electron microscope:  the chemical sensitivity of the Z-​contrast technique reveals the sharpness of the interface. Electrons accumulate in the STO side as a screening response to the polar discontinuity that occurs between the non-​polar (001) surface of STO and the polar LAO layer. Estimated from electric transport, the carrier density of this electron liquid falls in the bulk superconducting pocket and, indeed, the interface becomes superconducting below ∼300 mK (Reyren et al. 2007). For a complete description of this interface, we refer the reader to the several reviews that detail the properties of this system (Zubko et al. 2011, Hwang et al. 2012, Gariglio et al. 2015b). In the following, we focus on the relevant points concerning superconductivity. One longstanding argument on superconductivity in STO is the consequence of the multi-​orbital population on the coupling mechanism. For the interface system, Ti t2g electrons occupy dxy and dzx/​dzy orbital-​dominated bands. This multi-​ orbital electronic state can generate single-​gap or multi-​gap superconductivity. In bulk Nb-​doped STO/​In junctions, Binnig et al. (1980) reported the observation of two SC gaps:  measuring tunneling I−V and (dI/​dV)−V characteristics, they observed two distinct superconducting gaps when the doping exceeded a threshold value. Investigation of LAO/​STO interfaces, so far, has not revealed signatures of two gaps. Measurements of superfluid density by local magnetic susceptibility can be described by a single superconducting gap (Bert et al. 2012). This observation confirms a previous report on the temperature behavior of the critical magnetic field (applied perpendicular to the interface) which does not show anomalies due to the opening of a second gap (Reyren et al. 2009). Direct probing of the superconducting state by tunneling experiments also fits a single-​ gap scenario (Richter et al. 2013). Several effects could mask the clear observation of a second gap, hence the question of its existence remains open. One feature proper to the interface is that electrons are localized in an asymmetric confining potential. The electron liquid resides in STO, a material that has a large and electric-​ field-​ dependent dielectric constant:  this setting produces a confining potential with a complex profile (Gariglio et al. 2015a), which breaks the vertical inversion symmetry of the electronic states confined here. This breaking of inversion symmetry, also observed in 2D electron gases present in

  217

Superconductivity at the LaAlO3/​SrTiO3 interface 217 semiconductor heterostructures, is at the origin of the Rashba spin–​orbit interaction (Winkler 2003) observed in magnetotransport experiments (Caviglia et al. 2010). This interaction couples the spin state of an electron to its momentum and modifies the electronic band structure:  the spin degeneracy is removed and reversing the particle momentum also requires a flip of the spin state. In condensed matter, this coupling between particle motion and spin generates a fascinating set of phenomena such as the anomalous Hall effect (Nagaosa et al. 2010), the spin-​Hall effect (Hirsch 1999), and the spin-​Seebeck effect (Uchida et al. 2008). Concerning superconductivity, the Rashba interaction is expected to have dramatic effects on the superconducting properties of the system when the spin–​orbit coupling is larger than the superconducting gap (Gor’kov and Rashba 2001, Yip 2002).

7.4.2 The electronic structure In order to develop a model for the electronic structure of the oxide interfaces we use the fact that the key electronic states are formed by the titanium 3d-​shell of SrTiO3. Ignoring oxygen vacancies and charge transfer into SrTiO3, an oxygen valency of −2 and the +2 state of strontium imply a Ti4+ state. The atomic electron configuration [Ar]3d24s2 of Ti further yields that the 3d-​shell and the 4s-​states are emptied out by the charge transfer to oxygen. Reducing the oxygen content through vacancies will then leave a certain population of electrons in the 3d-​shell. This is responsible for the metallic behavior of bulk SrTiO3−δ and is known to be of importance in interface systems as well (Pavlenko et  al. 2012, 2013). In the case of the LaAlO3/​SrTiO3-​interface, the charge transfer into the Ti 3d-​ shell is believed to be governed, at least in part, by the so called polar catastrophe (Nakagawa et  al. 2006, Popović et  al. 2008). The alternating averaged charge of La3+O2− and Al3+O24− layers would amount to a voltage build-​up, increasing with the thickness of LaAlO3, similar to capacitors that are placed in series. A transfer of half an electron per unit cell from the topmost AlO2-​layer into the Ti 3d orbitals will relax this voltage (Nakagawa et al. 2006, Popović et al. 2008). Thus, beyond a threshold number of unit cells of LaAlO3, the build-​up voltage becomes larger than the energy required to transfer one electron into the Ti 3d states, resulting in an accumulation of 0.5 electrons per surface unit cell. This is consistent with the experimental observation of a threshold number of LaAlO3-​ layers for metallic behavior of the interface (Thiel et al. 2006). However, similar behavior in systems where the polar discontinuity is absent obtained by growing along the [110] direction (Herranz et al. 2012) or where LaAlO3 was replaced by amorphous Al2O3 (Fuchs et al. 2014), that is, where the straightforward polar catastrophe does not seem to apply, revealed that there are several related charge transfer mechanisms at work. Near the interface, the crystal field splitting of the 3d-​states is such that the bottom of the 3dxy-​orbital is about exy ≈ 200 meV lower than the degenerate

218

218  Surface and Interface Superconductivity 3dxz and 3dyz orbitals (Zhong et al. 2013, Berner et al. 2013b), while the two eg-​ orbitals are higher in energy by more than an electronvolt. Note, as pointed out by Fernandes et al. (2013), this order of the orbital splitting is different from the one in bulk SrTiO3. Confining ourselves to the lowest three d-​orbitals we obtain the kinetic energy. H kin = ∑ ψ †kα h k ψ kα ,



k ,α

(

(7.1)

)

where ψ †kα = d k†, xy , α , d k†, xz , α , d k†, yz , α combines the t2g -​orbitals. Furthermore, we use:



 k2  2m − ε xy − µ  l  h k =  − i δky    − i δkx 

i δky

    ηkx ky  . (7.2)   ky2 kx2 + − µ 2mh 2ml  i δkx

k2 kx2 + y −µ 2ml 2mh

ηkx ky

Here we adopted a continuum’s expansion of the dispersion near the bottom of the bands. It is straightforward to return to a proper lattice version, with lattice 2 1 constant a, by replacing kx , y a → 1 − cos kx , y a and kx , y a → sin kx , y a . The 2 mixing term between the dxz and dyz orbitals, caused by next-​nearest neighbor hopping between the two orbitals, is governed by η. Since the interface is not symmetric under z → –​z, the coupling between the dxy and dxz (dyz) orbitals through hopping along the y (x) direction leads to terms proportional to δ. In the tight-​ binding language

(



)

(

)

(

)

δ = E 0a ψ xy ,R z ψ xz ,R + ae y , (7.3)

where E0 is the electric field perpendicular to the surface caused by the polar environment and ψ j ,R is the tight-​binding wave function of orbital j and Bravais lattice point R. This overlap, together with atomic spin–​orbit interaction,



H so =

λ ∑ ψ †kα L ⋅ σαβ ψ kβ , 2 k

(7.4)

is crucial for the microscopic justification of the much-​discussed Dresselhaus–​ Rashba spin–​orbit interaction of the oxide interfaces. In Eq. 7.4, λ is the atomic  is the vector of the angular momentum operaspin–​orbit coupling strength and L tors projected to the three d-​orbitals under consideration. The three 3 × 3 matrices,  1 = λ 2, L  α are given in terms of the Gell–​Mann matrices (Georgi 1999) L  2 = − λ 5 L 2   γ,    and L 3 = λ 7 . It holds L = leff (leff + 1) with leff  =  1 and [ L α , L β ]− = −i eαβγ L

  219

Superconductivity at the LaAlO3/​SrTiO3 interface 219 that is, with an additional minus sign compared to a usual l  =  1 orbital state. Combining the orbital and spin degrees of freedom in the six-​component spinor ψ k = (ψ k↑ , ψ k↓ ), the full kinetic energy part of the Hamiltonian is given as H 0 = H kin + H so = ∑ ψ †k ˆ k ψ k ,



k

(7.5)



with the 6 × 6 matrix of band-​energies

λ   h + L  k 2 z ˆ k =   + iL y λ L 2 x

(



(

)

λ  y Lx − iL 2 λ h k − L z 2

)

.  

(7.6)

The resulting electronic structure is shown in Fig. 7.4 where we used the parameters given in the caption. It compares well with ARPES measurements on the hidden surface in LaAlO3/​SrTiO3-​heterostructures (Berner et al. 2013b) and on the surface of SrTiO3 (Santander-​Syro et al. 2014, King et al. 2014). Another issue worth mentioning is that the number of mobile carriers is significantly smaller compared to the electron density expected from the polar catastrophe or from the value estimated from high-​energy spectroscopy (Berner et  al. 2013a). One interpretation is that states in the dxy-​band, that are occupied first and that are more exposed to disorder at the surface given their spatial extend (see e.g. King et al. 2014), are localized. They do not contribute to the mobile carriers determined in transport measurements that are formed by the dxz-​ and dyz-​dominated bands. This picture is also consistent with the observation that the onset of metallic behavior and superconductivity is closely tied to the Fermi energy entering the dxz-​ and dyz-​dominated bands (Joshua et al. 2012, Fête et al. 2012). To demonstrate explicitly that the broken inversion symmetry, expressed in terms of δ, together with the atomic spin–​orbit coupling λ, is responsible for

(a) 100

xz

1.5

yz

0

1 0.5

ky a

εks/meV

(b)

xy

–100

0 –0.5 –1

–200

ky = 0 –1.5 –1 –0.5

xy 0

kx a

0.5

–1.5 1

1.5

–1.5 –1 –0.5

0

kx a

0.5

1

1.5

Fig. 7.4  A one-​dimensional cut (ky = 0) of the spectrum together with the predominant orbital character of the bands and the resulting Fermi surfaces of the three-​orbital Hamiltonian in Eq. 7.6 are shown in (a) and (b), respectively.We have set µ = 20 meV, ε xy = 200 meV , (2mla2)−1  =  300 meV, λ = 20 meV, δ/​a = 40 meV (Zhong et  al. 2013), mh/​ml  =  30 (Santander-​ Syro et  al. 2011), and η/​a2  =  140 meV (Joshua et al. 2012).

220

220  Surface and Interface Superconductivity Dresselhaus–​Rashba couplings we confine ourselves for the moment to the xz-​ and yz-​orbitals. A symmetry analysis for the point group C4u at the (001) interface together with time-​reversal symmetry reveals that the allowed spin–​orbit terms are to zeroth and first order in k of the type (Scheurer and Schmalian 2015) eˆk , so = βα 2σ3 + γ 0α 0(kx σ2 − ky σ1 ) + γ 1α1(kx σ1 − ky σ2 ) + γ 3α 3(kx σ2 + ky σ1 ).



(7.7)

Here σi are 2 × 2 Pauli matrices in spin space and αj corresponding 2 × 2 Pauli matrices in orbital space, respectively. The coupling constants β and γj can be obtained from the above Hamiltonian H0 if we integrate out the xy-​orbital. Since the energy splitting between dxy and the other two orbitals is larger than λ and δ, we can perform a perturbation theory, similarly to the analysis of Petersen and λ(ε xy +λ ) δλ Hedegård (2000), and obtain Eq. 7.7 with β =  and γ 0 = − γ 1 = − γ 3 = . ε xy ε xy

7.4.3 Pairing in oxide interfaces—​ theoretical considerations 7.4.3.1 Symmetry-​allowed pairing states In this section we give a brief summary of the allowed pairing states. We focus on the (001) termination of the interface. Cooper pairing is characterized by a corre k in orbital and spin space. The allowed order parameters sponding 6 × 6 matrix ∆ can then be classified according to the irreducible representations of the point group. For the (001)-​termination this is C4u, as indicated in Table 7.1.

Table 7.1  Irreducible representations of C4u, relevant for a (001) termination of the LaAlO3/​SrTiO3 interface, along with the corresponding behavior under time reversal. Irred. rep.

Pairing

Symmetry 1,x2

y2

Θ

A1

s-​wave

A2

g-​wave

xy (x2 − y2)

Y

B1

d x2 − y2 -​wave

x2 − y2

Y

B2

dxy-​wave

xy

Y

E(1,0)

e(1,0)

x

Y

E(1,1)

e(1,1)

x+y

Y

E(1,i)

e(1,i)

x + iy

N

+

Y

  221

Superconductivity at the LaAlO3/​SrTiO3 interface 221 Crucial for our argumentation is that the broken inversion symmetry, together with spin–​orbit interaction, lifts the Kramers degeneracy for a given momentum k that occurs in inversion-​symmetric situations. Thus, the band states

Hˆ k φks = ε ks φks (7.8)

above the superconducting transition temperature are, except for isolated high-​ symmetry points, singly degenerate. If we further assume that the normal state does not break time-​reversal symmetry, that is, that there is no magnetic order present, one can establish for the eigenstate of each band the relation

φks = e − i ϕks Θφ − ks (7.9)

between wave functions with momenta k and −k. Here Θ = TΘ is the time-​ reversal operator consisting of the complex conjugation  and a unitary operator TΘ . The property Θ2 = −1 of the time-​reversal operator for fermions implies for −iϕ −iϕ the phases ϕ ks that the condition e Ks = −e − ks must be obeyed. The matrix elements of the pairing operator in the eigenbasis read as

 kT φ . (7.10) ∆ ks ,s ′ = φ ks ∆ Θ ks ′

The pairing energy (∼40 µeV) and the associated superconducting transition temperature of the oxide interfaces are significantly smaller than the spin–​ orbit Rashba coupling (∼5 meV) of these systems. Therefore, it is natural to assume that the Cooper pairs are formed out of states within the same bands, implying that

∆ ks ,s ′ = δ s ,s ′ ∆ ks (7.11)

is diagonal in the band index. The resulting spectrum of the superconducting state is then given by

Eks = ± ε k2s + | ∆ ks |2 . (7.12)

Eq. 7.11 is valid for kBTc  E so and referred to as the weak-​pairing limit. In this limit we obtain with the help of Eq. 7.9 the following relation for the gap values under rotations by π:

∆ ks = + ∆ − ks . (7.13)

Thus, the combination of the weak-​pairing limit, the symmetry behavior under time reversal, and the broken inversion symmetry at the interface imply that only

222

222  Surface and Interface Superconductivity pairing states that are even under twofold rotations are allowed (Scheurer et  al. 2015). Since twofold rotations are elements of the point group, this has direct implications for the possible pairing states: pairing in the irreducible representation E, which is odd under this rotation, is forbidden. In particular, this implies that no time-​reversal symmetry-​breaking pairing state can occur via a second-​order phase transition (Scheurer et al. 2015). If we further use the experimental observation of a fully established gap (Richter et al. 2013), the only remaining allowed state is s-​wave pairing. While those are significant constraints for any microscopic theory of superconductivity, we will see in the next section that it excludes neither unconventional nor the much-​discussed topological pairing states for LaAlO3/​SrTiO3 interfaces. 7.4.3.2 Microscopic approaches Before we discuss various microscopic approaches for the oxide interfaces we stress that one of the peculiarities of LaAlO3/​SrTiO3 interfaces is their comparatively low carrier concentration, an effect that is even more pronounced in bulk SrTiO3 (Lin et  al. 2014). Thus, it is not obvious whether the usual adiabatic assumption that the Fermi energy is much larger than the characteristic boson frequency is fully justified. However, at least in the limit of weak pairing interactions the generalizations to the non-​adiabatic regime have recently been formulated (Gor’kov 2015, Chubukov et al. 2016). The identification of a microscopic pairing interaction is usually a highly non-​ trivial endeavor. Besides the isotope effect, the agreement between fine structures in tunneling spectra with the associated structures in the phonon DOS of Pb is generally considered convincing evidence for phonon-​induced pairing in conventional superconductors (McMillan and Rowell 1965). Alternatively, phase-​sensitive measurements in copper-​oxide high-​Tc superconductors, which revealed a pairing state with d x2 − y2 -​symmetry (Wollman et al. 1993, Tsuei et al. 1994), give strong support for an electronic pairing mechanism, where spin degrees of freedom play an essential role (Scalapino 1995, Chubukov et al. 2003). At this point it is not only an open issue whether the pairing mechanism in the oxide interfaces is conventional or not, even the sharp theoretical and experimental distinction between both regimes is not obvious at first glance. The symmetry of the pairing state is not always a good guiding principle (examples are the iron-​based superconductors or orthorhombic YBa2Cu3O7 where the pairing state transforms under the trivial representation). A possible theory-​based definition for an unconventional mechanism is one where one does not obtain the same pairing state if one adiabatically switches off the electron–​electron Coulomb interaction. In what follows we use this definition. Evidence for electron–​ phonon coupling in LaAlO3/​SrTiO3 interfaces was recently obtained using inelastic tunneling spectroscopy (Boschker et al. 2015). The emergence of fine structures in the tunneling conductance due to inelastic tunneling, that is, due to the excitation of real bosons, is a clear signature for electron–​phonon coupling (Jandke et al. 2016), while the method cannot be used to quantitatively determine the coupling constant. Due to the small value

  223

Superconductivity at the LaAlO3/​SrTiO3 interface 223 of the superconducting transition temperature, it was, however, not possible by Boschker et  al. (2015) to determine whether the observed shift of the phonon frequencies with isotope mass also yields a shift in Tc. Thus it seems premature to exclude unconventional pairing in these systems, in particular given that transition metal oxides are known for significant electronic interactions. The interaction with lattice vibrations as cause for superconductivity was discussed in detail by Klimin et al. (2014) with particular emphasis on the strongly polar nature of the heterostructures. Taking into account the interaction between electrons and optical phonons and including the screening of the Coulomb interaction in a layered polar interface, the authors determine a frequency-​dependent effective pairing interaction based on the approach by Takada (1980). This results in a superconducting transition temperature Tc, obtained from the linearized gap equation

∆ (ω ) = −

 ω′  1 ∞ dω′ ∆(ω ′ ), (7.14) K (ω , ω ′ )tanh  2 ∫− ε F ω ′  2Tc 

with pairing kernel K (ω , ω ′ ). Tc was then determined as a function of carrier concentration n. The key finding is that the interplay between the screening of the Coulomb interaction, as encoded in the pairing kernel K ( ω , ω ′), and the size of the Fermi energy ε F gives rise to a non-​monotonic dome-​shape of Tc (n), in qualitative agreement with experiment (Caviglia et al. 2008). Another interesting aspect of the theory by Klimin et al. (2014) is that the effective pairing interaction is always repulsive, that is, K ( ω , ω ′) > 0, an effect primarily caused by the rather moderate screening of the Coulomb repulsion at low carrier concentrations. A sizeable frequency dependence of K (ω, ω ′ ), which is least repulsive for small ω and ω ′ , does, however, allow for non-​trivial solutions of the superconducting gap equation. Such solutions should however come along with a significant variation of the gap function ∆(ω ) as a function of frequency, where one expects a sign change of ∆(ω ) for ω ≈ ε F , a behavior that might be visible in tunneling experiments. It would be interesting to explore whether such a sign change in ∆(ω ) in particular and diabatic pairing in general, impact the dynamic pair-​susceptibility discussed by Anderson and Goldman (1970) and Goldman (2006). A complementary theory for possible pairing states in LaAlO3/​SrTiO3 interfaces was recently discussed by Scheurer and Schmalian (2015). Key elements of the theory are the spin–​orbit–​split band structure and the fact that the mass anisotropy gives rise to significant nesting of the Fermi surface (see Fig. 7.4). The tendency towards nesting will then amplify interactions that couple almost parallel sheets of the Fermi surface, an effect that was accounted for using a fermionic renormalization group calculation. Beyond this, the theory did not make specific assumptions with regards to the microscopic nature of the pairing interaction. However, under rather general assumptions, including all symmetry-​ allowed interactions that couple nested Fermi surface sheets, two distinct pairing states emerged that are characterized by the relative sign of the gap function on the spin–​ orbit–​split sheets of the Fermi surface. In the case where the initial interaction

224

224  Surface and Interface Superconductivity between electrons is attractive, as is expected for a phonon-​interaction-​dominated pairing state, it was found that the two gap functions are in phase:  ∆1  =  +∆2. On the other hand, for repulsive interactions due to electron–​electron Coulomb interactions, the pairing state turned out to be of a kind where the two gap functions are out of phase: ∆1 = −∆2. The latter state is in its nature similar to other spin-​fluctuation-​mediated pairing states, where a pairing interaction that is repulsive at high energies becomes attractive at low energies, an effect that is caused by the exchange of particle–​hole excitations. Both pairing states are of s-​wave nature, giving rise to a full gap on the Fermi surface. The corresponding flow diagram and the nature of the pairing states are indicated in Fig. 7.5. While the theory of Scheurer and Schmalian (2015) could not distinguish between the electron–​phonon or electron–​electron interaction scenarios, it gave a well-​defined approach to discriminate between the two pairing states. Using the results of Qi et al. (2010) one finds that the relative sign of the two gap functions can be used to determine the topological invariant of the superconducting state. It follows that the s+− state, that is, the state with ∆1 = −∆2 that is due to an electron–​electron interaction, is topologically non-​trivial, while the s++ state with ∆1 = +∆2 due to an attractive interaction, is topologically trivial. The conclusion is that LaAlO3/​ SrTiO3 interfaces host a topological superconductor if the underlying mechanism is unconventional while superconductivity is topologically trivial in the case of an attractive electron–​phonon pairing interaction. Counter-​propagating Majorana modes at the edge could then, in principle, be used to discriminate between these two states. This connection between the topology and the mechanism of superconductivity has recently been generalized by Scheurer (2016) to other non-​centrosymmetric superconductors as well.

s+–

∆1 ∆2

ρF U

Repulsive

s++

∆1 ∆2

ky kx

Attractive

0

kx

ky

Attractive 0 Repulsive

ρF J

Fig.  7.5 In the middle panel, the flow of the dimensionless intra-​and inter-​band Coulomb interaction, ρFU, and pair hopping, ρF J, is illustrated. As shown by Scheurer and Schmalian (2015b), if the microscopic interaction is repulsive (attractive) the flow will start in the upper right (lower left) shaded region such that the coupling constants will eventually be driven to the strong-​coupling fixed point in the lower right (lower left) corner which is associated with the s+− (s++) state. The order parameter ∆1,2 on the two outermost Fermi surfaces in Fig. 7.4b of the s++ and s+− states is illustrated schematically in the left and right panels.

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Superconductivity at the LaAlO3/​SrTiO3 interface 225 Given these considerations it is an interesting question to combine the two theoretical approaches of Klimin et  al. (2014) and Scheurer and Schmalian (2015). As mentioned, the interaction due to optical phonons and including the screening of the Coulomb interaction yields a repulsive, yet frequency-​dependent pairing kernel K (ω, ω ′ ). Without the spin–​orbit splitting of the Fermi surface, this yields a conventional pairing state, albeit with a strongly frequency-​dependent gap function ∆(ω ). If one now includes the Dresselhaus–​Rashba splitting of the Fermi surfaces, superconductivity with ∆1 = −∆2 seems to be a viable candidate pairing state for the same K (ω , ω ′ ). In this case, the poorly screened Coulomb interaction together with the interaction with acoustic phonons would give rise to an unconventional topological superconductor, suggesting that the search for exciting novel physics in these 2D materials continues to be promising.

7.4.4 The properties of the LaAlO3/​SrTiO3 interface The electron system found at the LaAlO3/​SrTiO3 interface has several intriguing properties that set it apart from other 2D superconductors: (1) It is extremely diluted (Bert et al. 2012), with a superfluid density in the range of 1–​3 × 1012 cm−2. (2) The superfluid density and critical parameters (temperature, field, and current) can be tuned using the electrostatic field effect, allowing an on–​off switch of superconductivity using gate voltages (Caviglia et al. 2008, Bert et al. 2012). (3) The DOS exhibits a pseudogap around the Fermi energy in the non-​ superconducting phase (Richter et al. 2013). An interesting scenario attributes the pseudo-​gap to preformed electron pairs, and it was suggested that superconductivity might result from Bose–​ Einstein condensation (Cheng et al. 2015, van der Marel et al. 2011). (4) When compared with bulk strontium titanate, it becomes apparent that the Fermi surface of the LaAlO3/​SrTiO3 interface is reshaped by an orbital reconstruction that lifts the degeneracy of the 3d t2 g manifold. A sizeable spin–​orbit coupling also leaves its marks on the Fermi surface, leading to a spin-​splitting in the meV range, comparable to the Fermi energy (Caviglia et al. 2010, Diez et al. 2015). (5) The superconducting electrons coexist with localized magnetic moments (Bert et al. 2011, Li et al. 2011). (6) The electron system displays intrinsic spatial inhomogeneities below its ferroelastic transition, observed in the electrostatic landscape as well as current distributions. These are linked to crystallographic tetragonal domain boundaries (Honig et al. 2013, Kalisky et al. 2013).

226

226  Surface and Interface Superconductivity This (non-​exhaustive) list of unusual properties makes a compelling scientific case for pursuing nanoscale investigations of LaAlO3/​SrTiO3 interfaces and related oxide heterostructures: (1) The extremely low superfluid density is expected to lead to an unusually high kinetic inductance of superconducting circuits, which is a relatively unexplored regime for quantum electronics. (2) The electric-​field-​driven quantum phase transitions make it possible to design novel circuits in which local gates define on-​demand electrostatic interfaces between superconducting, metallic, and insulating regions. (3) The presence of preformed electron pairs above the superconducting critical temperature can be detected in nanoscale charge-​sensitive devices such as quantum dots. (4) The orbital reconstruction and spin–​orbit coupling suggest a possible non-​trivial symmetry of the superconducting gap in momentum space that can be investigated with nanoscale spectroscopy. (5) The local coexistence of superconductivity and magnetism reinforces the hypothesis of a non-​trivial form of electron pairing. (6) The intrinsic electronic inhomogeneities can lead to spontaneous 1D confinement along crystallographic domains that can be manipulated by electric fields and by strain gradients. It is no surprise that such an intriguing list of challenges has stimulated a number of groups worldwide to dig into the physics of oxide interfaces at the nanoscale. When probed at macroscopic scales, the underlying physics of these competing phases is washed out by spatial averaging. Probing the interfaces on scales smaller than the typical scales over which these phenomena vary brings about the exciting possibility of independently studying them. We will discuss the current experimental techniques used to create nanoscale electron transport devices at LaAlO3/​SrTiO3 interfaces with a focus on the unique nanolithographic approaches possible in this system (direct patterning, AFM writing, and electrostatic gating). We will also give specific examples of interesting results already obtained on superconducting devices.

7.4.5 Direct patterning Producing high-​quality nanoscale structures at the LaAlO3/​SrTiO3 interface has proven challenging due to inherent stoichiometric and structural intricacies associated with complex oxides. Oxygen vacancies and defects can be easily induced in the system, which can result in regions of undesired conductivity or degradation of the electronic properties (Liu et al. 2013).

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Superconductivity at the LaAlO3/​SrTiO3 interface 227 Currently available techniques for nanoscale patterning of the 2DES at the LaAlO3/​SrTiO3 interface can be divided into two categories: post-​deposition (rendering certain areas insulating after the growth) and pre-​deposition (preventing the formation of the 2DES in certain areas). The most commonly used techniques involve a post-​deposition approach adopted from thin film processing, which is based on the etching of the oxide in lithographically defined areas. In this process, a conducting LaAlO3/​SrTiO3 heterostructure is first grown and photo-​or e-​ beam-​lithography is used to define narrow regions that will ultimately be rendered insulating. To this end, chemical and physical etching treatments can be employed. Wet-​chemical etching in acidic solutions is often used to remove the exposed oxide regions. Chemical etching of LaAlO3 is however rather complicated as it is difficult to find durable materials that will resist the aggressive acids required for the etching of the film. Therefore, physical etching methods have been a preferred route in the post-​deposition patterning of LaAlO3/​SrTiO3 heterostructures:

•

Ar ion milling has been employed to completely remove the LaAlO3 layer in the exposed regions, rendering them insulating. However, the exposure to the ion beam often yields a highly conducting layer at the surface of the SrTiO3 substrate due to the formation of oxygen vacancies (Reagor and Butko 2005, Kan et al. 2005, Gross et al. 2011).

•

Conducting structures down to 100 nm in width have been achieved using reactive ion etching (RIE). Although the regions that were not protected by the resist remain insulating, this method was shown to induce side wall damage in the conducting channel (Minhas et al. 2016).

•

Argon ion beam irradiation was also successfully employed to create nanostructures at the LaAlO3/​SrTiO3 interface. This process relies on carrier localization arising from the defects created by the ion beam exposure, eventually producing an insulating ground state. Lateral dimensions of 50 nm were achieved; however, at the cost of an unexplained increase in resistivity of almost one order of magnitude (Paolo Aurino et al. 2013). This technique was also used to modify the interface conductivity without the initial patterning of the resist by using stencil masks (Mathew et al. 2013).

The pre-​deposition approach is rather uniquely suited for this interface system, making use of a distinctive feature of the 2DES at the LaAlO3/​SrTiO3 interface: a sharp transition from fully insulating to conducting behavior as a function of the thickness of the LaAlO3 film grown on SrTiO3. The precise transition occurs between 3 and 4 unit cells (uc). Conventional lithographic techniques can be used to laterally define structures by locally controlling the thickness of the crystalline LaAlO3 layer (Schneider et al. 2006). In some cases, two uc of LaAlO3 are first deposited epitaxially over the entire substrate to ensure that resist residues do not influence the conducting interface in the device region. The sample is then removed

228

228  Surface and Interface Superconductivity

5µ

m

5

µm

Fig. 7.6  Optical microscope image of the device (full scale: 1.5 mm × 1.2 mm).The inset shows an atomic force microscope (AFM) image of the bridge area. Polycrystalline/​amorphous LaAlO3, grown on the amorphous SrTiO3, has a lighter color, while the epitaxial LaAlO3 has a darker one. The 2DES is created only below the epitaxial LaAlO3. To measure the transport characteristics of the junction, current is driven from C to D, voltage is acquired between E and B while Hall voltage is probed between A and F. Reprinted with permission from Stornaiuolo et al. (2012), Copyright (2012) AIP Publishing LLC. from the growth chamber and undergoes a lithography step after which amorphous LaAlO3 is deposited and the remaining resist is lifted off. After this process, the areas protected by resist during amorphous LaAlO3 deposition contain only pristine crystalline LaAlO3, whereas the exposed regions are coated by amorphous LaAlO3. Further deposition of LaAlO3 under suitable growth conditions leads to a continuation of epitaxial growth to a thickness totaling > 4 uc in the areas where the epitaxial LaAlO3 surface is still open, thus the condition for a conducting interface is fulfilled. In regions covered by amorphous material, crystalline growth is inhibited and the interface is insulating. An analogous technique using amorphous SrTiO3 as a hard mask prior to growth of epitaxial LaAlO3 has been used to fabricate bridges as small as 500 nm (Stornaiuolo et al. 2012), as shown in Fig. 7.6. A similar technique has shown that 1D confinement can be obtained without the deposition of conducting LaAlO3/​SrTiO3 interfaces with thickness >4 uc. This conducting channel is formed at the boundary between two 2D non-​ conducting LaAlO3/​SrTiO3 interfaces with thicknesses of 1 and 3 uc, respectively (Ron and Dagan 2014). Quantized conductance steps have been reported in such nanowires, despite their lengths of several micrometers, much larger than the mean free path measured for 2D LaAlO3/​SrTiO3 interfaces. In a different approach, a self-​patterning process of the SrTiO3 substrate has been used to locally confine the 2DES to lateral sizes on the order of 100  nm by engineering the LaAlO3/​SrTiO3 interface atomic composition (Foerster et al. 2012). Initially TiO2-​terminated SrTiO3 substrates undergo a high-​temperature treatment that induces progressive SrO surface segregation, accompanied by a self-​assembly process at the step edges, which results in the spatial separation at the nanoscale of both SrO and TiO2 terminations (Bachelet et al. 2009). The chemical termination of the substrate is replicated by the LaAlO3 layer, resulting in conducting LaO/​TiO2 stripes embedded in an insulating AlO2/​SrO matrix. Despite

  229

Superconductivity at the LaAlO3/​SrTiO3 interface 229 the previously mentioned advancements towards 1D confinement of the 2DES at the LaAlO3/​SrTiO3 interface, the vast majority of systematic studies on the superconducting ground state have been reported in the bulk. Stornaiuolo and co-​ workers (Stornaiuolo et al. 2012) have carried out a systematic study of the superconducting properties of mesoscopic bridges with lateral dimensions of 500 nm, realized by the previously mentioned amorphous SrTiO3 mask technique. They have shown that the mobility and carrier density of such bridges are unaffected by the fabrication process. In the normal state, universal conductance fluctuations were observed, originating from quantum interference of phase-​coherent electron waves scattered by impurities in such samples, with dimensions comparable to the phase coherence length. In the millikelvin regime, these bridges exhibit superconducting behavior with critical current densities comparable to large-​scale devices.

7.4.6 Atomic force microscopy writing Ahn et al. (2004) proposed, in the early 2000s, to use the metallic tip of an atomic force microscope (AFM) in order to induce a local electrostatic field effect and manipulate the superconducting order at the nanoscale. This concept was demonstrated by Takahashi et al. (2006), when local switching of 2D superconductivity was achieved using the ferroelectric field effect. In this experiment heterostructures that stacked a ferroelectric Pb(Zr, Ti)O3 layer on top of superconducting Nb-​doped SrTiO3 were considered. An AFM was used to switch the polarization of the ferroelectric layer, thereby modifying the critical temperature of the underlying superconductor. While giving a convincing demonstration of the concept, this experiment did not explore the nanoscale limits of the technique. Moreover, in this particular material system, only the definition of planar interfaces between superconducting and metallic regions was possible. The AFM patterning of superconducting/​insulating interfaces remained elusive until the seminal work of the group of Jeremy Levy (Cen et al. 2008, 2009). Using a conducting AFM, Levy and collaborators applied a voltage to sub-​critical, macroscopically insulating samples (3.3 uc thick). They demonstrated the formation of conducting regions below the metallic tip as small as 2 nm. Piezoelectric force microscopy and electric force microscopy were used to characterize this patterning technique, highlighting its possible microscopic mechanism. The formation of conducting regions below the tip was attributed to a voltage-​induced addition or removal of OH− and H+ on the surface of LaAlO3. These conducting regions can be reverted multiple times to an insulating state and are stable in vacuum and cryogenic conditions.

7.4.7 Electrostatic gating The properties of oxide interfaces can also be tuned using metallic top gates. Such top-​gated devices have been used in semiconductor heterostructures for decades, but have also recently emerged as a viable and versatile route to locally manipulate

230

230  Surface and Interface Superconductivity the electronic state of the 2DES at the LaAlO3/​SrTiO3 interface. Since LaAlO3 has a large band gap (5.6 eV) and relatively high permittivity (r > 10), it is in fact an excellent dielectric for field effect devices. This was first experimentally demonstrated by Förg et al. (2012) and Hosoda et al. (2013) through the successful creation of top-​gated field effect transistors (FETs) that could be operated in a temperature range of 2–​373 K with relatively low leakage currents. While these initial studies were performed with large areas (several hundred square microns), it has recently been shown that such FETs can be scaled down to create integrated circuits with sub-​micron channel (and gate) lengths (Jany et  al. 2014, Woltmann et  al. 2015), an important step towards creating functional oxide-​ based electronics. In addition to their technological appeal, top-​gated devices add new functionality, which can be exploited to better understand the rich variety of electronic phases that emerge at the LaAlO3/​SrTiO3 interface. Bulk studies with dual (top and bottom) gated structures have allowed for more detailed studies of the spin–​orbit coupling (Hurand et al. 2015, Liu et al. 2015). It has also been shown that the superconducting properties of the interface can be tuned with top gates (Hurand et al. 2015, Eerkes et al. 2013). Recently, it has been demonstrated that by using appropriately designed top gates it is also possible to locally control superconductivity in nanoscale regions of the interface (Goswami et al. 2015, Bal et al. 2015). A big advantage of using the field effect to create confinement is that it avoids issues such as ion-​beam-​ induced damage or irreversibility, potential disadvantages of previously discussed methods of confinement. Furthermore, it provides a flexible platform to create intricate devices which require in situ tunability of the potential landscape. For example, the ability to create and locally tune nanoscale channels with a split-​ gate architecture (Goswami et al. 2015) as shown in Fig. 7.7 could allow one to systematically study a crossover of superconductivity from 2D to the 1D limit. Another exciting possibility is to create superconducting circuit elements such as Josephson junctions (JJs) through the local electrostatic manipulation of the critical temperature (see Fig. 7.8).

(a)

Fig. 7.7  Optical (a) and SEM (b) images of a split-​gate device which comprises a left (L)  and right (R)  gate separated by 200 nm. Source (S) and drain (D) contacts are used to inject current into the device while the voltage drop across the junction is measured at the probe contacts P1 and P2. Reprinted with permission from Goswami et  al. (2015). Copyright (2015) American Chemical Society.

L

Dev 3

S

P1 R

20 µm

P2 D

(b) L

R 1 µm

  231

Summary and outlook 231 250 0V –6 V

200 150

V (µV)

100 50 0 –50 –100 –150 –200 –250

–200

–100

0

100

200

I (nA)

Such gate-​defined JJs could form the basis of unique reconfigurable superconducting circuits, where the response would be fully controlled in situ by gate voltages. Another advantage of such circuits is that they are naturally devoid of any physical interfaces. Such interfaces between different materials (such as a superconductor and insulator/​normal metal) are known to degrade the performance of superconducting circuits based on more conventional JJs. In contrast, creating a circuit solely by electrostatic means ensures the use of just a single material (the LaAlO3/​SrTiO3 superconductor) with purely electrostatic interfaces between different elements. Since there is already some evidence that such a strategy to create JJs is indeed viable (Goswami et al. 2015, Bal et al. 2015), it provides a strong motivation to build on this work. It would be interesting to study the properties of these interfaces (e.g. transparency and electrostatic profile) in more detail and to compare the performance (e.g. noise characteristics) of such JJs with more conventional ones. It is also possible to envision more elaborate devices that exploit this local tunability of the superconductivity. For example, two such JJs could be incorporated in a superconducting loop to create a superconducting quantum interference device (SQUID). A SQUID at the LaAlO3/​SrTiO3 interface would allow one to directly probe the phase of the superconducting parameter, and thus provide a better understanding of the microscopic nature of superconductivity.

7.5 Summary and outlook Exciting results have been obtained thanks to the progress achieved both experimentally and theoretically to investigate the superconducting state at the atomic limit. The two examples discussed in this chapter, the single-​atom Pb layer on a semiconducting substrate and the metallic interface between a LaAlO3 layer and

Fig.  7.8 Voltage (V)–​current (I) characteristics of the constriction shown in Fig. 7.7 revealing a crossover from a superconducting state (black curve for VL,R = 0 V) to a resistive state (red curve for VL,R = –​6 V).

232

232  Surface and Interface Superconductivity a SrTiO3 substrate, epitomize the challenges overcome to prepare and study such systems, but also display the richness of the physics they hold. We have discussed the properties of these systems and how they differ from their bulk counterparts. The role of the phonon coupling between the substrate and the layer in addition to the new symmetry configuration are some of the distinctive features to remember. We anticipate that as the fabrication of atomically thin layers is becoming more widespread, the discovery of superconductivity at the 2D atomic limit is entering a new era, where novel phenomena are also expected, like the Bose phases (Wagenblast et  al. 1997, Spivak et  al. 2008). Moreover, the ability to combine superconducting layers with electronic structures with strong spin–​orbit interaction has been predicted to be pivotal to the formation of new quantum effects like Majorana states (Lutchyn et al. 2010, Oreg et al. 2010). Particularly fascinating is also the question of the parity of the superconducting state in these layers with broken inversion symmetry. The perspective of finding non-​trivial superconducting orders will be a driving force for some time.

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Part II Materials Aspects

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Mesoscopic Effects in Superconductor–​ Ferromagnet Hybrids G. Karapetrov Department of Physics, Drexel University, Philadelphia, PA 19104, USA

S.A. Moore, and M. Iavarone Department of Physics, Temple University, Philadelphia, PA 19122, USA

8 8.1 Theories underpinning S/​F hybrid structures 

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8.2 Domain wall and reverse domain superconductivity 

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8.3 Vortex behavior in planar S/​F hybrids  256 8.4 Conclusions and outlook 

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References (Chapter-​8) 

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For more than 50  years the coexistence of superconductivity and magnetism has been a widely studied topic within the condensed matter community. At first glance, the antagonistic nature of these materials makes superconductor/​ ferromagnet (S/​F) hybrid structures unlikely candidates for useful technological breakthroughs. In particular, the effects of the electromagnetic and exchange interactions from the ferromagnet tend to suppress the local value of the superconducting order parameter. Investigations into the coupling of these two competing ground states first evolved from the study of ferromagnetic superconductors (Matthias et  al. 1958, Matthias and Suhl 1960). Usually, in the case of strong ferromagnets the exchange field would dominate the system properties because it is much larger than the internal magnetic field. However, it was shown early on that in special circumstances superconductivity can be recovered by the application of sufficiently high magnetic fields (Jaccarino and Peter 1962). A thorough review of the early theoretical and experimental work involving both the orbital and exchange interactions in ferromagnetic superconductors has been provided by Bulaevskii et al. (1985). Although focused primarily on ferromagnetic superconductors, these early considerations opened the door towards the controllable manipulation of superconducting properties in systems involving both superconductivity and ferromagnetism. Recent advancements in materials fabrication techniques have allowed investigations into the interplay between superconductivity and ferromagnetism in artificially constructed hybrid structures with unprecedented control. Differently

G. Karapetrov, S.A. Moore, M. Iavarone, ‘Mesoscopic Effects in Superconductor-Ferromagnetic Hybrids’ in The Oxford Handbook of Small Superconductors. First Edition. Edited by A.V. Narlikar. © Oxford University Press 2017. Published in 2017 by Oxford University Press. DOI 10.1093/acprof:oso/9780198738169.003.0008

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242  Superconductor–Ferromagnet Hybrids from ferromagnetic superconductors, in S/​F hybrids the constituent materials can be physically separated. This separation, realized via a thin insulating layer, eliminates the effect of short-​range exchange interactions while still allowing the long-​range orbital interaction. By creating a purely magnetically coupled S/​F system, fundamental properties such as the nucleation length scales for superconductivity, the nucleation conditions for vortices, the dynamics of vortices, and the H–​T phase diagram can all be altered depending on the materials and geometries chosen for the system. This chapter focuses on local scanning probe measurements of the novel mesoscopic effects that emerge in magnetically coupled S/​F hybrid structures in the absence of proximity effects. Although global measurements provide a wonderful understanding of the underlying physics in S/​F, results continue to show that there are significant system properties occurring at the nanoscale which necessitate the need of measurements on the truly local scale.

8.1 Theories underpinning S/​F hybrid structures A non-​uniform magnetic field is an alternative way to confine the superconducting order parameter and study the resulting effects related to superconductivity and confinement in S/​F hybrid systems. Within the past few decades a number of important theoretical works have surfaced investigating many of the novel effects that can occur in magnetically coupled S/​F hybrid systems. The resulting body of work has focused primarily on the effects of the underlying magnetic template on vortex nucleation (Genkin et al. 1994, Erdin et al. 2001, Laiho et al. 2003, Chen et al. 2009, Piña et al. 2010, Milosevic et al. 2011), vortex dynamics (Bulaevskii et al. 2000, Ze et al. 2014, Bespalov et al. 2015), and on the nucleation of localized superconductivity above domains and domain walls (Aladyshkin et al. 2003, 2007, 2012, Aladyshkin and Moshchalkov 2006, Houzet and Buzdin 2006, Mironov and Mel’nikov 2012, Miguel et  al. 2014). This section will provide a brief overview of the concepts underlying the various mesoscopic phenomena that emerge in S/​F systems.

8.1.1 Vortex nucleation conditions and vortex behavior in S/​F hybrids One of the first functional aspects of S/​F hybrid systems involved the nucleation and control of superconducting vortices via the underlying magnetic template, first in planar and shortly thereafter in lithographically defined structures (Genkin et al. 1994, Erdin et al. 2001, Laiho et al. 2003, Marmorkos et al. 1996, Milošević and Peeters 2004). Particularly exciting is the possibility to increase substantially the vortex pinning due to this magnetic landscape, thereby increasing the critical

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Theories underpinning S/F hybrid structures  243 current by up to 100 times (Bulaevskii et al. 2000, Kayali and Pokrovsky 2004). However, the physics driving the nucleation of vortices can be complicated even within a theoretical framework, where the actual material properties are greatly simplified. 8.1.1.1 Planar S/​F hybrid structures—​magnetic “stripes” The treatment of vortex nucleation in S/​F systems revolves around the minimization of the total energy. For planar S/​F systems the total energy can be conveniently separated into individual energy terms Etotal = Esv + Evv + Evm + Emm + Edw, where Esv is the self-​energy of a single vortex, Evv is the energy due to vortex–​vortex interactions, Evm is related to the interaction of the vortex with the magnetic field of the ferromagnet, Emm is the self-​energy of the ferromagnet, and Edw is the surface energy of the domain walls (Erdin et al. 2001, Laiho et al. 2003, Genkin et al. 1994). As shown in Fig. 8.1, in planar S/​F systems close to Tc, two unique vortex states emerge from the solution of the energy minimization: straight vortices and vortex semi-​loops (Laiho et al. 2003). Straight vortices enter and exit at the two opposite superconductor interfaces, whereas vortex semi-​loops produce a loop inside the superconductor so that the vortex enters and exits through the same S/​F interface. Due to different relative energy contributions and barriers, semi-​loops escape the superconductor while straight vortices remain trapped as the temperature is lowered. Moreover, any signatures due to semi-​loops are not readily accessible to scanning probe measurements (Bobba et al. 2014) and as a result, our further considerations will focus only on straight vortices. What is most fascinating is that, under the appropriate conditions, both straight and semi-​loop vortices could be generated spontaneously in the superconductor by the underlying ferromagnet in the absence of an applied field. Naively speaking, it is not surprising that vortices are nucleated in the presence of the underlying magnetic field, but both theoretical (Laiho et al. 2003) and experimental works (Iavarone et al. 2011, 2015, Bobba et al. 2014) have conclusively shown that this nucleation is extremely sensitive to the unique system properties and spontaneous vortices are not a guaranteed consequence of the S/​F hybrid. In calculating the conditions for the appearance of spontaneous vortices, with respect to the

(a)

z FM

x

SC (b)

z FM SC

x

Fig. 8.1  Planar superconductor/​ferromagnet (S/F) bilayer. Different vortex structures are shown by thin arrows: straight vortices (a)  and semiloops (b).  The magnetization vectors in the domains are shown by thick arrows (Laiho et al. 2003).

244

244  Superconductor–Ferromagnet Hybrids total energy defined previously, many of the individual energy terms could be neglected. Particularly in the case of S/​F bilayers where the superconductor film thickness is less than the penetration depth (ds < λ), Edm and Emm, which determine the equilibrium domain structure of the ferromagnet, can be disregarded on the assumption that the superconductor does not influence the domain structure of the ferromagnetic film (Genkin et  al. 1994). Strictly speaking, as shown in magneto-​optical measurements of Nb on iron–​garnet films (Vlasko-​Vlasov et al. 2010, 2012, Tamegai 2011), this assumption is not always valid and care must be taken to ensure that the chosen ferromagnetic film is not substantially modified by magnetic interactions with the superconducting film. In general, selecting a “hard” ferromagnet with a Curie temperature much higher than the superconducting transition temperature (Tm >> Tc) can help to isolate the dynamics of the domain development in the ferromagnet from the superconductor. At low vortex densities, the vortex–​vortex interactions will also be negligible, and the Evv term’s contribution to the total energy could also be neglected. The remaining energy terms provide for the conditions for the nucleation of spontaneous vortices, as Etotal ≈ Esv + Evm < 0. In the limiting case where the superconducting film thickness and domain width are greater than the penetration depth (ds, w > λ), the self-​energy of the single vortex is



 Φ 20ds   λ  E sv =  ln  16π 2 λ 2   ξ 

and



 0.916  E vm ≈ −4 M0Φ 0w   π 2 

is the interaction energy between the vortex and the magnetic domain. As a result, vortices will only be energetically favorable if



ds  Φ    λ d  M0 >  0 2   ln ≈ 0.2  s  H c1 = Mc ,  64λ   0.916W   ξ   w

where M0 is the magnetization in the magnetic domain, H c1 = ( Φ 0 / 4πλ 2 )ln ( λ / ξ ) is the lower critical field of the superconductor, and Mc is the critical value of magnetization required to generate spontaneous vortices. Considering the complicated dependence of Mc on the S/​F system properties, it is not surprising that selecting a system in which there is the nucleation of vortices due to the ferromagnet, is actually a non-​trivial problem. It is important to note that the presence of an external magnetic field will introduce an additional energy contribution (±Hds Φ 0 / 4) with the sign depending on the relative orientation between the applied field and the magnetization of the magnetic domain. Under these external magnetic fields vortices can be nucleated, and vortex densities changed,

  245

Theories underpinning S/F hybrid structures  245 in systems otherwise not exhibiting spontaneous vortex characteristics due to effects of compensation of the applied field on the stray magnetic field of the ferromagnet. 8.1.1.2 Lithographically defined S/​F hybrids—​magnetic “dots” Defects, artificial or otherwise, have long been known as a suitable way to alter the vortex behavior in superconducting materials (i.e. increase pinning) (Larkin and Ovchinnikov 1979, Civale et al. 1991, Buzdin 1993, Metlushko et al. 1994), a fact which has clear benefits towards possible superconductor applications. Early magnetization measurements of ferromagnetic nanostructures fabricated on superconductors revealed similar matching effects to those found in superconducting systems with artificial pinning centers (Martín et al. 1997, Otani et al. 1994, Wernsdorfer et al. 1996, Jaccard et al. 1998), igniting theoretical studies in this field. A recent review by Aladyshkin et al. (2009) covers in detail the body of theoretical and experimental work of S/​F hybrids involving lithographically defined ferromagnetic structures, with special focus placed on global measurements such as magnetization. Here, we provide a brief overview of the underlying physics driving the characteristic behaviors seen in these S/​F systems. Although, compared to planar S/​F hybrids, the inclusion of lithography adds an extra step to the fabrication process, it also adds an extra degree of control with respect to the non-​uniform magnetic field distribution seen by the superconductor. The variety of shapes and arrays available make the number of possible S/​F hybrid configurations endless. However, before expanding into how arrays of ferromagnetic dots alter the superconducting properties it is first prudent to investigate the effects of a single dot (Milošević and Peeters 2003a, 2003b, Cabral et al. 2004). Fig. 8.2a shows the geometry of a ferromagnetic dot in close proximity to, but electrically isolated from, a superconductor. This scenario is well described within Ginzburg–​Landau (GL) theory (Gennes 1989, Abrikosov 1988), where for thin superconductors (ds < λ, ξ) it is convenient to average over the film thickness (Milošević and Peeters 2003b). Using the appropriate boundary conditions, the solutions to the GL equations for various magnetization values of the dot reveal the novel vortex configurations seen in Fig. 8.2b. A dot with a small radius (with respect to the coherence length)

hm

m

z0

z=0

d

Rd

dd

y/ξ

(b)

z

y/ξ

(a)

4 2 0 –2 –4 4 2 0 –2 –4

I

II

III

IV

V

VI

–4 –2 0 2 4 –4 –2 0 2 4 –4 –2 0 2 4 x/ξ x/ξ x/ξ

Fig.  8.2 (a)  Magnetic disk above a superconducting film:  oblique view of the system. (b)  Gray-​scale plot of the Cooper pair density (white—​ 1, dark gray—​ 0) for the ground state vortex configurations. A giant vortex is surrounded by antivortices, and the total vorticity is equal to zero (Milošević and Peeters 2003b).

246

246  Superconductor–Ferromagnet Hybrids will result in a multi-​quanta vortex state (giant vortex), which is surrounded by a ring of antivortices, making the total vorticity in the unit cell zero. Further increase of the magnetization of the dot will lead to additional rings of antivortices, while an increase of the dot size can lead to the giant vortex breaking up into multiple vortices within the same area. Obviously, the physics involving superconducting vortex states around even a single magnetic dot is quite interesting, and generalization to an array of magnetic dots allows for an even richer body of phenomena to explore. Fig. 8.3a gives an illustrative schematic of an array of ferromagnetic dots on top of a superconductor (Milošević and Peeters 2004), and the corresponding simulations of the vortex state for the array are shown in Fig. 8.3b. These simulations reveal many similarities to the behavior found for isolated dots, and additional investigations with respect to applied magnetic fields yield behaviors that could be compared directly with the experimental observations. Because of the minimization of energy, which is analogous to that found in planar S/​F structures, in applied fields that are parallel to the magnetization of the dots the vortices will nucleate at the site of the dot. Alternatively, for applied magnetic fields directed anti-​parallel to the dot magnetization, the vortices will nucleate in-between the magnetic dots. Moreover, it is found that at the so-​called matching fields there will be an integer number of vortices inside the unit cell of the artificial array. When this occurs the vortices are pinned the most strongly, and there will be a precipitous drop in bulk resistivity (Gomez et al. 2013). The effects of these periodic arrays on bulk measurements have been discussed in the aforementioned review article (Aladyshkin et al. 2009), and our considerations will instead be focused on local scanning probe measurements of the mesoscopic properties in the vortex state.

(a)

(b)

y/ξ

n=1

m Rd

d

a

y/ξ

n=2

hm

dd

n=3

y/ξ

Fig.  8.3 (a)  The superconducting film (with thickness d) with an array of magnetic dots (radius Rd, thickness dd, magnetic moment m, and period of the lattice a) on top of it: schematic diagram of the analyzed system. (b) Contour plots of the Cooper-​pair density (dark color–​ –l​ow density) showing the arrangement of external flux lines when pinned by magnetic dots (with saturation number ns = 1) for three different matching values of the external magnetic field (Hn) in two cases: (i) for parallel alignment between the magnetic moment of the dots and the external field (left), and (ii) antiparallel case (right). Dotted lines illustrate the magnetic dot lattice (Milošević and Peeters 2004).

40 36 32 28 24 20 16 12 8 4 0 40 36 32 28 24 20 16 12 8 4 0 40 36 32 28 24 20 16 12 8 4 0

Hn

H–n

0 4 8 12 16 20 24 28 32 36 40 0 4 8 12 16 20 24 28 32 36 40 x/ξ

x/ξ

  247

Theories underpinning S/F hybrid structures  247

8.1.2 Localization of superconductivity The treatment of the localized nucleation of superconductivity in an ideal S/​F system involves a ferromagnet with a uniform striped magnetic domain structure, as depicted schematically in Fig. 8.4a. In this configuration, the superconducting film experiences a non-​uniform magnetic stray field, which alternates in polarity in line with the magnetization of the underlying magnetic domains. Furthermore, for ferromagnetic films having a thickness much greater than the domain width (D >> w), this alternating magnetic field as seen by the superconducting film is step-​like and homogeneous in the regions above the domains. As a result, in these regions the stray field has a maximum value and the critical temperature for the nucleation of superconductivity is suppressed due to the orbital interaction. On the other hand, due to an energetically favorable rotation of the perpendicular magnetization towards the in-​plane direction at the domain wall (Chikazumi and Graham 2009), the stray field seen by the superconductor approaches zero locally. When the temperature of the S/​F system is decreased through Tc, due to a partial decrease in the orbital effect above this in-​plane region, the superconducting state will first emerge at the domain walls in a way that is reminiscent of surface superconductivity below Hc3 (Aladyshkin et al. 2003, Saint-​James et al. 1969). This localized superconductivity is the so-​called domain wall superconductivity (DWS) and is only expected to occur close to the Tc where the orbital interactions above the domain centers still suppress development of a more global superconducting state. While DWS in S/​F bilayers occurs in the absence of externally applied magnetic fields, the situation changes drastically when an external field is instead present. Due to compensation effect regions of the superconductor located above the domains having the opposite magnetization polarity to the external field will see a decrease in the effective stray field, while regions having the same polarity will instead see an increase. Upon lowering the temperature to below Tc the localized superconducting state that develops will be shifted away from the domain wall region, and towards the space with the lowest effective magnetic field. Increasing the applied field further will lead to continued shifting of the nucleation location until the value of the stray magnetic field is reached, at which point the localized superconducting state will be nucleated directly above the center of the magnetic domains with opposite polarity to the applied external field. This phenomenon is called reverse domain superconductivity (RDS) and any further increase in magnetic field will lead to a global weakening in the superconducting strength and lowering of Tc. Formally, theoretical considerations of DWS begin with the linearized GL equation (Abrikosov 1988, Shmidt et al. 1997, Tinkham 2004): 2



 2πi  1 −∇ + A Ψ= 2 Ψ Φ 0  ξ (T ) 

where A(r) is the vector potential, Φ0 is the flux quantum, ξ (T ) = ξ 0 / 1 − T / Tc 0 is the coherence length, and Tc0 is the critical temperature of the superconducting

248

248  Superconductor–Ferromagnet Hybrids film in the absence of the ferromagnetic layer. By choosing the proper gauge and neglecting any dependence of the order parameter in the perpendicular direction it is possible to calculate the critical temperature as a function of magnetic field, thereby mapping out the H–​T phase diagram for the S/​F system (Aladyshkin et  al. 2003). An H–​T diagram calculated using this approach, and exhibiting DWS and RDS, is shown in Fig. 8.4b where it is clear that the Tc of the system changes non-​monotonously under applied fields. The solution to such a problem depends intimately on the system properties, and especially important is the shift of critical temperature due to the orbital interaction ∆Tcorb = Tc 0ξ 20 / L2 where magnetic length L is related to the magnetic stray field through L2 = Φ 0 / 2πH stray . In the case that ∆Tcorb is too small it will be impossible to detect DWS or any related effects because the order parameter can not track the rapid variation of the local field over the surface of the superconductor, and as a result there appears only a global reduction in Tc. Another consideration that arises theoretically is the relationship between the superconducting nucleation length and the domain width. In the case of the idealized isolated domain wall, and sufficiently large ∆Tcorb, the DWS state will always be favorable. However, real S/​F systems usually have a periodic magnetic structure, and this periodicity imposes the other, perhaps more stringent, constraint that the domain width be larger than roughly twice the nucleation length at Tc (w >> ξ(Tc)). If this condition is not met, the localized superconducting regions above the domain walls overlap, again resulting in a global reduction in Tc and no DWS state. In S/​F systems where both ∆Tcorb and the domain widths are sufficiently large the nucleation of superconductivity will be located above the domain wall as shown in Fig. 8.4c, which is an experimentally

(

(a)

)

(b) z H

D

Fig. 8.4  (a) Superconductor–​ferromagnet (S/​F) bilayer. (b) The temperature dependence of the critical magnetic field for an S/​ F system with a thick ferromagnetic layer. The solid (dashed) line corresponds to the superconductivity nucleation at the domain boundary (far from the domain boundary). (c) The behavior of the ground-​ state wave function (solid line) localized at a domain wall in a periodic domain system.The magnetic field profile is shown by the dashed line (Aladyshkin et al. 2003).

1

S x

F

M w

|H|/B0

d

0

(c)

fk(x) (arb. units)

–1

Bz (x)/B0

1

0

–1

–2

0

2

x/(2w)

–0.6 (Tc – T∞)/∆T corb

0

  249

Domain wall and reverse domain superconductivity 249 verifiable phenomenon. A number of intricate scanning probe experiments have been performed to confirm these conditions in S/​F systems, which will be the focus of the following section.

8.2 Domain wall and reverse domain superconductivity This section deals with experimental measurements of the localized superconducting state in the cases of domain wall (DWS) and reverse domain superconductivity (RDS). Due to the mesoscopic nature of these phenomena, focus will be placed on local scanning probe measurements with the ultimate resolution given by scanning tunneling microscopy (STM), which can provide electronic information about the S/​F systems with atomic resolution. Fig. 8.5 provides a convenient illustration describing how the location of the nucleation of superconductivity is related to the applied magnetic field and system geometry. Assuming a step-​like magnetic landscape over the domains, in zero applied field, the stray magnetic field will go to zero above the domain walls. Close to Tc the stray fields generated by the underlying magnetic domains are sufficiently large compared to Hc2 and superconductivity is suppressed above the center of the domain. However, at the domain walls where the perpendicular stray magnetic field goes to zero, and close to Tc the condensate can develop in the localized space between the domains. Through the application of applied magnetic fields the stray fields from the magnetic domains can be compensated, eventually leading to a relocation of the zero effective field to the regions above the domains that are anti-​parallel to the applied field.

8.2.1 Local probes While mesoscopic effects happening at the nanoscale lead to the appearance of DWS and RDS, these phenomena have a strong influence on bulk system properties. In an early work on DWS and RDS, temperature-​dependent resistance measurements on an S/​F system of Nb/​BaFe12O19 were used to map the transition temperature as a function of applied magnetic field (Yang et al. 2004). The resulting H–​T diagram, shown in Fig. 8.5, is in excellent agreement with the behavior expected theoretically (Aladyshkin et al. 2003). A ferrimagnet having a complex magnetic domain structure was chosen as the magnetic template because of its appropriate domain size and stray magnetic field strength, which were well suited for the global transport measurements. However, the complicated nature of the magnetic field dependence of the domain structure in this type of magnetic material makes scanning probe measurements nearly impossible. Indeed, this difficulty is reflected in the relatively complicated interpretation of semi-​macroscopic low-​temperature scanning laser microscopy (LT-​SLM) measurements on the S/​ F system comprised of Nb/​PbFe12O19 (Fritzsche et al. 2006). Room temperature magnetic force microscopy (MFM) measurements of the domain structure are

250

250  Superconductor–Ferromagnet Hybrids (a) H DS

DS

z Ha

+Hc2

L

–Hc2

Fig.  8.5  Left—​The magnetic field distribution in the superconducting layer at different applied magnetic fields close to Tc and the applied magnetic field Ha:  (a)  Ha ≈ + Hs, (b)  Ha ≈ 0, and (c) Ha ≈ –​Hs, where Hs is the saturation field. DS denotes domain superconductivity, and L is distance in the plane of the thin film. Black dashed horizontal lines indicate the applied field Ha. Green dashed lines indicate the upper critical field Hc2. Red solid  lines  show the total magnetic field, H a + H d . Right—​ Schematic images of the superconducting areas (dark blue) in the superconductor/​ ferromagnet hybrid structure. F denotes ferromagnet with easy axis in the z-​ direction and with thickness much larger than the distance between domain walls. S stands for the superconducting Nb thin film. Dark arrows show the orientation of the magnetic moment in the magnetic domains (Yang et al. 2004).

S F

Ha ≈ +Hs

(b)

H

DWS

+Hc2 –Hc2

Ha

S

L

F

Ha ≈ 0

(c)

H DS

+Hc2 –Hc2

DS

L Ha

S F

Ha ≈ –Hs

shown in Fig. 8.6a, where it is clear that the domains arrange in a non-​periodic and unpredictable way as a function of magnetic field. Temperature-​dependent LT-​SLM measurements of the Nb nano-​bridge fabricated on top of the ferrimagnet are shown in Fig. 8.7b. Furthermore, the visualization of DWS in the same system was not possible due to this complexity in the magnetic template. The first measurements of DWS came from semi-​macroscopic LT-​SLM measurement on a 30-​μm Pb microbridge fabricated on top of a BaFe12O19 substrate (Werner et al. 2011). Differently from previous measurements, in this case the microbridge was placed on top of the substrate in such a way that only a single domain wall was running along the center of the bridge (Aladyshkin et al. 2011). As shown in Fig. 8.8, the results of this experiment quite conclusively reveal DWS and RDS as a function of temperature and applied magnetic field. Unfortunately, despite the beauty of this experiment, the electronic behaviors underlying the formation of DWS and RDS are still inaccessible at these length scales on the order of microns.

  251

15 10 4

H (k0e)

H (k0e)

5 0

2 0 –2 –4 7.85

–5

7.90

7.95

8.00

T (K)

Fig.  8.6 Diagram obtained from the R(T) curves by defining the critical temperature with three different resistance criteria. The inset shows an enlarged view of the H–​T phase diagram for the resistance criterion of Rcri  =  90% Rn (Yang et al. 2004).

90% 50% 10%

–10 –15 5

4

6 T (K)

7

8

(a)

(b) Normalized resistance

Transitions of the bridge at 200 and 290 mT 1.0 0.8

Temperatures of LT-SLM images

0.6

7.30 K

0.4

200mT

7.25 K

290mT 0.2

0.0 0.6

7.2 6.4

6.8

7.6

8.0

7.20 K

Temperature [K] 7.10 K

0

6.90 K

B

Single pa

thes of

6.60 K 6.53 K

dT

max

B forms rings

6.80 K 6.65 K

dR(r 2)/

7.35 K

B

7.25 K

B

6.85 K

A

First sig

ns of B µ0H = 2 0

0 mT

µ0H = 2

6.33 K 90 mT

6.20 K

Fig.  8.7  (a)  Room-​ temperature 40 × 40 μm2 MFM images of the PbFe12O19 substrate in external magnetic fields applied perpendicular to the basal plane. Bright and dark areas represent domains polarized reverse and parallel to the applied field, respectively. The separating domain walls are visible as thin dark lines. (b) The graph shows normalized resistance of the bridge vs temperature at 200 mT and 290 mT. Gray circles and diamonds indicate the temperatures at which the LT-​ SLM images were taken. Images in (b)  show temperature dependence of the  LT-​SLM dR(r )/​dT signal of the Nb bridge at 200 mT (left series) and 290 mT (right series) (Fritzsche et al. 2006).

252

252  Superconductor–Ferromagnet Hybrids

8.2.2 Scanning tunneling microscopy and spectroscopy (STM/​STS) The measurements described in the previous section confirm many of the properties expected of DWS and RDS, but unfortunately, these techniques are limited to a resolution of micrometers. Indeed, the system properties underlying the development of the DWS and RDS states evolve on mesoscopic length scales that are on the order of the nanometers, and therefore measurements of these properties at these distances require the use of truly microscopic probes. Recently, atomic resolution low-​temperature scanning tunneling microscopy (STM) and spectroscopy (STS) measurements confirming the prescribed conditions for the nucleation of DWS and RDS have been carried out on an S/​F system made of Pb/​Co–​Pd (Iavarone et al. 2014). Naturally, STM measurements are extremely sensitive to electronic properties, but they have no inherent sensitivity to magnetism, making possible investigations into S/​F systems problematic. However, other STM measurements on S/​F hybrids have shown that the structure

6

Hext = 0

(b)

1

2

5

6.5 3

3

4

2

0

6.0

T (K)

R (Ω)

4

1

(c) 7.0

5.0

6

7

8 5.5

(d) ∆V (µV)

T (K)

ES RDS

RDS

5.5 CS

4.0 –1000

7.0

–500

6.6 K 6.4 K simulation 6.3 K 6.2 K 6.1 K 6.0 K 5.7 K 5.0 K

0.6 0.4 0.2

4.5 6.5

1.0 0.8

ES

5.0

5

6.0

DWS

∆V (µV)

(a)

0

500

1000

0.0

0

10

Hext (Oe)

20

30

x (µm)

40

0

50 60

60

x (µm)

0.9 0.8

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

1

6.6 K

2

6.4 K

3

6.3 K

4

6.2 K

5

6.1 K

6

6.0 K

7

5.7 K

8

5.0 K

Optical image

Fig. 8.8  Evolution of DWS upon cooling through Tc and Hext−T phase diagram. (a) R(T) curve (I = 100 μA); dots indicate bias points of LT-​SLM voltage images 1–​8 in (d) and corresponding line scans in (c). (b) Hext−T phase diagram, constructed from experimentally determined values Tc0 = 7.25 K, B0 = 480 G, and Hc2(0) = 2.25 kOe.The phase diagram contains separate regions of DWS, extended superconductivity (ES), RDS, and complete superconductivity (CS). Dots label bias points for LT-​SLM data shown in (c), (d), and Fig. 8.3. (c) Line scans ΔV(x) across the bridge for different T, taken from voltage images in (d). Red dots show simulation results for T = 6.4 K. The position of the edges of the bridge is indicated by dashed gray lines. (d) Series of LT-​SLM voltage images ΔV(x,y) (1–​8 from left to right) taken at different T during cooling the Pb bridge through its resistive transition (I = 10 μA).White dashed lines indicate the position of line scans in (c). The graph on the right shows a corresponding optical LT-​SLM image (Werner et al. 2011).

  253

Domain wall and reverse domain superconductivity 253 2.5 nm (a)

100 nm

(b)

H=0 T = 1.5K

0 nm 1.6 (c)

100 nm

(d)

100 nm

(f)

H = +300 Oe 0.55 T = 5.5K

100 nm

H = –300 Oe T = 1.5K

H = +300 Oe 0.12 T = 1.5K 1.0 (g)

100 nm

H = –200 Oe T = 1.5K

H = +200 Oe 0.12 T = 1.5K 1.6 (e)

100 nm

100 nm

(h)

100 nm

H = –300 Oe T = 5.5K

of the underlying magnetic template can be distinguished by noting the location of superconducting vortices (Karapetrov et al. 2009, Iavarone et al. 2015). Because at low energies and temperatures STM measurements are directly proportional to the local density of states, conductance maps at zero bias (ZBC maps) allow for the visualization of vortices in STM experiments due to the systematic increase of states at zero bias (EF) when entering the normal vortex core. Respectively shown in Fig. 8.9a, b, the topography reveals large atomically flat terraces across the film surface, while the ZBC map in zero applied field shows the presence of spontaneous superconducting vortices at low temperature. These spontaneous vortices are generated by the stray field of the underlying magnetic template, the experimental

Fig. 8.9  (a) STM topography image of the Pb film in S/​F bilayer of [30 nm Pb]/​ [2 nm Co–​2 nm Pd]N = 200. Scan area is 450 × 450 nm2.The tunneling conditions are: V = −20 mV and I = 100 pA. (b–​h) Zero-​bias conductance maps acquired on the same area at different applied fields and temperatures.The white dashed lines indicate the approximate positions of the domain walls as inferred from the vortex configurations at different fields. The conductance maps are normalized to the conductance maps at V = −10 mV. (b) H = 0 Oe, T = 1.5 K; (c) H = +200 Oe, T = 1.5 K; (d) H = −200 Oe, T = 1.5 K; (e) H = +300 Oe, T = 1.5 K; (f) H = −300 Oe, T = 1.5 K; (g) H = +300 Oe, T = 5.5 K; (h) H = −300 Oe, T = 5.5 K. Scale bar is 100 nm for all images (Iavarone et al. 2014).

254

254  Superconductor–Ferromagnet Hybrids details of which are discussed in later sections of this chapter. Due to the insensitivity of STM to the direction of the magnetic field, it is not possible to tell a priori the polarity of the vortices, nor the locations of the underlying domains. However, due to compensation effects field-​dependent vortex maps shown in Fig. 8.9c–​f reveal the structure of the underlying magnetic template. When a magnetic field is applied to the S/​F system the density of vortices above the magnetic domain with the same polarity as the applied field will increase, and vice versa for the domains with opposite polarity. This provides for the unambiguous determination of the structure of the underlying magnetic template, which includes the relative polarity of the magnetic domains. While these particular field-​dependent ZBC maps of the vortex configuration have been carried out at low temperatures (T 2ξ(Tc)). The previously discussed STM work verifying RDS in two samples of Pb/​Co–​Pd also investigated the threshold conditions for the visualization of DWS, and found that the condition on the relative size of the domain width with respect to the coherence length must be satisfied in order to establish DWS at the nanoscale. For the S/​F systems, the authors kept the thickness of the superconducting 30  nm Pb film constant, but used ferromagnetic Co (2 nm)/Pd (2 nm) multilayer films with different domain widths, as seen in the room-​temperature MFM measurements shown in Fig.  8.10a–​ b. Local conductance spectroscopy close to T c on Sample A, having 200 bilayers of Co–​P d and a domain width of w ≈ 200 nm, shown in Fig. 8.12c reveals a spatially uniform superconducting state in zero field which does not vary with distance from the domain wall. Moreover, the spectra in the middle of a magnetic domain do not change in applied fields of up to 300 Oe (Fig. 8.12c). Under the same conditions, the situation is startlingly different in the case of Sample B, having 50 bilayers of Co–​Pd and a domain width of w ≈ 300 nm, where in zero field the superconductivity is absent and under applied fields the superconducting state re-​emerges above the magnetic domain (Fig. 8.12d). By extracting the temperature dependence of the energy gap it is confirmed that in Sample A  (Fig.  8.10e) the superconducting state is spatially uniform for the

  255

(a)

1.6°

(b)

1 µm

1 µm –1.5°

1.0

0.5

dI/dV

dI/dV

1.0

H = 0 Domain

0.5

H = 0 Domain H = 400 Oe Domain

H = 300 Oe Domain

5

(e)

1.0 0.5 0.0

H = 0 Domain wall H = 200 Oe Domain

0

2

T (K)

T = 5.0 K 0.0 –10 –5 0 V (mV) 1.5

(c) 10

∆(T )(meV)

∆(T )(meV)

T = 5.0 K 0.0 –10 –5 0 V (mV) 1.5

4

6

5

(d) 10 (f)

1.0 0.5 0.0

H = 0 Domain wall H = 400 Oe Domain

0

2

T (K)

4

6

Fig. 8.10  (a–​b) MFM phase images (scan area of 4 x 4 mm2, scale bar 1 mm) at room temperature of two ferromagnetic samples used in this experiment. (a) Sample A (200 bilayers of Co (2 nm)/​Pd (2 nm) showing a stripe domain pattern with average domain width w ≈ 200 nm. (b) Sample B (50 bilayers of Co (2 nm)/​Pd (2 nm)) showing a stripe domain pattern with average domain size w ≈ 300 nm. (c–​d) STM conductance spectra acquired on (a)  (Co–​Pd)/​Pb sample A  and (b)  (Co–​Pd)/​Pb sample B, at a position corresponding to the center of a magnetic domain. The two spectra were acquired in the absence and in the presence of an external field H at T = 5.0 K. The tunneling spectra have been normalized to the conductance value at V = –​10 mV. (e–​f) Temperature dependence of the gap values obtained from the conductance spectra acquired on (a) (Co–​Pd)/​Pb sample A and (b) (Co–​Pd)/​Pb sample B. The filled dots represent the temperature-​dependent gap values at the location above a magnetic domain wall in the absence of an external field. The open squares represent the temperature-​dependent gap values at a location corresponding to the middle of the magnetic domain in the presence of a compensating applied magnetic field. The gap values have been derived by the Bardeen–​Cooper–​Schrieffer (BCS) DOS fitting. The error bars represent the standard deviation obtained from the fit. The experimental points are compared with the BCS gap equation Δ(T) (solid lines) (Iavarone et al. 2014).

256

256  Superconductor–Ferromagnet Hybrids 800 600 H (Oe)

range of fields up to 300 Oe, whereas in Sample B (Fig. 8.10f) the strength of the superconducting state varies as a function of position and field. Mapping the highest local Tc as a function of field in the two samples (Fig. 8.11) reveals the expected H–​T behavior for DWS for Sample B, but not for Sample A. The phase diagram suggests that the sample magnetization conditions do not support DWS in Sample A. This is also confirmed by local spectroscopic investigations over a wide range of conditions.

Sample A Sample B

400 200 0

4.5

5.0

5.5 T (K)

6.0

6.5

Fig.  8.11 Phase diagram of the highest local critical temperature Tc versus applied magnetic field H derived from local tunneling spectroscopy measurements for the sample with magnetic domain width of 200  nm (Sample A) and the sample with magnetic domain width of 300 nm (Sample B) (Iavarone et al. 2014).

8.3 Vortex behavior in planar S/​F hybrids The controllable manipulation of the static and dynamic properties of superconducting vortices in S/​F hybrid structures holds the potential for applications, such as superconducting power transmission and computation. While the most useful properties exhibited in S/​F systems occur on the global scale, these same properties emerge from physics that develops on mesoscopic length scales. As was discussed in earlier sections, vortex behavior in S/​F hybrids can be largely divided into two subgroups depending on whether the S/​F system is planar or lithographically designed. Both types of systems alter the mixed state in the superconductor. The following section will focus on the experimental assessment of the vortex state in planar S/​F systems. Planar S/​F hybrids, in addition to lacking complicated lithographic steps, allow for the investigation of the most fundamental properties underlying the interaction between magnetically coupled superconducting and ferromagnetic layers. Introducing lithographically defined structures into the S/​F system introduces additional factors that must be taken into account when assessing the vortex behavior. In this section, we will focus directly on local scanning probe investigations of the vortex behavior in planar S/​F heterostructures.

8.3.1 Nucleation thresholds for superconducting vortices An overview of the theoretically predicted conditions for the nucleation of vortices in planar S/​F structures was presented in Sec. 8.1.1, from which we defined the general threshold magnetization for spontaneous vortices to be generated in the superconducting layer as Mc > Φ 0 / 64λ 2 (ds / 0.916w ) ln(λ /ξ ). For any given set of material-​dependent parameters, low-​temperature MFM and STM emerge as two ideal scanning probe methods for the investigation of vortex behavior at the nanoscale. These techniques each probe different length scales in the S/​F bilayer, and as a result each has its advantages and disadvantages with respect to investigations of the system properties. MFM measurements probe the variation of the relative magnetic

(

)

  257

Vortex behavior in planar S/F hybrids  257 field profile, which in the case of superconductors is directly related to the penetration depth (λ). This is particularly useful in S/​F systems because it also provides information regarding the polarity of vortices in the sample, as well as the capability to image the magnetic state of the underlying template. There are also drawbacks, however, when considering that the length scales over which λ evolves in typical type-​II superconductors are quite large. Whenever the density of vortices, or other magnetic features, is too large, there is a significant overlap of the magnetic signal, thereby limiting the spatial resolution of MFM in this regime. STM, on the other hand, measures variations in the electronic properties of the system being studied at the atomic scale. Since STM probes the local density of states in the superconductor, the coherence length (ξ) is the characteristic superconducting property investigated. The value of ξ is smaller than λ in type-​II superconductors, which means that STM can probe the vortex state with individual vortex resolution over a wider range of conditions, as shown in Fig. 8.12. To test the threshold conditions for the generation of spontaneous vortices, low-​ temperature MFM measurements were performed on Nb/​Permalloy(Py) systems as a function of Nb and Py thicknesses (Iavarone et al. 2011). For a fixed thickness of Py film (1 μm), attempts to see spontaneous vortices for Nb films of various thicknesses (100, 150, 200, and 360 nm) all failed, although an increase in the diamagnetic shielding was inferred due to the systematic decrease in the measurement contrast with increasing Nb thickness.Vortices, however, were successfully nucleated in small applied fields for the same systems, suggesting close proximity to the threshold conditions. Unfortunately, extremely thin Nb films are notoriously difficult to deal with experimentally due to oxidation and intrinsic pinning, which could obscure the results testing the threshold conditions for the nucleation of spontaneous vortices. Fortunately, as it turns out, the thickness of the Py film is already an ideal tuning parameter. Because of natural energy minimization processes, the configuration of magnetic domains in ferromagnetic Py is strongly dependent on the thickness, where above a critical thickness tc ≈ 200 nm there exist uniform stripe domains whose widths scale with thickness as w ∝ t (Chikazumi and Graham 2009, Murayama 1966). By tuning the Py thickness, the threshold condition for the nucleation of spontaneous vortices was met, as shown by the large area collage of low-​temperature MFM maps in Fig. 8.13a in the case of Nb (200 nm)/ Py (4 μm) films. Due to the inherent magnetic sensitivity of MFM the polarities of the spontaneous vortices/​antivortices are also distinguishable and, as shown in Fig. 8.13b, application of a magnetic field increases the density of vortices of one polarity while producing a reduction in the density of the vortices having the opposite polarity. More recently, similar low-​temperature MFM measurements expanded on these results by investigating a wider range of Nb and Py thicknesses where, below Tc, spontaneous vortices/​antivortices are additionally seen in Nb (200nm)/ Py (2 μm) films and Nb (120nm)/ Py (1.2 μm) films, as shown in Fig.  8.14a–​b and d–​e. While the previously described low-​temperature MFM measurement (Iavarone et  al. 2011)  showed no indication of spontaneous vortices for a Nb (100 nm)/Py (1 μm) sample, the later work does find spontaneous vortices for

258

258  Superconductor–Ferromagnet Hybrids (a)

(b)

1.3

0.15

(g)

(c)

(d)

(e)

(f)

1.1

0

100 150

m)

e (n

anc

Dist

50

–10

–5

0

+5

+10

V (mV)

0.4

Fig. 8.12  (a) Zero-​bias conductance map acquired at 2.0 K in zero applied field for S/​F bilayer of Pb (30 nm)/​[Co (2 nm)–Pd (2 nm)]N=50. The image shows spontaneous vortices produced by the underlying magnetic pattern. Scan area is 900 × 1,625 nm2. The image was obtained by acquiring several images of size 600 × 600 nm2. The inset shows the same image reported in (b). The image is rotated and superimposed to show the field of view of (b) on a larger map. The vertical dashed lines indicate the position of the individual vortices. The white dashed square is the scan area in (b). (b) Zero-​bias conductance map at T = 2.0 K and applied field H = 0 Oe in a smaller field of view. The scanning area is 600 × 600  nm2, scale bar, 100  nm. The dashed square in the image shows the field of view (300 × 300 nm2) for the zero-​bias conductance images shown in (c) T = 5.5 K, (d) T = 5.2 K, (e) T = 5.0 K, and (f) T = 4.88 K (scale bar, 100 nm for c–​f). The magnetic field is H = 0 Oe for all images. The white dashed line shows the approximate position of the domain wall. (g) Series of local tunneling dI/​dV spectra acquired across the dashed line in (d) at 5.2 K (tunneling conditions V = −10 mV and I = 100 pA). The conductance maps are normalized to the conductance maps at V = −10 mV (Iavarone et al. 2014). another sample fabricated under identical conditions. This comparison strongly underscores the fragility of this state. Because these samples are nominally the same, the authors attribute the difference in behavior to slight fluctuations in either the magnetic template or superconductor thickness, which drives the

  259

Vortex behavior in planar S/F hybrids  259 systems in opposite directions with respect to the necessary value of magnetization to generate spontaneous vortices as in Fig. 8.15c.

8.3.2 Equilibrium vortex configurations As a final point on the topic of vortices in planar S/​F hybrid systems, we would like to discuss the equilibrium vortex configurations in the case of a stripe domain structure, which differ significantly from the vortex lattice behavior in isolated type-​ II superconductors. Careful examination of the low-​ temperature MFM measurements of Nb/​Py in Fig. 8.13a–​b shows a transition from a linear chain of vortices confined above the domains in zero applied field to zigzag configurations of vortices when the field is increased. It may be expected that the inter-​ vortex spacing can be calculated from the well-​known formula av = 2Φ 0 / 3H (Tinkham 2004), thereby gaining quantitative insight into the stray field emanating from the underlying ferromagnetic domains. However, detailed STM measurements of vortex behavior on a NbSe2/​Py sample (Karapetrov et  al. 2009)  have shown that this is not exactly the situation due to the spatial confinement of the vortex lattice by the magnetic potential. In small applied fields those STM measurements, shown in Fig. 8.16a, revealed the development of a 1D chain of vortices. However, as the field increased this chain broke up into two separate chains, forming what looks like a zigzag organization of the vortices. With respect to the equilibrium vortex configuration, simulations of the system performed within GL theory reveal a delicate balance between the ferromagnetic film thickness, determined by the role of magnetic domain width, and the applied magnetic field. Sufficiently wide domains can support many chains of vortices with increasing field, whereas for

(a) H = 0 Oe frequency span 0.45 Hz

2 µm (b) H = +9.4 Oe frequency span 0.45 Hz

2 µm

Fig.  8.13 MFM images acquired on a Py/​ Nb bilayer magnetically coupled through a 10 nm layer of SiN.The thickness of Py is 4 μm; that of the Nb film is 200 nm; and the stripe’s width is 1.1 μm. (a)  Zero-​field-​cooled image showing vortex–antivortex pairs spontaneously created by the stray field of the Py. (b)  Field-​ cooled image obtained in an external applied field of H  =  +9.4 Oe. Images were obtained by scanning at a constant height z = 60 nm and a temperature T = 5.7 K. The scanning area for both images is 11.4 × 8.7 μm2 and the color scale represents a frequency span of 0.45 Hz full scale (Iavarone et al. 2011).

260

(a)

(b)

(c)

Frequency span 1.7 Hz (d)

Frequency span 1.7 Hz

(e)

(f)

Frequency span 2.4 Hz

Frequency span 2.8 Hz

Fig.  8.14  MFM maps of Nb (200  nm)/​Py (2 μm), 3.8 μm × 3.8 μm scan area at h = 180 nm. (a) T = 12 K. (b) T = 6 K. MFM maps of Nb (120 nm)/​Py (2 μm), 3.8 μm × 3.8 μm scan area at h = 180 nm. (c) T = 12 K. (d) T = 6 K. Below TS, both Nb (200 nm)/​Py (2 μm) and Nb (120 nm)/​Py (2 μm) samples show spontaneous vortices and antivortices. MFM maps of Nb(100 nm/Py (1 μm), 3.8 μm × 3.8 μm scan area at h = 130 nm and (d) T = 12 K, and at h = 30 nm and T = 6 K for the f case (Bobba et al. 2014).

z (nm)

∆ frequency 0* 4.0 µm

1.10

(b) Normalized ZBC

13

10* (a)

0

(c)

(d)

(e)

A B C

0.00 H=0

H = –500 Oe

H = +500 Oe

Fig. 8.15  (a) 20 μm × 20 μm room-​temperature MFM image taken in lift height mode showing the magnetic domain structure of the samples. (b) 1.13 μm × 0.57 μm low-​temperature scanning tunneling microscopy topography showing the morphology of the ultrathin Pb film. Tunneling conditions during scanning were V = −10 mV and I = 100 pA. (c–​e) 1.13 μm × 0.57 μm zero-​bias conductance maps taken in the same location as the topography in (b) as a function of applied fields showing the evolution of the vortex configuration, which reveals the nature of the underlying magnetic template. Points A, B, and C are points above the positive domain, the domain wall, and the negative domain, respectively. The white dotted lines indicate the approximate locations of the domain walls (Iavarone et al. 2015).

  261

Conclusions and outlook 261 (a)

100

200

(c)

600 500

inning stripes

Center of anti-p

occupied 7

itation"

rtex "lev

Lateral vo

–200

H (Oe)

400 –100

6 5

300

4

200

(b) –105 Oe

3 2

100 –165 Oe

N=1 0 0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

D (µm)

Fig. 8.16  (a) STM images of vortex configurations in NbSe2 at 4.2 K. Applied magnetic field values (in Oe) perpendicular to the surface of the superconductor are shown in the upper left corner. The white dotted lines show the underlying magnetic stripe domain boundaries.The scanning area is 0.7 × 0.7  μm2. (b) Cooper-​pair density plots, calculated at −105 and −165  Oe [in comparison with (a); see marked area], obtained for D = 1  μm (plot sizes are 1.087 × 2.176 and 1.087 × 2.193  μm2, respectively). (c) The equilibrium vortex phase diagram of the NbSe2 film at T = 4.2  K as a function of thickness of the underlying Py film D and external perpendicular field H (the domain structure of the Py film is assumed to be unperturbed by the applied field). N denotes the number of vortex chains along the positive magnetic domains (illustrated by |ψ|2 contour plots on the right, dark (light) color—​low (high) density). At high fields, vortices penetrate areas above negative magnetic domains (Karapetrov et al. 2009). small domains the density of vortices will quickly become too high, leading to the existence of “levitating” vortices.

8.4 Conclusions and outlook The field of magnetically coupled F/​S systems has matured in the last 10 years. As we described above, the magnetostatic interaction between the ferromagnet and the superconductor could lead to spontaneous formation of vortices and vortex–​antivortex pairs that are pinned to the magnetic domain structure. The formation of vortex–​antivortex lattices, vortex chains, and disordered vortex matter that was predicted theoretically has now been observed using local scanning probe techniques. Scanning probe microscopies have enabled us to observe well-​ established F/​S phenomena such as domain wall superconductivity and reverse domain superconductivity. The local nucleation of the superconducting order parameter in hybrid structures as the systems are cooled through the superconducting critical temperature has provided a very direct picture of the evolution of superconductivity in periodically modulated superconductors that has been theoretically known for some time.

262

262  Superconductor–Ferromagnet Hybrids Although many interesting phenomena have been experimentally confirmed so far, many outstanding issues still remain to be investigated. We believe confinement effects down to 0D and dynamic effects of vortex creation, annihilation, and pinning to be some of the most challenging to be accessed in experiments. Since many scanning probe techniques have a very high spatial resolution but they are intrinsically “slow,” the dynamic effects of vortex nucleation, vortex–​ antivortex annihilation, or vortex pinning and depinning at a magnetic dot or domain wall are still elusive. Better understanding and control of these processes could have a strong impact on basic physics and application of superconductors, especially in RF superconducting electronics applications (accelerator cavities, detectors).

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References (Chapter-8) 263 Chikazumi, S.S. and Graham, C.D. (2009) Physics of ferromagnetism, Oxford University Press, Oxford. Civale, L., Marwick, A.D., Worthington, T.K., Kirk, M.A., Thompson, J.R., Krusin-​Elbaum, L., Sun, Y., Clem, J.R. and Holtzberg, F. (1991) Phys. Rev. Lett. 67, 648. Erdin, S., Lyuksyutov, I.F., Pokrovsky, V.L. and Vinokur, V.M. (2001) Phys. Rev. Lett. 88, 017001. Fritzsche, J., Moshchalkov, V.V., Eitel, H., Koelle, D., Kleiner, R. and Szymczak, R. (2006) Phys. Rev. Lett. 96, 247003. Genkin, G.M., Skuzovatkin, V.V. and Tokman, I.D. (1994) J. Magn. Magn. Mater. 130, 51. Gennes, P.-​G.D. (1989) Superconductivity of metals and alloys, Addison-​Wesley, Redwood City, CA. Gomez, A., Gonzalez, E.M., Gilbert, D.A., Milošević, M.V., Kai, L. and Vicent, J.L. (2013) Supercond. Sci. Technol. 26, 085018. Houzet, M. and Buzdin, A.I. (2006) Phys. Rev. B 74, 214507. Iavarone, M., Moore, S.A., Fedor, J., Ciocys, S.T., Karapetrov, G., Pearson, J., Novosad, V. and Bader, S.D. (2014) Nature Comm. 5, 4766. Iavarone, M., Moore, S.A., Fedor, J., Novosad, V., Pearson, J.A. and Karapetrov, G. (2015) J. Supercond. Nov. Magn. 28, 1107. Iavarone, M., Scarfato, A., Bobba, F., Longobardi, M., Karapetrov, G., Novosad, V., Yefremenko, V., Giubileo, F. and Cucolo, A.M. (2011) Phys. Rev. B 84, 024506. Jaccard, Y., Martín, J.I., Cyrille, M.C., Vélez, M., Vicent, J.L. and Schuller, I.K. (1998) Phys. Rev. B 58, 8232. Jaccarino, V. and Peter, M. (1962) Phys. Rev. Lett. 9, 290. Karapetrov, G., Milošević, M.V., Iavarone, M., Fedor, J., Belkin, A., Novosad, V. and Peeters, F.M. (2009) Phys. Rev. B 80, 180506. Kayali, M.A. and Pokrovsky, V.L. (2004) Phys. Rev. B 69, 132501. Laiho, R., Lähderanta, E., Sonin, E.B. and Traito, K.B. (2003) Phys. Rev. B 67, 144522. Larkin, A.I. and Ovchinnikov, Y.N. (1979) J. Low Temp. Phys. 34, 409. Marmorkos, I.K., Matulis, A. and Peeters, F.M. (1996) Phys. Rev. B 53, 2677. Martín, J.I., Vélez, M., Nogués, J. and Schuller, I.K. (1997) Phys. Rev. Lett. 79, 1929. Matthias, B.T. and Suhl, H. (1960) Phys. Rev. Lett. 4, 51. Matthias, B.T., Suhl, H. and Corenzwit, E. (1958) Phys. Rev. Lett. 1, 449. Metlushko, V.V., Baert, M., Jonckheere, R., Moshchalkov, V.V. and Bruynseraede, Y. (1994) Solid State Commun. 91, 331. Miguel, A.Z., Tiago, S. and Clécio, C. de S.S. (2014) Supercond. Sci. Technol. 27, 055002. Milošević, M.V. and Peeters, F.M. (2003a) Phys. Rev. B 68, 024509. Milošević, M.V. and Peeters, F.M. (2003b) J. Low Temp. Phys. 130, 311. Milošević, M.V. and Peeters, F.M. (2004) Physica C 404, 246.

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264  Superconductor–Ferromagnet Hybrids Milosevic, M.V., Peeters, F.M. and Jank, B. (2011) Supercond. Sci. Technol. 24, 024001. Mironov, S.V. and Mel’nikov, A.S. (2012) Phys. Rev. B 86, 134505. Murayama, Y. (1966) J. Phys. Soc. Jpn 21, 2253. Otani, Y., Nozaki, Y., Miyajima, H., Pannetier, B., Ghidini, M., Nozières, J.P., Fillion, G. and Pugnat, P. (1994) Physica C 235–​240, Part 5, 2945. Piña, J.C., Zorro, M.A. and De Souza Silva, C.C. (2010) Physica C 470, 762. Saint-​James, D., Sarma, G. and Thomas, E.J. (1969) Type II superconductivity, Pergamon Press, New York, NY. Shmidt, V.V., Müller, P. and Ustinov, A.V. (1997) The physics of superconductors: Introduction to fundamentals and applications, Springer, New York, NY. Tamegai, T., Nakao, Y., Mohan, S. and Nakajima, Y. (2011) Supercond. Sci. Technol. 24, 24015. Tinkham, M. (2004) Introduction to superconductivity, Dover, Mineola, NY. Vlasko-​Vlasov,V., Buzdin, A., Melnikov, A.,Welp, U., Rosenmann, D., Uspenskaya, L., Fratello, V. and Kwok, W. (2012) Phys. Rev. B 85, 064505. Vlasko-​Vlasov, V., Welp, U., Kwok, W., Rosenmann, D., Claus, H., Buzdin, A.A. and Melnikov, A. (2010) Phys. Rev. B 82, 100502. Werner, R., Aladyshkin, A.Y., Guénon, S., Fritzsche, J., Nefedov, I.M., Moshchalkov, V.V., Kleiner, R. and Koelle, D. (2011) Phys. Rev. B 84, 020505. Wernsdorfer, W., Hasselbach, K., Sulpice, A., Benoit, A., Wegrowe, J.E., Thomas, L., Barbara, B. and Mailly, D. (1996) Phys. Rev. B 53, 3341. Yang, Z., Lange, M., Volodin, A., Szymczak, R. and Moshchalkov, V.V. (2004) Nat. Mater. 3, 793. Ze, J., Huadong, Y. and You-​He, Z. (2014) Supercond. Sci. Technol. 27, 105005.

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Theoretical Study of THz Emission from HTS Cuprate H. Asai Nanoelectronics Research Institute, The National Institute of Advanced Industrial Science and Technology, Tsukuba city, Ibaraki prefecture, 305-​8568, Japan

9 9.1 Intrinsic Josephson junction (IJJ) in HTS cuprate 

266

9.2 THz emitter utilizing IJJs 

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9.3 Temperature inhomogeneity in IJJ-​based THz emitter 

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9.4 THz emission from IJJs with temperature inhomogeneity 

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9.5 Summary 

286

References (Chapter-​9) 

287

A series of cuprate superconductors such as bismuth strontium calcium oxide and yttrium barium copper oxide are well known to have high critical temperatures (Tc) and are expected to be used for a wide variety of practical applications. Unlike conventional superconductors, these high-​temperature superconductors (HTS) exhibit exotic electromagnetic properties. One of the striking features of these materials is high anisotropy due to their layered structures. A single crystal of Bi2Sr2CaCu2O8-​x (Bi2212), for example, is comprised of stacks of superconducting CuO2 layers and insulating Bi–​Sr–​O layers along the c-​axis. These stacks behave as identical Josephson junctions on an atomic scale, and are referred to as intrinsic Josephson junctions (IJJs) (Kleiner et al. 1992). Recently, terahertz (THz) emitters utilizing the IJJs fabricated on the mesoscopic scale have attracted much attention. THz waves, which lie between microwaves and infrared waves, exhibit useful properties for various technological applications, for example ultra-​high-​speed wireless communication at TByte/​s, non-​destructive testing, bio sensing, and imaging for security control (Tonouchi 2007). However, the generation of high-​power electromagnetic waves in the frequency range 0.1~10 THz, known as the “terahertz gap,” is still a challenging problem. An IJJ-​based THz emitter is considered to be a promising candidate for a high-​power, practical THz emitter. In this chapter, we review the basic theory of IJJ and the mechanism of THz emission from an IJJ-​based THz emitter. In particular, we discuss the effect of the temperature inhomogeneity on the emission properties based on our theoretical studies. Moreover, we introduce a novel IJJ-​based THz emitter utilizing laser heating.

H. Asai, ‘Theoretical Study of THz Emission from HTS to Cuprate’ in The Oxford Handbook of Small Superconductors. First Edition. Edited by A.V. Narlikar. © Oxford University Press 2017. Published in 2017 by Oxford University Press. DOI 10.1093/acprof:oso/9780198738169.003.0009

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266  THz Emission from HTS Cuprate

9.1 Intrinsic Josephson junction (IJJ) in HTS cuprate Almost all HTS cuprates are composed of stacks of CuO2 layers and blocking layers which supply charge carriers to the CuO2 layers (note that some electron-​ doped cuprates (e.g. Ca1-​xSrxCuO2) do not have blocking layers). The superconducting CuO2 layers are separated by the insulating blocking layers and weakly coupled to the adjacent CuO2 layers by Josephson coupling. Namely, the crystal structures themselves naturally form Josephson junctions, referred to as intrinsic Josephson junctions (IJJs). Fig. 9.1 indicates the schematic of IJJs (Sakai et al. 1993, Tachiki et al. 1996, Kleiner et al. 2000, Tachiki et al. 2005). D and s are the distance between CuO2 layers and the thickness of CuO2 layers, respectively. Ψl (r ) = ∆ l (r ) e ϕl (r ) (r ≡ x , y ) is the two-​dimensional (2D) order parameter for the lth layer. In Bi2212, for example, D is 15 Å whereas the coherence length along the c//​z axis is of the order of ~1Å (Naughton et al. 1988), and thus, the individual order parameter Ψl (r ) is well defined. A phase difference between the (l + 1)th and lth layer, that is, the phase difference of the lth IJJ, is written as Pl (r ) = ϕl +1 (r ) − ϕl (r ) −



2π zl +1 dz Az (r ). φ0 ∫zl

(9.1)

Here, zl = Dl , Az (r ) is the z component of the vector potential. Similar to general Josephson junctions, phase differences of each IJJ have a bistable state for finite bias current Ib, that is, a zero-​voltage state and a resistive state. Thus, by ramping the current up and down repeatedly, we can see multiple branches in the current–​voltage (I–​V) curve which correspond to the different numbers of the resistive IJJs (Kleiner et al. 1992, Tsujimoto et al. 2012). Fig. 9.2 shows the schematic of multiply branched I–​V curves of the IJJs. An I–​V branch appears

Superconducting layer

S

Blocking layer Y l+1 (r) Pl (r)

D

Fig.  9.1 Theoretical model of intrinsic Josephson junctions (IJJs). Single crystals of high-​temperature superconductors (HTS) are comprised of superconducting CuO2 layers and insulating blocking layers.

Y l (r) z

Y l –1 (r) x

  267

Intrinsic Josephson junction (IJJ) in HTS cuprate 267 Nr = N Nr = 1 Nr = 2

Nr = N –1

Current

Ic

Fig.  9.2 Schematic figure of multiply branched I–​V curves of the IJJs. Nr is the number of IJJs in the resistive state, and N is the total number of IJJs.

Voltage

for each number of the resistive IJJs, N r . The outermost branch corresponds to N r = N , where N is the total number of the IJJs. In the following, we derive the partial differential equation for phase dynamics of each IJJ. The equation gives a solution for transverse plasma waves coupled with THz electromagnetic waves. First, for clarifying the relation between the phase differences and electromagnetic fields, we describe the currents flowing through IJJs and the charge accumulated into the IJJ layers using Pl (r ) and ϕl (r ).The inter-​layer current across the lth IJJ is given by

jz ,l (r ) = jc ,l (r ) sinPl (r ) + σc E z ,l (r )

(9.2)

where jc ,l is the critical current for the lth IJJ which is related to the amplitude ∆ l and ∆ l +1, and σc is the quasiparticle conductivity along the c-​axis. We assume the electric field is uniform in between the IJJ layers, and express the z-​component electric field for the lth IJJ as E z ,l . Regarding the in-​plane currents along the lth IJJ plane, we assume the following equations based on the London theory:

j x ,l (r ) = −

φ0 c   ∂ x ϕl (r ) ,  Ax (r , zl ) −  4πλ 2  2π

(9.3)



j y ,l (r ) = −

φ0 c    Ay (r , zl ) − ∂ y ϕl (r )  . 4πλ 2  2π

(9.4)

Here, Ax , Ay are the x and y components of the vector potential, and λ is the London penetration depth of the IJJ layer. For simplicity, we ignore in-​plane electric fields Ex, Ey that are much smaller than Ez. By assuming the Debye-​type screening, the charge accumulated into the lth IJJ layer is expressed as the following form (Anderson and Dayem 1964):

ρl (r ) = −

φ0 1   ∂t ϕl (r )  , (9.5)  A0 (r , zl ) −  4πµ 2  2πc

where µ is the Debye screening length and A0 is the scalar potential.

268

268  THz Emission from HTS Cuprate Next, by taking the time derivative of Eq. 9.1, we obtain 2πc zl +1 dz ( E z (r , z ) + ∂ z A0 (r , z )) φ0 ∫zl 2πc  φ0 φ  2πc   = ∂t ϕl +1 (r ) − A (r , zl ) − 0 ∂t ϕl (r )  A0 (r , zl +1 ) −  φ0  0  2πc φ0  2πc 2πcD 2πc 2 + −4πµ (ρl +1 (r ) − ρl (r )) + DE z ,l . E z ,l = φ0 φ0

∂t Pl (r ) = ∂t ϕl +1 (r ) − ∂t ϕl (r ) −

(

(9.6)

)

Eq. 9.6 and the Maxwell equation, ∇ ⋅ E = 4πρ (r ) / ε , yield the AC Josephson relation for IJJs: ∂t Pl (r ) =



2πcD 1 − α∆ (2) E z ,l , (9.7) φ0

(

)

with α = εµ 2 / (sD ). Δ(2) is the operator defined as ∆ (2) fl = fl +1 − 2 fl + fl −1 . On the other hand, the spatial derivative of Eq. 9.1 with respect to x gives, ∂ x Pl (r ) = ∂ x ϕl +1 (r ) − ∂ x ϕl (r ) 2π zl +1 − ∫ dz ∂ z Ax (r , z ) + (∂ x Az (r , z ) – ∂ z Ax (r , z )) φ0 zl 2π   φ φ    –  Ax (r , zl +1 ) − 0 ∂ x ϕl +1 (r ) +  Ax (r , zl ) − 0 ∂ x ϕl (r )  =    2π φ0   2π 2π D 8π 2 λ 2 2π D  j x ,l +1 (r ) − j x ,l (r ) + B y ,l (r ) . B y ,l (r ) = + φ0 φ 0c  φ0

(9.8)

 4π   ε From the Maxwell equation, ∇ × B (r ) =   j (r ) +   ∂t E , Eq. 9.8 is  c   c reduced to

∂ x Pl (r ) =

2π D 1 − η ∆ (2) B y , l ( r ) , (9.9) φ0

(

)

with η = λ 2 (sD ). Similar calculations with respect to y lead to



∂ y Pl (r ) =

8π 2 λ 2 2π D  j y ,l +1 (r ) − j y ,l (r ) − Bx ,l (r ) , φ 0c  φ0

∂ y Pl (r ) = −

2π D 1 − η∆ (2) Bx ,l (r ). φ0

(

)

(9.10)

(9.11)

As described in Eqs. 9.7, 9.9, and 9.11, the derivatives of the phase differences are directly connected with the electromagnetic fields in IJJs. Using Eqs. 9.8, 9.10,

  269

Intrinsic Josephson junction (IJJ) in HTS cuprate 269  4π   ε and the Maxwell equation, ∇ × B (r ) =   j (r ) +   ∂t E , the 2D Laplacian of  c   c Pl (r ) is calculated as (∂ 2y + ∂ 2x ) Pl (r ) =

8π 2 λ 2  ∂ x j x , l +1 (r ) + ∂ y j y , l +1 (r ) − ∂ x j x , l (r ) + ∂ y j y , l (r )   φ 0c  2 8π D  ε  + ∂t E z , l ( r )  . jc ,l (r ) sin Pl (r ) + σc E z ,l (r ) + 4π φ0c  

(

) (

)

(9.12)

From current conservation law for the lth IJJ layer, ∂t ρl (r ) + ∂ x j x ,l (r ) + ∂ y j y ,l (r ) + (1 / s ) ( jz ,l (r ) − jz ,l −1 (r )) = 0, and the Maxwell equation, ∇ ⋅ E = 4πρ (r ) ε , Eq. 9.12 becomes ( ∂ 2y + ∂ 2x ) Pl (r ) =

8π 2 λ 2 1 − η∆ (2)  jc ,l (r ) sin Pl (r ) + σc E z ,l (r ) φ 0c ε  + ∂t E z ,l (r ) 4π 

(

)

(9.13)

Finally, by using Eq. 9.7, we obtain the coupled Sine-​Gordon equation  for  Pl (r ) :



(

)

1 1 − η ∆ (2 ) × λ2  −1 β 1 1 − α∆ (2) ∂t Pl (r ) + 2 1 − α∆ (2)  Jc ,l (r ) sinPl (r ) + ωp ωp 

(∂ 2y + ∂ 2x ) Pl (r ) =

(

(

)

(

)

−1

 (9.14) ∂t2 Pl (r )  , 

)

where ω p = c / λ ε is the Josephson plasma frequency, and 2 2 β = 4πλσ / c ε , Jc ,l (r ) = jc ,l (r ) / jc 0 ( jc0 ≡ cφ0 / (8π λ D ) ) are the non-​ dimensional conductivity and the critical current density, respectively. As can be seen from the equation, the dynamics of lth phase difference, Pl (r ) , can be regarded as the nonlinear oscillator which is coupled with those of adjacent IJJs, Pl +1 (r ) and Pl −1 (r ) , with the coupling parameters α and η. As described in Eqs. 9.7, 9.9, and 9.11, these couplings are mediated by electromagnetic fields. Such coupled nonlinear oscillators, which are represented by the Kuramoto model (Kuramoto 1984, Sakaguchi and Kuramoto 1986), cause mutual synchronization, and the dynamics of the oscillators can be spontaneously identical even though the bare frequencies of the oscillators are different. If the coupling between IJJs results in the synchronized in-​phase state, in which all instances of Pl (r ) (l = 1, N ) are equal to the common phase difference P (r ) , the above Eqs. 9.7, 9.9, 9.11, and 9.14 are reduced to (Koshelev et al. 2008, Koyama et al. 2008, Asai et al. 2012):

( )



∂t P ( r ) =

2eVtot (r ) 2πcD E z (r ) = , (9.15) φ0 N

270

270  THz Emission from HTS Cuprate 2π D B y (r ) , (9.16) φ0



∂ x P (r ) =



∂ y P (r ) = −



(∂

2 y

)

+ ∂ 2x P (r ) =

2π D Bx (r ) , (9.17) φ0

 β 1  1 J r sin P (r ) + ∂t P (r ) + 2 ∂t2 P (r )  . 2  c( ) λ  ωp ωp 

(9.18)

E z (r ) , By (r ) , and Bx (r ) are electromagnetic fields, which are common in IJJs, N is the number of the IJJs, and Vtot (r ) is the total voltage applied to the IJJs. Here, we also assume the instances of Jc ,l (r ) are identical to the common critical current density Jc (r ) . These equations are analogous to those for a single Josephson junction (Tinkham 2004). If we assume small oscillation of the phase differences, P (r )  1, the equation has a plane wave solution propagating along the x–​y plane whose dispersion is given by ω ( k ) = ω p 1 + λ 2 k 2 , with k = kx , ky . As shown in Eqs. 9.15–​9.17, the appearance of the plane wave is associated with the electromagnetic wave propagation along the x–​y plane and called a transverse Josephson plasma wave (Tachiki et  al. 1994, Bulaevskii et al. 1996). The absorption of the electromagnetic waves originating from the excitation of the plasma waves has been verified in experiments (Tamasaku et al. 1992, Homes et al. 1993, Kakeya et al. 1998), and justifies the above IJJ model. As expected from Eq. 9.15, AC Josephson current flows through the IJJs under an applied voltage, and this current excites the transverse plasma waves whose frequencies are given by f J = (2e / h )Vtot / N . The plasma waves are coupled with the electromagnetic fields outside the IJJs, and a part of the plasma energy is converted to radiation field energy that propagates away from the IJJs. Thus, by applying the voltage to each IJJ, Vtot / N = 0.2 ~ 2 mV, IJJs can emit electromagnetic waves in a “terahertz gap” frequency range, 0.1 ~ 1THz. Bi2212 single crystals whose thicknesses are 1 ~ 2 μm, for example, contain 670~1340 IJJs, and IJJs can emit THz waves by applying 0.1 ~ 3 V to the crystal. As discussed above, the phase differences in IJJs can synchronize each other through electromagnetic coupling, and the in-​phase state is one simple solution which describes the transverse plasma waves inside IJJs. Furthermore, the THz emission power estimated from the in-​phase state is comparable to those in experiments as mentioned in Sec. 9.4. Thus, in this chapter, we focus on the analyses based on in-​phase approximation. Note that the plasma waves can also couple to the so-​called π kink state in which Pl (r ) has pairs of π solitons. For

(

)

  271

THz emitter utilizing IJJs 271 further details on the analysis based on the π kink state, see Lin and Hu (2008), Hu et al. (2008), and Koshelev (2008).

9.2 THz emitter utilizing IJJs Since the discovery of the intrinsic Josephson effect in Bi2212 (Kleiner et  al. 1992), various studies have investigated THz emission from Bi2212 from the viewpoints of both fundamental physics and technological applications. For obtaining coherent THz emission, synchronized motion of the phase differences coupled with Josephson plasma is crucial. Several studies tried to generate the synchronized phase motion accompanied by coherent dynamics of the Josephson vortex under applied magnetic fields (Irie et al. 1998, Krasnov et al. 1999, Kim et al. 2005, Bae et al. 2007). However, the emission power from such Bi2212 is limited to the pW range (Batov et al. 2006). On the other hand, Ozyuzer et al. (2007) have demonstrated ~μW range THz emission without an applied magnetic field. They fabricated a rectangular-​shaped pattern, which is called a mesa, on a Bi2212 single crystal by photolithography and Ar ion milling. Fig. 9.3 indicates the schematic figure of the Bi2212 mesa similar to their experimental setup. A DC voltage was applied between the Au contact attached to the top of the mesa and the base crystal, and thus, only the IJJs in the mesa region were subjected to the voltage. Therefore, the top electrode and the base crystal work as a “cavity resonator” that confines the electromagnetic waves excited in the mesa region. The electromagnetic standing waves induced by multiple reflections in the cavity enhance the synchronization of the phase differences. Ozyuzer et al. fabricated the mesas whose widths are comparable to the wavelength of the THz electromagnetic waves, 40~100 μm, and succeeded in coherent THz emission with frequencies ranging from 0.36 to 0.85 THz. They found that these emission frequencies f = f J almost satisfy the cavity resonance condition given by f = fC = c / (2 ε c w) that corresponds to the fundamental standing wave mode along the mesa width, where w is the mesa width and ε c is the dielectric constant along the c-​axis. After this pioneering work, THz emitters based on mesa-​structured Bi2212 have been

DC bias

“Cavity resonator”

Bi22

12 b

Electrode

ase

cr ys

tal

Bias voltage

Electrode IJJ mesa Base crystal

THz wave

Fig.  9.3 Schematic figure of a mesa-​ structured Bi2212 similar to the experimental setup (Ozyuzer et al. 2007). The top electrode and the base Bi2212 crystal work as a “cavity resonator.”

272

272  THz Emission from HTS Cuprate

1.2

Frequency (THz)

εc = 17.64 0.8

0.4

0

0

0.02 0.01 1/w (1/µm)

0.03

Fig. 9.4  Emission frequency of the Bi2212 mesa as a function of the inverse mesa width, 1/​w (after Kashiwagi et  al. 2011). The dotted line indicates the proportional relation given by cavity resonance condition 𝑓 = 𝑓C = 𝑐/​(2√𝜀c𝑤) with 𝜀c = 17.64.

extensively studied both experimentally and theoretically (Minami et  al. 2009, Kadowaki et al. 2010, Wang et al. 2010, Benseman et al. 2011, 2013, Kashiwagi et al. 2012, Sekimoto et al. 2013, Welp et al. 2013, Ji et al. 2014, Lin et al. 2008, Hu et al. 2008, Koshelev 2008, Koshelev et al. 2008, Koyama et al. 2009, 2011, Rakhmanov et  al. 2009, Savelev et  al. 2010, Klemm and Kadowaki 2010, Asai et al. 2012, 2012a, Asai and Kawabata 2014). Kashiwagi et al. (2012) investigated the relation between the mesa width and the emission frequencies for various sizes of the rectangular mesas. Fig. 9.4 indicates the emission frequency as a function of the inverse mesa width, 1/ w . The dotted line indicates the proportional relation given by the above resonance condition with ε c = 17.64 , and clarifies that the emission frequencies satisfy the cavity resonance condition. In addition to the studies on rectangular mesas, THz emission from various shapes of mesas has been observed such as square (Tsujimoto et  al. 2010), isosceles triangle (Delfanazari et  al. 2013), pentagonal (Delfanazari et  al. 2015), and cylindrical mesas (Tsujimoto et al. 2010, Guénon et al. 2010). These studies also verified that cavity resonance in the mesa plays a key role in intense THz emission. Of particular note is that some mesa-​structured THz emitters show relatively high tunability, 30~60% for the emission frequency (Wang et  al. 2010, Delfanazari et al. 2013, Kitamura et al. 2014), despite the cavity resonance frequencies being determined by the widths of the mesas. Several mechanisms for tunability have been considered, for example the appearance of plasma modes perpendicular to the IJJ plane (Benseman et al. 2011, Kakeya et al. 2015), the change of the effective length of the cavity because of hotspot formation (Wang et al. 2010), and the change of the kinetic inductance of the cavity with temperature. However, it is not entirely clear which mechanism is dominant and to what extent these mechanisms are additive. In any case, high tunability of the emission frequency is one of the advantages in practical application of the emitters. Here, we would like to stress the advantages of IJJ-​based THz emitters compared to emitters based on Josephson junctions with conventional superconductors. First, since the IJJs are junctions on an atomic scale, over 1,000 of the junctions exist even in a thin IJJ emitter whose thickness is around 2 μm. The coherent dynamics of the large number of Josephson junctions results in intense THz wave emission. Thus, IJJ-​based emitters can generate high-​power THz waves despite the device size being on the compact “mesoscopic” scale. At this stage, the emission power from an IJJ emitter observed in experiments has reached 30 μW (Sekimoto et al. 2013). Moreover, 600 μW power emission from an array of three IJJ emitters has been reported (Benseman et al. 2013). Second, HTS cuprate has a large superconducting gap of ~40 meV, thus IJJ emitters can generate high​frequency electromagnetic waves up to ~10 THz in principle. Recent experiments have reported up to 1.6 THz emission waves that are difficult to generate by conventional Josephson junctions (Kashiwagi et al. 2015). Third, the Tc of the HTS cuprate is larger than 77 K, and thus, IJJ emitters can operate around the boiling temperature of nitrogen (Kitamura et al. 2014, Hao et al. 2015). Hence, IJJ emitters can be compact and low-​priced by using a liquid nitrogen refrigerator.

  273

Temperature inhomogeneity in IJJ-based THz emitter  273 These unique advantages make the IJJ-​based THz emitter a promising candidate for practical THz emitters.

9.3 Temperature inhomogeneity in IJJ-​based THz emitter In the operation of IJJ-​based THz emitters, application of DC bias voltage is required in order to excite Josephson plasma waves in a IJJ mesa. Therefore, a strong Joule heating, which is typically 10~30 mW for Bi2212 mesas, occurs in the mesa region. Since the thermal conductivity of the IJJs is low, for example ~5 W/​m·K for the a–​b plane of Bi2212 (Crommie 1991), the Joule heating in the mesa region leads to a large inhomogeneity in temperature distribution. Consequently, a hot spot appears in the mesa wherein the temperature is locally higher than Tc during the THz emission process (Wang et al. 2009, Wang 2010, Yurgens et al. 2011, Gross et al. 2012). Wang et al. first observed formation of hot spots during THz emission by using low-​temperature scanning laser microscopy (LT-​SLM) (Wang et al. 2009). Interestingly, they reported intense THz emission when the hot spots appeared in the mesa, nevertheless the hot spots diminish the area of the superconducting state. Hence, temperature inhomogeneity such as the hot spot has been considered to play a crucial role in strong THz emission. Recently, temperature distribution of the IJJ mesa has been directly analyzed by observing luminescence of the fluorescent materials coated to the mesa surface (Benseman et  al. 2013a, Minami et  al. 2014, Tsujimoto et  al. 2014). Fig. 9.5 shows the temperature distribution of the mesa along the mesa length using a photoluminescence image of SiC (Minami et  al. 2014). We can clearly see the appearance of the hot spot in the mesa. In this manner, temperature inhomogeneity by Joule heating usually appears during THz emission. However, considering that the THz emission originates from the superconducting Josephson effect, the fact that temperature inhomogeneity by Joule “heating” is important for intense THz emission is counterintuitive. In the following section, we discuss the effect of temperature inhomogeneity on THz emission based on our theoretical works. 120 Tc

T (K)

80 40 0

0

100

200 Position (µm)

300

400

Fig.  9.5 Temperature distribution of the mesa along the mesa length (0 ~ 400 μm) using the photoluminescence image of SiC (after Minami et al. 2014). Copyright (2014) by the American Physical Society.

274

274  THz Emission from HTS Cuprate

9.4 THz emission from IJJs with temperature inhomogeneity In this section, we focus on emission in the high-​bias current regime where all IJJs are in a resistive state (Welp et al. 2013). In this regime, the I–​V curves are reversible for increasing and decreasing the bias, and thus, the emission can be reversible for the bias scan. Such reversibility is important for an application that requires stable emission. Moreover, a hot spot appearing in the high-​bias regime establishes a shunt resistor common to all IJJs. Theoretical studies demonstrate that, in addition to the mutual electromagnetic coupling between IJJs, such a shunt promotes the synchronization of the phase differences (Yurgens et al. 2011). Therefore, the in-​phase approximation discussed in Sec 9.1 can be appropriate in the high-​bias regime. The effects of temperature inhomogeneity on THz emission are discussed from the viewpoint of inhomogeneity of a critical current density Jc . A  temperature inhomogeneity causes inhomogeneous Jc distribution because the Jc depends on temperature. Since Jc gives the amplitude of AC Josephson oscillation in the THz frequency range, a change of the spatial Jc distribution strongly affects the THz emission properties. Hence, we simulate the effect of temperature inhomogeneity by changing the Jc distribution.

9.4.1 The effect of temperature inhomogeneity on THz emission properties Firstly, in order to clarify the effect of temperature inhomogeneity, we consider a rectangular mesa which includes a region where the temperature is locally high (Asai et al. 2012). As described above, such regions like “hot spots” are expressed as the spots where the Jc s are low. Fig. 9.6a indicates the three-​dimensional (3D) schematic view of the mesa. IJJs stack along the z-​axis, and the mesa is sandwiched by an infinite-​size substrate and an upper electrode whose geometry is the same as the mesa. We assume that the substrate and the electrode are perfect electric conductors. The uniform external current parallel to the z-​axis is injected from the upper electrode. The dimensions of the mesas are as follows:  width w = 0.48λ c, length l  =  0.72 λ c , and height h = 0.02λ c , where λ c is the magnetic penetration depth along the IJJ plane. If we take λ c = 100 μm, which is the typical value for the Bi2212 single crystal, the sizes of the mesas become similar to the experimental studies. In this study, we use a finite-​difference time-​domain method for calculating the electromagnetic field inside and outside of the mesa. We solve Eqs. 9.15–​ 9.18 for calculating the phase differences and electromagnetic fields in the mesa. Meanwhile, in the region outside of the mesa, we solve 3D Maxwell’s equations in free space. The electromagnetic fields in both the regions are directly connected to each other at the side edges of the mesa. For the outer boundary of

  275

THz emission from IJJs with temperature inhomogeneity 275 (a)

y

(b)

ϕ

w IJJ

l

y

Low Jc (Hot) spot

l

x

θ

Infinite substrate

w

z Mesa A

x

Mesa B

Ele

ct me rode sa

(xe, ye) (xs, ys)

Low Jc (Hot) spot

Mesa C

Low Jc (Hot) spot

Mesa D

the calculation region, we use the perfectly matched layer absorbing boundary condition. We consider four types of mesas with different Jc distribution (mesas A–​D). Mesa A  has homogeneous Jc distribution, whereas mesas B–​D have one low-​ Jc spot each. We take Jc (r ) = 1 − γ in the low-​Jc spot (xs < x < xe, ys < y < ye) and Jc (r )   =  1 in the other region. Here, positive values of γ indicate that the spot is hot compared to the other region, and the increase of γ corresponds to the increase of the temperature of the spot. The center of the mesa is located at the origin, and we choose xs = −0.24λ c , xe = −0.08λ c , ys = −0.12λ c , ye = 0.12λ c for mesa B, xs = −0.08λ c , xe = 0.08λ c , ys = −0.36λ c , ye = −0.12λ c for mesa C, and xs = −0.24λ c , xe = −0.08λ c , ys = −0.36λ c , ye = −0.12λ c for mesa D. Fig. 9.6b shows schematic figures of the geometries of these four mesas. We examine the emission power versus voltage (P–​V) curves of the mesas. Hereafter, the applied voltage Vtot is divided by the number of the IJJ layer, V = Vtot / N , in order to indicate the voltage applied to each IJJ layer. The emission powers are calculated from the time average of the surface integration of the Poynting vector given by (c / 4π ) ∫ ( E × H ) ⋅ n dS , where S indicates the outer S

boundary of the calculation region and n is the unit vector normal to the boundary. Fig. 9.7a–​d indicate the P–​V curves of mesas A–​D, respectively for γ = 0.1 ~ 0.3. Here, P0 = cφ20 / 16πD3, and Vp = ω p / 2e are the normalized power and voltage, respectively. In these figures, we can see sharp peaks at voltages where the AC Josephson frequency f J = (2e / h )V satisfies the cavity resonance condition for the transverse magnetic (TM) mode f J = fC = c

(m / 2w )2 + (n / 2l ) 2 /

ε c , where

m and n are arbitrary integers. The peak voltage V = 6.44Vp, for example, satisfies (m, n ) = (1, 0). If we take D = 1.2 × 10−7 cm, the peak powers become ~10 μW. These values are comparable to those reported in experimental works (Minami et al. 2009, Benseman et al. 2013a). As can be seen from these figures, the cavity

Fig.  9.6  (a)  Schematic view of the 3D mesa. (b)  Four types of mesas including the low-​ Jc spots (mesas A–​ D). Copyrighted by the American Physical Society.

276

276  THz Emission from HTS Cuprate

[10–6] 2

Power (P0)

(2,0)

(0,2) 5

6

7

8 9 10 Voltage (Vp)

(0,1)

1

(0,2)

11

γ = 0.1 γ = 0.2 γ = 0.3

12

Power (P0)

1

4 (c)

(b) [x10–6] 2

(2,0)

(1,0)

γ = 0.1 (2,0) γ = 0.2 γ = 0.3

(0,2)

1

(1,2) 4

13

5

6

(d) [10–6] 2 (0,1)

Power (P0)

Power (P0)

(a) [10–6] 2

7

8 9 10 Voltage (Vp)

(1,0)

11

12

γ = 0.1 γ = 0.2 γ = 0.3

1 (1,1)

13

(2,0)

(0,2) (1,2)

4

5

6

7

8 9 10 Voltage (Vp)

11

12

13

4

5

6

7

8 9 10 Voltage (Vp)

11

12

13

Fig. 9.7  Emission power versus voltage curve for (a) mesa A, (b) mesa B, (c) mesa C, and (d) mesa D (after Asai et al. 2012). Copyright (2012) by the American Physical Society. resonant peaks whose m and n are even appear in all mesas. Meanwhile, the peaks whose m is odd only appear when the y-​axis symmetry is broken (mesas B and D), and the peaks whose n is odd only appear when the x-​axis symmetry is broken (mesas C and D). In the mesa-​structured IJJs, the excitation of the electromagnetic mode comes from the AC Josephson current flowing through the mesa, and the distribution of the AC Josephson current has a symmetry corresponding to the mesa structure unless the solitonic states appear. Hence, in a homogeneous rectangular mesa such as mesa A, the odd-​numbered cavity modes whose electric fields are asymmetric with respect to the center of the mesa cannot be excited. Meanwhile, in mesas B–​D, the odd-​numbered cavity modes appear because the low-​Jc spots break the reflectional symmetries of the excitation in rectangular mesas. Furthermore, the increase of the asymmetric nature of the mesa enhances the mode excitation, as can be seen from the results for γ = 0.1 ~ 0.3 in Fig. 9.7b–​d. This result indicates that the increase of local temperature can lead to the increase of the THz emission power. Next, we investigate the internal mode of the mesa at peak voltages. Hereafter, we show the results of mesa D for γ = 0.2 because all cavity modes appear in this mesa. To investigate the oscillating part of the electric field, we take the Fourier transform of the electric fields E z in the mesa. Fig.  9.8a–​f show the amplitude maps of the oscillation part of E z at the emission frequencies for V = 4.77Vp,

  277

THz emission from IJJs with temperature inhomogeneity 277

0 0

(d)

(e) Amplitude (arb. unit)

1.5

0.36 0 –0.24

0 0

x (λc)

0.24 –0.36

y (λc)

0.36 0 –0.24

0 0

x (λc)

Amplitude (arb. unit)

x (λc)

0.24 –0.36

y (λc)

2

0.24 –0.36

y (λc)

Amplitude (arb. unit)

0.36 0 –0.24

(c)

0.36 3 0 0 –0.24 0

x (λc)

0.24

y (λc)

–0.36

(f)

0.36 2 0 0 –0.24 0

x (λc)

–0.36 0.24

y (λc)

Amplitude (arb. unit)

2

Amplitude (arb. unit)

(b)

Amplitude (arb. unit)

(a)

2

0.36 0 –0.24

0 0

x (λc)

0.24

–0.36

y (λc)

V = 6.44Vp, V = 8.07Vp, V = 8.77Vp, V = 11.3Vp, and V = 12.7Vp , respectively. These figures show that the standing waves corresponding to the cavity resonance modes appear. It should be noted that the amplitudes of the electric fields at the positions of the nodes have finite values. Since the shapes of the standing waves are clearly seen in these figures, this fact indicates the existence of background oscillating modes which are almost uniform in the mesas. If we assume the uniform background modes, the oscillating part of the electric field EOS in the mesa can be written as  mπ x mπ   n πx n π  +  EOS ( x , y , t ) = E BG sin (2π fct ) + ECV cos  +  cos   w 2  l 2  (9.19) × sin (2π fct + ϕ0 ) , where E BG and ECV are respectively the amplitudes of the uniform background mode and cavity resonance mode, fc is the cavity resonance frequency, and ϕ0 is the phase difference between these modes. Here, we examine the results at V = 6.44Vp which satisfy the TM(1,0) cavity resonance condition to see how well this “dual” mode model (Klemm and Kadowaki 2010) can describe the numerical results. Fig. 9.9 shows the time evolution of the oscillating part of the electric fields at three positions in the mesa: ( x , y ) = (0, 0), ( x , y ) = (0.24λ c , 0), ( x , y ) = ( −0.24λ c , 0). The open crosses, squares, and circles indicate the calculated values in our simulation using Eqs. 9.15–​9.18 and the lines indicate the EOS ( x , y , t ) obtained by Eq. (9.19) when m = 1, n = 0, E BG = 0.142E 0, ECV = 0.492E 0 with E 0 = ω p / 2eD, and ϕ0 = 0.5π . As seen in this figure, the calculated values are well described by Eq. 9.19. 2 2 2 Moreover, the amplitude distribution of EOS becomes ~ E BG + ECV sin ( πx / w )

Fig.  9.8  Amplitude maps of the oscillation part of Ez at the emission frequencies for (a)  V = 4.77Vp, (b)  V = 6.44Vp, (c)  V = 8.07Vp, (d)  V = 8.77Vp, (e) V = 11.3Vp, and (f)  V = 12.7Vp (after Asai et  al. 2012). Copyright (2012) by the American Physical Society.

278

278  THz Emission from HTS Cuprate

Electric Field ( ωp/2ed)

0.8

Fig.  9.9  Time evolution of the oscillation part of Ez at V = 6.44Vp. The crosses, squares, and circles indicate the calculated values in our simulation at (𝑥, 𝑦) = (−0.24 𝜆C, 0), (𝑥, 𝑦) = (0, 0), and (𝑥, 𝑦) = (0.24 𝜆C, 0), respectively. The lines indicate EOS at each position when 𝑚 = 1, 𝑛 = 0, 𝐸BG = 0.142𝐸0, 𝐸cv = 0.492𝐸0 and 𝜑0 = 0.5𝜋 (after Asai et  al. 2012). Copyright (2012) by the American Physical Society.

(x = 0, y = 0) (x = 0.24λc, y = 0) (x = –0.24λc, y = 0)

0.4

0

–0.4

–0.8

EOS(x = 0, y = 0) EOS(x = 0.24λc, y = 0) EOS(x = –0.24λc, y = 0)

1

3

2

4

Time (1/ωp)

when the phase difference ϕ0 = 0.5π , and this distribution agrees well with the amplitude distribution at V = 6.44Vp shown in Fig. 9.8b. Furthermore, for clarifying the relation between the internal modes and radiation patterns, we calculate the 3D radiation patterns at peak voltages. The analysis of the radiation pattern from samples is an effective approach to clarify the internal plasma mode in the mesa. Thus, the radiation pattern has been investigated experimentally (Kadowaki et  al. 2010, Tsujimoto et  al. 2010, Kashiwagi et  al. 2011)  and theoretically (Klemm and Kadowaki 2010). In this study, the far-​field radiation patterns are calculated from the equivalent electric and magnetic current along the surface of the calculation region (Luebbers et al. 1991). Fig. 9.10a shows the 3D plots of radiation intensity I (θ, φ) at V = 6.44Vp, and Fig. 9.10b shows the polar plots of I (θ, 0°) in the x–​z plane and I (θ, 90°) in the y–​z plane at V = 6.44Vp. We can see from these figures that the radiation pattern shows strong asymmetry with respect to the y–​z plane and is different from that

Fig.  9.10  3D plots of 𝐼 (𝜃,𝜙) at

V = 6.44Vp . (b)  The polar plots of 𝐼 (𝜃,

0°) in the x–​z plane and 𝐼 (𝜃, 90°) in the y–​z plane at V = 6.44Vp (after Asai et  al. 2012). Copyright (2012) by the American Physical Society.

I (θ, 90°) I (θ, 0°)

[arb.unit]

θ = 0°

z

θ = 90°

θ = –90°

y x

(a)

(b)

  279

THz emission from IJJs with temperature inhomogeneity 279 of the general patch antenna satisfying the TM (1,0) cavity resonance condition. This asymmetry is explained by the interference between electromagnetic waves emitted by two different radiation sources. As previously mentioned, the internal modes of the mesas are described by the sum of the background modes and cavity resonance modes. Hence, the total electromagnetic wave is also described by the sum of the electromagnetic waves emitted by these modes. If we assume a uniform background mode similar to the above discussion, the background mode emits the electromagnetic wave whose electric field is symmetric with respect to the y–​z plane. On the other hand, the TM (1,0) cavity resonance mode emits the electromagnetic wave whose electric field is antisymmetric with respect to the y–​z plane. Therefore, radiation intensity becomes strong by constructive interference on one side of the y–​z plane and weak by destructive interference on the opposite side because of the different symmetries of the radiation waves. In Fig. 9.11a–​b, we show I (θ, φ) , I (θ, 0°), and I (θ, 90°) at V = 12.7Vp which satisfies the TM (2,0) cavity resonance condition. The radiation intensity decreases to zero at θ  =  0◦ similar to the dipole antenna radiation. In contrast to the result in the TM (1,0) cavity resonance condition, the radiation pattern is almost symmetric with respect to the y–​z plane in the TM (2,0) cavity resonance condition. This is because the TM (2,0) cavity mode emits the electromagnetic wave whose electric field is symmetric with respect to the y–​z plane, similarly to the uniform background mode. In this case, interference between electromagnetic waves emitted by both modes results in similar radiation intensities on each side of the y–​z plane. The small asymmetric nature shown in Fig. 9.11a–​b reflects the fact that the background mode is slightly asymmetric because of the existence of the low-​Jc  spot. It is important to keep in mind that the diffraction at substrate edges also affects the total radiation pattern. In particular, in the case that the sizes of the substrates are comparable to the wavelength of the emission waves as in the experiments, the radiation patterns greatly change from those on the infinite ground plane (Huang 1983, Balanis 2005). Therefore, for the analysis of the experimental radiation patterns, we need to take into account the effect of the diffraction at substrate edges whose geometries are similar to the experimental setups. Fig.  9.12a, for example, shows

I (q, 90°) I (q, 0°)

[arb.unit]

q = 0°

z

q = 90°

q = –90°

y x

(a)

Fig.  9.11  3D plots of 𝐼 (𝜃,𝜙) at

V = 12.7 Vp. (b)  The polar plots of 𝐼 (𝜃,

(b)

0°) in the x–​z plane and 𝐼 (𝜃, 90°) in the y–​z plane at V = 12.7 Vp (after Asai et  al. 2012). Copyright (2012) by the American Physical Society.

280

280  THz Emission from HTS Cuprate (a)

(c)

Calculation (infinite substrate) θ = 0°

Calculation (finite-size substrate) θ = 0°

I (θ, 90°) I (θ, 0°)

[arb.unit]

(b)

I (θ, 0°)

[arb.unit]

Fig.  9.12 (a)  Experimental radiation pattern corresponding to the TM(2, 0) mode (after Kashiwagi et al. 2011). (b–​c) Calculated radiation patterns corresponding to the TM(2, 0) mode with an infinite-​ size substrate and a finite-​size substrate (after Asai et  al. 2013). Copyright 2013, with permission from Elsevier.

Experiment

θ = –90°

θ = 90°

θ = –90°

I (θ, 90°) I (θ, 0°)

θ = 90°

 

the experimental radiation pattern corresponding to the TM(2,0) mode (Kashiwagi et al. 2011). This figure indicates that the intensity peaks appear around ±60°. On the other hand, Fig. 9.12b–​c show the calculated radiation patterns corresponding to the TM(2,0) mode with an infinite-​size substrate and a finite-​size substrate whose dimensions are the same as the experimental setup (Asai et al. 2013). As shown in these figures, the intensity peaks appear around ±60° for the finite-​size substrates, similarly to the experimental results. Meanwhile, the intensity peak appears at around ±90° for the infinite-​size substrate. Finally, we briefly discuss the P–​V characteristic from the viewpoint of Fano resonance (Fano 1961). As shown in Fig. 9.7a–​d, the peaks corresponding to the even-​numbered cavity resonance mode show asymmetric Fano resonance-​like peaks. Fano resonance appears in systems where continuum and discrete states coexist such as the quantum dot (Kobayashi et al. 2002) in an Aharonov–​Bohm ring and the asymmetric nature of physical quantities with respect to the energy comes from the interference between these continuum and discrete states. In the in-​phase IJJs under DC bias voltages, the Josephson plasma mode which uniformly oscillates inside the IJJs appears for any voltage like the continuum mode, and this mode corresponds to the uniform background mode discussed above. Meanwhile, like the discrete mode, the Josephson plasma modes oscillating with electromagnetic standing waves appear only when the voltage satisfies the cavity resonance condition. Since the radiation power is related to the strength of the plasma mode excitation inside the IJJs, the asymmetric peaks in P–​V curves can be regarded as Fano resonance peaks coming from the interference between the continuum and discrete Josephson plasma modes. It should be noted that this asymmetric peak does not appear in the odd-​numbered cavity resonance condition. In the case of the odd-​numbered cavity resonance condition, the discrete Josephson plasma modes become antisymmetric with respect to the center

  281

THz emission from IJJs with temperature inhomogeneity 281 of the mesa, while the continuum modes are symmetric. Thus, the interference between the continuum and discrete plasma modes becomes very weak, and the Fano resonance peak disappears. In this manner, the precise measurement of the peak shapes of the P–​V curves gives further information for the internal plasma modes in the IJJs.

9.4.2 High-​power THz emission utilizing external local heating Although the IJJ-​based emitters are able to cover the frequency range of 0.3–​1 THz, the observed emission powers of the order of 30 μW are considerably lower than the 1 mW that is required for practical applications. Therefore, further investigations towards the realization of high-​power emission are important. Thus, in this subsection, we introduce a novel method for enhancing the THz emission power based on the knowledge in the previous subsection. As discussed in Sec. 9.4.1, an inhomogeneous temperature distribution greatly increases the THz emission power from an IJJ mesa. Thus, we consider an IJJ THz emitter locally heated by a focused laser beam as shown in Fig. 9.13 (Asai and Kawabata 2014). The IJJ mesa is fabricated on an IJJ base crystal, and the mesa is covered by an upper electrode. A focused laser beam is irradiated on the upper electrode and locally increases the temperature of the IJJ mesa beneath the electrode. The local temperature of the IJJ mesa is controlled by the laser heating. An artificial Jc distribution created by the local heating strongly enhances the THz emission. To validate this concept and investigate the optimum conditions for intense emission, we perform a numerical simulation based on a 2D model. The dimensions of the model are as follows:  the width of the electrode and the IJJ mesa wm = 60 μm, the width of the IJJ base crystal wb = 220 μm, the thickness of the electrode he = 1 μm, the thickness of the IJJ mesa hm = 1 μm, and the thickness of the IJJ base crystal he = 20 μm. A DC voltage is applied to the IJJ mesa region, and the base crystal is attached to an infinite ground plane. In this

Laser irradiation

z xL x

QL

Electrode

he

IJJ mesa

hm

wm

IJJ base crystal wb

Ground plane

hb

Fig. 9.13  Schematic figure of IJJ THz emitter locally heated by focused laser beam.The notation xL indicates the position of the heating spot. The DC voltage is applied to the IJJ mesa region. The external heating power is designated as QL (after Asai et  al. 2014). Copyright 2014, AIP Publishing LLC.

282

282  THz Emission from HTS Cuprate calculation, we systematically investigate theTHz emission of such a setup by solving Eqs. 9.15–​9.18 and thermal diffusion equations simultaneously. The thermal diffusion equation in IJJs is described as follows: 0=



∂  ∂T  ∂  ∂T  J2 κ ab (T ) κ c (T ) + + ex .     ∂x  ∂x  ∂z  ∂z  σc (T )

(9.20)

Here, T denotes the temperature, σc denotes the c-​axis conductivity in the IJJ mesa, Jex denotes the external current density injected into the IJJ mesa region, and κ ab (c ) denotes the thermal conductivity along the a–​b plane (c-​axis). In the calculation, we include a heat source QL in the electrode region that is indicated by the shaded area in Fig. 9.13 in order to model the local heating by laser irradiation. In this study, we assume that the spot size of QL is 5 μm, and the position of the heating spot is defined by the notation. We assume the temperature-​dependence of 𝜅(a,b)c, and 𝜎c adopted in Gross et  al. (2012). We impose the boundary condition 𝑇 = 𝑇bath at the boundary between the IJJ base crystal and the ground plane, and we take 𝑇bath = 0.3 𝑇c. THz emission as discussed below is also observed for different values of 𝑇bath < 𝑇c. The open boundary condition ∇𝑇 = 0 is used for other boundaries. The dynamics of the phase differences are calculated by using Eqs. 9.15–​9.18 for the x-​axis. Regarding the temperature dependence of Jc , we use the Ambegaokar–​Baratoff relation, 𝐽c(𝑇) = (Δ(𝑇)/​Δ(0))tanh(Δ(𝑇)/​𝑘B𝑇) where Δ(𝑇) denotes the BCS superconducting gap (Kleiner et al. 1992, Kashiwaya et al. 2000). In this study, we assume 𝐽c(0) = 400 A/​cm2. In this calculation, we investigate the THz emission power by varying the heating power QL and the heating-​spot position xL. First, for clarifying the optimum heating power, we vary QL from 0 to 0.55 W/​cm for a fixed heating-​spot position of xL = 13.3 μm. We decrease the external current from the critical current value of the mesa 2.4 A/​cm (= 𝐽c(0) × 𝑤m) to 0 A/​cm, and calculate emission power

(a)

20 10

Power (mW/cm)

1

0.15 W/cm 0.3 W/cm 0.45 W/cm

0

0.15 W/cm 0.3 W/cm 0.45 W/cm

0

Fig.  9.14 (a)  Current versus voltage curves, and (b)  emission power versus voltage curves for QL  =  0.15, 0.3, and 0.45 W/​cm with heating-​spot position of xL  =  13.3 μm (after Asai et  al. 2014). Copyright 2014, AIP Publishing LLC.

Current (A/cm)

2

(b)

0

1 Voltage (mV)

2

1.1

1.2

1.3

1.4

Voltage (mV)

1.5

1.6

  283

THz emission from IJJs with temperature inhomogeneity 283 P for each current. Fig. 9.14a–​b show the current I and the emission power P as a function of the voltage V for different values of heating power QL. We can see that the I–​V curve exhibits a negative differential resistance. This back-​bending feature comes from strong self-​heating in the high-​current region (0.3–​2.4 A/​cm) (Kurter et  al. 2010). The resistivity of the IJJs, 1/​𝜎c, decreases with increase in the mesa temperature (Yurgens 2011, Gross et al. 2012), and thus, the voltage across the IJJs is suppressed in the high-​current region. Importantly, in addition to the self-​heating, the local heating given by QL increases the mesa temperature and decreases the voltage. Hence, the I–​V curves shift towards the low-​voltage region with increase in QL as shown in Fig. 9.14a. At V = 1.23 mV, the I–​V curve exhibits a small kink, and the P–​V curve exhibits a sharp peak which indicates strong THz emission. As discussed in Sec. 9.4.1, this intense emission originates from the excitation of the cavity resonance mode in the IJJ mesa. A snapshot of the oscillating part of the electric field at I = 1.19 A, V = 1.23 mV for QL = 0.3 W/​cm is shown in Fig. 9.15a. As can be seen in the figure, a standing wave mode whose half-​wavelength is equal to the mesa width appears. Fig. 9.15b shows the frequency spectrum of the electromagnetic (EM) wave emitted by the mesa. It is seen that a sharp peak appears around 0.59 THz, which is cavity resonance frequency given by 𝑓c = 𝑐𝑚/​(2𝑤m√𝜀c), where m  =  1. The intense emission at V = 1.23 mV does not appear for QL > 0.55 W/​cm because the strong external heating always ensures that the voltage across the IJJs is less than the resonant voltage V = 1.23 mV. Fig.  9.16a shows the plot of the emission power as a function of QL; this plot is used to examine the optimum heating power for high-​power THz emission. Remarkably, the emission power P is dramatically enhanced by the external heat source QL in comparison to the case without the external local heating, as observed from this figure. Our results indicate that the use of external local heating is a powerful method to achieve high-​power THz emission. We have also confirmed that this enhancement is also observed for emission corresponding to other cavity resonance modes, e.g., for the mode m = 2.

(a)

(b)

Intensity (arb.unit)

Electric field (kV/m)

100

0 Heating Center of spot the mesa

10

0

−100 0

0.5 x/wm

1

0.5 Frequency (THz)

1

Fig.  9.15 (a)  Snapshot of the oscillating part of the electric field in the mesa, and (b)  frequency spectrum of emission from the mesa at 𝐼 = 1.19 A, 𝑉 = 1.23 mV for QL  =  0.3 W/​cm with heating-​ spot position of xL  =  13.3 μm (after Asai et al. 2014). Copyright 2014, AIP Publishing LLC.

284

284  THz Emission from HTS Cuprate (a)

Power (mW/cm)

20

10

0 0

0.2

0.4

0.6

QL (W/cm)

Temperature (Tc)

Fig.  9.16 Calculated results for fixed heating-​ spot position of xL  =  13.3 μm. (a)  Emission power as a function of QL. (b)  and (c)  Distribution of temperature T and critical current density Jc, respectively, in the mesa for QL = 0.1, 0.2, 0.3, 0.4, and 0.5 W/​cm (after Asai et al. 2014). Copyright 2014, AIP Publishing LLC.

0.1 W/cm 0.2 W/cm

0.8

0.3 W/cm 0.4 W/cm 0.5 W/cm

0.6

1 Critical current density (Jc(0))

(b)

1

(c) 0.5 W/cm 0.4 W/cm 0.3 W/cm

0.5

0.2 W/cm 0.1 W/cm

0 0

0.5 x /wm

1

0

0.5

1

x /wm

It is particularly noteworthy that the strongest emission is obtained around QL = 0.3 W/​cm. In order to study this behavior, we plot the spatial distribution of T and Jc in the mesa during intense emission, as shown in Fig.  9.16b and c, respectively. In the case of QL  =  0.3 W/​cm, the heating-​spot temperature is slightly lower than Tc, and thus a hot spot (T > Tc) is not formed. The change in Jc becomes significant when T is slightly below Tc, as expected from the temperature dependence of Jc (Kleiner et al. 1992, Tanaka et al. 2000). Therefore, the drastic Jc modulation via local heating strongly excites the THz Josephson plasma wave inside the IJJ mesa. On the other hand, for QL < 0.3 W/​cm, the hot-​ spot region is formed during THz emission. Since the hot-​spot region does not work as a THz oscillation source because Jc = 0 in this region, the formation of the hot spot results in reduction in the emission power. Conversely, for QL > 0.3 W/​cm, the mesa temperature becomes considerably lower than Tc. In this case, the Jc modulation becomes small compared to the case of T~Tc because Jc is nearly constant unless T is close to Tc. Consequently, the excitation of the THz plasma wave becomes weak. Therefore, we can conclude that local heating that

  285

THz emission from IJJs with temperature inhomogeneity 285 maintains the temperature of the heating spot slightly lower than Tc is preferable for high-​power emission. Next, for clarifying the optimum heating-​spot position, we calculate the emission power by varying xL. Here, we take QL  =  0.3 W/​cm, which yields the highest emission power in the above calculation. Fig.  9.17a shows the emission power as a function of xL. This figure indicates that the emission power increases as the heating spot approaches the edge of the mesa, and the maximum emission power at xL  =  0 is 33.8 mW/​cm. If we assume the 3D mesa length is comparable to the experimental mesas ~300 μm, the estimated emission power exceeds 1 mW. Fig. 9.17b–​c show the distributions of T and Jc in the mesa during the intense emission, respectively. Both distributions are asymmetric with respect to the center of the mesa (xL/​wm  =  0.5), and this asymmetric feature becomes remarkable with decrease in xL. Since the electric field corresponding to the m = 1 cavity resonance mode is antisymmetric with respect to xL/​wm = 0.5, as shown in Fig. 9.15a, the m = 1 mode is strongly excited by the asymmetric AC Josephson current. Therefore, intense THz emission is obtained by heating the mesa edge because this leads to a large asymmetric distribution with respect to the mesa center. It should be (a)

Power (mW/cm)

30 20 10 0

Temperature (Tc)

1

0.2 xL/wm

xL =25 µm 0.8

0.4 1

(b)

xL =15 µm

xL =5 µm

Critical current density (Jc(0))

0

(c)

xL =5 µm 0.5

xL =15 µm xL =25 µm

0

0.6 0

0.5 x/wm

1

0

0.5 x/wm

1

Fig. 9.17  Calculated results for the fixed heating power QL = 0.3 W/​cm. (a)  The emission power as a function of xL. The distribution of (b)  temperature T and (c) critical current density Jc for different values of xL (after Asai et al. 2014). Copyright 2014, AIP Publishing LLC.

286

286  THz Emission from HTS Cuprate noted that the optimum position of the heating spot depends on the resonance mode in the mesa. In the case of the m = 2 cavity resonance mode, for instance, the electric field is symmetric with respect to xL/​wm = 0.5. Hence, for the m = 2 mode, the local heating around xL/​wm = 0.5 is preferable for intense THz emission. In summary, the key points for designing intense THz emission are (a) control of the heating power to maintain the mesa temperature slightly lower than Tc and (b) control of the heating position corresponding to the symmetry of the resonance modes in the mesa. Regarding condition (a), our theoretical analysis is consistent with a recent experimental study that reported the strongest emission when T was slightly lower than Tc (Benseman 2013a). Moreover, several experimental studies have verified that the emission power can be enhanced by laser heating (Watanabe et al. 2015, Zhou et al. 2015). As discussed in this section, THz emission properties of mesa-​structured IJJs, such as emission power and radiation pattern, are strongly affected by temperature distribution in the mesa. Therefore, control of the temperature distribution in the mesa is crucial for realizing practical IJJ THz emitters.

9.5 Summary Crystal structures of HTS cuprates form stacks of superconducting layers and insulating layers, and these stacks behave as identical Josephson junctions, which are called intrinsic Josephson junctions (IJJs). The dynamics of the phase differences of IJJs is coupled with the electromagnetic fields in the IJJs. Thus, the oscillation of the phase differences results in the excitation of the Josephson plasma waves associated with the propagation of electromagnetic waves. By fabricating mesa-​structured IJJs on the mesoscopic scale ~100 μm, the Josephson plasma wave, whose frequency lies in a THz range, can be strongly excited, and this plasma excitation leads to intense THz emission from IJJ mesas. Therefore, IJJ mesas are considered to be a strong candidate for practical THz emitters. As discussed in Sec. 9.3 and 9.4, temperature distribution in the IJJ mesa greatly changes the emission properties of the IJJ mesa, for example emission power and radiation pattern. To put it the other way around, the properties of an IJJ-​based THz emitter can be designed by external heat control. Recent theoretical study has also reported that the circular polarization of the emission waves can be controlled by external heating (Asai and Kawabata 2016). Such tunability of the emission properties is a strong advantage in practical applications. Recently, the temperature distribution in the IJJ mesa has been directly and precisely observed in experimental studies (Benseman et al. 2013a, Minami et al. 2014). Moreover, control of the spatial 𝑇 has been experimentally achieved (Watanabe et al. 2015, Zhou et al. 2015). In this light, the IJJ-​based THz emitter has strong potential to be a next-​generation THz emitter realizing various practical uses.

  287

References (Chapter-9) 287

REFERENCES (CHAPTER-​9 ) Asai, H., Kadowaki, K. and Tachiki, M (2012) Phys. Rev. B 85, 064521. Asai, H., Tachiki, M. and Kadowaki, K. (2012a) Appl. Phys. Lett. 101, 112602. Asai, H., Tachiki, M. and Kadowaki, K. (2013) Physica C 491, 35. Asai, H. and Kawabata, S. (2016) IEEE Trans. Appl. Supercond. 26, 1800804. Anderson, P.W. and Dayem, A.H. (1964) Phys. Rev. Lett. 13, 195. Bae, M.H., Lee, H.J. and Choi, J.H. (2007) Phys. Rev. Lett. 98, 027002. Balanis, C.A. (2005) Antenna theory:  Analysis and design, 3rd ed., Wiley, Hoboken, NJ. Batov, I.E., Jin, X.Y., Shitov, S.V., Koval, Y., Müller, P. and Ustinov, A.V. (2006) Appl. Phys. Lett. 88, 262504. Benseman, T.M., Koshelev, A.E., Gray, K.E., Kwok, W.K., Welp, U., Kadowaki, K., Tachiki, M. and Yamamoto, T. (2011) Phys. Rev. B. 84, 064523. Benseman, T.M., Gray, K.E., Koshelev, A.E., Kwok, W.K., Welp, U., Minami, H., Kadowaki, K. and Yamamoto, T. (2013) Appl. Phys. Lett. 103, 022602. Benseman, T.M., Koshelev, A.E., Kwok, W.K., Welp, U., Vlasko-​Vlasov, V.K., Kadowaki, K., Minami, H. and Watanabe, C. (2013a) J. Appl. Phys. 113, 133902. Bulaevskii, L.N., Pokrovsky,V.L. and Maley, M.P. (1996) Phys Rev. Lett. 76, 1719. Crommie, M.F. and Zettl, A. (1991) Phys. Rev. B 43, 408. Delfanazari, K., Asai, H., Tsujimoto, M., Kashiwagi, T., Kitamura, T., Yamamoto, T., Sawamura, M., Ishida, K., Watanabe, C., Sekimoto, S., Minami, H., Tachiki, M., Klemm, R.A., Hattori, T. and Kadowaki, K. (2013) Opt. Express 21, 2171. Delfanazari, K., Asai, H., Tsujimoto, M., Kashiwagi, T., Kitamura, T., Yamamoto, T., Wilson, W., Klemm, R.A., Hattori, T. and Kadowaki, K. (2015) IEEE Trans. Terahertz Sci. Technol. 5, 505. Fano, U. (1961) Phys. Rev. 124, 1866. Gross, B., Guénon, S., Yuan, J., Li, M.Y., Li, J., Ishii, A., Mints, R.G., Hatano, T., Wu, P.H., Koelle, D., Wang, H.B. and Kleiner, R. (2012) Phys. Rev. B 86, 094524. Guénon, S., Grünzweig, M., Gross, B.,Yuan, J., Jiang, Z.G., Zhong,Y.Y., Li, M.Y., Iishi, A., Wu, P.H., Hatano, T., Mints, R.G., Goldbin, E., Koelle, D., Wang, H.B. and Kleiner, R. (2010) Phys. Rev. B 82, 214506. Hao, L.Y., Ji, M., Yuan, J., An, D.Y., Li, M.Y., Zhou, X.J., Huang, Y., Sun, H.C., Zhu, Q., Rudau, F., Wieland, R., Kinev, N., Li, J., Xu, W.W., Jin, B.B., Chen, J., Hatano, T., Koshelets, V.P., Koelle, D., Kleiner, R., Wang, H.B. and Wu, P.H. (2015) Phys. Rev. Applied 3, 024006. Homes, C.C., Timusk, T., Liang, R., Bonn, D.A. and Hardy, W.N. (1993) Phys. Rev. Lett. 71, 1645. Huang, J. (1983) IEEE Trans. Antennas Propag. 31, 649. Hu, X. and Lin, S.Z. (2008) Phys. Rev. B 78, 134510. Irie, A., Hirai, Y. and Oya, G. (1998) Appl. Phys. Lett. 72, 2159.

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288  THz Emission from HTS Cuprate Ji, M., Yuan, J., Gross, B., Rudau, F., An, D.Y., Li, M.Y., Zhou, X.J., Huang, Y., Sun, H.C., Zhu, Q., Li, J., Kinev, N., Hatano, T., Koshelets, V.P., Koelle, D., Kleiner, R., Xu, W.W., Jin, B.B., Wang, H.B. and Wu, P.H. (2014) Appl. Phys. Lett. 105, 122602. Kadowaki, K., Tsujimoto, M., Yamaki, K., Yamamoto, T., Kashiwagi, T., Minami, H., Tachiki, M. and Klemm, R.A. (2010) J. Phys. Soc. Jpn 79, 023703. Kakeya, I., Kindo, K., Kadowaki, K., Takahashi, S. and Mochiku, T. (1998) Phys. Rev. B 57, 3108. Kakeya, I., Hirayama, N., Omukai, Y. and Suzuki, M. (2015) J. Appl. Phys. 117, 043914. Kashiwagi, T., Yamaki, K., Tsujimoto, M., Deguchi, K., Orita, N., Koike, T., Nakayama, R., Minami, H., Yamamoto, T., Klemm, R.A., Tachiki, M. and Kadowaki, K. (2011) J. Phys. Soc. Jpn 80, 094709. Kashiwagi, T., Tsujimoto, M.,Yamamoto, T., Minami, H.,Yamaki, K., Delfanazari, K., Deguchi, K., Orita, N., Koike, T., Nakayama, R., Kitamura, T., Sawamura, M., Hagino, S., Ishida, K., Ivanovic, K., Asai, H., Tachiki, M., Klemm, R.A. and Kadowaki, K. (2012) Jpn. J. Appl. Phys. 51, 010113. Kashiwagi, T., Yamamoto, T., Kitamura, T., Asanuma, K., Watanabe, C., Nakade, K., Yasui, T., Saiwai, Y., Shibano, Y., Kubo, H., Sakamoto, K., Katsuragawa, T., Tsujimoto, M., Delfanazari, K., Yoshizaki, R., Minami, H., Klemm, R.A. and Kadowaki, K. (2015) Appl. Phys. Lett. 106, 092601. Kashiwaya, S. and Tanaka, Y. (2000) Rep. Prog. Phys. 63, 1641. Kim, S.M., Wang, H.B., Hatano, T., Urayama, S., Kawakami, S., Nagao, M., Takano, Y., Yamashita, T. and Lee, K. (2005) Phys. Rev. B 72, 140504. Kitamura, T., Kashiwagi, T., Yamamoto, T., Tsujimoto, M., Watanabe, C., Ishida, K., Sekimoto, S., Asanuma, K., Yasui, T., Nakade, K., Shibano, Y., Saiwai, Y., Minami, H., Klemm, R.A. and Kadowaki, K. (2014) Appl. Phys. Lett. 105, 202603. Kleiner, R., Steinmeyer, F., Kunkel, G. and Muller, P. (1992) Phys. Rev. Lett. 68, 2394. Kleiner, R., Gaber, T. and Hechtfischer, G. (2000) Phys. Rev. B 62, 4086. Klemm, R.A. and Kadowaki, K. (2010) J. Phys. Condens. Matter 22, 375701. Kobayashi, K., Aikawa, H., Katsumoto, S. and Iye, Y. (2002) Phys. Rev. Lett. 88, 256806. Koshelev, A.E. (2008) Phys. Rev. B 78, 174509. Koshelev, A.E. and Bulaevskii, L.N. (2008) Phys. Rev. B 77, 014530. Koyama, T., Matsumoto, H., Machida, M. and Kadowaki, K. (2009) Phys. Rev. B 79, 104522. Koyama, T., Matsumoto, H., Machida, M. and Ota, Y. (2011) Supercond. Sci. Technol. 24, 085007. Krasnov, V.M., Mros, N., Yurgens, A. and Winkler, D. (1999) Phys. Rev. B 59, 8463. Kuramoto, Y. (1984) Prog. Theor. Phys. Suppl. 79, 223.

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290  THz Emission from HTS Cuprate Watanabe, C., Minami, H., Kitamura, T., Asanuma, K., Nakade, K., Yasui, T., Saiwai, Y., Shibano, Y., Yamamoto, T., Kashiwagi, T., Klemm, R.A. and Kadowaki, K. (2015) Appl. Phys. Lett. 106, 042603. Welp, U., Kadowaki, K. and Kleiner, R. (2013) Nat. Photonics. 7, 702. Yurgens, A. (2011) Phys. Rev. B 83, 184501. Zhou, X.J., Yuan, J., Wu, H., Gao, Z.S., Ji, M., An, D.Y., Huang, Y., Rudau, F., Wieland, R., Gross, B., Kinev, N., Li, J., Ishii, A., Hatano, T., Koshelets, V.P., Koelle, D., Kleiner, R., Wang, H.B. and Wu, P.H. (2015) Phys. Rev. Appl. 3, 044012.

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Micromagnetic Measurements on Electrochemically Grown Mesoscopic Superconductors A. Müller, S.E.C. Dale, and M.A. Engbarth Department of Physics, University of Bath, Bath BA2 7AY, UK

10 10.1 Introduction 

291

10.2 Electrochemical preparation of β-​tin samples 

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10.3 Measurement techniques and sample preparation 

300

10.4 Summary and outlook 

317

References (Chapter-10) 

317

10.1 Introduction Although lead and tin were among the first superconductors discovered, especially in the mesoscopic regime, that is, where their dimensions are comparable to the characteristic length scales such as coherence length and penetration depth of superconductivity, these two materials are ideal samples to study the behavior of typical type-​I superconductivity (Engbarth 2010, Müller 2012).

10.1.1 Superconductivity in mesoscopic type-​I superconductors In the present section, the basic properties of mesoscopic superconductivity are outlined. A sample is said to be in the mesoscopic regime if its size is of a similar order as the superconducting length scales, namely the coherence length ξ and penetration depth λ. Both of these properties are temperature dependent and increase as the critical temperature, Tc, is approached, meaning that close to Tc even comparatively large samples may show mesoscopic behavior. For type-​I superconductors, the interface energy between normal and superconducting states is positive, hence N/​S interfaces are avoided and hence a sample tries to be either fully normal or fully superconducting, resulting in a sharp first-​order phase transition. However, under certain circumstances this “energy penalty” that has to be brought up to form the interface is lower than the energy

A. Müller, S.E.C. Dale, M.A. Engbarth, ‘Micromagnetic Measurements on Electrochemically Grown Mesoscopic Superconductors’ in The Oxford Handbook of Small Superconductors. First Edition. Edited by A.V. Narlikar. © Oxford University Press 2017. Published in 2017 by Oxford University Press. DOI 10.1093/acprof:oso/9780198738169.003.0010

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292  Electrochemically Grown Superconductors which is needed to keep the sample in the Meissner state. In that case, normal domains form inside the sample to minimize the overall energy. The energy which is needed to keep the sample field free depends on the geometry of the sample. The demagnetizing factor  takes the geometric effects into account. If a magnetic field H a is applied to a material, the sample is magnetized and hence generates a magnetization M . The magnetization in turn causes  surface currents which generate the shape-​ d ependent demagnetizing field H D which    opposes H a and M . The surface field on the inner surface H i in the sample is the superposition of these fields:

     H i = H a + H D = H a − ηM

with the demagnetizing factor η, where 0 < η < 1 depends on the shape and size of the sample.  The flux density B is defined by the internal field and the magnetization as

   B = µ0 H i + M

(

)

which, rearranged, leads to



   ηB + (1 − η) H i = H a µ0

 In the Meissner state, it is B = 0 for a superconductor and hence it is for the internal field



  H a Hi = 1− η

 The decisive  field for the retention of superconductivity is the internal field H i and if H i = H i > H c, that is, ignoring any supercooling and superheating at this moment, the Meissner state is, at least partially, destroyed. The Meissner state breaks down at H a = (1 − η) H c and the sample enters the intermediate state, which obviously has a lower magnetization than the Meissner state. For arbitrary shapes, the demagnetizing factor can be calculated analytically or numerically (Osborn 1945, Cronemeyer 1991). The usual approach is to calculate the demagnetizing factor for an ellipsoidal shape, which is analytically possible. For more complex shapes, the demagnetizing factor can be calculated, either by using an inscribed ellipsoid, or by trying to find an exact solution for a certain shape (Torre 2000). Joseph et al. used a series expansion to calculate the demagnetizing factor for a range of shapes (Joseph and Schlömann 1965, Joseph 1966, Joseph 1967), while Chen et al. found a method to analytically calculate the demagnetizing factor for long rectangular rods and rectangular prisms (Chen et  al. 2002, Pardo et  al. 2004). Any real sample has a demagnetizing

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Introduction 293 factor η > 0 and as a consequence enters the intermediate state at an applied field H a = (1 − η) H c where normal and superconducting regions coexist. In the normal regions the field has a strength Hn which is somewhat less than the critical field Hc. In contrast to the domains in a type-​II superconductor, the domains usually contain a large number of flux quanta Φ0, although the N/​S interface of these domains is accompanied by an energy penalty due to the positive surface energy of the interface. Although the demagnetizing factor accounts for the general shape of the sample, it fails to account for the effects due to corners of the sample, or for very flat samples. For fields lower than the penetration field Hp, only the corners of the sample are normal. If the applied field Ha is increased, the Meissner state becomes energetically unfavorable and a normal domain in the center of the sample would be energetically more favorable. However, a domain cannot simply jump into the center of the sample, but has to migrate into the center from one of the corners. Once a threshold field Hp is reached, the normal corners are joined and flux can enter the sample, forming a normal domain. By letting the flux enter, the field at the corners is relaxed and the corners separate again, giving rise to a new barrier for the next flux to enter. The fresh normal domain then migrates towards the center of the sample, minimizing the excess energy in the field around the sample. Sharp corners facilitate this barrier, as the field has to penetrate these and turn the corners normal before flux can enter the sample. This extra amount of energy is saved in a sample with rounded edges. The energy a flux line costs is proportional to its length. Hence in a cuboid, flat sample, the position of the domain is only controlled by the force due to the supercurrents, which drive it towards the center of the sample. In an elliptical cross-​sectioned sample, however, the domain has to grow in length as it moves inwards. This energy just compensates the effect caused by the supercurrents (Clem et al. 1973). This gives rise to an important difference in the behavior. Once a normal domain is trapped in the Meissner state in a sample with a rectangular cross-​section, the supercurrents surrounding it move the domain to the center of the sample and force it to keep its position. As this is the most stable state, additional energy is necessary to move the domain to the corner where it is able to leave the sample again. These two barriers cause a hysteresis in the magnetization behavior. For a sample with an elliptical cross-​ section, however, there is no metastable state for the domain at a given field and the domain leaves at the same field where it enters the sample. This is exactly the effect Provost et al. (1974) measured for increasingly flat elliptical disks. This effect in type-​I superconductors was originally called “edge-​pinning,” which changed after a similar effect was observed and described for type-​II superconductors and it is now mainly referred to as the “geometrical barrier” (Indenbom and Brandt 1994, Zeldov et al. 1994). The effect occurs for samples of any size and for many different geometries (Benkraouda and Clem 1996, Morozov et al. 1997). This is a major difference to superheating. Superheating occurs for almost perfect, defect-​free, interfaces and prevents

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294  Electrochemically Grown Superconductors any flux from entering the sample, while the above discussed geometrical barrier controls the behavior of macroscopic normal domains. For the case of very thin samples with thickness d dc, the minima are shifted towards the surfaces (Saint-​James 1965, Fink 1969, Schultens 1970). In a thin slab, one more dimension is omitted and the two geometries discussed above are basically combined in one single sample. With a magnetic field perpendicular to the long axis of the slab, the superconducting state is confined in both directions, perpendicular and parallel to the applied field. Additionally, in a real sample, the sharp, typically 90°, corners of the slab change the way the field can enter the sample. The supercooling field marks the onset of superconductivity in a decreasing external field in a type-​I superconducting slab which is limited by surface superconductivity. The value for Hc3 is valid for a semi-​infinite sample.

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 Introduction 295 This limitation is obviously not suitable for a thin slab with sharp corners. The problem is addressed by using the boundary conditions of a wedge-​shaped sample (van Gelder 1968, Fomin et al. 1998, Schweigert and Peeters 1999). Schweigert et al. found a numerical approximation for a wedge with a corner, having an angle α and a radius r >> ξ,λ, to be



H c3 3 0.746 α 2  = 1 + 0.14804 α 2 + 2  α  Hc2 α + 1.8794 

The value for a 180° wedge angle is identical to the value Hc3 for a semi-​infinite sample:

° H c180 = 1.695 H c 2 3

For a wedge with 90° angles, corresponding to a rectangular slab, the field is enhanced to a value

H c903 ° = 1.96 H c 2

which is even further increased for an equilateral triangular sample with 60° corners:

H c603 ° = 2.52 H c 2

An additional effect arises from the cross-​sectional area of the sample, which is also addressed in the same publication. For a circle with an area S < (2.33ξ)2 the maximum of the order parameter is in the center of the sample, comparable to the solution of a thin film mentioned earlier. For an area S < (2.33ξ)2 a vortex forms in the center which changes the nucleation field. It was shown that the behavior is qualitatively similar for a square and for triangular samples with different angles. These results indicate that the onset of superconductivity in a decreasing field takes place at larger fields if the samples have sharp corners. The superheating of the superconducting state, which is an effect of a microscopic surface barrier, is more pronounced in a mesoscopic slab, where the surface is more flawless than in large bulk samples. Numerical calculations of the superheating field for different slab widths have been performed by Landau and Rinderer (1995). Especially close to the crossover from a first-​to a second-​order phase transition, which occurs at the critical thickness dc = √5λ, the superheating field changes with the width of the sample. For the calculations, Landau and Rinderer used a symmetric form of ψ in a slab of infinite length. For very thin slabs (d ≈ 10λ), the dependence of Hsh on the GL parameter κ seems to vanish, and the calculations for two different values of κ join a common minimum of Hsh at d ≈ 4λ. Further decreasing the size results in an increasing superheating field which becomes identical to Hc for d < λ (Moshchalkov et al. 1995, Morelle et al. 2002, 2004).

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296  Electrochemically Grown Superconductors

10.1.2 Superconductors in close contact So far, all considered interfaces have been between a superconductor and a vacuum. The boundary condition for the order parameter ψ at these interfaces is simply the vanishing derivative of ψ. If, however, another material is in contact with the superconductor, the boundary conditions for ψ change, so that ψ either is suppressed or is enhanced close to the interface. Considering the case where ψ is enhanced at the interface, it is of major interest to investigate if superconductivity on one side of the interface can act as the seed for superconductivity in the other material. Usually the image of “wetting” within a first-​order phase transition is used to describe this process (Indekeu and van Leeuwen 1995, 1997). If one considers a drop of liquid on a solid surface, depending on the surface energies between liquid and solid, the liquid can either wet the material, or remain as a drop on the surface. This is analogous to the situation of a superconducting phase in contact with a normal phase. In type-​I materials with low κ, the superconducting phase spreads easily into the sample, which can be seen from the supercooling of a sample, which only occurs down to the surface field Hc3. In a decreasing field, as soon as the surface field Hc3 is reached, the whole sample turns superconducting. The interface between two different superconductors is theoretically explored in the framework of the GL equations by a number of groups, but experiments on these systems are very rare (Khlyustikov 1997, Montevecchi and Indekeu 2000, Yampolskii and Peeters 2000, Baelus et al. 2001, Marchenko and Podolyak 2005, Zha et al. 2006a, 2006b, Barba-​Ortega et al. 2011). The results of the simulations have in common that an increase in the critical field Hc, as well as in the critical temperature Tc, is possible and is predicted to different extents. When two superconductors with different critical temperatures are in close contact, for example tin with Tc = 3.72 K and lead with Tc = 7.2 K, Cooper pairs are able to pass from the superconducting lead into the tin, even if Ha > HcSn and/​or T > TcSn.

10.2 Electrochemical preparation of β-​tin samples Mesoscopic superconducting samples can be fabricated using a range of different methods (Xia et al. 1999, Shchukin and Bimberg 1999, Datta and Landolt 2000, Chernov 2001, Barth et al. 2005, Gates et al. 2005). A common technique is lithographic patterning, which can be used to get two-​dimensional samples (Martín et al. 1998, Kim et al. 2002). A lithographic pattern, which is exposed in some form of resist, is used to either etch or deposit material in the mask. The samples are usually polycrystalline thin films, forming a mesa of a given shape. The acuteness and smoothness of the sample are hereby limited by the wavelength of light and grain size of the material used in the process. Additionally, polycrystalline thin films are always type-​II superconductors. The use of focused

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Electrochemical preparation of β-​tin samples 297 ion beams (FIB) to deposit material in defined shapes is another technique which is used to grow mesoscopic samples (Hao et al. 2007). To deposit a metal with a FIB, a precursor is used which decomposes under the influence of the ion beam. It is almost impossible to remove all remains of the precursor, hence there is always some contamination of the precursor in the final sample. Although the FIB method can be used to grow three-​dimensional (3D) samples, the material is amorphous and contaminated with carbon. Electrochemistry is a technique where the growth of a crystal from a solution of dissolved metal ions can be controlled. It is hence ideal to controllably fabricate 3D single crystal samples with a wide range of sizes and shapes. Electrochemistry has been successfully used to fabricate superconducting and ferromagnetic samples (Pangarov 1964, Grujicic and Pesic 2004, Nasirpouri et al. 2011). The shapes that can be obtained from electrodeposition are however limited by the material and the setup used. However, electrodeposition can be used to deposit a metal onto a template, for example in the fabricated holes of another material to form nanotubes (Huczko 2000, Lai et al. 2006, Djenizian et al. 2008), or to plate another sample with a shell of controllable thickness (Kaltenpoth et al. 2003, Chen et al. 2004, Liu et al. 2007, Müller et al. 2011). Previous work had focused on the deposition of thin films (Santos and Bulhoes 2003) as well as dendritic growth (Wranglén 1960, Fukami et al. 2007) and the growth of nanostructures (Chou and Schaak 2008, Lizzul Rinne et al. 2002, Tian et  al. 2003, Torrent-​Burgués et  al. 2002). Gómez et  al. (1999) investigated the nucleation and electrochemical growth of microcrystallites with relatively poorly faceted habits. The techniques used in these experiments were refined to electrocrystallize samples with a better surface quality and a better controllability of their dimensions. The shapes, which can be grown from tetrafluoroboric Sn(II) solutions, range from square cuboids to complex dendritic shapes as Sn(II) concentration and growth potential are increased. All solutions used in these experiments had a concentration – of 1 M BF4​ and a Sn(II) concentration between 10 mM and 100 mM. The growth of tin crystals from Sn(BF4)2 follows the Volmer–​Weber mechanism as it is more favorable for the tin atoms to bind to an existing nucleus than to start growing a new nucleus. The number of active nucleation sites depends strongly on the material used for the working electrode. The number of crystals on boron doped diamond (BDD) is typically higher than on highly ordered pyrolytic graphite (HOPG), but the general size and shape distribution is similar on the two substrates. The growth mechanism of tin dendrites from SnCl2 in HCl was extensively investigated and described by Wranglén (1960). This work can be used and adopted to understand the growth and crystal orientation of the microcrystals prepared in this work. The dimensions of the crystals were accurately measured using scanning electron microscope (SEM) and atomic force microscope (AFM) images as well as using an optical microscope, where a micromanipulator was used to move and turn the crystals while observing them, which allowed a full 3D view of individual crystals.

298

10 mM

Fig.  10.1 Electron micrographs of tin crystals grown from boroflorate under different conditions. The overpotential increases from left to right, the electrolyte concentration from bottom to top, showing the wide variety of shapes and increasing complexity thereof. The scale bar is 5 µm in all cases.

25 mM

50 mM

100 mM

298  Electrochemically Grown Superconductors

Increasing overpotential

Tin crystallizes in two different configurations at room temperature, but only β-​tin is able to form large faceted crystals. β-​tin crystallizes in a body-​centered tetragonal lattice with a two-​atom basis and a = b = 5.82 Å and c = 3.17 Å. The second atom of the two-​atom basis is at ½ a and ¼ c. Wranglén observed that dendrites grow along the [110] direction and are formed from single pyramids whose tips point in the [001] direction. Due to the very low cohesion energy of tin (Gómez et al. 1999) and the low overpotentials used, the atoms were given plenty of time to rearrange on the crystal, forming the energetically most favorable shape. The evolution of crystal habits as a function of Sn(II) ion concentration and applied overpotential is shown in Fig. 10.1 (Müller et  al. 2010). The pyramid angle of the tin lattice can be seen in many of our crystals in Fig. 10.1. Here the end of the cuboid is cut at exactly this angle, while for other crystals in the figure the end is cut at an angle of about 30°. Despite the large variety of crystal shapes, the most interesting shapes for the measurement of superconducting properties in the mesoscopic regime are simple cuboids. Although the ideal crystal should have a square cross-​section, this is not always true for real crystals. Many crystals have a slightly lower “height” than “width” when grown flat on the electrode. Electrodeposition also offers an easy way to plate a crystal of one material with another material whose overpotential for plating is higher than that of the first material. This condition is true for the plating of Pb on Sn from fluoroboric acid solutions. The basic principle of growing a core and plating it with a shell is shown in Fig. 10.2. To plate the tin crystals with lead, but not reduce the tin from the already formed crystals, the potential for plating has to be larger than the potential for reducing tin. Good results were achieved for a plating potential slightly higher than the potential for the tin growth. After plating, the crystals had a rough look

  299

Electrochemical preparation of β-​tin samples 299 (a)

(b)

(c)

(d)

Electrolyte WE

Fig.  10.2 A  sketch of electrocrystallization and electroplating to form a core–​shell structure. An electrolyte with the ions for the core is used to grow a crystal on the working electrode (WE) (a–​b). After the growth process, the electrolyte is exchanged to another electrolyte (c) and the shell is plated on all but the bottom side of the existing crystals (d). Ideally no shell-​material will be deposited on the electrode. (see Fig. 10.3a–​c), indicating that the lead film is polycrystalline on top of the tin. The shell covers all exposed surfaces of the tin crystal. Only one side, where the crystal is attached to the electrode, is not completely covered in lead. If the bottom of the crystal is not completely attached to the electrode, a partial coverage of the bottom is possible. The thickness of the shell was estimated by comparing the charges of the tin and lead deposition. The total charge Q = ∫ I (t ) dt is a measure for the amount of material deposited and is directly proportional to the deposited mass Q ∝ M.

(a)

(b) HOPG Pb Pb

HOPG Sn

200 nm

Sn

4 µm

4 µm

(c)

(d)

FIB

4 µm

25 µm

SEM 52° 25 µm 0

0

0

y

x

Fig. 10.3  Two FIB-​milled cross-​sections of Sn–​ Pb core–​ shell structures (a–​ b) and an AFM scan of a cuboid Sn–​Pb core–​ shell structure, positioned at the center of a 1 µm wide Hall probe (c). The geometry of the dual-​beam FIB used to mill and image the crystals is shown in the sketch in (d).

300

300  Electrochemically Grown Superconductors The preparation of core–​shell structures is not limited to the growth of Sn–​ Pb core–​ shell structures. Pb–​ Ni and Sn–​ Ni superconducting–​ ferromagnetic core–​shell structures were prepared in a similar way by Sara E.C. Dale (Müller et al. 2011).

10.3 Measurement techniques and sample preparation For the micromagnetic measurements the sample needs to be positioned near to the center of one Hall cross. Usually, a single crystal is chosen from the working electrode and transferred to the Hall array. To manage the transfer of the sample from the electrode to the Hall probe a micro/​nano manipulator is used. The manipulator consists of a cantilever attached to a piezoelectric three-​axes positioner.

10.3.1 Critical state in mesoscopic pure tin cuboids The magnetization data were obtained using micro-​Hall probes and state-​of-​the-​ art Lock-​In techniques. The magnetization curves M(H) for three different rods (the largest, one intermediate sized, and the smallest sample) at a temperature of 3.35 K (T/​Tc = 0.9) are shown in Fig. 10.4. The most important dimension of the crystal is the width and hence the pyramidal cap of the largest rod is probably without any effect

5.38 × 2.06 × 2.33 µm3 2.25 × 1.18 × 0.68 µm3 0.87 × 0.70 × 0.85 µm3

20 16 × 0.5

Fig.  10.4 Magnetization data M(H) for three different-​ sized crystals at a temperature of T/​Tc = 0.9. The data for the largest rod were scaled for clarity.The field was swept from positive to negative field (right to left in the image), the values for the supercooled and superheated critical field clearly change for the different samples.

M [arb. unit]

12 8 4 0 –4 –8 –90

T/Tc = 0.90

Hc

–60

–30

H [Oe]

0

30

60

  301

Measurement techniques and sample preparation  301 upon the final results. The individual critical temperatures as measured from the experimental data are spread by ≈ 10 mK. At first sight it is clear that all transitions from the normal to the superconducting state and vice versa are perfectly type-​I-​like, with no hint of an intermediate state. Both critical fields, Hsc = Hc3, which is identified as the field where superconductivity nucleates in the sample (here at positive values of H), and Hsh, where superconductivity is destroyed in the sample (at negative values of H), are clearly size-​dependent at this temperature. Fig. 10.5 shows the values for Hsh and Hsc as a function of the reciprocal width w–​1 for four different temperatures. The top graph displays the superheating field, which is approximately independent of temperature and size. This is different from the supercooling field, which clearly exhibits a T-​dependence. In terms of the two superconducting length scales, ξ(T) and λ(T), samples enter the mesoscopic regime either upon decreasing the sample size, or increasing the temperature, which increases both lengths ξ and λ. The graph in Fig. 10.5 illustrates both effects for the supercooling field. A bulk sample where w >> ξ,λ and hence w–​1 → 0 enters the mesoscopic regime at a temperature extremely close to Tc, while smaller samples do so at lower temperatures. At low temperatures (T/​Tc Hcshell (a) and Hccore < Ha < Hcshell (b) and the resulting measured magnetization curve (c) for a negative sweep (cf. experimental data in Fig. 10.10).

310

310  Electrochemically Grown Superconductors both type-​I and type-​II regimes and yield a very low penetration field, Hp or Hc1, for the top of the shell. Hence diamagnetism due to the Meissner state is mainly restricted to the side walls, which are oriented parallel to the applied field. The demagnetizing factor for this geometry is much smaller. Flux is hence mainly screened out of the side walls, but can still penetrate through them and move into the thin top film. The fabricated shell thicknesses probably cover both type-​I and type-​II regimes, and hence qualitatively different phenomena are expected in different limits. Another important change in the magnetization behavior of the shell in the two regimes is the critical field up to which superconductivity exists. In the case of a thick film, the critical field for the destruction of superconductivity is defined by the bulk critical field extended by possible superheating effects as expected for type-​I superconductors. In very thin films, the GL parameter increases as well as the critical field H c 2 = 2κ H c. Deep in the type-​II limit for thin films, when Hc2 >> Hc, superconductivity in the shell is maintained at much larger fields than for thick films. Fig. 10.10 shows a compilation of magnetization curves for different core–​ shell structures at T ≈ 2 K.  The deposition ratio for the crystals was between QSn/​QPb ≈ 3/​2 for (a) and QSn/​QPb ≈ 2/​1 for (c) and (d). For a large crystal with a thick shell in Fig. 10.10a, the measured magnetization is generated by both core and shell in roughly equal parts. A signature of the core is seen as the Meissner state in it is destroyed and the core turns normal in one large jump. This is also

(a)

HcSn

–1500

(c)

M [arb. unit]

M [arb. unit] Sweep direction

0

1500

–1500

(d)

HcPb

5.6 µm × 1.1 µm × 0.9 µm t ≈ 70 nm

0

–100 –1500

HcPb

2.3 µm × 0.9 µm t ≈ 110 nm

–100

0 H [Oe] HcSn

100

HcSn 100

0 H [Oe] HcSn

100 M [arb. unit]

M [arb. unit]

0

–100

Fig.  10.10 Magnetization curves for various Sn–​Pb core–​shell samples at low temperature, T ≈ 2 K. The samples are different-​sized rods. The core dimensions are given in length × width × height (if known) as well as the estimated shell thickness t. The bulk critical fields of lead and tin are marked on the positive H-​ axis. The superposition of core and shell magnetization leads to very different, not yet fully understood, magnetic behavior of the individual samples.

(b)

HcPb

5.5 µm × 3.2 µm t ≈ 450 nm

100

1500

HcPb

2.6 µm × 0.9 µm × 0.7 µm t ≈ 50 nm

0

–100 0 H [Oe]

1500

–1500

0 H [Oe]

1500

  311

Measurement techniques and sample preparation  311 the case for smaller crystals with a slightly thinner shell (Fig. 10.10b). The shell is here clearly in the limit of t >> λeff and hence exhibits an intermediate state, as expected for a type-​I material. The large demagnetizing factor of the top and the low demagnetizing factor of the side walls give rise to a complicated magnetization behavior. This changes for thinner shells, where type-​II behavior is expected. With decreasing shell thickness, the core contribution to the measured magnetization becomes much more prominent as can be seen from the graph in Fig. 10.10c. A rather different behavior is seen in Fig. 10.10d, where a small core is plated with an even thinner shell. In contrast to the three samples discussed before, the shell is now thin enough to be well within the type-​II limit. This is reflected in the strongly enhanced critical field of the shell in the type-​II regime and much stronger vortex pinning (i.e. remanent magnetization). The core magnetization is completely hidden here by the shell magnetization, or possibly suppressed by it. From the magnetization data shown here, this cannot be unambiguously determined. If the dimensions of core and shell are chosen carefully, the crystals show a sharp step in magnetization when superconductivity in the core is destroyed. This transition is easy to distinguish at low temperatures. However, at higher temperatures, when the critical temperature of the core is approached, superconductivity in the core and hence diamagnetism in the core only occur at very small applied fields and are easily concealed by the magnetization of the shell. To overcome this problem and to get a strong signal from the core, one sample with a large core and a thin shell (core: 11.9 × 3.2 × 3.1 µm3, 40 nm shell) was prepared to measure the effect of the shell on the core. To confirm the enhancement of the core compared to a “bare” tin crystal, one pure tin crystal of comparable dimensions was measured alongside the core shell crystals.

10.3.5 Enhancement of the core Tc in lead/​tin core–​shell structures One of the effects which the shell has on the core is an enhancement of the critical temperature of the core due to proximity-​induced superconductivity. Here, the tin core is the “weak” superconductor, while the shell is much stronger and has a higher Tc and Hc(T) at a given temperature. The enhancement of the order parameter at the interface gives rise to an enhancement of the critical parameters of the core. For Tccore < T < Tcshell and Hccore < Ha < Hcshell the isolated core is in the normal state, when Cooper pairs cannot form. The shell, however, is superconducting and hence acts as a source for Cooper pairs which can diffuse across the interface and into the core. This was theoretically demonstrated by Baelus et al. (2001). Fig. 10.11 shows magnetization curves for a large Sn–​Pb core–​shell with a comparatively thin shell. The tin core has the dimensions 11.9 × 3.2 × 3.1 µm3, and is surrounded by an approximately 40 nm thick lead shell. A similar-​sized “bare” tin rod (13.5 × 3.3 × 2.4 µm3)

312

312  Electrochemically Grown Superconductors 20

100

3.64 K 3.88 K 4.07 K

2.48 K 2.94 K 3.41 K

0

10 M [arb. unit]

Fig.  10.11 Magnetization curves for a large Sn–​Pb core–​shell sample (core: 11.9 × 3.2 × 3.1 µm3, shell: ≈ 40 nm) and a Sn rod (13.5 × 3.3 × 2.4 µm3) measured simultaneously for T ≈ T Sn c which confirm the enhancement of Tc in the core.The inset shows magnetization curves at lower temperature, when the shell is practically invisible in the magnetization. The maximum fields of the sweep start and end point were chosen so that no shell magnetization was detected at these fields. It was |Hmax| = 500 G for the three curves close to T cSn.The data in the inset were recorded with |Hmax| = 2000 G. Two arrows mark the fields, H shC − S and H scC − S , plotted in Fig. 10.12.

HcSn –100 –250

0

250

0 c-s

Hsh

–10

–20 –75

c-s

Hsc

Pure tin rod for comparison

Sweep direction

–50

HcSn

–25

0

25

50

75

H [Oe]

was measured simultaneously. Data are shown for three different temperatures close to bulk TcSn. The black curve was recorded below the critical temperature of tin. The magnetization curve for the tin rod is shown, offset by –​10 G on the magnetization axis. It is evident that the critical field of the core is greatly enhanced for both “superheating” and “supercooling” branches. Although in the core–​shell structure, the core would not technically be superheated or supercooled, this notation for the fields of the destruction and onset of superconductivity is retained. If the temperature is raised above the critical temperature of bulk tin, the Hall probe under the bare tin rod does not detect any magnetization signal. In the core–​shell structure, however, the sharp transition of the core is still visible. Further, the slope of the M(H) curve in the core superconducting state has a similar slope below and above the critical temperature, suggesting that a similar amount of the core is superconducting above TcSn. Assuming that the whole core is superconducting below the critical temperature, this means that the whole core, and not just a surface sheath close to the shell, is still superconducting at T > TcSn. In Fig. 10.12 the critical fields of the onset and destruction of superconductivity in the core are shown as a function of temperature. Both values are greatly enhanced compared to the values for the pure tin rod. It is notable that the reference tin rod does not show any superheating, in fact superconductivity is destroyed even below Hc(t). This is probably due to the large size of these crystals; only the central part of the rod is measured by the Hall probe and the sample exhibits an intermediate state, as evidenced by magnetization jumps at low temperatures in an increasing field. Hence superconducting domains at the corners, which are about 5 µm away from the Hall probe, are hidden from our measurement. Close to the critical temperature, this effect is not observed.

  313

Measurement techniques and sample preparation  313

Hc [Oe]

200

100

0

Hsc, Sn-Pb core-shell Hsh, Sn-Pb core-shell Hsc, Sn rod Hsh, Sn rod 2

2.5

3

3.5

T [K]

TcSn

4

In the core–​shell structure, the supercooling field of the core, Hsc, at which superconductivity nucleates in the core, has a similar temperature dependence to Hsc of the tin rod at low temperatures. When the critical temperature of bulk tin is reached this behavior changes, and this critical field approaches zero much more gradually than before. The superheating field on the other hand approaches zero and has a value very close to zero at the highest temperatures where a core transition could be unambiguously identified. The values in the graph abruptly stop at T ≈ 4.3 K. This is where the signature of the core almost completely vanished into the background shell magnetization. If the superconductivity in the core is still there, it is not detectable in our experiment. It is not possible to estimate the actual critical temperature of the core from the data presented; only a lower limit can be estimated of Tccore > 4.3 K = 1.16 TcSn. It is interesting to note that superconductivity is restored in the core, in a decreasing field, even above the critical temperature of the core, while superconductivity is destroyed upon further sweeping to a reversed field with a lower absolute value above a certain temperature. The polycrystalline thin lead film that forms the shell is clearly in the type-​II limit in this sample and hence never exhibits a full Meissner state in a decreasing magnetic field, but contains vortices which are usually aligned parallel to the applied field. As the core stays in the Meissner state all the way to zero applied field, the trapped flux is still oriented in the same way, even after the sign of the applied field has changed. An explanation for the missing Meissner state after the applied field changed its direction might lie in the configuration of the trapped flux. The flux that stays in the lead shell is pushed into the center of the sample, as the shell walls become more superconducting in a decreasing applied field. When the critical field falls below the critical field of the core, the flux will be trapped at the interface between the tin core and the

Fig. 10.12  The critical field of the onset (H scC − S ) and destruction (H shC − S ) of superconductivity in the S–​S’ core and in the tin rod as a function of temperature.

314

314  Electrochemically Grown Superconductors lead shell. The supercurrents in the core, generated by the flux along its perimeter, have a particular direction of rotation. After changing the direction of the applied field, additional currents are necessary to keep the core field free. It is probable that, as soon as these currents start to form, superconductivity in the whole volume of the core is destroyed. This could be due to the fact that at this temperature the core is only superconducting if Cooper pairs are provided by the shell, which then sustain the proximity-​induced supercurrents in the core. However, it is not known exactly where magnetic flux is trapped in the sample and how this affects core superconductivity. Field cancellation effects of the shell magnetization and the applied field might also play a role, but without knowledge of the magnetic flux in the sample, these cannot be unambiguously determined. It is hence not currently possible to explain all the observed effects by existing theoretical models.

10.3.6 Little–​Parks-​like oscillations in the lead shell Little and Parks showed in their classic experiment that an oscillation of Tc of a cylindrical superconducting shell is observed when H is varied (Little and Parks 1962, Parks and Little 1964). Upon increasing the field, the temperature-​dependent resistance of the cylinder changed periodically and the periodicity of these so-​called Little–​Parks oscillations is connected with the flux quantum Φ0 and is a direct measure of the internal area A of the normal cross-​section. The periodicity in an applied magnetic field is



∆H =

Φ0 A

Similar oscillations are observed in the core–​shell structures in Fig. 10.13. In the original Little–​Parks experiment the oscillations were observed in a perfect “ring,” with the interior filled by an insulator. In the core–​shell structures of this work, the loop is formed of the four side walls perpendicular to the plane of the Hall probe sensor. The area surrounded by this frame (defined by the core) is simply A = l × w. The top lead film, perpendicular to the applied magnetic field, is ignored in this approximation for the reasons described above. As the shell is not filled with an insulator, but with another superconductor, the oscillations give a measure of how deep superconductivity spreads from the shell into the core. Fig. 10.13 shows magnetization curves recorded at different temperatures for one small core–​shell structure. For this analysis a small crystal is necessary as a smaller area yields a larger step size, ΔH, which is needed to be able to resolve the oscillations. The crystal measured here has a core of 2.6 × 0.9 × 0.7 µm3 and a shell of ≈ 50  nm. At low temperatures (T ≤ Tccore) the magnetization steps follow no clear resolvable pattern due to

  315

Measurement techniques and sample preparation  315 (a) 2.25 K 4.20 K 5.00 K

M [arb. unit]

100 50

6.50 K 7.00 K

Sweep direction

0 –50

–100 –1500 (b)

–1000

–500

0 H [Oe]

500

1000

20

1500

6.50 K 7.00 K

M [arb. unit]

10 0

–10 –20

–300

–200

–100

0 H [Oe]

100

200

300

strong diamagnetism and vortex pinning in the shell. Only for temperatures well above Tccore, for e­ xample 6.5 K, is a clear oscillatory pattern visible. For the two high temperatures, the position and height of these jumps are recorded in Fig. 10.14. The top panel shows the magnetization curves for 6.5 K and 7 K which clearly show a large number of steps which can be interpreted as Little–​ Parks-​like oscillations associated with closed superconducting loops around the perimeter of the crystal. From Fig. 10.14 it is clear that the steps occur for both sweep directions in a mirror-​inverted manner. The plots of ΔH and ΔM (Fig.  10.14b) show the values for both sweep directions. The data from the second sweep were mirror-​inverted to match the data of the first sweep, which are shown above in Fig. 10.14a. These plots convincingly confirm the reproducibility of the observed features. The periodicity of the jumps expected for a lead shell with an empty core,



∆H 0 =

Φ0 20.7G ⋅ µm 2 = = 8.8G A 2.6µm × 0.9µm

Fig.  10.13 Magnetization curves for a small Sn–​Pb core–​shell (core:  2.6 × 0.9 × 0.7 µm3, shell: 50 nm) at various temperatures (a). Oscillatory behavior is most prominent at high temperatures as seen in the expanded graph in (b).

316

316  Electrochemically Grown Superconductors (a)

M [arb. unit]

20 Sweep direction

10

0 –300

–200

–100

0

100

60

30

∆M [G]

∆H [Oe]

300

Negative sweep Inverted positive sweep

3 2 1 0

–300 –150

0

0

H [Oe]

∆H0

(c)

200

H [Oe]

(b)

–300

–200

150

300

–100

0

100

200

300

100

200

300

H [Oe] 400 300

t [nm]

Fig. 10.14  A close analysis of the magnetization jumps, (a), reveals that the effective area surrounded by superconductivity changes with temperature and applied field. The data for ∆H, (b), and ∆M (inset) are plotted for both sweep directions, with the data of the positive sweep direction mirror inverted to fit the magnetization data for the negative sweep shown on top. ∆H0 is the step size corresponding to the perimeter of the core. The thickness of the superconducting sheath t as derived from ∆H is shown in (c).

6.5 K 7.0 K

200 100 0 –100

–300

–200

–100

0 H [Oe]

is marked as a dotted line in the ΔH plot in Fig. 10.14b. This value is only reached for the last jumps at large, negative, decreasing applied fields and vice versa. At both temperatures a slow approach of ΔH to this value is visible as the magnitude of the field is increased. The higher value of ΔH corresponds to a smaller flux-​ containing area. This indicates that superconductivity exists to a certain extent in the core as well. Assuming a thickness t for the superconducting sheath, the surrounded area becomes A = (l–​2t)(w–​2t), with the core dimensions width, w, and length, l. The parameter t can be calculated from the magnetization jumps ΔH. This is shown in the bottom panel of Fig. 10.14. The value t  =  0 corresponds to the current flowing exactly around the perimeter of the core. The data show clearly that at low fields, superconductivity is able to penetrate deep into the core while, for increasing fields, superconductivity is increasingly restricted to the shell. The complex pattern at low applied fields, however, suggests a sheath of induced superconductivity in the core but is not fully understood. The data could also be complicated by the formation of an intermediate state in the tin core, which is

  317

References (Chapter-10) 317 able to form at fields and temperatures above the critical parameters of bulk tin. Further experiments will be required to fully understand this.

10.4 Summary and outlook In this chapter, which is mainly based on parts of the PhD theses of two of the authors, M.A. Engbarth and A. Müller (Engbarth 2010, Müller 2012), our focus was on the superconductivity in the typical pure type-​I superconductors lead and tin. Crystal growth by electrochemical techniques does not only allow researchers to fabricate very pure, close to ideal, mesoscopic samples with controllable size and shape of any of the materials, but also the combination of both materials in the form of core–​shell structures. By entering the mesoscopic regime, the border between type-​I and type-​II superconductivity is smeared out and fascinating new effects emerge. The combination of lead and tin in the form of core–​shell structures yields an interesting sample system for the observation of proximity effects between the two materials. In this chapter, the vortex states were observed using Hall-​ measurements of the net magnetization of the total structure; experiments with different measurement techniques might yield further insight into this fascinating family of core–​shell structures. It was already shown that the critical temperature of a tin core was raised to above the boiling point of normal, that is, not supercooled, 4He, which offers a range of new applications for superconducting tin. As was shown by Müller et al. (2011) the addition of a ferromagnetic shell is also possible. The switching behavior of the S–​S’ core–​shell structures would be even more pronounced with an added normal or ferromagnetic layer. Further experiments are necessary to study these cheap and easy-​to-​produce structures and examine possible uses thereof.

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318

318  Electrochemically Grown Superconductors Chen, D.-​X., Pards, E. and Sanches, A. (2002) IEEE Trans. Magn. 38, 1742. Chen, D.-​ X., Prados, C., Pardo, E., Sanchez, A. and Hernando, A. (2002) J. Appl. Phys. 91, 5254. Chen, Z., Zhan, P., Wang, Z., Zhang, J., Zhang, W., Ming, N., Chan, C. and Sheng, P. (2004) Adv. Mater. 16, 417. Chernov, A.A. (2001) J. Mater. Sci.: Mater. El. 12, 437. Chou, N.H. and Schaak, R.E. (2008) Chem. Mater. 20, 2081. Clem, J., Huebener, R. and Gallus, D. (1973) J. Low Temp. Phys. 12, 449. Cody, G.D. and Miller, R.E. (1968) Phys. Rev. 173, 481. Cronemeyer, D.C. (1991) J. Appl. Phys. 70, 2911. Datta, M. and Landolt, D. (2000) Electrochimica Acta 45, 2535. de Gennes, P. (1999) Superconductivity of metals and alloys, Advanced Book Program, Perseus Books, Reading, MA. Djenizian, T., Hanzu, I., Premchand, Y.D., Vacandio, F. and Knauth, P. (2008) Nanotechnology 19, 205601. Engbarth, M.A. (2010) Hall magnetometry of electrodeposited superconducting Pb mesostructures, PhD thesis, University of Bath. Fink, H.J. (1969) Phys. Rev. 177, 732. Fomin, V.M., Devreese, J.T. and Moshchalkov, V.V. (1998) EPL (Europhysics Letters) 42, 553. Fukami, K., Nakanishi, S., Yamasaki, H., Tada, T., Sonoda, K., Kamikawa, N., Tsuji, N., Sakaguchi, H. and Nakato, Y. (2007) J. Phys. Chem. C 111, 1150. Gates, B.D., Xu, Q., Stewart, M., Ryan, D., Willson, C.G. and Whitesides, G.M. (2005) Chem. Rev. 105, 1171. Gómez, E., Guaus, E., Sanz, F. and Vallés, E. (1999) J. Electroanal. Chem. 465, 63. Grujicic, D. and Pesic, B. (2004) Electrochimica Acta 49, 4719. Hao, L., Macfarlane, J., Gallop, J., Cox, D., Joseph-​Franks, P., Hutson, D., Chen, J. and Lam, S. (2007) IEEE T. Instrum. Meas. 56, 392. Huczko, A. (2000) Appl. Phys. A: Mater. 70, 365. Indekeu, J. and van Leeuwen, J. (1995 Physica C: Superconductivity 251, 290. Indekeu, J. and Van Leeuwen, J. (1997) Physica A 236, 114. Indenbom, M. and Brandt, E. (1994) Phys. Rev. Lett. 73, 1731. Joseph, R.I. (1966) J. Appl. Phys. 37, 4639. Joseph, R.I. (1967) J. Appl. Phys. 38, 2405. Joseph, R.I. and Schlömann, E. (1965) J. Appl. Phys. 36, 1579. Kaltenpoth, G., Himmelhaus, M., Slansky, L., Caruso, F. and Grunze, M. (2003) Adv. Mater. 15, 1113. Kaneko, S., Hiller, U., Slaughter, J., Falco, C., Coccorese, C. and Maritato, L. (1998) Phys. Rev. B 58, 8229. Khlyustikov, I. (1997) J. Exp. Theor. Phys. 85, 609. Kim, N., Hansen, K., Toppari, J., Suppula, T. and Pekola, J. (2002) J. Vac. Sci. Technol. B 20, 386. Lai, M., Martinez, J., Grätzel, M. and Riley, D. (2006) J. Mater. Chem. 16, 2843. Lambert, C. and Raimondi, R. (1998) J. Phys.: Condens. Mat. 10, 901.

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References (Chapter-10) 319 Landau, I.L. and Rinderer, L. (1995) J. Low Temp. Phys. 100, 219. Little, W.A. and Parks, R.D. (1962) Phys. Rev. Lett. 9, 9. Liu, M., Li, X., Imrane, H., Chen, Y., Goodrich, T., Cai, Z., Ziemer, K., Huang, J. and Sun, N. (2007) Appl. Phys. Lett. 90, 152501. Lizzul Rinne, C., Hren, J.J. and Fedkiw, P.S. (2002) J. Electrochem. Soc. 149, C150. Marchenko, V. and Podolyak, E. (2005) J. Exp. Theor. Phys. 100, 311. Martín, J., Jaccard,Y., Hoffmann, A., Nogués, J., George, J., Vicent, J. and Schuller, I. (1998) J. Appl. Phys. 84, 411. Miller, P.B. (1959) Phys. Rev. 113, 1209. Montevecchi, E. and Indekeu, J. (2000) Phys. Rev. B 62, 14359. Morelle, M., Bekaert, J. and Moshchalkov, V. (2004) Phys. Rev. B 70, 094503. Morelle, M., Teniers, G., Chibotaru, L., Ceulemans, A. and Moshchalkov, V. (2002) Physica C: Superconductivity 369, 351. Morozov, N., Zeldov, E., Konczykowski, M. and Doyle, R. (1997) Physica C: Superconductivity Appl. 291, 113. Moshchalkov, V.V., Gielen, L., Strunk, C., Jonckheere, R., Qiu, X., Haesendonck, C.V. and Bruynseraede, Y. (1995) Nature 373, 319. Müller, A. (2012) Designer 3D magnetic mesostructures, PhD thesis, University of Bath. Müller, A., Dale, S.E.C., Engbarth, M.A., Bending, S.J. and Peter, L.M. (2010) CrystEngComm 12, 2135. Müller, A., Dale, S.E.C., Engbarth, M.A., Bending, S.J., Peter, L.M., Knittel, A. and Fangohr, H. (2011) Adv. Funct. Mater. 21, 1874. Müller, A., Miloševic, M.V., Dale, S.E.C., Engbarth, M.A. and Bending, S.J. (2012) Phys. Rev. Lett. 109, 197003. Nasirpouri, F., Bending, S.J., Peter, L.M. and Fangohr, H. (2011) Thin Solid Films 519, 8320. Nasirpouri, F., Engbarth, M.A., Bending, S.J., Peter, L.M., Knittel, A., Fangohr, H. and Miloševic, M.V. (2011) Appl. Phys. Lett. 98, 222506. Osborn, J. (1945) Phys. Rev. 67, 351. Pangarov, N.A. (1964) Electrochimica Acta 9, 721. Pardo, E., Chen, D. and Sanchez, A. (2004) J. Appl. Phys. 96, 5365. Parks, R.D. and Little, W.A. (1964) Phys. Rev. 133, A97. Pearl, J. (1964) Appl. Phys. Lett. 5, 65. Provost, J., Paumier, E. and Fortini, A. (1974) J. Phys. F: Metal Physics 4, 439. Saint-​James, D. (1965) Phys. Lett. 16, 218. Santos, M.C. and Bulhoes, L.O.S. (2003) Electrochimica Acta 48, 2607. Schultens, H. (1970) Z. Phys. A Hadrons Nucl. 232, 430. Schweigert, V.A. and Peeters, F.M. (1999) Phys. Rev. B 60, 3084. Shchukin, V.A. and Bimberg, D. (1999) Rev. Mod. Phys. 71, 1125. Tai, P.C.L., Beasley, M.R. and Tinkham, M. (1975) Phys. Rev. B 11, 411. Tian, M., Wang, J., Snyder, J., Kurtz, J., Liu, Y., Schiffer, P., Mallouk, T.E. and Chan, M.H.W. (2003) Appl. Phys. Lett. 83, 1620.

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320  Electrochemically Grown Superconductors Tinkham, M. (2004) Introduction to superconductivity, Dover Publications, Mineola, NY. Torre, E. (2000) Magnetic hysteresis, IEEE Press, Piscataway, NJ. Torrent-​Burgués, J., Guaus, E. and Sanz, F. (2002) J. Appl. Electrochem. 32, 225. van Gelder, A.P. (1968) Phys. Rev. Lett. 20, 1435. Wranglén, G. (1960) Electrichimica Acta 2, 130. Xia, Y., Rogers, J.A., Paul, K.E. and Whitesides, G.M. (1999) Chem. Rev. 99, 1823. Yampolskii, S. and Peeters, F. (2000) Phys. Rev. B 62, 9663. Zeldov, E., Larkin, A.I., Geshkenbein, V.B., Konczykowski, M., Majer, D., Khaykovich, B., Vinokur, V.M. and Shtrikman, H. (1994) Phys. Rev. Lett. 73, 1428. Zha, G., Zhou, S. and Zhu, B. (2006a) Phys. Rev. B 73, 092512. Zha, G., Zhou, S., Zhu, B., Shi, Y. and Zhao, H. (2006b) Phys. Rev. B 74, 024527.

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Growth and Characterization of HTSc Nanowires and Nanoribbons

11 11.1 HTSc nanowires prepared by the template method 

321

M.R. Koblischka

11.2 HTSc nanowires prepared by electrospinning 

326

Experimental Physics, Saarland University, Campus C6 3, 66123 Saarbrücken, Germany

11.3 Use of HTSc nanowires as building blocks 

341

11.4 Summary and outlook 

342

References (Chapter-​11) 

343

Superconducting nanowires, fabricated by various methods like the template method or by electrospinning, represent an important family of mesoscopic structures. In general, superconducting nanowires enable a variety of studies concerning the basic properties of superconductivity within a spatially confined environment. In the literature, such nanowires of high-​temperature superconductors (HTSc) are fabricated by means of lithography, using thin film material as a base. In this way, electric transport measurements can be performed on such samples. However, there are two main problems as there is always the influence of the substrate on the HTSc nanowire, and furthermore, only electric transport measurements can be performed. With the advance of nanotechnology, different preparation methods of HTSc nanowires became possible, enabling the growth of long-​length, substrate-​free nanowires, which now can be accessed by various other characterization methods. This chapter focuses on the fabrication procedures of HTSc nanowires and their characterization by magnetic and electric transport measurements. The possibilities for employing substrate-​free HTSc nanowires as building blocks for new, nanoporous bulk superconducting materials for applications are discussed also.

11.1 HTSc nanowires prepared by the template method The template method using anodized alumina (AAO) templates is a very successful method to prepare nanowires of a large variety of materials (Masuda et al. 2000, Shingubara 2003). This includes nanowires of ZnO (Li et al. 2000,

M.R. Koblischka, ‘Growth and Characterization of HTSc Nanowires and Nanoribbons’ in The Oxford Handbook of Small Superconductors. First Edition. Edited by A.V. Narlikar. © Oxford University Press 2017. Published in 2017 by Oxford University Press. DOI 10.1093/acprof:oso/9780198738169.003.0011

322

322  HTSc Nanowires and Nanoribbons Liu et al. 2003) or magnetic Ni nanorods in ferrofluids (Monz et al. 2008, Bender et  al. 2011). The templates have to be fabricated individually by anodization, and even commercially available AAO templates can be used. Furthermore, the approach is not limited to alumina templates, as silica (Han et al. 2000) and TiO2 (Yang et al. 2006) templates can be produced as well. There are a variety of methods which can be employed to fill the nanopores. Among them are electrochemical reactions, chemical vapor deposition (CVD), and sol-​gel methods, which can be adapted to the fabrication of HTSc nanowires. The sol-​gel methods are the best-​suited techniques to produce the complicated ceramic structures of HTSc systems, so these up to now have often been used (Grigoryan et al. 2003, Ding et al. 2004, Zhang et al. 2006, Lai et al. 2008, Dadras and Aawani 2015). Depending on the following heat treatment, there is the opportunity to obtain single-​crystalline nanowires (Jian et al. 2004) and hollow nanotubes (Feng et al. 2009, Bae et al. 2008) instead of nanowires. A relatively simple method of producing YBa2Cu3Ox (YBCO) nanowires was presented recently (Li et al. 2005). The authors were using self-​fabricated alumina templates together with YBCO powder placed on top of the template. Using a temperature program in the furnace, where the YBCO is melting on top of the templates, the nanopores are filled with liquid superconductor material. Even though this method has some drawbacks, for example concerning the separation of the nanowires from the alumina templates, superconductivity of the nanowires could be proven. In Fig. 11.1, the scheme of the AAO template-​based nanowire fabrication is presented. Fig.  11.1a–​c illustrate the approach:  Firstly, pre-​reacted high-​Tc superconductor powder (indicated in dark grey) is placed on top of the template. Using an appropriate heat treatment, the powder is molten and subsequently fills the nanopores of the template Fig. 11.1c. A top layer of superconducting material remains also at the top surface of the template. Finally, a method to remove the nanowires from the template is required. Fig. 11.1d shows the situation for Fig.  11.1 (a–​ d) Schematic drawing of the steps in the template technique. (a)  and (b)  present the initial steps. The superconducting material is placed on top of the template (empty pores are indicated by white), and the pores are filled (indicated by dark gray) with the molten material. (c)  gives the situation for a commercial template. (d) shows the corresponding situation for a self-​made alumina template. Here, the untreated alumina bottom must be removed after the processing. Note that in both (c) and (d), superconducting material covers the top of the template.

(a)

(b) Superconducting material

Self-made template

(c)

Nanowires

Superconducting material

(d)

Nanowires Superconducting material

Alumina at bottom, to be removed by etching

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HTSc nanowires prepared by the template method  323 a self-​fabricated template. In this case, there is a big, unstructured bottom of alumina at the bottom of the template. This may also influence the way that the molten material enters and distributes itself within the nanopores. This bottom material must be removed from the template, for example by etching. The use of commercially available alumina templates is much closer to the basic scheme as illustrated in Fig. 11.1a–​c, eliminates the need to remove the bottom layer, and guarantees a proper thickness of the template during the entire process.

11.1.1 Synthesis and templates Firstly, a closer look at the templates is necessary. The commercially available AAO alumina templates (Whatman AAO anodisc templates) are commonly used as nanofilters. They are unframed and have an overall thickness of 50 µm and a diameter of 27 mm. The nanopores run through the entire thickness of the template. Following previous results concerning the analysis of commercial AAO templates (Rørvik et al. 2009), where PbTiO3 nanotubes were grown, a detailed investigation of the nanopore diameter distribution on the top (view from the top) and at the top part of the cross-​section, in the middle (cross-​section), and at the bottom of the templates (cross-​section) was performed (Koblischka et al. 2016). The results are illustrated by the SEM cross-​sectional images given in Fig. 11.2a–​c.

(a)

(b)

500 nm

3 µm (c)

(d)

3 µm (d)

500 nm (f)

3 µm

500 nm

Fig.  11.2 SEM cross-​sectional images of a commercial AAO alumina template with a nominal pore diameter of 100 nm. Panels (a), (c), and (e) show 10,000× magnification, whereas panels (b), (d) and (f) present 50,000× magnification. (a) and (b) show the top side, (c) and (d) a middle section, and (e)  and (f)  the bottom side. The arrow in (d) points to a constriction as typically found in the middle section of the templates.

324

324  HTSc Nanowires and Nanoribbons As superconducting material, we have employed pre-​ reacted powder of YBCO and NdBCO, ground to a particle size of about 1 µm by ball milling. The preparation of the powders is described elsewhere (Hari Babu et al. 2005). The powder was distributed on top of the template, which was kept on a round Al2O3 plate. A second Al2O3 plate was placed on top in order to fix the arrangement and to keep the template from deforming during the heat treatment. Eventually, an Y2BaCuO5 plate could be used as a reaction-​free underlayer as well. The entire arrangement was then placed into a standard laboratory furnace for the heat treatment, which was chosen similar to the standard single-​crystal growth process (Wolf et al. 1989), but with a longer holding time of the maximum temperature to ensure the complete melting of the powder. The sample was quickly heated up to 1050°C (YBCO)/​1100°C (NdBCO), kept there for 9 h, and then slowly cooled down to 900°C at 1 K/​min. From there on, the sample was cooled at 10 K/​min down to room temperature. Finally, either the filled templates or the separated nanowires were oxygen-​ annealed (T = 450°C, 12 h) to obtain superconducting samples. EDX and x-​ray analysis confirmed that the samples were stoichiometrically homogeneous, single-​ phase materials. Pure YBCO and NdBCO nanowires were obtained as proven by X-​ray and EDX measurements, but contamination by Al is always present, which may contribute to the crystal growth as long as the superconducting transition temperature is not affected (Thomsen et al. 1988, Sadowski and Scheel 1989).

11.1.2 Characterization Fig. 11.3 shows SEM images of the nanowires. In Fig. 11.3a, a cross-​section of a YBCO-​filled AAO template is presented. The alumina template is completely filled with superconducting material, and both the top and bottom Al2O3 plates are wetted by the melt (Koblischka et al. 2008). As expected from chemical considerations (McCallum et al. 1995), it turned out to be easier to perform nanowire fabrication from NdBCO as the powder is much faster to form a melt which can enter the nanopores than for YBCO. The resulting filled templates are very brittle, and pieces separate easily from them. The thickness of the template is 50 µm and the template is glued onto a Macor™ ceramic plate for easier handling in the characterization experiments. The black arrows mark the remaining layer of YBCO on top of the template, which has to be removed prior to the electric/​ magnetic characterization experiments. The yellow arrows mark a characteristic crack pattern which causes the breaking of the nanowires in the following steps. Fig. 11.3b shows a top view of the filled AAO pores. The YBCO material appears dark. Finally, Fig.  11.3c shows the nanowires separated from the template by chemical etching with a 4 mol/​l NaOH solution for at least 1 h.  The resulting nanowires do not have a homogeneous diameter, but the pieces of the nanowires obtained are free of cracks. Typically, the length of the resulting nanowires is about 2–​10 µm. Such a characteristic length scale was also obtained when treating filaments from superconducting tapes by ultrasound (Skov-​Hansen et al. 2001).

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HTSc nanowires prepared by the template method  325 (a)

(b)

Macor

50 nm (c)

YBCO-filled AAO template, 50 µm 5 µm

Further, the dimension of approx. 10 µm is similar to what is often observed in regular single-​crystal growth as crystal thickness (Wolf et  al. 1989, Sawada et al. 1995). In a next step, the nanowire samples have to be prepared for physical characterization measurements. To characterize the superconducting properties of the nanowires, pieces of the filled templates were fixed on a Macor™ piece using GE varnish, which can easily be removed by acetone. Mechanical polishing of both sides was performed in order to remove the remaining layers of the superconductor (Koblischka-​Veneva and Koblischka 2004). For this step, 3M Imperial Lapping films (1 µm, ¼ µm) and ethanol as a lubricant were employed. This is illustrated schematically in the insert of Fig. 11.4. The so-​treated sample can be employed for magnetic measurements and for electric characterization. In the latter case, gold electrodes were sputtered onto the top and bottom surfaces of the sample. Finally, the contacts were made by gold wires and silver epoxy glue (see Fig. 11.4). The superconducting transition temperatures, Tc,onset, were determined using resistance measurements (Fig. 11.4) on filled AAO templates. The data for Tc,onset proved to be quite high (YBCO—​90 K and NdBCO—​95 K), which is a consequence of the much easier oxygenation process of a nanowire as compared to a bulk superconducting sample. These measurements indicate that the superconducting properties of the nanowires are well developed, even though the data show that for YBCO the superconducting transition is quite broad with a secondary kink and not so well developed as for NdBCO. The resistance values measured for YBCO and NdBCO are quite similar to each other.

Fig.  11.3  (a)  SEM cross-​sectional view of a YBCO-​filled AAO template. For easier handling the AAO template was glued onto a Macor ceramic plate. The arrows mark the still remaining top layer of YBCO material. (b)  SEM top surface view of the filled template. The YBCO material appears dark. (c)  shows extracted YBCO nanowires from the templates after NaOH etching. The length of the nanowires is ranging between 2 and 10 µm.

326

326  HTSc Nanowires and Nanoribbons 1.2 10 mA

ρ/ρ(100K) [Ω]

1.0

Fig.  11.4 Resistance measurements on nanowire arrays. NdBCO shows a superconducting transition at 95 K, whileYBCO has a Tc of 88 K (indicated by dashed lines). The inset illustrates the arrangement of the samples and the electrical contacts for a four-​probe measurement.

0.8 YBCO NdBCO

0.6

Nanowires

Au-contacts, 200 nm

0.4

Al-template I/V contacts (4-point)

0.2 Sample holder

0.0 70

75

80

85

90

95

100

100

110

115

T [K]

The experiments performed show that the fabrication of HTSc nanowire (arrays) is possible using the template methods. The largest obstacle here is the required separation process from the template, as the chemical etching also attacks the surface of the HTSc nanowires. This is different for the case of metallic superconductors, so the template method seems to be better suited in this field. The commercially available templates do not offer a good way to produce nanowires due to their inhomogeneous distribution of pore sizes. The AAO templates with a nominal diameter of 20 nm have shown pore diameters of more than 200 nm in the remainder of the template (Koblischka et al. 2016), so no controlled fabrication of nanowires is possible in this way.

11.2 HTSc nanowires prepared by electrospinning In contrast to the template approach, electrospinning provides the possibility to grow long lengths of nanowires completely free of substrates or templates. This technique is fairly common to a large variety of materials including polymers, composites, and biomaterials (Agarwal et al. 2013, Li et al. 2004, Dzenis 2004, Huang et al. 2003, Law et al. 2004, Bhardwaj and Kundu 2010, Yang et al. 2014, Atchinson et al. 2015), but relatively new to the field of ceramic materials, especially superconducting and magnetic ones (Li et al. 2003, Wu and Pan 2012). In a previous work, it was shown that superconducting LSCO nanowires can be produced in this way (Li et al. 2011). The required apparatus for the electrospinning

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HTSc nanowires prepared by electrospinning 327 method can be arranged in a relatively cheap fashion, but due to the importance of the method in the field of polymers, also dedicated industrial types of apparatus are available (Atchinson et al. 2015).

11.2.1 Electrospinning Combined with a suitable thermal treatment, electrospinning is a potential method for inorganic nanowire fabrication. Fig. 11.5 presents a schematic drawing of the apparatus. As starting material, a precursor solution of the metallic acetates or nitrates in polyvinylalcohol (PVA) or a similar compound has to be prepared. In the experiment, the precursor solution is pushed by the pump. A liquid drop appears at the bottom of the needle. Due to the connection with a high-​voltage source, electrons move out quickly from the drop to the cable, leaving large amounts of positive charge in the drop itself. Finally, the Coulomb force overcomes the surface tension. Thousands of fibers spray out from the drop. Due to the electric field effect between the needle and the grounding area, the fibers are pulled down to the collection holder. On the collection holder, a non-​woven fabric of the nanowires is formed. This as-​g rown raw material requires then a subsequent heat treatment in order to remove the remaining solvent and to form the demanded phase of the ceramic material. The heat treatment must be chosen in such a way as to maintain the nanowire network structure, which is a severe limit for several types of ceramic materials. As a result, the final reaction temperature should be as low as possible to form the desired phase. According to this approach, the resulting nanowires are of the polycrystalline type.

Boost pump

Syringe

High-voltage source

Needle

Metal plate

Precursor: Acetate+ Ploymer Fig.  11.5 Schematic drawing of an electrospinning apparatus.

328

328  HTSc Nanowires and Nanoribbons

11.2.2 Synthesis of HTSc nanowires by electrospinning Several types of HTSc materials were prepared by the electrospinning technique, including the compounds (La1-​x,Srx)CuO4 (LSCO) (Li et  al. 2011, Zeng et  al. 2015), YBa2Cu3Ox (YBCO) (Duarte et al. 2015), and Bi2Sr2CaCu2Ox (Bi-​2212) (Duarte et al. 2013, Koblischka et al. 2016). As an example of the fabrication procedure, the preparation of LSCO and Bi-​2212 nanowires is described here in detail. 11.2.2.1 LSCO For the precursor, the following acetates were used:  lanthanum acetate [La(Ac)3*1.5H2O], strontium acetate [Sr(Ac)2*xH2O], and copper acetate [Cu(Ac)2*H2O] powders (Alfa Aesar) were weighted according to the molar ratio of 1.85: 0.15: 1, respectively and dissolved in 10 ml deionized water with 1.5 g polyvinylalcohol (PVA). LSCO nanowires and nanoribbons were fabricated via the electrospinning technique by employing the following conditions: a stable flow rate at 0.2 ml/​h, an applied voltage of 26 kV, and a tip–​collector distance of 25 cm. The samples were annealed according to the DTA (differential thermal analysis)/​ TGA (thermal gravimetry analysis) at 218°C for 1 h, at 486°C for 1 h, and at 700°C for 1 h, respectively. Considering the loss of the oxygen component during the heating process, a compensating heat treatment was applied at 500°C under flowing oxygen (50 cm3/​min) for 5 h, similar to previous experiments (Li et al. 2011). In order to remove the organic components and to maintain the fiber structure at the same time, the thermal treatment was performed in air, based on the results of TGA. There are two obvious mass-​loss processes accompanied by larger amounts of exothermal energy at 218°C and 486°C, respectively. So, the first thermal treatment was split up into three steps. The fabrication of the magnetic compound (La1–​x,Srx)MnO3 (LSMO) can be realized in the same way as for LSCO, and it is therefore tempting to try mixtures of nanowires of the two systems. Due to the interface effect, the respective properties (colossal magneto-​resistance—​ CMR or superconductivity) may be enhanced in such a system. 11.2.2.2 Bi-​2212 Bi-​2212 nanowire networks were grown by the electrospinning technique (parameters mentioned in Table 11.1), employing acetate powders of all constituents and PVP (polyvinyl pyrrolidone, MW 1,300,000) dissolved in propionic acid. After the electrospinning process, a heat treatment is required to remove all organic material from the as-​prepared nanowire networks and to form the superconducting phase. Finally, an oxygenation step in pure O2 is required. X-​ray analysis confirmed that the samples are pure Bi-​2212 phase with some residing carbon. The further details of the preparation procedure are described elsewhere (Koblischka et  al. 2016). An important issue of these nanowire networks is the determination of the superconducting volume. For this purpose, the weight of all samples fabricated was determined using a microbalance. Taking the theoretical density

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HTSc nanowires prepared by electrospinning 329 Table 11.1  Synthesis parameters for electrospinning of the as-​prepared BSCCO nanofibers. Parameter

Bi-​2212 + PVP

Applied voltage (kV)

20

Fly distance (mm)

190

Pump rate (ml

h–​1)

Travel speed (mm

0.1

s–​1)

50

Travel distance (mm)

100

Temperature (°C)

22

Relative humidity (%)

30–​45

of Bi-​2212 (6.4 g/​cm3) for the individual nanowires, we find the density of the present nanowire networks to be 0.0459 g/​cm3.

11.2.3 Microstructure 11.2.3.1 LSCO Fig. 11.6a–​d present a series of SEM images of the as-​spun LSCO nanofibers at two magnifications. It is important to note that the fiber structure shows two distinct shapes, that is, nearly round nanowires, and flat nanoribbons which can be observed using higher magnification. These structures were maintained during the whole thermal treatment. The average diameter of the nanowires was

(a)

10,000x

(b)

3 µm

(c)

10,000x

29,997x

900 nm

(d)

3 µm

10,000x

3 µm

Fig.  11.6 (a,b) SEM images of the LSCO nanoribbons. (a) as-spun, and (b) after the heat treatment. (c,d) SEM images of the LSCO nanowires, (c) ­as-spun, and (d) after the heat treatment.

330

330  HTSc Nanowires and Nanoribbons determined to be about 234 nm. The flat nanoribbons showed widths which are nearly 1 µm, but their corresponding thicknesses are only about 60–​80 nm. The total lengths of these fibers were found to be above 50 µm. Most of them were in the 100 µm range. These nanoribbons are extremely thin and transparent to the electron beam. The total lengths of the nanofibers are larger than 50 µm. Most of them are found in the vicinity of 100 µm. It is clearly a challenge to optimize the preparation conditions in such a way that either nanowires or nanoribbons can be prepared, having only a small diameter/​thickness variation. First experiments in this direction have revealed that the nanoribbons are formed when using a precursor with a high polymer content, that is, with a higher viscosity. A possible reason of this is due to the Coulomb force leading to a decomposition of the precursor droplet into nanowires with larger diameter. The competition between the Coulomb force and the surface tension leads to a transformation of the wires into nanoribbons but not to a separation into smaller nanowires as in the case of precursors with lower viscosity. Furthermore, the nanoribbons tend to locate at the edge part of the collection area which is far away from the projection area of the needle. This specific location of the nanoribbons in the collection area still requires further investigation. The reason for this distribution of the nanowires/​ ribbons will be important in order to reveal the mechanism of structure formation, which is the key for controllable fabrication of the nanoobjects. In Fig. 11.7a–​d, the results of a TEM (transmission electron microscope) analysis of the LSCO nanowires and nanoribbons are presented. The resulting images reveal clearly that the samples are polycrystalline. The nanoribbons (Fig. 11.7a–​b) (a)

0.5 µm

Fig. 11.7  LSCO TEM images. (a) and (b)  show nanoribbons, and (c)  and (d) present nanowires.

(b)

0.5 µm

(c)

(d)

20 nm

20 nm

  331

HTSc nanowires prepared by electrospinning 331 (a)

(b)

1 cm

1 cm

(c)

(d)

1 cm

1 cm

(e)

1 cm

consist of only one layer of grains, and show the presence of holes between the grains. The nanowires (Fig. 11.7c–​d) exhibit the same shape of the grains as found at the nanoribbons, with well-​defined grain boundaries. Grain size statistics obtained from several TEM images show a broad variation of the grain sizes and the average grain size is determined to be about 110 nm. One should notice here that the grain size of the nanoribbons is equal to the thickness. This indicates that the nanoribbons are formed by individual LSCO grains, linked to each other in a single layer. Fig. 11.8a–​e present optical images of a LSCO nanowire network sample at different steps of the thermal treatment. In the beginning, the sample looks nearly white with a light green color. The green color stems from the Cu2+ ions and the white background is the result of fibers crossing each other. After the heat treatment at 218°C, the sample appears brown because the PVA starts to decompose. During the 486°C heating, the sample goes through an intense polymer decomposition process, where a part of the organic component turns into carbon dioxide and water vapor, which is removed from the sample, and further, the LSCO phase starts to form. Thus, the sample turns slightly dark blue. Then, after the 700°C thermal treatment, all the organic components have been removed as there is nearly no mass loss above 700°C in the TGA spectrum. The sample looks completely dark blue, the same as observed for bulk LSCO materials. 11.2.3.2 Bi-​2212 Fig. 11.9 presents SEM images of the Bi-​2212 nanowires after the final heat treatment. In (a), a magnification of 1,000× is shown which gives an overview of

Fig. 11.8  Optical images of the LSCO nanofiber samples at different steps of thermal treatment and the DTA/​ TGA measurement. (a)  as-​ prepared, (b)  218°C, (c)  486°C, (d)  700°C, and (e) 500°C in flowing O2.

332

332  HTSc Nanowires and Nanoribbons (a)

1,000x

30 µm

(b)

Fig.  11.9 (a)  SEM image of the nanowire network at low magnification (1,000×). (b)  High magnification SEM image (10,000×), revealing the polycrystalline character of the nanowires and details of the elongated Bi-​2212 grains.

10,000x

3 µm

an entire section of the nanowire network, revealing the long lengths of the electrospun nanowires, and also the numerous interconnects between them. This overall structure is created during the electrospinning process, and survives all subsequent heat treatments. Fig. 11.9b shows an image with high magnification (10,000×). Here, the arrangement of the elongated, platelet-​like Bi-​2212 grains can be seen. The average grain size of the Bi-​2212 grains is determined to be ~37  nm. The interconnection between the individual grains is an important issue for the fabrication of conductors from this material, and the field dependence of these contacts governs the resulting magnetic properties. Fig. 11.10 presents TEM images of the Bi-​2212 nanowires. In Fig. 11.10a, a larger section of the nanowire network is chosen. The image clearly reveals the elongated character of the Bi-​2212 grains, which represent an ideal shape for elongated conductors (Kametami et al. 2015). The nanowires always consist of several such elongated grains. The dark positions reveal the shape of the interconnects between the individual filaments. In Fig. 11.10b–​c, individual nanowires are shown. Characteristic for the growth of Bi-​2212 is the formation of several needle-​like outgrowths, which are also seen in the SEM image of Fig. 11.9b.

11.2.4 Magnetic and electric characterization The magnetic properties of the nanowires may reveal important information about high-​Tc superconductivity in a confined geometry (Bezryadin 2013,

  333

HTSc nanowires prepared by electrospinning 333 (a)

(b)

0.2 µm

(c)

0.2 µm

Altomare and Chang 2013). However, the amount of material to be measured is very small, so even magnetometry using superconducting quantum interference devices (SQUID) comes to its limits when measuring a piece of a nanowire network, not to speak of individual nanowires only. Therefore, most measurements published in the literature up to now concern data of nanowire networks or from powders of nanowires. A solution to this problem will only be possible with novel magnetometry methods like cantilever magnetometry (Rossel et al. 1996, Brugger et al. 1999, Weber et al. 2012, Adhikari et al. 2012). In Fig. 11.11a, an M(T) measurement of a LSCO nanowire network sample is presented when field-​cooling the sample. A magnetic field of 2 mT is applied perpendicular to the sample surface. The sample shows an onset of superconductivity at about 29.3 K, while a sharp decay of the susceptibility begins at 13.9 K. Both values are considerably lower than the superconducting transition of the bulk material (37 K), which may be related to changes in the vibration spectrum as determined by Raman measurements (Li et al. 2011). Fig. 11.11b gives the temperature dependence of the magnetization M(T) of a Bi-​2212 nanowire network at an applied magnetic field of 1 mT (perpendicular to the sample surface). The onset of superconductivity is found at 76.3 K, while the onset of irreversibility at this field is at about 66 K. The Tc obtained here is somewhat lower than the bulk Tc of the Bi-​2212 phase (85 K), which was also observed previously (Duarte et al. 2013).

Fig. 11.10  TEM images of the Bi-​2212 nanowires. (a)  gives a larger section of the nanowire network, revealing also the interconnects between individual nanowires. (b) and (c) show single nanowires.

334

334  HTSc Nanowires and Nanoribbons (a)

(b)

Tc onset

0.00

FC

–0.02

–0.2

–0.4

m (10−3 emu)

m (×10−6 emu)

0.0

La1.85Sr0.15MnO7.5 nanoribbons 2 mT

Irreversibility onset

–0.04

1 mT

–0.06

ZFC

–0.08

–0.6 0

25

50

75 T (K)

100

120

150

–0.10

0

20

40

60

80

100

Fig. 11.11 (a) M(T) of LSCO. Magnetization as function of temperature measured when cooling the sample in an applied magnetic field of 2 mT, revealing an onset of superconductivity at 29.3 K. A second step is observed at 13.9 K. The dashed lines are used to extract the onset temperature. Note that there is a diamagnetic background stemming from the Kaptan tape fixing the sample in the sample holder. (b) M(T) of Bi-​2212 in FC and ZFC mode. Tc,onset is here at about 76 K, which is lower than the bulk Tc. The irreversibility sets in at 62 K (merger of the ZFC (zero-​field cooling) and FC curves). Fig. 11.12 presents a comparison of the M(T) data (indicated by symbols) of the Bi-​2212 nanowire network with resistance data measured on the same sample in a separate experimental run. The resistance is measured in a quasi-​ four point measurement configuration using gold contacts. Magnetic fields of 0.1 T, 1 T, and 10 T were applied, perpendicular to the sample surface. The resistance data reveal an onset of superconductivity at 85 K (which corresponds to the bulk Tc, as indicated in the figure), and the superconducting transition broadens on increasing the applied magnetic field. The magnetic data are characterized by two temperatures, the Tc,onset at 76 K, and the 1.6

0.02

62 K 76 K

R (Ω)

ZFC 0.8

–0.04 0T 0.1 T 1T 10 T

0.4

Fig.  11.12  M(T) (symbols) measured in ZFC mode plotted together with the R(T) data (full and dashed lines) of a Bi-​2212 nanowire network sample.

0.0

–0.02

0

20

40

60 T (K)

80

–0.06 –0.08 –0.10 100

M (10–3 emu)

0.00

1.2

  335

HTSc nanowires prepared by electrospinning 335 secondary step at 62 K. The secondary step in M(T) coincides with a slope change of the resistance data. An interesting feature is the observation that the magnetically determined critical temperature is lower than the one measured in resistance measurements on the same samples (Koblischka et al. 2016), which is the opposite of what one would expect from the common knowledge of bulk superconductors (Murakami 1992), but which was already observed in thin film samples (Naugle 1967, Tinchev 2010). Possible reasons for this behavior may be surface superconductivity, the influence of the metallic layer, or an influence of the temperature dependence of the superconducting parameters λ and ξ, which both rise towards the critical temperature, Tc. This will be further investigated in the future. Fig. 11.13 presents M(H) measurements on a Bi-​2212 nanowire array sample at temperatures between 5 K and 30 K. The magnetic fields are always applied perpendicular to the sample surface. The magnetization loops reveal clearly the signatures of granular samples (Senoussi 1992). At elevated temperatures, the magnetization loops become fully asymmetric, which does not allow the use of the classical Bean model (Bean 1962) to calculate the flowing critical currents. Therefore, a different approach to evaluate the critical currents in such samples is required. In order to solve this problem, a modelling of the entire M(H)-​loop is required. To obtain the critical current densities from the magnetization loops, one can employ the approach of Senoussi of a three-​current model (Senoussi 1992), where the entire magnetization is composed of three contributions of the grains, the Josephson contacts between them, and the shielding of the entire sample. This model can reproduce the changing shape of the measured magnetization loops, as found in an earlier publication (Koblischka et al. 2016). A more sophisticated approach is based on an extended critical state model (ECSM) developed by Gokhfeld (Gokhfeld et al. 2011, Gokhfeld 2014). On this base, the critical current densities can be evaluated from the magnetization loops, yielding a critical current density of about 2×107 Acm-​2 at 2 K, which is considerably high for this type of superconducting material. This value shows that the nature of the interconnects is very important for the performance of this material. The nature of these interconnects will be investigated in the future. The experimental M(H)-​loops demonstrate the presence of an additional diamagnetic contribution at high temperatures and high magnetic fields, which can be expressed following MD = –​KDH. The temperature dependence of this coefficient KD is shown in Fig. 11.14. The coefficient KD appears to increase quickly with temperature which is atypical for diamagnetic atoms. There are several scenarios, for example, one can interpret this feature as a contribution of a paramagnetic phase at low T, a superconducting screening of an additional diamagnetic phase, or self-​diamagnetism of normal cores of Abrikosov vortices. Effects of flux creep and flux flow (Yeshurun et  al. 1996)  are other important characteristics of a superconducting material. Therefore, we have measured

336

0.050

0.06

5K

0.02 0.00 –0.02 –0.04 –0.06

0.000

–0.025

–6

–4

–2

0 µ0H2 (T)

2

4

6

–6

8

0.00

–0.02

–0.04

–0.050 –8

15 K

0.02 M (emu/cm3)

0.025 M (emu/cm3)

M (emu/cm3)

0.04

0.04

10 K

–4

–2

0 µ0H2 (T)

2

4

6

–6

–4

–2

0

µ0H2 (T)

2

4

6

4

6

0.04

0.04

0.02

20 K

25 K

0.02

0.00

30 K 0.01 M (emu/cm3)

M (emu/cm3)

M (emu/cm3)

0.02

0.00

–0.02

–0.02

–0.04

–0.04

0.00 –0.01 –0.02

–6

–4

–2

0

µ0H2 (T)

2

4

6

–6

–4

–2

0

µ0H2 (T)

2

4

6

–6

–4

–2

0

µ0H2 (T)

2

Fig. 11.13  M(H) measurement of a Bi-​2212 nanowire network sample at various temperatures as indicated in the figure.The data are denoted by symbols, fits using the ECSM are indicated by solid lines.

  337

HTSc nanowires prepared by electrospinning 337

Kd (10–3 emu/cm3/T)

3.4 3.2 3.0 2.8 2.6 2.4 0

10

20

30

40

T (K)

50

60

70

Fig.  11.14 Temperature dependence of the diamagnetic contribution, Kd(T), for the Bi-​2212 nanowire network sample.

magnetization loops using various field sweep rates to obtain data on dynamic flux creep (Jirsa et al. 1993, van Dalen et al. 1996). Fig. 11.15 presents magnetization loops measured with various field sweep rates dB/​dt = 2 mT/​s, 5 mT/​s, and 10 mT/​s at temperatures of 5 K and 20 K. For clarity, the diamagnetic contribution

1.2

10 mT/s 5 mT/s 2 mT/s

0.6 0.0

M (emu/g)

–0.6 –1.2

5K

0.6 0.3 0.0 –0.3 –0.6 –8

20 K –4

0 µoHa (T)

4

8

Fig. 11.15  M(H) measurements at 5 K (upper graph) and 20 K (lower graph) performed using different sweep rates of the externally applied magnetic field (2, 5, and 10 mT/​s).

338

338  HTSc Nanowires and Nanoribbons 8 –0.38

M (emu/g)

70 K

H irr (T)

6

4

–0.40 –0.42 –0.44 6.0

2

Fig.  11.16 Irreversibility line, Hirr(T), for the Bi-​ 2212 nanowire network between 20 K and 70 K. The dashed line is a fit to the data, see text.

0

6.5

Data Fit 20

30

7.0

µoHa (T)

40

50

60

70

T (K)

was removed from the M(H)-​data. One can see from the loops that the distance between the curves is largest at the zero-​field peak and in the area of low fields. At high fields, all three curves merge together. According to previous work (van Dalen et al. 1996), a dynamic relaxation rate Q can be calculated via Q = d ln js/​d ln(dBe/​dt). Of course, this procedure requires the determination of the critical current densities from the magnetization loops. A  first estimation of the creep rate reveals by far less flux creep than observed on measurements of Bi-​2212 single crystals (van Dalen et al. 1996) at the same temperatures. This is due to the much higher density of grain boundaries in these nanocrystalline nanowires as compared to a well-​grown single crystal. As a result, the flux pinning within the nanowires is much stronger than in single crystals, which is most important at high magnetic fields. Another important parameter of a HTSc is the temperature behavior of the irreversibility field, Hirr(T). From the M(H)-​loops, Hirr was determined directly from the magnetization loops without invoking a current criterion. This procedure works well with the exception of the highest temperatures, and at low temperatures below 20 K, Hirr is outside of the measurement range. The result is presented in Fig. 11.16. It can be seen that the irreversibility field decreases monotonously on rising temperature. The data are fitted well (dashed line) by the function Hirr = A × (1–​T/​Tc)3. This behavior was also observed by various authors on Bi-​2223 and Bi-​2212 samples (Koblischka and Sosnowski 2005). By means of the fitting, the critical temperature is determined to be 74.4 K, corresponding to the result from the M(T) curve. At temperatures below 20 K, the available magnetic field of 7 T is not strong enough to determine the irreversibility field, Hirr, and above 60 K, it is difficult to determine the closing of the loops as the M(H)-​data run practically parallel to each other, which is illustrated in the inset to Fig. 11.16. Fig. 11.17 presents the resistance temperature measurement results at different fields ranging from 0 T to 10 T.  The applied current was 50 µA.

  339

HTSc nanowires prepared by electrospinning 339 1.6

R (Ω)

1.2

0T 0.01 T 0.1 T 1T 2T 6T 10 T

0.8 10 T

0T

0.4

0.0

0

20

40

60 T (K)

80

100

In general, the sample shows the electric transition behavior well agreed with that of the superconductor nanowires. Two steps of resistance drops are observable. The first step is always described as the superconducting transition of electrodes in the case of measurements on metal nanowires or wires as bridges prepared from bulk (Tinkham et  al. 2003, Bollinger et  al. 2004). However, such an explanation is not valid in the present case as there is no bulk contact present. Compared to the magnetic measurement as shown in Fig. 11.12, the onset transition temperature around 76 K in the zero field R(T) curve is consistent with the superconducting transition temperature obtained from the zero-​field cooling (ZFC) M(T) curve, which indicates that the resistance drop of the first step can be attributed to the superconducting transition of the sample. The non-​zero resistance around Tc provides evidence of the existence of thermally activated phase slips. The second drop in resistance corresponds to the behavior of the intergrain weak links inside the individual nanowires (Bollinger et al. 2004), as the resistance behavior is found to change dramatically with even low fields in this region. Even at low temperatures near 4 K, the resistance is small but non-​zero, which can be related to the scattering of charge carriers at the grain boundaries. Fig. 11.18 shows the U(I) measurement of the nanowire network at various temperatures ranging from 18–​100 K. It can be found that below the Tc of the sample (18–​71 K), a transition from the superconducting to the normal state regime occurs, even though the superconducting regime shows a fairly high residual resistance. This may also be attributed to the high amount of grain boundaries inside the nanowire network which correlates to the observations of the R(T) data. The critical current for this transition increases on decreasing the temperature from 0.06 A (71 K) to 0.18 A (18 K). The phase slip behavior is based on the BCS (Bardeen–​Cooper–​Schrieffer) theory, which cannot provide a complete explanation of the mechanism of high-​temperature

Fig.  11.17 Resistance R as function of temperature T measured in magnetic fields ranging from 0 T up to 10 T.

340

340  HTSc Nanowires and Nanoribbons 0.4

U (V )

0.3

0.2

100 K 71 K 60 K 50 K 33 K 18 K

100 K

18 K

0.1

Fig.  11.18  U/​I characteristics of the Bi-​ 2212 nanowire network measured at various temperatures between 18 K and 100 K.

0.0 0.00

0.05

0.10

0.15

0.20

I (A)

superconductivity. However, the result of our Bi-​2212 nanowire network sample at least proves the statement that cuprate superconductivity is weak-​coupling BCS behavior. In Fig. 11.19, a single nanowire prepared for the electric measurements is shown. In Fig. 11.19a, the entire contact pattern is presented, which enables four contact points to the sample, which is cut from the nanowire network by means of FIB (focused ion beam) milling and subsequent nanomanipulation. In Fig.  11.19b, the nanowire is shown with the electrical contacts made to the pads by means of Pt-​precursor evaporation in the FIB environment. The problem to be solved is currently the overly high resistance of this arrangement. The resulting metallic Pt is polycrystalline with a high number of grain boundaries leading to a relatively high resistance of the contacts. Therefore, more work is required to achieve the direct electric measurement of an individual HTSc nanowire.

(a)

Fig.  11.19 Preparation for measurements of a single nanowire. (a) Contact pads with mounted sample, and (b) close view of the electrically contacted Bi-​ 2212 nanowire.

mag HV HFW det tilt WD 5.00 KV 500 x 258 µm ETD 0° 3.8 mm

(b)

100 µm

mag HV WD HFW det tilt 5.00 KV 8 500 x 19.7 µm ETD 52° 3.8 mm

5 µm

  341

Use of HTSc nanowires as building blocks 341

11.3 Use of HTSc nanowires as building blocks While the HTSc nanowires are of fundamental interest for the investigation of superconducting properties in a confined environment (Mohanty et  al. 2004, Bezryadin 2013, Altomare and Chang 2013, Chen et  al. 2014, Belkin et  al. 2015), there are also several applications for such nanowires including sensors (Bonetti et al. 2012, Arpaia et al. 2015) and electrodes (Xu and Heath 2008, Liu et  al. 2014). The superconducting nanowire networks (i.e. non-​woven superconducting fiber fabrics) form an entirely new class of HTSc material, without predecessors among conventional superconductors. Another interesting field of application of such nanowires is their possible use as building blocks to fabricate bulk superconducting samples with nano-​engineered microstructure (Mukherjee et al. 2010). In this approach, a large amount of nanowires must be prepared. Thus, a simple, straightforward approach with a high yield is required for such a task. The lengths of the nanowires obtained via the electrospinning technique range up to several hundreds of µm, which is considerably different from what can be obtained employing the template approach, where the nanowire length turned out to be always smaller than the template thickness (Koblischka et al. 2008). This is also consistent with observations by others (Jian et al. 2004, Zhang et al. 2006, Lu et al. 2006), where the sol-​gel technique was applied together with the alumina templates. In contrast to the template approach, the electrospinning method offers the possibility of producing a large amount of long nanowires, which is required for using them as building blocks for bigger structures. The nanowires in the as-​grown fabric exhibit already a large number of contacts between each other due to their considerable lengths, and these contacts transform into electrically useful contacts to provide the flow of supercurrents through the entire sample perimeter (Koblischka et al. 2016a). This is essential for the possible applications of such nanowire networks. The envisaged bigger structures can be fabrics made from non-​woven nanowires in contrast to the superconducting fabrics produced using woven Y2O3 clothes (Reddy et al. 2000, Reddy and Schmitz 2002, Noudem et  al. 2002), thick films on various substrates including Si, base materials for ink-​jet printing of superconducting circuitry (Feys et al. 2012, Vilardell et al. 2013), hybrid materials together with, for example, polymers (Haupt et al. 1993, Gwak et al. 2004), and even nanoporous bulk samples (Reddy et  al. 2005). The latter materials would offer interesting properties such as a simpler and easier oxygenation process via the nanopores while maintaining good mechanical properties. In Fig. 11.20, optical images of nanowire network “fabrics” are presented, mounted on a sample holder for electric characterization measurements. The sample in Fig. 11.20a is contacted using silver paste without a gold cover, while

342

342  HTSc Nanowires and Nanoribbons (a)

(b)

Fig.  11.20  Optical images of Bi-​2212 nanowire network samples on the measurement holders. (a)  Pure sample, and (b) sample with gold coating. the sample in Fig. 11.20b is covered with 20–​100 nm gold film sputtered on the sample surface. These nanowire fabrics are quite brittle due to the low density of the material, but as seen in the electric and magnetic measurements, the numerous interconnects between the individual nanowires enable the flow of reasonably high transport currents. Fig. 11.21 presents an image of completely heat-​treated nanowire fabric material, which can be used as a base to prepare nano-​engineered bulk samples. Note the scale bar in the centimeter range, which indicates that it is possible to produce a large amount of nanowire fabric material via the electrospinning technique. The feasibility of large-​scale sample production represents the largest advantage of the electrospinning method, and will certainly be used in the future for new generations of HTSc material.

11.4 Summary and outlook In this chapter the focus was on the preparation of HTSc nanowires by two different methods, the templating technique and the electrospinning method, and the

Fig.  11.21 Optical image of heat-​ treated nanowire fabrics as produced by electrospinning.

1 cm

  343

References (Chapter-11) 343 respective characterization of the nanoproducts. Although with both techniques it is not possible to generate nanowires with diameters close to the superconducting coherence length, ξ, the present diameters in the range of 100–​250 nm enable measurements of the superconducting properties of such nanowires without the influence of a substrate, and especially close to the critical temperatures, where λ and ξ are considerably larger as at low temperatures. Furthermore, the possibility to produce large amounts of nanowires and, most importantly, non-​ woven nanowire networks, enables a variety of new applications of HTSc superconductors. The fabrication of the nanowires further enables the study of grain boundary effects on the flow of transport currents in a direct fashion, and many more details about the specific nature of HTSc materials may be revealed by studying nanowire samples.

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344  HTSc Nanowires and Nanoribbons Ding, D., Wang, J. and Chen, S. (2004) Supercond. Sci. Technol. 17, 438. Duarte, E.A., Quinteiro, P.A., Meisel, M.W. and Nino, J.C. (2013) Physica C 495, 109. Duarte, E.-​A., Rudawski, N.G., Quintero, P.A., Meisel, M.W. and Nino, J.C. (2015) Supercond. Sci. Technol. 28, 015006. Dzenis, Y. (2004) Science 304, 1917. Feng, W., Huang, H. and Yang, S. (2009) J. Eur. Ceram. Soc. 29, 1387. Feys, J., Vermeir, P., Lommens, P., Hopkins, S.C., Granados, X., Glowacki, B.A., Baecker, M., Reich, E., Ricard, S., Holzapfel, B., Van Der Voort, P. and Van Driessche, I. (2012) J. Mater. Chem. 22, 3717. Gokhfeld, D.M. (2014) Phys. Solid State 56, 2380. Gokhfeld, D.M., Balaev, D.A., Petrov, M.I., Popkov, S.I., Shaykhutdinov, K.A. and Valkov, V.V. (2011) J. Appl. Phys. 109, 033904. Grigoryan, S., Manukyan, A., Hayrapetyan, A., Arzumanyan, A., Kuzanyan, A., Kafadaryan, Y. and Vardanyan, E. (2003) Supercond. Sci. Technol. 16, 1202. Gwak, J., Ayral, A., Rouessac, V., Cot, L., Grenier, J.C., Jang, E.S. and Choy, J.H. (2004) Mater. Chem. Phys. 84, 348. Han, Y.-​J., Kim, J.M. and Stucky, G.D. (2000) Chem. Mater. 12, 2068. Hari Babu, N., Reddy, E.S., Shi, Y., Withnell, T.D. and Cardwell, D.A. (2005) IEEE Trans. Appl. Supercond. 15, 3090. Haupt, S.G., Riley, D.R. and McDevitt, J.T. (1993) Adv. Mater. 5, 755. Huang, Z.M., Zhang, Y.Z., Kotaki, M. and Ramakrishna, S. (2003) Compos. Sci. Technol. 63, 2223. Jian, X., Xiaohe, L. and Li, Y. (2004) Mat. Chem. Phys. 86, 409. Jirsa, M., Pust, L., Schnack, H.G. and Griessen, R. (1993) Physica C 207, 85. Kametami, F., Jiang, J., Matras, M., Abraimov, D., Hellstrom, E.E. and Larbalestier, D.C. (2015) Sci. Rep. 5, 8285. Koblischka, M.R., Koblischka-​ Veneva, A. and Skumryev, V. (2008) in Superconductivity:  Recent developments and new production technologies, edited by M. Muralidhar, NOVA Science Publishers, New York, p. 1. Koblischka, M.R. and Sosnowski, J. (2005) Eur. Phys. J. B 44, 277. Koblischka, M.R., Zeng, X.L. and Hartmann, U. (2016) Physica status solidi (a) 213, 1067. Koblischka, M.R., Zeng, X.L., Karwoth, T., Hauet, T. and Hartmann, U. (2016) IEEE Trans. Appl. Supercond. 26, 1800605. Koblischka-​Veneva, A. and Koblischka, M.R. (2004) in Magneto-​optical imaging, edited by T.H. Johansen and D.V. Shantsev, Kluwer Academic Publishers, Dordrecht, The Netherlands, p. 242. Lai, S.H., Hsu, Y.C. and Lan, M.D. (2008) Solid State Commun. 148, 452. Law, M., Goldberger, J. and Yang, P.D. (2004) Ann. Rev. Mater. Res. 34, 83. Li, D., Herricks, T. and Xia, Y.N. (2003) Appl. Phys. Lett. 83, 4586. Li, D. and Xia, Y. (2004) Adv. Mater. 16, 1151. Li, J.M., Zeng, X.L., Mo, A.D. and Xu, Z.A. (2011) CrystEngComm 13, 6964.

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346  HTSc Nanowires and Nanoribbons Vilardell, M., Granados, X., Ricart, S., van Driessche, I., Palau, A., Puig, T. and Obradors, X. (2013) Thin Solid Films 548, 489. Weber, D.P., Rüffer, D., Buchter, A., Xue, F., Russo-​Averchi, E., Huber, R., Berberich, P., Arbiol, A., Fontcuberta i Morral, A., Grundler, D. and Poggio, M. (2012) Nano Lett. 12, 6139. Wolf, T., Goldacker, W., Obst, B., Roth, G. and Flükiger, R. (1989) J. Cryst. Growth 96, 1010. Wu, H., Pan, W., Lin, D. and Li, H. (2012) J. Adv. Ceram. 1, 2. Xu, H. and Heath, J.R. (2008) Nano Lett. 8, 3843. Yang, L., Cai, Q. and Yu, Y. (2006) Inorg. Chem. 45, 9616. Yang, M., Shuai, Y., Zhou, G., Mandal, N., Zhu, L. and Mao, C. (2014) ACS Appl. Mater. Interfaces 6, 13782. Yeshurun, Y., Malozemoff, A.P. and Shaulov, A. (1996) Rev. Mod. Phys. 68, 911. Zeng, X., Koblischka, M.R. and Hartmann, U. (2015) Mater. Res. Exp. 2, 095022. Zhang, G., Lu, X., Zhang, T., Qu, J., Wang, W., Li, X. and Yu, S. (2006) Nanotechnology 17, 4252.

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Mesoscopic Structures and Their Effects on High-​Tc Superconductivity H. Zhang Materials Physics Laboratory, State Key Laboratory for Mesoscopic Physics, Department of Physics, Peking University, Beijing 100871, China

12 12.1 Introduction and motivation 

348

12.2 Model 

350

12.3 Calculating results and discussion  352 12.4 Strain between two blocks and its effect on superconductivity  356 12.5 Carrier-​compensated system and mesoscopic structures  359

High-​ temperature superconductors (HTSc) have been known for the last 30  years or so, but their mechanism for superconductivity has still not been understood. One of the key reasons is that their crystal structures are not simple. Broadly, the structure of HTSc is unique in having some mesoscopic features. For example, the structures of a majority of cuprite superconductors are composed of two structural blocks, perovskite and rock salt, stacked along the c-​direction. The perovskite block is the active block where the Cu(2)-​O planes are located and the charge carriers concentrated. The rock salt block corresponds to the charge-​reservoir block, which supplies the carriers to the Cu(2)-​O planes. These mesoscopic features are closely linked with the superconductivity. A close understanding of these mesoscopic features in relation to superconductivity, we believe, would be relevant for knowing the mechanism of the high Tc. We have developed a model to calculate the interaction between the two blocks in the form of what we call the combinative energy to study its relationship with the value of the critical temperature Tc. We have studied various Bi-​based superconductors:  Bi2Sr2CuOy, Bi2Sr2CaCu2Oy, and Bi2Sr2Ca2Cu3Oy; Hg-​based superconductors:  HgBa2CuOy, HgBa2CaCu2Oy, HgBa2Ca2Cu3Oy, HgBa2Ca3Cu4Oy, and HgBa2Ca4Cu5Oy; Tl-​based superconductors:  Tl2Ba2Can-​1CunO2n+3 (with five superconductive compounds) and Tl2Ba2Can–​1CunO2n+4 (with four superconductive compounds); Y-​based superconductors, with different oxygen deficiencies; and La2–​xMxCuO4 (M = Ba, Sr), with different M concentrations. For all these superconductors we have found a close relationship existing between the superconducting critical temperature and the combinative energy, and in particular, the higher the combinative energy, the lower is the Tc value. Furthermore, our x-​ray analysis and Raman spectroscopic studies have revealed the structure of Cu(2)-​O planes located at the boundary of the two blocks to be quite robust

12.6 Existence of fixed triangle (local mesoscopic structure) by x-​ray diffraction 

363

12.7 The existence of the fixed triangle (local mesoscopic structure) demonstrated by Raman spectroscopy 

365

12.8 Low wave number evidence about mesoscopic structure 

368

12.9 Discussions 

369

12.10 Summary and outlook 

375

References (Chapter-12) 

375

H. Zhang, ‘Mesoscopic Structures and their Effect on High-Tc Superconductivity’ in The Oxford Handbook of Small Superconductors. First Edition. Edited by A.V. Narlikar. © Oxford University Press 2017. Published in 2017 by Oxford University Press. DOI 10.1093/acprof:oso/9780198738169.003.0012

348

348  Mesoscopic Structures and Their Effects in the sense that their bond lengths and angles hardly changed and they form a so-​called “fixed triangle.” This fixed triangle, we believe, may be responsible for the strong anisotropy of the high-​Tc superconductors. In order to explore how important the electron–​phonon coupling is in high-​Tc superconductors, we calculated the phonon frequencies at the gamma point of the YxPr1–​xBa2Cu3O7 system by considering a special local structure in the YxPr1–​xBa2Cu3O7, that is, a “fixed triangle.” It is found that a low-​frequency mode is possibly caused by the “fixed triangle” which roughly matches with the Raman spectrum and suggests the coupling between electrons and phonons to be nontrivial for this system. It is suggested that because of the very strong anisotropy, not only the energy, but also the anisotropy of phonons should be taken into account. Under this strong anisotropy, the state of the carriers may be changed or polarized which determines the superconductivity. The strong anisotropy of phonons may be the main difference between the conventional superconductors and the high-​Tc superconductors. The results show that the lattice dynamics or electron–​phonon interaction are still of great importance for high-​Tc phenomena.

12.1 Introduction and motivation With the exception of recently discovered (Drozdov et  al. 2015)  pressurized hydrogen sulphide with Tc exceeding 200 K, the cuprates are the only superconducting materials that have transition temperatures above the boiling point of liquid nitrogen, with a maximum Tc of 162 K under pressure. Their structure is layered, with one to several CuO2 planes, and upon hole doping, their transition temperature follows a dome-​shaped curve with a maximum of Tc. In the underdoped regime, there exists a pseudogap, and there are two gaps with the same Tc. In the superconducting state, Cooper pairs are formed that exhibit coherence length on the order of Cu–​O distance within the CuO2 plane and about one order of magnitude smaller along the perpendicular direction (ξab ~1.5 nm, ξc ~ 0.3 nm). Because the high-​Tc superconductors demonstrate strong two-​dimensional (2D) characteristics, the coupling along the c-​direction in the crystals is of great importance, and this aspect has been studied extensively by Kleiner et al. (1992). In most of these studies, the high-​Tc superconductors are regarded as a multilayer of superconducting and non-​superconducting layers, and the coupling between the superconducting layers is considered. However, the role of the different layers in superconductivity is not very clear. As is well known, there are many layers along the c-​direction in the high-​Tc superconductors. Instead of considering the individual coupling between them, in the present study, for simplicity, we first distributed them into a few blocks based on their structural features and available experimental facts and examined the interaction between such blocks. A lot of work has been done towards understanding high-​temperature superconductivity from different perspectives. One such important perspective is possibly through the cohesive energy. Billesbach and Hardy (1989) calculated the

  349

Introduction and motivation 349 lattice instability using a rigid-​ion model. Torrance and Metzger (1989) studied the effect of Madelung energy on the hole conductivity in the high-​Tc superconductors. Muroi and Street (1995) calculated the cohesive energy as a function of different Cu-​O planes, and found the cohesive energy had some correlation with the hole concentration in the Cu-​O planes. Zhang et al. (2000) used the cohesive energy to successfully explain the change of Tc of Y-​doped superconductors. Ohta and Maekawa (1990) studied the Madelung energy of a PbBi-​based superconductor, and found some relations between the energy and carrier concentration. Mizuno et  al. (1997) described some relationship between the Madelung energy and carriers in a La-​based superconductor. Mueller (1991) emphasized the likely importance of Madelung energy in high-​Tc superconductivity. Despite these extensive studies, any quantitative relationship between the cohesive energy and superconductivity has, to our knowledge, not been proposed to date. Most of the high-​Tc superconductors are structurally composed of two different blocks, the perovskite and the rock salt (Park and Snyder 1995) (see Fig. 12.1), as local mesoscopic structures which are important for superconductivity. There are two ways to stack perovskite unit cells. One way is to put a second cell on top of the first. In this way, two unit cells share a single

(a)

(b)

Rock Salt layer

X

(c) Y (a+b)/2

Rock Salt block

Ruddlesden-Popper

Fig.  12.1 Structure of the high-​ Tc superconductors, stacked by rock salt and perovskite blocks along the c-​direction. (a)  and (b)  are two different ways of stacking. (c) Typical perivskite structure.

350

350  Mesoscopic Structures and Their Effects metal–​oxygen plane, as depicted in Fig. 12.1a. The second way is to put the second cell on top of the first without sharing a single plane. The second cell has an a/​2 + b/​2 displacement relative to the first, as shown in Fig.  12.1b. In the whole structure of superconductors, the perovskite unit cell is the active block where the Cu(2)-​O planes are present and the charge carriers are located. The rock salt block corresponds to the charge-​reservoir block, which supplies the carriers to the Cu(2)-​O planes. The absence of either of the two blocks is detrimental to high Tc or even to superconductivity. Based on this fact, we believe that the lattice dynamics is important for the high-​Tc superconductivity, and accordingly we have developed a model to calculate the cohesive energy of different superconducting systems. When treating the cell as two blocks:  perovskite and rock salt, it is found that the combinative energy between the two blocks is closely related with the value of Tc. This result supports our point of view that the interaction between the two mesoscopic blocks is of immediate relevance to superconductivity. Further, as also mentioned earlier, the Cu(2)-​O plane, located in the boundary of the two blocks, forms a local mesoscopic structure in the form of a “fixed triangle,” which is of prime importance to the strong anisotropy of the high-​Tc superconductors. It is suggested that because of the very strong anisotropy, not only the energy, but also the anisotropy of phonons should be taken into account. Under this strong anisotropy, the state of carriers may be changed or polarized to determine the superconductivity.

12.2 Model According to the classical theory of crystals, the cohesive energy En is made up of Madelung energy, repulsive energy of ions, and electron affinity energy:

En = E m + E r + E a (12.1)

which can be derived by the following formulae:

E m = 1 / 2α ∑ ei e j /rij (12.2)



E r = ae − r/ ρ (12.3)



E a = ∑ ∑ εi j (12.4)

Here Em, Er, and Ea represent the Madelung energy, repulsive energy, and electron affinity, respectively. ei, ej are the electric charges of different atoms in the cell, εij is the ith ionization energy of the jth atoms in the cell, r is the distance between the positive and negative ions, and a, ∝ are the coefficients. We discard

  351

Model 351 the electron affinity energy, because once the atom becomes an ion, the ion has a closed outer shell, and the electron affinity energy will not strongly affect other electrons or vacancies anymore. We use the ionic model to simplify this problem. Some authors (Billesbach et  al. 1989, Torrance and Metzger 1989, Muroi and Street 1995, Zhang et al. 1990, Ohta and Maekawa 1990, Mizuno et al. 1997, Mueller 1991, Park and Snyder 1995) have demonstrated that the ionic model can be used to deal with the HTSc. According to Pauling’s rule (Pauling 1987)  it is reasonable to consider that the high-​Tc superconductors are ionic compounds. But, obviously, they have some covalent character. In the Cu(2)-​O plane, Cu3d and O2p orbits hybridize, giving rise to the charge carriers. In order to compensate for the deficiency of the ionic model for the high-​Tc superconductors, we directly put some holes in the Cu(2)-​O plane, the number of which depends on the oxygen deficiency. The whole cell is kept electrically neutral. This method is consistent with the experimental fact that the holes are mainly concentrated on the Cu(2)-​O plane. In this way, the covalence is approximately considered, which makes the calculation more precise and the model more reasonable. To calculate the Madelung energy, we use the standard (Evjen 1932) method. In this way, the distribution of charges in a cell is balanced and the summation is highly convergent. In the calculation of the repulsive energy we use a Bohr approximation. To test the accuracy of this program we calculated several samples and found that the calculated results matched the experimental results very well (Zhang et al. 2000). Besides the calculation of the cohesive energy of the whole cell, the energy of the different parts in a single cell was calculated for the consideration of the interaction of the different blocks mentioned above. To calculate the energy difference between different parts of a single cell (hereafter, in order to differentiate it from the cohesive energy of a whole cell, it is called combinative energy) there are two ways. The first way is to separate out all the Cu-​O planes of the cell and leave aside the rest. This method is demonstrated in Fig. 12.2a. In this way, all the Cu-​O planes are considered equally. All the Cu-​O planes are separated from the cell into an independent plane, leaving out some discrete parts. After calculating, we get the average combinative energy between each Cu-​O plane and the rest. The combinative energy indicates the strength of interaction between each Cu-​O plane and its adjacent planes. Another way is to treat the cell as two different blocks, so called perovskite and rock salt, instead of some independent planes. Fig. 12.2b demonstrates this process. Unlike the first method, in this way the perovskite and rock salt blocks are considered as a “packaged unit.” Then the combinative energy calculated will mainly indicate the interaction between the two blocks in the cell. In fact, we do find an interesting outcome in this way, which does not appear in the first method. The relationship among these parameters is: the total cohesive energy of the cell (En), the cohesive energy of the perovskite block (Epb), the cohesive energy of

352

352  Mesoscopic Structures and Their Effects (b)

(a)

RockSalt remain part

Ion CuO Ion

Ion CuO Ion

Fig. 12.2  (a) The structure is separated into layers; (b)  the structure is divided into two blocks: perovskite and rock salt bocks. Copyrighted by the American Physical Society.

Cu-O Planes

RockSalt

Perovskite

the rock salt block (Erb), and the combinative energy (Ecb) between the two blocks which can be written as

En = E pb + E rb + E cb (12.5)

Bi-​based and Hg-​based superconductors (Zhang et  al. 2000), two Tl-​based systems, namely TlBa2Can–​1CunO2n+3 and Tl2Ba2Can–​1CunO2n+4 (Yang et  al. 2001, Han et al. 2007) with different numbers of Cu-​O planes, Y-​based superconductors (Zhang et al. 2000a, Poulsen et al. 1991, Jorgensen et al. 1990) with different oxygen deficiencies, and La2–xMxCuO4 (M = Ba, Sr) (Wang and Zhang 2003), with different M concentrations, have been studied with this model. The detailed results can be known from the related references. The following provides some brief results.

12.3 Calculating results and discussion With the first method described in the model, that is, separating the structure into layers, we calculated the cohesive and combinative energies of the Bi-​ based superconductors, Bi2Sr2CuOy (2201), Bi2Sr2CaCu2Oy (2212), and Bi2Sr2Ca2Cu3Oy (2223), Hg-​ based superconductors, HgBa2CuOy (1201), HgBa2CaCu2Oy (1212), HgBa2Ca2Cu3Oy (1223), HgBa2Ca3Cu4Oy (1234), and HgBa2Ca4Cu5Oy (1245), and Y-​based superconductors, with different oxygen deficiencies. The calculated results show that there is no obvious correlation among the cohesive energy, combinative energy, the Tc values, and the number of Cu-​O planes. For example, in the Bi-​based system, the combinative energy for

  353

Calculating results and discussion 353 1

2

3

Combinative energy Tc

120

40

80

30

40

Tc (K)

Combinative energy (eV)

50

20

1

2 n (number of Cu-O planes)

3

0

the 2201 phase is 46.83 eV, the 2212 phase 38.28 eV, and the 2223 phase 51.37 eV. We cannot find any regular pattern among these parameters. These results show that separating the structure into different layers is not reasonable. The second method shows a different result from the first one. We recalculated the same parameters as we did using the first model. Fig. 12.3 illustrates the relationship among the combinative energy between the two blocks, the values of Tc, and the number of Cu-​O planes in the Bi-​based system. Clearly, there exists an obvious correlation between the combinative energy and the value of Tc. As the value of Tc reaches the maximum in the three Cu-​O planes, the combinative energy between the two blocks reaches a minimum. The values of Tc and combinative energy demonstrate very good correspondence. In order to confirm our calculation and the correlation among the value of Tc, the combinative energy between the two blocks, and the number of Cu-​O planes, the parameters described above were further calculated for the Hg-​system superconductors by the two different methods. The Hg-​based system has five superconducting phases:  1201, 1212, 1223, 1234, and 1245. The relationship between the value of Tc and the number of Cu-​O planes shows a clear dome-​like behavior. If a correlation such as that in the Bi-​system exists in the Hg-​system, it will be more reliable and important. In the Hg-​based system, as the number of Cu-​O planes goes up, the value of Tc reaches a maximum (~133 K) in the 1223 phase having three Cu-​O planes and then it decreases with further increase of Cu-​O planes. For the 1234 and 1245 phases, which carry more than three Cu-​O planes, their Tc values are less than 133 K. So far, there has not been any satisfactory explanation of this behavior. The following results may provide a possible way to understand it. Fig. 12.4 illustrates the relationship of the combinative energy between the two blocks, the value of Tc, and the number of Cu-​O planes in the Hg-​based system. Clearly, there exists an obvious correlation between the combinative energy and Tc. As the value of Tc rises to its maximum in the three Cu-​O planes, the combinative energy between the two blocks drops to its minimum. The value of Tc and

Fig.  12.3 The combinative energy and the value of Tc in the Bi-​system. There exists a close relationship between the energy and the value of Tc. Copyrighted by the American Physical Society.

354

354  Mesoscopic Structures and Their Effects 140 Tc

Fig.  12.4 The combinative energy and the value of Tc in the Hg-​system, with 5 Cu-​ O planes. Copyrighted by the American Physical Society.

Ece

130

18 120 16 110

Tc (K)

Combinative energy (eV)

20

14 100 12 1

2

3

4

90

5

n (number of Cu-O planes)

combinative energy demonstrate a very good correspondence. For the TlBa2Can–​1 CunO2n+3 series, the highest Tc value appears for n = 4, but for the Tl2Ba2Can–​1 CunO2n+4 series, the maximum is reached for n = 3 (see Fig. 12.5–​12.6). There is no corresponding relationship between Tc and n, but a relationship does exist between Tc and the combinative energy. In the case of YBa2Cu3O7–​δ, we have considered the effect of changing the δ value, from 0.07 to 0.62. Fig. 12.7 demonstrates the combinative energy calculated for ortho I and II (when δ < 0.35, it is ortho I and when δ > 0.35, ortho II), respectively. The difference between the three curves is very small. The results demonstrate a very close relationship between the Tc value and the combinative energy. As the Tc value increases, the combinative energy decreases. At about 60 K, the Tc value shows a plateau, and the combinative energy shows a plateau at about δ ~ 0.45 too (corresponding to Tc  =  60 K). This result further 130

18 En Tc

Fig.  12.5 The combinative energy and the value of Tc in TlBa2Can–​1CunO2n+3 (n  =  1, 2, 3, 4, 5). Copyrighted by the American Physical Society.

120 110

16

100

15

90

14

80 70

13

60

12 11

50 1

2

3

4

Number of Cu-O layers

5

40

Tc (K)

Combinative energy(eV)

17

  355

Calculating results and discussion 355 130

En Tc

20

125 120 115

18

110 105

16

Tc (K)

Combinative energy (eV)

22

100 95

14

90 12

1

2

3

85

4

Number of Cu-O layers

Fig.  12.6  The combinative energy and the value of Tc in Tl2Ba2Can–​1 CunO2n+4 (n = 1, 2, 3, 4). Copyrighted by the American Physical Society.

30 25 20 15

100 Orth II Orth I Orth I to II Tc

90 80 70 60 50

40 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 δ

Tc (K)

Combinative energy (eV)

demonstrates that the interaction between the two blocks is of great importance for superconductivity. Fig.  12.8–​12.9 respectively show the relationship between Tc and the combinative energy in La2–​xSrxCuO4 and La2–​xBaxCuO4 systems. It is easy to see that there is an obvious relationship between the combinative energy and the Tc values versus Sr concentrations in La2–​xSrxCuO4 when there are holes at position 1 (the definition of the positions of the holes is shown in Fig. 12.8). A similar correlation also exists in La2–​xBaxCuO4 as displayed in Fig. 12.9. When Tc gets to its maximum, the combinative energy also reaches its maximum. It indicates that the interaction between the two blocks does contribute to the change of Tc in these systems. The combinative energy begins decreasing with increasing Ba concentration when 0.08 < x < 0.125, 0.17 < x < 0.24, and Sr concentration above x = 0.15 as Tc decreases. The results indicate that the increasing of the interaction between the two blocks does have something to do with the suppression of superconductivity in these systems. To sum up, the method considers the cell as two relatively independent blocks and the combinative energy between those blocks mainly represents the

Fig.  12.7 The variation of Tc and the combinative energy with different oxygen deficiency d. The results are obtained for different phases in YBa2Cu3O7–​d. Copyrighted by the American Physical Society.

356

356  Mesoscopic Structures and Their Effects 45 (Tc) (Combinative energy) 40

–32

3 4 2

–34

5

30 1

–36

25 20 15

–38 0.00

35

Tc (K)

Fig.  12.8 The combinative energy and the value of Tc versus Sr concentration in La2–​xSrxCuO4 with holes at position 1 (0.5, 0.5) in the CuO2 planes. To guide the eyes, the scale of the combinative energy is overturned in Fig. 12.8 and Fig. 12.9, relative to other figures about the relationship between the Tc and the combinative energy. Copyrighted by the American Physical Society.

Combinative energy (eV)

–30

0.05 0.10 0.15 0.20 0.25 0.30 Sr concentration in La2–xSrxCuO4

10 0.35

strength of the interaction between the blocks. It seems that the interaction of these blocks, the perovskite and the rock salt, plays an important role in superconductivity. This result supplies some important clues for understanding the mechanism of high-​Tc superconductivity. How does the interaction between the blocks affect crystalline structure and then the superconductivity? The following result (Sec. 12.6) about the stable Cu(2)-​O plane or “fixed triangle” will give some interesting insights.

12.4 Strain between two blocks and its effect on superconductivity We just discussed the effect of combinative energy on superconductivity. To discuss the interaction between the two blocks, the strain between the blocks

Fig.  12.9 The combinative energy and the value of Tc versus Ba concentration in La2–​xBaxCuO4 with holes at position 1 (0.5, 0.5) in the CuO2 planes. Copyrighted by the American Physical Society.

–19.5

% (Tc) % (Combinative energy)

30

20

–20.0 –20.5

10

–21.0 –21.5 0.00

0 0.05

0.10

0.15

0.20

Ba concentration in La2–xBaxCuO4

0.25

Tc (K)

Combinative energy (eV)

–19.0

  357

Strain between two blocks and its effect on superconductivity 357 would be another approach. Here we try to study the strain between the blocks in relation to superconductivity. Series of samples of Bi2–​xPbxSr2CaCu2O8+y and Bi2–​xPbxSr2Ca2Cu3O10+y with x changing from 0 to 0.8 were studied as examples. The combinative energy between the perovskite block and the rock salt block can be calculated by subtracting the cohesive energy of each structural block from the total cohesive energy of the lattice. According to classical theory of crystals, the cohesive energy E c is made up of the Madelung energy Em, repulsive energy of ions Ep, and electron affinity energy Ei:

E c = E p + E m + E i (12.6)

which can be derived by the following formulae: 2

ZZe 1 ∑ i j (12.7) 2a i , j rij



Em =



E p = b ∑ (1 + i,j



Zi Z j ( ri + rj )/ ρ − rij / ρ + )e e (12.8) ni n j

E i = ∑ ε ij (12.9) i,j

where Zi , Z j are the electric charges of different atoms in the cell, ε ij is the j th ionization energy of the i th atoms in the cell, and ri rj and ni , n j are the radius and the number of outer shell electrons of the i th and the jth ions, respectively. rij is the distance between the positive and negative ions and b is the coefficient. We discard the electron affinity energy because once the atom becomes an ion, the ion has a closed outer shell and the electron affinity energy will not strongly affect other electrons or vacancies any more. We use the ionic model to simplify this problem. Some authors (Zhang et al. 2000, 2000a, Poulsen et al. 1991, Jorgensen et al. 2000) have demonstrated that the ionic model can be used to deal with HTSc. For a stable structure, the cohesive energy should be at a minimum. Accordingly, the following equations are used to determine the constant b:

 ∂E  = 0 (12.10)   ∂l  l = l 0



 ∂2 E   ∂l 2  > 0 (12.11) l = l0

where l represents the structural character.

358

358  Mesoscopic Structures and Their Effects Assume that there is a very small change in the structure, that is, the distance between the structural blocks can be treated as elastic. We calculate the cohesive energy under a series of structural changes. Then the data can be fitted to: Ec = E0 +



1 K ∆ 2 (12.12) 2

and we have the modulus of elasticity:

ε=



Kc (12.13) NA A

where k is the coefficient of elasticity, ∆ the change of the distance between the blocks, C the lattice parameter, and A the cross-​section area of the cell. In order to determine the range in which the displacement between the structural blocks can be treated elastically, we calculated the combinative energy in a rather wide range of displacement: the displacement between the two structural blocks in the range δ = –​0.6–​1.5 Å. The calculated data points are fitted to the following formula:    

E c = −268.926 + 0.0137 ∆ + 184.481∆ 2 − 169.331∆ 3 + 82.547 ∆ 4 − 24.199∆ 5 + 2.846 ∆ 6 + 2.757 ∆ 7 − 0.056 ∆ 8 − 1.733∆ 9

(12.14)

 the change of from which it can be estimated that within the range of ±0.05A, cohesive energy is mainly determined by the second-​order polynomial. Strain within this range can be treated elastically. Fig. 12.10 shows the modulus and Tc in three Bi-​system superconductors (three different Cu-​O plane numbers), which have different numbers of Cu-​O planes, from one to three.

100

120 Tc (K) Modulus

100

95

90

60

85

40 80 20

Fig.  12.10  Tc and modulus changing with the number of Cu-​O planes. Copy­ righted by the American Physical Society.

75 0

1

2

Number of Cu-O planes

3

Modulus

Tc (K)

80

  359

Carrier-compensated system and mesoscopic structures  359 76.2

110

Tc (K) Modulus

76.0 75.8 75.6

106

75.4 104

75.2

Modulus

Tc (K)

108

75.0

102

74.8 100

74.6 0.12

0.14

0.16

0.18

0.20

Oxygen content

Fig. 12.11 shows the modulus, Tc, and oxygen content in Bi-​2223 phase. The correlation among them is very clear and uniform. The highest modulus corresponds to the lowest Tc. The results demonstrate that the strain between the two blocks clearly affects superconductivity. How does the strain affect superconductivity? We believe that it makes the Cu-​O plane more rigid. Because the strain between the two blocks is directional, the atoms at their boundary have constraints in terms of moving or vibrating as atoms in other positions. This will be discussed in Sec. 12.6.

12.5 Carrier-​compensated system and mesoscopic structures According to the electronic phase diagram, the carrier concentration determines the superconductivity. Is the lattice dynamics still important to the superconductivity? We studied a compensated system to demonstrate it. In the system of Y1–​xCaxBa2–​yLayCu3Oz, partial Y3+ is replaced by Ca2+, and partial Ba2+ by La3+ (see Fig. 12.12). In this way, the carrier concentration in the system is not changed when x = y. If the Tc is changed, it is attributed to the change of the lattice and the interaction between the structural blocks. The YBa2Cu3Oy (YBCO) system has been investigated by many research groups in the manner of individual calcium or lanthanum doping (Ha et al. 2000, Hatanda and Shimizu 1998, Awana and Narlikar 1994, Giri et al. 2005, Mohan et al. 2007, Sedky et al. 1998, Wang et al. 1991, Tokiwa et al. 1998, Westerholt et al. 1989, Mazumder et al. 1988, Ha 1998, Manthiram et al. 1988, Bornemann and Morris 1991, Wu et  al. 1991, Sun et  al. 2004). Generally speaking, cation doping in YBCO depresses Tc. Partial substitution of Ca2+ at the Y3+ site will supply hole carriers, hence the carrier concentration or charge distribution in

Fig.  12.11  Tc and modulus changing with oxygen content in Bi2–​xPbxSr2Ca2 Cu3O10+y. Copyrighted by the American Physical Society.

360

360  Mesoscopic Structures and Their Effects this system will be altered. The Ca substitution also leads to the reduction of total oxygen content, a little change in the effective copper valence, and the depression of Tc. On the contrary, La3+ replacing Ba2+ will provide electronic carriers. The La substitution at the Ba site makes the oxygen content increase, the effective copper valence slightly decrease, and the overall effect is to cause a Tc depression. The samples were synthesized by the solid-​state reaction method. Pure oxides of Y2O3, CaCO3, BaCO3, La2O3, and CuO were mixed according to the stoichiometric formula of Y1–​xCaxBa2–​yLayCu3Oz with both x and y ranging from 0 to 0.5 in intervals of 0.1. The synthesis process is similar to the preparation of a self-​compensating Y1–​xCaxBa2–​yLayCu3Oz (y = x ) system (Sun et al. 2004). The crystalline structures of the samples were characterized by x-​ray diffraction (XRD) using an X’pert MRD diffractometer with Cu Kα radiation. Detailed structural parameters were obtained by the Rietveld refinement method, using the X’pert Plus software. The goodness of fit (GOF = Rwp /​Rexp) of all the samples is less than 1.7, demonstrating that the refinement results are reliable. The Tc of each sample was determined by DC magnetization measurement at a 10 Oe field in the temperature range of 30–​100 K, using the Quantum Design MPMS system. For detailed results see Jin et al. (2009). To show clearly the Tc dependence on the content of Ca or La, respectively, the Tc is plotted as functions of both x and y for the Y1-​xCaxBa2–​yLayCu3Oz system in Fig. 12.12, which represents a three-​dimensional (3D) phase diagram.

91.6

84

100

80

90

89

70

81.3 57.5

T c (K

50

)

40

87.9

77.6

52

60

67

81.9

87

80

69.7 80.6

55

59.7

84.9

51 54

38

45

30 20

84.5

80.9

77.5

88.5

57.8

48 48.5

78.5

10 0 0.0

52.5

80.5

58.4 82.8 0.0

0.1 0.1

0.2

Ca

t ten

0.2

0.3

0.3 0.4

0.4

t

en

nt

co

Fig.  12.12  Three-​dimensional phase diagram, depicting the Tc-​dependence of the Ca and La content simultaneously in the Y1–​xCaxBa2-​yLayCu3Oz system. © IOP Publishing. Reproduced with permission. All rights reserved.

78

85.2

78

0.5

0.5

La

n co

  361

Carrier-compensated system and mesoscopic structures  361 As an interesting phenomenon, the maximum Tc values are evidently along the diagonal of the basal plane, which correspond to the optimally doped samples, that is, the self-​compensated samples. But it is clearly shown that even if the sample is optimally doped or self-​compensated, with the increase of x and y values, the Tc is depressed gradually. The highest transition temperature is about 92 K for pure YBCO, which corresponds to the optimum oxygen content of 6.93 and the average copper valence of +2.28 (Sedky et  al. 1998). If assuming that in the system, the average copper valence and the optimum oxygen content are both the same as those of pure YBCO (in fact, all the samples were fully oxygenated at 450℃ for 24 h in flowing oxygen), the optimum-​doped condition is y = x according to the electric neutrality condition. It is consistent with the experimental result, that is, the self-​compensated Y1–​xCaxBa2–​yLayCu3Oz (y = x ) samples have higher Tc values compared with the samples that are uncompensated (y ≠ x ). In the Y1–​xCaxBa2–​yLayCu3Oz system, Ca2+ (x) replacing Y3+ will provide more hole carriers and depress the Tc in the manner of over-​doping. But La3+ (y) substituting for Ba2+ will supply electronic carriers, which neutralize part of the holes caused by the Ca doping. When y < x and the system remains overdoped, the Tc increases with the increase of La content at first. Once the content of La exceeds the optimum doping value, that is, y > x, the excess electrons will neutralize more hole carriers and the sample becomes underdoped. As a result, the Tc decreases with the further addition of La. In other words, with x staying constant and y increasing, the system Y1–​xCaxBa2–​yLayCu3Oz undergoes a transition from overdoped to optimum-​doped at y = x, and finally to underdoped. Accordingly, the Tc values exhibit a roof shape in the 3D phase diagram, where the ridge represents the Tc of the optimum-​doped sample. From Fig. 12.12, it is clearly shown that in the self-​compensated Y1–​xCaxBa2–​y LayCu3Oz (y = x ) samples, the Tc value is still depressed. It is reasonable to attribute this depression to lattice dynamics. Besides the interaction of the two blocks, this evidence demonstrates that the lattice is still of importance to superconductivity. Furthermore, the relationship between the Tc and the bond lengths of the Cu(2)-​O(4) and the Cu(1)-​O(4) is investigated and illustrated in Fig. 12.13–​ 12.14. It is evident that the dependence of Tc on the Cu(2)-​O(4) bond length depicts a better linearity than on that of the Cu(1)-​O(4). It is quite interesting that the relationship between the Tc and the bond lengths is different for the underdoped (x < y), optimal-​doped (x = y), and overdoped (x > y) samples. In the underdoped region, Tc increases linearly with the lengthening of the Cu(2)-​O(4) bond and the constriction of the Cu(1)-​O(4) bond. This may be attributed to the charge transfer mechanism. The variations of the two bond lengths indicate that the bridging atom O(4) moves far from the Cu(2)-​O plane and accordingly the hole carriers transfer toward the Cu(2)-​O plane. From the result of the electronic phase diagram and in the underdoped region, the larger hole carrier concentration on the Cu(2)-​O plane implies a higher Tc. But after

362

362  Mesoscopic Structures and Their Effects 100 90

Tc (K)

80 70 60 50

Underdoped (x < y) Overdoped (x > y) Optimal-doped (x = y)

40

Fig.  12.13 The correlation between Tc and the bond length of the Cu(2)-​O(4). © IOP Publishing. Reproduced with permission. All rights reserved.

30

2.16

2.20

2.24

2.28

2.32

2.36

Cu(2)-O(4) bond length (angstrom)

the highest Tc value, the behavior is completely the opposite. It seems to show that the carrier concentration could not respond to it. From this fact, although we can consider that the carrier transfer affects the Tc value, on the other hand, we also can think that the change of the bonds represents the change of the lattice. In the compensated region the change of bonds affects the superconductivity although the carrier concentration is kept constant. The structural change may be more basic than the carrier concentration. This hypothesis is being further studied.

90 80

Tc (K)

70 60 50

Underdoped (x < y)

Fig.  12.14 The correlation between Tc and the bond length of the Cu(1)-​O(4). © IOP Publishing. Reproduced with permission. All rights reserved.

Overdoped (x > y)

40

Optimal-doped (x = y) 30

1.80

1.84

1.88

1.92

Cu(1)-O(4) bond length (angstrom)

1.96

  363

Existence of fixed triangle (local mesoscopic structure) by x-ray diffraction  363

12.6 Existence of fixed triangle (local mesoscopic structure) by x-​ray diffraction Calculation of the bond angles and lengths demonstrates that Cu(2)-​ O(2), Cu(2)-​O(3), and O(2)-​O(3) form a stable triangle in the Y1–​xCaxBa2–​yLayCu3Oz system, called a “fixed triangle” and independent of the doping level in the system. The four fixed triangles form a stable Cu(2)-​O plane. Fig. 12.15 shows the results. Similar studies in other superconducting systems also found the same phenomena. Fig. 12.16 shows the results in the YBa2–xLaxCu3–xAlxOz and YBa2-xLaxCu3-xZnxOz systems. As the content of the dopant changes, these bond angles and lengths in the Cu(2)-​O plane remain nearly invariant, but other bond angles and lengths are noticeably changed. For the sake of simplicity only 2.8 2.7

Bond length (Å)

2.6

Cu(2)-O(2) Cu(2)-O(3) O(2)-O(3)

Ca content = 0.5

2.5 2.4 2.3 2.2

O(2) Cu(2) O(3)

2.1 2.0 1.9

0.0

0.1

0.2

0.3

0.4

0.5

La content (a) 2.8 2.7

Bond length (Å)

2.6

La content = 0.1

Cu(2)-O(2) Cu(2)-O(3) O(2)-O(3)

2.5 2.4

O(2)

2.3 2.2

Cu(2)

2.1

O(3)

2.0 1.9

0.0

0.1

0.2

0.3

Ca content (b)

0.4

0.5

Fig.  12.15 The bond lengths between each other in the triangle formed by Cu(2), O(2), and O(3) versus the La content in Y0.5Ca0.5Ba2–​yLayCu3Oz (a) and the Ca content in Y1–​xCaxBa1.9La0.1Cu3Oz (b). © IOP Publishing. Reproduced with permission. All rights reserved.

364

364  Mesoscopic Structures and Their Effects (a) Bond lengths (angstrom)

2.8 2.7

YBa2–xLaxCu3–xAIxOz

2.6 2.5 2.4

Cu(2)–O(2) Cu(2)–O(3) O(2)–O(3) Cu(1)–O(4) Cu(2)–O(4)

2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 0.00

0.05

0.10

0.15

0.20

0.25

0.30

Doping content X

Fig.  12.16 Changes of some bond lengths with doping in (a) YBa2-x LaxCu3-xAlxOz and YBa2-xLaxCu3-x ZnxOz systems.

Bond lengths (angstrom)

(b)

2.8 2.7

Cu(2)–O(2) Cu(2)–O(3) O(2)–O(3) Cu(1)–O(4) Cu(2)–O(4)

2.6 2.5 2.4 2.3

YBa2–xLaxCu3–xZnxOz

2.2 2.1 2.0 1.9 1.8 1.7 0.00

0.05

0.10

0.15

0.20

0.25

0.30

Doping content X

the bond lengths are given; the bond angles are not shown. We have studied 10 doped systems of YBCO where the fixed triangle or stable Cu(2)-​O plane exists in them all. It can be concluded that the fixed triangle is a special local mesoscopic structure in these high-​Tc superconductors. Because the Cu(2)-​O plane is just located at the boundary of the two blocks and it has been demonstrated above that the interaction between the perovskite block and rock salt block has a close relationship with the superconductivity, it is reasonable to think that the stable Cu(2)-​O plane results from the interaction of the two structural blocks. We (Du et al. 2000) once pointed out that the interaction between the two blocks resulted in the strain between the two blocks, which followed from the combinative energy between them. The strain or combinative energy affects the superconductivity. How does it affect the superconductivity? We try to suggest it makes the Cu-​O plane steady. Because the strain between the two blocks has direction, the atoms in the boundary of the blocks cannot move or vibrate as can the atoms in other positions. This may be the reason of the formation of the stable Cu(2)-​O planes. This stable Cu(2)-​O plane induces very strong

  365

The existence of the fixed triangle (local mesoscopic structure) demonstrated by Raman spectroscopy  365 anisotropy in the crystal. The phonons produced in the Cu(2)-​O plane are very strongly anisotropic too. In this case, the interaction between carriers and phonons becomes different from that in conventional superconductors, where just the energy and not the momentum of phonons is considered. In fact, the Fermi surface measured by XPS (X-​ray photoelectric spectroscopy) depicts a “kink” in momentum space, showing a strong anisotropy of electron–​phonon interaction. This will be discussed later in more detail.

12.7 The existence of the fixed triangle (local mesoscopic structure) demonstrated by Raman spectroscopy The existence of the fixed triangle is further corroborated by Raman spectroscopy. The Raman spectra of the polycrystalline La-​doped samples were obtained at room temperature with a Horiba-​Jobin Yvon HR800 spectrometer equipped with a microscope and a CCD (charge-​coupled device) detector. The 785  nm laser line was used for Raman excitation. The laser was focused onto a spot approximately 2 μm in diameter. The detailed results were published in Jin et al. (2012). As is pointed out by Cava et al. (1988), the off-​stoichiometry of the oxygen content in the La-​doped YBCO system is (no more than 0.05 with x ≤ 0.4) similar to our doping region except x = 0.5. Therefore, most structural and spectroscopic changes are due to the cation substitution effects. Hence, we did no chemical analysis to get the exact oxygen content values of different samples. Since the Cu(2)-​O planes have already been widely recognized as the key ingredient for HTSc, the variation of the local structure at the Cu(2)-​O planes with doping deserves more attention. Three bond lengths and three bond angles related to the Cu(2)–​O(2)–​O(3) triangle were calculated and are shown in Fig. 12.15, respectively. It’s evident that the “fixed triangle” reported previously by us (Yu et al. 2008, 2009) is confirmed again in this La-​doped YBCO system, where all six parameters constituting the Cu(2)–​O(2)–​O(3) triangle are relatively stable. The Raman spectra of the YBa2–​xLaxCu3Oz system with different La content after background corrections are displayed in Fig. 12.17. As is known, 13 atoms in the Y123 unit cell give rise to a total of 39 phonon modes, 15 of which are Raman-​ active with symmetries of 5Ag+ 5B2g+ 5B3g (orthorhombic notation) (Ambrosch-​ Draxl et al. 2002, Feile 1989). The spectra depict four Ag modes corresponding to the c-​direction vibrations of Ba, Cu(2), O(2)-​O(3) (out-​of-​phase), and O(4), visible around 115 cm–​1, 150 cm–​1, 337 cm–​1, and 505 cm–​1, respectively. However, the c-​direction O(2) + O(3) (in-​phase) mode around 440 cm–​1 is too weak to be identified. Furthermore, several modes around 190  cm–​1, 210 cm–​1, 300 cm–​1, 390 cm–​1, 534 cm–1, and 575 cm–​1 are discernible in the background-​corrected

366

366  Mesoscopic Structures and Their Effects In-plane bond-stretching mode

Intensity (arb. unit)

Fig.  12.17  The background-​corrected Raman spectra of polycrystalline YBa2–​x LaxCu3Oz with x  =  0.1, 0.2, 0.3, 0.4, and 0.5 at room temperature. The vertical dashed and dotted lines mark the O(2)–​O(3) out-​of-​phase bond-​buckling modes and the in-​ plane Cu(2)–​ O(2) bond-​ stretching modes, respectively. © IOP Publishing. Reproduced with permission. All rights reserved.

Out-of-phase bond-buckling mode

x = 0.5 x = 0.4 x = 0.3 x = 0.2 x = 0.1

0

100

200

300

400

500

Raman shift

(cm–1)

600

700

800

spectra. The 210 cm–​1 and 300 cm–​1 modes are identified as the B2g (along the a-​axis) vibration and B3g (along the b-​axis) vibration of the apical O(4) atom, respectively (Strach et al. 1995, McCarty et al. 1990). The 534 cm–​1 mode is very weak and sharp, supporting its origin as the in-​plane Cu(2)–​O(2) bond-​stretching B2g mode (Liu et al. 1988). In addition, at high doping levels, a mode below 100  cm–​1 evolves, which could not be fully explained by Fano asymmetry. We speculate that it is caused by the vibration of La impurity since this mode lies near the Ag mode of Ba and its intensity increases dramatically with the La content. The 390  cm–​1 and 575  cm–​1 modes are quite broad and their intensities increase dramatically with the addition of La, thus they could be attributed to IR modes which become Raman-​active through the loss of local inversion symmetry associated with the increased doping level (Kourouklis et al. 1987, Galinetto et al. 2001). The weak 190 cm–​1 mode is not usually seen in the Raman spectra of the YBCO system although it was once reported by Burns et al. (1988) and no assignment has been given so far. We note that this frequency is in accord with the c-​direction vibration of the Y atom, which should be IR-​active and of B1u symmetry (Cardona et al. 1987). Thus it is possible that the disorder induced by the cation dopants mentioned above breaks the perfect centrosymmetry about Y which activates this mode as Raman-​active. The profiles of all the peaks could be well fitted to the Lorentzian lineshape. The doping dependence of phonon frequencies of all the intrinsic (not disorder-​induced) modes obtained by fitting is shown in Fig. 12.18. The doping dependence of phonon linewidths of 4 Ag modes and the in-​plane Cu(2)–​O(2) bond-​stretching B2g mode, placed around 534  cm–​1 (the other two in-​plane modes around 210 cm–​1 and 300 cm–​1 are relatively flat and the fittings about their linewidths are not reliable), are plotted in Fig. 12.18. As Fig. 12.17 shows, the Ag phonon modes of Ba, Cu(2), and O(4) all harden evidently with the addition of La, which is naturally caused by the contraction of the c-​axis since the bond length markedly influences the corresponding phonon frequency.

  367

The existence of the fixed triangle (local mesoscopic structure) demonstrated by Raman spectroscopy  367 540

in-plane Cu(2)-O(2) bond-stretching B2g mode

530 520 O(4) Ag mode

510 500 340

c-axis out-of-phase O(2)-O(3) bond-buckling Ag mode

Phonon frequency (cm–1)

330 310 300 290

O(4) B3g mode

280 270 230 220

O(4) B2g mode

210 160 Cu(2) Ag mode

150 120 110

Ba Ag mode 0.1

0.2

0.3

0.4

0.5

La content

In contrast, it is noticeable that the frequencies of the c-​axis O(2)–​O(3) out-​ of-​phase bond-​buckling Ag mode around 335 cm–​1 and the in-​plane Cu(2)–​O(2) bond-​stretching B2g mode around 534 cm–​1 are almost independent of La content. The insensitivity of these two modes related to the CuO2 subunit to the doping level reflects the stiffness of the in-​plane Cu(2)–​O(2) bond and O(2)–​O(3) bond, and thus provides the direct evidence for the stability of the fixed triangle structure, that is, the local mesoscopic structure. In addition, we notice that when the YBCO system is doped by Ca or Co impurities, there were similar observations in Raman spectroscopy that the frequency of the out-​of-​phase bond-​buckling mode around 335  cm–​1 remained

Fig. 12.18 The peak frequencies of 4 Ag (along c-​axis) phonon modes and 3 in-​ plane phonon modes appearing in the Raman spectra of the YBa2–​xLaxCu3Oz system as a function of the content of La. The error bars are denoted by the vertical solid lines.The dashed lines are linear fits to the data.

368

368  Mesoscopic Structures and Their Effects invariant with increased doping (Palles et  al. 1998, Kakihana et  al. 1989). It is generally accepted that Ca substitutes at the Y site, Co at the Cu(1) site, and La at the Ba site in our case. Therefore, the insensitivity of this phonon frequency and the stability of the CuO2 triangle should be universal to the YBCO family, although the local environment around the CuO2 plane might change it. To sum up, our Raman spectroscopic studies on YBCO have confirmed the existence of a “fixed triangle” as a mesoscopic structure in Cu(2)-​O planes. All chemical bonds and angles constituting the O(3)–​Cu(2)–​O(2) triangle are almost invariant with doping. The frequencies of the out-​of-​plane c-​axis O(2)–​O(3) buckling modes around 335 cm-​1 and the in-​plane Cu(2)–​O(2) bond-​stretching modes around 534 cm-​1 are both independent of the doping level, providing the direct evidence for the stability of this fixed triangle. It was revealed previously that just these two phonon modes couple strongly with the antinodal and nodal electronic states, respectively, resulting in an anisotropic electron–​phonon interaction in the cuprates (Devereaux et al. 2004, Johnston et al. 2010). We believe that the stability of the CuO2 triangle calls for more attention for its possible role in the electron–​phonon interaction and in inducing the d-​wave pairing.

12.8 Low wave number evidence about mesoscopic structure In order to explore how important the electron–​phonon coupling is in high-​Tc superconductors, we calculate the phonon frequencies at the gamma point of the YxPr1–​xBa2Cu3O7 system by considering the role of the special mesoscopic structure in the form of the “fixed triangle.” It is found that a low frequency mode is possibly caused by the “fixed triangle” which is of importance in this system. It is supposed that this fixed triangle may be equivalent to a large effective mass and have some collective vibrations. If so, they may possibly show some vibrating modes at very low frequency in the Raman spectra. We tried to infer some insight about this in two ways. One was through the results of Raman spectra and the other through calculations of the phonon spectrum. The Raman spectra are shown in Fig. 12.19. It shows a low-​frequency mode at about 113 cm–​1 for which we presently attempt to propose a different interpretation. The phonon eigenmodes and corresponding eigenfrequencies of YPBCO were calculated by ABINIT software based on the density functional perturbation theory, using pseudopotentials and a planewave basis. Unfortunately, the special mode in the spectrum was not discovered. This result may possibly be because we did not consider the collective vibration of the fixed triangle. We subsequently tried to use a simple method and considered the contribution of the fixed triangle. If we use a harmonic approximation for the interaction between atoms in the unit cell, then it is known that the phonon frequency is inversely proportional to the square root of the effective mass for the atoms

  369

Discussions 369 YPBCO YPCO 148

Intensity (a.u.)

113

50

100

150

200

Raman shift (cm–1)

involved, that is, ω ph = ∝

1 . We note that the frequency of the Cu(2) Ag meff

mode lies near 150  cm–​1 and an atomic group of CuO4 with a much larger effective mass forms due to steady cohesion induced by the fixed triangles. We can estimate the frequency of the c-​axis mode for CuO4 according to the ωCuO 4 mCu( 2 ) equation = = 0.706, which gives a frequency ~106 cm–​1. Although mCuO 4 ωCu( 2 ) it is a little lower than the 113 cm–​1 observed experimentally we feel it would be reasonable to attribute this low-​frequency mode to the c-​axis vibration of the CuO4 atom group. Because the collective vibration does not exist in traditional superconductors and also has previously not been reported in high-​Tc superconductors, this and the electron–​phonon interaction, we feel, call for special attention.

12.9 Discussions 12.9.1 Electronic–​phonon interaction and lattice dynamics In early studies of the high-​Tc superconductivity (HTSc), the role of electron–​ phonon interaction (EPI) in cuprates was questioned by researchers because some important properties of the cuprates could not be explained by conventional BCS (Bardeen–​Cooper–​Schrieffer) theory in which the EPI is the key factor responsible for Cooper pairing (Bardeen et  al. 1957). For instance, the isotope effect in the cuprates is not evident as in conventional superconductors.

Fig.  12.19 The Raman spectra of YPBCO and YBCO. The low frequency mode at 113 cm–​1 may possibly be caused by the fixed triangle.

370

370  Mesoscopic Structures and Their Effects Many theories were proposed to account for the mechanism of HTSc, such as resonating valence bond (RVB) theory (Anderson 1987), strong coupling theory (Zeyher and Zwicknagl 1990), the exciton model (Hayden et al. 2004), the t-​J model (Zhang and Rice 1988), and so on. But it was soon realized that these new theories were inadequate to explain the mechanism of HTSc. At the same time, substantial electron–​phonon interaction became visible in the cuprates. Lanzara et al. (2001) observed for the first time that there was an abrupt change of the electron velocity at 50–​80 meV in different families of the cuprates using angle-​resolved photoemission spectroscopy (ARPES), which was referred to as the “kink” in the electronic dispersion, and the most plausible candidate responsible for such a kink is believed to be a strong interaction between electrons and phonons. Meevasana et al. (2006) interpreted this kink as originating from the coupling between electrons and special phonons with some collective behavior. Subsequently, it was found that the isotope effect in high-​Tc superconductors was, in fact, nontrivial. For example, Khasanov et  al. (2004) directly observed an evident oxygen-​isotope (16O/​18O) effect in the in-​plane penetration depth of YBCO film. Iwasawa et al. (2008) found a distinct oxygen-​isotope shift near the kink in the electronic dispersion of the BSCCO system, which demonstrated the dominant role of the EPI in the cuprates. Recently, a few review articles emphasizing the significance of the EPI in the mechanism of HTSc have been published (Kresin and Wolf 2009, Johnston et al. 2010, 2010a). Of course, a lot of researchers did not agree with the EPI scenario. For instance, Allen (2001) thought the EPI in the cuprates could not corroborate their d-​wave symmetry. Alexandrov (2001, 2008)  and Alexandrov and Bratkovsky (2010) proposed a bipolaronic model to explain how the EPI could explain high-​temperature superconductivity of cuprates. An interesting experimental result, that revealed the velocity of longitudinal ultrasonic waves along the a–​b plane in a Bi-​2212 single crystal (Chang et al. 1993) was larger by a factor of two than along its c-​axis, was cited to indicate the strong anisotropy of phonons in the cuprates. Although this argument could not fully substantiate the EPI in the HTSc, it was suggested that quasi-​2D charge carriers weakly coupled with the anisotropic phonons undergo a quantum phase transition from conventional s-​wave symmetry to unconventional d-​wave symmetry. Accordingly, the anisotropic phonons and resulting anisotropic EPI are possibly responsible for the d-​wave pairing in cuprates. In spite of this, the anisotropy of phonons must arise from the anisotropy of the crystalline structure of HTSc among the conventional oxide superconductors, such as SrTiO3 and BaPbBiO3, which are known to exhibit anisotropy. It is natural to ask why the anisotropy of phonons is indeed so pronounced in HTSc cuprates. The answer to the above question, we believe, lies in the role of mesoscopic structure in the form of the Cu(2)–​O(2)–​O(3) “fixed triangle” in the Cu(2)-​O planes. On the other hand, by calculating the electron–​phonon coupling constant, Devereaux et al. (2004) had concluded that the out-​of-​phase O(2)–​O(3) buckling

  371

Discussions 371 mode and the in-​plane Cu–​O stretching mode respectively couple strongly with the antinodal and nodal electronic states, resulting in the anisotropic EPI and corresponding band renormalizations observed in the ARPES (Philip 2001). Combined with our recent discovery of the suppression of the Tc with doping in different doped-​YBCO systems being closely correlated with the amplitude of fluctuations of the CuO2 fixed triangle (Guo et al. 2013), it is reasonable to speculate that the stability of the Cu–​O2 triangle might be favorable for achieving the most appropriate anisotropic EPI between the electrons and the phonons, especially the modes 335  cm–​1, 534  cm–​1, and 113  cm–​1 associated with this subunit, and thus favorable for keeping the superconductivity and sustaining a high Tc. The stability of the Cu(2)-​O plane should be paid more attention and its role in the electron–​phonon interaction and in inducing the d-​wave pairing might be of considerable significance.

12.9.2 Hints from the block model and boundary superconductors According to the above results, it is believed that the interaction between the perovskite block and the rock salt block is crucial for the high-​Tc superconductivity. Most recently, superconductivity was observed on the interface between insulating oxides (Reyren et  al. 2007)  and the interface between metallic and insulating copper oxides (Gozar et al. 2008). Illustrations of the multilayer superconductivity are shown in Fig. 12.20. Each single layer is not superconducting, but at the boundary of two layers, superconductivity exists. These results clearly show that the interaction between the different crystalline structures can induce the superconductivity, which is strong circumstantial evidence for our block model published in 2000 (Zhang et al. 2000). The interaction of the blocks may result in the formation of the fixed triangle and then a stable Cu(2)-​O plane. This stable plane is responsible for the strong anisotropy of phonons, the actions of which are different from the phonons in conventional superconductors because their anisotropy should be taken into account.

12.9.3 Effect of the local mesoscopic structure of the Cu-​O plane on superconductivity Although it is well known that the coupling between electrons and phonons results in the formation of Cooper pairs in conventional superconductivity, the pertinent mechanism remains elusive for the HTSc. Cohen (1990) proposed a model that was capable of explaining the small oxygen-​isotope effect in YBa2Cu3O7 within the framework of phonon-​mediated electron pairing. Lanzara et al. (2001) observed a dispersion anomaly (“kink”) from an ARPES experiment. They considered it was direct evidence for ubiquitous strong electron–​phonon coupling in HTSc, but some people did not agree with them (Philip 2001). Meevasana et al. (2006) found that the weight of the self-​energy in the overdoped Bi-​2201 system

372

372  Mesoscopic Structures and Their Effects

Normalized Intensity

(a)

LaAIO3

(b)

1.5 nm from interface SrTiO3 substrate

0

Normalized intensity

525

SrTiO3

2 nm

Fig.  12.20 Illustrations of the superconductivity in a multilayer. Each single layer is not superconducting, but at the boundary of two layers, superconductivity exists. (a) High-angle annular dark field image. (b) O-K EELS spectra of SrTiO3. (c) Small changes of the Ti-L2,3 fine structure (courtesy by D.  Mueller and I.  Bozovic) (Reyren et  al. 2007, Gozar et al. 2008).

(c)

1.5 nm from interface SrTiO3 substrate

0 455

465 460 Energy Loss (eV)

470

La2CuO4

La1.55Sr0.45CuO4

10 nm

545

LaSrAIO4

Intensity (a.u.)

2 nm

535 540 530 Energy Loss (eV)

530 540 Energy loss (eV)

shifted to higher energies, which was related to a change in the coupling between electrons and c-​axis phonons. The above results indicate that it is hard to exclude the electron–​phonon coupling from HTSc. But how the coupling works in HTSc and how much it is really relevant in HTSc are still very ambiguous.

  373

Discussions 373 From our experimental result, it is firmly hypothesized that the mystery of the HTSc lies in the particularity of the phonon mode due to the structural specialty of the high-​Tc cuprate, that is, the stable Cu(2)-​O plane formed by four fixed triangles. It is intelligible that the atoms in the vital superconducting Cu(2)-​O plane probably behave collectively since Cu(2), O(2), and O(3) atoms form a “fixed triangle.” Such an atom group with colossal effective mass is sure to represent a special kind of phonon mode different from a single atom. Furthermore, the coupling between electrons in the unit cell and that exceptional phonon might differ from that in conventional superconductors. The vibration of the “fixed triangle” should be much weaker than of individual atoms. The interaction between electrons and it should also be much weaker, and the potential energy in the system should be lower. The fixed triangle is strongly coincident with the theory of C.M. Varma (1997, 2006, Aji and Varma 2007), who believes that there exists a loop current among Cu(2), O(2), and O(3) atoms and this highly determines the HTSc. The location of the loop is exactly at the fixed triangle (Fig. 12.21). Here we make a guess about a small isotopic effect by the fixed triangle. If the atoms in the fixed triangle vibrate together to some extent, like a “heavy atom,” the small isotopic effect could be understood. Replacing copper or oxygen by their isotopes, it just changes a part of the mass of this heavy atom. In this case, the isotopic effect should be smaller than expected. To better elucidate the possible significance of the fixed triangle, the electron–​ phonon interaction term is separated from the Hamiltonian of the system and expressed as: H ep = H ep1 + H ep 2 (12.15)



where H ep is the total electron–​phonon interaction Hamiltonian, H ep1 the interaction Hamiltonian between electrons and the fixed triangle on the Cu(2)-​O plane, where the state of the carrier may be changed and potential energy is slightly lowered, and H ep2 the interaction Hamiltonian between electrons and other phonons in the crystal. The physical process of the interaction of electrons and phonons in the YBCO unit cell is illustrated in Fig. 12.22b. Assuming (a)

0

– 0

+

0

+ Cu

– 0

–

+

0

– 0

+

0

– (b)

0

+ Cu

– 0

– 0

+ Cu

+

O(2)

+ 0

Cu

– 0

Cu(2) O(3)

Fig.  12.21 Similarity in geometry of the loop current among Cu(2), O(2), and O(3) atoms (a) and fixed triangle (b).

374

374  Mesoscopic Structures and Their Effects (b) q’

Fig.  12.22 Possible physical origin of the two-​gap phenomenon associated with the lattice scattering (a). Lattice scattering of electrons in the YBCO unit cell (b). Dark plane represents all the other phonons. © IOP Publishing. Reproduced with permission. All rights reserved. Reprinted by permission from Macmillan Publishers Ltd. Reprinted with permission from the AAAS.

(a)

” – k

Fermi surface

Pseudogap Superconducting gap

Hep1 – Hep2

q

k’

O(2)

Cu(2) O(3) –

k

that the fixed triangle behaves as a phonon mode  with a collective wave vector  a wave vector k is scattered by it to another q , and one electron (hole) with   electronic state denoted by k´, in this process the energy of the system may be lowered and forms a pseudogap (Fig.  12.22a). When it goes through the  Cu(2)-​O plane, it is scattered to the third electron state with a wave vector k´´   by absorbing other phonons (which are all included in a wave vector q´) in the unit cell. In this process, it is likely to induce the formation of Cooper pairs, which may have something to do with the superconducting gap and the HTSc. Here we make the conjecture clearer. It is well known that there are pseudogaps and gaps in the HTSc (Feile 1989), but so far, a last word about these two kinds of gaps has not been made. As illustrated in Fig.  12.22a, the Fermi surface of the electrons is supposed to lower in two stages due to lattice scattering. Firstly, it lowers (as the red curve shows) after the Hamiltonian H ep1 when scattered by the fixed triangle, and the pseudogap is formed. Secondly, it lowers further due to the Hamiltonian H ep2 when scattered by other phonons in the unit cell, and superconductivity occurs. It should be mentioned that in the conjecture (Eq. 12.15), H ep1 could be due to the scattering by other phonons and H ep2 could be due to the fixed triangle phonon. It should be pointed out that it is reasonable that the stable Cu(2)-​O plane may result in the pseudogap. It has been found that in the La–​Sr–​Mn–​Cu–​O system, which is not a superconductor, there exists a pseudogap (Mannella et al. 2005). Where is this pseudogap from? The answer is not clear. Most recently, we found some evidence that this pseudogap may be also from a stable metal–​ oxygen plane (Guo et al. 2015). From that point of view, the final mechanism of the HTSc may possibly be approached. If the conjecture is correct, then the exceptional nature of the collective phonon mode and its coupling with electrons might be represented in various experimental measurements such as electronic quasiparticle dispersion expressed in ARPES, an isotopic effect, the temperature dependence of the resistivity, the Raman spectrum, and so on.

  375

References (Chapter-12) 375

12.10 Summary and outlook The structure of HTSc is unique in having some mesoscopic features. The structures of cuprite superconductors are composed of two structural blocks, perovskite and rock salt, stacked along the c-​direction. The interaction of the two blocks is closely related to high-​Tc superconductivity. To understand this interaction well would be relevant for knowing the mechanism of the high Tc. Furthermore, our x-​ray analysis and Raman spectroscopic studies have revealed the structure of Cu(2)-​O planes located at the boundary of the two blocks to be quite robust in the sense that their bond lengths and angles hardly changed and they form a so-​called “fixed triangle.” This fixed triangle, we believe, may be responsible for the strong anisotropy of the high-​Tc superconductors. In order to explore how important the electron–​phonon coupling is in high-​Tc superconductors, we calculated the phonon frequencies at the gamma point of the YxPr1–​xBa2Cu3O7 system by considering a special local structure in the YxPr1–​xBa2Cu3O7, that is, the “fixed triangle.” It is found that a low-​frequency mode is possibly caused by the “fixed triangle” which roughly matches with the Raman spectrum and suggests the coupling between electrons and phonons to be nontrivial for this system. It is suggested that because of the very strong anisotropy, not only the energy, but also the anisotropy of phonons should be taken into account. Under this strong anisotropy, the state of the carriers may be changed or polarized which determines the superconductivity. To study the mesoscopic structures and their effects on superconductivity is a way to understand the role of the electron–​phonon coupling and then to understand the mechanism of high-​Tc superconductivity.

Acknowledgments The authors are grateful for the support of State Key Laboratory for Mesoscopic Physics at Peking University. During nearly the past 20 years, many graduate students have worked in my lab, and contributed to this work. Here I  sincerely thank them for their works, although some of them have lost contact.

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378  Mesoscopic Structures and Their Effects Poulsen, H.F., Andersen, N.H., Andersen, J.V., Bohr, H. and Mouritsen, O.G. (1991) Nature 349, 594. Reyren, N., Thiel, S., Caviglia, A.D., Fitting Kourkoutis, L., Hammerl, G., Richter, C., Schneider, C.W., Kopp, T., Rüetschi, A.-​S., Jaccard, D., Gabay, M., Muller, D.A., Triscone, J.-​M. and Mannhart, J. (2007) Science 317, 1196. Sedky, A., Gupta, A., Awana, V.P.S. and Narlikar, A.V. (1998) Phys. Rev. B 58, 12495. Strach, T., Ruf, T., Schönherr, E. and Cardona, M. (1995) Phys. Rev. B 51, 16460. Sun, X.F., Rui, X.F., Wang, F., Zhang, L. and Zhang, H. (2004) J. Phys.: Condens. Mat. 16, 2065. Tokiwa, A., Syono,Y., Kikuchi, M., Suzuki, R., Kajitani, T., Kobayashi, N., Sasaki, T., Nakatsu, O. and Muto, Y. (1998) Jpn. J. Appl. Phys. 27, L1009. Torrance, J.B. and Metzger, R.M. (1989) Phys. Rev. B 63, 1515. Varma, C.M. (1997) Phys. Rev. B 55, 14554. Varma, C.M. (2006) Phys. Rev. B 73, 155113. Wang, N.L., Tan, M.C., Wang, J.S., Zhang, S.Y., Jiang, H.B., Sha, J. and Zhang, Q.R. (1991) Supercond. Sci. Technol. 4, S307. Wang, S.X. and Zhang, H. (2003) Phys. Rev. B 68, 012503. Westerholt, K., Wueller, H.J., Bach, H. and Stauche, P. (1989) Phys. Rev. B 39, 11680. Wu, X.S., Wang, F.Z., Liu, J.S., Jiang, S.S. and Gao, J. (1991) Physica C 320, 206. Yang, K., Yu, L.P., Han, S.H. and Zhang, H. (2001) Physica C 357, 305. Yu, J., Zhao, Y. and Zhang, H. (2008) Physica C 468, 1198. Yu, J., Jin, W.T., Zhao, Y. and Zhang, H. (2009) Physica C 469, 967. Zeyher, R. and Zwicknagl, G. (1990) Z. Phys. B 78, 175. Zhang, F.C. and Rice, T.M. (1988) Phys. Rev. B 37, 3759. Zhang, H., Cheng, L.L., Qin, X.C. and Zhao, Y. (2000) Phys. Rev. B 61, 1618. Zhang, H., Cheng, L.L., Qin, X.C. and Zhao, Y. (2000a) Phys. Rev. B 62, 13907. Zhang, H., Zhao, Y., Zhou, X.Y. and Zhang, Q.R. (1990) Phys. Rev. B 42, 2253.

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Magnetic Flux Avalanches in Superconducting Films with Mesoscopic Artificial Patterns M. Motta Departamento de Física, Universidade Federal de São Carlos, 13565-​905, São Carlos, SP, Brazil

A.V. Silhanek

13 13.1 Avalanches in superconductors  380 13.2 Artificial pinning centers in superconducting films 

391

13.3 Effects of the antidot geometry and lattice symmetry in flux avalanches 

399

13.4 Summary and outlook 

406

References (Chapter-13) 

407

Département de Physique, Université de Liège, B-​4000 Sart Tilman, Belgium

W.A. Ortiz Departamento de Física, Universidade Federal de São Carlos, 13565-​905, São Carlos, SP, Brazil

During the past few decades, vortex matter in type-​II superconductors has been one of the main topics of research in superconductivity. One of the most fascinating issues in the study of vortex matter in superconducting films is the occurrence of abrupt flux invasions, called flux avalanches. They present a startling dendritic morphology in thin films (Leiderer et al. 1993, Durán et al. 1995) which was discovered via magneto-​optical imaging (MOI) experiments, a technique allowing for direct observation of the magnetic flux distribution across the sample. When such events are triggered, flux bursts penetrate into the sample at speeds as large as 100 km/​s (Leiderer et al. 1993, Bolz et al. 2003), and the otherwise smooth vortex penetration is disrupted. From the applications point of view, this unpredictable phenomenon is certainly undesirable, since it degrades the current-​carrying capability of the specimen, being therefore likely to cause a direct impact on the stability of a device. It is thus of practical interest to understand the causes for the occurrence of flux avalanches, as well as the main mechanisms governing these self-​propelled events. In some instances, it is crucial to be able to avoid them (Baziljevich et al. 2002, Colauto et al. 2010, 2013), whereas, in some others, it

M. Motta, A.V. Silhanek, W.A. Ortiz, ‘Magnetic Flux Avalanches in Superconducting Films with Mesoscopic Artificial Patterns’ in The Oxford Handbook of Small Superconductors. First Edition. Edited by A.V. Narlikar. © Oxford University Press 2017. Published in 2017 by Oxford University Press. DOI 10.1093/acprof:oso/9780198738169.003.0013

380

380  Flux Avalanches in Superconducting Films might be desirable to tune—​if possible—​certain environmental parameters, so as to control their initiation and propagation throughout the film. A very common strategy employed to enhance the critical current density (J c) consists in inserting structures of pinning centers (PCs), with sizes ranging from nano-​to micrometers. Besides promoting an increase of the capacity of carrying currents without losses, artificially lithographed structures in superconducting thin films have also revealed new features of the interaction between vortices and PCs. For instance, superconductors decorated with a regular arrangement of artificial pinning centers exhibit commensurability effects between the periodic pinning landscape and the vortex lattice (Harada et al. 1996), resulting in local maxima in magnetization loops at low fields and high temperatures. These maxima, indicative of an enhanced pinning capability at the corresponding values of the applied field, signify that the arrangement of pinning wells has peaks of efficiency when the “vortices per pinning center” ratio approaches some special numbers. On the other hand, an experimental fact, known for a long time, is that flux avalanches only occur in films having large enough critical current densities (Gheorghe et al. 2008), so that samples with larger critical current densities are expected to exhibit avalanches. Therefore, the insertion of arrays of pinning centers not only enhances J c but also promotes avalanches, in the sense that the instability region in the field–​temperature (HT ) diagram occupies a larger area as compared to the equivalent for a pristine film. This chapter deals with the occurrence and the morphology of flux avalanches in superconducting thin films with arrays of mesoscopic through holes (antidots, ADs). The influence of different AD geometries, as well as of the lattice symmetry, on the guidance and consequently the branching of flux avalanches, will be discussed. The chapter is organized as follows. Sec. 13.1 gives a brief introduction to magnetic flux avalanches in superconductors, with special attention to the thermally driven avalanches. The effects of inserting artificial pinning centers in superconducting films on the vortex dynamics, including their features in flux avalanches are treated in Sec. 13.2. Sec. 13.3 offers a rich scenario of avalanche morphologies provoked by their interactions with different AD geometries and lattice symmetries caused by current crowding effects. Finally, a summary is presented in Sec. 13.4.

13.1 Avalanches in superconductors 13.1.1 Vortex avalanches When a hard type-​II superconductor is submitted to an external magnetic field, flux penetrates into the sample in a smooth way, establishing a gradual distribution with the maximum on the edge of the superconductor. The so-​called critical

  381

Avalanches in superconductors 381 state models, as firstly presented by Charles Bean in the early 1960s (Bean 1962, 1964), constitute a useful framework to explain the existence of a large hysteresis in the magnetic properties of superconducting materials, as a consequence of the existence of inhomogeneities acting as pinning centers. These models are capable of providing not only qualitative but, in many instances, quantitative descriptions of the critical current of the specimen, which is the central property for applications. From a macroscopic point of view, currents are induced in the region of the sample where the flux penetrates, to counteract the change in the internal magnetic field, obeying Maxwell’s equations. This is true for both superconducting bulk samples and thin films, though for the latter, in the perpendicular geometry, currents flow not only in the flux-​penetrated periphery, but rather throughout the whole specimen, due to strong demagnetization effects (see sections 8.4–​8.6 in Brandt 1995, Baziljevich et al. 1996). In the flux-​penetrated region, the macroscopic current corresponds to the maximum current which the superconductor can carry without any loss—​the critical current—​which justifies the term “critical state.” Although the critical state model is successful in determining J c from magnetic measurements, it is unable to provide a detailed description of the vortex distribution or any indication about the dynamics inside the type-​II superconductors. Microscopically, the critical state models can be described considering that quantum units of flux, so-​called vortices, nucleate at the edges and penetrate the sample. Since the vortices are captured by PCs, the pinning force acts—​ up to its maximum capability, ultimately determined by the nature of the pinning center—​against the Lorentz-​like force (Chen et al. 1998) generated by the shielding currents. Disregarding other forces, the balance between those two leads to a metastable state:  while trying to keep its center free from vortices, the superconducting specimen employs all means—​namely, its highest possible lossless current—​to avoid flux penetration, resisting the external magnetic pressure. However, once penetrated, a vortex is pushed toward the center, being possibly anchored by the nearest available PC. Following this reasoning one can easily conclude that the vortex density decreases from the edge to the center of the sample. In order to build up a picture of the magnetic dynamics of the Bean model, de Gennes (1966) proposed an analogy between critical state models and a sand pile. When sand is added to the tip of a sand pile, the angle between the horizontal plane and the cone generatrix will increase until its critical value—​namely, the angle of repose (see Altshuler and Johansen 2004 and references therein). If more grains are added to the pile tip, a readjustment occurs (i.e. an avalanche), so that the angle is maintained at its critical value. In superconductors, small adjustments of the current to maintain it at the critical value lead to vortex movement when the applied magnetic field is ramped slowly. As in sand piles, or even in snowslides, vortex avalanches with small amounts of flux lines occur in order to maintain the critical state in force.

382

382  Flux Avalanches in Superconducting Films Nb thin film

–

=

0.7 mm T = 8 K and H = 6 Oe

T = 8 K and H = 5 Oe

Flux motion

Fig. 13.1  Direct visualization of the flux distribution as described by the critical state models using MOI technique in a 140  nm thick Nb thin film. The left and center images were taken after a zero field cooling (ZFC) procedure at 8 K and applying a perpendicular field of 6 Oe and 5 Oe, respectively. Brighter regions in left and center panels indicate areas where magnetic flux is stronger. Right panel:  differential MO image obtained by subtracting the central image from the left one; although a non-​zero difference is seen throughout the whole sample, the highest intensity, close to the center, indicates that the highest difference occurs at the flux front.

Fig. 13.1 presents the magnetic flux landscape recorded by means of MOI in a Nb thin film with area 2 × 2 mm2 and thickness of 140 nm. The left and center panels show magneto-​optical (MO) images after cooling the film down to 8 K in zero field—​below its critical temperature (Tc ) of 9 K—​and then applying a perpendicular field of 5 Oe (center) and, subsequently, 6 Oe (left). Regions where magnetic flux is absent appear as dark, whereas the maximum local field corresponds to the brightest intensity. Both images show flux distributions as predicted by critical state models for superconducting films (Jooss et al. 2002): due to the large demagnetization factor, the intensity is the highest near the border of the sample, decreasing toward its center. As expected, the flux front penetrates deeper for larger applied fields. Also seen on the images are fanlike penetration patterns, caused by local defects at the right and bottom edges (Schuster et al. 1994, Vestgården et al. 2007, Brisbois et al. 2016). On the right panel, the differential MO (Soibel et al. 2000), created by subtracting the central image from the left one, evidences the overall effect of the small vortex avalanches, that occur in response to the field increase, a mechanism through which the flux distribution is self-​adjusted in order to maintain the critical state regime. Since the local temperature of the region where such small avalanches occur is essentially constant, we refer to them as isothermal avalanches. The most prominent feature of the image is the brighter rim near the flux front, indicating that flux was reaching regions which thus far had been free from vortices. As a matter of fact, in length scales comparable to the vortex size, the flux penetration process develops in a discontinuous way, through small bursts

  383

Avalanches in superconductors 383 of vortices invading the sample toward its center. In order to register a map of those randomly distributed isothermal avalanches, one would require an experimental setup with spatial resolution capable of resolving individual vortices (Bending et al. 2004), which is rarely the case in existing facilities throughout the world. In reality, Fig. 13.1 represents the result of a large number of such isothermal avalanches which occurred throughout the film as the applied magnetic field was increased from 5 Oe to 6 Oe. Besides these isothermal avalanches occurring to maintain the critical state, there is a second type of avalanches in superconductors which are the subject of this chapter. These avalanches occur suddenly during nearly adiabatic thermal runaways and, for this reason, are called thermally driven avalanches (Altshuler and Johansen 2004) or, very frequently, simply avalanches. They destroy locally the critical state and sometimes may even be harmful for the maintenance of the superconducting state.

13.1.2 Flux jumps and flux avalanches Since the early 1960s huge jumps have been recorded in the magnetization curves of superconducting bulk samples. Kim and co-​workers (1963) have named them flux jumps, highlighting that such odd events can destroy partially or even completely the superconducting state when the magnetic field is ramped. The very first observation of these abrupt penetrations was made seven years earlier by Schawlow (1956), when he was studying the surface energy of the intermediate state of several pure metals with different geometries using the Bitter decoration technique* (Bitter 1931, 1932). In 1967, Wertheimer and Gilchrist (1967) reported images of flux jumps in Nb and NbZr disks using MOI. Upon increase of the applied field, the occurrence of non-​standard flux jumps in NbZr disks was reported. Bubbles were observed in the helium bath, indicating that a significant amount of heat was released during the abrupt penetration of magnetic flux. Similar penetrations also appear on imaging experiments of thin films, which are manifestations of flux invasions in the form of dendritic morphology† and will be discussed later in this chapter. More recently, J.I. Vestgården and co-​authors (2011) have found by numerical simulations that the temperature during the evolution of an avalanche can become momentarily higher than Tc along the paths of the flux invasion in a thin film. This seems to be in agreement with early observations of Durán et al. (1995) who showed a disordered vortex lattice similar to that obtained in a vortex liquid. Thus, to be on the safe side—​considering that vortices do not exist in the normal state—​one should replace the term “vortex avalanche” by “flux avalanche” when referring to thermally driven avalanches in thin films. Fig. 13.2 shows flux avalanches for a Nb thin film with thickness of 140  nm, as captured by MOI. At low temperatures and low fields, the flux jumps are generally fingerlike with

*  The Bitter decoration technique consists of spreading out a ferromagnetic or a superconducting powder on the specimen in the state (temperature and magnetic field) that one wants to observe. After that, the images of the powder distribution are generally taken by means of a scanning electron microscope or optical microscope. † The term “dendritic” comes from the Greek word for tree (“déndron”) and means having a branching structure similar to a tree or having dendrites.

384

384  Flux Avalanches in Superconducting Films (a)

(b) Nb film

0.7 mm

T=3K H = 30 Oe

T=4K H = 47.5 Oe

Fig.  13.2  Flux avalanches observed by MOI in a 140  nm thick Nb thin film. (a) Image of fingerlike dendrites taken at T  = 3 K and H  = 30 Oe. (b) At a higher temperature and applied field, T  = 4 K and H  = 47.5 Oe, a huge multi-​branched avalanche takes place, after fingerlike dendrites triggered at lower fields. The tooth-​ saw-​like magnetic domain walls are artifacts due to in-​plane magnetization domains of the MO layer.

few ramifications, as illustrated in Fig. 13.2a. Fig. 13.2b presents an image for which the field and the temperature are higher; on the right edge, one can see a huge and multi-​branched avalanche, a feature which appears typically at higher temperatures and fields. The fingerlike jumps on the other edges were triggered at lower fields. Also noticeable at the borders is the subtle advance of a front of smooth penetration. This hypothesis of thermal origin for the flux jumps had already been addressed by Kim et  al. (1963). Five years later, a detailed treatment of the flux jump problem was carried out by Swartz and Bean (1968) and, later on, by Mints and Rakhmanov (1981) and by Wipf (1991), all of them considering the stability conditions in type-​II superconductors. The mechanism for the occurrence of flux jumps is based on the fact that the heat generated by the viscous movement of vortices in a small region of the superconductor can result in a thermomagnetic instability (TMI) extending to much larger regions of the superconductor, which could even transit to its normal state, depending on the relative importance between diffusivities of the magnetic flux (Dm ) and heat (Dt ) (Mints and Rakhmanov 1981). A dimensionless parameter τ can be defined as:

τ=

Dt µ 0κ 0σ = Dm c

(13.1)

where κ 0 is the thermal conductivity, σ is the electrical conductivity, and c is the specific heat of the bulk superconductor. When τ  1, the heat diffusion is faster than the magnetic flux diffusion which means that the specimen has enough time to assimilate the heat generated by the flux motion. Then, the superconductor remains stable and a smooth penetration, as described by the

  385

Avalanches in superconductors 385 critical state model, takes place. Microscopically speaking, dynamically driven isothermal avalanches build the gradient flux profile as in the critical state-​like regime. However, if τ  1, that means, when the magnetic flux diffusion is faster than the thermal diffusion, the sample has no time to assimilate the heat generated by the flux motion, like in a nearly adiabatic process. As a consequence, thermomagnetic instabilities can trigger a flux jump due to a positive feedback loop, as illustrated in Fig. 13.3. Then, the Joule heating increases the local temperature T of the sample by a small amount δT , resulting in a reduction of the critical current density by δ J c , since J c (T ) is a decreasing function of the temperature. This, in turn, reduces the pinning force locally by δ F p as well. More flux comes into that region due to the decrease of the screening currents and the Faraday law assures that an electric field E is induced locally. Thus, more heat is generated, leading to an additional temperature increase and a further reduction of the critical current density, and so on. Depending on the specific circumstances—​sample parameters and external conditions—​ this process can be nearly adiabatic, leading to a local flux jump, or even to an extended jump, embracing the entire volume of the sample. MOI opened up an opportunity to investigate flux avalanches, including the morphology and dynamics of such catastrophic events. Whereas bulk superconductors have limited morphologies for the flux jumps, thin films present unusual and impressive dendritic morphologies. The occurrence of flux avalanches seems to be a quite general feature of superconductors in the thin film geometry, and has been observed for several materials: YBCO (Leiderer et al. 1993, Baziljevich et  al. 2014), Nb (Durán et  al. 1995), MgB2 (Johansen et  al. 2001, 2002), Pb (Menghini et al. 2005), Nb3Sn (Rudnev et al. 2003), YNi2B2C (Wimbush et al. 2004), NbN (Rudnev et  al. 2005), and more recently amorphous Mo79Ge21 (a-​MoGe) (Motta et  al. 2013)  and a-​MoSi (Colauto et  al. 2015). In the next

When Dm >> Dt :

T’ = T + δT

Jc’ = Jc – δJc Fp’ = Fp – δFp

Thermomagnetic instabilities (TMI) can lead to abrupt penetrations

Positive feedback

Q α E.Jc’

adiabatic process B’ = B + δB

Trigger: local flux movement

E = v x B’

Q: heat T: temperature Fp: pinning force Jc: critical current density B: magnetic induction E: electric field v: flux velocity

Fig.  13.3  Schematic diagram for the occurrence of flux jumps. When Dt  Dm , a smooth penetration occurs, whereas for Dm  Dt sudden penetrations take place as a result of thermomagnetic instabilities.Vortex motion generates slowly spreading heat, as in a nearly adiabatic process with a positive feedback, which can be evoked as the main cause for instabilities leading to flux avalanches (after Wipf 1991).

386

386  Flux Avalanches in Superconducting Films section, MOI, a technique that allows one to explore different regimes of flux penetration, is discussed.

13.1.3 Magneto-​optical imaging (MOI)

‡ The

general stoichiometric formula of garnets is X3Y2(ZO4)3.

As already cited above, MOI is a quite important technique allowing one to obtain an ample and direct view of the macroscopic flux distribution in superconductors. An equally relevant feature of MOI is the possibility to observe, in real time, events consisting of relocation of magnetic flux. This technique is based on the magneto-​optical Faraday effect, in which the polarization plane of a linearly polarized beam of light is rotated when passing through a Faraday-​ active material under a magnetic field applied parallel to the incident beam (Jooss et al. 2002). Since the superconducting materials do not present a significant Faraday effect, Faraday-​active crystals have been used to visualize the flux distribution in superconducting materials. The most currently employed indicators are ferrimagnetic films of complex structure—​so-​called garnet‡—​exhibiting in-​plane spontaneous magnetization. These indicators show a large Faraday effect in a broad temperature range—​from cryogenic up to 400 K—​with field response nearly temperature-​independent below 150 K (Jooss et  al. 2002). Since the 1980s, the most used MOI sensor has been the bismuth-​substituted yttrium iron garnets (Bi:YIG) with a thickness between 2 and 5 µm (in our case 5 µm), grown by liquid phase epitaxy on gadolinium gallium garnet (Gd3Ga5O12, GGG) substrates with [100] growing direction. The in-​plane spontaneous magnetization (M s ) exhibited by these indicators tilts out of the plane by an angle φ due to the perpendicular component of the magnetic field which might, for instance, be due to the stray field of a nearby superconductor. The Faraday rotation is proportional to the out-​of-​plane component (Mz ). A schematic representation of the indicator film is shown in Fig. 13.4a as well as the stack of layers and their thicknesses for the Bi:YIG films employed in the experiments presented here. A MO setup is essentially composed by a polarized light microscope, a cryostat with an optical window, electromagnetic coils to apply a magnetic field, and a camera to record the images. The superconducting thin film is carefully positioned on the cold finger with a tiny amount of vacuum grease to assure a good thermal contact between them. The indicator film is directly placed on top of the superconducting film to reveal the magnetic flux distribution. A scheme of the experimental assembly is illustrated in Fig. 13.4b. In our case, non-​magnetic nuts are added around the assembly to give mechanical support and to avoid any undesired movement of the indicator film. Once the sample-​plus-​indicator assembly is mounted on the cold finger, the whole set is placed into the cryostat. As illustrated in Fig.  13.4c, a light source emits a visible light beam of high intensity, which passes through a polarization filter (polarizer). Then, the

  387

Avalanches in superconductors 387 (a)

(c)

φ GGG

0.5 mm 5 µm

Bi:YIG

Mz Bi:YIG

Image

Ms

Analyzer

H

AI mirror 100 nm

Ms: In-plane spontaneous magnetization H: Applied magnetic field

Polarizer

2θF

2θF

Light source

(b)

Beam splitter

Cold finger

Vacuum grease

Coils

Superconductor

Indicator film

Indicator film Non-magnetic nuts

Superconducting thin film

Magnetic field

Fig. 13.4  (a) Schematic representation of the indicator and the MOL with spontaneous magnetization in plane. A component out-​of-​plane (Mz ) arises under the influence of the perpendicular magnetic field. Scheme (b) sketches the grease-​superconducting thin film-​indicator assembly on the cold finger (after Colauto 2008)  and (c)  outlines the MO setup. Panels (a) and (c) adapted with permissions from Johansen et al. (1996) and from Altshuler and Johansen (2004), respectively. Both copyrighted by the American Physical Society.

polarized beam is deflected by a beam splitter, reaching the indicator film. The light passes through the GGG substrate, which has a low light absorption coefficient, and passes through the magneto-​optical layer (MOL), rotating an angle θ F at places where the MOL has a nonzero Mz component. After that, the mirror layer reflects back the light which returns to the MOL, rotating θ F again, in the same sense. Thus, the total rotation angle of the polarization plane is 2θ F , as presented schematically in Fig. 13.4c. Then, the beam passes again through the beam splitter and reaches the analyzer filter. This analyzer is set at an angle close to 90° relative to the polarizer,§ allowing only rotated light to go through. A CCD (coupled charged device) camera captures the intensity of the rotated light, producing a map of light intensities proportional to the rotation angle, forming an image of the flux distribution landscape of the superconductor (Jooss et al. 2002, Altshuler and Johansen 2004). The image at the top of Fig. 13.4c illustrates the initial stage of smooth penetration in a superconducting film, where brighter intensities correspond to more intense local fields. The MOI technique presents a typical magnetic field sensitivity of around 10–​5 T, and a spatial resolution on the order of 0.8 μm. Among other techniques employed to map magnetic flux distributions, it offers the possibility to go from large fields of view, on the order of centimeters, down to single vortex resolution (Goa et  al. 2001). Moreover, the temporal resolution, which reaches 10 ns, is

§  As a matter of fact, the field sense (up/​ down) can be discriminated when the analyzer and polarizer are slightly uncrossed.

388

388  Flux Avalanches in Superconducting Films another relevant characteristic of the MOI technique, allowing, for instance, Bolz and co-​authors (2003) to measure flux avalanche velocities of about 100 km/​s in YBCO thin films. In view of such characteristic features, dynamical processes can be investigated in a wealth of details by means of MOI, since it is possible to obtain images with a resolution of a few microns in a short time, allowing one to observe in real time changes of the magnetic flux distribution in superconducting materials. These features make MOI a very powerful technique through which flux avalanches of different morphologies are observed in superconducting films. Subsequent thorough studies paved the way for improvement of a theoretical model which is not only capable of describing in detail the general features of this fascinating phenomenon, but even predicting some of its distinctive characteristics.

13.1.4 Thermomagnetic model The thermal origin of the instabilities leading to abrupt penetration of magnetic flux into a superconductor is a well-​established idea. Several theoretical studies have been developed in order to map the dynamics of the flux avalanches (Maksimov 1994, Baggio et  al. 2005, Denisov et  al. 2006). The most successful description has been produced via the Thermomagnetic (TM) model, which allows comparisons with experiments with a high degree of correspondence, as will be discussed later. Considering a more general description, which includes a superconducting thin film subject to a magnetic field perpendicular to the film plane, the TM model is written using the Ampère and Faraday laws and the heat diffusion equation:

cT = κ 0∇2T − d −1h0 (T − T0 ) + d −1 j.E (13.2)

where h0 is the coefficient for heat transfer to the substrate held at the temperature T0, d is the thin film thickness, E is the electric field, and j is the sheet current related to the current density J by j = J d . This equation provides the temperature distribution of the specimen and the second term on the right side refers to the thermal link between the superconducting thin film and the substrate. In addition, the nonlinear current–​voltage relation of type-​II superconductors, which is conventionally approximated by a power law, is also considered:



E=

ρ0  j  d  jc 

n −1

j

(13.3)

where j = j , n is the creep exponent, and ρ0 is a resistivity constant. The heat diffusion equation is solved together with Eq. 13.3 and Maxwell’s equations. The time evolution of the magnetic properties inside the sample is obtained by inverting the Biot–​Savart law (Vestgården et al. 2011). In this model, the heat capacity,

  389

Avalanches in superconductors 389 the thermal conductivity, the thickness, the critical current density, and the heat transfer to the substrate strongly influence the occurrence of thermomagnetic instabilities. One of the most remarkable predictions of the TM model is the existence of a threshold flux penetration depth (l * ) above which a thin superconducting stripe with a width of 2w experiences the onset of dendritic flux avalanches, that is, a condition for the instability onset. This critical value l * means that if the flux penetration depth (l ) from the edge of the sample is smaller than l * when magnetic field is applied starting from zero, the sample is stable and a critical state penetration takes place. However, when l ≥ l * , the model establishes that there is enough penetrated flux for the sample to become unstable, and TMIs trigger flux avalanches (Denisov et al. 2006a). As a consequence of the existence of l * , it is possible to identify an onset field to trigger flux avalanches, the so-​called threshold magnetic field (H th ). For a long stripe with width of 2w obeying the Bean model,



H th =

J cd  w  acosh   w − l *  (13.4) π

where w − l * is the position of the flux front measured from the center of the sample. The critical penetration depth l * also depends on J c, which allows one to obtain the stability/​instability limit in an applied magnetic field versus critical current density diagram. Fig. 13.5a is an H × J c diagram showing the threshold line (thick continuous), H th ( J c ), and three isothermal curves (thin), J c ( H ), representing the typical field dependence of the critical current density at different temperatures. At lower values of T (e.g. T1), J c ( H ) crosses the threshold curve twice, defining a lower and an upper limit for the occurrence of avalanches, H1th and H 2th , respectively. As the temperature increases, the difference between these threshold fields decreases, there being a limiting temperature, T th , above which J c ( H ) and H th ( J c ) do not cross each other (T2), so that no avalanches can occur. One can then envisage that the portion of the HT diagram where thermomagnetic instabilities occur consists of an enclosed region, as depicted in Fig. 13.5b: at small enough values of T (e.g. T1), as the field H is increased from zero in an isothermal process, avalanches start to occur when the threshold penetration depth is achieved; however, l * increases with H [see Eq. 1 in Denisov et al. (2006a)] and eventually exceeds the lateral dimensions of the film so that avalanches become unviable in this upper limit, at which l * = w, H th diverges (Yurchenko et al. 2007), and J c = J cth , the threshold value of the critical current, below which no avalanches occur. The upper limiting temperature on the instability region, T th , represents the case for which the curves H th ( J c ) and J c ( H ) (Fig. 13.5a) touch each other in a single point of tangency. In order to shed light on the morphology of flux avalanches, Denisov et  al. (2006) have shown that the dendritic avalanches are more likely to occur in thin films than in bulks as a consequence of the demagnetization effects, requiring

390

390  Flux Avalanches in Superconducting Films (a)

Jc(H)

T1 < T th < T2

(b) T1

th

H2

Jc(H)

T2 T th

H Instability region

T1 H th(Jc)

H1th Jcth

Jc

Hc2 (T)

T th

Instability region

H

Fig.  13.5  Schematic diagrams of the region at which flux avalanches occur. Panel (a): H × J c (after Yurchenko et al. 2007). Panel (b):  H × T (after Colauto et al. 2007).

T2

Hc1 (T) T

treatment using non-​ local electrodynamics. For bulk samples, the flux jump shape depends on the electric field background, there existing a threshold value (E c ) separating uniform (below E c ) from fingering jumps (see above). In the case of thin films, the heat transfer coefficient between the superconducting film and the substrate is also important, in the sense that, above a certain critical value, avalanches appear in dendritic patterns, independent of E . Moreover, temperature plays a crucial role on the morphology of the flux avalanches in films. At lower temperatures, the flux patterns are small, narrow, with few branches, and sometimes occur as in fingerlike shapes. However, at higher temperatures, close to T th , the avalanches are larger and highly branched (Johansen et al. 2002). Recently, Vestgården et  al. (2013, 2015)  have found by numerical simulations that varying some parameters, such as the thickness, the Joule heating, the ­thermal diffusion, and the heat transfer to the substrate, one can also modify the branching and size of the flux avalanches. For a thicker sample, the field to trigger the very first jump is higher, and the jumps become larger and with fewer ramifications than for thin samples. When the Joule heating, related to the positive feedback, is increased the avalanches are larger and more branched. For larger thermal diffusion, the dendritic behavior smears and the avalanches adopt a thick fingerlike morphology. Improving the heat transfer to the substrate provokes a decrease in the avalanche size, with fewer and thinner branches. Nevertheless, an experimental confirmation of these findings is still lacking, since it is very hard to control most of those parameters in a systematic way. Once again, it is worth mentioning that simulations have shown that the temperature at the core of the flux avalanche can be higher than Tc during its evolution through the superconducting material. Besides that, Vestgården et al. (2011) have found values for the velocity of an avalanche which are compatible with those so far measured in real experiments (Leiderer et al. 1993, Bolz et al. 2003), reaching tens of km/​s. Also, the avalanche velocity is not constant, being larger at the initial steps of the propagation process, and decreasing markedly at later stages. Early investigations of non-​dendritic flux jumps in thick Nb samples by Wertheimer and Gilchrist (1967) show more modest average flux front velocities strongly dependent on the thickness of the Nb (for instance, around 50 m/​s for

  391

Artificial pinning centers in superconducting films 391 thickness of 100 µm). The authors also report a fast exponential increase of the flux velocity at the early stages of the avalanche formation, followed by a progressive slowing down as it further evolves toward the center of the sample. The TM model has provided consistent information about the flux avalanche regime and has also been an important aid to the understanding of some experimental data. Since the purpose of this chapter is to describe avalanches in decorated superconducting films, the following section discusses the influence of arrays of artificial pinning centers in the superconducting properties of thin films.

13.2 Artificial pinning centers in superconducting films 13.2.1 Enhancement of the critical current density The pinning force is caused by virtually any kind of inhomogeneity or defect distributed throughout the material. These inhomogeneities, such as impurities, vacancies, stoichiometric defects, grain boundaries, dislocations, voids, thickness variation of a film, etc., suppress the superconducting properties at that point or region. They reduce the condensation energy related to the nucleation of the normal core of the vortex and act as attractive potential wells holding the vortices in place and avoiding their viscous motion, thus minimizing the free energy of the system. Such naturally created pinning centers, which are randomly distributed, cause distortions in the Abrikosov lattice (Abrikosov 1957) due to the landscape of forces acting on the vortex system and destroying the translational long-​range order. This mechanism of pinning, called core pinning and first described by P.W. Anderson (1962), is more effective when the dimensions of the inhomogeneities approximately match with the coherence length ξ of the superconducting material. Besides the intrinsic pinning, artificial pinning centers can be introduced in type-​II superconductors to optimize and trap the vortex lines. This strategy may increase the critical current density and consequently, the potential technological functionality of the material. One way to insert these defects is by means of irradiating the specimen with high-​energy heavy ions (Civale et  al. 1991), creating randomly distributed cylinders—​known as columnar defects—​of non-​ superconducting material of diameter ∼ ξ , having a distribution of pinning strengths.** This approach has been largely used in the study of vortex matter ** Columnar defects running across the full thickness of a sample would be able to fully pin flux lines; on the other hand, blind columns, that is, defects not going through the sample, provide less efficiency in vortex pinning. Thus, a set of columns presenting a range of depths corresponds in practice to a collection of trapping centers with a distribution in the pinning strength (see Niebieskikwiat et al. 2001).

392

392  Flux Avalanches in Superconducting Films

††  The term “antidot” has been taken from

similarly structured thin films of semiconductors (Moshchalkov et al. 1996).

in the high-​temperature superconductors where ξ is typically of the order of a few nanometers, similar to the typical size of defects that can be created by such irradiation schemes (Banerjee et al. 2003). Although the understanding of the pinning strength of individual imperfections for single vortices is important, the collective effect involving many pinning centers and many vortices is essential to immobilize the vortices efficiently (Campbell and Evetts 1972). Bearing this in mind, fabrication of periodic arrays of pinning centers in superconducting films became a goal to be achieved and, indeed, its realization brought to the scene a rich diversity of new vortex phenomena such as commensurate pinning effects (Harada et al. 1996), vortex rectification (see Silhanek et al. 2010 and references therein), manifestation of multiquanta vortices (Metlushko et  al. 1994, Moshchalkov et  al. 1996), and composite flux-​line lattices (Baert et  al. 1995, Silhanek et al. 2005). In fact, recent advances in micro-​and nanofabrication have allowed the preparation of superconducting thin films decorated with arrays of artificial pinning centers. Among the fabrication procedures, lithography techniques, such as electron beam lithography, have made it possible to attain distribution of pinning sites with sizes and separations comparable to the superconducting characteristic length scales, that is, the temperature dependent magnetic penetration depth λ (T ) and coherence length ξ (T ) , ranging from micro-​to nanometers. As a consequence, there is a two-​dimensional (2D) lateral modulation—​due to the presence of the defect array—​which leads to a confinement of the flux lines and, as is well known in quantum mechanics, quantization effects (Moshchalkov and Fritzsche 2011). Besides the size and the distance among the imperfections, micro-​and nanofabrication also permit one to control the distribution, the geometry, as well as the type of the defect. There are basically three different types of defects:  the antidots†† (Baert et  al. 1995, Moshchalkov et  al. 1996), the blind holes (partially drilled holes) (Bezryadin et al. 1996, Raedts et al. 2004), and the magnetic dots (magnetic material either on top of or below the superconductor) (Vélez et al. 2008). Another mechanism allowing one to trap vortices is so-​called electromagnetic pinning which takes place due to the perturbation of screening currents of the vortex near the edge superconductor/​defect—​there they must flow parallel to the hole edge—​as imposed by the boundary conditions. Buzdin and Feinberg (1993) have studied this interaction by the method of images, in which a vortex interacts with an antivortex in the hole. Therefore, the empty AD always acts as an attractive potential for the vortices. Although a multiquanta vortex is energetically unfavorable in continuous media, and even in randomly distributed defects, relatively large artificial pinning centers placed in periodic arrays can favor its appearance. The saturation number ns, that is, the maximum

  393

Artificial pinning centers in superconducting films 393 number of vortices that a cylindrical antidot in a lattice can pin, can be roughly estimated‡‡ by:

ns 

rAD 2ξ (T ) (13.5)

where rAD is the radius of the AD. When 1 < n < nS , n is the number of flux quanta, the flux nφ0 is captured by the defect and creates a potential barrier for other vortices far from the AD. Increasing the magnetic pressure, for instance, vortices close to the hole can overcome this barrier and enter the hole. For n > nS , the additional vortices start to occupy interstitial positions in the superconducting region. The interesting problem of optimum AD size for maximizing the pinning was addressed theoretically by Takezawa and Fukushima (1994) and experimentally by Moshchalkov et al. (1998). The interplay between artificial pinning centers arranged periodically, resulting in periodic pinning forces, and the repulsive interaction between vortices leads to commensurability effects (Daldini et  al. 1974). It means that, when the lattice constant of the pinning center mesh and that of the vortex array are identical, a local maximum in pinning force is reached and, consequently, a local maximum in the critical current density occurs. The magnetic fields at which this matching occurs are called matching fields, being defined as (Moshchalkov and Fritzsche 2011):

Hn = n

φ0 A (13.6)

where A is the unit-​cell area of the lattice of ADs. At the first matching field H1, for instance, the equilibrium configuration is one in which each hole is, on average, occupied by a single flux quantum. Fig. 13.6 shows the matching fields obtained by dc magnetization and ac susceptibility measurements for different samples. Fig. 13.6a shows magnetization versus magnetic field curves for a W0.77Ge0.33 thin film with holes of radius 0.17 μm in a square lattice of 1 μm, prepared by electron beam lithography; the matching fields are multiples of 20.7 Oe. In the same graph one can see the absence of matching fields in the magnetic response of a reference plain film. Fig. 13.6b shows both components of the ac susceptibility for a Pb thin film with square holes of 0.8 μm in a square lattice of 1.5 μm and blind holes with the same characteristics, both with matching fields being multiples of 9.2 Oe. By observing Fig. 13.6a, one can see that the effectiveness of vortex anchorage by the lattice of ADs appears as a substantial enhancement of the critical current at temperatures close to Tc . It is evidenced by a wider and taller loop for the sample decorated with ADs as compared to the plain film. An important variable for an array of PCs is its spatial arrangement. Recently, Misko and Nori (2012)

‡‡ This

equation deduced by Mkrtchyan and Shmidt (1972) was later corrected by Nordborg and Vinokur (2000). The number of vortices trapped by a pinning center can also depend on magnetic field, density of holes, and symmetry of the lattice and the defect geometry. See for instance, Doria and Zebende (2002), Bezryadin et al. (1996), Buzdin (1993), and Kramer et al. (2009).

394

394  Flux Avalanches in Superconducting Films (a)

(b) T = 4.45 K

4

0.0

2

0

–2

–4

T = 7.09 K

0.2

Ac susceptibility (SI)

Magnetization (10–6 emu)

Fig.  13.6  Matching fields measured using different techniques. (a)  M ( H ) curve for a patterned W0.77Ge0.33 and a plain film and (b) in-​phase and out-​of-​ phase components of the ac susceptibility for Pb thin films with arrays of through holes (open symbols) and blind holes (filled symbols) (after Raedts et al. 2004). Panel (a) adapted with permission from Moshchalkov et al. (1996). Copyrighted by the American Physical Society.

WGe films Tc = 4.53 K

Plain Antidots –60 –40 –20

0

20

Magnetic field (Oe)

40

60

–0.2 –0.4 –0.6 –0.8 –1.0

Pb films Tc = 7.22 K

Blind holes Antidots –8

–6

–4

–2

0

2

4

6

8

Magnetic field/first matching field

have predicted that a graded distribution of ADs would increase the pinning capacity of the sample. Such gradual arrangement presents a maximum density on the edges, decreasing toward the center of the sample, mimicking the vortex distribution described by critical state models for a partially penetrated sample started from the virgin state. This has been experimentally confirmed by several studies: Motta et al. (2013), Wang et al. (2013, 2016), and Guénon et al. (2013), and further theoretical efforts have been done by Ray et al. (2013, 2014) and by Reichhardt et al. (2015). In a particular case, Motta et al. (2013) verified that at low fields, a linearly graded array of ADs provides more efficient pinning than a uniform distribution, that is, the critical current density for the sample patterned with a graded distribution is enhanced as compared to a homogeneous distribution of ADs. Concerning the improvement of the critical current density, nanostructured superconducting films have made it possible to attain the depairing current density, J dp. This is the ultimate limit, that is, the highest dissipationless current achievable, being reached when the kinetic energy due to the supercurrents becomes larger than the binding energy of the Cooper pairs. In practice, the critical current is limited by the vortex depinning process in most of the samples and consequently, J c is orders of magnitude smaller than J dp. Conceivably, J c = J dp can be reached in a superconducting strip of width smaller than 4.4ξ (T ) , so that vortices cannot fit inside the sample (Likharev 1979). Xu et al. (2010) claim to have achieved J dp in superconducting nanomesh films with dimensions smaller than both λ and ξ. The authors argue that the supercurrents in such mesh can find alternative paths, avoiding resistive regions caused by thermal fluctuations and thus keeping J dp as the upper current limit for a larger temperature window. By using resistivity measurements, Córdoba et al. (2012) have described a reentrance of the superconducting state for low temperatures and higher fields in thin films decorated with a square lattice of holes; according to the authors, the sample supports applied currents as high as J dp. Arguably, these findings are of

  395

Artificial pinning centers in superconducting films 395 practical importance for technological applications, allowing for development of more efficient devices. To summarize this section, we have reviewed that nanostructured superconducting films have attracted much attention as a test bench for improving some superconducting critical parameters, most specially the current carrying capacity. A variety of new vortex phenomena has arisen due to the introduction of neatly fabricated and well-​controlled arrays of artificial pinning centers in films, enabling one to understand the interaction of the vortex lattice with such arrangements of pinning centers at the microscopic level. The effects of underlying arrays of antidots on the abrupt flux penetration due to thermomagnetic instabilities, as well as the guidance of flux penetration, are discussed in the next section.

13.2.2 Guidance and flux avalanches Apart from the above-​described effects related to interactions among vortices and arrays of PCs, flux avalanches are also influenced by the existence of an underlying landscape of pinning centers. Such interaction is quite strong, resulting in the occurrence of flux avalanches in a larger window of temperatures and magnetic fields and, furthermore, in the existence of guidance of the intriguing ramifications in preferential directions, which is described below. Firstly, the existence of the antidot lattice increases the instability region in the magnetic field versus temperature diagram. Fig.  13.7a shows the region where the avalanches take place for 25 nm thick amorphous Mo79Ge21 films

(a) Magnetic field (Oe)

Tc ~ 6.7 K

300

Smooth penetration

200 100 10 1

Flux avalanches Plain Antidots

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Temperature (K)

Magnetic moment (10–4 emu)

(b) a-Mo79Ge21 films

400

0.0

0.0

–0.2 th

H2

a-Mo79Ge21 films

–0.4

T=2K Plain Antidots

0

100

H1th

–0.2

–0.4 0

200 300 400 Magnetic field (Oe)

10

20

30

500

Fig. 13.7  Occurrence of flux avalanches in a-​MoGe films. (a) HT diagram showing the region dominated by thermomagnetic instabilities for a plain 25 nm thick a-​MoGe and for a sister film decorated with a square array of square ADs. (b) Flux jumps in the magnetic moment against field curves for each of the samples, taken at 2 K, where one can identify the upper threshold limit (H 2th ). The inset shows the lower threshold field (H1th ) needed to trigger the first avalanche. Panel (a)  adapted with permission from Motta et al. (2013). Copyright 2013, AIP Publishing LLC.

396

396  Flux Avalanches in Superconducting Films with and without a square lattice of square ADs having a lattice parameter of 1 μm and AD side of 0.5 μm. It is evident from the figure that the area in the diagram referring to avalanche activity is larger for the patterned film (light gray) as compared to the plain film (dark gray), with upper limiting temperatures of 2.5 K and 4.9 K, respectively. For Pb films with and without ADs, such difference in T th is smaller, around 0.5 K (Menghini et al. 2005), probably due to the fact that the intrinsic pinning is stronger in Pb than in a-​MoGe and hence, the presence of AD arrays impacts the latter more considerably (Motta et al. 2014). In order to obtain the boundary between smooth entrance and abrupt penetration illustrated in Fig. 13.7a, the field-​dependence of the magnetic moment [m ( H )] was measured at different temperatures. Fig. 13.7b shows m ( H ) taken at 2 K for those a-​MoGe samples, where one can see that both curves present a jumpy behavior, which is a fingerprint of flux avalanches when observed using this technique. The boundaries are traced by taking the first jump, that is, H1th, highlighted in the inset, and the last flux jump (H 2th ) for several temperatures (Colauto et al. 2007), as illustrated in the graphs. As already mentioned above, the magnetic field range of such jumps at this temperature is larger for the patterned sample, evidencing that arrays of antidots are inducers of flux avalanches (Hébert et al. 2003, Silhanek et al. 2004, Menghini et al. 2005, Motta et al. 2011, 2013). Another aspect seen in the inset to Fig. 13.7b is the fact that the jump heights are smaller in the decorated film as compared to the equivalent feature for the plain specimen. The experimental fact that the region in the HT diagram corresponding to avalanches occasioned by thermomagnetic instabilities is larger for the patterned film can also be interpreted in the light of the TM model. The plain film has a lower critical current density and its J c ( H ) curve in Fig. 13.5, even at low temperatures, crosses the H th ( J c ) limiting curve relatively close to its vertex. For T > T th , J c ( H ) does not cross the H th ( J c ) line and, consequently, avalanches do not happen anymore. When an array of ADs is inserted in the film, the critical current density is enhanced and then the J c ( H ) curves shift to the right, as also illustrated in Fig. 13.5a. Thus, the temperature needed to bring J c ( H ) outside the instability region has to be higher than for a plain film, resulting in an increase of the region area in the HT diagram where the avalanches occur. Such an increase of the avalanche activity has been also observed when arrays of magnetic dots in superconducting films are magnetized parallel to the applied magnetic field, increasing the pinning force on the vortices and, consequently, leading to higher critical currents (Gheorghe et al. 2008). Moreover, arrays of antidots induce easy directions, which depend on the lattice symmetry and the AD geometry, as demonstrated in the next section. Flux entering the sample experiences guided motion, regardless of the invasion being smooth or abrupt. Also known as vortex channeling, it has been observed by Pannetier et  al. (2003), using MOI in an YBa2Cu3O7-​ δ thin film disk with a square array of circular ADs. Under conditions of smooth penetration, the

  397

Artificial pinning centers in superconducting films 397 authors observed that flux fronts form a macroscopic square, which is nothing but an imprint of the fourfold symmetry of the lattice of ADs. It occurs since the vortices are guided preferentially along the channels parallel to the principal axis of the array, that is, along the rows of ADs, leading to an anisotropic penetration into the disk. Besides that, a square sample patterned with a square array of ADs whose main axes are parallel to the sample edges exhibits channels and a deeper penetration as compared to a sister sample with rows of ADs aligned along the sample diagonal—​that is, rotated 45° from the edges—​for which no channels are seen. Therefore, the vortex channeling reduces the shielding capacity of patterned thin films. Fig. 13.8a illustrates the smooth penetration regime taken at T  = 5 K and H  = 2 Oe for a Nb thin film with thickness of 20 nm decorated with a square array of ADs with lattice parameter of 1.5 μm and AD size of 0.4 μm. Both panels (Fig. 13.8a–​b) are fair examples of vortex channeling along the AD rows perpendicular to the edges of the sample, both in the critical state regime (a) and in the avalanche regime (b). Similar results were obtained by Silhanek et  al. (2003) by using magnetotransport measurements. By controlling the orientation of the current in-​plane, they have concluded that the vortex velocity follows the principal axis of the lattice even when the Lorentz force, induced by the applied current, is misaligned up to 15°. Along the easy direction, the vortex velocity is higher than along the diagonals (hard direction) and hence the mobility shows an angular dependence. Furthermore, the anisotropic penetration disappears gradually at temperatures close to Tc since the pinning efficiency vanishes. In a rectangular array of PCs, guided vortex motion is facilitated along the short side of the array, whereas pinning is stronger along the long side. Therefore, a critical current anisotropy is observed, which leads to a lower limit value along the short side of the lattice, thus resulting in anisotropic flux penetration (Welling et  al. 2004, Vélez et  al. 2008). Interestingly, a similar result, that is, anisotropic

(a)

0.8 mm

(b)

T=5K H = 2 Oe

T=3K H = 2 Oe

Fig. 13.8  MO images of the flux penetration in a 20 nm thick Nb thin film patterned with a square array of ADs with lattice parameter of 1.5 μm and AD size of 0.4 μm. (a)  Channeling in the smooth penetration regime, along the main axes of the array, taken at 5 K and 2 Oe; and (b) flux avalanches along the main axis of the array, taken at 3 K and 2 Oe.

398

398  Flux Avalanches in Superconducting Films critical current density, was found for a square array of rectangular ADs, in which the critical current is higher along the long side of the hole, as compared to the equivalent short side (Gheorghe et al. 2006). Superconducting thin films decorated with arrays of magnetic dots have also presented channeling and an anisotropic critical current density, depending on the magnetic state of the dots as well as on the orientation of the array relative to the edges of the sample. Gheorghe et al. (2008) have studied a variety of Pb films patterned with square arrays of Co/​Pt magnetic dots. Of particular interest here are two decorated samples of square shape, one with rows parallel to the sample borders (Ss) and the other with rows aligned to the diagonals (Sd). A clear contrast in the flux penetration behavior appears when the magnetization of the dots is aligned with the applied field: sample Ss exhibits channeling, whereas sample Sd does not. Such difference occurs due to the fact that, in the latter case, vortex motion is inhibited, since the Lorentz forces are perpendicular to the edges. Besides that, an anisotropic distribution of currents was observed in the flux front penetration of a circular sample. When the dots were demagnetized, there was no influence of the pinning centers and the flux fronts were circular. However, when the dots were fully magnetized, that is, parallel to the applied field, the flux front exhibited a fourfold symmetry. Thus, the lattice symmetry and magnetic state of the underlying array of magnetic dots influence the flux penetration in such patterned superconducting films. In the flux avalanche regime, penetration along preferential directions has been observed by several authors using MOI (Vlasko-​Vlasov et  al. 2000, Menghini et al. 2005, Gheorghe et al. 2006, 2008, Motta et al. 2011, Zadorosny et al. 2013). Vlasko-​Vlasov et  al. (2000) have pointed out that in a Nb film patterned with a lattice of holes, the main trunk and the secondary branches are guided along the AD rows for increasing and decreasing applied fields. Different experiments conducted under the same conditions presented the same overall patterns, that is, dendritic formations along the easy axes of the AD lattice, although randomly distributed across the sample area. Thus, even though guidance results from the presence of the array of PCs, the stochastic nature of the process is maintained, just as in the case of plain films. Analogous results were also obtained in Pb and also Nb samples with different AD sizes by Menghini et  al. (2005), Gheorghe et al. (2006), Motta et al. (2011), and Zadorosny et al. (2013). Fig. 13.8b presents an example of AD-​guided dendrites following the principal axis of the lattice taken at T  = 3 K and H  = 2 Oe for a Nb thin film decorated with a square array of ADs with lattice parameter of 1.5 μm and AD size of 0.4 μm. Menghini et  al. (2005) have described the evolution with applied field and temperature of the dendrite morphology in Pb films decorated with square patterns of ADs. At isothermal conditions and increasing fields, once an avalanche is triggered, its shape is frozen while new avalanches may appear elsewhere. Such a feature has already been discussed in plain films by Johansen et  al. (2001). At low temperatures, the flux avalanche morphology is essentially fingerlike (not branched) and following the direction toward the center of the sample. At higher temperatures, the dendrites become larger and treelike, that is, more branched.

  399

Effects of the antidot geometry and lattice symmetry in flux avalanches  399 At the upper border of the instability regime (see Fig. 13.5b), T ≈ T th , a single, large, treelike avalanche takes place. Therefore, even when penetration occurs in the form of sudden invasions of flux bundles—​which is a far more complex scenario than for a single vortex motion—​the underlying pinning landscape also plays a non-​trivial role in the morphology of the penetrated flux distribution. Although the guidance of flux avalanches was reported more than a decade ago, the essential ingredient to understand the resulting peculiar morphology, which accounts for the subjacent AD distribution, has not been discussed in depth. This ingredient, the current crowding effect, is presented in more detail in the next section.

13.3 Effects of the antidot geometry and lattice symmetry in flux avalanches A typical event in nature exhibiting dendritic morphology is an atmospheric electric discharge, which leads the system cloud–​air–​ground to decrease its electrostatic energy. In a sense, flux avalanches in superconducting thin films belong to a similar type of complex systems and can be interpreted as its magnetic equivalent, that is, a flux avalanche corresponds to a magnetic discharge. This section compiles a comprehensive set of experiments involving magnetic avalanches in sharply patterned films. It is worth noting that the implications of the experimental findings reported in this chapter could reach domains beyond flux avalanches in superconductors, such as electric discharges in controlled non-​uniform media (Kayaba and Kikkawa 2008). Even though introducing arrays of ADs is beneficial to increase the critical current density, it is detrimental for applications of superconducting thin films at low temperatures and low fields, since it favors the occurrence of stronger and more abundant flux avalanches. This is a problem of fundamental impact on the designing of devices using superconducting films. It is thus highly desirable to be able to detect and understand magnetic flux avalanches in superconducting films decorated with different arrays of antidots which might, in turn, be influential for the triggering and guidance of the abrupt magnetic events. Of particular ­interest is the role of lattice symmetry and AD geometry on the morphology of the avalanches. Let us start by discussing the intriguing format assumed by the branches of avalanches occurring in patterned films (Motta et  al. 2011a, 2014). Fig.  13.9a presents a MO image captured at T  = 3 K for a patterned a-​MoGe film submitted to a perpendicular applied dc field of 0.7 Oe. The lattice parameter is 1.5 μm and the lateral dimension of the square ADs is 0.8 μm, with the principal axes completely aligned with the edges of the sample. Noticeably, the main trunk of each avalanche is perpendicular to the sample edges where it was rooted, following,

400

400  Flux Avalanches in Superconducting Films (a)

(b) 0.5 mm 1 mm

Fig.  13.9  MO images of flux avalanches in (a)  a-​MoGe film patterned with a square array of 1.5 μm with square ADs of 0.8 μm taken at T  = 3 K and H  = 0.7 Oe, and in (b) Nb film decorated with a square array of 4 μm with square ADs of 1.5 μm taken at T  = 4.5 K and H  = 1.7 Oe.

0.8 µm 1.5 µm

1.5 µm 4 µm

as discussed earlier, the main axis determined by the AD rows. However, the secondary ramifications always emerge at 45° relative to the main trunk, which confers to the avalanche an aspect similar to a Christmas tree. Fig. 13.9b shows that avalanches occurring in a Nb thin film also decorated with a square array of square ADs exhibit exactly the same morphology, with the secondary branches growing at 45°. Although the dimensions of lattice parameters and ADs are different from one film to the other, the overall morphology of the dendrites is reproduced. Moreover, Nb is well known for having strong intrinsic pinning, whereas MoGe, contrarily, is quite weak in this sense (Van Baarle et al. 2003). Therefore, such similarity in the avalanche morphology is not material-​related, being thus directly attributable to the geometric features of the underlying pinning landscape (Motta et al. 2014). In order to explain this tendency of the ramifications to follow the diagonal direction of the square AD array, it is sufficient to rely on recent papers on the current crowding effect by Clem and Berggren (2011), Hortensius et al. (2012), Henrich et al. (2012), Clem et al. (2012), Adami et al. (2013), and Semenov et al. (2015). Together they provide theoretical predictions and experimental results which allow one to recognize the main ingredient of this peculiar morphology. In fact, the current crowding effect, anticipated by Clem and Berggren, was observed experimentally by Adami and coauthors (2013) in superconducting corner-​shaped microstrips of aluminum, that is, an Al film in the format of a 90º angle, in which an asymmetry on the vortex dynamics was detected. It means that flux penetration from the inner concave angle is easier than from the outer convex angle. Since the Al sample consists of a 3.3 μm wide transport bridge embracing

  401

Effects of the antidot geometry and lattice symmetry in flux avalanches  401 a 90º angle, as presented in Fig. 13.10a, comparable to the distance between the ADs in the samples presented in Fig. 13.9, each of our ADs could perfectly be envisaged as a square void surrounded by four such samples, positioned in such a way as to form a closed superconducting loop. Fig. 13.10b depicts the case of an AD surrounded by superconducting microstrips, composed by four 90º bridges placed in each quadrant. Apart from the screening current component which circulates around the whole sample, the streamlines are closed loops of screening currents around the AD. The dashed thick line in the superconducting microloop represents the middle current streamline and emphasizes the current crowding occurring close to the inner corners. If magnetic flux reaches such a central portion of the sample coming from an edge, entering the AD as a consequence of the local magnetic pressure, it will leave the AD as in the current crowding effect experiment reported by Adami and co-​authors (Adami et al. 2013), that is, entering the superconducting microstrip preferentially through the inner concave angle. Moreover, in a superconducting film with an array of ADs, the total current flow splits around the AD, leading to

(a)

(b)

90 degree corner shape of AI

AD

L = 3.3 µm t = 67 nm

Screening currents only due to the AD

Current crowding

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Fig.  13.10 (a)  AFM image of a 90º microstrip of aluminum. (b) Combina­ tion of four 90º strips to form a square AD, showing only the screening currents around it. (c)  Scheme of the total current around a square AD and the current crowding effect in its corners. (d)  Evolution of a flux avalanche in a sample with a square array of square ADs. The white ADs are those already penetrated by flux and the arrows on the corners indicate where the Lorentz force is maximum (after Motta 2013). Panel (a) adapted with permission from Adami et  al. (2013). Copyright 2013, AIP Publishing LLC.

402

402  Flux Avalanches in Superconducting Films current crowding effects on the AD corners. In other words, the current crowding effect can be understood as a conglomeration of the current streamlines around an obstacle, in this case, the AD corners. Vortices subject to the maximum Lorentz force (FLmax) are those in regions of maximum density of current streamlines, that is, the tip of the ADs, as depicted by thick arrows in Fig. 13.10c. Regarding the avalanche progression, Fig. 13.10d illustrates how the main trunk and its 45º ramifications evolve in a sample with a square array of square ADs. The main trunk develops mainly due to screening currents parallel to the sample edge, which circulate throughout the film and push the flux toward the center. In turn, the secondary branches grow as a consequence of the current crowding effect, resulting in a maximum Lorentz force at the corners of the square AD. The 45º branching in a superconducting film patterned with a square array of square ADs has also been reproduced by numerical simulations based on the TM model (discussed in Sec. 13.1.4). Fig. 13.11 illustrates a Christmas tree-​like avalanche as captured by the distribution of the magnetic flux density (a) and the sheet current (b). Panel (a) shows that flux rushes into the sample preferentially in the direction along the diagonal of the ADs. One can notice that flux is pushed away from the tip even in the main trunk, where brighter tones are present, maintaining clearly the habit to propagate at 45º. Shown in panel (b) are the current streamlines, which present the same tendency: paths around the square AD reveal the sources of flux/​antiflux in that neighborhood and, thus, the preferential direction of the Lorentz force pushing flux along the direction of the AD diagonals. Panel (c) shows a zoomed-​in image of the region marked in (b), in which one can see the streamlines agglomerated on the tip of the AD in the forefront of the avalanche branches. Another suggestive observation, which also emphasizes the role of the AD geometry on the morphology of avalanches, has been reported by Motta et  al. (2014). Fig. 13.12 depicts results for a Nb film patterned with circular ADs on the left side, and square ADs on the right, as illustrated in panel (a). Panel (b) presents a dendrite triggered on the right side (square AD half) of the sample,

Fig. 13.11  Flux avalanche in a square array of square ADs reproducing the Christmas tree at the reduced temperature (T / Tc ) of 0.4 and applied field of 5.3 Oe. (a) Distribution of the magnetic flux density, and (b)  the normalized induced sheet current. (c)  Zoom-​ up of the region marked on panel (b). Figure adapted with permission from Motta et  al. (2014). Copyrighted by the American Physical Society.

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(b)

B(mT)

–2

0

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  403

Effects of the antidot geometry and lattice symmetry in flux avalanches  403 (a) 4 µm

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with 45º branches. When branches reach the left half of the film (circular ADs), the ramifications modify their directions due to the different AD geometry and follow a fingerlike pattern (on the left of the white dashed line), with some small 90º branches. The image was captured at a temperature of 6 K, after increasing the field up to 48 Oe and then decreasing it down to 14 Oe. The fact that one had to “prepare the state” in order to see both habits in a single image was attributed by the authors as due to the importance of intrinsic, randomly distributed pinning centers in Nb, which play a far more important role in this sample than in a-​MoGe films. The current crowding effect for square ADs displayed in a square lattice, introduces a second set of main axes, rotated by 45º from the primary axes dictated by lattice symmetry. For circular ADs, the main axes for flux guidance remain those originally set by the lattice symmetry, and the avalanches are essentially fingerlike sticks with minor ramifications perpendicular to the main branch. To further emphasize the effects of the antidot geometry and of the lattice symmetry on the flux avalanche morphology, a Nb film patterned with a square lattice of triangular ADs was prepared, and results are presented in Fig. 13.13. Panel (a) shows the dimensions of the square lattice (4 μm) and of the triangular ADs (1.5 μm) placed in the sample. As the lateral dimensions are larger than the field of view of the microscope, partial MO images were captured in different portions of the sample, two of which are illustrated in (a). Panel (b) presents the flux distribution on the top edge of the sample, which faces the base of the triangular ADs, leading to straight fingerlike trunks with 90º secondary branches induced by current crowding effects at the tips of the ADs. When the flux avalanches invade the sample from the opposite edge, as shown in panel (c), their morphology is in the form of a Christmas tree, as in a square array of square ADs. Again, the orientation of the tips of the triangular AD relative to the flux front induces this pattern, due to current crowding, as illustrated in Fig. 13.13c and discussed above for square ADs. The effect of the lattice symmetry on the avalanche morphology is evident by the 45º secondary branches on avalanches rooted at the bottom edge. After entering the AD, flux has the tendency of leaving it from the triangle base at 30º from the sample edge (maximum Lorentz force), as presented in Fig. 13.13d. However, the triangle tip acting as its first attractive neighbor is located along the diagonal of the lattice and, consequently, 45º branches are

Fig.  13.12  Nb sample with circular ADs on the left side and square ADs on the right. (a)  SEM image with the AD size and the lattice parameter. (b)  MO image taken at T  = 6 K and H  = 14 Oe after increasing the magnetic field up to 48 Oe. Figure adapted with permission from Motta et  al. (2014). Copyrighted by the American Physical Society.

404

404  Flux Avalanches in Superconducting Films (b)

(d) Edge J

(a)

FLmax

1.5 µm 500 µm

Fig.  13.13 Flux avalanche in a Nb film patterned with a square array of triangular ADs. (a)  A  scheme of the decorated sample (not to scale) and the regions where the MO images presented in (b) and (c) were taken at T = 3 K and H  =  3 Oe. Panels (d)  and (e)  show the total screening current and the maximum Lorentz force (boldface arrows) around triangular ADs placed close to the top and bottom edges (after Motta 2013).

(c) 4 µm

(e)

J Edge

the natural solution of the branching. This case emphasizes that the avalanche morphology results from compromise between the AD shape and the lattice symmetry. (Menghini et al. 2007). It is worth noticing that the avalanche morphology is strongly influenced by the geometric quality of the AD, including the sharpness of the interface between the AD and superconductor. Fig. 13.14 shows microstructural images, obtained by AFM (atomic force microscopy) and by SEM (scanning electron microscopy), and MO images for two films decorated with ADs of 0.4 μm displayed in a square array of 1.5 μm, for (a)  a-​MoGe and (b)  Nb films. AFM images illustrate the difference between both samples:  whereas the AD geometry for the a-​MoGe film is sharply square, the ADs on the right (Nb film) are clearly rounded. This is also confirmed by the SEM images, in which the profiles taken along one row of ADs, as indicated in the bottom panel, show clearly a sharp AD/​superconductor interface for the a-​MoGe sample, different from the gradual crossover for the Nb film. Therefore, these representative images, taken by SEM and AFM techniques, provide a consistent picture reinforcing that the 45º ramifications are due to the sharp corners of the square ADs in a-​MoGe, presented on the top left of Fig. 13.14a, while the 90º branched avalanches, shown on the top right of Fig. 13.14b, are byproducts of the round ADs in Nb. This result reinforces the notion that the avalanche morphology is not material-​dependent, but intimately related to an interplay among the AD geometry and lattice symmetry.

  405

Effects of the antidot geometry and lattice symmetry in flux avalanches  405 a-MoGe film

(a)

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T = 4.5 K – H = 0.9 Oe

T = 3 K – H = 3 Oe

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Fig.  13.14 MO, AFM, and SEM images for two structured films with ADs of 0.4 μm in square arrays of 1.5 μm. Left side (a):  a-​ MoGe film; right side (b): Nb film. Upper left and right panels show avalanches taken at T = 4.5 K and H = 0.9 Oe (a-​MoGe film) and at T = 3 K and H = 3 Oe (Nb). Central pictures are AFM images and, just below, SEM profiles taken along the rows of ADs marked by the dashed lines in the very bottom images (after Motta 2013).

As a final example of how symmetry and geometry combine to guide the avalanche dendrites, Fig. 13.15a shows sudden penetrations on a specially patterned a-​MoGe film. Its AD lattice consists of a centered rectangular 2D Bravais lattice, prepared by shifting every row by one half of the repetition length in its own direction and by a full length in the orthogonal direction, as detailed in Fig. 13.15b. As expected, the avalanches follow the angles θ1 ≈ 63° and θ2 ≈ 53° along the window of temperatures and fields in which avalanches take place. Besides that, avalanches that occur in the top and bottom edges have main trunks perpendicular to the border, which are, however, absent in the left and right edges. This difference in morphology is a consequence of the fact that every vertical AD row is perpendicular to the horizontal borders (and guides the avalanches into the sample with

(a)

(b) θ2

2 µm 3 µm 400 µm

θ1

6 µm

Fig.  13.15  (a)  MO image captured at T   =  4 K and H   =  1.1 Oe for an a-​MoGe film decorated with a centered rectangular lattice depicted (not to scale) in (b), with θ1 ≈ 63° and θ2 ≈  53°.

406

406  Flux Avalanches in Superconducting Films main trunk and inclined branches), whereas there is no row at all perpendicular to the side borders, giving rise to a “doubly degenerate” main trunk. This remarkable result emphasizes that the avalanche morphology depends on the geometric features of the ADs and on how they are structured in the film.

13.4 Summary and outlook Superconducting thin films submitted to magnetic fields in the perpendicular geometry may become unstable when temperature and field values are within certain restricted intervals. Such instabilities, of thermomagnetic origin, generate practical conditions for the triggering of flux avalanches, which may invade the film at extremely high speeds. These avalanches represent a problem of multiple importance, given the richness of the physics involved in their nucleation and propagation and, not less significant, their negative impact on the current-​ carrying capability of the specimen. Furthermore, inserting arrays of submicrometric antidots (ADs) is a customary strategy in the pursuit of increasing critical currents but, as a counterpart, the presence of such arrays induces avalanches. The focus of the present chapter is on the effects on flux avalanches—​ triggering conditions, propagation, and morphology—​ caused by the presence of arrays of artificial pinning centers in the superconducting media, particularly in thin film geometry. At temperatures close to Tc, effects related to the insertion of lattices of artificial pinning centers become evident, such as the matching fields and the accompanying enhancement of the critical current which, in some cases, can reach the depairing current. At lower temperatures, the presence of arrays of ADs increases the avalanche activity, enlarging the temperature and magnetic field ranges within which dendrites are triggered. The effects of lattice symmetry and AD geometry on the morphology of flux avalanches can be thoroughly understood in terms of the so-​ called current crowding effect, which takes place at the corners of the ADs. Such ADs, when displayed in 2D arrangements are, in fact, the building blocks of a toy-​model with which one can understand—​and even anticipate—​the specific morphology adopted by avalanches of magnetic flux, occurring in superconducting films with mesoscopic artificial patterns. A feature of the problem still deserving further consideration is that, for 2D arrangements of ADs, the preferential axes for flux propagation—​and, as a consequence, the avalanche morphology—​are ultimately defined by the AD geometry and the lattice symmetry. Conceivably, for fixed AD format and lattice symmetry, one can vary the ratio between AD size and lattice parameter, so as to tune the relative importance of the current crowding effect at the corners of the ADs for the avalanche morphology. Thus, for large enough values of the lattice parameter, the avalanches would predominantly be fingerlike, with small secondary branches orthogonal to the main trunk, regardless of the AD geometry.

  407

References (Chapter-13) 407 Another aspect to be stressed here is the noticeable similarity between the problem treated in this chapter, that is, the overall features of magnetic avalanches in patterned superconducting films, and the equally important problem related to the occurrence of electric discharges in controlled non-​uniform media: both studies are likely to benefit from advances on the counterpart problem.

Acknowledgements The authors would like to thank F. Colauto, J.I. Vestgården, and T.H. Johansen for fruitful discussions, and D.  Carmo, J.  Brisbois, and O.-​A. Adami for critical reading of the manuscript. This work was partially supported by the CAPES Foundation, the Brazilian National Council for Scientific and Technological Development (CNPq), the São Paulo Research Foundation (FAPESP), the Fonds de la Recherche Scientifique (F.R.S.-​FNRS), the program for scientific cooperation F.R.S.-​FNRS-​CNPq, and NanoSC–​COST Action MP1201. The work of A.V.S. is partially supported by “Mandat d’Impulsion Scientifique” MIS F.4527.13 of the F.R.S.-​FNRS.

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410  Flux Avalanches in Superconducting Films Menghini, M., Wijngaarden, R.J., Silhanek, A.V., Raedts, S. and Moshchalkov, V.V. (2005) Phys. Rev. B 71, 104506. Menghini, M., Van de Vondel, J., Gheorghe, D.G., Wijngaarden, R.J. and Moshchalkov, V.V. (2007) Phys. Rev. B 76, 184515. Metlushko, V.V., Baert, M., Jonckheere, R., Moshchalkov, V.V. and Bruynseraede, Y. (1994) Solid State Commun. 91, 331. Mints, R.G. and Rakhmanov, A.L. (1981) Rev. Mod. Phys. 53, 551. Misko, V.R. and Nori, F. (2012) Phys. Rev. B 85, 184506. Mkrtchyan, G.S. and Shmidt, V.V. (1972) Sov. Phys. JETP 34, 195. Moshchalkov, V.V., Baert, M., Metlushko, V.V., Rosseel, E., Van Bael, M.J., Temst, K., Jonckheere, R. and Bruynseraede, Y. (1996) Phys. Rev. B, 54, 7385. Moshchalkov, V.V., Baert, M., Metlushko, V.V., Rosseel, E., Van Bael, M.J., Temst, K., Bruynseraede, Y. and Jonckheere, R. (1998) Phys. Rev. B 57, 3615. Moshchalkov, V.V. and Fritzsche, J. (2011) Nanostructured superconductors, World Scientific, Singapore. Motta, M., Colauto, F., Zadorosny, R., Johansen, T.H., Dinner, R.B., Blamire, M.G., Ataklti, G.W., Moshchalkov, V.V., Silhanek, A.V. and Ortiz, W.A. (2011) Phys. Rev. B 84, 214529. Motta, M., Colauto, F., Ortiz, W.A., Vestgården, J.I., Johansen, T.H., Cuppens, J., Moshchalkov, V.V. and Silhanek, A.V. (2011a) Arxiv. Available from:  http://​ arxiv.org/​pdf/​1109.2532v1.pdf. [Accessed: December 18, 2015]. Motta, M. (2013) PhD thesis, Federal University of São Carlos (Universidade Federal de São Carlos), Brazil. Motta, M., Colauto, F., Ortiz, W.A., Fritzsche, J., Cuppens, J., Gillijns, W., Moshchalkov, V.V., Johansen, T.H., Sanchez, A. and Silhanek, A.V. (2013) Appl. Phys. Lett. 102, 212601. Motta, M., Colauto, F., Vestgården, J.I., Fritzsche, J., Timmermans, M., Cuppens, J., Attanasio, C., Cirillo, C., Moshchalkov, V.V., Van de Vondel, J., Johansen, T.H., Ortiz, W.A. and Silhanek, A.V. (2014) Phys. Rev. B 89, 134508. Niebieskikwiat, D., Silhanek, A., Civale, L., Nieva, G., Levy, P. and Krusin-​ Elbaum, L. (2001) Phys. Rev. B 63, 144504. Nordborg, H. and Vinokur, V.M. (2000) Phys. Rev. B 62, 12408. Pannetier, M., Wijngaarden, R.J., Fløan, I., Rector, J., Dam, B., Griessen, R., Lahl, P. and Wördenweber, R. (2003) Phys. Rev. B 67, 212501. Raedts, S., Silhanek, A.V., Van Bael, M.J., Jonckheere, R. and Moshchalkov, V.V. (2004) Physica C 404, 298. Ray, D., Olson Reichhardt, C.J., Jankó, B. and Reichhardt, C. (2013) Phys. Rev. Lett. 110, 267001. Ray, D., Reichhardt, C. and Olson Reichhardt, C.J. (2014) Phys. Rev. B 90, 094502. Reichhardt, C., Ray, D. and Olson Reichhardt, C.J. (2015) Phys. Rev. B 91, 184502. Rudnev, I.A., Antonenko, S.V., Shantsev, D.V., Johansen, T.H. and Primenko, A.E. (2003) Cryogenics 43, 663.

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References (Chapter-13) 411 Rudnev, I.A., Antonenko, S.V., Shantsev, D.V., Johansen, T.H. and Primenko, A.E. (2005) App. Phys. Lett. 87, 042502. Schawlow, A.L. (1956) Phys. Rev. 101, 573. Schuster, T., Indenbom, M.V., Koblischka, M.R., Kuhn, H. and Kronmüller, H. (1994) Phys. Rev. B 49, 3443. Semenov, A., Charaev, I., Lusche, R., Ilin, K., Siegel, M., Hübers, H.-​W., Bralović, N., Dopf, K. and Vodolazov, D.Y. (2015) Phys. Rev. B 92, 174518. Silhanek, A.V., Van Look, L., Raedts, S., Jonckheere, R. and Moshchalkov, V.V. (2003) Phys. Rev. B 68, 214504. Silhanek, A.V., Raedts, S. and Moshchalkov, V.V. (2004) Phys. Rev. B 70, 144504. Silhanek, A.V., Van Look, L., Jonckheere, R., Zhu, B.Y., Raedts, S. and Moshchalkov V.V. (2005) Phys. Rev. B 72, 014507. Silhanek, A.V., van de Vondel, J. and Moshchalkov, V.V. (2010) in Nanoscience and engineering in superconductivity, edited by V.V. Moshchalkov, R. Wördenweber, and W. Lang, Springer, Berlin, p. 1. Soibel, A., Zeldov, E., Rappaport, M., Myasoedov, Y., Tamegai, T., Ooi, S., Konczykowski, M. and Geshkenbein, V.B. (2000) Nature 406, 282. Swartz, P.S. and Bean, C.P. (1968) J. Appl. Phys. 39, 4991. Takezawa, N. and Fukushima, K. (1994) Physica C 228, 149. Van Baarle, G.J.C., Troianovski, A.M., Nishizaki, T., Kes, P.H. and Aarts, J. (2003) Appl. Phys. Lett. 82, 1081. Vélez, M., Martín, J., Villegas, J., Hoffmann, A., González, E., Vicent, J. and Schuller, I.K. (2008) J. Magn. Magn. Mater. 320, 2547. Vestgården, J.I., Shantsev, D.V., Galperin, Y.M. and Johansen, T.H. (2007) Phys. Rev. B 76, 174509. Vestgården, J.I., Shantsev, D.V., Galperin, Y.M. and Johansen, T.H. (2011) Phys. Rev. B 84, 054537. Vestgården, J.I., Shantsev, D.V., Galperin, Y.M. and Johansen, T.H. (2013). Arxiv. Available from: http://​arxiv.org/​pdf/​1309.6463.pdf. [Accessed: October 22, 2015]. Vestgården, J.I., Shantsev, D.V., Galperin, Y.M. and Johansen, T.H. (2015). Arxiv. Available from: http://​arxiv.org/​pdf/​1212.3412.pdf. [Accessed: October 22, 2015]. Vlasko-​Vlasov, V., Welp, U., Metlushko, V. and Crabtree, G. (2000) Physica C 341–​348 (Part 2), 1281. Wang, Y.L., Latimer, M.L., Xiao, Z.L., Divan, R., Ocola, L.E., Crabtree, G.W. and Kwok, W.K. (2013) Phys. Rev. B 87, 220501. Wang, Y.L., Thoutam, L.R., Xiao, Z.L., Shen, B., Pearson, J.E., Divan, R., Ocola, L.E., Crabtree, G.W. and Kwok, W.K. (2016) Phys. Rev. B 93, 045111. Welling, M.S., Wijngaarden, R.J., Aegerter, C.M., Wördenweber, R. and Lahl, P. (2004) Physica C 404, 410. Wertheimer, M.R. and Gilchrist, J. le G. (1967) J. Phys. Chem. Solids 28, 2509. Wimbush, S.C., Holzapfel, B. and Jooss, C. (2004) J. Appl. Phys. 96, 3589. Wipf, S. (1991) Cryogenics 31, 936.

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412  Flux Avalanches in Superconducting Films Xu, K., Cao, P. and Heath, J.R. (2010) Nano Lett. 10, 4206. Yurchenko, V.V., Shantsev, D.V., Johansen, T.H., Nevala, M.R., Maasilta, I.J., Senapati, K. and Budhani, R.C. (2007) Phys. Rev. B 76, 092504. Zadorosny, R., Colauto, F., Motta, M., Johansen, T.H., Dinner, R.B., Blamire, M.G., Ataklti, G.W., Moshchalkov, V.V., Silhanek, A.V. and Ortiz, W.A. (2013) J. Supercond. Nov. Magn. 26, 2285.

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Part III Device Technology

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Superconducting Spintronics and Devices M.G. Blamire and J.W.A. Robinson Department of Materials Science, University of Cambridge, Cambridge, CB3 0FS, UK

The field of conventional spin electronics or spintronics is generally recognized as having been initiated by the discovery of giant magnetoresistance (GMR) by Grünberg (Grünberg et  al. 1986)  and Fert (Baibich et  al. 1988)  in the 1980s. This discovery was itself initiated by the evolution of the precision growth of multilayers of magnetic and non-​magnetic metals. Spintronics has developed into an area of vast technological importance, underpinning much of modern data storage technology through its role in read-​out from magnetic discs. It has also been proposed as an eventual replacement for standard complementary metal-​ oxide-​semiconductor (C-​MOS) data processing; here the technological outlook is less promising because, although the basic physics seems understood, the practicalities of creating transistor-​like devices with much lower power consumption than C-​MOS seem intractable. Stimulated by the discovery of GMR, in the 1990s multilayer growth capabilities began to be applied to ferromagnet/​superconductor heterostructures. In general, superconductivity and ferromagnetism are highly incompatible. For example magnetic impurities strongly suppress conventional superconductivity and a conventional proximity effect between superconductors and ferromagnets is extremely short-​ranged with superconductivity only penetrating the ferromagnet over several nanometers (Buzdin 2005). Nevertheless, these initial experiments demonstrated that at very short length scales there was a complex dependence of critical temperature on ferromagnet layer thickness which was the first experimental evidence for predicted interactions of Cooper pairs and the exchange field in a ferromagnet. This work led to an explosion of experimental and theoretical activities which has seen a series of remarkable discoveries over the past 15 years. In this chapter we will review these results, with a particular emphasis on those which appear to signal that it will eventually be possible to create a superconducting equivalent of conventional spintronics which might provide a pathway to genuinely low-​power data processing.

14 14.1 Conventional spintronics 

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14.2 The rationale for superconducting spintronics 

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14.3 S/​F proximity effects and Josephson junctions 

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14.4 Spin transport in the superconducting state 

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14.5 Superconducting spintronic memory  423 14.6 Superconducting spintronic logic  424 14.7 Superconductor/f​ erromagnet thermoelectric devices 

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14.8 Materials and device structures  425 14.9 Summary and outlook 

427

References (Chapter-14) 

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M. Blamire J. Robinson, ‘Superconducting Spintronics and Devices’, in The Oxford Handbook of Small Superconductors. First Edition. Edited by A.V. Narlikar. © Oxford University Press 2017. Published in 2017 by Oxford University Press. DOI 10.1093/acprof:oso/9780198738169.003.0014

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14.1 Conventional spintronics Spintronics is distinguished from conventional electronics by the key role of the electron spin as a degree of freedom in device operation. GMR (Baibich et al. 1988, Grünberg et  al. 1986), the first spintronic technology identified, relies on differential spin-​scattering of electrons within a ferromagnet, or at a ferromagnet/​non-​magnet interface, and as such can be analyzed within a framework in which the two spin populations can be treated as independent. The original device structure was a ferromagnet/​non-​magnet multilayer in which interlayer exchange coupling imposed an equilibrium antiparallel alignment between neighboring magnetic layers: in this configuration the majority spin carriers from one ferromagnetic layer were scattered at the interface with the next layer because this spin direction corresponded to the minority spins with a different Fermi surface. Applying sufficient field to overcome the exchange coupling aligns all the ferromagnetic layers, with the result that the majority spin electrons can freely propagate between layers, leading to a much lower resistance. It was quickly discovered that the effect could be reproduced in a simpler trilayer structure of two ferromagnets sandwiching a non-​magnetic layer provided that one of the magnetic layers was prevented from reversing in field by exchange coupling it to an antiferromagnet (Dieny et al. 1991). In this spin-​valve geometry, the structure can be switched between high and low resistance states by fields as low as a few Oe. The large resistance change and the sensitivity to small fields led to the rapid exploitation of spin valves as the read sensors in hard disc drives and the rapid growth in recording density made possible by their enhanced sensitivity compared with anisotropic magnetoresistance sensors. The tunnel junction equivalent (tunnel magnetoresistance, TMR) was shown to provide even larger resistance changes (Moodera et al. 1995) and TMR read heads with very large MR are now standard in hard disc drives (Yuasa et  al. 2004). Arrays of TMR devices are also used to directly store data in magnetic random access memory (MRAM). In comparison to other memory technologies MRAM has both the advantages of a fast write time and non-​volatility. Hard disc read heads and MRAM are passive devices which react to an external stimulus (magnetic field) by changing their magnetic state and hence their resistance. True spin electronics requires that active devices can be created so that spin information can be used for output state control. GMR can be analyzed in terms of the flow of a spin current between ferromagnetic layers:  in such structures the spin current is carried by the spin-​polarization of the charge current and at the interface between ferromagnetic and non-​magnetic materials changes in the polarization of the available states at the Fermi energy give rise to a spin accumulation (Valet and Fert 1993). As well as giving rise to the resistance changes associated with GMR, the spin accumulation in a ferromagnet gives rise to a spin transfer torque (STT) whereby spin angular momentum is transferred to the bulk magnetism and, for sufficiently large values, can drive a precession of

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The rationale for superconducting spintronics 417 the moment or a controllable reversal of the magnetic state (Myers et al. 1999, Kiselev et al. 2003). A spin accumulation is a splitting of the chemical potential for spin-​up spin-​ down electrons which drives spin diffusion in a non-​magnetic material independently of the path of the charge current flow and hence can give rise to a spatial separation of the spin and change current in a multi-​terminal structure. This was demonstrated in the so-​called non-​local geometry in which a spin-​polarized charge current injected from a ferromagnet into a non-​magnetic material generates a spin-​accumulation in the non-​magnetic material which diffuses away from the contact and can be detected at significant distances away in materials, such as Cu, with long spin-​diffusion lengths (Jedema et al. 2001). More recently, it has been demonstrated that spin-​orbit scattering of charge currents in non-​magnetic metals can give rise to the spin Hall effect which is effectively a spin current orthogonal to the charge current direction (Valenzuela and Tinkham 2006). STT is now sufficiently well understood that it is being developed as a means of switching MRAM data cells (rather than by applying local magnetic fields) and of moving domain walls in shift-​register memory devices (race track memory— ​Hayashi et  al. 2008). However, large charge current densities of the order of 1010 A/​m2 are required to drive STT and so the potential for controlling currents which could themselves switch other devices without large energy dissipation seems small. More importantly from the point of view of future logic applications, there is still no viable transistor-​like device capable of switching or amplifying a spin signal. The original Johnson spin switch/​transistor (Johnson 1993)  simply uses the spin as a means of distinguishing two carrier types (by analogy with the semiconductor bipolar transistor) and the output is set via the relative orientation of different magnetic layers. This feature is shared by a variety of spin-​transistor-​ like devices (e.g. Jansen 2003, Datta and Das 1990, Dennis et al. 2003). Several devices in which the magnetic state of the device is controlled by the spin input signal have been proposed (Chiba et  al. 2013, Acremann et  al. 2008)  but not experimentally realized.

14.2 The rationale for superconducting spintronics Interest in alternative logic and memory architectures is driven by the approach of conventional C-​MOS semiconductor technology to the physical limits of its scaling and power dissipation. Although conventional spintronics has frequently been proposed as a replacement technology for parts or the entirety of C-​MOS, at the time of writing there has been no convincing demonstration that it can meet all of the performance requirements for this to happen.

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418  Superconducting Spintronics and Devices Superconducting computing has also periodically been proposed as a replacement for semiconductor logic. Superconducting device switching times and power dissipation are both orders of magnitude lower than the semiconductor equivalents. Although the cryogenic cooling requirements present problems, these are much more surmountable in a centralized “cloud computing” scenario in which the processors of an entire data center can be maintained at low temperatures. However, existing superconducting architectures, such as rapid single flux quantum (RSFQ) logic (Likharev and Semenov 1991), do not scale well to the device densities required to be competitive with semiconductor chips; memory cells are particularly large and poorly scalable. There is therefore an interest in exploring the potential for synergies between spintronic and superconducting architectures (Holmes et al. 2013) which has been stimulated by a rapid series of important discoveries regarding the interaction between magnetic and superconducting systems. The remainder of this chapter is organized as follows: we will review the basic interactions between superconductivity and magnetism including the potential for spin-​polarized supercurrents opened up by the discovery of triplet proximity effects; this will be followed by defining the methods by which superconducting spintronic memory could be realized and the requirements for this to be compatible with non-​magnetic RSFQ logic. Ideas for realizing fully superconducting spintronic logic are still in their infancy, but we will review the concepts so far proposed and suggest ideas for the future.

14.3 S/​F proximity effects and Josephson junctions Conventional superconductivity is mediated by singlet Cooper pairs of electrons which have antiparallel spins. The proximity effect between a superconductor (S) and a ferromagnet (F) tends to strongly suppress superconductivity because the internal exchange field of the latter acts to align electron spins and so destroys pairs. Consequently the critical temperature (Tc) of a thin superconductor is strongly suppressed by contact with a ferromagnet. Similarly, the Josephson critical current (Ic) between superconductors through a ferromagnetic barrier is much lower than through the non-​magnetic (N)  barrier of equivalent thickness. Nevertheless, in the thickness range in which it is still finite, Ic can be shown to oscillate in sign owing to a distance-​dependent phase-​shift between the electrons in the pair (Oboznov et al. 2006, Robinson et al. 2006); a weaker oscillation of Tc in S/​F bilayers (Jiang et al. 1995) is a consequence of the same effect. A  negative Ic corresponds to a ground state π-​phase difference across an S/​F/​S junction (Ryazanov et  al. 2001, Frolov et al. 2004) which has direct application as a phase battery in quantum circuits (Feofanov et al. 2010).

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S/F proximity effects and Josephson junctions  419 Within a S/​F heterostructure it is possible to vary the net exchange energy by changing the relative magnetization direction of two or more magnetic layers. Thus in an F/​S/​F trilayer (the equivalent of the spintronic spin valve) in which the S layer is thinner than the singlet pair coherence length, the Tc suppression can be controlled by summing or subtracting the exchange fields of the magnetic layers via the relative alignment of their moments (Tagirov 1999). For conventional singlet Cooper pairs, the Tc suppression should be greatest for a parallel configuration of magnetizations and vice versa. This superconducting spin-​valve effect was first demonstrated by Gu et al. albeit with a measured difference in Tc (ΔTc) between parallel and antiparallel moments of only approximately 10 mK. According to quasiclassical theory much larger values of ΔTc can be obtained and, for particular parameter combinations, superconductivity should be completely suppressed and so the original predictions for such devices (Tagirov 1999, Oh et al. 1997) suggested that it should be possible to switch the critical temperature from a value close to that of the bulk superconductor and zero. The idea of creating a stable, non-​volatile mechanism for switching superconductivity on and off is attractive for a range of applications and so extensive studies have been carried out by various groups on such spin-​switch structures over the past decade (Flokstra et al. 2015, Leksin et al. 2012, Zhu et al. 2013, Li et al. 2013, Moraru et al. 2006, Gu et al. 2002). Although these results have fallen short of the original predictions in that the relative values of ΔTc are a small fraction of Tc and comparable to the resistive transition width, there has been much theoretical work (Baladié et al. 2001, Fominov et al. 2010, Leksin 2015) which has demonstrated an improved understanding of what should be achievable. Recent examples of experimental progress include epitaxial rare-​earth/​superconductor heterostructures which exhibit much larger ∆Tc values than transition metal systems (Gu et al. 2015) and EuS/​Al/​EuS spin valves where the ferromagnetic insulator EuS can generate large ∆Tc values together with infinite magnetoresistance (MR)—​ complete magnetic switching between fully superconducting and normal states (Li et al. 2013). Similarly, Ic of a Josephson junction containing a spin-​valve barrier can be strongly dependent on the magnetic state (i.e. the two magnetic layers being either parallel or antiparallel)—​see Fig. 14.1 (Bell et al. 2004). Since Ic (left axis) is measured at zero field after removal of the applied setting field, the device is non-​volatile and shows a much larger Ic switching than the normal state GMR (right axis). This structure is therefore the equivalent of the conventional spintronic magnetic random access memory (MRAM) storage node. Recently several groups have begun developing this type of structure (Bell et al. 2004, Robinson et al. 2010a) specifically for cryogenic MRAM applications and have optimized the layer structure so that the effect of the injected magnetic flux (Blamire et al. 2013)  is accounted for (Baek et  al. 2014)  and the transition between 0 and π-​states can be exploited (Niedzielski et al. 2015). The effects discussed above are all associated with singlet pairing, but there have been specific predictions that spin mixing processes can occur such that triplet pairs can be formed at S/​F interfaces (Bergeret et al. 2001). This process

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420  Superconducting Spintronics and Devices 55

26.4

45

26.3

40

Resistance (mΩ)

Fig.  14.1 Resistance (continuous line) and zero-​ field critical current (points) from a Nb/​Py/​Cu/​Co/​Nb nanopillar junction plotted versus the applied field or the set field, respectively.The inset diagrams illustrate the magnetic configuration of the device in different field ranges (after Bell et al. 2004).

Zero field Ic (µA)

50

35 26.2 30

–30

–20

10 –10 0 µ0H, µ0HSET (mT)

20

30

involves the conversion of singlet pairs into triplet pairs and vice versa due to the presence of a magnetically inhomogeneous S/​ F interface (a spin-​ mixing interface), which results in a combination of spin-​mixing and spin-​rotation processes. Here we briefly describe the pair conversion process. In the presence of an aligned ferromagnetic exchange field, singlet pairs gain a net momentum via the spin-​mixing process and this results in the generation of zero-​spin triplet pairs. A  triplet pair can exist in several different forms depending on the orientation of the magnetization axis. If a zero-​spin triplet pair encounters a second ferromagnetic layer that is misaligned to the magnetization orientation of the first ferromagnet where it formed, the different triplet components transform into each other and spin-​polarized triplet pairs are projected with a net polarization that is chiefly determined by the polarization of the ferromagnet layer. The first and clearest proof that spin-​polarized triplet pairs exist was experimental observation of supercurrents in Josephson junctions with half-​metallic CrO2 barriers (Keizer et al. 2006), but the first definitive confirmation of the underlying theories (Houzet and Buzdin 2007, Eschrig and Löfwander 2008) was obtained in 2010 (Robinson et al. 2010b, Khaire et al. 2010). Thus, an inhomogeneous magnetic interface can result in an unconventional S/​F proximity effect mediated by triplet rather than singlet pairs. A triplet Josephson current in which the Cooper pairs consist of parallel rather than antiparallel (singlet) electrons can propagate through much thicker ferromagnet barriers than the equivalent singlet current (Khaire et  al. 2010, Robinson et  al. 2010b) and also carry spin, meaning that control of the charge supercurrent necessarily enables control of spin currents in the superconducting state. Currently, there are two established methods of generating triplet pairs—​using intrinsically inhomogeneous magnetic materials (Robinson et  al. 2010b, 2012,

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Spin transport in the superconducting state 421 2014)  and non-​collinear magnetic heterostructures (Khaire et  al. 2010, Anwar et  al. 2010), respectively. The latter has the potential for varying the degree of inhomogeneity by altering the misalignment of the mixer layers and the primary ferromagnet (Klose et  al. 2012, Banerjee et  al. 2014a). Further potential spin mixer structures such as domain walls (Baker et al. 2014) have been identified which may mean larger triplet supercurrent densities can be generated. In particular spin–​orbit effects at the S/​F interface (Bergeret and Tokatly 2014)  have been predicted to offer efficient singlet–​triplet coupling.

14.4 Spin transport in the superconducting state Although the pair condensate in a singlet superconductor has zero net spin, this is not necessarily true for the population of quasiparticle excitations. Indeed, there are circumstances in which the quasiparticle spin-​decay time (Yang et al. 2010) and length (Quay et al. 2013) in the superconducting state are much longer than in the normal state and a very large effective spin polarization can be induced even by unpolarized current injection (Quay et al. 2013). This happens because Zeeman splitting of the quasiparticle density of states in the superconductor gives rise, at certain bias voltages in tunnel junctions, to a much larger density of states for one spin sign compared with the other (see Fig. 14.2) and hence a much larger tunneling probability; this can be used to tune the spin polarization of injected carriers. Similarly large values of the spin polarization should be possible using ferromagnetic insulator tunnel barriers which act as spin filters—​for example EuO (Santos et al. 2008) and GdN (Senapati et al. 2011).

S

2∆ 2Ez

I

N

EF

dl/dV

2Ez/e

eV

V0

V

Fig.  14.2 Left, schematic diagram of tunneling between an exchange-​split superconductor and a normal metal. Right, a schematic diagram of the corresponding conductance–​voltage characteristic. The tunnel current is strongly spin-​polarized when the normal metal Fermi energy (EF) is aligned with the lowest-​energy gap edge in the superconductor (positions marked by arrows in the right figure).

422

422  Superconducting Spintronics and Devices As in the normal state, a quasiparticle spin accumulation gives rise to a diffusive spin current which can be detected by a splitting of the chemical potential for spin-​up and spin-​down carriers (Quay et  al. 2013). Aluminum is favored for these experiments because of its long spin–​orbit scattering time; other superconductors such as Nb tend to have much shorter lifetimes and so act as spin absorbers (Ohnishi et  al. 2010, Wakamura et  al. 2014). This has been exploited to detect quasiparticle spin currents via the inverse spin Hall effect (Wakamura et al. 2015). However, in the superconducting state any quasiparticle spin currents must be diffusive and independent of the (zero-​ spin) charge supercurrent, meaning that many of the familiar concepts of conventional spintronics such as GMR do not have a direct quasiparticle spin equivalent. Although the transport of spin via supercurrents can be inferred from junctions with thick ferromagnetic barriers (for which singlet pair transport is blocked), particularly those involving half-​metals (Anwar et  al. 2010, Keizer et al. 2006) for which only electrons of one spin sign are available for transport, for practical spintronics it is essential that triplet pair propagation is sensitive to the magnetic orientation of ferromagnets with less than 100% spin polarization. This has been inferred from superconducting spin valves containing Ho layers as spin mixers (Banerjee et al. 2014b) for which the parallel alignment has the higher Tc, and S/​F/​F triplet spin switches based on metallic materials (Wang et  al. 2014)  and a combination of metallic and half-​metallic materials (Singh et al. 2015). In the CrO2/​Cu/​Ni/​MoGe devices reported by Singh et al. the largest suppression of Tc exceeded 1 K and was achieved by rotating the Ni so its magnetization was directed perpendicular to that of the CrO2. Theoretically (Fominov et al. 2010), this configuration should favor the maximum conversion of singlet pairs into triplet pairs and since CrO2 is a half-​metal, processes that flip the spin of an electron do not occur and so it was inferred that the triple pairs with the correct sign of spin travel much further away from the superconducting (MoGe) layer. Through such a process the superconducting layer is efficiently drained of Cooper pairs, which translates to a large suppression of Tc. However, direct measurements of spin transfer and selectivity via supercurrents have yet to be reported. Although the large spin polarization in excess of 90% reported for spin filter Josephson junctions (Massarotti et al. 2015, Senapati et al. 2011) tends to imply that the supercurrents in such devices must be spin polarized, there should still be a residual singlet current in such devices (Bergeret et al. 2012) and so proving triplet transport will require further forms of measurement. The most relevant of these is the observation of a pure 2nd harmonic current–​phase relation for devices showing strong spin polarization which is absent for devices with less magnetic barriers (Pal et  al. 2014); this has been predicted to be a signature for triplet pairing in ballistic S/​F/​S devices with a spin mixer at only one interface (Trifunovic 2011).

  423

Superconducting spintronic memory 423

14.5 Superconducting spintronic memory The generic concept underlying any approach to spin-​based memory is to use different magnetic states to provide non-​volatility and to use a superconducting architecture to reduce the switching energies and times to levels which are compatible with other, possibly non-​spintronic, data processing technologies. The basic magnetic cryogenic random access memory (cMRAM) concept requires that the magnetic state of an element can be reversibly changed by an appropriate on-​chip input signal. The two (or possibly more) magnetic states should then be able to deliver an output signal to provide a means of reading the data. The most developed methods for reading the magnetic states are via spin-​ valve Josephson junctions (Baek et  al. 2014, Bell et  al. 2004, Niedzielski et  al. 2015) and superconducting spin switches (Li et al. 2013). In either case, the most obvious method of writing data is via STT switching—​indeed this has already been demonstrated (Baek et al. 2015), albeit with current densities such that the entire device is driven into the normal state. Since current density required for STT is dependent on the anisotropy and magnetization (Sun 2000), it is probable that careful optimization of the magnetic properties of the free layer can help to reduce the current densities required. The minimum energy required for switching a magnetic element (for example the free layer in a spin valve) is the anisotropy energy barrier required to prevent thermally activated reversal: for memory applications this is typically taken to be 50 kBT, where kB is Boltzmann’s constant, and so for 4 K operation this corresponds to only 3 × 10–​21 J. Alternatively, it has been predicted that triplet pair exchange can induce an effect exchange interaction between ferromagnetic layers (Waintal and Brouwer 2002, Zhao and Sauls 2008) conceptually similar to the Ruderman–​ Kittel–​Kasuya–​Yosida (RKKY) interaction (Bruno et al. 2011) which underlies conventional GMR. Since this coupling is controlled via the phase difference across the structure (Robinson and Linder 2015) it should be possible to switch the free layer via the superconducting phase 𝜙 provided that the Josephson energy U = (φ0 I c / 2π ) (1 − cos φ) is large enough to overcome the magnetic reversal energies. Regardless of the method of coupling between the spin currents and the magnetic state, an eventual aim for a cMRAM technology is for a device which can be switched by single flux quantum (SFQ) logic (Likharev and Semenov 1991) voltage pulses. The energy and pulse width of a single SFQ pulse are of the order of 10–​19 J and 5 ps respectively; the former is significantly greater than the magnetic anisotropy energy required for thermal reversal discussed above and a small number of pulses could cover potential magnetic switching times. The primary

424

424  Superconducting Spintronics and Devices research questions therefore relate to the best means of reading and writing magnetic data via superconducting states.

14.6 Superconducting spintronic logic One of the long-​term goals of conventional spintronics is the creation of active three-​terminal transistor-​like elements (Bernstein et al. 2010). Although several concepts have been proposed (Behin-​Aein et  al. 2010, Bernstein et  al. 2010, Datta and Das 1990, Bauer et al. 2003, Sugahara and Nitta 2010) nothing has been demonstrated which replicates the functions of the semiconductor field-​ effect transistor (FET). As an alternative, there is a growing interest in exploiting high-​speed memory to generate memristor-​type coupled memory logic (Behin-​ Aein et al. 2010). From a superconducting perspective, non-​equilibrium transistor-​like structures have been investigated in the past (Faris et al. 1983), but the time constants for such devices have generally been considered to be too slow to compete with RSFQ logic. However, it is possible that experiments on non-​equilibrium spin devices may identify ways in which their operation can be speeded up.

14.7 Superconductor/​ferromagnet thermoelectric devices Finally we note briefly the potential application of superconductor/​ferromagnetic insulator devices as thermoelectric systems in low-​temperature electronic circuits. Controlling temperature and heat flow in electronic circuits that operate at low temperatures is currently a major challenge as most thermoelectric materials are semiconductors which have negligible thermoelectric effects at cryogenic temperatures (Tritt and Subramanian 2006). Conventional superconductors cannot solve this problem as their near-​ perfect electron–​ hole symmetry at the Fermi energy means that they cannot display noticeable thermoelectric effects. Theoretically, however, a thin film superconductor proximity coupled to a ferromagnetic insulator with a significant Zeeman-​split density of states can achieve a significant thermoelectric effect as thermal effects are coupled with a so-​called thermoelectric figure of merit that can approach unity (Machon et  al. 2013, Ozaeta et  al. 2014, Kawabata et  al. 2013)  near Tc and larger values can be obtained by further cooling. For interferometer setups involving superconductor/​ferromagnetic insulator layers the figure of merit is predicted to exceed a value of ten for high values of spin-​polarization (greater than 95%) of the ferromagnetic insulators and is phase-​tunable via external magnetic flux (Giazotto et al. 2014).

  425

Materials and device structures 425

14.8 Materials and device structures To create superconducting memory and logic devices, a detailed understanding of basic materials properties will be essential, as these will determine device performance. The types of devices that must be created will necessarily involve combining several different materials and so identifying the best materials combinations for a particular device operation will be important. In this section we briefly discuss the key materials properties that need to be considered and the generic types of device structures that will need to be explored in order to advance superconducting spintronics. As with the development of non-​superconducting spintronics a basic requirement for superconducting spintronics will necessarily involve the efficient transmission of spin. This can be achieved in various ways in the superconducting state via triplet pairs, but also in combination with non-​superconducting electron spin currents involving quasiparticle (non-​superconducting) currents, spin-​polarized charge currents, or charge-​less pure spin currents. In all cases, it must be possible to accumulate spin and be able to transfer spin between different circuit elements via normal (non-​magnetic) metal layers or potentially via the superconducting state through superconducting layers (Quay et al. 2013, Yang et al. 2010). The key materials parameter for efficient transmission of spin is a long spin-​diffusion length and for this, metals such as Al and Cu are ideal because their spin-​diffusion lengths are known to be hundreds of nanometers at cryogenic temperatures (Blum et al. 2004). Furthermore, Al has the additional advantage that below its superconducting transition the spin-​diffusion length significantly increases over the normal state (Quay et  al. 2013, Yang et  al. 2010)  and this behavior can be exploited in devices involving the injection and detection of spin-​currents in the superconducting state. Owing to their relatively long mean free paths for both charge and spin-​flip scatter, elemental ferromagnets Co, Fe, and Ni in combination with normal (Cu or Al) and superconducting metals (Nb or Al) have proven to be efficient materials combinations for spin-​mixers (Khaire et al. 2010, Klose et al. 2012, Wang et al. 2010, Banerjee et al. 2014a), for controlling triplet current densities in Josephson junctions containing magnetization switchable F/​N/​F multilayers (Banerjee et al. 2014a), or for controlling the transition temperature of superconducting spin valves (Banerjee et  al. 2014b, Leksin et  al. 2012, Wang et  al. 2014). There are predictions that suggest that a triplet supercurrent can STT switch ferromagnetic elements in S/​F structures via the ac-​Josephson effect (Shomali et al. 2011, Linder et  al. 2012). In this case, ferromagnets with short spin-​flip lengths are preferable in order to achieve maximum transfer of spin. A  further important consideration relates to minimizing current switching energies for STT, which can be achieved by minimizing magnetocrystalline anisotropy in combination with the selection of ferromagnets with high spin-​polarizations and low magnetic moments. While anisotropy can be controlled in the elemental ferromagnets via

426

426  Superconducting Spintronics and Devices crystallinity, much of the early S/​F/​S work involved the use of weakly ferromagnetic alloys such as CuxNi1–​x and PdxNi1–​x in which magnetic moments can be tuned via stoichiometry (Ryazanov et al. 2001, Oboznov et al. 2006, Kontos et al. 2002). The recent superconducting spin-​valve work at NIST (Baek et al. 2014, 2015)  used Ni0.7Fe0.17Nb0.13 because of its lower magnetization than conventional permalloy (NiFe). The spin-​polarization of transition metal ferromagnets is approximately 40% (Bass and Pratt 2007) and this must set a limit to the achievable polarization of spin-​ polarized triplet supercurrents. For superconducting spintronic applications where higher spin-​polarizations are required in combination with metallic materials, ferromagnetic insulators are a highly attractive alternative. The primary application of ferromagnetic insulators is as tunnel barriers for which the inbuilt magnetic exchange splitting gives rise to different barrier heights for different spin signs and hence a large spin-​filtering effect can be achieved. For example, spin filtering in GdN-​based devices in the superconducting state can achieve polarizations that exceed 90% (Pal et al. 2014) and this material can be grown in combination with other nitrides such as AlN (insulator) and NbN (superconductor). However, GdN cannot easily be combined with elemental materials such as Al which would be beneficial as spin–​orbit coupling in NbN drastically reduces its spin-​diffusion length (Wakamura et al. 2015). Nevertheless, this problem may be overcome by investigating ways to avoid interdiffusion of N into Al which may result in an unwanted insulating barrier of AlN. Alternatively, Eu-​based chalcogenides such as EuO or EuS ferromagnetic insulators can be explored. Such materials can achieve high spin-​polarizations in excess of 90% (Santos et al. 2008) and have successfully been combined with Al in S/​F spin valves (Li et al. 2013). Theoretically and experimentally the conditions required for generating spin-​polarized triplet pairs are fairly well understood, particularly in S/​F devices involving equilibrium proximity effects and Josephson supercurrents. The majority of experiments involving supercurrent measurements have involved devices where the supercurrents flow perpendicular to the plane (Fig.  14.3a) with the notable exception of the work where planar devices (Fig. 14.3b) were fabricated in order to pass supercurrents into the half-​metal CrO2 (Keizer et al. 2006, Anwar et al. 2010, 2012). For devices in which the charge and spin degrees of freedom are addressable and in which magnetism can be controlled, non-​local multi-​terminal device geometries should be engineered so that non-​equilibrium transport involving dynamic interactions of triplet pairs and magnetic layers is made possible (Linder and Yokoyama 2011, Teber et al. 2010). Non-​local transport experiments involving triplet pairs have not so far been investigated experimentally although non-​local S/​F devices (see Fig. 14.3c) have been created in order to demonstrate long spin-​lifetimes of quasiparticle spin currents in superconducting Al (Quay et al. 2013, Yang et al. 2010), the absorption of pure spin currents into a layer of superconducting Nb (Wakamura et al. 2015), and charge accumulation in the normal metal N spacer of an S/​N/​S multi-​terminal device (Crosser et al. 2008).

  427

References (Chapter-14) 427 (a)

(b) S FI2 FI1 S

V

(c) V

V S

S FI1

FI1

S

FI2

Fig. 14.3  (a) Generic superconductor/​ferromagnet heterostructure junction; two ferromagnetic layers are shown. (b) Lateral superconductor/​ferromagnet/​superconductor junction. (c) Non-​local device in which spin accumulation generated by spin injection from the left superconductor/​ferromagnet junction can be detected in the right junction.

14.9 Summary and outlook Spin transport in the superconducting state has been demonstrated, and so there is real potential for the development of active devices based on spin currents. The key requirements for this to be realized are (a)  an efficient means of coupling spin currents to the magnetization states of individual circuit elements and (b) a method for amplifying spin signals so the circuit gain and signal propagation are possible. Although both of these have proved impossible to realize in the context of normal spintronics, there are extra ingredients available in the superconducting state which give grounds for optimism that superconducting spintronics might become a reality.

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432

15

Barriers in Josephson Junctions: An Overview M.P. Weides

15.1 Josephson effect 

433

15.2 Tunnel barriers 

443

15.3 Metallic barriers 

449

15.4 Semiconducting barriers 

450

15.5 Magnetic barriers 

451

15.6 Summary and outlook 

453

References (Chapter-15) 

454

Institute of Physics, Karlsruhe Institute of Technology, Karlsruhe, Germany and Materials Science in Mainz, Johannes Gutenberg University, Mainz, Germany

Quantum mechanics is most often used to describe matter on the scale of a few atoms, or less. Interestingly, some quantum properties are displayed on a much larger, that is, macroscopic scale. A classic example of such macroscopic quantum phenomena is the Josephson effect: a current flowing indefinitely long without any voltage applied across a barrier of superconducting electrodes. The unique properties of superconducting devices like ultralow power consumption, long coherence, good scalability, smooth integration with state-​of-​the-​art electronics, and outstanding magnetic field sensitivity suit them well for a wide range of applications. The issues relevant to fundamental and applied disciplines, like material characterization, particle detectors, ultralow-​loss microwave components, medical imaging, and potentially scalable quantum computers are commonly addressed using superconducting devices. For most applications, besides the superconducting material itself the key element is a superconducting junction. The main characteristic of such junctions, also called Josephson junctions (JJs), is the local reduction or even suppression of the critical current Ic in the barrier. These barriers affect the static and dynamics properties of JJs such as coupling strength, ground state, phase damping, and tunability of the critical current. This chapter aims to provide an overview on JJs. Following a general introduction to the field and the fundamental physics of JJs a variety of weak link elements (tunneling dielectric, metallic, semiconducting, ferromagnetic, constriction) will be systematically discussed. The present chapter will draw attention to various comprehensive reviews and books covering this field by Likharev (1979) on the theory of superconducting weak links, Barone and Paterno (1982) on the physics and application of the Josephson effect including an excellent review of device materials, Golubov et  al. (2004) on the theoretical basis and experiments of the current–​phase-​relation in JJs, Schäpers (2001) on semiconducting junctions, and

M.P. Weides, ‘Barriers in Josephson Junctions: An Overview’, in The Oxford Handbook of Small Superconductors. First Edition. Edited by A.V. Narlikar. © Oxford University Press 2017. Published in 2017 by Oxford University Press. DOI 10.1093/acprof:oso/9780198738169.003.0015

  433

Josephson effect 433 Buzdin (2005) on junctions with a ferromagnetic barrier. Our focus in this chapter is on JJs made by conventional superconducting films, that is, metallic films displaying superconducting properties below the critical temperature Tc of a few Kelvin. The most striking property of superconductors is their vanishing electrical resistance when cooled below the transition (or critical) temperature Tc, as discovered by Kamerlingh-​Onnes (1911). This effect was later called superconductivity. The ideal diamagnetic response to an applied magnetic field (Meissner effect) and the quantization of the magnetic flux through a hole are two further unusual and fundamental characteristic properties of superconductors. According to the microscopic theory by Bardeen, Cooper, and Schrieffer (BCS-​ theory) (de Gennes 1999), developed in 1957 for so-​called low-​Tc superconductors, the conduction electrons are interacting with phonons of the crystal lattice. If the temperature is below Tc this interaction gives rise to an effective attraction between electrons and leads to the formation of Cooper pairs where two electrons with opposite spins and momenta in their most frequent appearance act as singlet pairs (Eschrig 2010). This Cooper pair singlet has no center of mass momentum as the momenta from both spins cancel out. The pairs are correlated due to the Pauli exclusion principle for the electrons. The total spin of each pair is an integer. Therefore the Cooper pair has a bosonic character and below Tc all Cooper pairs are condensed into a common electronic ground state of the superconductor. In order to break a pair, the energies of all other pairs have to be changed, that is, there is an energy gap ∆ (T = 0) = 1.764kBTc for single-​particle excitation (kB being the Boltzmann constant). The energy gap is highest at low temperatures but vanishes at the transition temperature Tc when superconductivity ceases to exist. Quasiparticles are broken Cooper pairs and considered as an excitation carrying an energy above the ground state. The bosonic nature of the Cooper pair enables the description of the superconducting state by an effective macroscopic wave function Ψ = n e i θ with an amplitude related to the Cooper pair density n and phase θ. A remarkable phenomenon is the quantization of magnetic flux through a superconducting ring. The electromagnetic induction effect (Lenz’s law) and the vanishing electrical resistance generate a persistent current whose magnetic flux is frozen. In analogy to atomic physics, where the wave function of the orbital electron has to contain an integral number of the wavelength, such a fluxoid can only assume integral multiples of the magnetic flux quantum h Φ0 = = 2.07 × 10−15 Tm 2 as confirmed experimentally by Deaver and Fairbank 2e (1961) and Doll and Näbauer (1961). Its existence is required by the single-​ valuedness of the wave function Ψ along a superconducting ring.

15.1 Josephson effect The static and dynamic properties of JJs will be discussed, and properties of two coupled superconductors will be given. The coupling is meditated by the

434

434  Barriers in Josephson Junctions: Overview Ψ(x)

Fig.  15.1 Two superconductors separated by a barrier. The proximity effect leads to an overlap of the wave functions Ψ, i.e. their coupling, and the Josephson effect. θ1, θ2 are the phases of superconducting wavefunctions Ψ1, Ψ2 of each superconducting condensate.

Ψ1 θ1

Ψ2 θ2

SC 1

Barrier

X

SC 2

barrier layer, whose specific properties (dielectric, normal, or ferromagnetic metallic, semiconducting) and geometry affect the coupling strength. Typical implementations are SIS or SNS-​type sandwiches (I: insulator, N: normal metal, S:  superconductor), point contacts (1D), “Dayem” (2D) or variable thickness (3D) constrictions, and normal or semiconducting nanowires (Likharev 1979). The dynamics of charge carriers and of electromagnetic fields in these JJs are related to the phase difference γ of the wave functions Ψ of each superconducting condensate. We will derive the Josephson equations (Josephson 1962) following Feynman (1964) by solving the Schrödinger equation for two coupled quantum mechanical systems with indices i, j and coupling constant K related to overlap of wave functions in Fig. 15.1: ∂Ψ i − i = E i Ψi + K Ψ j h ∂t

(



)

Each wave function Ψi = ni e i θi is interpreted as the square root of the Cooper pair density ni with phase θi. Inserting the wave function leads to

. . −i ni i θi iθ e + i ni e i θi Ψi = E ni e i θi + K n j e j  2 ni

(

)

Separating this equation into the real and imaginary parts, we obtain .





ni K n j sin θ j − θi (15.1) =  2 ni

(

.

i ni θi = −

)

i E ni + K n j cos θ j − θi  (15.2)  h  .

(

.

)

Considering conservation of charge, θi = θ j , and two identical superconductors (ni = nj), we find

ni =

2K ni sin θ j − θi = − n j h

(

)

  435

Energy UJ

Josephson effect 435

Supercurrent Is

EJ

Ic –2π

2π

Phase γ

Now we introduce the gauge-​invariant phase difference



γ = θ j − θi −

2π ∫ Adl Φ0 (15.3)

For vanishing magnetic fields the vector potential A can be set to zero. Rewriting the temporal change of particle density n as current across the barrier, Eq. 15.1 leads to the first Josephson equation I s = I c sin ( γ ), (15.4)



describing a DC Josephson current proportional to the sine of the phase difference across the barrier, see Fig. 15.2. This current may take values between −Ic and Ic. From Eq. 15.2 we obtain the temporal change of the phase difference γ. Setting 2π E j − E i = 2eV yields the second Josephson equation γ = V or, as commonly given Φ0 Φ V = 0 γ (15.5) 2π where V is the voltage across the junction. With a fixed voltage V the phase γ will vary linearly with time. According to Eq. 15.4 this generates an oscillating current with 2eV amplitude Ic and frequency , that is, the JJ is a voltage-​to-​frequency converter . h A complementary approach to visualize the dynamical behavior of the Josephson equations is seen by taking the temporal derivative of Eq. 15.4 and rewriting γ with Eq. 15.5: .



I s = I c cos ( γ )

2π V Φ0

Recalling that an inductance L obeys V = LI we define the Josephson inductance

Fig. 15.2  Current–​phase relation I s ( γ ) (Eq. 15.4) and corresponding energy–​ phase U J ( γ ) relation (Eq. 15.7).

436

436  Barriers in Josephson Junctions: Overview LJ =



Φ0 1 2π I c cos ( γ ) (15.6)

with several remarkable properties: 1. Non-​linear in phase γ due to the 1/​cos(γ) term 2. Its absolute value varies between

Φ0 and infinity 2π I c

3. For π/​2 < γ < 3π/​2 the inductance LJ is negative. As with any inductance, the JJ can store the energy Φ Φ I U J = ∫ IVdt = ∫ I c sin ( γ ) 0 γ dt = 0 c ∫ sin ( γ ) d γ = E J 1 − cos ( γ ) 2π 2π    (15.7) Φ0I c , see Fig. 15.2. In the last equation we have 2π calculated the integral from the ground state γ = 0 to the final phase γ. As with any potential energy, U J ( γ ) depends only on the current state of the junction, but not on history or velocities. U J ( γ ) has a minimum equal to zero for the ground state γ = 2π · n where n is any integer. In the quantum regime, the characteristic energy EJ together with the e2 charging energy of the junction, EC = (junction capacitance C), deter2C mine whether the classical treatment of phase dynamics is a good approximation (E J  EC ) with large fluctuations in charge, or if the charge is a good quantum number (E J  EC ). In this chapter, we will consider JJs operating in the phase regime, where the phase can be treated as a quasiclassical variable. with the Josephson energy E J =

15.1.1 Critical current The critical current Ic through a tunnel junction and its temperature dependence were calculated from microscopic theory by Ambegaokar and Baratoff (1963a, 1963b) as

I c Rn =

 ∆ (T )  π ∆ (T ) tanh   (15.8) 2 e  2kBT 

where Δ(T) is the temperature-​dependent energy gap of the (identical) superconductors. This is a remarkable result since the critical current Ic through the barrier depends directly on the normal electron transparency Rn of the junction. For barrier types other than tunnel barriers the product Ic Rn is constant as well, and depends on their density of states, geometry, and transport regime (ballistic, diffusive) (Likharev 1979).

  437

Josephson effect 437

15.1.2 Resistively and Capacitively Shunted Junction Model We will review the properties of the phase dynamics in a standard JJ, which can be described within the framework of the Resistively and Capacitively Shunted Junction (RCSJ) model introduced by Stewart (1968) and McCumber (1968). This simple model of equivalent circuits for JJs describes the I–​V characteristic for bias currents below and above the critical current Ic. At zero voltage bias, the JJ acts as a nonlinear inductor (Eq. 15.6) shunted with a self-​capacitance C, leading to an anharmonic LC oscillator. The effect of damping due to intrinsic effects of the environment is described by a shunt resistance R. According to the Kirchhoff circuit law equations, the total current flowing through such a system is the sum of the currents in each of the three paths, that is, the Josephson-​current Is, the quasiparticle current Iq, and the displacement current Id (see Fig. 15.3). Assuming that the resistance R is voltage-​independent I q =V / R (only valid for small tunnel currents) and the total current I can be expressed as I = I s + I q + I d = I c sin ( γ ) +



V + CV R

with capacitor C = I d / V carrying the displacement current and ohmic shunt resistance R for the normal electron current. The voltage Vc = IcR is the characteristic voltage of the JJ. Inserting the second Josephson relation (Eq. 15.5) yields a differential equation for phase difference γ: Φ .. 1 Φ0 0 = − I + I c sin ( γ ) + γ + C 0 γ (15.9)    R 2π 2π



∂U / ∂γ

(b) Iq

Is

Id

LJ

C

R

Energy UJ

(a)

Phase γ

Fig. 15.3  (a) Equivalent circuit in the RCSJ model. R and C denote the shunt resistor and capacitor, respectively.The ideal Josephson junction with Josephson inductance LJ is depicted by the cross. Indicated are the quasiparticle Iq, Josephson Is, and displacement Id currents. (b) Potential of Josephson junction under current bias. The plasma oscillation is indicated.

438

438  Barriers in Josephson Junctions: Overview The term −I + Ic sin(γ) can be understood as ∂U J / ∂γ where the potential  I  energy from Eq. 15.7 is rewritten as E J  − − cos ( γ ) due to the applied bias  Ic  current I. Under current bias, this energy has the form of a statically tilted washboard potential (Fig. 15.3b). Normalizing the time to the inverse of the plasma frequency t = ω −p10 τ, we obtain from Eq. 15.9 the equation of motion for the Josephson phase 0 = − γ + sin ( γ ) +



where the plasma frequency is given by ω p 0 =

.. 1 γ + γ (15.10) βc

1 = L JC

2π I c . Eq. 15.10 is a nonΦ 0C

linear second-​order differential equation and identical to that describing a driven and damped pendulum. The dimensionless, material-​dependent parameter

βc =



2πI c R 2C 2πI c R = RC Φ0 Φ0

is called the Stewart–​McCumber parameter. It is the square root of the quality L factor Q = R of the LCR resonator and describes the damping of the plasma C oscillations in the junction. For tunnel junctions, the IcRn is almost independent of the barrier height, thickness, or area, and the product RC independent of the tunnel barrier area. Junctions with dielectric barriers such as AlOx have a high RC product, that is, a large βc, since the damping is weak. In the current–​voltage characteristic, shown in Fig. 15.4, this manifests as a large hysteresis with the particle continuously overcoming the local tunnel barriers (γ ∝ V ≠ 0) for the current-​biased mode measurements. Junctions with a metallic interlayer instead of the tunnel barrier,

Underdamped Rn

Current bias

Overdamped

Fig. 15.4  Current–​voltage characteristic for over-​and underdamped current-​ biased junctions. Indicated are normal state Rn and subgap-​resistance Rsg.

Ic

Rsg 0 0

2∆/e

Voltage

  439

Josephson effect 439 that is, SNS-​type junctions, typically have βc below 0.7 and are overdamped (see Fig. 15.4) due to the reduced C and R values compared to SIS junctions. After reaching the voltage state, and in the low damping limit (βc 1), the JJ can still be in the voltage state for bias currents I < Ic and switches back to the zero-​voltage state at the retrapping current Ir. For strongly underdamped junctions βc can be calculated from measured Ic and Ir as β 1

2

c  4 I  βc ≈  c   π Ir  By coupling the JJ to high-​frequency photons, the DC current–​voltage characteristic exhibits discrete steps in voltage at nhf/​2e (Shapiro steps) due to the response of the supercurrent with integer n. Applying a fixed voltage V across the junctions, the phase varies linearly with time and the current will be an AC 2e current with frequency V , that is, the junction can act as an exactly reproduch ible voltage-​to-​frequency converter to be used in precision metrology.



15.1.3 Magnetic field dependence The phenomenological theory of superconductivity given by Ginzburg and Landau (1950), developed prior to the BCS theory, gives a very good description of the macroscopic properties at T ≤ Tc by using Ψ as the pseudowavefunction acting as a complex order parameter. Two characteristic lengths can be introduced:  at the vicinity of a superconductor surface or near the interface to an ordinary conductor, the order parameter Ψ varies from its bulk value over the coherence length ξ. The second characteristic length is the London penetration depth λL, which describes the fall-​off of magnetic fields and screening currents inside a superconductor (see Fig. 15.5a). More generally, the penetration depth essentially plays the role of skin depth in a superconductor down to zero frequency where the Meissner effect occurs. The magnetic field dependence of the supercurrent is a powerful tool to check the actual degree of uniformity of the Josephson barrier. The aim of this section is to derive the influence of an in-​plane magnetic field on the supercurrent through the junction (see Fig. 15.5). For the sake of simplicity, the magnetic field of the bias current is neglected. For the magnetic field penetration of the junction, the effective magnetic thickness caused by screening currents  t   t  is Λ = d + λ L tanh  1  + λ L tanh  2  (Weihnacht 1969). It is defined by the  2λ L   2λ L  London penetration depth λ L , the barrier thickness d, and the thicknesses of the superconducting electrodes t1 and t2. For thick superconducting electrodes (t1 , t2  λ L ) this reduces to Λ = d + 2λ L . The characteristic length scale for spatial Φ0 variation of γ(x) is the Josephson penetration depth λ J = . This length 2πµ 0d ′jc

440

440  Barriers in Josephson Junctions: Overview (a) z

(b)

j

y w x

(c) Ic/Ic0

λL d λL

λL d

1

4

2

3 –3

dx

λJ

–2

–1

0

1

L

2

3

Φ/Φ0

µ0H

Fig. 15.5  (a) Distribution of the screening currents for a Josephson tunnel junction in magnetic field H, (b)  closed integration path across tunnel barrier, and (c)  critical current vs an applied in-​plane magnetic field produces a Fraunhofer diffraction pattern, i.e. periodic oscillations of the critical current Ic as function of applied flux Φ = 2λL · L · µ0H. Non-​zero minima or non-​periodic modulations indicate inhomogeneities in the barrier.

indicates how far a magnetic field can penetrate into an extended JJ of length L  λ J . The quantity d ′ is given by

 t  t  d ′ = d + λ L coth  1  + λ L coth  2   λL   λL 

and reduces to d ′ = d + 2λ L if t1 , t2  λ L . The Josephson penetration depth λ J is inversely proportional to the square root of the critical current density jc, that is, the normalized junction length = L / λ J can be controlled by jc. If either the junction length L or width w is much larger than λ J the so-​called long Josephson junction is obtained where the Josephson phase can be a function of one or two spatial coordinates. On the contrary, the short Josephson junction is characterized by the Josephson phase γ(t), which is only a function of time. The magnetic field H parallel to the layers is related to the gauge-​invariant phase difference γ as introduced in Eq. 15.3. The phase difference for two different points separated by dx at coordinate x is then (Fig. 15.5) 4 2π  1 γ ( x + dx ) − γ ( x ) = A ( x + dx ) − ∫ A ( x ). The enclosed flux in the path 1234 ∫  2 3   Φ0 (Fig. 15.5b) is calculated as

2

3

4

1

δΦ = ∫ µ 0 HdS = ∫ο Adl = ∫ Adl + ∫ Adl + ∫ Adl + ∫ Adl (15.11) S 2 4 1   3   0

0

The second and fourth terms in Eq. 15.11 are vanishing when the horizontal parts of the integration path are considerably deeper inside the superconductor than λL. For thin electrodes of thickness ≈ λ L , the reader is referred to Weihnacht

  441

Josephson effect 441 (1969). Taking the enclosed magnetic flux Φ into account one gets for a differentially small section dx of the junction

γ ( x + dx ) − γ ( x ) 2π = γ x = δΦ = Λµ 0 H dx Φ0

where µ0 is the vacuum permeability and Λµ0H is the magnetic flux per unit length. Therefore, the magnetic field induces a gradient of the Josephson phase γ across the JJ. Assuming that L ≤ λJ, the magnetic field penetrates uniformly, and integration leads to

γ=



2π Λµ 0 Hx + γ 0 , Φ0

and the current density as

  2π Λµ 0 Hx + γ 0  . js ( x ) = jc sin ( γ ( x )) = jc sin   Φ0 

The local supercurrent density jc is oscillating sinusoidally along the junction. The total supercurrent is given by I c ( H , γ 0 ) = ∫ js ( x ) dx . The maximum supercurrent for a given field is calculated by maximizing I c ( H , γ 0 ) with respect to γ 0 :



 2π  Λµ 0 Hx + γ 0  dx = I c I c ( H ) = ∫ jc ( x ) sin  Φ  L

0

 Φ sin  π   Φ0  Φ π Φ0

assuming at the last equal sign a homogeneous critical current density jc, rectangular junction area with length L, Ic = L2jc, and Φ = µ0HΛL as flux induced by H on the total JJ cross-​section. This results holds only for short JJs (L ≤ λJ) and if the self-​fields of the bias current are negligible. The sinc-​type critical current diffraction pattern is often referred to as the Fraunhofer pattern in analogy to the diffraction of light through a slit (Eq. 15.5). The critical current diffraction pattern is a powerful tool to determine the uniformity of jc across the barrier and the presence of shorts, leading to modifications in oscillation fields and a non-​zero critical current baseline, respectively.

15.1.4 Superconducting electrodes In general, superconducting electrodes with relatively high transition temperatures Tc are required to facilitate operation at convenient accessible temperatures like that of boiling helium at atmospheric pressure, 4.2 K.  Besides, a high-​Tc material is favorable since it exhibits a large energy gap Δ = 1.76kBTc. The coherence length ξ determines the material’s sensitivity to variations in physical

442

442  Barriers in Josephson Junctions: Overview microstructure or chemical composition, whereas the London penetration depth λL is of importance for magnetic field applications, or high kinetic inductance circuits (i.e. in superconducting films where the inertial mass of mobile charge carriers in alternating electric fields is manifested as an equivalent series inductance) (Beasley 1982). Of relevance are mechanical properties such as low intrinsic stress of the thin films to reduce destructive tension on underlying films, and low thermal contraction as well as high mechanical strength in order to avoid damage and device failure during thermal cycling. Last, low substrate temperature during deposition reduces the requirements on the thermal stability of underlying films. The superconducting electrodes can be divided into soft and hard metals and their alloys. Typical substrate materials are sapphire, silicon, silicon-​oxide, and silicon-​nitride. Successful deposition of films depends on the proper substrates offering a good lattice match and a clean, even surface. Soft superconductors (typical candidates are lead, tin, or indium) set weak requirements on the deposition technique since thermal evaporation can be used. Of larger concern are hazardous waste products and low thermal stability under repetitive thermal cycling, leading to sample degradation. Deposition (temperature, substrate, growth, and rate) and storage conditions have to be chosen carefully. To overcome the mechanical and chemical restrictions, superconducting lead alloys, for example with bismuth or indium, were developed up to the mid-​1980s. Lead-​based junctions were used widely; however, lead oxide tends to develop defects in the barrier (called pinholes) that short-​circuit the electrodes when the device is thermally cycled between cryogenic temperatures and room temperature. Besides, lead is one of the oldest known work and environmental hazards, and has to be avoided wherever possible. In consequence, lead is no longer widely used and was replaced by niobium-​based junctions. Another relatively soft material is aluminum featuring simple processing and patterning, self-​limited oxidation, and good reproducibility. Its low gap limits its use to applications in the milliKelvin temperature range, for example for quantum circuits operating at frequencies f around 10 GHz ≈ kB/​h 0.5 K. Such frequencies are low enough to not cause dissipation (i.e. quasiparticles) in the superconductor, and high enough to prevent thermally populated states when operating in conventional dilution cryostats at T = 10 mK. On the contrary, hard superconductors such as the transition metals in the d-​block of the periodic table are mechanically hard, and provide further advantages such as chemical stability and high critical temperature. These elemental superconductors feature good adherence to the substrate, high stability of oxides, and no or weak degradation during thermal cycling. They tend to incorporate water vapor and oxygen from the gas phase, which negatively affects their Tc. From a technological point of view, high-​quality films of transition metals require ultralow background pressure, or fast deposition rates to reduce the incorporation of oxide defects. Exemplary materials are vanadium, tantalum, or niobium. Today, mostly niobium-​based tunnel junctions with AlOx, AlN, or MgO tunnel barriers are used when higher temperatures, larger gap, or increased critical current density jc are required, demanding optimized substrate

  443

Tunnel barriers 443 Table 15.1  Typical values for bulk low-​Tc superconductors (Beasley 1982, Kleiner and Koelle 2004). Tc

λL

ξ

λL/​ξ

(K)

(nm)

(nm)

2Δ

2Δ μ0Hc

(meV)

(T)

Niobium

9.25

39

38

1.02

3.0

0.21

Aluminum

1.17

16

1600

0.01

0.34

0.01

Lead-​bismuth

8.3

202

20

0.01

3.4

–​

7.2

237

83

–​

2.7

0.08

17.3

300

4

–​

6.4

0.19

Lead Niobium nitride

and deposition conditions. Their high reactivity limits the deposition of thin films with high transition properties as a low background pressure with minimized out-​gassing rate of the vacuum chamber and a high deposition temperature are required. See Table 15.1 for an overview.

15.2 Tunnel barriers In a superconductor–​insulator–​superconductor tunnel junction (short superconducting tunnel junction), the currents flow through the insulating layer via the process of quantum tunneling. JJ devices based on tunnel barriers have the widest range of application due to excellent parameter control and very weak phase damping as the Stewart–​McCumber parameter βc can be high. Moreover, the theoretical analysis of a tunnel junction leads to the simplest case where the current–​phase relationship is always sinusoidal: I s = I c sin γ . The critical current Ic of the tunnel junction is determined by the area and thickness of the tunnel barrier, and by the properties of the superconductors on either side of the barrier. For a junction with identical superconductors, the critical current Ic is related to the superconducting gap Δ and the normal state resistance Rn of the tunnel junction via the Ambegaokar–​Baratoff equation (Eq. 15.8). For JJs, the two most characteristic properties are (a) the critical current Ic, being directly related to the barrier transmission ∝ 1/​Rn, and (b) the junction capacitance C (related to phase damping βc). For a tunnel junction this calculation is straightforward. Barrier transmission and specific capacitance:  The quantum tunneling rate across the barrier depends on (a) the number of electrons reaching the barrier, (b) the tunneling probability across the barrier, and (c) the number of unoccupied energy levels on the other side. It can be calculated using the Wentzel–​Kramers–​ Brillouin approximation for a tunnel barrier. The normal state tunnel resistance Rn is proportional to exp

(∫

t 0

)

2m (Φ ( x ) − E )dx , where m denotes the electron

444

444  Barriers in Josephson Junctions: Overview effective mass, Φ(x) the barrier potential, and E the potential barrier at the beginning and the end t of the barrier (i.e. at x = 0 and x = t). Introducing a mean bar-

(

(

)

)

rier height Φ , the tunnel resistance can be approximated as exp t 2m Φ − E dx . Reducing the barrier potential height Φ(x) while keeping the barrier thickness t constant decreases Rn and the Josephson inductance LJ, and increases the resonance frequency (or plasma frequency) f = 1 / 2π L JC =

1 2eI c cos ( γ ) , 2π C

where C is the shunting capacitance. Tunnel barriers based on conventional AlOx dielectrics have a barrier height Φ ≈ 1.5–​2.0 eV and a thickness of 1–​2 nm. Another characteristic property, the specific capacitance cs of a tunnel junction, can be modeled as a parallel plate 1 capacitor cs = εε 0 . For the most relevant junction type nowadays, Nb–​AlOx–​Nb t junctions, van der Zant et al. (1994) determined a specific capacitance between 40 and 110 fF/​µm2 for critical current densities jc between 0.3 and 20 kA/​cm2, respectively. A somewhat weaker dependence on jc was found by Maezawa et al. (1995) with values between 40 and 70 fF/​µm2. Like the potential height, the dielectric constant of the AlOx barrier is not a well-​defined property with typical values ε = 4.5–​8.9 at around room temperature (Gloos et al. 2003), smaller than that of bulk Al2O3. Superconductor–​tunnel barrier–​superconductor sandwich structures are fabricated from natural (e.g. oxide grown out of the bottom electrode) or artificial (deposition of dielectric films) barriers. In general, the first method provides better results, although some high-​quality junctions with artificial barriers have been obtained. The exponential dependence of normal state resistance Rn on thickness t requires barrier films on the order of a few nanometers, rendering the junctions very sensitive to even slight variations in thickness. Here, the magnetic field dependence of the critical current (see Fig. 15.5c) is a simple and reliable tool to check the uniformity of the barrier. Oxide materials are used frequently as dielectric tunnel barriers. A nonporous oxide protects the underlying metallic superconductor against further oxidation. The metal-​oxidation reaction depends on the change in free energy. For some elements such as alkali and alkaline earth metals (for example magnesium) the oxides require less volume than the consumed metal, leading to porous films and ongoing oxide formation including contractive stress. To the contrary, protective oxides form a homogeneous film where either positively charged metal or negatively charged oxygen ions are transported across the oxide films. This ionic diffusion is governed by an electric field E across the barrier which leads to self-​limitation of the oxide formation for increasing oxide thickness. The time dependence of the oxide growth depends on the method and target range of thickness. Cabrera–​Mott theory of  oxidation:  The basic theoretical framework for thin film oxidation was elaborated by Mott (1939, 1940, 1947)  and Cabrera and Mott (1949) for room-​temperature oxidation of very thin films of nanometer

  445

Tunnel barriers 445 dimensions. The oxidation of thin metal films is a complex phenomenon involving many parallel processes participating simultaneously. The motions of ions and electrons through the disordered oxide layer are considered independently. They depend on oxidation kinetics and the chemical and physical nature of the metal films as the metal species diffuse and not the oxygen. After the initial oxide formation the oxidation of the metal film continues via electric field-​assisted diffusion of the electronic and ionic particles. This electric field across the barrier is related to the contact potential difference of the metal film and the adsorbed atoms on the surface of the oxide. Oxidation techniques:  The oxide growth is affected by the method used to provide oxygen molecules and ions, see Fig. 15.6. Typical methods for formation of tunnel barriers are (a)  thermal, (b)  UV-​assisted, and (c)  plasma-​assisted oxidation. The activation energy for diffusion within the oxide depends on temperature. Thermal oxidation is governed by the Arrhenius equation for the chemical reaction rate k ∝ exp  − E a / (kBT ) with the activation energy Ea. The oxidation of soft metals like aluminum or magnesium is done generally between room temperature and up to a few hundred degrees Celsius. For fixed oxidation temperature, the critical current density jc shows a nearly universal dependence on the product of oxygen partial pressure and oxidation time, as observed for Nb-​Al–​AlOx–​Nb JJs by Kleinsasser et  al. (1995b). While the self-​limiting oxide growth is beneficial to achieve the required uniformity, only tunnel dielectrics of low thickness can be obtained via thermal oxidation. For thicker barriers, ultraviolet light stimulates oxidation in an oxygen atmosphere since the rate of oxygen incorporation can be enhanced dramatically compared to thermal oxidation, which in turn leads to a significant improvement in the oxide film quality (Tsuchiya et al. 2009), see Fig. 15.6c. First, UV light dissociates molecular oxygen to form O and O3. Both are highly reactive and thus lead to the faster formation of an oxide layer. Second, the photoemission

(a)

(b)

(c)

Electric field E Conduction band A Fermi level EF

A+B

B

Metal

Oxide

Adsorbed oxygen O2

hf Oxygen ions O–

Oxygen

Fig.  15.6  Schematic band levels for oxidation of metals before (a)  and after (b)  equilibrium is reached. Tunneling of electrons from the metal/​oxide interface lifts the lowest unoccupied level of adsorbed O2, initially below the Fermi level of aluminum, and creates oxygen ions O at the oxide–​gas interface. The electric field E assists the metal ions in moving through the oxide and forming additional oxides at the oxide–​gas interface. In (c) photoemission of electrons from the metal film increases the electronic equilibrium of the adsorbed O levels, the electric field E, and finally the limiting thickness of the oxide film.

446

446  Barriers in Josephson Junctions: Overview of electrons from the metal film due to the incident UV light enhances the oxidation rate, as the induced high electric field E increases the ionic currents within the oxide film (Cabrera and Mott 1949). The thick oxide results in very low critical current densities jc down to 1 A/​cm2 as observed for Nb-​Al–​AlOx–​Nb JJs (Fritzsch et al. 1998). To oxidize hard materials like niobium (Greiner 1971, Kleinsasser and Buhrman 1980), or to form ultra-​thick barriers from soft metals, an oxygen plasma is created by applying either DC or RF fields. The externally applied electric field enhances the migration of ions across the barrier by providing more oxygen ions and kinetic energies, thereby much thicker oxide layers are possible. The experimental scene of the most important types of tunnel junctions is now discussed.

15.2.1 Al–​AlOx–​Al junctions Aluminum-​based tunnel junctions were explored first by Giaever (1960) and Fisher and Giaever (1961) to study electron tunneling between superconductors and normal metals, respectively. The low critical temperature of aluminum (Tc = 1.1 K) limits their potential to milliKelvin applications such as quantum circuits, detectors, or metrology. Fig. 15.7 provides an overview on several deposition techniques that have been developed, ranging from cross-​shaped junctions with orthogonal leads to in situ, shadow-​evaporated junctions offering ultra-​small size and less contamination. Shadow evaporation requires oblique angle thin-​film deposition such as thermal or electron-​beam evaporation, and cannot be used for isotropic deposition techniques such as sputtering. For the low pressures of Cross junction

Niemeyer–Dolan junction

90º rotated

Al1

Substrate

AlOx

Al2

1st evaporation Insensitive Resist Sensitive

2nd evaporation

Resist

Bridge-free junction (after 1st & 2nd evaporation) Insensitive resist Sensitive resist

Fig.  15.7 Schematic cross-​sectional diagrams for cross (Fisher and Giaever 1961), Niemeyer–​ Dolan (Dolan 1977, Niemeyer 1974, Niemeyer and Kose 1976), and bridge-​free junctions (Lecocq et al. 2011) using single or bilayer resists. The first technique suffers from ex situ sample handling between the two steps (requiring extensive cleaning before oxidation). The second method facilitates in situ sample handling, and the third method provides increased mechanical robustness of the resist mask and the accessible range of the junction size due to the absence of bridges, allowing patterning of junction areas ranging from the nanometer to thousands of micron squares. Both Niemeyer–​Dolan and bridge-​free techniques are based on bilayer resists where the lower resist has an undercut due to higher sensitivity to exposure.

  447

Tunnel barriers 447 a typical evaporation process, the mean free path of particles is very large (e.g. 100 m for a pressure of 10−6 mbar) and the evaporated particles do not scatter before reaching the substrate. For sufficiently large distances to the evaporation source a unidirectional beam of particles is deposited on the substrate. For JJs of low capacitance the lateral dimensions of the tunnel barrier have to be kept in the order of 100 nm. This is below the optical resolution limit and thus, optical lithography is not applicable. Instead, an electron beam lithography process is used to reach feature dimensions below 10 nm. The simplest geometry of JJs is the cross-​shaped junction consisting of crossing narrow superconducting strips with a width corresponding to one dimension of the junctions (see Fig. 15.7). Before deposition of the second strip a cleaning and reoxidation process is included to form the tunnel barrier. No side-​wall isolation is required. The conventional (so-​called Niemeyer–​Dolan) shadow evaporation technique (Dolan 1977, Niemeyer 1974, Niemeyer and Kose 1976), utilizes a resist mask in the shape of a free-​hanging bridge to interrupt the deposited aluminum films. Two successive depositions are carried out from two different angles to provide two electrodes which are only connected by a small overlap. Again, the surface of the first aluminum film is oxidized before the second deposition to create a tunnel junction. This technique, while employed widely, relies on a suspended resist bridge, which collapses sometimes during plasma cleaning and sets a limit on the maximal junction size. The bridge-​free technique (Lecocq et al. 2011) increases the mechanical robustness of the resist and allows one to extend the junction size to more than 104 µm2 without having to modify the resist stack height and evaporation angles.

15.2.2 Amorphous Si:H barrier JJs with grown silicon barriers were studied intensively in the 1980s (Bradley et al. 1989, Jillie et al. 1982, Kroger et al. 1979, Shinoki et al. 1981, Smith et al. 1982). Some high-​quality junctions have been obtained; however, they could not match the strongly underdamped, large capacitance (50 fF/​µm2), and very homogeneous AlOx tunnel barriers developed at the same time (Gurvitch et al. 1983).

15.2.3 MgO barrier A further improvement in critical current density is achieved by replacing the electrodes with niobium nitride, which offers a superconducting energy gap about 1.7 times larger than niobium, with a critical temperature about 17 K. For instance, such a SIS mixer could operate at frequencies above 1 THz, and the circuit operations can be extended to higher temperatures up to around 10 K. Niobium-​nitride based trilayers with native oxides, such as AlOx, show a reduced gap for the upper electrode, as oxygen atoms from the barrier interact with the freshly deposited NbN. Improved stability and gap are offered by aluminum nitride or magnesium oxides as an artificial barrier since they exhibit high thermodynamic stability during the deposition of the counter electrode. MgO and

448

448  Barriers in Josephson Junctions: Overview NbN both have a cubic crystal structure with a lattice mismatch of only a few percent, and epitaxial MgO is employed as a high-​quality tunnel barrier. Grown first by Shoji et al. (1985) as a trilayer structure, magnesium oxide barriers have been implemented as edge-​junctions for very small junction areas (Hunt et  al. 1989)  or overdamped junctions with non-​hysteretic current–​voltage curves by adding a sidewall shunt (Senapati and Barber 2009). Nowadays, epitaxial NbN/​ MgO/​NbN trilayers with good tunneling characteristics and a large range of critical current densities are realized (Kawakami et al. 2001).

15.2.4 AlN barrier The operating speed and integration level of superconducting electronic circuits depend primarily on the critical current density jc, and are improved by using an ultra-​thin and homogeneous barrier layer. The conventional AlOx barriers are formed by oxidizing a thin aluminum overlayer on the niobium base electrode. As the reactivity of aluminum with oxygen is high, a relatively thick barrier is formed as precise control of the oxidation process is difficult at low levels of oxygen exposure. On the contrary, its reactivity with nitrogen is very low, and nitride barrier formation is done commonly using a plasma process to control the level of nitridation precisely (Kleinsasser et al. 1995a). Moreover, aluminum nitride is refractory (as niobium or niobium nitride), stable, and chemically and structurally compatible with the nitride electrodes. Compared to Nb-​Al–​AlOx–​Nb junctions, requiring only the deposition of metals and their oxidation, the growth of nitride metals and dielectrics is more complicated since the tunneling properties depend strongly on the quality of the superconducting electrodes and the barrier–​electrode interface properties (Wang et al. 1999).

15.2.5 Nb-​Al–​AlOx–​Nb Niobium-​based JJs can be operated at liquid helium temperatures as the critical temperature is about 9 K. The most obvious approach to create niobium-​based junctions, implementing niobium oxide as the tunnel oxide barrier, is challenging as the native oxide has both a large dielectric constant and problematic chemistry at the Nb/​NbOx interface where insulating and metallic niobium oxides are formed. In 1982 it was discovered that Nb-​Al–​AlOx–​Nb junctions are of high quality, see Geerk et al. (1982) and Huggins and Gurvitch (1985). In these devices a 4–​10 nm thick Al film is sputtered on Nb and subsequently exposed to oxygen. The barrier is nearly free of defects because the thin aluminum layer wets the niobium surface so that AlOx is formed homogeneously over the whole junction area. The junction area is defined by a selective niobium etching process (SNEP) (Gurvitch et al. 1983), selective niobium anodization process (SNAP) (Kroger et al. 1981), selective niobium overlap process (SNOP) (Villegier et al. 1985), or derivatives thereof, from a Nb-​Al–​AlOx–​Nb trilayer film covering the whole substrate. These self-​aligned fabrication processes require fewer steps and better alignment than the conventional technology for trilayer stack pattering. The Nb-​Al–​AlOx–​Nb technology has been widely accepted for superconducting electronics.

  449

Metallic barriers 449

15.3 Metallic barriers At the superconducting–​metallic interface electric charge is exchanged, leading to the reduction of Cooper pair density n within the superconductor and the induction of Cooper pairs into the metal N via the proximity effect. The Cooper pairs maintain their singlet nature while penetrating across the superconductor–​metal interface at some distance inside the metal. Within the metal, the Cooper pair density decays with a characteristic length scale ξN. In the clean limit, where the electron mean free path  is larger than the superconducting coherence length ξ = 2hvF/​πΔ as calculated from the BCS theory (Fermi velocity vF), the coherence length within the normal metal layer is



ξ N (T ) =

vF 2πkBT

For the opposite case, the dirty limit where   ξ:



ξ N (T ) =

D 2πkBT

where D = vF  / 3 is the diffusion constant of the electrons in the normal conductor. Approaching 0 K, both equations diverge, meaning that, in principle, the Cooper pair density can extend deeply into the normal metal. However, at the interface itself the charge transmission can change drastically, thereby reducing significantly the induced density. In superconductor–​normal metal–​superconductor junctions, ξN determines the characteristic length scale of current transport. The critical current density jc drops with the thickness of the junction dN approximately as exp(−dN /​ξN), whereas the normal state resistance Rn increases only linearly with dN. The IcRn product falls exponentially in dN if the barrier thickness is not thin compared to ξN. SNS junctions have a negligible capacitance C and a low normal state resistance RN, leading to strong damping of the plasma oscillation (βc < 0.7) and a non-​hysteretic current–​voltage dependence. Such junctions are of interest—​ for instance—​for Rapid Single Flux Quantum (RSFQ) logic circuits which encode, process, and transport digital information based on magnetic flux quanta. Since junctions with non-​hysteretic behavior allow changing the output voltage through control of the bias current, overdamped Josephson arrays are employed for example for arbitrary time-​varying quantum standards. Whereas the standard tunnel junction has a sinusoidal current–​phase relation and the critical current Ic is affected by junction area and barrier thickness only (correlated with oxygen exposure dose, see Kleinsasser et al. 1995b), the situation for non-​tunnel JJs is multifaceted: the current–​phase relation can be non-​sinusoidal and multi-​valued. For instance, if the geometric constriction is sufficiently narrow, the Josephson current is a function of the transport regime (clean, dirty), junction geometry (2D, 3D), constriction length (short, long), temperature, magnetic field, and the constriction cross-​section dimensions (Likharev 1979). Constriction junctions were analyzed intensively

450

450  Barriers in Josephson Junctions: Overview decades ago (Golubov et  al. 2004, Likharev 1979), and revisited recently for NanoSQUIDs (Foley and Hilgenkamp 2009). They have a non-​sinusoidal current–​ phase relation (Jackel et al. 1976, Troeman et al. 2008) and reduced non-​linearity depending on the bridge geometry (2D or 3D) (Vijay et  al. 2009, 2010). Their relatively strong coupling leads to rather high critical currents (Granata et al. 2009, Hao et al. 2008, Hasselbach et al. 2002, Troeman et al. 2007, 2008, Voss et al. 1980). Currently, metal-​doped silicon-​based Josephson barriers are regaining attention for voltage standards (Olaya et al. 2010), for example SixNb1-​x (Hertel et al. 1983), a metal–​insulator alloy of two different interdiffusible materials in contact with each. Metal-​doping of the amorphous silicon barriers causes the transition from underdamped to the desired overdamped junction behavior. Such silicon-​based barriers of > 10 nm thickness show critical current densities up to jc ≈ 100 kA/​cm2 (Baek et  al. 2006, Kroger et  al. 1979). Long SNS nanowire junctions in the diffusive limit have a characteristic semiclassical energy, the so-​called Thouless energy ETh ∝ L−2 (L normal metal length), which is determined by the normal metal thickness dN. The Josephson coupling of two superconductors through the normal metal is governed by either the superconducting gap Δ or the energy scale ETh, whichever is lower. Very interestingly, SNS nanowire junctions offer unique properties absent in constriction or SIS junctions. Earlier experiments on SNS junctions contacted as four-​(Baselmans et  al. 1999)  or three-​terminal (Huang et  al. 2002)  devices showed tunable supercurrent and coupling as function of a control current, that is, they formed a mesoscopic SNS transistor (Wilhelm et al. 1998). Alternatively, the supercurrent can also be reduced by raising the effective electron temperature (Morpurgo et al. 1998).

15.4 Semiconducting barriers Initiated by the first proposal of Josephson field-​ effect transistors (JFET) by Clark, Prance, and Grassie, compare Clark et  al. (1980), a large variety of Josephson barriers comprising a semiconducting weak link have been realized. The JFETs can be used as cryogenic voltage amplifiers and low-​dissipative, fast digital elements. For example, junctions based on silicon (Becker et  al. 1995), bulk III–​V semiconductors (Kleinsasser 1991, Takayanagi and Kawakami 1985), or on two-​dimensional electron gases in III/​V semiconductor heterostructures have been realized. Technically, a major challenge is the poor coupling of the semiconductor to the superconductor due to the formation of a Schottky barrier, particularly in silicon-​based junctions (Heslinga et al. 1990). This problem can be overcome by using semiconductors where the Fermi level is pinned in the conduction band. By that a surface accumulation layer is formed, as is the case for indium arsenide (Kleinsasser 1991, Takayanagi and Kawakami 1985). The clean limit can be reached by employing a high-​mobility two-​dimensional electron gas in a semiconductor heterostructure, for example InAlAs/​In(Ga)As (Takayanagi and Akazaki 1995, Takayanagi et al. 1995) or InP/​InGaAs (Schäpers et al. 2003). Here, interesting quantum effects such as critical current oscillations

  451

Magnetic barriers 451 or supercurrent quantization have been observed. More recently, one-​dimensional structures were introduced as a conducting link in a JJ. Supercurrent was observed in junctions based on InAs nanowires (Doh et  al. 2005, Günel et  al. 2012), Ge/​Si core/​shell wires (Lauhon et  al. 2002, Xiang et  al. 2006), carbon nanotubes (Jarillo-​Herrero et  al. 2006), or InN nanowires (Frielinghaus et  al. 2010). Semiconducting nanowire-​based JJs are particularly interesting as they are prepared by the so-​called bottom-​up approach, allowing one to realize nanoscale structures without elaborate lithographic means. For indium nitride or indium arsenide nanowires, the surface accumulation layer provides a transparent contact to the superconductor, and its large diffusion constant leads to the relatively high Thouless energy of 0.15 meV. InAs nanowires in the short to intermediate junction limit showed large Thouless energies, see Günel et al. (2012) and Spathis et al. (2011). The significantly lower carrier concentration of InAs compared to InN enables gate control of the carrier concentration. This feature allows one to realize a gate-​tunable JJ (Doh et al. 2005, Günel et al. 2012). Recently, signatures of Majorana fermions (fermions that are their own antiparticle) were detected by transport measurements in semiconductor/​superconductor structures using InSb (Mourik et al. 2012) or InAs (Das et al. 2012, Finck et al. 2013) nanowires.

15.5 Magnetic barriers In a singlet superconductor–​ferromagnet structure the situation differs from the superconductor–​metal interface as the electron energy depends on the spin orientation along the magnetic exchange field Eex. The superconducting correlations carried by opposite-​spin pairs penetrate into the ferromagnet over only a short distance of the order of the magnetic coherence length. The up-​spin electron decreases its energy by Eex, while the down-​spin electron energy increases by the same value. This variation in energy is compensated by an increase or decrease of their kinetic energies, respectively. Thereby the Cooper pair acquires a center-​of-​mass momentum 2kF = 2E ex / vF , that is, the order parameter oscillates perpendicular to the interface with periodicity vF / E ex (Buzdin 2005). In general, the resulting wave vector kF is complex, leading to a complex coherence length ξ F−1 = ξ F−11 + i ξ F−12 with decay length ξ F 1 and oscillation length ξ F 2 of the superconducting correlations inside the ferromagnet. In the clean limit these vF v characteristic length scales read ξ F 1 = and ξ F 2 = F , and in the dirty limit 2πkBT 2E ex

ξF 1 = ξF 2 =

D . Considering magnetic (spin–​orbit and spin-​flip) scattering on E ex

the order parameter, the characteristic length scales have to be modified, leading to a decrease of the coherence length ξ F 1 and an increase of the oscillation length ξ F 2 (Buzdin 2005, Demler et al. 1997). Qualitatively, inside the ferromagnet the Cooper pair wave function decays and acquires a spatially dependent oscillatory phase. This damped oscillatory behavior results in novel and interesting physics, such as a non-​monotonic dependence

452

452  Barriers in Josephson Junctions: Overview

Fig.  15.8 Characteristic properties of a SFS junction:  Non-​ monotonic (a) F-​layer thickness dF and (b) temperature T dependences of the critical current Ic provide indirect detection for a change in the Josephson coupling sign.

(b) Ic

Ic

(a)

0 coupled

π coupled 0 coupled

π coupled

dF

Tc

T

of critical current Ic on temperature T or ferromagnetic layer thickness dF, or the realization of π coupling in JJs with a ferromagnetic interlayer, compare Fig. 15.8. Features like Josephson phase inversion or generation of spontaneous flux (Weides et al. 2006b) render the π junctions valuable phase-​shifting elements for utilization in superconducting circuits. Exploiting their unique properties, π Josephson junctions have been suggested as central elements for new classes of superconducting devices such as RSFQ architecture (Feofanov et al. 2010, Ortlepp et al. 2006, Ustinov and Kaplunenko 2003)  or quiet qubits, where a phase shift of π is necessary to produce a degenerate double-​well potential (Ioffe et al. 1999, Yamashita et  al. 2005, 2006). Owing to their weak dissipation, superconducting single-​flux-​quantum digital circuits offer unique features such as high-​speed operation (beyond 100 GHz) with low power (10 nW) per gate operation. Their drawbacks of large master cell size and the lack of dense, fast, energy-​efficient memory can be solved by including phase-​tailored JJs such as π, 0-​π, or ϕ (junction with non-​zero Josephson phase ϕ across the barrier in the ground state) coupled junctions with large IcRn products (Goldobin et al. 2013, Sickinger et al. 2012). In JJs based on s-​wave superconductors the coupling of phases of the superconducting electrodes can be shifted by π when using a ferromagnetic barrier, that is, SFS or SIFS-​type junctions (I: insulating tunnel barrier) for certain F-​layer thicknesses. These SFS (SIFS) junctions are characterized by an intrinsic phase-​shift of π in the current–​phase relation or, in other words, a negative critical current: I s ( γ ) = I c sin ( γ + π ) = − I c sin ( γ ). Since 1999 the properties of such π-​coupled JJs have been studied and considerable progress in experiment and theory has been achieved. In particular, the π coupling was demonstrated by changing the temperature (Bannykh et al. 2009, Ryazanov et al. 2001, Sellier et al. 2003, Veretennikov et al. 2000, Weides et al. 2006a) or the F-​layer thickness (Bannykh et al. 2009, Blum et al. 2002, Kontos et al. 2002, Weides et al. 2006a) or determining the current–​phase relation of π junctions incorporated into a superconducting loop (Bauer et al. 2004, Guichard et al. 2003, Ryazanov et al. 2003). As ferromagnetic barrier material the alloys CuNi, PdNi, CoFe, Cu2MnAl Heusler-​ alloy, and NiFe, and pure metals (Ni, Co, and others) have been investigated experimentally. To explore the Josephson dynamics and to obtain higher IcRn products an additional tunnel barrier, SIFS-​type junctions, is required, yielding a lower critical current density (Kontos et al. 2002) with typical jc ≈ 10 A/​cm2 or

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Summary and outlook 453 lower (Bannykh et al. 2009, Kontos et al. 2002, Weides et al. 2006a, Wild et al. 2010). This is at least two orders of magnitude lower than the 1–​4 kA/​cm2 of critical current density (corresponding to a Josephson penetration depth of < 5 µm) used for conventional SIS technology. High jc, π-​coupled, and underdamped junctions are of considerable interest as their large characteristic voltage Vc = IcRn will improve substantially superconducting classical and quantum circuits. Such junctions were realized for the first time in 2012 on SIsFS multilayers, see Larkin et al. (2012). SIsFS contacts feature a high IcRn product due to the presence of a tunnel barrier “I.” In the case of a rather thick intermediate layer “s” it resembles a conventional SIS junction where the sFS junction can turn the SIsFS structure into the π-​state. At the same time, the higher IcRn product is preserved. The modes of operation of SIFS and SIsFS devices as well as dependencies of their characteristics on temperature, materials, and geometrical parameters have been investigated theoretically (Bakurskiy et al. 2013a, 2013b, Vernik et al. 2013). Fabrication of high IcRn π Josephson junctions opens perspectives for application of SIsFS technology in complementary logic circuits (Terzioglu and Beasley 1998), in RSFQ with active π junctions, and for the investigation of fractional vortices. The exponentially decreasing damping for T→0 (Weides et al. 2006a) makes SIsFS JJs promising devices for the observation of macroscopic quantum effects using fractional vortices (macroscopic spins) and for qubits, cf. Goldobin et al. (2005) and Kato and Imada (1997).

15.6 Summary and outlook In this chapter the focus was on Josephson junctions and their barriers. The unique properties of superconducting devices like ultralow power consumption, long coherence, good scalability, smooth integration with state-​of-​the-​art electronics, and outstanding magnetic field sensitivity suit them well for a wide range of applications. The key element besides the superconducting material itself is the superconducting junction. The main characteristic of these junctions, also called Josephson junctions, is the local reduction or even suppression of the critical current Ic across the barrier. JJ barriers form the most intriguing part of the junction as they serve as the seat of mesoscopic static and dynamics properties such as coupling strength, ground state, phase damping, and tunability of the superconducting device. A review of the main barrier types of JJs was given. Following an introduction to the physics of barriers, the focus was on junctions and devices based on “conventional” low-​Tc superconductors. Of particular interest were barriers made from insulators, metals, magnets, semiconductors, and nanowires. Selected examples of intriguing developments were discussed. In general, superconducting electronics may provide an alternative to today’s semiconducting electronics which are limited in speed and heat dissipation. To a large extent, the performance of superconducting electronics operating classically or quantum mechanically is defined by the quality of the Josephson barrier. Each application has its own requirements, for example high critical current densities

454

454  Barriers in Josephson Junctions: Overview jc and large energy gaps Δ for microwave mixers, low shunting capacitance for Josephson voltage standards, or low intrinsic electric dipole density for quantum circuits, to name a few. Beside improvements on the established standard barriers AlOx and AlNx, new concepts of integrating semiconducting or ferromagnetic barriers are being explored actively, so far mostly on single chip scales. By adopting and contributing to material and interface concepts of conventional electronics and spintronics, superconducting barrier research, which was started by Giaever around 1960, will continue thriving.

Acknowledgments The author is grateful to Edward Goldobin, Reinhold Kleiner, Dieter Kölle, Hermann Kohlstedt, John Martinis, Hannes Rotzinger, Valery Ryazanov, and Alexey Ustinov for helpful discussions at various stages of this work.

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Hybrid Superconducting Devices Based on Quantum Wires K. Grove-​Rasmussen, T.S. Jespersen, A. Jellinggaard, and J. Nygård Niels Bohr Institute, Center for Quantum Devices and Nano-​Science Center, University of Copenhagen, DK-​2100 Copenhagen, Denmark

16.1 Introduction Quantum wires coupled to superconducting electrodes constitute highly tunable mesoscopic superconducting systems that can readily be studied by electron transport. We focus in this chapter on submicron devices based on semiconductor nanowires or carbon nanotubes. The choice of device geometries, contact materials, and gate electrode configurations makes it possible to address vastly different regimes. Weakly coupled wires yield quantum dots (QDs) dominated by charging effects and a discrete energy spectrum resulting in intriguing sub-​gap state physics. In strongly coupled systems, the superconducting proximity effect is effective, leading for example to quantized supercurrents in one-​dimensional (1D) wires. With the proper choice of materials, topological superconductivity is within reach. This chapter describes the experimental progress in quantum wire-​ based hybrid devices. We present a series of recent examples which illustrate the key phenomena that have allowed detailed investigations of important scenarios ranging from individual impurities on superconductors to proximitized systems that may hold Majorana quasiparticles. The chapter is organized as follows. In Sec. 16.2 we describe experimental aspects of hybrid devices, including materials and fabrication techniques. In subsequent sections we introduce QDs with superconducting leads (Sec. 16.3) and review experiments on superconductivity-​enhanced QD spectroscopy (Sec. 16.4), sub-​gap states (Sec. 16.5), and non-​local signals in Cooper pair splitter devices (Sec. 16.7). What distinguishes these different examples is to a large extent the strength of the electronic coupling between the superconductor and

16 16.1 Introduction 

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16.8 Summary and outlook 

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References (Chapter-​16) 

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K. Grove-Rasmussen, T.S. Jespersen, A. Jellinggaard, J. Nyård, ‘Hybrid Superconducting Devices Based on Quantum Wires’, in The Oxford Handbook of Small Superconductors. First Edition. Edited by A.V. Narlikar. © Oxford University Press 2017. Published in 2017 by Oxford University Press. DOI 10.1093/acprof:oso/9780198738169.003.0016

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460  Hybrid Devices Based on Quantum Wires the quantum wire. We finally present a new direction in the field addressing this coupling via epitaxially grown superconductor–​ semiconductor interfaces in nanowires (Sec. 16.8) and end with a summary and outlook (Sec. 16.9). We firstly list a number of reviews that complement this chapter. In particular we refer to introductory papers covering aspects of hybrid devices (De Franceschi et al. 2010, Frolov et al. 2013, Ferrier et al. 2010), a more in-​depth review (Martin-​Rodero et al. 2011), and textbooks (Heikkila 2013, Nazarov and Blanter 2009, Schäpers 2001). Aspects pertinent to topological superconductivity and Majorana modes in wires are addressed in recent reviews (Qi and Zhang 2011, Beenakker 2013, Leijnse et  al. 2012a, Alicea 2012, Stanescu et  al. 2013, Elliot et al. 2015, Das Sarma et al. 2015). Topological devices will not be covered in this review. While we focus solely on hybrid devices based on 1D quantum wires, some of the phenomena addressed in this chapter have also recently been realized by attaching superconducting contacts to semiconducting nanoparticles (Buizert et al. 2007, Shibata et al. 2007, Deacon et al. 2010a, 2010b, 2015, Katsaros et al. 2010, Kanai et al. 2010, 2012, Kim et al. 2011, Baba et al. 2015), molecules (Winkelmann et al. 2009), and to two-​dimensional (2D) semiconductor heterostructures (Schäpers 2001, Takayanagi et al. 1995, Bauch et al. 2005, Irie et al. 2014, Shabani et al. 2015, Amado et al. 2013). Early work on nanoscale hybrid junctions was based on metallic nanoparticles acting as superconducting QDs (Ralph et al. 1995, Tinkham 2004). This chapter does not provide a comprehensive review of the field but is intended to point out essential experimental aspects by providing a handful of representative examples originating from work of the authors and their co-​workers. We focus on phenomena related to QD behavior in the quantum wires and we include citations of most other experiments in the field, but only sporadic references to the theoretical literature. We also discuss recent developments in epitaxial hybrid nanowire material showing greatly improved proximity effect, crucial for topological superconductivity and devices with Majorana zero modes.

16.2 Experimental aspects of hybrid devices A key asset of wire-​based devices is the ability to attain strong quantum confinement due to their nanoscale transverse dimensions while ease of fabrication is given by their extended linear geometry. The transverse confinement is determined by growth and with standard nanofabrication techniques submicron electrode spacings can be routinely achieved, making it straightforward to contact wires that are typically several micrometers long. Fig. 16.1 shows an archetypical nanowire device drawn to scale, illustrating that the minimal lithographic features (bottom gates) are considerably smaller than the wire. The second attractive feature is the electrostatically tunable channel characteristics offered both by

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semiconducting nanowires and by single-​wall carbon nanotubes. For some applications the properties of the chosen quantum wires are important such as spin–​ orbit interactions or transverse confinement (subband splittings). The future of the field may lie in extended systems based on branched wires or hybrid networks that are currently being demonstrated (Plissard et al. 2013). Whether to choose a semiconductor nanowire or a carbon nanotube as the active element of the device depends on the specific aim of the experiment. For optimized fabrication recipes, nanowire devices yield more reliable metal–​semiconductor contacts and the resulting hybrid devices are more easily tuned, either electrostatically as in most experiments (Sec. 16.2–​16.7) or by doping (Günel et  al. 2012, Paajaste et  al. 2015). The recently developed molecular beam epitaxy (MBE)-​ based superconductor–​ semiconductor contacts (see Sec. 16.7) are a further step in the direction of reproducible contacts, which also has been shown to greatly suppress quasiparticle poisoning of the superconducting gap (Krogstrup et al. 2015, Chang et al. 2015, Higginbotham et al. 2015). A challenge for carbon nanotube-​based devices is a larger spread in contact resistance due to a microscopic contact area and diverse electronic properties (metallic, small gap semiconductor, or semiconductor) which cannot be controlled in the common chemical vapor deposition growth process. However, unlike semiconductor nanowires, a key benefit of carbon nanotubes is their structural perfection and symmetries leading to reproducible and well-​understood electronic-​level structures of carbon nanotube QDs. A single subband gives rise to fourfold degenerate shells and the spectrum including details of the spin–​orbit coupling can be modeled quantitatively (for an extensive review, see Laird et al. 2015). Multi-​wall carbon nanotubes usually do not possess this regularity and fewer experiments on hybrid devices have been reported for this material (Buitelaar et  al. 2002, 2003, Gräber et  al. 2004, Haruyama et  al. 2004, Tsuneta et  al. 2007, Pallechi et al. 2008).

Fig.  16.1 Schematic rendering of a tunable N–​S junction based on a single-​ crystal nanowire channel and two source-​drain contacts, here one superconducting (S)  and one normal (N)  metal electrode. The oxide-​ covered insulating substrate holds several fine gate electrodes (1–​9) that couple electrostatically to the wire. The device is drawn to scale with 100  nm thick electrodes and a 70 nm diameter wire.

462

462  Hybrid Devices Based on Quantum Wires We note that standard nanofabrication techniques can be employed to manufacture superconducting junctions based on quantum wires. In most cases, electron beam lithography is used to define submicron electrode patterns in which the superconducting materials are deposited onto segments of the wires by sputtering or evaporation from solid targets. This standard fabrication scheme allows for a large choice of superconducting electrodes (e.g. Al-​, V-​, Pb-​, or Nb-​based thin films) and even two different superconducting materials can be applied to the same nanowire or nanotube, resulting in asymmetric junctions (Günel et al. 2014, Kumar et al. 2014). In a few reported experiments, alternative routes for contacting have been reported such as nanotube growth on top of prefabricated electrodes (Schneider et al. 2012), electrophoretic assembly of nanowires (Montemurro et  al. 2015a), nanotubes confined in nanopores (Haruyama et al. 2003, 2004), and nanotube laser welding (Kasumov et al. 1999, Reulet et al. 2000). The latter approach was used for investigating superconductivity in suspended ropes of nanotubes (Kociak et  al. 2001, Kasumov et  al. 2003). Superconductivity has also been addressed in double-​wall nanotubes (Shi et  al. 2014)  and multi-​wall nanotubes (Takesue et  al. 2006). Recently, nanowire devices incorporating high-​ temperature superconductors have also been pursued (Montemurro 2015b). In addition to the superconducting source/​drain electrodes, one or more gate electrodes are included in the device design, in the form of either a global (substrate) back gate or multiple local gates defined by lithography (bottom/​ side/​top gates) as in Fig. 16.1. In hybrid QD devices, these gates are crucial, both for controlling the states of the dot and for tuning the essential couplings between the different elements of the device. Finally, the lithography-​based contacting readily allows for integration with other electrode patterns such as superconducting quantum interference devices (SQUID) or superconducting loops to control the phase of the superconducting junctions (van Dam et  al 2006, Cleuziou et  al. 2006, 2007a, 2007b, Pillet et  al. 2010, 2013, Spathis et al. 2011, Giazotto et al. 2011, Schneider et al. 2012, Maurand et al. 2012, Chang et al. 2013, Basset et al. 2014, Delagrange et al. 2015, 2016, Kim et al. 2016). The first hybrid devices based on carbon nanotubes were reported by Kasumov et al. (1999) and Morpurgo et al. (1999), while InAs nanowires were employed firstly by Doh et al. (2005). The range of semiconductor nanowires in hybrid devices has subsequently been expanded to other III–​V materials, including InSb (Mourik et al. 2012, Deng et al. 2012), InP (Doh et al. 2008), InN (Frielinghaus et  al. 2010), and InAs/​InP (Lee et  al. 2012), as well as Si/​Ge (Xiang et al. 2006) and PbS (Kim et al. 2016), while current efforts also include ternary alloys and new core–​shell heterostructures in order to optimize certain transport properties. Hybrid devices based on topological insulators have also appeared, but will not be addressed here. We note that nanotubes have also been employed as templates for the manufacture of superconducting metallic nanowires, but here the nanotubes act merely as passive scaffolds

  463

Superconducting junctions with normal quantum dots 463 (a) 18 Self-assembled quantum dots

Number of publications

16

Carbon nanotubes

14

Nanowires

12 10 8 6 4 2

19

9 20 9 0 20 0 0 20 1 0 20 2 0 20 3 0 20 4 0 20 5 0 20 6 07 20 0 20 8 0 20 9 1 20 0 1 20 1 1 20 2 1 20 3 1 20 4 1 20 5 16

0

Year (b) 80 Number of publications

70 60 50 40 30 20 10

V CO YB

Ta

N N b bT Pb iN ,P b/ ln Re

M l oR e N bN

Ep

i-A

Al

0

Superconducting electrode material

(for a review, see Bezryadin 2012). Recently, superconductivity in branched 1D systems was also addressed experimentally in nanowire crosses (Plissard et  al. 2013). While mesoscopic superconductivity was already an established field before the advent of nanowire and nanotube devices (Beenakker 1997, Lambert and Raimondi 1998), these 1D systems have indeed offered a wide range of new possibilities for designing and tuning hybrid quantum devices as the following sections will show. Fig. 16.2 illustrates the number of publications to date on hybrid devices based on nanowires, nanotubes, and self-​assembled QDs (a) as well as the specific superconducting electrode materials (b).

16.3 Superconducting junctions with normal quantum dots In the following we provide a series of examples that illustrate the prospects offered by coupling superconducting reservoirs with confined states in quantum

Fig.  16.2 (a)  Number of experimental publications involving hybrid devices based on nanotubes, semiconductor nanowires, and self-​assembled quantum dots. (b)  Number of reports on different superconducting electrode materials.

464

464  Hybrid Devices Based on Quantum Wires wire-​based QDs. In QDs individual electrons and charging effects are important—​for an introduction to QD physics in general, see Kouwenhoven et  al. (1997), Hanson et al. (2007), and Ihn (2010). Most notably, in the Coulomb blockade regime current is fully blocked since the single electron charging energy U = e2/​C is the dominant energy scale. Current can only flow for particular configurations of the chemical potentials μ (usually tuned by gate potentials) and here transport is mediated by sequential tunneling of individual electrons. When coupling such dots to superconductors, the macroscopic quantum state of the latter exhibits an interplay with the discrete states and individual electrons of the QDs. Fig.  16.3a–​d illustrate schematically four generic two-​and three-​terminal hybrid device geometries with normal QDs: S–​QD–​S, S–​QD–​N, S–​QD–​S with a normal tunnel probe, and a Cooper pair splitter double QD geometry. In most instances, we represent the superconductor by the Bardeen–​ Cooper–​Schrieffer (BCS) type density of states (DOS) with peaks in quasiparticle DOS at the edges of the superconducting gap ±∆. The energy diagram for an S–​QD–​N device is shown in Fig. 16.3e, indicating relevant QD energy scales and couplings, and the superconducting and the normal lead DOS. On a coarse scale, superconductor–​quantum dot junctions (S–​QD–​S) exhibit similar superconducting phenomena as SNS junctions with a normal (N) central

(a)

S

S

(e) µN+1,N

(b)

S

N

Γ

S S

ΓN

µN,N–1

2∆

N

(c)

(d)

ΓS U

N

eVsd

S

S

N

N

Fig.  16.3 (a–​d) Examples of different hybrid device geometries based on quantum dots: (a) Josephson quantum dot junction S–​QD–​S, (b) S–​QD–​N, (c) Josephson junction probed by a normal tunnel probe, and (d) a Cooper pair splitter device. (e) Schematic energy diagram of a S–​QD–​N device. Transport (in the normal state) through the QD takes place via single electron tunneling by adding and removing the Nth electron. The energy for adding the (N + 1)th electron relative to adding the Nth is given by the charging energy U.The resonance in the bias window has a width of Γ = ΓS + ΓN due to coupling to the reservoirs. In the superconducting state, the S electrode has an energy gap Δ in the density of states. The Andreev reflection processes, where two electrons tunnel into the superconductor from the normal region, and related phenomena are not shown. A small bias of eVsd is applied between the S and N contacts.

  465

Superconducting junctions with normal quantum dots 465 region, thus resembling Josephson junctions (JJ) that are conventionally realized in narrow superconducting bridges (weak links) or in superconductor–​insulator (SIS) junctions. If the transmission (coupling) Γ of the SNS junction is sufficiently large and the QD is not dominated by charging, that is, Γ > ∆ and Γ > U, the device will indeed behave as a (gateable) JJ and support a supercurrent flowing at vanishing bias and modulated by quantum interference effects (Fabry–​Perot resonances), except in the case of perfect transmission. This regime has been realized both in carbon nanotubes (Jarillo-​Herrero et al. 2006, Jørgensen et al. 2006, 2009, Cleuziou et al. 2007a, Tsunata et al. 2007, Zhang et al. 2008, Pallecchi et al. 2008, Wu et al. 2009, Liu et al. 2009, Schneider et al. 2012) and in semiconductor nanowire junctions (Doh et al. 2005, Sand-​Jespersen et al. 2009, Nishio et al. 2011, Nilsson et al. 2012, Günel et al. 2012). If the coupling is weak, the supercurrent is strongly suppressed and quasiparticle tunneling will dominate transport (at a bias eV > 2∆ as illustrated in Fig. 16.4a) (Krstíc et al. 2003, Doh et al. 2008, Grove-​Rasmussen et al. 2009, Hao et al. 2011).1 The JJ and quasiparticle tunneling regimes are the two extreme cases; with a QD present in the junction for intermediate coupling, additional regimes exist depending on the choice of parameters (Γ, U, Δ) and the available quantum states of the dot (Jarillo-​Herrero et  al. 2006, De Franceschi et  al. 2010). Some of these regimes will be treated in Sec. 16.5–​6, reflecting the recently emerged understanding of the intricate intermediate coupling regime in S–​QD–​S and S–​QD–​N junctions. However, we will firstly in Sec. 16.4 address transport in the weak coupling regime where the electronic structure of the QDs is not strongly affected by the superconducting correlations. We will not discuss the JJ regime further in this chapter but just point out that nanowire-​or nanotube-​based devices have been used to demonstrate remarkable features such as supercurrent reversal controlled by the charge state of the QD (van Dam et al. 2006, Cleziou et al. 2006, Jørgensen et al. 2007), sub-​ gap states (Grove-​Rasmussen et al. 2009, Deacon et al. 2010a, Pillet et al. 2010, Lee et al. 2014, see Sec. 16.5), the interplay between Kondo physics and superconductivity (Buitelaar et al. 2002, Vecino et al. 2004, Gräber et al. 2004, Grove-​ Rasmussen et  al. 2007, Eichler et  al. 2007, 2009, Sand-​Jespersen et  al. 2007, 2008, Buizert et al. 2007, Wu et al. 2009, Deacon et al. 2010b, Kanai et al. 2010, Maurand et al. 2012, Luitz et al. 2012, Lee et al. 2012, Kim et al. 2013, Pillet et al. 2013, Delagrange et al. 2015, 2016), Shapiro steps (Doh et al. 2005, Cleuzio et al. 2007, Nishio et al. 2011), and multiple Andreev reflections (Buitelaar et al. 2002, 2003, Xiang et al. 2006, Sand-​Jespersen et al. 2007, Nilsson et al. 2012, Günel et al 2012, Abay et al. 2013, 2014). In short nanowire junctions the critical current can be very high (Frielinghaus 2010, Roddaro et al. 2011, Abay et al. 2012, 2014, Günel et al. 2014, Paajaste et al. 2015) and a stepwise increase in the supercurrent is correlated with the quantized normal state conductance (Xiang et al. 2006, Abay et al. 2013). Conversely, longer disordered junctions can be dominated by mesoscopic conductance fluctuations that are enhanced in the superconducting 1 The suppressed supercurrent in this regime (if carefully measured) does reveal an interesting interplay with charging effects (see below).

466

466  Hybrid Devices Based on Quantum Wires (a)

(b) 2

Vsd (mV)

eVsd = 2∆

Eadd

N

N+1

N+2

1 0 –1 –2 0.25

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Vgate (V)

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0.50

2

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(d)

0.1

0 –2

–1

0

1

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Vsd (mV)

Fig. 16.4  Transport spectroscopy of carbon nanotube QDs with superconducting leads. (a) Schematic illustration of elastic cotunneling of quasiparticles from source to drain via a QD level. The addition energy is given by Eadd = U + ΔE, where ΔE is the level spacing. (b)  Bias spectroscopy plot of differential conductance dI/​dV as a function of bias Vsd and gate voltage Vgate. N indicates the integer number of electrons on the QD. (c) Cross-​section taken at the dashed line in (b), with quasiparticle (QP) cotunneling peaks and resonances involving sequential transport through ground (GS) and excited state (ES). (d) Inelastic cotunneling process involving two QD levels spaced by δ. One level (thick) is more strongly coupled to the leads than the other (thin), giving rise to gate-​ dependent shifts of the resonances. (e) Bias spectroscopy of a second device. (f) Cross-​ sections at the dashed lines in (e). All data were taken at zero magnetic field and at a temperature of 0.3 K, i.e. below the critical temperature TC ~ 1.7 K of the Nb-​based electrodes. Adapted from Grove-​Rasmussen (2009). state (Sand-​Jespersen et  al. 2009, Doh et  al. 2008, 2009, Günel et  al. 2012). Finally, nanotube JJ wideband superconducting charge detectors (Häkkinen et al. 2015), tuning of the Josephson current by injection of hot electrons (Roddaro et al. 2011), and a SQUID-​based Josephson quantum electron pump have been reported (Giazotto et al. 2011). New directions may emerge combining hybrid devices with nanoelectromechanical systems (Schneider et al. 2012).

  467

Superconductivity-enhanced spectroscopy of quantum dots 467

16.4 Superconductivity-​enhanced spectroscopy of quantum dots In this section we will demonstrate how superconducting electrodes can increase the resolution of tunneling spectroscopy measurements on QDs. The peaks at the gap edges of the superconducting DOS are used to probe the electronic states of a carbon nanotube QD with a resolution that is enhanced compared to the case of normal metallic electrodes with a nearly constant DOS. The principle is known from earlier work on tunneling spectroscopy with superconductors but here we see how the technique can be used to enhance otherwise featureless cotunneling processes in QDs. Fig. 16.4b shows a map of the differential conductance dI/​dVsd for a nanotube QD as a function of gate potential Vgate and bias Vsd applied to the superconducting (niobium-​based) source-​drain electrodes. The diamond-​like features reflect the characteristic Coulomb blockade charging pattern for QDs where the number N of electrons confined on the dot increases in a stepwise manner as a function of Vgate. The charging energy U ~ 5 meV (diamond height) exceeds by far the superconducting gap ∆ ~ 0.225 meV for this device. Within the bias window |eVsd | < 2∆ the conductance is suppressed due to the gap in the DOS of the electrodes. At eVsd = ±2∆ sharp horizontal ridges are seen in all diamonds. Here quasiparticles from the leads can take part in elastic cotunneling processes via quantum states of the dot as illustrated schematically in Fig. 16.4a, where the DOS peaks in the two electrodes are exactly aligned (eVsd = 2∆). Two tunneling events each constitute such a cotunneling process. While the QD transport process is elastic (the dot is left in its initial state), the resonances appear at finite bias 2∆ due to the gapped leads. At higher bias the contribution from these processes diminishes as the DOS of the electrodes is reduced away from the gap edges. Fig.  16.4c shows a cut slightly off-​resonance, at the position indicated by the dashed line in Fig.  16.4b. The quasiparticle cotunneling peaks (QP) are sharp and are followed by resonances at higher bias resulting from sequential tunneling via ground (GS) and excited states (ES) of the dot. Multiple Andreev reflections that could lead to sub-​gap currents are suppressed due to the weak coupling to the leads. The sharp QP peaks at fixed energy 2∆ throughout Fig. 16.4b are thus a clear result of the superconducting DOS singularities (Grove-​Rasmussen et al. 2009). Without superconducting leads elastic cotunneling would prevail for all biases and only result in an overall, featureless background conductance within the diamonds at all energies (one exception is the case of Kondo resonances that dominate for strongly coupled nanotube dots (Nygård et al. 2000), occasionally coexisting with superconductivity (Buitelaar et al. 2002)). Fig. 16.4e shows the spectroscopic map of a second device for equivalent bias and gate ranges. Again the charging energy U~12 meV is much greater than the gap, but the QD is more strongly coupled to the superconducting leads. In addition to the horizontal QP cotunneling lines seen for the previous device, sharp

468

468  Hybrid Devices Based on Quantum Wires resonances are also found at higher bias Vsd, away from the gap (see cross-​section Fig. 16.4f). These can be explained by inelastic cotunneling processes occurring for superconducting electrodes biased by eVsd = 2∆ + δ, where δ is the effective spacing between two levels of the QD. One such process is depicted in Fig. 16.4d. Superconductivity ensures that the resonances are sharp peaks and not just steps in the conductance. The gate-​dependent (i.e. non-​horizontal) shift of these resonances is due to tunnel-​renormalization and different couplings of the two levels involved in the process (Holm et  al. 2008). Superconductivity enhances such spectroscopic features in QD transport experiments. However, the superconducting correlations do not as such influence the intrinsic electronic structure of the dots in the data shown here (except an indication of a sub-​gap state excitation shifted into the gap for the rightmost diamond in Fig. 16.4e). In the following section we will encounter a much more intricate interplay between the quantum wire states and superconductivity leading to clear sub-​gap states. We end by noting that superconducting probes have also been implemented for spectroscopy studies of N–​CNT–​N devices, that is, in geometries similar to Fig. 16.3c with N and S interchanged (Dirks et al. 2009, Chen et al. 2009, Bronn and Mason 2013).

16.5 Sub-​gap states in hybrid quantum dots In this section, we continue our study of hybrid phenomena focusing on superconductor–​quantum dot–​normal devices (S–​QD–​N). For weak coupling between the superconducting electrode and the QD (Γ > U

2

4 1

1,1 1,0

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Singlet

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3.30

3.32

3.34

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(h)

6

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9.30

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D

–∆ 0.2

N

(g)

Singlet

D

0

Vsd

S

(f)

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2,0

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Singlet

∆ > U (Meng et al. 2009) and U >> Δ (Kiršanskas et al. 2015). The two models yield similar results and even though the data in Fig. 16.5e–​g are in the regime of U > Δ, we start in the low U regime due to the simplicity of this model. Fig. 16.5i shows the energy spectrum in this so-​called superconducting atomic limit (Meng et al. 2009, Bauer et al. 2007), where the gap is larger (Δ = ∞) than the charging energy. In this case, the zero-​and two-​electron charge states on the quantum couple via Andreev reflections. The resulting energy diagram shows the energy of the zero, one, and two electron states versus gate voltage. Thus for small (large) gate voltages, the zero (two) electron state marked by 0 (2) is the ground state and the anticrossing between these states is proportional to the coupling ΓS. For appropriate choice of ΓS/​U as shown, the ground state changes from singlet to doublet and back to singlet as the gate voltage is swept. The excitations below the gap are plotted in Fig. 16.5h. The model qualitatively resembles the data in Fig. 16.5e–​f even though the higher excited state is not observed. In the case of larger ΓS, the one-​electron energy does not cross the singlet state (not shown), keeping the singlet ground state for all gate voltages, consistent with the phase diagram and data in Fig. 16.5c and Fig. 16.5g, respectively. We note that this model also fails to capture the finite energy gap observed in the data. In the opposite regime U > Δ (Koerting et al. 2010, Kiršanskas et al. 2015), the two-​electron singlet state is more favorably formed with a quasiparticle above the gap in the superconductor than by two electrons on the dot. Thus states with 2  Note that the transition can also be obtained by tuning the coupling to S, where an interplay between Kondo physics and superconductivity becomes important.

472

472  Hybrid Devices Based on Quantum Wires an additional quasiparticle at energy Δ are also included (Kiršanskas et al. 2015, Jellinggaard et al. 2016). Fig. 16.5k shows the energies of combinations of zero, one, and two electrons on the dot and a single quasiparticle above the gap (ignoring all other quasiparticle states for simplicity). In this case states with the same parity connected via a single tunneling event are coupled. For example, the zero-​electron state couples to the states having one electron and one quasiparticle (assuming opposite spin). Similar to the model above, we choose parameters where the ground state of the superconductor–​QD system makes a doublet–​singlet–​doublet transition versus gate voltage. The resulting ground to excited state transition below the gap is shown in Fig. 16.5j. Despite the simplification of considering only the lowest-​energy quasiparticle state in the lead, this model qualitatively describes the behavior shown in the experiments in Fig. 16.5e–​f. In contrast to the model in the superconducting atomic limit, only one excitation (twofold degenerate for a singlet ground state) exists below the gap and this excitation is limited by a finite gap. For more precise modeling, more complex methods such as numerical renormalization group calculations have been employed (Wilson 1975). Recent theoretical predictions even indicate that the normal electrode also plays a role in the ground state phase diagram for finite coupling (Žitko et al. 2015). Sub-​gap states are currently a very active topic in condensed matter physics. One of the challenges lies in understanding and demonstrating coupled YSR states. The simplest extension in this direction is a YSR double dot (Žitko 2015) or molecule (Yao et al. 2014), but chains of YSR states have also already been examined theoretically (Pientka et  al. 2013). Furthermore, we point out that hybrid devices are predicted to host Majorana bound states (yet another sub-​gap state) for appropriate spin–​orbit coupling and effective magnetic fields. The development in this field is outside the scope of this chapter, but Majorana physics in hybrid devices is currently under intense experimental and theoretical focus. Finally, the experiments utilizing ABS in superconducting break junctions as qubits constitute an interesting frontier (Janvier et al. 2015).

16.6 Non-​local signals in hybrid double quantum dots In the previous sections we have primarily focused on hybrid phenomena observed in devices with only two electrodes contacting a QD nanostructure. Similarly to conventional QDs, where complex multi-​ dot behavior has been extensively mapped out during recent decades (van der Wiel et al. 2003, Chang et  al. 2009), novel hybrid phenomena are also expected to appear in complex superconductor circuits. One of the simplest extensions is a hybrid double dot with three electrodes:  two QDs are coupled to a central superconducting electrode and two normal electrodes as shown schematically in Fig. 16.6a. This device is called a Cooper pair splitter due to its intended operation, where Cooper pairs are injected from the central superconductor into the two dots. Ideally, one

  473

Non-local signals in hybrid double quantum dots  473 electron tunnels into each dot, thereby splitting the Cooper pair into the two branches of the device. The Cooper pair splitting process is expected to generate non-​local spin-​entangled electrons in the two dots, since Cooper pairs in the superconductors are ordered in spin singlets (Recher et al. 2001, Lesokovik et al. 2001, Bouchiat et al. 2003). Such devices may allow tests of Bell’s inequalities for Einstein–​Podolsky–​Rosen (EPR) pairs of electrons as a solid-​state spin equivalent of Aspect’s experiment with entangled photons (Aspect et  al. 1982, Bell 1964, Einstein et al. 1935). Electron transport below the superconducting gap in Cooper pair splitter devices may be discussed in terms of Andreev reflections. Two important tunneling processes must be identified in order to understand typical measurements and device considerations. The first process involves tunneling of a Cooper pair through only one QD via a local Andreev reflection at the superconductor interface (reverse if considering holes). Ideally, this process is suppressed by the charging energy of the QD, but experimentally this process is often significant or even dominant due to a relatively small superconducting gap (U >> Δ ~ Γ) (Hofstetter et al. 2009). The second process is inverse Cooper pair splitting, also called crossed Andreev reflection, where two electrons, one in each dot, tunnel into the superconductor to form a Cooper pair. This process is not suppressed by charging energy since it only involves a single electron tunneling on each side. Several other factors, however, influence the strength of the Cooper pair splitting amplitude. First, the superconducting electrode should not be wider than the average extent of a Cooper pair, since tunneling is to occur simultaneously on each side of the superconducting electrode. This is easily achieved with, for example, electrodes of aluminum, where the coherence length (reflecting the spatial extent of a Cooper pair) is in the micrometer range. For distances larger than the coherence length, the Cooper pair splitting is predicted to be exponentially suppressed (Recher et al. 2001). Second, destructive interference between quasiparticle paths in the bulk of a three-​dimensional (ballistic) superconductor is predicted to add a prefactor proportional to ( λ F / d ) where λ F is the Fermi wavelength and d is the separation between the tunneling points to the two QDs (Leijnse et al. 2013). The second factor is much less elucidated and depends on the dimensionality and material properties of the superconducting injector; in the 1D case, the prefactor does not carry a dependence on the Fermi wavelength. Finally, the tunneling rates to the normal (ΓNL, ΓNR) and the superconducting electrodes (ΓSL, ΓSR) also play an important role, where subscript L and R refer to the left and right dots, respectively (see a schematic for a single dot configuration in Fig. 16.3e). The ideal Cooper pair splitter regime is predicted to occur when the tunneling rates from the superconductor to the dot are much smaller than from the dot to the normal electrode (Recher et al. 2001). For bias conditions larger than temperature, the dots are thus predominantly empty, realizing the optimal configuration of two empty dots for the Cooper pair splitter process. Cooper pair splitting has been revealed experimentally by charge correlation measurements (Hofstetter et  al. 2009, 2011, Herrmann et  al. 2010, 2012,

474

474  Hybrid Devices Based on Quantum Wires Das et  al. 2012a, Schindele et  al. 2012, 2014, Fülop et  al. 2014, 2015), while entanglement detection still remains a challenging task.3 The first observations of Cooper pair splitting were obtained by Hofstetter et al. (2009) and Herrmann et al. (2010). In the former work, the Cooper pair splitting was identified as an additional signal in, for example, the left dot conductance for appropriate alignment of the right dot resonance. Such a signal is non-​local in the sense that the conductance in the left “sensor” dot depends on the electrochemical potential (resonance) of the right “control” dot. In the device, the local Cooper pair tunneling was still dominating due to non-​optimal parameters. Cooper pair splitting has been reported in nanowires (above example), nanotubes, InAs islands (Deacon et  al. 2015), and graphene (Tan et  al. 2015). The highest efficiency of 90% was seen in a nanotube sample (Schindele et al. 2012), while other reports investigate the effect of a finite bias between the normal leads (Hofstetter et  al. 2011)  and the interplay with sub-​gap states (Schindele et  al. 2014). Recently, more advanced devices with bottom gating technology have been fabricated allowing tuning of the couplings to the superconducting and normal leads. Studies reveal that tuning the couplings can alter the non-​local signals from positive to negative in both dots (Fülop et al. 2014). Moreover, shot noise correlation measurements have also confirmed Cooper pair splitting (Das et al. 2012a). To illustrate the essential features of Cooper pair splitting, we present an example of non-​local measurements in Fig. 16.6–​16.7. The device, schematically shown in Fig. 16.6b, consists of an InAs nanowire device fabricated in the Cooper pair splitter geometry. The central superconducting electrode (Al/​Ti: 95/​5 nm) is around 100 nm wide and the distance to the normal metallic electrodes (Au/​ Ti: 90/​5nm) is around 350 nm. To control the electrostatic potential of the two QDs, bottom gates (Au/​Ti: 12/​5 nm) have been fabricated below the nanowire with a 24 nm Hafnium oxide layer acting as a dielectric barrier. In these measurements the gates g4 and g12 are used as the left and right QD plunger gates, respectively. The other gates are adjusted to form QDs on both sides, identified by Coulomb blockade peaks in the conductance versus plunger gate voltage as shown in Fig. 16.6c–​d. A small magnetic field of 140 mT is applied to drive the superconductor in the normal state. Stability diagrams of the two rightmost peaks (solid circles) in each QD reveal Coulomb blockade diamonds with charging energies on the order of 2 and 3 meV (see Fig. 16.6e–​f). Fig. 16.6g–​h show bias spectroscopy plots for a low bias range in the superconducting state of the same charge states as the stability diagrams above. On the left side features related to the superconducting gap can be observed at Δ ~ 120 µeV, while it is less apparent for the right dot due to the relatively low conductance configuration.4 The resonances observed below the gap are ascribed to local pair tunneling, which is not completely suppressed despite Coulomb blockade. The left dot is in this case tuned to have a stronger coupling to the superconducting electrode than 3  Here we only review the Cooper pair splitters made in QD devices. 4  The gap is visible in the right dot for stronger coupling to the superconductor.

  475

Non-local signals in hybrid double quantum dots  475 (a) NL

(b) S

NR N

QDR

QDL

VL (g4)

G (e2/h)

0.5 (c)

N

S

VR (g12)

(d)

0.4 0.3 0.2 0.1

Vsd (mv)

2

–40

–20

0

20

40

(e)

–40

–20

0

20

40

(f) dI/dVsd (e2/h)

0

1 0 –1

0.5

–2

0.2

(g)

(h)

Vsd (mv)

0.1 0 –0.1 0

–0.2 20

25

30

35

∆VL (mV)

40

45

50 10

15

20

25

30

35

∆VR (mV)

to the normal lead (ΓSL > ΓNL), as revealed by the smaller slope of the sub-​gap resonances on the left side versus the plunger gate voltage. The right dot is in the opposite regime (ΓSR < ΓNR). As discussed above, this regime is not optimal for Cooper pair splitting according to theory, but non-​local signals are still resolved. A non-​local signal is defined as an additional signal added to the local pair tunneling when both dots are at resonance. Fig. 16.7a, c show the conductance through the left and right dots when sweeping across a sub-​gap resonance in the right dot at a small finite bias of Vsd = 40 µV. The left plunger gate voltage

Fig.  16.6 (a)  Schematics of a Cooper pair splitter device. Cooper pairs may be split by one electron tunneling into each dot as shown by dashed arrows. (b) Schematic representation of a bottom-​ gated Cooper pair splitter device. (c–​d) Conductance of the left and right QD in a Cooper pair splitter in the normal state (B = 140 mT). Coulomb blockade and filling of spin-​degenerate states are observed. (e–​f) Stability diagram in the normal state showing Coulomb diamonds corresponding to the (left and right) Coulomb peaks indicated by solid circles in (a–​b). (g–​h) Bias spectroscopy revealing a superconducting energy gap due to the central superconducting electrode. The left side shows more resolved sub-​gap states, which indicate the following couplings:  ΓSL > ΓNL, ΓSR < ΓNR. The dashed line shows the bias relevant for the measurements of Fig. 16.7. All measurements are performed at 40 mK.

476

476  Hybrid Devices Based on Quantum Wires VL = 4.779 V

0

(d)

0.2

0.1

30

31

∆VR (mV)

∆GR

0.1

0.05

0 29

(b)

Left dI/dVsd (e2/h)

Right dI/dVsd (e2/h)

(c)

VR = 3.0599 V

∆GL

Left dI/dVsd (e2/h)

0.1

Right dI/dVsd (e2/h)

(a)

0.05

0

0 0.2

0.1

0

43

45

47

∆VL (mV)

Fig. 16.7  Measurements of differential conductance dI/​dVsd versus gate voltages of the right and left QD at Vsd = 40 μV (see Fig. 16.6g–​h). (a) Conductance measured in the left QD almost on resonance versus the gate voltage of the right dot.The gate voltage for the left dot is chosen as indicated by the square in (d) resulting in a finite conductance, while the right dot is swept through a resonance as shown in (d). An additional conductance ΔGL (non-​local signal) is observed in the left QD (a), when the right dot is at resonance (c). Similarly, a non-​local signal is observed in the right QD ΔGR (b), when the left QD is at resonance (d). In this case the gate voltage for the right dot is kept constant corresponding to the open circle marked in (c).The conductance traces are averages of 20 sweeps and cross-​capacitances are negligible. is fixed such that the left QD has a finite conductance, as indicated by the open square in Fig. 16.7d. Comparing Fig. 16.7a, c, an additional positive signal ΔGL is observed in the left dot when the right dot is at resonance. Similarly, if the left electrochemical potential is swept while the right dot is (partly) on resonance, a non-​local signal ΔGR appears at the same gate voltages as the left dot resonance (see Fig. 16.7b, d). In this case, a positive bias is applied to the superconducting electrodes, which results in electron tunneling from the normal leads to the superconductor. For negative bias, the non-​local signals are not as clear, but these results represent a typical example at finite bias of non-​local signals in Cooper pair splitter devices (Schindele et al. 2014). The next generation of experiments involves demonstrating entanglement of the non-​local Cooper pairs. Several routes exist, all involving the integration of spin detection in the device. One proposal relies on the intrinsic spin–​orbit coupling in a bent nanotube geometry, where the intrinsic magnetic fields, due to

  477

Epitaxial superconducting contacts to nanowires  477 the spin–​orbit coupling and an external field, can be used to configure the spin-​ components of the quantum states (Hels et  al. 2016)  to act as spin-​filters for entanglement detection based on Bell’s inequality (Braunecker et al. 2013, Mazza et al. 2013, Burset et al. 2011). Other schemes involve fabricating spin-​filtering ferromagnetic contacts, using fields from micromagnets or coupling the Cooper pair splitter to microwaves and/​or an on-​chip circuit quantum electrodynamics (QED) resonator (Cottet 2012a, 2012b). Furthermore, the predicted spin-​blockade of the inverse Cooper pair splitting is still to be observed in experiments. Here, loading a triplet from the normal leads should suppress the current since the superconductor only accepts singlets (Eldridge et al. 2010). The crossed Andreev reflection and spin control in the two dots are also prerequisites for observing poor man’s Majorana modes (Leijnse et al. 2012b) and exotic superconductivity (Sothmann et al. 2014) in these types of devices. For strong coupling to the superconductor, the Cooper pair splitter device may also host a YSR molecule (Yao et al. 2014). Interestingly, even incremental extensions of the hybrid devices generate a wealth of experimental and theoretical possibilities and challenges.

16.7 Epitaxial superconducting contacts to nanowires The characteristics of the superconducting gap induced in the normal side of S–​N structures and the Josephson coupling of SNS Josephson junctions rely on the electrical contact between the superconductor and the normal material. In the case of semiconductor–​superconductor nanoscale hybrids, achieving transparent electrical contacts poses a significant experimental challenge. Contact problems associated with Shottky barriers at the semi–​super interface can be avoided by using semiconductor nanowires of InAs, InSb, or InN where the Fermi level at the surface is pinned in the conduction band, providing a natural surface electron accumulation layer. Even for such semiconductors, contacting remains a challenge due to the native oxide formed at the surface of the nanowires which needs to be removed prior to deposition of the superconductor. In the first generation of nanowire devices the oxide was removed by a brief etch in hydrofluoric acid (Doh et al. 2005, Samuelson et al. 2004) followed by rapid transfer to the evaporation system. The uncontrollable regrowth of the oxide resulted in irreproducible contacts, and techniques for self-​terminating etching and passivation were developed by Suyatin et al. (2007) based on an ammonium polysulfide treatment. Also techniques for removing the oxide by in situ ion milling have been developed. This can result in low resistance contacts, however, with potential damage also to the semiconductor (Sourribes et al. 2013). Using these techniques, key experiments in quantum transport were realized using both normal and superconducting contacts (Doh et al. 2005, van Dam et al. 2006, Hofstetter et al. 2009, Mourik et al. 2012, Das et al. 2012b, Deng et al. 2012, Lee et al. 2014). Following the experiments addressing Majorana zero modes in the tunnel spectroscopy of strong

478

478  Hybrid Devices Based on Quantum Wires spin–​orbit nanowires contacted to superconductors (Mourik et  al. 2012, Das et  al. 2012b, Deng et  al. 2012, 2014, Churchill et  al. 2013, Finck et  al. 2013), increased attention was directed to the characteristics of the proximity-​induced superconductivity. The induced gap showed a considerable softness with values of ~5 for the ratio of the tunnel conductance above the gap to the conductance below the gap (see also Lee et al. (2012) for discussion on the soft gap). Such sub-​gap states will significantly limit the performance of proposed topological Majorana-​based qubits. In an attempt to explain the gap softness, Stanescu et al. (2014) emphasized the softening due to quasiparticle poisoning by normal metals in S–​InSb–​N structures and Takei et al. (2013) considered the consequences of microscopic disorder in the semiconductor–​superconductor interface also being a source of gap softening. These findings suggest that care must be taken to achieve not only low-​resistance contacts, but also spatially uniform contacts. To this end, a strategy for fabrication of semiconductor/​superconductor nanowire hybrid devices was introduced (Krogstrup et  al. 2015)  based on direct in situ growth of the complete hybrid structure in a single MBE growth.5 The steps of the nanowire growth process are illustrated in Fig. 16.8a–​c. A key element of the process is that after growth of the nanowires, the sample remains inside the MBE reactor under ultrahigh vacuum until the superconductor has been grown. As the nanowire surfaces are thus never exposed to ambient conditions this results in a perfect oxide-​free interface. Moreover, despite the difference in crystal structures, certain material combinations have the possibility of forming epitaxially matched interfaces (Pilkington and Missous 1999), thus providing the ultimate limit of spatial uniformity. As an example, Fig. 16.9a shows transmission electron microscopy (TEM) images of hybrid structures consisting of aluminum (fcc crystal structure) grown on the (1-​100) surface of an InAs nanowire in the wurtzite crystal structure with the [0001] direction along the wire axis. The TEM analysis shows that the Al phase is oriented with the [112] direction out-​of-​plane from the (1-​100) surface of the InAs. As is apparent from the image the two phases have the same periodicity and simulating the atomic arrangement at the interface (Fig. 16.9c–​d) shows that with a domain size of (1[111]/​1[0001], 3[1-​10]/​2[11-​20]) the two materials can be joined with only a 0.3% domain mismatch. It is important to note that depending on the surfaces of the nanowire and the orientation of the superconductor there are many different possibilities for the epitaxial matching at the surface. The orientation realized in the growth is determined by different factors including the interface energies due to strain, the thickness of the superconductor, the energies of the free surfaces, energies associated with possible dislocations at wire corners, etc. Predicting the combinations of semiconductors and superconductors which can be used to form expitaxial hybrids is thus a challenge; however, based on a number of simplifying assumptions interesting ­candidates have been identified (Krogstrup et  al. 2015)  such as InAs/​Au (2/​3 5  The unique morphologies of nanowires make the approach different from earlier attempts to optimize 2D superconductor–​semiconductor interfaces (Taboryski et al. 1996, Kutchinsky et al. 1999) in III–​V growth with in situ epitaxial aluminum on GaAs (Cho et al. 1978, Missous et al. 1993).

  479

Epitaxial superconducting contacts to nanowires  479 MBE reactor. Ultrahigh vacuum

Super Semi

Au Semi

(e) Al InAs (g)

(a)

(b) (c) (d)

(f)

(g)

2 µm

InAs

Al

50 nm

Fig. 16.8  (a–​c) Illustration of the basic steps in the growth of epitaxial semiconductor–​ superconductor nanowire hybrids. Initially (a–​b) semiconductor nanowires are grown using standard recipes. For InAs NWs the typical growth temperature is ~420oC. Subsequently the temperature of the sample is lowered to below 0oC and the superconductor is grown. Depending on the orientation of the nanowire facets with respect to the superconductor source, either two or three facets half-​shell hybrids can be grown (the three-​facets case is illustrated). Continuous rotation results in full-​shell hybrids. (d) Tilted scanning electron microscopy image of an array of InAs/​Al half-​shell hybrids. (e) Low-​resolution transmission electron micrograph of a single InAs/​Al hybrid clearly showing the semiconductor core and the aluminum half-​shell. (f) TEM micrograph of a ~100 nm thick cross-​sectional slice cut using a microtome.The hexagonal shape is clear, and the thin aluminum two-​facet coverage is faintly visible at the top and top-​r ight facets. (g) A zoom of the TEM image in panel (e) emphasizing the interface. Panels (e–​g) adapted from Krogstrup et al. (2015). domain ratio, 1.0% strain), InAs/​V (1/​2 domain ratio, 0.3% strain), and InSb/​Nb (1/​2 domain ratio, 1.8% strain). These combinations are important for superconducting contacts with high critical temperature and high critical magnetic field to strong spin–​orbit semiconductors, but have not been realized so far. The concept of using the epitaxial semi/​super structures as a basis for hybrid electrical devices

480

480  Hybrid Devices Based on Quantum Wires Fig.  16.9 (a), (b)  High-​resolution transmission electron microscopy looking along the interface of an InAs nanowire grown in the [0001] and [112] directions, respectively. The crystal directions are indicated and the corresponding atomic arrangements are represented by the spheres, emphasizing the epitaxial match. (c) Top and side views of the simulated atomic arrangement for the orientation in (a)  showing how with this particular orientation the aluminum remains epitaxially matched to the InAs also between neighboring facets. The two different grain orientations are labeled α and β. Grain boundaries appear if grains of different orientation meet  along the nanowire ((c) bottom panel). Adapted from Krogstrup et al. (2015).

(a)

(c) FCC [11-2] α

β

2

C3 β

α

2

C6

α

β

WZ [1–100]

(b) FCC [111]

β

α

α

β

ZB [111]B

reverses the challenge of device fabrication; instead of having to achieve open and uniform electrical contacts, processes are required for locally removing the superconductor which is already in perfect contact with the semiconductor along its entire length. To this end, for the Al/​InAs hybrid, standard wet etching processes based on hydrofluoric acid or commercial aluminum etchants can be employed for selective etching of the aluminum. These are compatible with standard lithographic processes and etched regions can routinely be defined with a ~100 nm spatial resolution depending on the thickness of the aluminum. Also, while the semiconductor/​superconductor interface is not prone to oxidation using the epitaxial approach, native oxide will appear on the free surface of the superconductor. In order to make electrical contact to the superconducting shell this oxide can be removed by ion milling, for example. Since this process does not influence the sensitive semi/​super interface and making low-​resistance metal/​ metal contacts is usually trivial, this extra step is not problematic. Fig. 16.10a–​c illustrate schematic and scanning electron microscope (SEM) micrographs of epitaxial S–​nanowire–​N devices used for measuring the gap induced in epitaxial hybrid devices. Fig.  16.10d–​e show results of tunneling spectroscopy of the induced superconducting gap in the semiconductor measured at 20 mK with a comparison to the best results obtained from devices fabricated by the conventional method with removal of the native oxide and subsequent evaporation of the superconductor. The hardness of the induced gap is clearly improved by at least a factor of ~100. This number is reproducibly achieved over many devices, and the epitaxial nanowire heterostructures discussed here have also been employed as a platform for studying quasiparticle relaxation (Higginbotham

  481

Epitaxial superconducting contacts to nanowires  481 (a) Epitaxial interface

Evaporated superconductor

Normal

Epitaxial superconductor

Gate Semiconductor

(c)

(b)

1 µm

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2∆*

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et  al. 2015), electrostatically gateable superconducting transmon qubits (gatemons) (Larsen et  al. 2015, Casparis et  al. 2016), single-​ electron transistors (Taupin et al. 2016), and Majorana zero modes (Albrecht et al. 2016, Deng et al. 2016). Recently, there have also been studies of 2D versions of the epitaxial hybrids (Shabani et al. 2015) and the technique should also be compatible with the branched nanowire structures (Plissard et al. 2013) needed in most schemes for Majorana braiding towards topological quantum information (Alicea et  al. 2011, Hyart et al. 2013, Aasen et al. 2015). We note that in the context of Majorana devices, the superconductivity of the epitaxial shells needs to survive the magnetic fields required to enter the topological regime. For typical values of the

Fig.  16.10 (a)  Schematic illustration of an electrical device for studying the properties of induced superconductivity in epitaxial hybrids. (b)  Scanning electron micrograph of an actual device in the same configuration. The normal metal is gold and the semiconductor wire is InAs, coated with Al which is superconducting. Using the side-​gates the device can be operated close to pinch-​off of the semiconducting channel, providing tunneling spectroscopy of the induced gap as illustrated in (c). (d)  Tunnel conductance measured at 20 mK of an epitaxial device and a control device where the superconducting contact was fabricated using conventional cleaning of the semiconductor and metal evaporation. The hard gap of the epitaxial devices is clearly seen, and further enhanced in panel (e) where the same data is plotted on a logarithmic scale.  Panels (b–​e) adapted from Chang et al. (2015).

482

482  Hybrid Devices Based on Quantum Wires g-​factor of InAs, the necessary field is in the range 100–​700 mT (Das et al. 2012b, Krogstrup et al. 2015, Albrecht et al. 2016), which is larger than the bulk critical field of aluminum; however, for thin epitaxial aluminum films the critical field can exceed 1.5 Tesla as also expected from studies of planar aluminum films (Meservey et al. 1971).

16.8 Summary and outlook Even though the attachment of superconducting electrodes to quantum wires is in principle a minor upgrade compared to standard two-​terminal field-​effect-​ transitor (FET)-​like device geometries, it induces a wealth of new phenomena of which a few examples are described in this review. We have focused on quantum dot regimes (Sec. 16.3) where superconducting electrodes both enhance existing transport features (Sec. 16.4) and induce new ones such as sub-​gap resonances related to bound states (Sec. 16.5). The proposals on topological superconductivity and Majorana fermion modes in hybrid nanowire devices have greatly stimulated the field and already led to a series of exciting results. Experimental efforts in this direction now cover developments of new hybrid materials (Sec. 16.7) as well as highly advanced device geometries to allow for creation, manipulation (braiding), and detection of the new quasiparticles. The latter developments include integration with charge sensors, high-​frequency detection schemes, and superconducting cavities (Larsen et al. 2015, De Lange 2015 et al., Nichol 2015). Yet other schemes are needed to demonstrate entanglement of EPR-​like pairs generated in Cooper pair splitters (Sec. 16.6). Interestingly, the hybrid wire devices have stimulated ideas closely related to deep quantum concepts such as non-​trivial exchange statistics and non-​locality. If the fundamental tests based on quantum wires are successful, one may in turn expect a transfer of schemes from the bottom-​up grown wire-​like materials to planar hybrid technologies that are compatible with top-​down, large-​scale integration. For instance, the insight gained from nanowires has already led to progress in synthesis of 2D epitaxial hybrids that can form the basis for more complex devices in the near future. Yet, the combination of 1D quantum wires and superconducting materials has overall proven to be a powerful platform for basic science and potentially disruptive quantum technologies.

Acknowledgments We are grateful to our many co-​workers at the Niels Bohr Institute, Center for Quantum Devices and Nano-​Science Center, University of Copenhagen, as well as international collaborators who contributed through discussions or joint projects. In the area of hybrid devices and materials we wish to thank in particular

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References (Chapter-16) 483 R. Aguado, S. Albrecht, B.M. Andersen, A. Baumgartner, B. Braunecker,W. Chang, A.  Cottet, S.  Csonka, K.  Flensberg, M.C. Hels, A.  Higginbotham, E.  Johnson, H.I. Jørgensen, G.  Kiršanskas, V.  Koerting, T.  Kontos, P.  Krogstrup, A.  Levy Yeyati, P.-​E. Lindelof, M.H. Madsen, T. Marangoni, C.M. Marcus, J. Martinek, T. Novotný, J. Paaske, M. Polianski, P. Recher, C. Schönenberger, C. Strunk, and C.B. Sørensen.

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490  Hybrid Devices Based on Quantum Wires Schindele, J., Baumgartner, A. and Schönenberger, C. (2012) Phys. Rev. Lett. 109, 157002. Schindele, J., Baumgartner, A., Maurand, R., Weiss, M. and Schönenberger, C. (2014) Phys. Rev. B 89, 045422. Schneider, B.H., Singh, V., Venstra, W.J., Meerwaldt, H.B. and Steele, G.A. (2012) Nat. Comm. 5, 5819. Schäpers, T. (2001) Superconductor/​semiconductor junctions, Springer Tracts in Modern Physics vol. 174, Springer-​Verlag, Berlin, Germany. Shabani, J., Kjaergaard, M., Suominen, H.J., Kim, Y., Nichele, F., Pakrouski, K., Stankevic, T., Lutchyn, R.M., Krogstrup, P., Feidenhans’l, R., Kraemer, S., Nayak, C., Troyer, M., Marcus, C.M. and Palmstrøm, C.J. (2015) Phys. Rev. B 93, 155402. Shi, W., Wang, Z., Zhang, Q., Zheng, Y., Ieong, C., He, M., Lortz, R., Cai, Y., Wang, N., Zhang, T., Zhang, H., Tang, Z., Sheng, P., Muramatsu, H., Kim, Y.A., Endo, M., Araujo, P.T. and Dresselhaus, M.S. (2012) Sci. Rep. 2, 625. Shiba, H. (1968) Prog. Theor. Phys. 40, 435. Shibata, K., Buizert, C., Oiwa, A., Hirakawa, K. and Tarucha, S. (2007) Appl. Phys. Lett. 91, 112102. Sothmann, B., Weiss, S., Governale, M. and König, J. (2014) Phys. Rev. B 90, 220501. Sourribes, M.J.L., Isakov, I., Panfilova, M. and Warburton, P.A. (2013) Nanotechnology 24, 045703. Spathis, P., Biswas, S., Roddaro, S., Sorba, L., Giazotto, F. and Beltram, F. (2011) Nanotechnology 22, 105201. Stanescu, T.D. and Tewari, S. (2013) J. Phys. Condens. Mat. 25, 233201. Stanescu, T.D., Lutchyn, R.M. and Das Sarma, S. (2014) Phys. Rev. B 90, 085302. Suyatin, D.B., Thelander, C., Bjoerk, M.T., Maximov, I. and Samuelson, L. (2007) Nanotechnology 18, 105307. Taboryski, R., Clausen, T., Bindslev Hansen, J., Skov, J.L., Kutchinsky, J., Sørensen, C.B. and Lindelof, P.E. (1996) Appl. Phys. Lett. 69, 656. Takayanagi, H., Akazaki, T. and Nitta, J. (1995) Phys. Rev. Lett. 75, 3533. Takei, S., Fregoso, B.M., Hui, H-​Y., Lobos, A.M. and Das Sarma, S. (2013) Phys. Rev. Lett. 110, 186803. Takesue, I., Haruyama, J., Kobayashi, N., Chiashi, S., Maruyama, S., Sugai, T. and Shinohara, H. (2006) Phys. Rev. Lett. 96, 057001. Tan, Z.B., Cox, D., Nieminen, T., Golubev, D., Lesovik, G.B. and Hakonen, P.J. (2015) Phys. Rev. Lett. 114, 096602. Taupin, M., Krogstrup, P., Nguyen, H., Mannila, E., Albrecht, S.M., Nygård, J., Marcus, C.M. and Pekola, J.P. (2016) arXiv:1601.01149. Tinkham, M. (2004) Introduction to superconductivity, 2nd ed., Dover Publications, Mineola, NY. Tsuneta, T., Lechner, L. and Hakonen, P.J. (2007) Phys. Rev. Lett. 98, 087002.

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17

Superconducting Nanodevices J. Gallop and L. Hao

17.1 The drive to the nanoscale 

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17.2 Types of Josephson junction 

494

17.3 NanoSQUIDs imply improved energy sensitivity 

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17.4 Applications 

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17.5 Future developments 

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17.6 Summary and outlook 

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References (Chapter-​17) 

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National Physical Laboratory, Hampton Rd., Teddington TW11 0LW, UK

This chapter concerns nano-​superconducting devices, treating only the very smallest superconducting circuits, at the extreme opposite end from superconducting magnets, motors, or generators. A dominant theme of the chapter is that of nano-​superconducting quantum interference devices (nanoSQUIDs; Foley and Hilgenkamp 2009)  but there is a wide range of other nanoscale superconducting devices to be considered. There are some compelling reasons why superconducting electronics has progressed strongly ever closer to the true nanometer scale over recent decades. First the need for more complexity, with an array of a very large number of similar or identical devices. This naturally leads to a requirement to minimize the size of each individual element. Second, the need for a detection device to maximize its sensitivity to some small signal to be measured. This frequently means that the sensor must be brought as close as possible to the item being sensed. This becomes particularly important in those cases where the stand-​off distance is comparable with the lateral dimension of the object being sensed. A third reason for requiring miniature measuring or sensing devices is to ensure that the perturbation of the source of the drive should be minimized.

17.1 The drive to the nanoscale 17.1.1 The capture versus matching trade-​off Detection at its most general is a combination of search and find. If the entity to be detected is not localized (it could be a particle with an unknown or variable position) or delocalized (a field, distributed across a large spatial volume) then it is important to have a large capture volume. As we have seen above, to maximize the sensitivity to the presence of the detected entity it is sometimes important

J. Gallop, L. Hao, ‘Superconducting Nanodevices’, in The Oxford Handbook of Small Superconductors. First Edition. Edited by A.V. Narlikar. © Oxford University Press 2017. Published in 2017 by Oxford University Press. DOI 10.1093/acprof:oso/9780198738169.003.0017

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The drive to the nanoscale 493 to match the size of the detector to the entity. Thus there is often a necessary compromise between capture area and sensitivity which is clearly displayed by the example of a nanoSQUID detecting a nanoparticle dipole moment. In this case if the relative position of the dipole and the nanoSQUID can be controlled it is optimal to make the SQUID detection loop as small as possible, to approach the size of the dipole. This will also allow the energy sensitivity of the SQUID to be maximized, as discussed below. However, if the entity is not localized (position unknown or is a distributed field) then the sensing volume should be maximized, subject to the limit of threshold detection. In a quantum detection process the detector can convert a field into the detection of a particle, in the sense that the wave function of the field can be collapsed to a specific localized state as in the detection of a single photon. Nanoscience and nanotechnology are driving the needs for measurement at the physical scale of 100  nm and below. The growing requirements extend beyond this simple scale change but are also frequently associated with a change from purely classical behavior (as observed on the macro scale) to quantum-​ dominated properties. This is not just in the area related to quantum computing and quantum information processing (QIP) but also feeds in to what happens to “classical” variables at high sensitivity and short length scales. Thus, although current is a continuous variable for macroscopic electrical circuits, this is no longer true when capacitors in the range 1 aF to 1 fF (corresponding to length scales in the range 10–​100 nm) are incorporated in the circuit. Then the quantized nature of electric charge becomes apparent through the effects of Coulomb blockade. Similar changes happen with other physical parameters such as magnetic dipole moment or optical intensity. These changes lead to additional measurement challenges and opportunities.

17.1.2 Necessity for “weak-​link” response and the Josephson effects Any advantages which nano-​superconducting circuits and devices garner over non-​superconducting equivalents generally relate to the presence of macroscopic quantum behavior, coupled with the ability to influence the superconducting order parameter by way of externally applied stimuli. The macroscopic quantum nature of superconductivity is not always apparent but is usually implicit in the behavior, as will be discussed below in the context of the nanowire superconducting particle detector. Weak superconductivity may be defined as the situation when the macroscopic wave function can be influenced by external parameters including applied electric or magnetic fields, applied currents both direct and alternating, and temperature. The superconducting wave function has a characteristic length scale over which it may undergo a sensible change: this is characterized by the (temperature-​dependent) coherence length ξ(T). The Bardeen–​Cooper–​Schrieffer (BCS) theory of

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494  Superconducting Nanodevices conventional metallic superconductors can be related to the temperature-​dependent superconducting order parameter Δ(T) by the following expression:



ξ(T ) ~

ξ(0)∆ ∆(T )

It is only when the size of a superconductor is reduced to this scale that we are able to manipulate the amplitude and the phase of the order parameter. This scale can be applied to the cross-​sectional size of a wire-​like conductor, whether isolated or acting as the coupling region between larger pieces of superconductor. We require to know the typical values of the coherence lengths of superconductors in order to understand the requirements for nanoscale superconducting weak links; so, for example ξ(0) is around 1.6 µm for Al compared with only 38 nm for Nb. Importantly, note that as the temperature T approaches the transition temperature Tc from below, the coherence length diverges.

17.1.3 Smallest superconductor Recent experiments are in agreement with theory predictions that superconductivity is suppressed in nanoparticles so small that the electronic energy spacing exceeds the value of the energy gap in the bulk material. For example for Al (Tc 1.3 K in the bulk and Δ(0)  =  0.35 meV), superconductivity is suppressed if the particle diameter is less than about 10 nm (Ralph et al. 1995, 1997, Black et al. 1996, Bennemann and Ketterson 2008). The predicted and now observed quantum phase slips in narrow wires are the process which leads to suppression of superconductivity in restricted geometries (Bezryadin et al. 2000). For this reason our discussion of nano-​superconducting devices is limited to an approximate scale range from 5 nm to around a few hundred nanometers. In the next section we summarize in brief the fabrication techniques which have made these length scales accessible over the last two decades.

17.2 Types of Josephson junction In this chapter we have somewhat arbitrarily decided to treat only those superconducting devices with a maximum linear dimension below around 1 micrometer (1 μm) as nanodevices, not quite at the conventional nanoscale ( 1 / k B . •

(17.8)

This relationship has not been widely experimentally tested to date, yet it will become of increasing importance in a future nanoscale world. To test it we need to define how δE is connected to a system’s spatial scale. To be relevant to this chapter the most significant question is “at what length scale does the conventional definition of temperature fail?” An important contribution to this debate came

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Future developments 517 from Hartmann et al. (2004). They considered a linear chain of atoms interacting with coupling energy ε between nearest neighbors. The effect of this coupling is to split the initially degenerate energies of the isolated particles. Depending on the accuracy with which a temperature is to be assigned to the chain a minimum length may be estimated, depending on the ratio of T to Q, the chain’s Debye temperature. For cryogenic temperatures it turns out that the minimum chain length for temperature to be realized is of the order of micrometers, representing a chain of thousands of atoms. This may be profoundly significant when considering a useful or appropriate definition of the temperature of nanoscale objects. 17.5.2.1 Thermodynamic fluctuations Superconducting detectors may have an influence here, in allowing the scale at which temperature may be well defined to be extended downwards. This relates to random fluctuations in the thermodynamic temperature of an object with small heat capacity, first identified by Landau and Lifschitz (1995) and derived from the fluctuation dissipation theorem (Day et al. 1997):



∆T 2 =

kBT 2 C

(17.9)

This appears to lead to a minimum detectable energy change given by the expression

∆E = C ∆T = (CkBT 2 )1/ 2

(17.10)

However, single-​photon detectors based on transition edge sensors have demonstrated how this limit may be breached (Miller et al. 2003). If the dimensionless rate of change of the transition edge sensor with temperature α = (T/​R.dR/​dT) then it has been shown that the energy sensitivity is reduced: 1/ 2



 Ck T 2  ∆E = C ∆T =  B   α 

(17.11)

α can have values as high as 100 and it is this response which makes superconducting single-​photon detectors so successful and in this case the uncertainty in temperature can be suppressed in a similar way, to around 1/ 2



k T2 ∆T =  B   αC 

(17.12)

so that the chain length scale at which temperature can be defined will probably be reduced to no more than 100 nm. Further work and confirmation of the above equations would be an important contribution to understanding nanoscale thermodynamics.

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17.5.3 Advanced readout methods for high frequencies Conventional dc SQUIDs of whatever physical scale are read out using relatively low-​frequency low-​noise amplifiers, typically operating below 100 kHz. The group of Schurig at PTB (see for example Drung et al. 2007) have demonstrated very high bandwidths, up to 350 MHz, by using a 16 element Series SQUID Array (see Sec. 17.4.6) with a cold flux locked loop electronics positioned in close proximity to the SQUID array. This demonstrated high-​frequency operation close to the quantum limit but still limited by the noise of the first amplifying stage. SQUID performance where the subsequent amplifier noise is less than the quantum limit was demonstrated by Hatridge et al. (2011) using a dispersive readout, in which a dc SQUID is coupled to a lumped circuit capacitor, giving a resonant circuit with a relatively low Q factor (~30). The main point about this LC circuit is that the resonant frequency can be periodically modulated between around 7 GHz and 4 GHz by applying a dc magnetic flux, while the junctions remain in the superconducting state. Thus there is no Johnson noise associated with the shunt resistance of the junctions (a little like the ISTED described in Sec. 17.2.2 and 17.2.3). By applying a drive frequency at the resonant frequency and measuring the reflected signal from the resonant circuit there will be a phase change of the reflected signal when the flux applied to the SQUID is changed. In addition, the SQUID inductance becomes nonlinear in response to increasing power and this permits parametric response to an input signal. Although in the low power limit the system is limited by the noise temperature of the microwave amplifier which follows the reflected signal, if the drive power is sufficiently increased the input gain grows until the noise temperature of this amplifier is overwhelmed and quantum-​limited operation, even at 50 mK, has been demonstrated. In this way bandwidths in the MHz region have been achieved.

17.5.4 Kinetic inductance: sinner or saint? We have already seen how important the magnetic inductance L of the SQUID loop is, in determining the signal to noise level achievable with a particular design. Conventional treatments of SQUID theory generally assume the inductive component comes purely from the geometric properties of the loop; that is, if a current I is circulating the geometric inductance L is just the integral over a cross-​section of the loop of the magnetic flux density perpendicular to this surface. There is an additional contribution to the inductance which derives from the inertia of the superconducting pairs carrying any circulating supercurrent and it becomes particularly significant when the thickness of the superconductor becomes less than the London penetration depth λ. For conventional metallic superconductors such as Nb this is around 50 nm. For conventional SQUID devices significant kinetic inductance is a disadvantage since it adds to the geometric inductance of the loop

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Future developments 519 without in any way increasing the available signal. So the general rule is to design junctions to have very low kinetic inductance. 17.5.4.1 Kinetic inductance detector An interesting application of kinetic inductance to provide a direct basis for magnetic field detection has been described by Luomahaara et  al. (2014). When a loop of thin superconducting film (with thickness less than the London penetration depth λ) is subject to a circulating supercurrent Is (such as would arise if a magnetic field B is applied perpendicular to the loop area), the kinetic inductance becomes non-​linear in Is as Is approaches the critical current of the loop. If the loop is included as part of a high Q RF resonant frequency, small changes in B (or Is) will give rise to easily detectable changes in the resonant frequency. The high accuracy of frequency measurements makes this a sensitive technique and the authors predict that, for macroscopic loops (up to say 10 mm in diameter) sensitivities of fT/​(Hz)1/​2 should be achievable. This is not as yet a nanoscale device but the principle of operation could be applied at this scale, for example to detect the change in circulating current induced in a nanoscale loop by the nearby movement of another superconducting element (e.g. a nanomechanical resonator).

17.5.5 Unconventional superconductors There is often a need to operate nanoSQUIDs at lower temperatures than the conventional liquid helium boiling point (1–​4 K). Although Josephson (SIS) tunnel junctions operate over a wide range of temperature since it is only close to Tc that the critical current is strongly T dependent, this is not the case for microbridge junctions. Here the operating temperature range may be much smaller so that junctions must be designed for a chosen temperature. Bilayer devices can be useful when an operating temperature below 1 K is required and neither Nb nor Al microbridges are suitable. A combination of a thin sandwich of a superconductor and a normal metal can have a Tc which is adjustable by varying the relative thicknesses. An example is the Ti–​Au bilayer junctions, patterned by EBL, reported by Blois et al. (2013). A dominant feature of the past 20 years of the development of the century-​old study of superconductivity has been the appearance of many families of novel superconducting materials. These range from organic metals with relatively low transition temperatures, through many unexpected ternary or quaternary compounds (especially of perovskite structure) and including unexpectedly high-​Tc metallic superconductors such as MgB2. The highest cuprate superconductors extend operating temperatures to over 100 K. The most widely used cuprate superconductor for SQUIDs is YBCO. The Josephson junctions used for these devices are typically based on weak links formed where there is a grain boundary within the material and the conductor is patterned to an appropriate width and thickness. Good low-​noise performance of YBCO nanoSQUIDs has been reported (Arpaia et al. 2014) using nanowire

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520  Superconducting Nanodevices junctions some 200 nm long, patterned by EBL. Flux modulation occurs up to an operating temperature of 83 K but optimum low-​noise performance (1 μΦ0/​Hz1/​2) is observed. Effective MgB2 SQUIDs using microbridge junctions have also been reported (Mijatovic et al. 2005) operating up to 37 K though the noise performance is not yet comparable with Nb devices. Further improvements can be expected.

17.6 Summary and outlook As with many other aspects of condensed matter physics, superconducting devices have evolved rapidly over the past decade or so, as they begin to enter the nanoscale. Fabrication methods which have been developed for superconducting nanowires, nanoscale Josephson junctions based on a variety of barrier materials, and microbridge weak-​link methods have been described in this chapter. The unique properties of the devices fabricated by these means have been outlined. Considerable progress in understanding the device fabrication, limitations, and operation means that the quantum limit can be approached over a temperature range up to close to 10 K and up to frequencies of 100 MHz. If nano-​superconductivity is to make a breakthrough into the quantum information-​processing field even greater improvements in performance will be required, especially in the area of reproducibility of devices. However, there are indications of how this may proceed so the future remains bright and challenging.

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References (Chapter-17) 521 Blois, A., Rozhko, S., Hao, L., Gallop, J.C. and Romans, E.J. (2013) J. Appl. Phys. 114, 233907. Bose, S., Banerjee, R., Genc, A., Raychaudhuri, P., Fraser, H.L. and Ayyub, P. (2006) J. Phys. Condens. Matter 18, 4553. Bose, S., Raychaudhuri, P., Banerjee, R. and Vasa, P. (2005) Phys. Rev. Lett. 95, 147003. Bouchiat, V., Faucher, M., Thirion, C., Wernsdorfer, W., Fournier, T. and Pannetier, B. (2001) Appl. Phys. Lett. 79, 123. Clarke, J. (1996) SQUID fundamentals, in SQUID sensors:  Fundamentals, fabrication and applications, Vol. 329 of the NATO ASI Series, pp. 1–​62. Clarke, J. and Braginski, A.I. (Eds.) (2004) The SQUID handbook: Applications of SQUIDs and SQUID systems, vol. 1, Wiley-​VCH, Weinheim. Cox, D. (2015) Focused ion beam—​Materials science applications, IOPScience, Morgan & Claypool Publishers. Crommie, M.F., Lutz, C.P. and Eigler, D.M. (1993) Science 262, 218. Day, P., Hahn, I., Chui, T.C.P., Harter, V., Rowe, D. and Lipa, J.A. (1997) J. Low Temp. Phys. 107, 359. Devoret, M.H. and Schoelkopf, R.J. (2013) Science 339, 1169. Drung, D., Aßmann, C., Beyer, J., Kirste, A., Peters, M., Ruede, F. and Schurig, T. (2007) IEEE Trans. Appl. Supercond. 17, 699. Etaki, S., Poot, M., Mahboob, I., Onomitsu, K., Yamaguchi, H. and van der Zant, H.S.J. (2008) Nature Physics 4, 785. Etaki, S., Konschelle, F., Blanter, Y.M., Yamaguchi, H. and van der Zant, H.S.J. (2013) Nat. Comm. 4, 1803. Fink, H.J. and Grunfeld, V. (1986) Phys. Rev. B. 33, 6088. Foley, C.P., and Hilgenkamp, H. (2009) Supercond. Sci. Technol. 22, 064001. Galer, S., Hao, L., Gallop, J.C., Palmans, H., Kirkby, K. and Nisbet, A. (2010) Radiat. Prot. Dosimetry 143, 427. Gallop, J.C. and Radcliffe, W.J. (1985) IEEE Trans. Magn. 21, 602. Gallop, J.C. (2003) Supercond. Sci. Technol. 16, 1575. Gol’tsman, G.N., Okunev, O., Chulkova, G., Lipatov, A., Semenov, A., Smirnov, K., Voronov, B., Dzardanov, A., Williams, C. and Sobolewski, R. (2001) Appl. Phys. Lett. 79, 705. Granata, C., Esposito, E., Vettoliere, A., Petti, L. and Russo, M. (2008) Nanotechnology 19, 275501. Granata, C., Vettoliere, A., Russo, R., Fretto, M., De Leo, N. and Lacquaniti, V. (2013) Appl. Phys. Lett. 103, 102602. Granata, C. and Vettoliere, A. (2015) ArXiv:1505.06887.pdf Hadfield, R.H. (2009) Nature Photonics 3, 696. doi:10.1038/​nphoton.2009.230 Hagedorn, D., Dolata, R., Buchholz, F.-​I. and Niemeyer, J. (2002) Physica C 372–​376, 7. Hao, L., Gallop, J.C., Gardiner, C., Josephs-​Franks, P., Macfarlane, J.C., Lam, S.K.H. and Foley, C. (2003) Supercond. Sci. Technol. 16, 1479.

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522  Superconducting Nanodevices Hao, L., Macfarlane, J.C., Lam, S.K.H., Foley, C.P., Josephs-​ Franks, P. and Gallop, J.C. (2005) IEEE Trans. Appl. Supercond. 15, 514. Hao, L., Macfarlane, J.C., Gallop, J.C., Cox, D., Beyer, J., Drung, D. and Schurig, T. (2008) Appl. Phys. Lett. 92, 192507. Hao, L. (2011) J. Phys.: Conf. Ser. 286, 012013. Hao, L., Aßmann, C., Gallop, J.C., Cox, D., Ruede, F., Kazakova, O., Josephs-​ Franks, P., Drung, D. and Schurig, T. (2011) Appl. Phys. Lett. 98, 092504. Hao, L., Cox, D.C., Gallop, J.C., Chen, J., Rozhko, S., Blois, A. and Romans, E. (2013) IEEE Trans. Appl. Supercond. 23, 1800304. Hao, L., Cox, D.C., Gallop, J.C. and Chen, J. (2015) IEEE J. Sel. Top. Quant. 25, 9100108. Hartmann, M., Mahler, G. and Hess, O. (2004) Phys. Rev. Lett. 93, 080402. Hatridge, M., Vijay, R.D., Slichter, H., Clarke, J. and Siddiqi, I. (2011) Phys. Rev. B 83, 134501. Hazra, D., Pascal, L.M.A., Courtois, H. and Gupta, A.K. (2010) Phys. Rev. B 82, 184530. Hazra, D., Kirtley, J.R. and Hasselbach, K. (2014) Appl. Phys. Lett. 104, 152603. Jamet, M., Wernsdorfer, W., Thirion, C. and Mailly, D. (2001) Phys. Rev. Lett. 86, 4676. Jamieson, D.N., Yang, C., Hopf, T., Hearne, S.M., Pakes, C.I., Prawer, S., Mitic, M., Gauja, E., Andresen, S.E., Hudson, F.E., Dzurak, A.S. and Clark, R.G. (2005) Appl. Phys. Lett. 86, 202101. Josephson, B.D. (1962) Phys. Lett. 1, 251. Kirkby, K.J., Grime, G.W., Webb, R.P., Kirkby, N.F., Folkard, M., Prise, K. and Vojnovic, B. (2007) Nucl. Instrum. Meth. B 260, 97. Kirtley, J.R. and Wikswo, J.P. (1999) Ann. Rev. Mater. Sci. 29, 117. Kirtley, J.R. Gibson Jr., G.W., Fungy, Y.K.K., Klopfer, B., Nowack, K., Kratz, P.A., Molz, J.M., Arpesz, J., Forooghiz, F., Huberx, M.E., Bluhmz, H. and Moler, K.A. (2013) Proc. ISEC 2013, pp. 1–​2. Korneev, A., Kouminov, P., Matvienko, V., Chulkova, G., Smirnov, K., Voronov, B., Gol’tsman, G.N., Currie, M., Lo, W., Wilsher, K., Zhang, J., Słysz, W., Pearlman, A., Verevkin, A. and Sobolewski, R. (2004) Appl. Phys. Lett. 84, 5538. Lam, S.K.H. and Tilbrook, D.L. (2003) Appl. Phys. Lett. 82, 1078. Lam, S.K.H. (2006) Supercond. Sci. Technol. 19, 963. Lam, S.K.H., Yang, W., Wiogo, H.T.R. and Foley, C.P. (2008) Nanotechnology 19, 285303. Landau, L.D. and Lifshitz, E.M. (1980) Statistical physics, 3rd ed., Pergamon, London. Luomahaara, J., Vesterinen, V., Gronberg, L. and Hassel, J. (2014) Nature Commun. DOI: 10.1038/​ncomms5872 McDonald, D.G. (1987) Appl. Phys. Lett. 50, 775. Mijatovic, D., Brinkman, A., Veldhuis, D., Hilgenkamp, H., Rogalla, H., Rijnders, G., Blank, D.H.A., Pogrebnyakov, A.V., Redwing, J.M., Xu, S.Y., Li, Q. and Xi, X.X. (2005) Appl. Phys. Lett. 87, 192505.

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References (Chapter-17) 523 Miller, A.J., Nam, S.W., Martinis, J.M. and Sergienko, A.V. (2003) Appl. Phys. Lett. 83, 791–​793. Mirza, M.M, MacLaren, D.A., Samarelli, A., Holmes, B.M., Zhou, H., Thoms, S., MacIntyre, D. and Paul, D.J. (2014) Nano Lett. 14, 6056. Mohammad, M.A., Muhammad, M., Dew, S.K. and Stepanova, M. (Eds.) (2012) in Nanofabrication techniques and principles, edited by M. Stepanova and S. Dew, Springer, Dordrecht, ­chapter 2. Nagel, J., Kieler, O.F., Weimann, T., Wolbing, R., Kohlmann, J., Zorin, A.B., Kleiner, R., Koelle, D. and Kemmler, M. (2011) Appl. Phys. Lett. 99, 032506. Neocera scanning SQUID microscope:  http://​www.neocera.com/​magma/​ Products.html Pla, J.J., Tan, K.Y., Dehollain, J.P., Lim, W.H., Morton, J.J.L., Jamieson, D.N., Dzurak A.S. and Morello, A. (2012) Nature 489, 541. Ralph, D.C., Black, C.T. and Tinkham, M. (1995) Phys. Rev. Lett. 74, 3241. Ralph, D.C., Black, C.T. and Tinkham, M. (1997) Phys. Rev. Lett. 78, 4087. Romans, E.J., Rozhko, S., Young, L., Blois, A., Hao, L., Cox, D. and Gallop, J.C. (2011) IEEE Trans. Appl. Supercond. 21, 404. Rozhko, S., Hino, T., Blois, A., Hao, L., Gallop, J.C., Cox, D.C. and Romans, E.J. (2013) IEEE Trans. Appl. Supercond. 23, 1601004. Ruede, F., Bechstein, S., Hao, L., Aßmann, C., Schurig, T., Gallop, J., Kazakova, O., Beyer, J. and Drung, D. (2011) IEEE Trans. Appl. Supercond. 21, 408. Skocpol, W.J., Beasley, M.R. and Tinkham, M. (1974) J. Appl. Phys. 45, 4054. Sobolewski, R., Verevkin, A., Golt’tsmann, G.N. and Lipatov, A.L. (2003) IEEE Trans. Appl. Supercond. 13, 1151. Tettamanzi, G.C., Pakes, C.I., Lam, S.K.H. and Prawer, S. (2009) Supercond. Sci. Technol. 22, 064006. Tilbrook, D.L. (2009) Supercond. Sci. Technol. 22, 064003. Vasyukov, D., Anahory, Y., Embon, L., Halbertal, D., Cuppens, J., Neeman, L., Finkler, A., Segev, Y., Myasoedov, Y., Rappaport, M.L., Huber, M.E. and Zeldov, E. (2013) Nature Nanotechnology 8, 639. Villégier, J.C., Delaet, B., Feautrier, P., Frey, L., Delacour, C. and Bouchiat, V. (2006) J. Phys.: Conf. Ser. 43, 1373. Wernsdorfer, W. (2009) Supercond. Sci. Technol. 22, 064013. Wölbing, R., Nagel, J., Schwarz, T., Kieler, O., Weimann, T., Kohlmann, J., Zorin, A.B., Kemmler, M., Kleiner, R. and Koelle, D. (2013) Appl. Phys. Lett. 102, 192601.

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18 18.1 Introduction: superconducting qubits  524

Superconducting Quantum Bits of Information—​Coherence and Design Improvements

526

J. Bylander

18.3 Decoherence. Characterization and mitigation of noise 

529

Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-​412 96 Gothenburg, Sweden

18.4 Superconducting qubits 

539

18.5 Circuit quantum electrodynamics (c-​QED) 

553

18.6 Second-​generation superconducting qubits 

557

18.7 Summary and outlook 

561

References (Chapter-18) 

562

18.2 Single-​qubit Hamiltonians and reference frames 

18.1 Introduction: superconducting qubits Qubits—​quantum bits of information—​can be realized in many natural or engineered quantum systems. By taking advantage of superposition states and entanglement, researchers can explore physics and develop new quantum technology, notably for quantum information processing. Superconductors are naturally well suited for making quantum devices. They dissipate very little energy, in principle enabling long lifetimes of the quantum states. Devices with nanoscale features can be made on a microchip, using lithographic techniques, and can be scaled to a large number of qubits in an integrated circuit. This chapter contains a description of qubit dynamics and decoherence, an introduction to the basic device concepts for superconducting qubits, and a discussion of the key advances that have led to modern, improved devices.

18.1.1 Context The criteria for quantum computer hardware were concisely formulated by DiVincenzo (2000)—​essentially prescribing a scalable architecture (to tens of thousands of coupled qubits), enabling the application of high-​accuracy computational gates and measurements, and a protocol for error correction throughout the course of a computation. The review by Ladd et  al. (2010) describes the progress and challenges for several physical platforms. Over the last few years,

J. Bylander, ‘Superconducting Quantum Bits of Information—Coherence and Design Improvements’, in The Oxford Handbook of Small Superconductors. First Edition. Edited by A.V. Narlikar. © Oxford University Press 2017. Published in 2017 by Oxford University Press. DOI 10.1093/acprof:oso/9780198738169.003.0018

  525

Introduction: superconducting qubits  525 advances have been tremendous—​perhaps in particular with superconducting qubits—​making it an exciting time to be working in this field. A number of review articles contain detailed descriptions of the first-​generation devices—​charge, flux, and phase qubits (Averin 2000, Makhlin et al. 2001, Martinis and Osborne 2004, Devoret and Martinis 2004, Devoret et al. 2004,You and Nori 2005, Wendin and Shumeiko 2007, Clarke and Wilhelm 2008, Oliver 2013, Tsai 2016). The reviews by Schoelkopf and Girvin (2008) and You and Nori (2011) treat quantum optics and atomic physics using microwave photons and superconducting qubits. Houck et al. (2012) reviewed quantum simulation. The work on materials improvements for superconducting qubits, identifying and eliminating the fluctuators within the materials that lead to decoherence, was reviewed by McDermott (2009), Siddiqi (2011), Oliver and Welander (2013), and Martinis and Megrant (2014). A comprehensive review of quantum noise, measurement, and amplification was written by Clerk et al. (2010). Future outlooks of the field, along with overviews of the progress on coherent manipulations, quantum-​ state measurement, and materials improvement, have been published in several reviews (Siddiqi 2011, Devoret and Schoelkopf 2013, Oliver and Welander 2013).

18.1.2 Scope of this chapter The present chapter focuses on the characterization and mitigation of decoherence—​topics less comprehensively treated in the above-​mentioned reviews. We first describe qubit dynamics mathematically (Sec. 18.2); we describe how decoherence originates with noise and how to characterize and mitigate this noise using magnetic-​resonance-​type pulse sequences (Sec. 18.3). We then introduce the first-​generation superconducting qubits (Sec. 18.4) and the now-​ dominant circuit-​ QED (quantum electrodynamics) architecture with qubits coupled to microwave resonators (Sec. 18.5); we finally review the evolution of the experimental state of the art with improved qubit designs (Sec. 18.6). Out of necessity, the author has chosen to cite a limited number of works relevant to the topic—​ decoherence and qubit improvements—​ rather than a comprehensive list; several relevant works were consequently left out. A  number of important and modern topics in this rapidly evolving and diverging field have had to be omitted altogether, for example, qubit couplings; measurement methods and amplification; optimal control; nonadiabatic transitions; microwave quantum optics; quantum-​ coherent phase slips; Andreev bound states; qubits based on Majorana fermions; qubit-​based sensors; and hybrids involving superconductors and other systems such as semiconductors, surface-​acoustic waves, spin-​ensemble quantum memories, and others. The currently exploding research on architectures for quantum computing, quantum error correction, analog and digital quantum simulation, and quantum adiabatic annealing—​now also funded by big corporations—​is also not discussed. As should be obvious

526

526  Superconducting Quantum Bits of Information from this (non-​exhaustive) list, this field has grown almost out of control since the first demonstration of quantum coherence in an artificial atom (Nakamura et al. 1999).

18.2 Single-​qubit Hamiltonians and reference frames 18.2.1 Laboratory frame The model Hamiltonian for a quantum-​mechanical two-​state system is  Hˆ = − ω (t ) ⋅ σˆ (18.1) 2



where  is Planck’s constant and σˆ = ( σˆ x , σˆ y , σˆ z ) are Pauli operators (see e.g. Abragam 1961). The static Hamiltonian is conveniently written as  Hˆ 0 = − [ ∆σˆ z + εσˆ x ] , (18.2) 2



here with parameters Δ and ε expressed in units of angular frequency. In the presence of an external harmonic control field with amplitude Arf, angular fre quency ωrf, and phase η, we add one term:  Hˆ rf = − Arf cos (ω rf t + η) σˆ x . In the 2 laboratory frame of reference (x,y,z), these parameters correspond directly to externally applied fields. Fig. 18.1 shows the archetypal two-​state system—​a spin-​1/​2 particle. ω

ω 1

B z

1/2

x

ωq y

∆ 0 0

ε

Fig. 18.1  A quantum two-​state system encodes a physical qubit. A spin-​1/​2 particle in an external magnetic field, B, constitutes the archetypal two-​state system. A strong field along the laboratory negative z-​axis, Bz = Δ/​2g, where g is the gyromagnetic ratio, causes Larmor precession of the spin’s magnetic moment around the –​z axis at the angular frequency Δ. Setting ε = 0, in the Hamiltonian Ĥ0 (Eq. 18.2), the magnitude of this quantizing field consequently defines the level-​splitting frequency, ωq = Δ, between spin-​up and spin-​down. (Note that the Hamiltonian is here written with the prefactor containing a minus sign, in which case the ground state is spin-​up, |+ z〉; sometimes the opposite convention is used, with a plus sign, in which case the ground state is spin-​down, |–​z〉.)

  527

Single-qubit Hamiltonians and reference frames  527 The physical meanings of the control parameters differ, depending on the particular qubit modality (see Sec. 18.4.3); for example, in a superconducting flux qubit, the energy ε is proportional to the circulating current around a loop of wire, and ∆ is a function of the tunnel coupling between opposing directions of circulation. (In this example the “laboratory frame” is clearly an abstract construction, because one of the basis vectors, ∆σˆ z , does not have a spatial direction, but instead represents a coupling energy.)

18.2.2 Unitary rotations and Bloch-​sphere representation ϑ ϑ   A general, pure qubit state, Ψ〉 = e i χ  cos 0〉 + e i φ sin |1〉 , is fully described by  2 2  the two phase angles ϑ and ϕ (and a global phase χ), and can be visualized on the surface of the Bloch sphere—​see Fig. 18.2. We can write its density matrix as ρˆ = Ψ〉〈 Ψ =



(

)

1 ˆ 1 + r ⋅ σˆ (18.3) 2

where r is the Bloch vector. The simplest qubit dynamics arises from constant Hamiltonians, Ĥ1  =  − Aσˆ n , here applied in an arbitrary direction n. We can then easily solve the 2 d Schrödinger equation for the state vector, i  Ψ〉 = Hˆ 1 | Ψ〉, in the eigenbasis dt of the Hamiltonian, using a unitary rotation,

0

z

r

t

with the angle δ = ∫ dt ′A (t ′ ) , which can be visualized on the Bloch sphere.

φ x

0

18.2.3 Qubit frame (eigenbasis) By applying a bias field, ε, the qubit eigenbasis is being tilted away from the laboratory frame by a mixing angle θ  =  arctan(ε/​Δ)—​see Fig. 18.3—​and the qubit’s level splitting increases to ωq  =  (ε2 + Δ2)½—​see Fig. 18.1. The transformations



σˆ x ′ = cosθ σˆ x − sinθ σˆ z , σˆ y ′ = σˆ y , (18.5) σˆ z ′ = cosθ σˆ z + sinθ σˆ x ,

bring the Hamiltonian into the tilted frame (x’,y’,z’),

y

ϑ

δ δ   δ  Rn (δ ) = exp  i n ⋅ σ , (18.4)  = cos 1 + i sin n ⋅ σ  2 2 2

1

Fig.  18.2 The Bloch sphere represents all the possible states of a qubit. A general qubit state is parameterized by the phase angles ϑ and ϕ. The Bloch vector, r, has unit length for pure states, |r| = 1, and less than unit length for mixed states, |r| < 1. We identify the spin-​up and spin-​down states with the quantum-​ logical |0〉 and |1〉 ket vectors, respectively (see Abragam 1961).

528

528  Superconducting Quantum Bits of Information Laboratoty frame

Qubit frame

Rotating frame

z

z’

Z

z’

∆

Fig. 18.3  Laboratory frame (x,y,z), qubit eigenframe (x’,y’,z’), and rotating frame (X,Y,Z). (Figure adapted from Yan et al. 2013.) The Pauli spin matrices have  representations

θ

^ H0

Y

y

^ Hrf

^ H0 y’ X

ω rf t ε x’

x

2ω R cos(ω rf t + η )

x’

–h ∆ω 2

Y ^ ~ H –h ω 2 R

X

 0 1 1 0   0 −i  σˆ X =   , σˆ Y =  . , σˆ Z =   1 0  i 0   0 −1



 Hˆ ’ = − ω q σˆ z ′ + Arf (t ) cos θ cos (ω rf t + η) σˆ x ′ 2 (18.6) + Arf (t ) sin θ cos (ω rf t + η) σˆ z ′  .

The qubit’s free-​evolution dynamics under the static Hamiltonian is naturally described in this frame: with the control field turned off, Arf = 0, the spin (Bloch vector) precesses around the –​z’ axis (quantization axis) at angular frequency ωq.

18.2.4 Rotating frame One means of driving transitions between the qubit states is by applying a weaker field, Arf, nearly resonant with the level splitting, ωrf ≈ Δ, and perpendicular to the quantization axis (i.e. to the qubit energy eigenbasis)—​see Eq. 18.6. The transformations



σˆ X = cos(ω rf t ) σˆ x ′ + sin(ω rf t )σˆ y ′ , σˆ Y = cos(ω rf t )σˆ y ′ − sin(ω rf t )σˆ x ′ , (18.7) σˆ Z = σˆ z ′ ,

bring the Hamiltonian into the frame (X,Y,Z) rotating around the axis z’ at the drive angular frequency ωrf—​again see Fig. 18.3. Under weak driving, Arf  ω rf , we can invoke the rotating wave approximation (RWA) and omit terms oscillating at ωrf and 2ωrf in the rotating frame. The Hamiltonian is then given by

  H = −  ∆ωσˆ Z + ω R (cos ησˆ X + sin ησˆ Y ) , (18.8) 2

where ω R = (1 2) Arf cos θ is the Rabi angular frequency under resonant driving, Δω = ωq−​ωrf = 0. Under detuned driving, this Hamiltonian describes an effective field directed along a tilted axis, with the effective Rabi angular frequency ωR,eff  =  (ωR2 + Δω2)½ ≈ ωR + Δω2/​2ωR. The qubit’s driven evolution is conveniently described in this rotating frame. For resonant driving, with a choice  of phase η = 0, the rotating-​frame Hamiltonian becomes H = − (  2) ω R σˆ X , and

  529

Decoherence. Characterization and mitigation of noise  529 describes a pseudospin quantized along –​X with a level splitting ωR. Note that the Rabi nutation around –​X, in the rotating frame, is analogous to the free precession around  –​z’ in the tilted frame (qubit frame), Hˆ ′ = − (  2) ω q σˆ z ′ (where “free” precession means the absence of a driving field).

18.3 Decoherence. Characterization and mitigation of noise The Hamiltonians in Sec. 18.2 describe the ideal, coherent qubit dynamics in the absence of noisy control parameters. In reality, decoherence of a pure quantum superposition state arises from the interaction between the constituent system and its environment. These interactions act to entangle the qubit with uncontrolled degrees of freedom in the environment or “bath,” resulting in a mixed state for the single qubit. A mixed state is represented by a Bloch vector with less than unit length, |r| < 1, that is, located inside the Bloch sphere—​see Fig. 18.2. In the Bloch–​Redfield description, two characteristic times—​T1 and T2—​ phenomenologically describe the dissipative dynamics of two-​state systems—​ see an illustration in Fig. 18.4 and explanations throughout Sec. 18.3. With the Hamiltonian Hˆ ′ = − (  2) ω ′ (t ) ⋅ σˆ ′ , where primes denote the qubit frame, the Bloch equations take the form

Energy relaxation 0

Dephasing

z’

0 y’

z’ y’

Noise

Noise x’ 1

x’ 1

Fig.  18.4 Energy relaxation (T1 relaxation or depolarization or mixing) is due to noise, at the qubit frequency, that is transverse to the qubit’s quantization axis (eigenstates). Dephasing (with characteristic time scale Tϕ) is due to low-​frequency noise that is longitudinal to the quantization axis. During free evolution, the Bloch vectors of an ensemble of qubits fan out—​their phases ϕ(t) differ—​due to the varying fields, resulting in an average reduction in the transverse polarization.The times T1 and Tϕ combine to form the coherence time, 1/​T2 = 1/​2T1 + 1/​Tϕ (see Abragam 1961).

530

530  Superconducting Quantum Bits of Information

r = r × ω ′ (t ) −

r xˆ ′ + ry yˆ ′ rz − rz0 zˆ ′ − x , (18.9) T1 T2

where rz0 is the steady-​state polarization. T1 is the longitudinal energy-​relaxation or depolarization time toward a state of thermal equilibrium of the diagonal elements (populations) of the density matrix. T2 is the transverse relaxation or coherence time characterizing the decay of the off-​diagonal elements (coherences) of the density matrix; this occurs due to a combination of depolarization (T1) and pure dephasing, which has a characteristic time Tϕ, 

1 1 1 = + . (18.10) T2 2T1 Tφ

This nomenclature was originally developed to describe magnetic-​resonance experiments, in which an ensemble of precessing spins produce an aggregate signal that induces a current in a detection coil (see Abragam 1961). Field inhomogeneities across the sample then cause effectively different energy-​splittings among the constituent spins, thereby broadening the spectroscopic resonances. Spins precessing at different rates cause dephasing of the net transverse magnetization. For a single artificial (or natural) spin, on the other hand, such as a superconducting qubit, the inhomogeneities should be understood as a temporal fluctuation of the control field, and the decay is in the average taken over repeated measurements on identically prepared qubits. Below we explain first how to treat the fluctuations of a qubit’s control fields (Sec. 18.3.1) and how the noise is characterized (Sec. 18.3.2); then how the noise and sensitivities affect the coherence times (Sec. 18.3.3–​4); and how the decohering effects of noise can be mitigated, both in the qubit’s eigenframe and in the rotating frame (Sec. 18.3.5–​6).

18.3.1 Fluctuating Hamiltonian. Sensitivity to noise Fluctuations naturally have their physical origins in the laboratory frame, and we can write the control parameters as a sum of a static part and a fluctuating part,

∆ = ∆ 0 + δ∆ (t ) ,

ε = ε 0 + δε (t ) (18.11)

For now we do not explicitly consider noise in the applied driving field, δArf(t), which is often not as significant as the noise in the static Hamiltonian. In the laboratory frame the fluctuating part of the Hamiltonian becomes

 δHˆ = − [δ∆σˆ z + δεσˆ x ] (18.12) 2

where, for simplicity, we omitted to explicitly mention the time dependence of δΔ(t) and δε(t).

  531

Decoherence. Characterization and mitigation of noise  531 In the qubit (tilted) and rotating frames, the fluctuating terms undergo the same geometric transformations as in Sec. 18.2.2 (Eqs. 18.5 and 18.7) and become

 δHˆ ′ = − [( δ∆ cosθ + δε sin θ) σˆ z ′ + ( − δ∆ sinθ + δε cos θ) σˆ x ′ ] (18.13) 2

and  = −  [( δ∆ cosθ + δε sin θ)σ  Z + ( − δ∆ sinθ + δε cos θ)cos (ω rf t ) σ X δH    2  Y ], + (δ∆ sin θ − δε cos θ) sin (ω rf t ) σ (18.14) respectively. We can make a connection between decoherence of the quantum state and fluctuations in a control parameter, λ  =  λ0 + δλ, where λ represents a physical noise source such as electric charge, magnetic flux, or the critical current of the Josephson junction (see Sec. 18.4). We expand the unperturbed Hamiltonian in the qubit’s eigenframe to second order in the fluctuation, δλ,

   ∂ 2 H 0′  δλ 2  ∂H 0′  + … ⋅ σˆ . (18.15) Hˆ ′ =  H 0′ ( λ 0 ) +  δλ +   2    ∂ ∂ 2 λ λ    

The sensitivity to longitudinal noise (along z’) is related to pure dephasing whereas the sensitivity to transverse noise (⊥, which may include both σˆ x ′ and σˆ y ′ ) is related to energy relaxation (see Sec. 18.3.3–​4). We adopt the notation used by Ithier et al. (2005) for the sensitivities to fluctuations of λ,

Dλ ,k = −

2 ∂ k H 0′ ( where k = 1 or 2). (18.16)  ∂λ k

The Hamiltonian becomes



 Hˆ ’ = − ω q σˆ z ′ + δω q σˆ z ′ + δω ⊥ σˆ ⊥  2    δλ 2  ˆ ˆ = − ω q σˆ z ′ +  Dλ1, z ′ δλ + Dλ 2, z ′ + σ δω σ  ⊥ ⊥ , z′ 2  2  

(18.17)

where the longitudinal and transverse sensitivities are Dλ1, z ′ =

Dλ 2, z ′ = Dλ1, ⊥

∂ω q

, ∂λ ∂ 2ω q

∂λ ∂ω ⊥ = . ∂λ 2

−

Dλ21, ⊥

ωq

, (18.18)

532

532  Superconducting Quantum Bits of Information The effect of fluctuations in the physical parameters is now expressed in the qubit’s eigenbasis by an expansion using the chain rule,

Dλ1, z ′ =

∂ω q ∂ε ∂ω q ∂∆ + , (18.19) ∂ε ∂λ ∂∆ ∂ λ

and the sensitivities to ε and Δ noise have a geometrical dependency on the quantization angle in Fig. 18.3, ∂ω q

ε = sinθ, ωq (18.20) ∆ = = cosθ. ∂∆ ω q

∂ε ∂ω q

=

The sensitivities of ε and Δ to the physical noise sources, ∂ε ∂λ and ∂∆ ∂λ, depend on the qubit device parameters, which will be explained in Sec. 18.4.

18.3.2 Noise power spectral density (PSD) and noise spectroscopy Fluctuations of a parameter λ are characterized by the noise power spectral density (PSD), S(ω)



Sλ ( ω ) =

∞

∫ dt 〈 λ (t ) λ (0)〉e

− i ωt

, (18.21)

−∞

1/ω (1/f )

σ2 ω low

ω high

ω

Fig.  18.5 The variance of the noise is obtained by integrating the noise-​ PSD over the experimental bandwidth, i.e. from ωlow to ωhigh. Inverse-​frequency noise ( 1/ f α ), with α close to 1 dominates the flux noise for superconducting qubits, from millihertz to gigahertz; also the charge noise and critical-​ current noise have 1/​ f-​type PSD, due to an ensemble of two-​state fluctuators, with different coupling strengths to the qubit (afterYan et al. 2012).

which is the (type-​1) Fourier transformation of the autocorrelation function, cλ(t) = 〈λ(t) λ(0)〉. This is the unsymmetrized, bilateral PSD, which has units [λ2] × (rad) s-​1. Conversely, we obtain the variance of the noise, in units of [λ2] × s–​2, by taking the t  =  0 inverse Fourier transform. Experimentally, we integrate over the finite bandwidth, Δω, determined by the experimental protocol—​see Fig. 18.5—​and obtain

σ2λ = cλ (0) = 2 ∫ d ω Sλ (ω ). (18.22) ∆ω

The quantum PSD is not symmetric: positive frequencies correspond to qubit deexcitation and the (stimulated and spontaneous) emission of a quantum into the environment, whereas negative frequencies correspond to qubit excitation and the absorption of a quantum from the environment. In thermal equilibrium, at temperature Θ, the excitation and deexcitation rates are related by the detailed  ω  balance condition, Γ ↑ = exp  − q  Γ ↓ , which also implies, for the quantum noise  kBΘ   ω  spectral density, S (ω ) = exp  q  S ( − ω ).  kBΘ 

  533

Decoherence. Characterization and mitigation of noise  533 On the other hand, for low-​frequency noise causing dephasing, the asymmetry between positive and negative frequencies can be ignored, and the noise can be treated classically. We can infer the noise PSD of the various fluctuators present in the qubit’s environment by measuring the decays, characterized by T1 and T2, which these fluctuators effectively cause. For example, in a flux qubit, λ = ε corresponds to equivalent flux noise, with sensitivity ∂ε/​∂Φ, whereas for λ = Δ, the noise is parameterized as critical-​current noise, with sensitivity ∂ε/​∂ic, where ic = δIc/​Ic. (Here “parameterized” means that the inferred fluctuation, δλ, is equivalent to—​and indistinguishable from—​the fluctuation of a certain physical quantity (in this case ic, although it could possibly also be a charge fluctuation); this is a practical way of characterizing the noise even in cases when it may not be possible to unequivocally determine the physical noise source.) The following few sections make the connections between the observed coherence times and the PSDs of quasi-​static, Gaussian noise.

18.3.3 Longitudinal relaxation (T1). Relation to noise PSD Energy relaxation involves the exchange of a quantum of energy with the environment, and is consequently due to noise at the qubit frequency, Sλ(±ωq), that couples to the transverse component of the Hamiltonian—​see Fig. 18.4. The matrix element for the 0 → 1 transition is

(

)

mλ = 1 ∂Hˆ ′ / ∂λ δλ 0 , (18.23)

which we can write as mλ = Dλ1, ⊥ δλ , using the transverse noise sensitivity Dλ1,⊥ from Sec. 18.3.1. Explicit expressions for the interaction Hamiltonians’ susceptibilities to δλ -​noise, ∂H ′ / ∂λ, relevant to superconducting qubits, are given in Sec. 18.4. Fermi’s golden rule yields the energy-​deexcitation rate, due to fluctuations of the parameter λ,



Γ (↓λ ) =

2π 2 mλ ρDoS + ω q ,  (18.24)

(

)

where ρDoS(ωq) is the environmental density of states (DoS) at angular frequency ωq. Combining this DoS with the fluctuation δλ into a noise PSD, Sλ(ωq) = 2πℏ δλ2 ρDoS(ωq), with dimensions of [λ]2/​s, we obtain the relation

Γ (↓λ ) =

1 2 Dλ1, ⊥ Sλ + ω q , (18.25) 4

(

)

which is a useful expression for experimental noise spectroscopy—​see Sec. 18.3.2. (Note that Sλ is the unsymmetrized PSD, and that the prefactor ¼ appears because of the way the Hamiltonian and the noise sensitivity are defined—​see Sec. 18.3.1.)

534

534  Superconducting Quantum Bits of Information The total inversion-​recovery time is given by summing the rate contributions of the different fluctuators:  1/​T1  =  Γ1  =  ∑λ Γ↓(λ). In general, depolarization occurs due to excitation as well as deexcitation, 1/​T1 = Γ1 = Γ↑ + Γ↓, where Γ (↑λ ) = 1 4 Dλ21, ⊥ Sλ ( − ω q ). However, superconducting qubits are operated at low temperature, so that we can approximate Γ↑ ≪ Γ↓. It is clear from Eq. 18.25 that there are two means available for improving the T1 time:

• improving the device designs to reduce the sensitivity to noise, Dλ • improving the device materials to reduce the number of fluctuators, Sλ(ω).

We will show later in this chapter how researchers have made enormous progress on both fronts, ever since the first demonstration of coherent oscillations in a superconducting qubit (Nakamura et al. 1999).

18.3.4 Transverse relaxation (T2) and dephasing (Tϕ). Relation to noise PSD Transverse relaxation is a combination of depolarization (Sec. 18.3.3) and pure dephasing processes. The latter is the loss of information about the relative phase in a superposition, with characteristic time Tϕ, due to slow fluctuations in the longitudinal (quantizing) field—​see Fig. 18.4. During free precession of a pure quantum state, 2 2 ρˆ (t ) = α 0〉〈0 + β 1〉〈1 + αβ* 0〉〈1 e i φ (t ) + α *β 1〉〈0 e − i φ (t ) ,

2

(18.26)

2

with α + β = 1, the phase evolves in time according to t



φ (t ) = ∫dt ′ω q = 〈 ω q 〉t + δφ (t ) , (18.27) 0

where 〈ω q 〉 is the average Larmor angular frequency. The fluctuation of the phase due to fluctuations in the parameter λ, t



δφ (t ) = Dλ1, z ′ ∫dt ′δλ (t ′ ) , (18.28) 0

now depends on the longitudinal noise sensitivity, Dλ1,z′. Under the assumption that the stochastic process δλ(t) is Gaussian-​distributed, the ensemble average can be evaluated as

〈e i δφ (t ) 〉 = e

1 − 〈 δφ2 (t ) 〉 2

= e − χ (t ) . (18.29)

  535

Decoherence. Characterization and mitigation of noise  535 The function χ is called the coherence integral, and the coherences of the density matrix fall off as

ραβ (t ) = αβ *e

− i 2 π 〈 ω q 〉t − χ (t )

. (18.30)

Because this decay is generally not exponential, we identify the dephasing time t = Tϕ as the time for which χ(t) = 1. The functional expression for the coherence integral is

χ(λ ) (t ) =

∞

t2 2 Dλ1, z ′ ∫ dωSλ (ω ) g N (ω , t ) , (18.31) 2 −∞

where we explicitly write the dependence on the parameter λ, because the dephasing fluctuations may have several sources. Here g N is a dimensionless weighting function that depends on experimental realization; it acts as a frequency-​domain filter of the noise PSD, Sλ(ω). For example, during free-​induction decay (as in a Ramsey interference fringe), the control field is turned off (constant) for a time t; consequently, the frequency-​domain filter is a function g0 = sinc 2( ωt 2), that is, its main lobe is centered at ω = 0, acting as a low-​pass filter for the noise reaching the qubit. In superconducting qubits, the noise PSD is generally of the 1/​f-​type, that is, low-​frequency noise dominates: this is why the observed free-​induction decay time, denoted T2∗, is much smaller than the T1 energy-​relaxation time (the relation is then written 1/​T2∗ = 1/​2T1 + 1/​Tϕ∗). Sec. 18.3.5 further treats dephasing and mitigation of low-​frequency noise. The Bloch–​Redfield approach, allowing for a simple addition of rates, is strictly only valid if the noise is weakly coupled to the qubit and is generated by a large number of fluctuators, and therefore has a short correlation time compared to the qubit dynamics (e.g. white noise in the relevant frequency range). Superconducting qubits usually exhibit energy relaxation that is exponential in time (or sometimes a sum of exponentials with faster and slower time scales). Pure dephasing, on the other hand, is oftentimes nonexponential due to the dominant 1/​f-​type noise, which is not accounted for in the Bloch–​ Redfield description; under reasonable assumptions, relaxation and dephasing processes nevertheless factorize (Makhlin and Shnirman 2003), and the decay law then depends on the particular noise realization. Then Tϕ is taken as the 1/​e decay time.

18.3.5 Mitigation of low-​frequency noise. Dynamical decoupling (DD) Dephasing of the quantum state, due to slow fluctuations of the longitudinal effective field, can be mitigated by dynamical decoupling (DD) of the qubit from the noise—​see Fig. 18.6. The application of open-​loop control sequences of qubit

536

536  Superconducting Quantum Bits of Information Dynamical decoupling of the noise by the CPMG sequence (a) Pulse: x’π/2 |0 z’

z’

y’π

x’π/2

z’

Free evolution + fanout due to noise

y’

Refocusing

x’

|1 (b)

x’π/2

Sequence repeated N times

y’π

y’π

y’π

y’π

y’π

...

x’π/2

1

(c) t

N π-pulses

(d)

1

N=0 1

0 0

0

(e) 1

N=1

2

6

2

10

4 Frequency (MHz)

0.25

0.5

6

0

8

0

2

4 Frequency (MHz)

6

8

Fig.  18.6  Dynamical decoupling and noise spectroscopy. (a)  Spin-​echo refocusing consists in applying a pulse along σˆ x′ or σˆ y′ , of duration τπ  =  π/​ωR (i.e. a π-​pulse), in the middle of the free evolution interval of length τ. The π-​pulse rotates the Bloch vector to the opposite side of the Bloch sphere. The accrued phase for each spin, k, is then –​Δϕk(1) during the first half and +Δϕk(2) during the second half. As long as the field remains unchanged over the time τ, then –​Δϕk(1) +Δϕk(2) = 0 for all spins k, i.e. a refocusing of the average of the Bloch vectors occurs. (b) CPMG refocusing pulse sequence. (c) An illustration using the filter function to sample the noise PSD for a particular value of N, and with τ corresponding to an angular frequency ω′. The PSD, S(ω), is assumed constant within the filter’s bandwidth, Δω. (d) CPMG filter functions for a total pulse-​sequence length Nτ. (e) Single π-​pulse (N = 1) filter function for various pulse lengths, τ. (Figure adapted from Bylander et al. 2011.) rotations can reverse dephasing, and thereby refocus the Bloch vector at the end of the sequence. Perhaps the simplest DD method is spin-​echo refocusing with a single π-​pulse (Hahn 1950)—​see Fig. 18.6a. The frequency-​domain filter function for the noise depends on the number of pulses applied, N, and their distribution,

g N (ω, τ ) =

1

(ωτ )2

1 + ( −1)

1+ N

e

i ωτ

N

2

+ 2∑ ( −1) e j =1

j

i ωδ j τ

 τ  cos  ω π  , (18.32)  2

  537

Decoherence. Characterization and mitigation of noise  537 where δj ∈ [0, τ]×1/​τ indicates the normalized position of the jth π-​pulse during the free-​evolution time τ, and τπ is the duration of each π-​pulse. For one π-​pulse, this function is g1 = sin 2 (ωt 4) sinc 2 (ωt 4), which peaks at a frequency ω > 0, that is, it acts as a band-​pass filter for the noise. This is advantageous for noise that decreases with increasing frequency, such as the dominant 1/​f noise in superconducting qubits, and enhances the coherence time compared to that of the free-​induction decay, T2(echo, N=1) > T2∗. (We saw in Sec. 18.3.4 that g0 peaks at ω = 0.) The inclusion of a greater number of π-​pulses, with a fixed time separation τ’ = τ/​N, sharpens the filter g N so that it asymptotically peaks at ω’ = π/​τ’. For a fixed number of pulses, N, a decreased time separation moves the filter’s peak to a higher frequency, thereby providing more efficient refocusing of the low-​frequency noise. A  multitude of DD control sequences have been implemented, notably CP and CPMG (Carr and Purcell 1954, Meiboom and Gill 1958), but also more recently—​see Bylander et al. (2011) and references therein. In general, the optimal refocusing protocol depends on the particular noise realization. In some qubit realizations, the refocused T2 has been demonstrated to be limited only by energy relaxation, T2(echo)  =  2T1. (Note that this applies to single qubits; quantum error correction in multi-​qubit systems is outside of the scope of this chapter.)

18.3.6 Relaxation, decoherence, and DD in the rotating frame (T1ρ and T2ρ) The T1 and T2 times serve to characterize decoherence during free evolution. The Hamiltonian Hˆ ′ + δHˆ ′ (Eqs. 18.6 and 18.13) describes this situation, with the qubit quantized along z’ (Sec. 18.2.3). Under driven evolution,  + δ H  the Hamiltonian is more conveniently written in the rotating frame, H (Eqs. 18.8 and 18.14), with the new quantization axis X (Sec. 18.2.4). Relaxation in the rotating frame is described by generalized Bloch equations (Geva et  al. 1995, Smirnov 2003, Yan et  al. 2013), with corresponding coherence times denoted T1ρ and T2ρ—​see Fig. 18.7. For resonant driving and within the RWA, these equations take on the form

rX = − Γ X rX + vX , rY = − ΓY rY + vY + ω R rZ , (18.33) rZ = − Γ Z rZ + vZ − ω R rY .

where the variables vX,Y,Z define the steady-​state values of rX,Y,Z (Smirnov 2003). If the driving is strong enough, ω R  Γ X ,Y , Z , then vY = vZ = 0; in addition, whenever ω R  kBQ, then also vX = 0. Using the simplified notation Sx ′ = Dλ1, ⊥ Sλ and Sz ′ = Dλ1, z ′ Sλ, the relaxation rates in the rotating frame become

538

538  Superconducting Quantum Bits of Information Spin locking

Rabi

Z

Precession

Z

Y

^ ~ H

r

Y

^ ~ H

X

X

Fig. 18.7  Relaxation and dephasing in the rotating frame (T1ρ and T2ρ), when the qubit is quantized along X by a strong driving field, are analogous to relaxation and dephasing in the qubit frame (T1 and T2), with quantization along x’ by a static field. T1ρ can be measured by spin locking. The experiment consists in a preparation of the spin in state |+X〉, the excited state in the effective eigenbasis, by means of a π/​2-​pulse. Then the spin is locked to the X-​axis by a field applied along σˆ X , i.e. longitudinal. For spin locking at a typical Rabi frequency for superconducting qubits, ωR/​2π ≈ 10–​100 MHz; both Γ↓ρ and Γ↑ρ rates contribute to the polarization, because ℏωR ≪ kBΘ. On the other hand, T2ρ is ideally identical to TR, but this holds only in the absence of inhomogeneous fields. (Figure adapted fromYan et al. 2013.)

1 1 Sx ′ ω q + ω R + Sx ′ ω q − ω R  + Sz ′ (ω R ) ,  2 8 1 1 (18.34) ΓY = Sx ′ ω q + Sz ′ (ω R ) , 4 2 1 1 Γ Z = Sx ′ ω q + ω R + Sx ′ ω q − ω R  + Sx ′ ω q . 8 4

ΓX =

(

)

(

)

)

(

)

( ) (

( )

Longitudinal relaxation in the rotating frame, T1ρ, is due to a combination of transverse Y-​and Z-​noise. Note that z’  =  Z, so that z’-​noise translates directly into Z-​noise, whereas x’-​noise translates into X-​and Y-​noise due to the mixing in the frame transformation, that is, SY (ω R ) is derived from Sx ′ ω q ± ω R . We can further make the approximation ω q + ω R ≈ ω q, yielding

(



1 1 = Γ X = Γ1 + Γ ω with T1ρ 2

Γω =

)

1 Sz ′ (ω R ). (18.35) 2

Transverse decay in the rotating frame, T2ρ, is a combination of relaxation and dephasing. It is ideally identical to the decay of the Rabi oscillations,

1 1 1 3 1 = Γ R = (ΓY + Γ Z ) = Γ X + Γ φρ = Γ1 + Γ ω . (18.36) T2ρ 2 2 4 2

  539

Superconducting qubits 539

( )

The pure-​dephasing rate, Γ φρ = (1 4) Sx ′ ω q = (1 2) Γ1, here involves noise at the qubit frequency that is mixed down to zero frequency and acts as longitudinal quasi-​static noise, that is, SX (0) is derived from Sx ′ ω q . These expressions also show that the decay time constant of Rabi oscillations is ultimately limited by TR = (4 3) T1. Inhomogeneities in the driving field, ωR and ΔωR, are not included in this simplified analysis, and would lead to non-​exponential decay of the Rabi oscillations—​see the work by Yan et al. (2013). The above relations show that spin locking in the rotating frame is analogous to inversion recovery in the qubit frame, and Rabi precession around an effective field, ωR,eff, directed along X, is the rotating-​frame analogy of the Ramsey free induction around the quantizing field, ωq, directed along z’, in the qubit frame, see Sec. 18.3.3–​4. Furthermore, the driven-​evolution analogy to spin echo is the rotary echo, in which the spin’s direction of precession is flipped between –​X and +X by a change of phase of the driving field. Rotary echoes mitigate low-​frequency noise in the driving field, just like spin echoes mitigate low-​frequency noise in the Larmor frequency, and the decay time can exceed the Rabi decay time, T2ρ(echo) > TR, up toward the limiting (4/​3)T1 (Gustavsson et al. 2012).

( )

18.4 Superconducting qubits The Josephson junction—​a sandwich structure of superconducting metals separated by an insulating tunnel barrier—​is the basic circuit element for superconducting quantum hardware. Its dissipationless inductance and geometric capacitance form an LC oscillator, which is rendered anharmonic by the Josephson nonlinearity. Aluminum deposited on silicon oxide or on sapphire, with amorphous aluminum oxide as a tunneling barrier dielectric, have for long been the materials of choice for making Josephson junctions for quantum circuits. In recent years, as researchers in the field have increasingly worked to minimize decoherence due to defects in the materials, other materials have come to the fore—​see for example the reviews by Siddiqi (2011), Oliver and Welander (2013), and Martinis and Megrant (2014). Together with superconducting capacitors, inductors, and interconnects, Josephson junctions enable flexible device designs. The circuit variables that describe the workings of these devices—​currents and voltages, or electric charges and magnetic fluxes—​are macroscopic quantities, involving many microscopic degrees of freedom. Remarkably, these macroscopic variables exhibit atomic-​like, discrete excitation spectra—​hence the designation “artificial atom”—​and the lowest-​lying levels constitute the basis states of a physical qubit.

540

540  Superconducting Quantum Bits of Information Since the quantum states involve many electrons, which can interact with a large number of atoms and also defects in the host materials, the qubits are prone to decoherence. Very active research is devoted to the problem of reducing the noise in the device materials and environment, and that of isolating the qubits from the residual unavoidable noise, while at the same time enabling coherent control by external fields. In this section we will introduce the originally realized superconducting quantum circuits. We will see how the Hamiltonians of the different superconducting qubits differ due to the shape of their nonlinear potentials, how they map onto the qubit Hamiltonian of Sec. 18.2, what types of noise and decoherence limit their performance, and how the quantum information can be read out. In following sections we introduce the strong coupling of qubits to microwave resonators (Sec. 18.5), and then describe modern device developments with improved performance (Sec. 18.6).

18.4.1 The Josephson junction Josephson junctions are treated at length in the chapter by Weides in this volume, and in numerous textbooks. The quantization of electronic circuits, via the Lagrangian and Hamiltonian formalisms, was described in detail by Devoret (1997). Here we introduce the necessary notation for the present chapter. The first Josephson relation describes how the junction supports a dissipationless current—​a supercurrent—​up to a critical current, I0, due to a coherent coupling of the superconducting phase difference, φ, between the two superconductors, I = I 0 sin ϕ. (18.37)



The second Josephson relation describes the proportionality between the voltage across the junction and the time-​rate of change of the phase, V = ϕ0



dϕ . (18.38) dt

The flux quantum is defined as Φ 0 = h 2e, and ϕ0 = Φ 0 2π. The Josephson relations can be combined into V = L J dI dt , where, in the small-​current limit, we define the Josephson inductance as

LJ =

L J0 ϕ0 = , (18.39) 2 I 0 cos ϕ 1 − (I / I0 )

with L J0 = ϕ0 I 0 . The inductive Josephson energy stored in the junction yields, for a finite voltage, E J = ∫ dt IV = ϕ0 I 0 .

  541

Superconducting qubits 541 A direct-​current superconducting quantum interference device (dc-​SQUID, Clarke and Braginski 2004) consists of two Josephson junctions placed in parallel, forming a loop. The magnetic flux, Φext, penetrating this loop modulates the critical current, and consequently the inductance. For a small loop inductance, β L = L loop / L J 0 < 1, the SQUID can be seen as a flux-​tunable Josephson junction, and assuming identical junction, the current–​phase relation (18.37) is modified into I = I 0 cos π



Φ ext sin ϕ. (18.40) Φ0

18.4.2 Prehistory: quantum behavior of the superconducting phase and charge 18.4.2.1 Macroscopic quantum tunneling of the phase The first indications of quantum coherence of a state variable in an engineered device were obtained in the early 1980s, with the observation of macroscopic quantum tunneling (MQT) of the superconducting phase difference across a Josephson junction (Voss and Webb 1981, Devoret et al. 1985)—​see Fig. 18.8. The MQT experiment was a remarkable demonstration of quantum-​mechanical behavior of a macroscopic quantity, φ. This variable is the phase of the Ginzburg–​ Landau order parameter, or wavefunction, Ψ  =  ns1/​2 eiφ, that describes the

Ib

V

U

I > I0 Γ

JJ 2π

Fig. 18.8  Macroscopic quantum tunneling (MQT) of the fictitious phase particle of a Josephson junction (JJ) in the underdamped, hysteretic regime (i.e. with the Stewart–​ McCumber parameter βc > 1, where β1c/ 2 = Q 2 = ( I 0 ϕ0 ) R 2C , R is the damping, and C is the junction capacitance). Devoret et al. (1985), in Berkeley, measured the current at which the current-​biased junction switched out of its zero-​voltage state, and inferred, from the switching current, the escape rate of the phase from that state. As they decreased the temperature, they observed a crossover from a regime of thermal activation over the junction’s washboard potential barrier to one of quantum tunneling through the barrier. Absent quantum tunneling, one would have expected the escape rate to continue to zero as the temperature was decreased, but instead there remained a finite, quantum escape rate even at the lowest temperatures. (See Clarke et al. 1988, and review articles listed in Sec. 18.1.1.)

542

542  Superconducting Quantum Bits of Information superconducting state, which contains a macroscopic number of Cooper pairs with density ns. In the microscopic theory, the phase factor eiφ is identical for each of the Cooper pairs comprising the Bardeen–​Cooper–​Schrieffer (BCS) ground state. The demonstrations of MQT of the phase and of “quasi-​bound states” with a discrete anharmonic spectrum in the Josephson potential (Devoret et al. 1984, Martinis et al. 1985, 1987) portended the invention of the phase qubit (Martinis et al. 2002). One should note here, that the Josephson effects are said to be “classical” (or “semiclassical”), which is meant in the sense that the phase difference φ is a classical variable, and the Josephson relations constitute its classical equations of motion; quantum fluctuations of the phase are not taken into account. This use of the word classical does not take away from the quantum origin of the Josephson coupling, which is due to the tunneling of Cooper pairs across a barrier; the very existence of the Cooper pairs and the BCS ground state certainly have a quantum-​mechanical origin. Moreover, the superconducting quantum interference device (SQUID) is, despite including the word “quantum” in its name, understood in terms of wave interference resembling that in classical optics. 18.4.2.2 Single-​charge tunneling Around the same time, high-​resolution lithographic techniques had enabled the reproducible fabrication of circuits in the “Coulomb-​blockade” regime (Grabert and Devoret 1992). When a metallic grain (called an “island”) is separated from a ground reservoir by a tunnel junction of capacitance CJ and a shunt capacitance C, its total capacitance is C∑ = CJ + C. This capacitance can be made sufficiently small that the electrostatic Coulomb charging energy required to add a single electron onto the island, EC = e2/​2C∑, is much larger than the thermal energy, kBΘ, when cooling the device in a dilution refrigerator. An externally applied gate voltage polarizes the island, and can control the tunneling of single charges through a barrier separating the island from a ground reservoir. The demonstration of 2e-​ quantization of charge on a superconducting island—​a “single-​Cooper-​pair box,” abbreviated CPB or SCB (Bouchiat et  al. 1998)—​heralded the first observation of quantum coherence in a charge qubit (Nakamura et al. 1999), a result that instigated in this field a most vigorous activity which remains in full force today. The charging energy tends to localize the charge on the islands of a circuit, at the expense of a delocalization of the phase, whereas the Josephson energy makes the superconducting phase difference well defined at the expense of the charge. This is understood because the superconducting phase difference across the junction, ϕˆ , and the charge on the junction capacitor, expressed in units of Cooper pairs, nˆ = Qˆ / 2e, are canonically conjugate operators,

[ϕˆ , nˆ ] = i (18.41)

or equivalently, for the magnetic flux and the charge,

  543

Superconducting qubits 543 ˆ ˆ Φ  ,Q  = i . (18.42)



This duality leads to a phase–​number (or flux–​charge) uncertainty relation: whenever the phase is precisely determined, the conjugate number has large quantum fluctuations and vice versa.

18.4.3 First-​generation qubits—​charge, flux, and phase Different modalities, or “flavors,” of superconducting qubits can be realized, depending on the choice of device parameters—​see Fig. 18.9. The first-​generation superconducting quantum circuits came in essentially three modalities—​charge, flux, and phase qubits—​characterized by the ratio 4EC/​EJ which measures the relative importance of the Coulomb-​charging and Josephson-​ coupling energies of their constituent Josephson junctions—​see Table 18.1. The nonlinearity of the Josephson inductance enables different potential shapes—​ cosine for the charge qubit, quartic for the flux qubit, and cubic for the phase qubit. The bound states in these potentials have strongly anharmonic spectra for the charge and flux qubits, and a slightly anharmonic spectrum for the phase qubit. The simplest single-​junction qubits—​the charge and phase qubits—​are special limiting cases of the circuit, which allow us to understand the behavior of charge and phase across the junction in terms of the impedance of the shunting circuitry. At the qubit transition frequency, the purely capacitive shunt of the CPB yields a large, reactive impedance, compared to the quantum resistance, RQ = h/​(2e)2 ~ 6.5 kΩ, that is, |Z(ω)| = (ωCg)–​1 ≫ RQ. Consequently, the charge is well defined, whereas the quantum fluctuations of the phase are

Z (ω) JJ

Ib

Vg

L

JJ

Cg C

Vg

Fig. 18.9 The general equivalent circuit for all superconducting qubits is a Josephson junction, JJ, shunted by a capacitor, C, and an inductor, L. The Josephson inductive energy, EJ = φ0/​I0, and the charging energy, 4EC = 4e2/​2CΣ (with a factor 4 because of the 2e charge of a Cooper pair), were introduced in Sec. 18.4.1–​2. Similarly, the energy scale of the linear inductor is EJ = φ02/​2L. These three terms each contribute one term to the Hamiltonian (Eq. 18.68) of the shunted Josephson junction, Ĥφ = 4EC (n̂ –​ ng)2 –​ EJ cosφ̂ + EL (φ̂ –​ Φext/​φ0)2 (after Koch et al. 2009, adapted with permission).

544

544  Superconducting Quantum Bits of Information Table 18.1  First-​generation superconducting qubits, approximate parameters. Name

Eigenstate

EJ/​4EC

EL/​EJ

Charge qubit

Island charge number

4EC, opposite circulating persistent currents around a loop constitute the qubit states, and the environmental impedance is in between those of the charge and phase qubits. Although the first-​generation charge and phase (and to some extent flux) qubits are not used much anymore in attempts to develop quantum computing hardware due to their sensitivities to noise, they provide a conceptual background and a history of the development of the field.

18.4.4 Charge qubit (Cooper-​pair box, CPB) 18.4.4.1 Device principle The Cooper-​ pair box charge qubit comprises a nanoscale superconducting island, coupled to a ground reservoir via a Josephson tunnel junction of capacitance CJ, and to a gate-​voltage source via a capacitance Cg (normally Cg ≪ CJ)—​see Fig. 18.10. The qubit is represented by adjacent number states, E/EC

n Reflection readout

Cg C

n=1

Vg

4 0 –1

n = –1

n=0

n=0

0

1

ng

Fig. 18.10  Cooper-​pair box (CPB) charge qubit. The boxed cross symbolizes a small Josephson junction, where the charging and Josephson energies are on the same order of magnitude. The energy bands are charging parabolas, with avoided crossings ∆ ≈ E J between the ground and first-​excited states. Note that the spectrum is strongly anharmonic. The charge on the island, Q0 = –​2en, can be read out by microwave reflection techniques. (Bouchiat et al. 1998; see also the review articles listed in Sec. 18.1.1.)

  545

Superconducting qubits 545 n〉 and n + 1〉, corresponding to the number of Cooper pairs that have tunneled onto the island, referenced to the initially charge-​neutral island (n  =  0). Such charge states are well defined in a regime known as Coulomb blockade of single-​ charge tunneling, which has two requirements. First, the total capacitance of the island must be small, typically C∑ = CJ + Cg ~ 1 fF for a 0.01 μm2 junction, yielding a charging energy EC ~ 100 μeV. This is roughly equivalent to the photon energy at 20 GHz or the thermal energy at 1 K. By placing the device in a 50-​mK dilution refrigerator, the thermal fluctuations of the island charge can therefore be suppressed. Second, the junction’s tunneling resistance, RT, must 2 exceed the quantum resistance, RQ = h (2e ) ≈ 6.5 kΩ (which is easy to achieve for a small junction area); moreover, the capacitive connection to the circuit environment ensures high impedance at gigahertz frequencies, Z = 1/​iωCg. This high-​impedance environment suppresses the quantum fluctuations of the charge on the island: the total charge on the island is therefore discrete, Q0 = –​2en. The gate voltage, Vg, controls the amount of charge on the sum capacitance of the island; this charge, Q = Q0 + Qg, on the other hand, is a continuous polarization charge, due to the displacement of the electronic gas against the fixed background of the positive ions, usually expressed in units of the equivalent number of Cooper pairs on the gate capacitor, ng = CgVg/​2e. 18.4.4.2 Hamiltonian The static Hamiltonian of the circuit has two terms. First, the electrostatic energy acts as a kinetic energy for the charge. For n excess Cooper pairs on the island, it is given by the parabola 4EC (n –​ ng)2—​see Fig 18.10. Here, the factor 4 appears due to the 2e-​charge of the Cooper pair. The second term is the inductive energy stored in the Josephson junction; it acts as a potential, –​EJ cos φ. In the charge-​ qubit regime, EJ < EC, typically with EJ/​h ~ 5 GHz; it can be made tunable by an external magnetic flux by splitting the Josephson junction. We can now write down the Hamiltonian,

(

Hˆ CPB = 4 EC nˆ − n g



)

2

− E J cos ϕˆ . (18.43)

Since the phase and the number of charges are conjugate variables, the expres n 〉 =|n + n 〉 represents a translation of the number state—​ sion exp in0 ϕ this 0 explains how the Josephson term coherently couples the charge–​number states—​  = (1 2) ∑ n + 1〉〈n + n 〉〈n + 1 . Working in the charge basis, and we obtain cos ϕ

( )

n

(

)

{|n〉}, and restricting the treatment to n = {0, 1}, we can now write the Hamiltonian in the form

(

(

Hˆ CPB = 4 EC ng2 0〉〈0 + 1 − ng

)

2

1〉〈1

) − E2 ( 1〉〈0 + 0〉〈1 ), (18.44) 2

J

546

546  Superconducting Quantum Bits of Information which elucidates the role of the Josephson coupling in the mixing of the charge states. Alternatively, in the phase basis, it takes the form 2

 i∂  Hˆ CPB = 4 EC  − − ng  − E J cos ϕˆ , (18.45)  ∂ϕˆ 



which accentuates the periodic potential. The energy eigenvalues of the CPB Hamiltonian form periodic, parabolic bands, E 0,1 = E 00 + E ± , where

(

E 00 = EC + EC 1 − 2ng

)

2

(

and E ± = ± 4 EC2 1 − 2ng

)

2

+ E J2 4 , that are split by EJ at

charge degeneracy, ng = 1 2—​cf. Fig. 18.1, 18.10. At this bias point, the phase difference across the Josephson junction, φ, is well defined (but the charge number is indeterminate) and the eigenstates are the superpositions ( 0 ± 1 ) / 2 , whereas away from ng = 1 2, the island charge, n, is a good quantum number (but the phase is indeterminate) and the eigenstates are the diabatic states, 0 and  1 . We can choose a basis and map this two-​level Hamiltonian onto that of a spin-​½ particle in a magnetic field (Eq. 18.2), familiar from Sec. 18.2,

 Hˆ = − ( ∆σˆ z + εσˆ x ). (18.46) 2

Working near the charge-​degenerate point, it is convenient to choose a new reference for the energy and to make the associations ε = 4 EC 1 − 2ng and ∆ = E J . The Hamiltonian is then diagonalized in its energy eigenbasis, with quantization axis σˆ z, when biased at ng = 1 2 (ε = 0). We further denote the mixing angle by θ = arctan ( ∆ / ε ), and write the eigenstates as Ψ − 〉 = cos (θ 2) 0〉 + sin (θ 2) 1〉 and  Ψ+ 〉 = − sin (θ 2) 0〉 + cos (θ 2) 1〉. Transitions between the qubit states can be induced via dipole coupling, by applying a microwave voltage to a capacitively coupled gate, with a frequency near resonance, ω ≈ ω 0 = ε 2 + ∆ 2 . This driving term can be written as ε (t ) = ε 0 + Ω (t ) cos (ωt + ϑ ), where Ω (t ) is a pulse envelope; it appears in the Hamiltonian as a σˆ x term for the phase value ϑ = 0, and as a σˆ y term for ϑ = π 2 (cf. Eq. 18.6). Recalling the transition rates and noise sensitivities introduced in Sec. 18.3.1 and 18.3.3, the matrix element (Eq. 18.23) for this transversely coupled charge is mδn = 4 ECδn .

(

)

18.4.4.3 Decoherence The CPB is affected by two of the most severe defects in superconducting qubit materials: charged two-​level fluctuators and nonequilibrium quasiparticles. Two-​level fluctuators produce the dominant 1/​f noise affecting the CPB, with a typical value SQ ( f = 1 Hz) ≈ 1 me / Hz1/ 2 ; quasiparticles prevail below the superconducting gap, although ideally suppressed exponentially by temperature—​ they can change the gate charge by a charge e, providing a strong decoherence

  547

Superconducting qubits 547 mechanism. Decoherence due to charge fluctuations, δn g (t ), is therefore strong, even at optimal bias, ng = 1 2. We recall from Sec. 18.3.4 that at this point, sometimes called the “sweet spot,” the longitudinal term in the qubit’s eigenbasis is, to first order, unaffected by the noise. 18.4.4.4 Readout The charge qubit is usually operated near ε = 0 (i.e. ng = ½), where its sensitivity to noise, δε(t), is minimum. At that bias point, however, its eigenstates are superpositions of charge states, with expectation value 〈n 〉 = 1 2. A determination of the qubit state therefore requires the charge to be brought, by means of an adiabatic pulse, away from ε = 0, before measurement. The CPB can be capacitively coupled to an rf-​SET (radio-​frequency single-​ electron transistor, Schoelkopf et al. 1998); the charge on the CPB island then modulates the impedance of the SET, so that the reflection coefficient of an applied voltage wave yields distinguishable reflected output signals, depending on the qubit state. 18.4.4.5 History and state of the art The CPB qubit grew out of work on single-​charge tunneling; see for example Grabert and Devoret (1992). In Saclay, Bouchiat et al. (1998) demonstrated 2e-​quantization of the CPB; at NEC, Nakamura et al. (1997) did spectroscopy of the energy levels, and then Nakamura et  al. (1999, 2001)  used nonadiabatic driving to demonstrate the first coherent oscillations in any superconducting qubit, with nanosecond T2 coherence time, and resonant driving to demonstrate Rabi oscillations. Other early work was done at Yale, by Lehnert et al. (2003), and at Chalmers, by Duty et al. (2004). At NEC, Astafiev et al. (2004) studied energy relaxation and found that quantum charge noise, with an ohmic spectrum, was the dominant contributor to spontaneous emission. Wallraff et  al. (2004) used a CPB to demonstrate strong-​coupling circuit-​ QED (see Sec. 18.5). Kim et  al. (2011) used a CPB and measured a long (0.2 ms) T1 energy-​relaxation time in a device that was effectively decoupled from its high-​frequency environment; however, charge noise made it hard to obtain a stable enough biasing to measure a significant T2 coherence time. Due to its sensitivity to noise, the CPB, in its original design, has been largely abandoned as a qubit, but lives on in its higher reincarnations discussed in Sec. 18.6.

18.4.5 Quantronium The “quantronium” circuit—​see Fig. 18.11—​was developed in Saclay (Vion et al. 2002, Ithier et al. 2005), based on the understanding that decoherence is minimal at the “sweet spot”—​where Dλ1, z ′ = 0 for the noisy parameter λ (recall Eq. 18.18 in Sec. 18.3.1). This circuit is a CPB shunted by a large Josephson junction, so

Switching readout

Vg Ib

Fig. 18.11 The quantronium qubit has a doubly noise-​insensitive point, for both charge and phase bias. (Figure adapted from Vion et al. 2002.)

548

548  Superconducting Quantum Bits of Information that the qubit can be read out while staying biased at the charge-​degenerate point. The flux through this loop controls an additional flux degree of freedom, and consequently also an additional noise channel, but with a sweet spot for the phase (controlled by the external flux) at ϕ = π. Readout is done by shifting the flux away from the degeneracy point, which maps the qubit states onto different values of the circulating current in the loop formed by the CPB and the Josephson junction. This change in current, in turn, modifies the switching current of the larger Josephson junction, and can be read out by applying a current pulse and then measuring the presence or absence of a voltage indicating that the junction has switched. Alternatively, the state-​dependent inductance of the CPB can be determined by measuring the phase-​shift of a reflected signal. The coherence time of the quantronium, when biased at the charge-​and-​flux degeneracy point, was improved to T2 ≈ 500 ns, so by more than ten times compared to that of the CPB.

18.4.6 Flux qubit or persistent-​current qubit 18.4.6.1 Device principle The flux qubit is reminiscent of the rf-​SQUID (rf, radio-​frequency), that is, a Josephson junction placed in an inductive loop—​see Fig. 18.12. States of persistent current (supercurrent) circulate around the loop, controlled by an externally applied magnetic flux, Φ ext , flowing through the loop. A typical persistent current is Ip ≈ 0.2 μA. The ground and excited states are represented by opposite persistent currents, which give rise to magnetic fluxes pointing in opposite directions—​note, however, that the difference between the fluxes is much less than one flux quantum, due to the phase drop across the loop. In fact, the loop usually consists of three or four Josephson junctions, which serve to provide inductance in a compact way.

ISQ

IΦ

Φext

Φext = Φ0/2

U Φ = Φ /2 ext 0

E (arb. un.)

U

Φext

0.49

0.50

0.51 Φ0

Fig.  18.12 A  three-​junction persistent-​current flux qubit, with switching readout by a dc-​SQUID. The right and left potential wells correspond to the adiabatic states represented by clockwise and counter-​ clockwise circulating currents, with slopes dE/​dΦ  =  Ip. Note that the spectrum is strongly anharmonic (Mooij et  al. 1999, Orlando et al. 1999; see also the review articles listed in Sec. 18.1.1.)

  549

Superconducting qubits 549 Here we briefly describe the 3-​junction flux qubit; for details, see Orlando et al. (1999). Two of the junctions typically have Josephson and charging energies EJ/​h ≈ 300 GHz and EC/​h ≈ 4 GHz, respectively, whereas the third junction is smaller by a factor α ≈ 0.75 (≈ 0.5 for the 4-​junction qubit), and consequently has smaller Josephson energy and capacitance, αEJ and αC. A typical energy ratio is EJ/​EC ≈ 50–​100, that is, the Josephson energy dominates, but the charging energy also plays an important role to determine the coupling strength between the persistent-​current states. 18.4.6.2 Hamiltonian The potential energy is the sum of the Josephson energies, U = ∑ E Ji (1 − cos ϕi ). i

Φ Using the fluxoid quantization condition, ϕ1 − ϕ2 + ϕ3 = − ext , we can eliminate ϕ0 the variable ϕ3. Two new variables, ϕ ± = (ϕ1 ± ϕ2 ) / 2, now define the main axes of the two-​dimensional potential which is shaped akin to an egg carton:

 Φ  U = E J 2 + α − 2 cos ϕ + cos ϕ − − α cos  ext + 2ϕ −   . (18.47)  ϕ0   

A fictitious particle, representing the phase, inhabits this potential landscape. When the external flux is close to half a flux quantum, Φ ext ≈ Φ 0 2, the potential within a unit cell assumes a tunnel-​coupled double-​well profile along the ϕ − axis. The particle stays confined to the double well since the next-​nearest-​neighbor barriers are high. The ground states in the right and left wells correspond to the clockwise and counterclockwise circulating persistent-​current diabatic states, and a change in the external flux makes the double well asymmetric. The kinetic energy corresponds to the electrostatic energy stored in the capacitors,

T=

1 T −1 Q ⋅ C ⋅ Q , (18.48) 2

where Q is the vector of island charges and C is the capacitance matrix. The Hamiltonian becomes where

  Φ Hˆ = Tˆ − E J 2 cos ϕˆ + cos ϕˆ − + α cos  ext + 2ϕˆ −   , (18.49) ϕ     0

the kinetic term, in the phase basis, is 2 2  1    ∂  1 ∂  Tˆ = 2EC   −i ∂ϕˆ  + 1 + 2α + γ  −i ∂ϕˆ   , assuming identical gate γ 1 +   + − cap­acitances for all islands, Cg = γC (used to model the effect of offset charges).

550

550  Superconducting Quantum Bits of Information The left-​and right-​well states are hybridized by tunnel coupling, which can  EJ  1.3 be calculated in the WKB approximation, ∆ ≈ E J EC exp  −0.64  ; the  EC   coupling is exponentially sensitive to the size of the junction capacitance, which contributes to a certain experimental variability between devices. Restricting the treatment to the coupled diabatic states, we obtain the two-​level Hamiltonian

 Φ 1 1 Hˆ = − 2I pΦ 0  ext −  σˆ x + ∆σˆ z  . (18.50) 2   Φ0 2  

Just like for the charge qubit, we can again map this expression onto the qubit Hamiltonian (Eq. 18.2), now using the substitution

Φ 1 ε = 2I pΦ 0  ext −  . (18.51)  Φ0 2 

Again recalling the transition rates in Sec. 18.3.1 and 18.3.3, the matrix elements (Eq. 18.23) for transversely coupled flux noise, δΦ, and reduced critical current  Φ  Φ δI noise, δi = 0 , are mδΦ = I 0 sin  ext + 2ϕ −  δΦ and mδi = − E J cos  ext + 2ϕ −  δi , I0  ϕ0   ϕ0  respectively. 18.4.6.3 Decoherence The flux qubit is very sensitive to magnetic-​flux noise, δΦ ext , which was also determined to have a 1 / f α -​type spectrum, over the very large frequency range from millihertz to gigahertz (Yan et al. 2012, Sank et al. 2012), with a geometry-​ and temperature-​dependent α-​value between 0.7 and 1.1 (Anton et  al. 2012b, 2013), and a “universal” strength SΦ ( f = 1 Hz) ≈ 1 µΦ 0 / Hz1/ 2. This dominating flux noise seems to originate in surface spins on the superconducting metal, but there is no consensus to date on its precise microscopic mechanism (McDermott 2009, Oliver and Welander 2013). The flux-​noise sensitivity necessitates operation at flux degeneracy, Φ ext = Φ 0 2, which has enabled a free-​induction decay (Ramsey) time T2* = 2.5 µs. This falls short of the T1 relaxation time, due to noise in the Δ parameter of the Hamiltonian (Yan et  al. 2012), which can be parameterized as critical-​current noise, investigated by Anton et al. (2012a), charge noise, or second-​order flux noise. The T2-​time can be brought up to the T1 limit by dynamical decoupling techniques, with inferred pure dephasing exceeding 100 μs (Bylander et al. 2011). Although long T1 energy-​relaxation times, up to 12 μs, have been demonstrated even in first-​generation designs (Bylander et al. 2011), flux qubits tend to have substantial variation, likely due to the presence of random two-​level

  551

Superconducting qubits 551 fluctuators in the junction dielectric. This has been remedied in newer designs—​ see Sec. 18.6. 18.4.6.4 Readout Just like the charge qubit, the flux qubit is usually operated near ε = 0, where its sensitivity to noise, δε(t), is minimum, and where its eigenstates are superpositions of persistent-​current states, with expectation value 〈 I 〉 = 0. A determination of the qubit state therefore requires the flux to be brought, by means of an adiabatic pulse, away from ε = 0, prior to measurement. The flux-​qubit loop can be inductively coupled to a dc-​SQUID; the persistent current then modulates the magnetic flux through the SQUID loop, thereby causing the SQUID’s switching current to depend on the qubit state. A sample-​and-​hold current pulse with amplitude near the switching current then causes a switching event if the qubit is in one of its states, but not the other, and the associated voltage across the SQUID can be measured. 18.4.6.5 History and state of the art Early work on the flux qubit (Mooij et  al. 1999, Orlando et  al. 1999, van der Wal 2000, Friedman et  al. 2000, Berkley et  al. 2003)  led to the demonstration of coherent dynamics, in Delft, by Chiorescu et al. (2003). Bertet et al. (2005), Yoshihara et al. (2006), and the NEC/​MIT collaboration (including this author) did extensive studies of coherence, noise spectroscopy, dynamical decoupling, single-​qubit gate fidelities, and nonadiabatic transitions (see e.g. Bylander et al. 2011, Gustavsson et al. 2011, 2012a, 2012b,Yan et al. 2012,Yoshihara et al. 2014, Jin et al. 2015).

18.4.7 Phase qubit The demonstration of discrete states for the fictitious phase particle in the wells of the tilted-​washboard potential of a current-​biased junction was discussed in Sec. 18.4.2. For a bias current, I, just below the critical current, the anharmonicity is approximately cubic, with a plasma oscillation frequency 1



ωp (I ) = 24

I0 ϕ0C

 I  41 1 − I  . (18.52) 0

For realistic experimental parameters, the well can accommodate a few discrete states with progressively smaller frequency spacings for the higher states, and the two lowest can be in the range 5–​20 GHz, and be used as a qubit. In an improved circuit—​see Fig. 18.13—​the junction was decoupled from the current leads by placing it in a loop of inductance L, which can be biased by a static or pulsed magnetic flux, thereby still current-​biasing the junction. The potential then takes on the form

552

552  Superconducting Quantum Bits of Information ISQ

U

IΦ ∼Φ0

Fig.  18.13 Phase qubit. Current-​biased Josephson junction in a loop. The qubit states are represented by the lowest and first-​excited levels in the Josephson potential (Martinis et al. 2002, adapted with permission; see also the review articles listed in Sec. 18.1.1.)

U = E J (1 − cos ϕ ) + E L (ϕ − ϕext ) , (18.53) 2

with the externally controlled phase defined as ϕext =

Φ ext . ϕ0

The Hamiltonian becomes

 1  Hˆ = −  ω q σˆ z + ∆I circ (σˆ x + χσˆ z ) , (18.54) 2  2ω qC 

where ∆I circ is the variation of the circulating current from the nominal bias curω q 1 rent, and χ = ≈ for the experimental parameters used. 3∆U 4 This Hamiltonian cannot be cast in the same form as those for the charge and flux qubits (Eq. 18.2), but transitions between levels (σˆ x ) can still be induced by applying a resonant signal, ∆I circ (t ), and σˆ z rotations can be induced by changing the current. Here the transition rate (Eq. 18.23) for transversely coupled critical-​ 1  current noise is mδi = δI circ δi . Note also that the phase qubit has no noise-​ 2 2ω qC insensitive “sweet spot,” which makes it suffer from dephasing at all bias points. In order to distinguish between the qubit states, one takes advantage of the difference in tunneling rates out of the potential well. The |0〉 state is metastable, the |1〉 state is relatively long-​lived, and |2〉 tunnels rapidly. A  pulse resonant with the |1〉 →|2〉 transition causes the junction to switch into the voltage state only if the qubit was in the|1〉 state but not if it was in the|0〉 state. In the improved design (Fig. 18.13), where the junction is coupled in a flux-​biased, superconducting loop, instead of connected to current leads, the potential energy forms two asymmetric wells, with the qubit states residing in the upper well. A short adiabatic pulse that reduces the potential barrier then causes the flux to transition into the readout well if the qubit is in the|1〉 state, but not if it is in the|0〉 state. The resulting difference in flux, by approximately Φ0, can be detected by a SQUID magnetometer with reduced energy dissipation compared to the switching-​junction readout.

  553

Circuit quantum electrodynamics (c-QED)  553 The phase qubit was developed by Martinis’s group, first at NIST and then in Santa Barbara (Martinis et al. 2002, 2005, Steffen et al. 2006, Martinis 2009). Initially, losses in the dielectric barrier of the rather large Josephson junction were detrimental to the coherence of phase qubits. Reducing the junction area, while increasing the critical-​current density to keep I0 the same, and shunting the junction by an external capacitor with a less lossy dielectric, brought about an improvement.

18.4.8 Dissipative readout of superconducting qubits The readout methods for first-​generation qubits, discussed in Sec. 18.4.4–​7, yield good qubit-​state discrimination within a short enough time. However, they dissipate heat and generate nonequilibrium quasiparticles that need time to relax, which limits the reset time to about 0.5 ms. Moreover, these readout methods exert significant backaction on the qubits, which leads to dephasing, if not relaxation, and are hardly QND (quantum nondemolition) in character. Better performance is currently obtained with dispersive techniques and a following quantum-​limited amplifier—​see Sec. 18.5.

18.5 Circuit quantum electrodynamics (c-​QED) In quantum optics, the strong coupling of an atom to a mode of the electromagnetic field—​cavity quantum electrodynamics—​was an early model system for studying quantum information; see for example Raimond et al. (2001). This field was carried over into the microwave domain, with solid-​state artificial atoms and transmission-​line or lumped superconducting resonators (Blais et al. 2004, Wallraff et  al. 2004, Chiorescu et  al. 2004). Denoted circuit quantum electrodynamics (c-​QED), this on-​chip architecture has proven useful for QND qubit readout; as a qubit–​qubit coupling bus; and as a spectral filter for the environmental density of states, reducing qubit relaxation by the Purcell effect. C-​QED allows for very strong atom–​photon coupling, afforded by the large effective electric dipole moment—​for a Cooper-​pair box it can be 104 times larger than that of an alkali atom and 10 times larger than that of a Rydberg atom—​and the large vacuum field strength—​about 100 times larger than in cavity QED, due to the small mode volume of about 10-​6 cubic wavelengths.

18.5.1 Jaynes–​Cummings model The Jaynes–​Cummings model describes the interaction of matter and radiation at the quantum level (Walls and Milburn 2004). Here we limit the description to the

554

554  Superconducting Quantum Bits of Information single-​mode case, in which the photonic part is described by the Hamiltonian of a harmonic oscillator of frequency ωr,

1  Hˆ r = ω r  aˆ †aˆ +  , (18.55)  2

where aˆ † and aˆ are the bosonic creation and annihilation operators with the comˆ and the photon-​number operator is nˆ = aˆ †aˆ . The mutation relation aˆ , aˆ †  = 1, transverse electric and magnetic field operators are given by

(

)

Eˆ x = E 0 aˆ + aˆ † sin (kz ) , (18.56) Bˆ y = −iB0 aˆ − aˆ † cos (kz ) ,

(

)

where E0 = (ℏωr/​ε0V)½ and B0 = (ℏωrμ0/​V)½ are the rms field fluctuations “per photon,” V is the effective cavity volume, ε0 is the electric permittivity, and μ0 is the magnetic permeability. The atomic system, starting from the Rabi model with static part Hˆ 0 and a quantized interaction term, has the Hamiltonian

(

)

Hˆ = Hˆ 0 − E 0 aˆ + aˆ † sin (kz ) dˆσˆ x , (18.57)

with the dipole operator dˆ = −erˆ ⋅ Eˆ ( r , t ), in the laboratory frame. We define the coupling by g = − E 0 sin (kz ) dˆ , and arrive at

(

)

Hˆ = Hˆ 0 + g σˆ x aˆ + aˆ † . (18.58)

Since we can write σˆ x = σˆ + + σˆ −, the coupling consists of four terms; invoking the RWA, we can disregard the terms aˆσˆ − and aˆ † σˆ + that do not preserve the number of excitations, and arrive at the Jaynes–​Cummings Hamiltonian for the coupled atom–​photon system:

 1   Hˆ JC = − ω q σˆ z + ω r  aˆ †aˆ + 1ˆ  + g aˆσˆ + + aˆ † σˆ − . (18.59)  2 2 

(

)

The uncoupled spectrum (g = 0) consists of harmonic-​oscillator ladders shifted by the atomic energy difference, ℏωq, between ground state, |f〉 (for fundamental), and excited state, |e〉. Fig. 18.14 shows the frequency spectrum of the bare states,

1 1  ωn ,e / f =  n +  ω r ± ω q . (18.60)  2 2

(In order to differentiate from the photon-​number states, |n〉, where n = 0, 1, 2, …, we here denote the qubit states by |f〉 and |e〉 rather than by |0〉 and |1〉; in the qubit frame, |f〉 = |+ z’〉 and |e〉 = |–​z’〉.) Due to the RWA, the interaction couples only the states |f, n+ 1〉 with |e, n〉, and the Hamiltonian becomes 2 × 2 block

  555

Circuit quantum electrodynamics (c-QED)  555

2g (n+1)1/2 n

n–1

n

2

1

2

0

1

2g

1

n –1 1 0

0

0

0

0 f

e

diagonal, with off-​ diagonal terms

f

(

e

†

)

 + + a σ  − f , n + 1 = g n + 1 e , n g a σ

(and equally for the conjugate). In this approximation the Hamiltonian becomes

 1  Hˆ n = − ∆ωσˆ z + ω r  n +  1ˆ + g n + 1σˆ x , (18.61)  2 2

which clearly maps onto the qubit Hamiltonian (Eq. 18.8), with the photon-​ number dependent eigenfrequencies

1 1  ωn , ± =  n +  ω r ± ∆ω 2 + 4 g 2 (n + 1). (18.62)  2 2

For n = 0, we denote by Ω V = ∆ω 2 + 4g 2 the vacuum Rabi splitting. The new, hybridized eigenstates are called the “dressed states” (the atom is “dressed” by the photons),

θn θ f ,n + 1 + cos n e ,n , 2 2 (18.63) θn θn e ,n , n , − = cos f ,n + 1 − sin 2 2

n , + = sin with the mixing angle

  2g θn = arctan  n + 1 . (18.64)  ∆ω 

The vacuum Rabi splitting can be observed when the coupling strength is much higher than both the loss rate of the resonator and the decoherence rate of the qubit, g  Γ r , 1 T2 , so that an excitation can be exchanged coherently between the photonic and the qubit parts of the system without decaying through other channels.

Fig. 18.14  Jaynes–​Cummings ladders for the resonant and dispersive regimes. See for example Walls and Milburn (2004).

556

556  Superconducting Quantum Bits of Information

18.5.2 Dispersive regime,  ∆ω

g

When the detuning is large, ∆ω  g, the resonator field cannot induce transitions between the eigenstates. The coupling now renormalizes the eigenenergies of the system. A perturbation calculation in the small parameter ∆ω g yields the effective dispersive Hamiltonian

 Hˆ disp = − ω q + χ σˆ z +  (ω r − χσˆ z ) aˆ †aˆ, (18.65) 2

(

)

where

χ=



g2 (18.66) ∆ω

In the first term, ω q + χ represents the Lamb-​ shifted atomic energy levels (due to the vacuum). In the second term, − χσˆ z aˆ †aˆ represents the ac Stark shift of the resonator:  this is a “dispersive” frequency shift dependent on the qubit state. This Hamiltonian is diagonal in the basis { f , n , e , n }, and as shown in Fig. 10.14, the eigenenergies become

ωn , f / e = 

1 ω q + χ + n (ω r  χ ). (18.67) 2

(

)

The dispersive regime offers the advantage that it suppresses the relaxation rate of the qubit into the resonator (the Purcell effect). The resonator effectively acts as a band-​pass filter of the available electromagnetic modes, in which the qubit must emit a photon during relaxation (Houck et al. 2008, Reed et al. 2010, Jeffrey et al. 2014, Sete et al. 2015).

18.5.3 Dispersive qubit-​state readout The dispersive frequency shift provides a means for reading out the qubit state by probing the resonator with homodyne or heterodyne detection of the reflection or transmission coefficient—​see Fig. 18.15. Note that when the resonator is coupled to a multi-​level artificial atom—​such as the transmon (Sec. 18.6.2)—​the observed dispersive shift obtains a correction  g2  EC from the transition between the first and second excited states, χ = . ∆ω  ∆ω − EC  The linear dispersive shift occurs only for moderate probe strengths, up to n ≈ 100, because a strong probe disturbs the qubit (Gambetta et al. 2006, Boissonneault et  al. 2009). In order to obtain a signal-​to-​noise ratio (SNR) adequate for state discrimination, the output signal must therefore be amplified by a low-​noise amplifier, and be averaged over several measurements. It is possible to obtain SNR > 1, enabling state discrimination without averaging—​in a “single shot”—​by using “quantum-​limited” amplifiers. These are nonlinear

  557

Second-generation superconducting qubits 557 Transmitted microwave signal Magnitude

ng Φext C-shunted flux qubit

Resonator Transmon Transmon spectrum E (arb. un.)

E/EC

4 0 –1

0

1

ng

0.49

0

π/2

Flux qubit spectrum

8

1

Phase

~

–π/2

0.50

2χ ωr Probe frequency

Φext 0.51 Φ0

Fig.  18.15  Circuit-​QED architecture for dispersive readout. The qubit states can be distinguished by measuring the phase or magnitude of the transmitted signal. The solid lines represent |f〉 and |e〉; the dashed line represents the bare resonator without qubits. The resonator can be lumped or distributed (as in the commonly used coplanar waveguide resonator, here a half-​wavelength resonator). The resonator further serves as a coupling bus and Purcell filter for the qubits. The dispersion relations for the transmon and C-​shunted flux qubits (Blais et al. 2004,You et al. 2007, adapted with permission) are shown for comparison with those of the CPB and standard flux qubit in Fig. 18.10 and Fig. 18.12. and/​or parametric devices of various kinds, which may provide gain while adding only the minimal amount of noise allowed by quantum mechanics. A thorough description of quantum-​limited amplifiers is outside the scope of this chapter; see Clerk et al. (2010) for a review of quantum noise, amplification, and detection. We note that dispersive readout is, in principle, of QND character, since the qubit’s Hamiltonian commutes with the interaction term in the dispersive Hamiltonian.

18.6 Second-​generation superconducting qubits This section introduces several improved designs of superconducting qubits, based on lessons learned from experiments with the first-​generation qubits (Sec. 18.4) and using the c-​QED architecture (Sec. 18.5). Coherence times have been hugely improved by minimizing the sensitivity to fluctuating impurities (Sec. 18.3.3) and the coupling to external modes.

558

558  Superconducting Quantum Bits of Information Improvements of the materials used to fabricate the qubits also play a key role. A large number of studies on qubits, SQUIDs, and microwave resonators have helped identify loss mechanisms, fabrication methods, and useful materials; see for example Sage et al. (2011), Anton et al. (2012a, 2012b, 2013), Sandberg et al. (2013), and Chang et al. (2013). However, this chapter does not have room for a thorough review; the reader is referred to, for example, McDermott (2009), Siddiqi (2011), Oliver and Welander (2013), Martinis and Megrant (2014), and Wang et al. (2015). These combined design and materials research efforts have, incredibly, enhanced the energy-​relaxation time by six orders of magnitude from the first demonstration in 1999 until today, and this has been dubbed a “Moore’s law for quantum coherence” (Steffen 2011). Long coherence times, compared to the gate-​operation time, now enable gate fidelities exceeding the predicted fault-​ tolerance threshold for certain types of quantum error-​ correcting codes (Chow et al. 2012, Barends et al. 2014). Nevertheless, it is important to make further improvements, in order to bring down the required overhead resources for quantum computing from daunting to merely extremely challenging. While “sweet-​spot”-​operation of the quantronium, flux, and charge qubits was quite successful in minimizing dephasing (Sec. 18.3.4)—​if only at one bias point for the gate charge and external flux—​and dynamical-​decoupling methods kept the coherent states alive until the T1 limit (Sec. 18.3.5–​6), the T1 relaxation time was not improved by these methods. However, the demonstration of first-​order noise insensitivity inspired researchers to attempt to minimize the energy dispersion for all bias points—​to make the energy bands flat with respect to the noisy control parameters: see Sec. 18.6.2–​4 below.

18.6.1 Dispersive readout The flat dispersion has implications for qubit readout. For example, the CPB’s discrete charge states differ by a voltage (cf. the slopes in Fig. 18.10: dE dQ = V ); on the other hand, the eigenstates of the transmon qubit are superpositions of many charge states and cannot be distinguished by charge. Instead, dispersive readout is the currently dominant method, with qubits coupled to resonators in the c-​QED architecture (Sec. 18.5). Table 18.2  Modern superconducting qubits, typical parameter values. Name

EJ/​4EC

EL/​EJ

Transmon

100

0

Fluxonium

1–​100

0.05–​0.5

C-​shunt flux qubit

10

0.25

  559

Second-generation superconducting qubits 559

18.6.2 Transmon The currently most popular and commonly used qubit is the “transmon,” invented at Yale (Koch et al. 2007, Houck et al. 2008, 2009)—​see Fig. 18.15. It is a derivative of the CPB, with the same structure and same Hamiltonian, only with its Josephson junction shunted by a large capacitor in order to decrease the charging energy, EC. Consequently, the charge dispersion becomes essentially flat, which suppresses the sensitivity to charge noise exponentially in the ratio 8E J / EC . Typically, E J / EC ≈ 30. The transmon’s transition frequency is approximated as ω q ≈ 8E J EC − EC, where 8E J EC /  is the plasma frequency of the Josephson junction. The spectrum is much less harmonic than that of the CPB, with absolute and relative anharmonicity (for the first level) α abs = ω12 − ω q ≈ − EC ≈ 200 MHz and ω − ωq EC α rel = 12 ≈− , respectively. ωq 8E J

18.6.3 Reduction of surface losses. 3D cavities, planar designs, and materials Early work had identified the qubit materials’ oxide dielectrics and surfaces as a major source of dielectric loss (McDermott 2009). For the transmon, a viable strategy to reduce this detrimental effect was to reduce the participation ratio of the Josephson junction to the total capacitance between the islands. By reducing the electric field strength in the surface oxide, a greater proportion of the electric field goes through a vacuum, which has no dielectric loss. A breakthrough was achieved through three-​dimensional (3D)-​cavity QED (Paik et al. 2011, Rigetti et al. 2012, Wang et al. 2015), with T1 times up to 140 μs. The cavity supports transverse electric modes, interacting with the atomic dipole of the transmon qubit. The mode volume is significantly increased compared to planar resonator geometries, thereby much reducing the electric-​field strength in the lossy dielectrics. 3D cavities also helped improve flux qubits, enabling reproducible relaxation times in the 6–​20 μs range and pure dephasing between 3 and 10 μs (Stern et al. 2014). A transmon made of titanium nitride superconductor achieved T1  =  60 μs (Sandberg et al. 2013), indicating that defects in or near the surfaces contribute to decoherence, and that nitrides may be a better choice than oxides. Other design improvements also help decrease the dielectric losses in planar geometries, with coplanar-​waveguide or lumped-​element resonators, for example, placing the interdigitated-​capacitor fingers further apart or etching away part of the insulating substrate. One variety is the “Xmon” qubit (Barends et al. 2013)—​a tunable transmon made with epitaxial aluminum, with one island connected to ground via a split Josephson junction, and embedded in a ground plane. This device reproducibly reached T1 > 40 μs and T2* > 10 μs, due to a reduction

560

560  Superconducting Quantum Bits of Information of the loss from the surface oxides and interfaces. A reduction of the surface participation ratio minimized the coupling of electric fields to the dipole moments of incoherent two-​level systems in the dielectric.

18.6.4 Fluxonium The “fluxonium” qubit—​ Fig. 18.16—​ consists of a single, small Josephson junction shunted by a huge inductance, typically L ≈ 500 nH, but with small capacitance (Koch et al. 2009, Manucharyan et al. 2009, Manucharyan 2013). It is reminiscent of the CPB (Sec. 18.4.4) as well as the rf-​SQUID, with the Hamiltonian

(

Hˆ ϕ = 4 EC nˆ − n g



)

2

2

 Φ  − E J cos ϕˆ + E L  ϕˆ − ext  , (18.68) ϕ0  

ignoring phase slips and quasiparticle tunneling. Typical parameter values are EJ/​h = 10 GHz, 4EC/​h = 5 GHz, and EL/​h = 0.5 GHz—​see also Table 18.2. The single junction is a phase-​slip center, allowing for the shuttling of flux in and out of the loop. The inductance provides the desired high impedance at the qubit frequency, enabling weak coupling to the electrodynamic environment and therefore a small relaxation rate. At the same time, the shunt provides very low impedance at low frequencies (dc), consequently neutralizing the dephasing effect of offset charges, because the junction charge can be offset by an arbitrary amount by rearrangement of the electrons in the shunt. Since the circuit has ˆ has a continuous spectrum, and the no isolated islands, the number operator, n, phase, ϕˆ , is defined on the entire real axis (the potential is not periodic in ϕˆ , as the cosine has a parabolic envelope). Varieties of the fluxonium include the “L-​shunt flux qubit” and the “metastable flux qubit” (Kerman 2010). The fluxonium supports a host of bound and semi-​bound states in its potential, depending on the parameters. Suitable qubit states are represented by the bound states in separate wells of the cosine potential, associated with large charge oscillations across the Josephson junction. States in adjacent wells overlap minimally with one another, leading to avoided level crossings of typically Δ/​2π = 50 kHz, which is much smaller than the thermal energy, requiring that the qubit be operated far from degeneracy between two states. U

Fig.  18.16 The fluxonium qubit has low sensitivity to charge and flux noise. (Adapted from Manucharyan 2013 and Pop et al. 2014).

~ ϕ

  561

Summary and outlook 561 Indeed, this small coupling between the qubit states and to external modes, and a careful study of the coherent suppression of quasiparticle dissipation, allowed the demonstration of record T1 relaxation times exceeding 1 ms (Catelani et al. 2011, Pop et al. 2014). Whereas the first-​generation persistent-​current flux qubit (Sec. 18.4.6) is very sensitive to dephasing flux noise, when operated far from degeneracy, the large inductance of the fluxonium makes it 100 to 1,000 times less sensitive and allows for T2 coherence times in excess of 100 μs with projected performance even much higher.

18.6.5 Capacitively shunted flux qubit While the transmon and also the fluxonium qubits can be thought of as derivatives of the charge qubit, there is also an important, noise-​insensitive descendant of the flux qubit. Decreasing the persistent current of the flux qubit (flattening the bands), by decreasing the EJ/​EC ratio, decreases the sensitivity to flux noise but increases the sensitivity to charge noise. A capacitive shunt of the small Josephson junction (the “α junction”) reduces the effects of charge noise (You et al. 2007, Steffen et al. 2010), with the added benefit of decreasing the participation ratio of the dielectric loss of the junction, compared to that of the external capacitor, and minimizing parasitic fringing fields. Furthermore, the C-​shunted flux qubit has broad frequency tunability and strong anharmonicity (500 MHz). These changes considerably improve relaxation times: Yan et al. (2015) showed reproducibly T1 > 40 μs, accounted for with a model involving ohmic charge noise, 1/​f flux noise, resonator loss, and quasiparticle tunneling. Dephasing was also improved, in particular when biased away from the flux-​degenerate point, Φext = Φ0/​2, and T1-​limited decoherence was obtained via dynamical decoupling. This group also demonstrated that dephasing at the flux-​insensitive point is dominated by residual thermal photons in the readout resonator, which limits T2.

18.7 Summary and outlook The ongoing improvements in superconducting qubit performance—​ due to improved device designs, elimination of noise sources, and implementation of control methods—​are truly astounding. Six orders of magnitude in 16 years. The complexity of quantum hardware has only just started to increase, but primitive experiments with several quantum-​coherent coupled qubits have already been performed. Ideas of what useful—​or merely interesting—​experiments one can do with a special-​purpose quantum machine are starting to emerge. This evolution cannot be stopped. It might be the case that the basic ingredients are almost in place—​long-​lived qubits enabling high-​fidelity single-​and coupled-​gate

562

562  Superconducting Quantum Bits of Information operations and measurement—​so that the next 16 years will see improvements mainly in systems engineering. Look, then, in Sec. 18.1.2, at the list of topics left out of this chapter, and hope that research funding will come and let us explore.

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564  Superconducting Quantum Bits of Information Hahn, E.L. (1950) Phys. Rev. 80, 580. Houck, A.A., Schreier, J.A., Johnson, B.R., Chow, J.M., Koch, J., Gambetta, J.M., Schuster, D.I., Frunzio, L., Devoret, M.H., Girvin, S.M. and Schoelkopf, R.J. (2008) Phys. Rev. Lett. 101, 080502. Houck, A.A., Koch, J., Devoret, M.H., Girvin, S.M. and Schoelkopf, R.J. (2009) Quant. Inf. Proc. 8, 105. Houck, A.A., Türeci, H.E. and Koch, J. (2012) Nature Phys. 8, 292. Ithier, G., Collin, E., Joyez, P., Meeson, P.J., Vion, D., Esteve, D., Chiarello, F., Shnirman, A., Makhlin, Y., Schriefl, J. and Schön, G. (2005) Phys. Rev. B 72, 134519. Jeffrey, E., Sank, D., Mutus, J.Y., White, T.C., Kelly, J., Barends, R., Chen, Y., Chen, Z., Chiaro, B., Dunsworth, A., Megrant, A., O’Malley, P.J., Neill, C., Roushan, P., Vainsencher, A., Wenner, J., Cleland, A.N. and Martinis, J.M. (2014) Phys. Rev. Lett. 112, 257402. Jin, X.Y., Gustavsson, S., Bylander, J., Yan, F., Yoshihara, F., Nakamura, Y., Orlando, T.P. and Oliver, W.D. (2015) Phys. Rev. Appl. 3, 034004. Kerman, A.J. (2010) Phys. Rev. Lett. 104, 027002. Kim, Z., Suri, B., Zaretskey, V., Novikov, S., Osborn, K.D., Mizel, A., Wellstood, F.C. and Palmer, B.S. (2011) Phys. Rev. Lett. 106, 120501. Koch, J., Yu, T.M., Gambetta, J., Houck, A.A., Schuster, D.I., Majer, J., Blais, A., Devoret, M.H., Girvin, S.M. and Schoelkopf, R.J. (2007) Phys. Rev. A 76, 042319. Koch, J., Manucharyan, V., Devoret, M.H. and Glazman, L.I. (2009) Phys. Rev. Lett. 103, 217004. Ladd, T.D., Jelezko, F., Laflamme, R., Nakamura,Y., Monroe, C. and O’Brien, J.L. (2010) Nature 464, 45. Lehnert, K.W., Bladh, K., Spietz, L.F., Gunnarsson, D., Schuster, D.I., Delsing, P. and Schoelkopf, R.J. (2003) Phys. Rev. Lett. 90, 027002. McDermott, R. (2009) IEEE Trans. Appl. Supercond. 19, 2. Makhlin, Y., Schön, G. and Shnirman, A. (2001) Rev. Mod. Phys. 73, 357. Makhlin, Y. and Shnirman, A. (2003) JETP Lett. 78, 497. Manucharyan, V.E., Koch, J., Glazman, L.I. and Devoret, M.H. (2009) Science 326, 113. Manucharyan, V.E. (2013) Fluxonium qubit, in Lecture Notes 44th IFF Spring School 2013, Quantum Information Processing, edited by D.P. DiVincenzo, Forschungszentrum Jülich, Jülich. Martinis, J.M., Devoret, M.H. and Clarke, J. (1985) Phys. Rev. Lett. 55, 1543. Martinis, J.M., Devoret, M.H. and Clarke, J. (1987) 35, 4682. Martinis, J.M., Nam, S., Aumentado, J. and Urbina, C. (2002) Phys. Rev. Lett. 89, 117901. Martinis, J.M. and Osborne, K. (2004) arXiv:cond-​mat/​0402415v1 Martinis, J.M., Cooper, K.B., McDermott, R., Steffen, M., Ansmann, M., Osborn, K.D., Cicak, K., Ho, S., Pappas, D.P., Simmonds, R.W. and Yu, C.C. (2005) Phys. Rev. Lett. 95, 210503.

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References (Chapter-18) 565 Martinis, J.M. (2009) Quant. Inf. Proc. 8, 81. Martinis, J.M. and Megrant, A. (2014) arXiv:1410.5793v1 Meiboom, S. and Gill, D. (1958) Rev. Sci. Instrum. 29, 688. Mooij, J.E., Orlando, T.P., Levitov, L., Tian, L., van der Wal, C.H. and Lloyd, S. (1999) Science 285, 5430. Nakamura, Y., Chen, C.D. and Tsai, J.S. (1997) Phys. Rev. Lett. 79, 2328. Nakamura, Y., Pashkin, Y.A. and Tsai, J.-​S. (1999) Nature 398, 786. Nakamura, Y., Pashkin, Y.A. and Tsai, J.-​S. (2001) Phys. Rev. Lett. 87, 246601. Oliver, W.D. (2013) Superconducting qubits, in Lecture Notes 44th IFF Spring School 2013, Quantum Information Processing, edited by D.P. DiVincenzo, Forschungszentrum Jülich, Jülich. Oliver, W.D. and Welander, P.B. (2013) MRS Bull. 38, 816. Orlando, T.P., Mooij, J.E., Tian, L., van der Wal, C.H., Levitov, L.S., Lloyd, S. and Mazo, J. (1999) Phys. Rev. B 60, 15398. Paik, H., Schuster, D.I., Bishop, L.S., Kirchmair, G., Catelani, G., Sears, A.P., Johnson, B.R., Reagor, M.J., Frunzio, L., Glazman, L.I., Girvin, S.M., Devoret, M.H. and Schoelkopf, R.J. (2011) Phys. Rev. Lett. 107, 240501. Pop, I.M., Geerlings, K., Catelani, G., Schoelkopf, R.J., Glazman, L.I. and Devoret, M.H. (2014) Nature 508, 369. Raimond, J.-​M., Brune, M. and Haroche, S. (2001) Rev. Mod. Phys. 73, 565. Reed, M.D., DiCarlo, L., Johnson, B.R., Sun, L., Schuster, D.I., Frunzio, L. and Schoelkopf, R.J. (2010) Phys. Rev. Lett. 105, 173601. Rigetti, C., Gambetta, J.M., Poletto, S., Plourde, B.L.T., Chow, J.M., Córcoles, A.D., Smolin, J.A., Merkel, S.T., Rozen, J.R., Keefe, G.A., Rothwell, M.B., Ketchen, M.B. and Steffen, M. (2012) Phys. Rev. B 86, 100506. Sage, J.M., Bolkhovsky, V., Oliver, W.D., Turek, B. and Welander, P.B. (2011) J. Appl. Phys. 109, 063915. Sandberg, M., Vissers, M.R., Ohki, T., Gao, J., Aumentado, J., Weides, M. and Pappas, D.P. (2013) Appl. Phys. Lett. 102, 072601. Sank, D., Barends, R., Bialczak, R.C., Chen, Y., Kelly, J., Lenander, M., Lucero, E., Mariantoni, M., Megrant, A., Neeley, M., O’Malley, P.J.J., Vainsencher, A., Wang, H., Wenner, J., White, T.C., Yamamoto, T., Yin, Y., Cleland, A.N. and Martinis, J.M. (2012) Phys. Rev. Lett. 109, 067001. Schoelkopf, R.J., Wahlgren, P., Kozhevnikov, A.A., Delsing, P. and Prober, D.E. (1998) Science 280, 1238. Schoelkopf, R.J. and Girvin, S.M. (2008) Nature 451, 664. Sete, E.A., Martinis, J.M. and Korotkov, A.N. (2015) Phys. Rev. A 92, 012325. Siddiqi, I. (2011) Supercond. Sci. Technol. 24, 091002. Smirnov, A.Y. (2003) Phys. Rev. B 67, 155104. Steffen, M., Ansmann, M., McDermott, R., Katz, N., Bialczak, R.C., Lucero, E., Neeley, M., Weig, E.M., Cleland, A.N. and Martinis, J.M. (2006) Phys. Rev. Lett. 97, 050502. Steffen, M., Kumar, S., DiVincenzo, D.P., Rozen, R.R., Keefe, G.A., Rothwell, M.B. and Ketchen, M.B. (2010) Phys. Rev. Lett. 105, 100502.

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566  Superconducting Quantum Bits of Information Steffen, M. (2011) Physics 4, 103. Stern, M., Catelani, G., Kubo,Y., Grezes, C., Bienfait, A., Vion, D., Esteve, D. and Bertet, P. (2014) Phys. Rev. Lett. 113, 123601. Tsai, J.-​S. (2016) Achievements and outlook of research on quantum information systems using superconducting quantum circuits, in Lecture Notes in Physics, Principles and Methods of Quantum Information Technologies, edited by Y. Yamamoto and K. Semba, Springer, Tokyo, Japan, vol. 911, p. 477. van der Wal, C.H., ter Haar, A.C.J., Wilhelm, F.K., Schouten, R.N., Harmans, C.J.P.M., Orlando, T.P., Lloyd, S. and Mooij, J.E. (2000) Science 290, 773. Vion, D., Aassime, A., Cottet, A., Joyez, P., Pothier, H., Urbina, C., Esteve, D. and Devoret, M.H. (2002) Science 296, 886. Voss, R.F. and Webb, R.A. (1981) Phys. Rev. Lett. 47, 265. Walls, D. and Milburn, G. (1994) Quantum optics, Springer-​Verlag, Berlin. Wallraff, A., Schuster, D.I., Blais, A., Frunzio, L., Huang, R.-​S., Majer, J., Kumar, S., Girvin, S.M. and Schoelkopf, R.J. (2004) Nature 431, 162. Wang, C., Axline, C., Gao, X.Y., Brecht, T., Chu, Y., Frunzio, L., Devoret, M.H. and Schoelkopf, R.J. (2015) Appl. Phys. Lett. 107, 162601. Wendin, G. and Shumeiko, V. (2007) J. Low Temp. Phys. 33, 724. Yan, F., Bylander, J., Gustavsson, S., Yoshihara, F., Harrabi, K., Cory, D.G., Orlando, T.P., Nakamura, Y., Tsai, J.S. and Oliver, W.D. (2012) Phys. Rev. B 85, 174521. Yan, F., Gustavsson, S., Bylander, J., Jin, X.Y.,Yoshihara, F., Cory, D.G., Nakamura, Y., Orlando, T.P. and Oliver, W.D. (2013) Nature Commun. 4, 2337. Yan, F., Gustavsson, S., Kamal, A., Birenbaum, J., Sears, A.P., Hover, D., Gudmundsen, T.J., Yoder, J.L., Orlando, T.P., Clarke, J., Kerman, A.J. and Oliver, W.D. (2015) arXiv:1508.06299v1. Yoshihara, F., Harrabi, K., Niskanen, A.O., Nakamura, Y. and Tsai, J.S. (2006) Phys. Rev. Lett. 97, 167001. Yoshihara, F., Nakamura, Y., Yan, F., Gustavsson, S., Bylander, J., Oliver, W.D. and Tsai, J.S. (2014) Phys. Rev. B 89, 020503. You, J.Q. and Nori, F. (2005) Phys. Today 58, 42. You, J.Q., Hu, X., Ashhab, S. and Nori, F. (2007) Phys. Rev. B 75, 140515. You, J.Q. and Nori, F. (2011) Nature 474, 589.

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NanoSQUIDs Applied to the Investigation of Small Magnetic Systems M. J. Martínez-​Pérez, R. Kleiner, and D. Koelle Physikalisches Institut and Center for Quantum Science (CQ) in LISA+, Universität Tübingen, Auf der Morgenstelle 14, D-​72076 Tübingen, Germany

19 19.1 SQUID basics 

567

19.2 NanoSQUIDs 

572

19.3 Measurement techniques using nanoSQUIDs 

590

19.4 Particle positioning 

592

19.5 Applications 

596

19.6 Summary and outlook 

600

References (Chapter-​19) 

600

Studies on the properties of magnetic nanoparticles (NPs) and single molecule magnets (SMMs) have fueled the development of new magnetic sensors for single-​particle detection with improved performance. Many of the recent advances in this field include the development of magneto-​optical techniques based on nitrogen vacancy centers in diamond (Schäfer-​Nolte et  al. 2014)  or the use of carbon nanotubes (CNTs) as spin detectors (Ganzhorn et al. 2013). Alternatively, miniature magnetometers, based on either microHall bars (Lipert et al. 2010) or micro-​and nano-​superconducting quantum interference devices (SQUIDs), provide direct measurement of the stray magnetic fields generated by the particle of study, making the interpretation of the results much more direct and simple. Unfortunately, the field sensitivity deteriorates rapidly when Hall sensors are reduced to the submicron size. SQUID-​based sensors, on the other hand, can theoretically reach quantum-​limited resolution. SQUIDs constitute, still at present, the most sensitive magnetic sensors in the solid state (Clarke and Braginski 2004, Kleiner et al. 2004).

19.1 SQUID basics A SQUID consists of a superconducting ring, intersected by one or two Josephson junctions (JJs), in the case of the radio-​frequency (rf) or direct current (dc) SQUID, respectively. Its operation is based on two fundamental phenomena in superconductors, the fluxoid quantization and the Josephson effect. The former arises from the quantum nature of superconductivity, as the macroscopic wave function describing the whole ensemble of Cooper pairs shall not interfere

M. J. Martínez Pérez, R. Kleiner, D. Koelle, ‘NanoSQUIDs Applied to the Investigation of Small Magnetic Systems’, in The Oxford Handbook of Small Superconductors. First Edition. Edited by A.V. Narlikar. © Oxford University Press 2017. Published in 2017 by Oxford University Press. DOI 10.1093/acprof:oso/9780198738169.003.0019

568

568  NanoSQUIDs and Small Magnetic Systems destructively. This fact leads to the quantization of the magnetic flux Φ threading a superconducting loop (London 1950), in units of the magnetic flux quantum Φ0 = h/​2e ≈ 2.07 × 10−15 Vs. The Josephson effect, on the other hand, was predicted by B.D. Josephson (1962) and demonstrated experimentally by P.W. Anderson and J.M. Rowell (1963). A finite coupling energy results from the overlap of the macroscopic wave functions between two weakly connected superconductors forming the JJ. The supercurrent Is through the weak link and the voltage drop U across it satisfy the Josephson relations

I s (t ) = I 0 sin δ (t )

(19.1)



U (t ) = (Φ0 / 2π ) ∂δ / ∂t ,

(19.2)

with the gauge-​invariant phase difference δ between the macroscopic wave functions of both superconductors and the maximum attainable supercurrent I0. The simple sinusoidal current–​phase relation (CPR), Eq. 19.1, is found for many kinds of JJs. However, some JJ types exhibit a non-​sinusoidal CPR, which can even be multivalued (Likharev 1979).

19.1.1 Resistively and capacitively shunted junction model To describe the phase dynamics of JJs, a very useful approach is the resistively and capacitively shunted junction (RCSJ) model (Stewart 1968, McCumber 1968, Chesca et al. 2004). Within this model, the current across the JJ flows through three parallel channels, as a supercurrent Is, described by Eq. 19.1, a dissipative quasiparticle current Iqp = U/​R across an ohmic resistor R, and a displacement current Id  =  C ∂U/​∂t across the junction capacitance C (see Fig.  19.1a). For a finite temperature T, one includes thermal noise of the resistor as a current noise source IN. Using Kirchhoff’s law and Eq. 19.2, one obtains the equation of motion for the phase difference δ:

I + I N = I 0 sin δ + (Φ0 / 2πR ) ∂δ / ∂t + (Φ0C / 2π ) ∂2δ / ∂t 2

(19.3)

This is equivalent to the equation describing the mechanical motion of a point-​ like particle moving in a tilted washboard potential (see Fig. 19.1b):

U J = E J {(1 − cosδ ) – (i + iN ) δ} ,

(19.4)

with the Josephson coupling energy EJ = I0Φ0/​(2π) and normalized currents i = I/​I0, iN  =  IN/​I0. In this analogy, the mass corresponds to the capacitance, the friction coefficient to the conductance, the driving force, which tilts the potential, to the bias current I, and the velocity to the voltage U. Hysteresis in the current

  569

SQUID basics 569 (a)

(b)

I

i=0

2EJ

UJ Id

Iqp C

R

Is I0

IN

i = 1/2 U i=1 0

1 δ/2π

2

Fig. 19.1  (a) Equivalent circuit of the resistively and capacitively shunted junction (RCSJ). (b)  Tilted washboard potential for different normalized bias currents.

voltage characteristics (IVC), that is, bias current I versus time-​averaged voltage V = , can then be well understood as a consequence of the particle’s inertia: the dissipative state (𝜕δ/​𝜕t ∝ U ≠ 0) is achieved once the bias current I is large enough, so that the metastable minima disappear (at I ≥ I0). If I is decreased after that, the particle becomes retrapped at the retrapping current Ir < I0, leading to a hysteretic IVC. This behavior can be quantified by the Stewart–​McCumber parameter

βC = (2π / Φ0 ) I 0R 2 C .



(19.5)

In order to obtain a non-​hysteretic IVC, βC must be kept below ~1. This can be achieved by means of, for example, an additional shunt resistor, parallel to the JJ.

19.1.2 The dc SQUID The dc SQUID (Jaklevic et  al. 1964)  consists of two JJs connected in parallel in a superconducting loop with inductance L (see Fig.  19.2a). When the loop is threaded by an externally applied magnetic flux Φ, the fluxoid quantization links the phase differences δ1 and δ2 of the two JJs to the total flux in the SQUID ΦT = Φ + LJ via

δ1 − δ 2 + 2πn = (2π / Φ0 ) (Φ + LJ ).



(19.6)

Here, J is the current circulating in the SQUID loop and n is an integer (Kleiner and Koelle 2004). In the limit of a negligibly small screening parameter (b) 2

I

δ2

Φ L I

1 0

∆Ic/2I0

J δ1

(c)

βL

5

Ic /I0

(a)

1 0.01

0

0.5

1.0 Φ/Φ0

1.5

2.0

100 ∝1/βL 10–1 10–2 10–2 10–1

10–0 10–1 βL

Fig. 19.2  The dc SQUID. (a) Schematic representation. (b)  Calculated critical current vs applied magnetic flux (for T  =  0 and identical JJs) for different values of the screening parameter βL. (c) Dependence of normalized Ic modulation on βL (for T = 0 and identical JJs).

570

570  NanoSQUIDs and Small Magnetic Systems

β L = 2LI 0 / Φ0

(19.7)

that is, βL«1, the contribution of LJ to the total flux in Eq. 19.6 can be neglected, and by assuming for simplicity identical critical currents in the two JJs, the maximum supercurrent (critical current) Ic of the SQUID can be easily obtained as

I c = 2I 0 cos ( πΦ / Φ0) .

(19.8)

This simple relation (see Fig.  19.2b for βL«1) summarizes the principle of operation of a dc SQUID. The pronounced dependence of Ic on Φ can be used to probe tiny changes in applied magnetic flux. No analytical expression for Ic(Φ) can be obtained when the effect of a finite SQUID inductance L  =  Lg + Lk (and hence, finite βL) is included. Here, the geometric inductance Lg relates the induced flux LgJ to the current J circulating in the SQUID loop. The kinetic inductance Lk can often be neglected; however, it becomes significant when the width and/​or thickness of the SQUID ring are comparable to or smaller than the London penetration depth λL. The finite L leads to a monotonic decrease of the critical current modulation (ΔIc/​2I0) with increasing βL (see Fig. 19.2.b–​c). For example, Ic modulates by 50% for βL = 1. This effect allows us to estimate L from the measurement of Ic(Φ). In practice, it is much more convenient to operate the dc SQUID in the dissipative state. This is done by biasing the device slightly above Ic, leading to a Φ0-​periodic modulation of the voltage V(Φ), which is often sinusoidal. For this mode of operation, however, one needs non-​hysteretic IVCs, that is, βC ≲ 1. An applied flux signal δΦ causes then a change in the SQUID voltage δV, which for small enough signals is given by δV = (∂V/​∂Φ)δΦ. Hence, the dc SQUID operates as a flux-​to-​voltage transducer. Usually, the working point (with respect to bias current I and applied bias flux) is chosen such that the slope of the V(Φ) curve is maximum, which is denoted as the transfer function VΦ = (∂V/​∂Φ)max.

19.1.3 dc SQUID noise As happens in many physical systems, SQUIDs exhibit at low frequency f a characteristic 1/​f spectral density of voltage noise power SV, which becomes flat at high frequencies in the thermal white noise region. As we are interested in SQUIDs’ capabilities as a flux detector, it is more convenient to focus on the spectral density of flux noise power given by SΦ = SV/​VΦ2 or the rms flux noise √SΦ, which is given in units of Φ0/​√Hz (see Fig. 19.3a). The white noise is mainly produced by Johnson–​Nyquist noise associated with the dissipative quasiparticle current in the JJs or the shunt resistors. This can be analyzed quantitatively by using the so-​called Langevin approach. In this model, thermal noise is included as two independent fluctuating noise terms in the coupled equations describing

  571

SQUID basics 571

1/2 S1/2 ) Φ (Φ0/Hz

(a)

(b) C407–R42 C401–C43 C509–S33 C509–U33

10

1 4.2 K 13 mK 0.1

100

102

104

0.01

ƒ (Hz)

0.1

1

10

T (K)

two parallel-​coupled JJs within the RCSJ model. From numerical simulations it is possible to obtain expressions of SΦ in the white noise region for specific values of βL, βC, and the noise parameter Γ = kBT/​EJ = 2πkBT/​(I0Φ0). For βC ≲ 1, βL > 0.4, and ΓβL < 0.1, one finds (Chesca et al. 2004):

SΦ ≈ 4(1 + β L ) Φ0k ≤ TL / ( I 0R ).

(19.9)

For βL ≤ 0.4, SΦ increases again with decreasing βL. Typically, SQUIDs are designed to give βL ≈ 1.  Under these circumstances Eq. 19.9 reduces to SΦ ≈ 16kBTL2/​R (Tesche and Clarke 1977). Linear scaling of SΦ with temperature is found, however, to saturate in the sub-​Kelvin range (see Fig.  19.3b) due to the hot-​electron effect stemming from limited electron–​phonon interaction at low T (Wellstood et al. 1994). We note that √SΦ scales linearly with L (for fixed βL ≈ 1), meaning that small loop inductances yield lower white flux noise levels. Other sources of white noise are shot and quantum noise, lying usually below the Johnson–​Nyquist term. For the case βL = 1, the former is given by SΦ ≈ hL (Tesche and Clarke 1977), whereas the latter arises from zero point quantum fluctuations giving SΦ ≈ hL/​π (Koch et al. 1981). 1/​f noise is, on the other hand, ascribed to in-​phase and out-​of-​phase critical current fluctuations in the JJs or thermally activated hopping of Abrikosov vortices in the superconducting film. However, a complete description of the different sources of 1/​f noise is still missing. This is especially dramatic in the case of the high transition temperature (high-​Tc) cuprate superconductors, where strong excess noise at low frequencies is typically found (Koelle et  al. 1999). The 1/​f component has also been ascribed to universal flux noise arising from fluctuating spins at the surface of the devices (Koch et al. 2007). This is supported by the observation of a paramagnetic signal following a Curie-​ like T-​dependence (Sendelbach et al. 2008, Bluhm et al. 2009, Martínez-​Pérez et al. 2011).

Fig.  19.3 Rms flux noise of Nb thin film SQUIDs with Nb/​Al-​AlOx/​Nb JJs (a)  √SΦ(f) at 4.2 K and 13 mK (after Martínez-​Pérez et  al. 2010). (b)  High-​ frequency (white) noise, measured at different temperatures on different sensors. The white noise depends on T as expected from theory (SΦ ∝ T) down to ∼100 mK when it saturates.

572

572  NanoSQUIDs and Small Magnetic Systems

19.1.4 dc SQUID readout The voltage response to magnetic flux can be linearized to obtain a larger dynamic range. This can be achieved by operating the SQUID in the so-​called flux locked loop (FLL) mode (Drung and Mück 2004). Here, the SQUID is biased at an optimum working point and behaves as a null-​detector of magnetic flux. A small variation δΦ of the external magnetic flux leads to a variation δV of the total voltage. This small deviation from the working point is amplified at room temperature, integrated, and fed back to the SQUID via a feedback resistor and a feedback coil, which is inductively coupled to the SQUID. The output voltage across the feedback resistor is then proportional to the flux signal δΦ. The dynamic response in FLL mode is limited by the slew rate. This quantity gives the speed at which the feedback circuit can compensate for rapid flux changes at the input. Under optimum conditions, the total bandwidth of the FLL should be only limited by propagation delays between the room-​temperature feedback electronics and the SQUID. For a typical distance between these elements of ~1 m, one finds a theoretical limit of ~20 MHz. Since the flux-​to-​voltage transfer function VΦ is typically small (several tens or hundreds of µV/​Φ0), the voltage noise at the FLL output can easily be dominated by the room-​temperature amplifier noise. There are various ways of circumventing this problem. One can use a flux modulation scheme with a step-​up transformer to increase the SQUID signal at low temperature. This is usually used in commercial SQUID magnetometers. The V(Φ) curve is flux-​modulated by an ac signal of peak-​to-​peak amplitude Φ0/​2 and frequency fm ~ 100 kHz. The resulting modulated SQUID response is amplified and lock-​in detected, using fm as reference frequency. In this configuration, the output SQUID impedance and input preamplifier impedance have to be matched for optimum noise performance. This intermediate matching circuit is tuned at the modulation frequency, thereby limiting the operable bandwidth of the system. Using suitable electronics, a bandwidth of up to 100 kHz can be achieved. Commercial SQUID magnetometers are usually limited to ~10 kHz. On the other hand, there are different possible ways of increasing VΦ at low temperatures, enabling direct readout of the signal (Drung and Mück 2004). One possibility is the use of additional positive feedback (APF), which induces a distortion of the V(Φ) characteristics and hence an increased VΦ at the positive slope. Alternatively, a low-​noise SQUID or serial SQUID array (SSA) amplifier can be used to amplify the front-​end SQUID voltage at low temperatures in a two-​stage configuration.

19.2 NanoSQUIDs The idea of using nanometric JJs to improve the energy resolution ε = SΦ/​(2L) of SQUIDs is attributed to Voss et al. (1980). Four years later, Ketchen et al. (1984)

  573

NanoSQUIDs 573 presented the first SQUID microsusceptometer devoted to detecting the tiny signal produced by nanoscopic magnets. Throughout the 1990s the use of miniaturized SQUIDs with constriction-​type JJs gained much popularity. Such SQUIDs were first produced by W. Wernsdorfer et al. (1995) and allowed the experimental observation of magnetization reversal of nanometric particles. Since then much effort has been dedicated to the further miniaturization of SQUID devices and to the optimization of their noise characteristics (Granata and Vettoliere 2016). Different approaches have been followed based on the use of different kinds of JJs exhibiting very promising capabilities. These approaches are surveyed in the present section. We will start by motivating the use of nanoSQUIDs (Sec. 19.2.1) and analyzing the importance of their size and geometry (Sec. 19.2.2). We will then describe micro-​and nanoSQUIDs made of various types of JJs (Sec. 19.2.3–​19.2.6).

19.2.1 NanoSQUIDs: motivation In commercial SQUID magnetometers, the stray magnetic field created by the sample is usually sensed by a gradiometric pickup coil which is inductively coupled to the SQUID sensor (see Fig. 19.4a). NanoSQUIDs were conceived to boost the sensitivity of these devices down to the single spin. For this purpose, the use of strongly miniaturized SQUID loops and JJs is based on the following ideas: (a) Coupling the magnetic signal from a NP directly to the SQUID nanoloop. No intermediate pickup coils are used, as this degrades the overall coupling. Here, single SQUID loops (Fig. 19.4b) can be used to detect the magnetic moment µ of a NP, or gradiometric configurations (Fig. 19.4.c–​ d) enable measurements of the magnetic ac susceptibility χac. (b) Reducing the cross-​section (width and thickness) of the superconducting thin film forming the loop, as this provides better coupling to nanoscopic samples placed in close vicinity to it (see Sec. 19.2.2). (c) Shrinking the dimensions of the sensor, and hence its inductance, to reduce the white magnetic flux noise (see Sec. 19.2.2). (d) Reducing the dimensions of the JJs and the loop makes the nanoSQUID less sensitive to the application of external magnetic fields Bext, necessary to perform magnetization measurements (see Sec. 19.5.1). (e) Reducing the loop size and the SQUID-​to-​sample distance has a major influence on the spatial resolution for scanning SQUID microscopy applications. NanoSQUIDs can be applied to the study of magnetization reversal of small nanomagnets placed nearby. For this purpose a dc external magnetic field Bext is swept back and forth while recording changes in the magnetization M of the sample upon spin reversal (see Fig. 19.4b). Single-​domain nanomagnets usually

574

574  NanoSQUIDs and Small Magnetic Systems (a)

Pickup coil

M

(b) Bext

SQUID magnetometer

µ

(c)

nanoSQUID magnetometer

(d)

Bac µ

Series-gradiometer

µ

Bac µ

Parallel-gradiometer

Fig. 19.4  Schematic layouts of various SQUID sensors. (a) Conventional magnetometer based on a gradiometric pickup coil inductively coupled (via a mutual inductance M) to a SQUID sensor. (b–​d) NanoSQUIDs which incorporate no intermediate pickup coils; the stray field created by a NP with magnetic moment μ is directly sensed by the SQUID loop. Dc magnetization measurements can be performed by applying an external magnetic field Bext in the nanoloop plane (a). The frequency-​dependent magnetic ac susceptibility χac can be sensed by using series-​(c) or parallel-​planar gradiometers (d); a homogeneous ac excitation magnetic field Bac is applied perpendicularly to the gradiometer’s plane through on-​chip excitation coils. exhibit hysteretic M(Bext), arising from the existence of an energy barrier created by the magnetic anisotropy. The inspection of such hysteresis loops reveals much information on the reversal mechanisms, such as quantum tunneling of the magnetization, coherent rotation, domain wall nucleation and propagation, or the formation of topological magnetic states like vortices. Depending on the particle’s anisotropy, this requires the application of relatively large magnetic fields, a difficult task when dealing with superconducting materials. Measurements are usually done by carefully aligning Bext with respect to the nanoSQUID, so as to minimize the magnetic flux coupled to the loop and the JJs. The maximum Bext will be limited by the upper critical field of the superconducting material, for example ~1 T for Nb films, unless ultrathin films are used, which however increases significantly Lk and hence the flux noise (see Sec. 19.2.4). Even more demanding, nanoSQUID sensors can also be used to quantify the response of a nanomagnet to an oscillating magnetic field Bac = B0cos(ωt), that is, its frequency-​dependent magnetic susceptibility χac  =  χre + iχim, where χre is the real part going in-​phase with Bac and χim is the imaginary or out-​of-​ phase part. These quantities bear much information on the dynamic behavior of spins and the relaxation processes to thermal equilibrium, the interaction between spins, and the ensuing magnetic phase transitions. These measurements

  575

NanoSQUIDs 575 can be performed by means of SQUID-​based susceptometers which are usually designed in a gradiometric fashion so as to be insensitive to homogeneous external magnetic fields, but sensitive to the imbalance produced by a magnetic sample located in one of the coils (see Fig. 19.4c–​d). χre and χim are directly accessible by applying a homogenous Bac via on-​chip excitation coils and lock-​in detecting the output response of the nanoSQUID. Alternatively, the rms flux noise √SΦ can be measured, as it is directly related to χim through the fluctuation-​dissipation theorem (Reim et al. 1986). The detection of χac demands large sensitivity, as the net oscillating polarization induced in the sample is, by far, much smaller than the total saturation magnetization. At best, broad-​band frequency measurements must be performed which provide, on the other hand, an easy way to filter out the 1/​f noise of the SQUIDs, therefore improving the effective sensitivity of the sensor. Frequencies are usually restricted to ~1 MHz, mainly limited by the room-​temperature amplifiers and the FLL circuit.

19.2.2 NanoSQUIDs: design considerations The figure of merit for nanoSQUIDs applied to the detection of small magnetic moments is the spin sensitivity √Sμ = √SΦ/​ϕμ which is given in units of µB/​√Hz; µB is the Bohr magneton. √Sμ expresses the minimum magnetic moment that can be resolved per unit bandwidth. Here, the coupling factor ϕµ = Φ/​µ is defined as the amount of magnetic flux Φ coupled to the SQUID loop by a magnetic dipole, divided by its magnetic moment µ = |µ|. The magnitude of ϕμ depends on the SQUID geometry, particle position rµ (relative to the SQUID), and orientation êµ = µ/​µ of its magnetic moment. Optimum nanoSQUID devices shall therefore meet the condition of maximizing ϕµ while minimizing SΦ. Ketchen et al. (1989), to the best of our knowledge, were the first who gave an estimate of the coupling factor. Assuming that a magnetic dipole lies at the center of an infinitely thin loop with radius a, with êµ along the loop normal, the simple expression ϕμ = µ0/​(2a) was found.1 This yields a spin sensitivity √Sμ = (2a/​µ0 )√SΦ.2 With the definition of the classical electron radius re = (µ0/​4π)·(e2/​me) ≈ 2.8 fm, and with Φ0 = h/​2e and µB = eh/​(4πme), one finds µB/​Φ0 = 2re/​µ0. Hence, one can rewrite these results obtained by Ketchen et al. (1989) as ϕμ = (re/​a)·(Φ0/​ µB) ≈ (2.8 µm/​a)·(nΦ0/​µB) and √Sμ = (a/​re)·µB·√SΦ/​Φ0 ≈ (a/​2.8 µm)·µB·√SΦ/​nΦ0. Intuitively, it is easy to understand that better coupling would result if the particle were located close to the loop’s banks (Bouchiat 2009). A quantitative estimate of ϕµ might become, however, a difficult task in this near-​field regime (Tilbrook 2009). Under these circumstances, the cross-​section of the SQUID banks and 1  Ketchen et al. (1989) derived the expression for the flux Δϕ coupled by a magnetic moment m to the loop with radius a as Δϕ  =  2πm/​a in cgs units. In SI units this yields Δϕ = µ0m/​(2a) and hence ϕμ = Δϕ/​m = µ0/​(2a). 2  Ketchen et al. (1989) introduced the spin sensitivity S , which relates to our definition n of the spin sensitivity as Sn = √Sμ/​µB, that is, Sn has the units of number of spins (of moment µB) per √Hz. The Ketchen result Sn = a√SΦ/​(2πµB) in cgs units yields Sn = 2a√SΦ/​(µ0µB) and √Sμ = (2a/​µ0)√SΦ in SI units.

576

576  NanoSQUIDs and Small Magnetic Systems the flux focusing effect caused by the superconductor itself must be taken into account. ϕμ can be quantified by calculating the magnetic field at the position of the SQUID, originating from a magnetic moment μ at position rµ, and hence, the magnetic flux coupled to the SQUID. These calculations appear much easier if one exploits the fact that sources and fields can be interchanged. That is, one can also solve the equivalent problem of evaluating the magnetic field BJ(rµ), created by a circulating current J through the SQUID loop, at the position rµ of the magnetic dipole. Here, the normalized quantity bJ = BJ/​J does not depend on J. Following this approach one finds (Bouchiat 2009, Nagel et al. 2011)

φµ (rµ , êµ ) = êµ • bJ (rµ ).

(19.10)

This means, with the knowledge of BJ/​J, which depends on the SQUID geometry via the spatial distribution of the supercurrent density js circulating in the SQUID loop, one can easily calculate the coupling factor for any position and orientation of the magnetic dipole in three-​dimensional (3D) space.3 The field BJ can be calculated from js. This has been done for various types and geometries of nanoSQUIDs by numerically solving the London equations in equidistantly spaced stacks of 2D sheets, in order to take into account the finite thickness of the thin film SQUID structures (Nagel et  al. 2011, 2011a, 2013, Schwarz et al. 2013, 2015, Wölbing et al. 2013, 2014). According to these calculations, typical coupling factors ϕμ  =  10–​20 nΦ0/​µB can be achieved for magnetic dipoles at 10  nm distance from a constriction (~100–​200  nm wide and thick) in the SQUID loop.4 Fig. 19.5 shows examples of such simulation results calculated for YBa2Cu3O7 (YBCO) and Nb nanoSQUIDs described in Schwarz et al. (2013) and Nagel et al. (2011a), respectively. The direct experimental determination of ϕμ is quite difficult, as it requires ideally a point-​like particle with a well-​defined magnetic moment. However, the numerical simulation results mentioned above have been confirmed by measurements on spatially extended systems, such as a Ni nanotube (Nagel et al. 2013) or a Fe nanowire (Schwarz et al. 2015), by comparing the measured flux coupled to nanoSQUIDs with the calculated flux signals obtained by integrating the coupling factor over the finite volume of the nanotube or nanowire. Small SQUID sensors developed by Josephs-​Franks et al. (2000) and Gallop et al. (2002) were also coupled to a ferromagnetic iron STM tip, which was scanned over the sensor’s surface while 3 The current J through an infinitely thin wire, forming a loop with radius a in the x–​y plane and centered at the origin, induces a field BJ = µ0J/​(2a)·êz, at the center of the loop. Hence, for a magnetic dipole placed at the origin rµ = 0 and pointing in z-​direction, êµ = êz, Eq. 19.10 yields ϕμ = êz·BJ(rµ)/​J = µ0/​(2a), the same result as derived by Ketchen et al. (1989). 4 We note that taking into account realistic SQUID geometries, the coupling factor depends significantly on the width of the SQUID loop, film thickness d, and λL. For example for a magnetic dipole centered at a circular SQUID loop with inner radius a = 500 nm, outer radius R = 2 µm, and d = λL = 100 nm one finds ϕμ = 3.5 nΦ0/​µB, a factor 1.6 smaller ϕμ as obtained from the Ketchen et al. (1989) result (with R = a = 500 nm); ϕμ decreases further with decreasing ratio d/​λL.

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NanoSQUIDs 577

Fig. 19.5  Calculated coupling factor ϕμ vs position of a magnetic dipole pointing in the x-​direction. Left: YBCO nanoSQUID, with Au layer on top. (a) SEM image showing SQUID hole and constriction (not acting as a JJ) in the (x,y) plane. Dashed line indicates location of (x,z) plane for which data are shown in (b); it also indicates the position of the linescan ϕμ(x) shown in (c). (b) Contour plot of ϕμ vs dipole position (x,z). Dashed lines indicate position of the linescans shown in (c) and (d). The black and gray rectangles indicate the YBCO and Au films, respectively. (c) Horizontal linescan ϕμ (x) at a distance of 10 nm above the Au layer. Right graph: Vertical linescan ϕμ(z) at the center of the constriction. Reprinted with permission from (Schwarz et al. 2013). Copyright (2013) American Chemical Society. Right: Nb nanoSQUIDs. Main graphs show contour plots ϕμ(x,z) for (a) a magnetometer and (b) a gradiometer. Nb structures are indicated by black rectangles; dashed lines indicate position of linescans ϕμ(x) [above (a)] and ϕμ(z) [right graphs]. Insets show SEM images. Reprinted with permission from (Nagel et al. 2011a). Copyright [2011], AIP Publishing LLC. recording the SQUID output in open-​loop configuration (Josephs-​Franks et al. 2003). In this way, they were able to detect experimentally, for the first time, the SQUID response as a function of the position of the magnetic sample. The thermal white flux noise, on the other hand, depends on geometrical parameters through the loop inductance L, but also on junction parameters such as I0, R, and C. The SΦ(L) dependence (Eq. 19.9) implies that SΦ can be simply reduced by decreasing the loop dimensions. However, apart from considering the constraints on βC and βL by doing so (which will affect the choice of junction parameters), one also has to take care of a possible increase in the kinetic inductance Lk when the thickness and width of the loop structures are reduced to a length scale comparable to or even smaller than λL. Hence, to improve the spin sensitivity one has to find a compromise between improved coupling and deterioration of flux noise (via an increased Lk) upon shrinking the cross-​section of the SQUID loop. A detailed investigation of this problem has been reported for YBCO nanoSQUIDs based on grain boundary JJs (Wölbing et al. 2014).

578

578  NanoSQUIDs and Small Magnetic Systems

19.2.3 NanoSQUIDs based on sandwich-​like junctions This category includes both SIS junctions (S: superconductor, I: thin insulating barrier, oxidized metal typically) and SNS junctions (N: normal metal barrier). SIS junctions have rarely been adapted to the fabrication of submicrometric JJs. Main drawbacks include their large intrinsic and parasitic capacitance due to their large area and the surrounding metallic layers, respectively. Additionally, the operation of these JJs in large magnetic fields is only possible with careful alignment of the field perpendicular to the junction plane, as an in-​plane field in the 1–​10 mT range can easily suppress the critical current due to the Fraunhofer-​ like modulation of Ic(B). On the other hand, this technology, in particular for Nb/​Al-​AlOx/​Nb trilayers, has the advantage of being very well developed and reproducible, leading to high-​quality JJs with controllable critical current densities. Additionally, metallic layers are usually used to shunt these junctions, for lowering βC to yield non-​hysteretic IVCs, albeit at the cost of also lowering the characteristic voltage Vc  =  I0R. The absence of hysteresis offers the advantage of operating the SQUID as a flux-​to-​voltage converter, using advanced readout techniques such as the FLL. The first SQUID device designed to measure the magnetic field created by magnetic NPs was based on micrometric Nb–​NbOx–​Pb edge junctions (Ketchen et al. 1984). These JJs (with I0 ~ 15 µA, R ~ 3 Ω, βC ~ 0.1) were connected in parallel through two oppositely wound loops, giving rise to a microgradiometer. The flux noise of this susceptometer (with L ~ 80 pH, βL ~ 1.3) was measured in the white noise region at 4.2 K, giving 0.84 µΦ0/​Hz1/​2. This susceptometer was operated attached to the mixing chamber of a dilution refrigerator. The output signal was measured in open-​loop configuration and amplified by an rf SQUID preamplifier circuit. Magnetic susceptibility measurements on small spin systems were reported and will be reviewed in Sec. 19.5.2. Very similar devices based on trilayer Nb/​Al-​AlOx/​Nb technology (√SΦ = 0.7 µΦ0/​Hz1/​2 at 4 K) were adapted for use in scanning SQUID microscopes by Gardner et al. (2001) and Huber et al. (2008). Sensors incorporated two oppositely wound pick up loops, field coils, and modulation coils so that the SQUID could operate in a FLL. The sensor’s substrate was cut by polishing, leading to ~20 µm loop-​to-​tip distance. A capacitive approach control was used to monitor the probe-​to-​substrate distance. These microsusceptometers were largely improved by using a terraced cantilever obtained through a multilayer lithography process. In this way the pickup loop stands above the rest of the structure, lying at just 300 nm above the substrate. Additionally the pickup loop diameters were reduced down to 600 nm using focused ion beam (FIB) milling (Koshnick et al. 2008). Another interesting example is SQUID sensors fabricated at the PTB Berlin (Drung et al. 2007) and commercialized through Magnicon. Different families of sensors are available, based on single-​and two-​stage readout configurations, all based on Nb/​Al-​AlOx/​Nb JJ technology. All of them include APF to increase VΦ and modulation coils for FLL operation. In the two-​stage versions the input signal is coupled to a voltage-​biased front-​end SQUID inductively coupled to a

  579

NanoSQUIDs 579 SSA. High-​input inductance sensors incorporate an intermediate double transformer loop with gradiometric design that provides an overall input inductance ~1 μH. This contrasts to 2 nH for devices without an intermediate loop, for which the input signal is directly coupled to the front-​end SQUID via four single-​turn gradiometric coils connected in parallel. SQUIDs are non-​hysteretic down to sub-​ kelvin temperatures (βC ~ 0.25) and exhibit √SΦ = 800 nΦ0/​Hz1/​2 at T = 4.2 K. Broad-​ band SQUID microsusceptometers (Martínez-​ Pérez et  al. 2009, 2010)  were fabricated from PTB sensors by using focused ion beam induced deposition (FIBID) of superconducting material with W(CO)6 as precursor gas (Sadki et al. 2004). This technique allowed converting the intermediate double transformer loop into a susceptometer inductively coupled to the front-​ end SQUID and modifying the gradiometric microSQUID itself for directly coupling the sample’s response (Martínez-​Pérez et al. 2011) (Fig. 19.6a). SQUID-​based microsusceptometers with improved reflection symmetry were later produced by the PTB (Drung et al. 2014, Schurig 2014). The sensitivity was boosted by defining a nanoloop (450  nm inner diameter, 250  nm linewidth) by FIB milling in one of the pickup coils (Fig. 19.6b). These sensors exhibited an extremely wide bandwidth of operation (1 mHz–​1 MHz). They were operated inside the mixing chamber of a dilution refrigerator at temperatures 0.013–​5 K for the investigation of microscopic crystals of SMMs and magnetic proteins. These measurements will be reviewed in Sec. 19.5.2. Submicrometric Nb/​AlOx/​Nb JJs in a cross-​type design were recently used by Schmelz et al. (2012, 2015) for fabricating miniaturized SQUIDs. The key advantage of cross-​type JJs over conventional window-​type JJs is the elimination of the parasitic capacitance surrounding the JJ, which becomes increasingly important upon reducing the JJ size. At T = 4.2 K, 0.8 × 0.8 µm2 JJs (I0 ~ 30 µA, C ~ 40 pF) show non-​hysteretic IVCs, if they are shunted with a AuPd layer (R ~ 24 Ω, yielding βC ~ 1). Sensors are also produced with an integrated Nb modulation coil. Square-​shaped washer-​type SQUIDs with minimum inner size of 0.5 µm have an inductance of a few pH (βL ~ 0.06) and exhibit a peak-​to-​peak modulation Vpp ~ 0.3 mV. SQUIDs are operated in liquid helium and read out with a low-​noise SQUID preamplifier, yielding √SΦ = 66 nΦ0/​Hz1/​2 in the white noise region. (a)

Field coils

(b)

JJs

100 µm

100 µm

Fig. 19.6  (a) SEM image of a SQUID-​ based microsusceptometer fabricated through FIB milling and FIBID of W (after Martínez-​ Pérez et  al. 2011). (b) SEM image of a SQUID microsusceptometer in which a nanoloop is patterned in the pickup coil (inset) through FIB milling. SEM images courtesy of J. Sesé.

580

580  NanoSQUIDs and Small Magnetic Systems A technique to produce submicrometer SNIS JJs and nanoSQUIDs was presented by Granata et al. (2013). This technique uses a Nb/​Al-​AlOx/​Nb trilayer to pattern a 3D SQUID loop (0.2 µm2) using FIB milling and anodization. The resulting JJs have an area of approximately 0.3×0.3 µm2 and are intrinsically shunted by the relatively thick (80 nm) Al layer, leading to non-​hysteretic IVCs with Ic ~ 80 µA, R ~ 1.8 Ω. The smallness of the SQUID loop leads to L = 7 pH (βL ~ 0.5). Measurements at 4.2 K yield VΦ ~ 3 mV/​Φ0 and √SΦ ~ 0.68 µΦ0/​Hz1/​2 in the white noise region. A different approach is based on the use of SNS JJs. Using a Nb/​HfTi/​Nb trilayer process, submicrometric (200 × 200  nm2) junctions can be obtained through e-​beam lithography and chemical-​mechanical polishing (Hagedorn et al. 2006). This process has been used to produce nanoSQUIDs (Nagel et al. 2011a, Wölbing et al. 2013) where bottom and top Nb wires (250 nm linewidth; 200 and 160  nm thick, respectively) are connected by two JJs with a 24  nm thick HfTi barrier. The latter can be varied to modify the desired critical current density. The junctions (Ic ~ 100 µA, Vc ~ 50–​60 µV) are intrinsically shunted, showing non-​hysteretic IVCs at 4.2 K.  The resulting SQUIDs offer much flexibility in terms of design. Both series-​and parallel-​gradiometers and single SQUID loops have been realized (Nagel et al. 2011a, Wölbing et al. 2013, Bechstein et al. 2015). Devices have been patterned in a washer-​type geometry (inner hole size 500 nm × 500 nm) or in a microstrip-​type geometry (inner hole size 1.6 µm × 225 nm), for which the loop plane lies parallel or perpendicular to the junction’s (substrate) plane, respectively (Fig. 19.7). The loop inductance exhibits typical values of a few pH (βL ~ 0.2). A key advantage of the microstrip-​type SQUIDs is the fact that a magnetic field applied in the plane of the SQUID loop is perpendicular to the JJ (and substrate) plane; in this way the field-​induced suppression of Ic can be avoided. It has been shown that magnetic fields up to 0.5 T can be applied while degrading only marginally the performance of the JJs (Wölbing et al. 2013). On-​chip flux biasing of the devices is easily possible, in order to operate them in a FLL. The SQUID transfer function reaches values within 100–​200 µV/​Φ0. The SQUIDs are typically operated at 4.2 K and the output signal is read out using a commercial SQUID amplifier or a SSA. Flux noise in the white noise region as low as ~110 nΦ0/​Hz1/​2 is obtained. Based on numerical solutions of the London Fig.  19.7 SEM images of Nb nanoSQUIDs with Nb/​HfTi/​Nb JJs (200 × 200 nm2; indicated by white rectangles). (a)  Square-​shaped SQUID loop in the substrate plane, with inner loop size 500 × 500  nm2. (b)  Microstrip-​type SQUID loop perpendicular to the substrate plane. Arrows indicate flow of bias current Ib and modulation current Imod. SEM images courtesy of O.F. Kieler.

(a)

(b)

Ib Ib

Imod 1 µm

Imod 1 µm

  581

NanoSQUIDs 581 equations for ϕµ, this yields a spin sensitivity of just ~10 for a magnetic dipole 10 nm away from the SQUID loop. Magnetization measurements on magnetic nanotubes have been performed successfully and will be summarized in Sec. 19.5.1. By combining three mutually orthogonal nanoSQUID loops, a 3-​axis vector magnetometer has been realized very recently (Martínez-​Pérez et al. 2016). Such a device allows to simultaneously detect the three components of the vector magnetic moment μ of a magnetic NP placed at a specific position, and subjected to an applied magnetic field along the substrate normal (z-​direction) for magnetization reversal measurements. Two microstrip-​type Nb nanoSQUIDs, as described above, are sensitive to fields in the x-​ and y-​directions, respectively. A third SQUID with gradiometric layout is sensitive to the local stray field from a magnetic NP along the z-​direction. Simultaneous operation of all three nanoSQUIDs in such devices in a FLL has been demonstrated at 4.2 K in fields up to 50 mT, with a white flux noise below 250 nΦ0/​Hz1/​2. Numerical simulations of the coupling factors showed that the three orthogonal components of the magnetic moment of a magnetic NP (placed in the center of one of the gradiometer loops) can be detected with a relative error flux below 10%. Such a device can provide important information on the magnetic anisotropy of a single magnetic NP. µB/​Hz1/​2

19.2.4 NanoSQUIDs based on constriction-​like junctions In superconducting films, Josephson coupling arises along a constriction having dimensions below or similar to the temperature-​dependent coherence length ξ of the superconductor (Likharev 1979). These kinds of junctions were first investigated by Anderson and Dayem (1964) and, for this reason, receive also the name Dayem bridges. NanoSQUIDs based on constriction-​like junctions (cJJs) typically have a simple planar configuration. These SQUIDs exhibit some drawbacks, such as their low reproducibility. Moreover, cJJs usually have hysteretic IVCs due to the heat dissipated for bias currents above Ic, which impedes SQUID operation in the conventional current-​biased mode as a flux-​to-​voltage transducer. To avoid hysteresis, cJJs can be operated within a limited temperature range close to Tc, where Ic is reduced. Additionally, cJJs exhibit unconventional and multivalued CPRs if the length of the constriction is larger than ξ. This makes it difficult to optimize the SQUID performance by resorting to results obtained from the RCSJ model. Moreover, the large kinetic inductance of the constriction can dominate the total inductance L of the SQUID, which prevents improving the flux noise via reducing the size of the SQUID loop. On the other hand, cJJ-​based nanoSQUIDs can be fabricated relatively easily from thin film superconductors, for example, Al, Nb, or Pb, through one-​step electron-​beam or FIB nanopatterning techniques. Additionally, the use of nanometric-​thick films and the smallness of the constriction make these SQUIDs quite insensitive to in-​plane magnetic fields and yield large coupling factors if magnetic particles

582

582  NanoSQUIDs and Small Magnetic Systems (a) Bext

Fig.  19.8  cJJ-​based nanoSQUIDs. (a)  Schematic view; a magnetic NP with magnetic moment µ is located close to one constriction where the coupling is maximum. (b)  SEM images of a Nb microSQUID (left) and a Nb nano SQUID (right). The left graph shows the microSQUID with a Ni wire on top and the nanoSQUID on the same scale, for comparison. Graph (b) Reproduced with permission from (Wernsdorfer 2009). All rights reserved. © IOP Publishing [2009].

µ

(b)

2000 nm

200 nm

are placed close to the constriction. It is exactly the small size of cJJs that makes them promising candidates for the fabrication of nanoSQUIDs with single-​spin sensitivity (see Fig. 19.8a). The fabrication of nanometric cJJ-​based SQUID sensors was first reported in the 1980s (Voss et al. 1980). With the goal of reducing the shot noise characteristic of SIS junctions, Voss et al. fabricated Dayem bridges with linewidths down to 30 nm using e-​beam lithography on Nb. The first nanoSQUIDs, with loop size ~1 µm2, exhibited Ic = 30 µA, a quite large L = 150 pH, and flux noise ~370 nΦ0/​ Hz1/​2 at 4.2 K. The production of nanoSQUIDs devoted to the investigation of small magnetic systems was pioneered by Wernsdorfer et al. (1995); for reviews see Wernsdorfer (2001, 2009). Fig.  19.8b shows examples of a cJJ-​based Nb micro-​and nanoSQUID. These SQUIDs (Hasselbach et al. 2000) were produced from Nb and Al films using e-​beam lithography. Typical geometric parameters were 1 µm × 1 µm inner loop area, 200 nm minimum linewidths, and film thicknesses ~30 nm. Due to the small film thickness, the kinetic inductance for Nb SQUIDs dominates the total inductance. The size of the constrictions was approximately 30  nm wide and 300 nm long, so for Nb significantly exceeding ξ. This fact leads to a highly non-​ideal CPR (Likharev 1979, Faucher et  al. 2002)  and therefore non-​ideal Ic(Φ) dependence with strongly suppressed critical current modulation depth for Nb cJJ SQUIDs. Furthermore, the kinetic inductance of the constrictions can be a few hundred pH, dominating the overall inductance of the devices (Faucher

  583

NanoSQUIDs 583 et  al. 2002). Impressively large magnetic fields could be applied parallel to the nanoSQUID loops up to 0.5 and 1 T for Al and Nb, respectively. The total flux noise was calculated by measuring the critical current noise giving ~40 µΦ0/​Hz1/​2 and ~100 µΦ0/​Hz1/​2 for Al and Nb, respectively (Hasselbach et al. 2000). Due to hysteresis these nanoSQUIDs were operated as flux-​to-​current converters (see Sec. 19.3). These sensors allowed the vastest realization of true magnetization measurements (Sec. 19.5.1) and were also implemented into probe tips to perform scanning SQUID microscopy (Hasselbach et al. 2000, Veauvy et al. 2002). For the latter purpose the sensor’s substrate was cut using a dicing machine and a mesa was defined by means of reactive ion etching so that the SQUID was only 2–​3 µm away from the apex of the mesa, which leads to a smallest vertical distance between the SQUID and a sample surface of ~0.26 µm. Similar devices with a SQUID-​to-​sample distance of 420 nm have been demonstrated to work in a setup with 40 mK base temperature (Hykel et al. 2014). These devices were combined with a force microscope, based on the use of a mechanically excited quartz tuning fork. Following the success of the approach described above, many groups have tried to improve the performance of constriction-​based nanoSQUIDs. Thermal hysteresis can be efficiently suppressed by covering the device with a normal metallic layer, which provides resistive shunting and acts as a heat sink. Lam and Tilbrook (2003) developed ultra-​small nanoSQUIDs by e-​beam lithography on a 20 nm thick Nb film covered by a 25 nm thick Au layer. The loop size was 200 nm × 200 nm and the constriction widths were in the range 70–​200 nm. The inductance L ~ 15 pH was determined from ΔIc/​Ic ~ 15% (Ic = 460 µA). Thanks to the Au layer, these devices did not show any hysteresis at temperatures above 1 K, allowing their operation in the conventional current bias configuration. Flux noise measured at 4.2 K with commercial SQUID readout electronics gave ~5 µΦ0/​Hz1/​2, increasing by about 15% when operating in a magnetic field of 2 mT (Lam 2006). Field operation up to a few hundred mT was improved by reducing the hole dimensions down to 100 nm × 100 nm and the largest linewidths down to 250 nm (Lam et al. 2011). Preliminary experiments were performed on ferritin nanoparticles attached to the nanoSQUID constrictions (Vohralik and Lam 2009). However, the magnitude of the flux change observed in some cases (up to 440 µΦ0) was larger than that expected for a ferritin NP located at optimum position (up to 100 µΦ0). Hao et  al. (2008) produced low-​noise nanoSQUIDs by FIB milling of a 200 nm thick Nb layer covered by 150 nm thick amorphous W. This technique led to the definition of a nanoloop with 370 nm inner diameter, intersected by two nanobridges (65 nm wide and 60–​80 nm long), and critical currents in the range 50–​300 µA. The nanoSQUIDs were non-​hysteretic at 5–​9 K, allowing their operation with current bias in open-​loop or FLL configuration. Flux noise measured using an SSA gave 200 nΦ0/​Hz1/​2 at 6.8 K. Recently, the same group investigated the possibility of extending the temperature operation range down to the sub-​Kelvin level (Blois et al. 2013). This was achieved by using superconducting

584

584  NanoSQUIDs and Small Magnetic Systems Ti films, inversely proximized by Au layers to reduce Tc. These SQUIDs (with 40 nm wide and 120 nm long constrictions) exhibited no hysteresis within 60 mK < T < 600 mK. Experimental values of VΦ = 27 µV/​Φ0 and white flux noise 1.1 µΦ0/​Hz1/​2 (L = 212 pH) were reported. These devices allowed the detection of the magnetic signal produced by a 150 nm diameter FePt nanobead having 107 µB at 8 K (Hao et al. 2011). The nanobead response was investigated by applying magnetic fields up to 10 mT perpendicularly to the nanoSQUID loop operated in open-​loop configuration. Measurements exhibited a clear shift coming from the magnetic hysteresis of the sample. This hysteresis was not observed in the absence of the nanobead. Unshunted nanometric SQUID loops (down to 150 nm inner hole size) were also fabricated using FIB milling. Troeman et al. (2007) reported nanopatterning of Dayem bridges (80 nm wide, 150 nm long) in a 50 nm thick Nb layer. It was observed that the Ga implantation depth could reach values of 30 nm, suppressing the superconducting properties of Nb. At T = 4.2 K, devices with Ic < 25 µA were non-​hysteretic and exhibited VΦ up to 160 µV/​Φ0. These nanoSQUIDs could be operated in a conventional current-​bias mode and their white flux noise achieved ~1.5 µΦ0/​Hz1/​2. Further examples of unshunted and hysteretic devices can be found in Granata et al. (2009). Nanoloops with inner diameters ~200 nm and Dayem bridges with sizes down to 280 nm long and 120 nm wide were produced by e-​beam lithography onto Nb layers. More recently, Hazra et al. (2013) demonstrated that hysteretic nanoSQUIDs made of Al–​Nb–​W layers (2.5 µm inner loop size; 40 nm wide, 180 nm long cJJs) could be operated with oscillating current bias and lock-​in readout at T < 1.5 K. In this configuration Ic is considerably reduced due to the inverse proximity effect of W on Nb. The typical non-​sinusoidal CPR of cJJs comes closer to the sinusoidal CPR of ideal point contacts when reducing their dimensions down to ξ (Likharev 1979). This effect might be strengthened by implementing nanobridges of variable thickness. Here, the thicker superconducting banks should serve as phase reservoirs, while the variation in the superconducting order parameter should be confined to the thin part of the bridges, yielding enhanced nonlinearity of the CPR (Vijay et al. 2009). Bouchiat et al. (2001) reported on the implementation of cJJ-​based nanoSQUIDs by local anodization of ultrathin (3–​6.5  nm thick) Nb films using a voltage-​biased atomic force microscope (AFM) tip. This technique allowed producing constrictions (30–​100 nm wide and 200–​1000 nm long) and variable thickness nanobridges by further reducing the constriction’s thickness down to a few nanometers (within a ~15 nm long section of the constriction). The latter exhibited ΔIc/​Ic twice as large as the former, evidencing a behavior closer to that resulting from an ideal CPR. Vijay et al. (2010) produced Al nanoSQUIDs based on 8 nm thick and 30 nm wide cJJs with variable length (l = 75–​400 nm). The cJJs were either connected to superconducting banks of the same thickness (“2D devices”) or to much thicker (80  nm) banks (“3D devices”). For 3D devices with l ≤ 150 nm ≈ 4ξ, the measured Ic(Φ) curves indicate a CPR which is close to that for an ideal short metallic weak link. Both 2D

  585

NanoSQUIDs 585 and 3D devices were shown to be fully operative up to in-​plane magnetic fields of 60 mT (Antler et al. 2013). Such nanoSQUIDs are operated as flux-​dependent resonators (Hatridge et al. 2011, Levenson-​Falk et al. 2013). This measurement setup will be described in Sec. 19.3.4. An alternative approach to realize nanoSQUIDs with variable thickness bridges has been reported recently by Hazra et  al. (2014). Here, suspended Al nanobridges (25 nm thick, 233 nm long, 60 nm wide) connect to Nb(30 nm)/​ Al(25  nm) bilayer banks to form a nanoSQUID (2.5 µm-​in-​diameter loop). These devices have the advantage of using cJJs from a material (Al) with a relatively large ξ, while maintaining relatively high Tc and critical magnetic field in the superconducting banks forming the SQUID loop. A very interesting issue regarding the measurements of magnetic systems is the capability of sensors to work upon the application of large magnetic fields. For the application of nanoSQUIDs to the investigation of magnetization reversal of magnetic NPs, the nanoSQUIDs have to be operated up to the switching fields of the NPs, which can be in the Tesla range (see Sec. 19.5.1). When working with SQUIDs based on constrictions, the applied magnetic field is limited by the superconductor’s critical field. For fields applied along any direction within the plane of the SQUID loop, the use of very thin superconducting layers can increase the effective critical field. Following this idea 3–​5 nm thick cJJ Nb nanoSQUIDs were fabricated. Supporting in-​plane fields up to 10 T, these sensors proved to be well suited for measuring magnetization curves of microcrystals of Mn12 SMMs (Chen et al. 2010). These SQUIDs exhibit, however, pretty large kinetic inductances, leading to large flux noise (~10−​4 Φ0/​Hz1/​2). More promising is the use of new materials with large upper critical fields, such as boron-​doped diamond (Mandal et al. 2011). Micrometric SQUIDs based on 100 nm wide constrictions in 300 nm thick films were demonstrated to operate up to impressive fields of 4 T applied along any direction. These devices were, however, hysteretic due to heat dissipation. Flux sensitivity was determined from the critical current uncertainty, giving 40 µΦ0/​Hz1/​2. So far we have reviewed some examples of nanoSQUIDs made from low-​ Tc superconductors. The fabrication of Dayem bridges from high-​Tc superconductors such as YBCO is much more challenging due to its reduced ξ ~ 1 nm. However, Arpaia et  al. (2014) recently succeeded in the fabrication of YBCO Dayem bridges with 50 nm × 50 nm cross-​section and 100–​200 nm length. These JJs exhibit large critical currents of a few mA at 300 mK. NanoSQUIDs based on this technology were fabricated and preliminary measurements showed an impressive white magnetic flux noise of 700 nΦ0/​Hz1/​2 at 8 K. Finally, an important breakthrough in the field of nanoSQUIDs applied to scanning microscopy was achieved recently with the implementation of the SQUID-​on-​tip (SOT) (Finkler et al. 2010, Vasyukov et al. 2013). This device is based on the deposition of a nanoSQUID directly on the apex of a sharp quartz pipette (see Fig. 19.9). Al, Nb, or Pb nanoSQUIDs based on Dayem bridges are shadow-​evaporated in a three-​angle process, without requiring any lithography

586

586  NanoSQUIDs and Small Magnetic Systems (a)

Quartz tube

(b)

Nb ∅238 nm

Au

Fig.  19.9  SQUID-​on-​tip (SOT). (a) Schematic of a sharp quartz pipette with superconducting leads, connecting to the SOT at the bottom end; inset shows magnified view. (b)  SEM image of a Nb SOT having a diameter of 238 nm. Reprinted by permission from Macmillan Publishers Ltd:  Nature nanotechnology (Vasyukov et al. 2013), copyright (2013).

Nb

Nb or Pb

200 nm Bridges

or milling steps. Special care must be taken for fabricating the so far most sensitive Pb sensors, which require the use of a He cooling system during deposition. This procedure led to the smallest nanoSQUIDs fabricated so far with effective nanoloop diameters down to 50 nm. The fact that the nanoSQUIDs are located on a sharp tip reduces minimum probe-​to-​sample distances to below 100 nm, boosting enormously the spatial resolution of the microscope. Although these nanoSQUIDs exhibit hysteretic IVCs, operation with voltage bias and readout with an SSA enable the detection of the intrinsic flux noise of the devices. The SOTs can be operated in large magnetic fields up to ~1 T (limited by the upper critical fields of the superconducting materials). So far, flux biasing to maintain the optimum working point during continuous external field sweep is not possible. By adjusting the external magnetic field to values which yield large transfer functions, these devices exhibit extraordinary low flux noise levels down to 50 nΦ0/​Hz1/​2. For a magnetic dipole located at the center of the loop with orientation perpendicular to the loop plane (assuming an infinitely narrow width of the loop, the approximation used by Ketchen et al. 1989), this translates into a spin sensitivity of 0.38 µB/​ Hz1/​2, the best spin sensitivity reported so far for a nanoSQUID. A device capable of distinguishing in-​plane and out-​of-​plane magnetic signals was also reported (Anahory et  al. 2014). This is achieved by using a V-​shaped apex of a pipette with θ-​shaped cross-​section to form a three-​JJ SQUID (3JSOT) with two oblique nanoloops in parallel. Field components can be decoupled by tuning the sensitivity of this device to the two orthogonal components of the magnetic field.

19.2.5 NanoSQUIDs based on proximized structures Due to the proximity effect, a normal metal put into good contact between superconducting electrodes acquires some of their properties. This leads to the

  587

NanoSQUIDs 587 opening of a minigap in the density of states of the normal metal. Andreev pairs can propagate along relatively long distances at low temperatures, carrying information on the macroscopic phase in the superconductor. In the long (short)-​ junction regime, when the Thouless energy of the metal is larger (smaller) than the superconducting energy gap, the junction properties will be governed by the properties of the normal metal (superconductor). The first dc SQUID built with long proximized junctions was based on the use of a CNT intersecting an Al ring (Cleuziou et al. 2006). A gate-​modulated supercurrent was demonstrated and flux-​induced modulation of the critical current (a few nA) was observed at mK temperatures. The goal was to exploit the small cross-​section of the CNT itself (~1 nm2) to provide an optimal coupling factor for molecular nanomagnets attached to it. An experimental proof-​of-​principle of such a CNT-​based magnetometer is, however, still missing. A  micrometric dc SQUID built with graphene proximized junctions (50 nm long, 4 µm wide) was also reported (Girit et  al. 2009). Flux-​induced Ic modulation was observed but, as in the previous example, no noise performance of the device was reported. Micrometric dc SQUIDs containing two normal metal bridges as weak links have also been reported. Nb/​Au/​Nb and Al/​Au/​Al-​based devices showed IVCs with a pronounced hysteresis, arising from the heat dissipated through the normal metal after switching (Angers et al. 2008). SQUIDs with shorter Cu nanowires (280–​370 nm long, 60–​150 nm wide, 20 nm thick) enclosed within a V ring were non-​hysteretic and exhibited VΦ ~ 0.45 mV/​Φ0 at 240 mK (Ronzani et al. 2013). NanoSQUIDs based on proximized InAs nanowires (~90  nm diameter, 20 or 50 nm long) were also reported (Spathis et al. 2011). The resulting JJs were in the intermediate length regime, and these nanoSQUID devices showed VΦ ~ 45 µV/​Φ0 at 244 mK (see Fig. 19.10a). A different kind of interferometer consists of a superconducting loop interrupted by a normal metal island. A  magnetic field applied to the loop varies the phase difference across the normal metal wire, allowing flux-​modulation of the minigap. This behavior can be probed using an electrode tunnel-​coupled to the normal metal island (see Fig. 19.10b) providing, therefore, a flux-​modulated electric response similar to that obtained with conventional dc SQUIDs.

(b)

(a)

S Clean barriers

Tunnel barrier

1 µm

N

200 nm

Fig. 19.10  (a) SEM image of a SQUID sensor consisting of a proximized highly doped In As nanowire enclosed within a V ring (after Spathis et al. 2011). (b) Scheme of a SQUIPT device. The inset shows a SEM image of the SQUIPT core; a normal metal probe is tunnel-​ connected to a proximized Cu island enclosed within an Al ring. SEM images courtesy of F. Giazotto and S. D’Ambrosio.

588

588  NanoSQUIDs and Small Magnetic Systems This device received the name Superconducting Quantum Interference Proximity Transistor (SQUIPT), for being the magnetic analog to the semiconductor field-​ effect transistor. SQUIPTs were pioneered by Giazotto et  al. (2010) using Al loops and Cu wires (~1.5 µm long, ~240 nm wide). These magnetometers were further improved by reducing the length of the normal metal island down to the short-​junction limit, leading to a much larger minigap opening. By choosing proper dimensions of the normal metal island, such sensors do not exhibit any hysteresis down to mK temperatures (Jabdaraghi et al. 2014, Ronzani et al. 2014) and can be voltage-​or current-​biased, providing impressive VΦ values of a few mV/​Φ0. SQUIPTs are in an early stage of development, still showing a very narrow temperature range of operation limited to the sub-​Kelvin level. On the other hand, they exhibit record low dissipation power of just ~100 fW (Ic ~ pA, Vout ~ 100 mV) and should achieve flux noise levels of just a few nΦ0/​Hz1/​2. The latter has not been determined experimentally yet because it is limited by the intrinsic voltage noise of the room-​temperature amplifiers.

19.2.6 NanoSQUIDs based on YBCO grain boundary junctions Due to the extremely short ξ of high-​Tc superconductors, the fabrication of Dayem bridges or trilayer JJs based on thin insulating layers is very difficult. Fortunately, Josephson coupling arises naturally at grain boundaries (GB). Grain boundary junctions (GBJs) can be fabricated, for example, by epitaxial growth of high-​Tc cuprate superconductors on bicrystal substrates or biepitaxial seed layers (Hilgenkamp and Mannhart 2002, Tafuri and Kirtley 2005, Tafuri et al. 2013). Although micrometric SQUIDs based on GBJs have been produced (Koelle et  al. 1999), the miniaturization of high-​quality GBJs is challenging, because of degradation of the material due to oxygen loss. NanoSQUIDs made of high-​Tc GBJs are, on the other hand, very attractive due to their large critical current densities (~105 A/​cm2 at 4.2 K) and huge upper critical fields (several tens of T). YBCO nanoSQUIDs based on GBJs were recently fabricated by FIB milling (Nagel et  al. 2011, Schwarz et  al. 2013, 2015). Devices consist of 50–​300  nm thick YBCO epitaxially grown onto a SrTiO3 bicrystal (24° misorientation angle) and covered by typically 60 nm thick Au serving as resistive shunt and to protect the YBCO film during FIB milling. Typical sizes are 200–​500 nm for the inner hole and 100–​300 nm GBJ width (Fig. 19.11a). The high quality of the junctions leads to large Ic up to ~1 mA (Vc ~ 0.5–​2 mV) at 4.2 K. Devices are non-​hysteretic and can be operated from sub-​Kelvin temperatures up to ~80 K. Even more interesting, large magnetic fields can be applied perpendicular to the GBJ in the substrate plane, leading to a minimum degradation of the Ic modulation for fields up to 1 T, and a ~30% reduction at 3 T (Schwarz et al. 2013). In addition, via a narrow constriction (width down to ~50 nm) patterned in the loop, a modulation current Imod can be applied to operate the devices at their optimum working point

  589

NanoSQUIDs 589 GB

Ib

130 nm

I mod 90 nm H

Ib

130 nm

(b) 102 1/2 S1/2 ) Φ (µΦ0/Hz

(a)

101 100

10–1

I mod

Experiment Fit T = 4.2 K

45 nΦ0/Hz1/2

10–2 100 101 102 103 104 105 106 107 ƒ (Hz)

Fig. 19.11  YBCO nanoSQUID. (a) SEM image showing SQUID loop (400  × 300 nm2), intersected by 130 nm wide grain boundary (GB) JJs; the GB is indicated by the vertical dashed line. Arrows indicate flow of bias current Ib across the GBJs and modulation current Imod across the 90 nm wide constriction (not acting as a JJ). (b) Rms flux noise of optimized YBCO SQUID, measured in open-​loop mode. Dashed line is a fit to the measured spectrum with white noise as indicated by the horizontal line. [after Schwarz et al. (2013, 2015), copyright (2013) American Chemical Society.] in FLL mode. The currents flowing through the constriction are kept below its critical current, so the constriction is not acting as a weak link. The constriction is also the position of optimum coupling of a magnetic NP to the SQUID loop (see Fig. 19.5a). Numerical simulations based on London equations for variable SQUID geometry provided expressions for L and ϕµ (via Eq. 19.10) for a magnetic dipole 10  nm above the constriction, as a function of all relevant geometric parameters. Together with RCSJ model predictions for the white flux noise at 4.2 K, an optimization study for spin sensitivity has been performed. For films with λL = 250 nm an optimum thickness dopt = 120 nm is found. For smaller d, the increasing kinetic inductance contribution to the flux noise dominates over the improvement in coupling. For optimum βL ~ 0.5 and d = dopt, the spin sensitivity decreases monotonically with decreasing constriction length lc (which fixes the optimum constriction width wc). For lc and wc of several tens of nm, an optimum spin sensitivity of a few µB/​Hz1/​2 is predicted in the white noise limit (Wölbing et al. 2014). For an optimized device with small inductance L ~ 4 pH (d  =  120  nm, lc  =  190  nm, wc  =  85  nm, width wJ ~ 180  nm, and length lJ  =  350  nm of the bridges straddling the GB, loop size lc × lJ), direct readout measurements of the magnetic flux noise at 4.2 K gave 50 nΦ0/​Hz1/​2 at 7 MHz (close to the intrinsic thermal noise floor), which is amongst the lowest values reported for dc SQUIDs so far (see Fig. 19.11b). With a calculated coupling factor ϕµ = 13 nΦ0/​µB, this device yields a spin sensitivity of 3.7 µB/​Hz1/​2 at 7 MHz and 4.2 K (Schwarz et al. 2015). Due to the extremely low white noise level, 1/​f-​like excess noise dominates the noise spectrum across the entire bandwidth of the readout electronics.

590

590  NanoSQUIDs and Small Magnetic Systems Bias reversal can only partially eliminate this excess noise, which deserves further investigation.

19.3 Measurement techniques using nanoSQUIDs NanoSQUIDs were conceived to directly sense the stray magnetic field created by a nanoscale magnetic sample. Changes in its magnetization or magnetic susceptibility lead to a change in the magnetic flux threading the nanoloop that must be quantified. In the following we provide a summary of different nanoSQUID readout methods that can be used to accomplish this goal.

19.3.1 NanoSQUID as a flux-​to-​voltage transducer In the simplest approach, nanoSQUIDs are current-​biased so that they operate in the dissipative state. In this way, the output voltage is a Φ0-​periodic function of the magnetic flux that can be linearized using the FLL scheme as described in Sec. 19.1.4. This method requires, however, the use of SQUIDs exhibiting non-​hysteretic IVCs. There are only a few examples in the literature in which nanoSQUIDs, operated as true flux-​to-​voltage transducers, have been applied to the investigation of small magnetic systems. See for instance the experiments described in Sec. 19.5 (Awschalom et al. 1990, Mártinez-​Pérez et al. 2011a, Buchter et al. 2013).

19.3.2 NanoSQUID as a switching current detector SQUIDs based on unshunted Dayem bridges usually exhibit hysteretic IVCs, making the previously described operation scheme unfeasible. In this case one can exploit the fact that the critical current Ic is directly related to the change of magnetic flux threading the loop, that is, the nanoSQUID operates as a switching-​ current detector. Measurements are based on ramping a current until the SQUID switches to the dissipative state, producing a voltage drop. At this point the current is rapidly turned off and the flux-​dependent Ic is calculated considering the duration of the ramp (Wernsdorfer 2009). Combined with on-​chip modulation coils, this technique can also be used with a FLL scheme (Wernsdorfer 2009, Russo et al. 2012, Granata et al. 2013a). Sensitivity is in this case constrained by the accuracy in determining Ic, that is, the switching to the dissipative state, which is described by the escape of a particle from a potential minimum. Such a process can be thermally activated or quantum driven and is strongly influenced by electronic noise. For these reasons, a large number of switching events are needed to obtain sufficient statistics, limiting enormously the sensitivity.

  591

Measurement techniques using nanoSQUIDs 591

19.3.3 NanoSQUID as a threshold sensor In order to avoid Joule heating in the dissipative state, the nanoSQUID might be operated as a threshold sensor. For this purpose the nanoSQUID is current-​ biased very close to the switching point. If the magnetic flux threading the loop changes abruptly, due to the reversal of the magnetization of a NP placed nearby, the nanoSQUID is triggered to the dissipative state and a voltage drop will be measured (Wernsdorfer 2009). This and the previous technique were successfully applied by Wernsdorfer and co-​workers in combination with the application of sweeping magnetic fields parallel to the SQUID loop; for a review see Wernsdorfer (2009). Measurements up to large magnetic fields applied along any direction in space were also possible. For this purpose, the measurement procedure was divided into three steps. First, Bext is applied to saturate the particle’s magnetization along any desired direction. Second, the magnetic field was swept along the opposite direction up to a given value (Btest) and back to zero. To ascertain whether the latter procedure had led to the particle’s magnetization reversal a third check measurement was performed. For this purpose a conventional in-​plane field sweep was used. In this way, if the particle’s magnetization reversal is (not) detected in the third probing step one can conclude that Btest was above (below) the switching magnetic field Bsw. These steps can then be repeated several times to determine Bsw precisely. Notice that the second step can be performed at temperatures above the critical temperature of the nanoSQUIDs. Rather than tracing out full M(H) loops, this technique can be used to trace out the dependence of Bsw on the field direction and temperature (Jamet et al. 2001).

19.3.4 NanoSQUID with dispersive read-​out So far, we have discussed nanoSQUIDs operated in the voltage state or close to it. Such an operation scheme entails the dissipation of joule power that can eventually affect the state of the magnetic system under study. An elegant way to circumvent this problem is the operation of the SQUID as a flux-​dependent resonator (Hatridge et al. 2011, Levenson-​Falk et al. 2013). This approach has also the advantage of increasing enormously the bandwidth up to ~100 MHz. For this purpose, the nanoSQUID remains in the superconducting state and is treated as a flux-​dependent inductance connected in parallel to a capacitor. The resonance frequency of the circuit will depend on the total flux threading the nanoSQUID loop. This can be read out by conventional microwave reflectometry giving a direct flux-​to-​reflected phase conversion. These devices were operated in the linear regime, that is, using low-​power driving signals, and the spectral density of flux noise was estimated. For this purpose, the overall voltage noise of the circuit was estimated and scaled with the transduction factor dV/​ dΦ giving 140 nΦ0/​Hz1/​2 for a bandwidth going from dc up to 0.6 MHz and 290 nΦ0/​Hz1/​2 for frequencies up to 20 MHz. The noise performance can also be boosted considerably by taking advantage of the CPR non-​linearity, operating

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592  NanoSQUIDs and Small Magnetic Systems the nanoSQUID as a parametric amplifier. For this purpose, the driving power is increased so that the resonance peak is distorted, giving a much sharper dependence of the reflected phase on the flux threading the nanoloop. Under such circumstances impressive flux noise values of 30 nΦ0/​Hz1/​2 for a 20 MHz bandwidth were reported. To date, two different nanoSQUIDs have been operated as flux-​ dependent resonators. First, a device made of Al/​AlOx/​Al junctions (Hatridge et  al. 2011) and, second, a true nanoSQUID based on 3D cJJs (Levenson-​Falk et al. 2013) were produced by shadow mask evaporation. The latter device is expected to provide a much better magnetic coupling for magnetic NPs located close to the constrictions.

19.4 Particle positioning One of the key issues that still need to be improved concerns the particle manipulation close to the sensor’s surface. This point is particularly important since the magnetic signal coupled to any form of magnetometer strongly depends on the particle location with respect to the sensor itself. Although conceptually very simple, this problem has prevented the realization of true single-​particle magnetic measurements so far. This problem was originally solved by depositing small droplets containing suspended NPs over a substrate with many nanoSQUIDs. After solvent evaporation some of the NPs happen to occupy positions of maximum coupling. This is the so-​called drop-​casting method and was successfully applied by Wernsdorfer et  al. (1997) to measure 15–​30  nm individual Co NPs. The drawbacks of this technique are obvious; it requires the use of substrates containing hundreds of nanoSQUIDs. Many strategies have been developed to improve control of the positioning of magnetic NPs or SMMs on specific areas of nanoSQUID sensors. Among them, the use of scanning probes has emerged as a very promising approach. Other attractive possibilities are micro-​and nano-​fabrication techniques, for example, e-​beam lithography, electrodeposition, STM lithography, manipulation via an AFM tip, dip pen nanolithography (DPN), and focused electron beam induced deposition (FEBID), or methods involving chemical functionalization of the NP or the substrates (Bellido et al. 2012). Alternatively, tiny magnetic molecules such as SMMs can be handled exploiting “larger” carriers that are easily visible and manipulated in conventional scanning electron microscopes (SEMs). CNTs appear as promising tools for this purpose. SMMs have indeed been successfully grafted over or encapsulated inside CNTs, which were later used to infer their magnetic properties (Ganzhorn et al. 2013). Finally, we note that scanning SQUID microscopes could in principle be applied to the study of magnetic NPs deposited on surfaces (Kirtley 2009). The latter approach circumvents the problematic of locating magnetic systems close

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Particle positioning 593 enough to the sensor and allows, on the other hand, performing in situ reference measurements. Scanning SQUID microscopes with nanometric resolution have been applied to the study of edge currents in topological insulators (Nowack et al. 2013) or surface magnetic states, for example in oxide interfaces (Bert et al. 2011) or unpaired spins in metals (Bluhm et al. 2009). However, their use for the investigation of magnetic molecules or nanoparticles arranged on surfaces is still awaited. Particularly promising is the application of the SQUID-​on-​tip (SOT) to the study of such systems, due to its sensitivity, large range of magnetic field operation, the fact that magnetic coupling between the nanoSQUIDs and the nanoparticles can be easily optimized, and the possibility of performing reference measurements.

19.4.1  In situ nanoparticle growth and grafting NP grafting onto the nanoSQUID sensor itself was successfully applied by Wernsdorfer et al. (1995a). Magnetic NPs based on Co, Fe, or Ni were sputtered using low-​energy cluster beam deposition techniques onto substrates containing a large number of microSQUIDs. Alternatively, NP and Nb deposition was realized simultaneously so as to produce superconducting substrates in which nanometric clusters were embedded (Jamet et al. 2000). Similarly to the drop-​casting method, these techniques are not practical as many tens of nanoSQUIDs must be patterned and characterized. Nanometric control over the particle position can be achieved by means of nanolithography methods. Wernsdorfer et al. (1995) used this approach to define Co, Ni, TbFe3, and Co81Zr9Mo8Ni2 NPs with the smallest dimension 100 nm × 50 nm × 8 nm. Alternatively, the use of FEBID of high-​purity cobalt allows the definition of much smaller particles (sub-​10 nm) and arbitrary shapes located at precise positions with nanometric resolution. Here, a beam of accelerated electrons is focused on the surface of a sample. When a precursor gas, for example Co2(CO)8, is injected close to the sample, the electrons can decompose the molecules adsorbed on the surface, leading to local deposition of cobalt (Córdoba et  al. 2010). This technique has been successfully applied to the integration of amorphous Co nanodots onto YBCO nanoSQUIDs (see Fig. 19.12a).

19.4.2 Scanning probe-​based techniques A scanning probe, for example the tip of an AFM, can be used to manipulate very precisely the position of the NP. For this purpose, an area containing the particles of interest is first scanned in non-​contact mode to determine the position of the clusters. The microscope is switched then to the contact mode and the tip is used to literally “push” the NP to the desired position (Martin et al. 1998, Pakes et al. 2004). This technique was applied to improve the coupling between a nanoSQUID and Fe3O4 NPs (15 nm diameter) deposited via the drop-​casting method (Wernsdorfer 2009).

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594  NanoSQUIDs and Small Magnetic Systems (a)

Fig. 19.12  (a) SEM image of a YBCO nanoSQUID having a FEBID-​Co nano­ particle deposited on top of the constriction. (b) SEM image of a SiNi cantilever used to hold a nanodot over a Nb nanoloop. SEM images courtesy of J. Sesé.

(b)

1 µm 200 nm

The use of micro-​and nano-​manipulators installed inside SEM facilities also offers many possibilities. For instance, Hao et al. (2011) used a sharpened carbon fiber probe attached to a micromanipulator, to deposit a ~0.15 µm diameter single FePt particle onto a nanoSQUID. Alternatively, larger carriers can be used to manipulate the position of small NPs. As an example, microscopic SiNi cantilevers containing the NP of interest can be moved using a micromanipulator (Gella 2015) (see Fig. 19.12b). A very promising approach is dip pen nanolithography (DPN), called such in analogy with the traditional dip pen or fountain pen writing method. DPN uses an AFM tip as a “nib,” a solid-​state substrate as “paper,” and a solution containing the NP of interest as “ink.” The AFM tip is first coated with the desired solution and then brought into contact with the surface. Capillarity transport of the NPs from the tip to the surface via a water meniscus enables the successful deposition of small collections of molecules in submicrometer dimensions (Piner et al. 1999). Bellido et al. (2010) showed that this technique can be successfully applied to the deposition of dot-​like features containing monolayer arrangements of ferritin-​based NPs onto microSQUID sensors (Fig. 19.13a), allowing detection of their tiny magnetic susceptibility (Martínez-​Pérez et al. 2011a, see Sec. 19.5.2). Ferritin is a ubiquitous protein which transports and stores iron in the form of a nanometric (