Probing Unconventional Transport Regimes in Delafossite Metals (Springer Theses) 3031142438, 9783031142437

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Springer Theses Recognizing Outstanding Ph.D. Research

Philippa H. McGuinness

Probing Unconventional Transport Regimes in Delafossite Metals

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

Theses may be nominated for publication in this series by heads of department at internationally leading universities or institutes and should fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder (a maximum 30% of the thesis should be a verbatim reproduction from the author’s previous publications). • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to new PhD students and scientists not expert in the relevant field. Indexed by zbMATH.

Philippa H. McGuinness

Probing Unconventional Transport Regimes in Delafossite Metals Doctoral Thesis accepted by the University of St Andrews, St Andrews, Scotland

Author Dr. Philippa H. McGuinness Physics of Quantum Materials Max Planck Institute for Chemical Physics of Solids Dresden, Sachsen, Germany

Supervisor Prof. Dr. Andrew P. Mackenzie Physics of Quantum Materials Max Planck Institute for Chemical Physics of Solids Dresden, Sachsen, Germany School of Physics and Astronomy University of St Andrews St Andrews, UK

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-031-14243-7 ISBN 978-3-031-14244-4 (eBook) https://doi.org/10.1007/978-3-031-14244-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

For Arthur Clay (1933–2011) a real knackler and a true gentle man

Supervisor’s Foreword

In this excellently written thesis, Philippa H. McGuinness takes a two-pronged approach to investigating the chemical physics of two-layered metals, PdCoO2 and PtCoO2 , from the delafossite family. These remarkable materials have some of the highest electrical conductivities ever measured, something that could be due either to suppressed electron backscattering resulting from special features of the electronic structure or to an astonishing level of crystalline perfection in the host lattice. A virtuoso set of experiments described in one of the main chapters of the thesis proves that the latter explanation is correct: the conducting planes of the best crystals contain only one defect per hundred thousand atoms. Obtaining this important result required a number of innovations both in device fabrication for the experiments and in the analysis of the data, all of which are described in the thesis with great clarity. Philippa then describes the use of the crystals in a set of ingenious experiments studying non-local transport in microfabricated squares of varying size and purity. Again, the experimental results, surprising at first sight, are combined with elegant analysis that brings out the key underlying physics. Throughout her time in the group, perhaps the most impressive aspect of Philippa’s work has been the independence with which she has tackled some of the key conceptual issues that arose from her experiments. She thinks deeply about the underlying physics of what she does and has a sense of how to distil the essence of a problem that people usually only develop when they have far more experience. Using those skills, she turns excellent experimental results into outstanding pieces of work, as evidenced by the insights that she presents in this thesis. She is also an excellent team player, always willing to help her colleagues. We have been fortunate to have her in Dresden and St Andrews! St Andrews, UK August 2022

Andrew P. Mackenzie

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Abstract

This thesis describes investigations into the origins and nature of the remarkable electrical transport of the delafossite metals PtCoO2 and PdCoO2 using focused ion beam-based microstructuring techniques. These compounds are amongst the highest conductivity materials known, but questions remain regarding the origin of their ultralow resistivity and the effects of their properties on transport in unconventional regimes such as the ballistic regime. In the initial introductory chapters, I will review the key properties of both delafossite metals and the application of focused ion beam microstructuring to transport measurements within low resistivity materials. The experimental findings are split into two chapters. Initially, I will describe an investigation into the origins of the high conductivity by introducing defects to PtCoO2 and PdCoO2 through high-energy electron irradiation and observing the changes to the resistivity. These measurements demonstrate that the ultralow resistivity of the delafossite metals is the result of an extreme purity of up to 1 defect in 120,000 atoms, rather than backscattering suppression. In addition, I will report the effects of the defects on the electrical transport more broadly. Here, by examining the difference before and after irradiation, insight is gained into the origins of the unconventional magnetotransport of PtCoO2 . The other study uses PtCoO2 and PdCoO2 as test systems for the investigation of the effects of a non-circular Fermi surface on the transport within four terminal, square-shaped junctions inside the ballistic regime. These junction devices have been shown to be a sensitive probe of this regime in other materials, and I will demonstrate that the nearly hexagonal Fermi surface of the ultrapure delafossite metals results in not only strongly ballistic behaviour, present at a scale multiple times the mean free path, but also novel phenomena which are not seen with a circular Fermi surface.

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Acknowledgements

Working towards a Ph.D. is never truly a solo pursuit and this thesis would not have been completed without the support of many others in a wide variety of ways, times and places, to whom I would now like to express my gratitude. The first thanks go to my supervisor, Andy Mackenzie. I remember thinking on my first visit to Dresden and Swiss Saxony in a cold February that this Ph.D. would be an adventure in both physics and life and, thanks in no small part to the group culture Andy has established, both adventures have been very rewarding. Despite a very busy schedule, he was always very willing to find time to discuss physics with me and those discussions, even when they were during a hike in damp conditions, proved to be extremely motivating and valuable, especially in seeing the big picture when I was buried in the details. I also greatly appreciate all of his pastoral support and compassion over the years. Modern science is nearly always a collaborative venture and the electron irradiation studies described in this thesis would not have been possible without the hard work of a number of people, to all of whom I am thankful. In particular, I am grateful to Marcin Konczykowski and the fellow members of the Laboratory for Irradiated Solids with whom we have most closely worked to build up a rewarding collaboration, including Jeremie Lèfevre and Romain Grasset, all of whom worked extremely hard to facilitate some challenging experiments. Excellent and tireless technical support was provided by Olivier Cavani. Our understanding of the energetics of defect formation was greatly aided by the calculations of Cyrus Dreyer and careful TEM studies by Celesta Chang, in the group of Dave Muller, provided highly useful information on the frequency and nature of growth defects in delafossite materials. My initial introduction to the world of experimental condensed matter physics was via the capable hands of Veronika Sunko, and I appreciate not only her clear explanations but also her patience and dedication to answering any question I had on any physics topic, no matter how misguided it may have been. Our collaborative work to develop the initial electron irradiation protocols on delafossite materials involved many hours of testing solutions to significant challenges in a number of areas, including during beam times, and I greatly appreciated both Veronika’s efforts

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during these times and our joint establishment of the excellent tradition of postbeamtime wine and steak. I would also particularly like to thank Elina Zhakina, whose attitude to life has both made the hard days of my Ph.D. years easier and the joyful days even better. We shared countless hours discussing, measuring together and jointly developing protocols for making bespoke ballistic microstructures suitable for irradiation, for making the delafossite squares and for making demanding samples for a proposed collaboration with the group of Eli Zeldov. We also set up and performed multiple beam times together, in collaboration with local staff, for irradiation of samples discussed in this thesis and many others. The experimental work described throughout the thesis benefited hugely from this collaboration, but I also hope my future workdays involve as much shared laughter as my days working alongside Elina did. Seunghyung Khim’s fantastic work growing pure crystals was vital to all of the experiments described within this thesis and I appreciate both his work and his genial and collaborative attitude. Many of these experiments also benefited massively from the fantastic technical support available at the institute. Renate Hempel-Weber, Sebastian Seifert, Heike Rave, Markus König and the late Christophe Klausnitzer were all very keen not only on working with me to sort any equipment issues but also helped me in trying out a variety of new ideas, for which I am grateful. Complicated travel was aided greatly by Uta Prautzsch. I am also thankful to Burkhard Schmidt for the help with navigating theoretical physics, Apple computers and German bureaucracy. I spent a year of my Ph.D. studies in beautiful St Andrews and had a fantastic time in room 120 with Nat Mica, Steffi Matern, Matt Trott and Conor Jackson as well as enjoyable evenings with many others including Rhea Stewart, Matt Neat and Scott Taylor. I have many fond memories of coffee runs to Taste and Friday night drinks in Aikman’s. Igor Markovi´c introduced me to some great parts of Scotland and some great whiskies. I always felt very included as part of the department, despite not being attached to a lab there, and that is thanks to the friendliness of many of the staff and students, for which I am very grateful. Thanks must also go to the CM-CDT administrative staff past and present including Julie Massey, Wendy Clark, Debra Thompson and Christine Edwards, along with Chris Hooley. There was never a question which they were unable to answer and they worked hard to make my life as simple as possible, even with the complications of a Ph.D. based in two countries. In Dresden, I had a really great time due to the fantastic atmosphere in the group and especially its truly international nature. From canoeing to multicultural Friendsgiving dinners to pizza roulette and lots more, I have many fantastic memories with Maja Bachmann, Mark Barber, Dorsa Fartab, Cliff Hicks, Markus König, Kim Modic, Nabhanila Nandi, Hilary Noad, Ekta Singh, Veronika Sunko, Eteri Svanidze, Po-Ya Yang, Elina Zhakina and Haijing Zhang amongst lots of others. I’ve also had many fun moments with my officemates over the years: Jack Bartlett, Alexander Steppke, Toni Helm, Belén Zúñiga and Edgar Morales. To my latest officemate, Fabian Jerzembeck, thank you for introducing me to Karneval, driving on inferior roads and enduring conversations about places such as Whitehorse and Spa-Francorchamps.

Acknowledgements

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Two truly fantastic secondary school teachers, Ian Brockbank and Travis Holgate, opened my eyes not only to the joys and depths of higher level maths and physics but also to the value in challenging yourself and persevering when it gets tough. Without their gentle encouragement, I would not be writing these acknowledgments. I will always be very grateful to them both. All of this thesis was written during the COVID-19 pandemic and so I am especially grateful for the support of those friends and family who have been there through this particularly difficult year for us all. Many of those people are key workers and I would like to recognise the immense efforts of themselves and others in keeping everything going in truly exceptional times. You are all heroes. To Bridget, thank you for the daily support, coffee chats and cat photos. These have propelled me through many tough moments over the past 5 years. To Char and Bekka, we might not be in rainy Manchester anymore, but I know you are always just a message away. Thank you for all the laughter, compassion and understanding. I owe you all lots of coffee and cake when we can meet again. To Dexter and Lola, thank you for being by my side during the writing-up process. Finally, I am endlessly grateful to my family including my parents, Ally and Tony, and my brother Cameron, for their truly unconditional support. They have always encouraged me to pursue my dreams, no matter what they were, which sacrifices they had to make in their own lives or even if they involved them having to attend yet another physics lecture. One of the few silver linings of the pandemic for myself was unexpectedly being able to spend a significant amount of time with them, and I enjoyed that time greatly, even though I remain the only family member who will ever actually read the instructions. I also acknowledge the financial support granted as a student of the Scottish Condensed Matter Centre for Doctoral Training under grant no. EP/L015110/1 from the Engineering and Physical Sciences Research Council and from the Max Planck Society. The electron irradiation studies were supported by the EMIR&A French network (FR CNRS 3618) at their SIRIUS facility. The underpinning research data for this thesis are accessible at: https://doi.org/ 10.17630/f464e1b5-3f24-4ca8-bdca-180478f4b932.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 The Ultrapure Delafossite Metals PdCoO2 and PtCoO2 . . . . . . . . . . . . 2.1 Synthesis of Single Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fermi Surface and Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Electrical Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 11 12 14 17

3 Creation and Measurement of Microstructures of Ultrapure Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Focused Ion Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Basic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Ion Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Milling with the FIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Deposition of Materials Using the FIB . . . . . . . . . . . . . . . . . . 3.2 Interaction of FIB Ions with Materials . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Collision Cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Mechanisms of Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Estimating the Scale of Damage . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Mitigating Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Device Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Current Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Handling of the Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Typical Device Production Process . . . . . . . . . . . . . . . . . . . . . 3.4 Measurement of Low Resistivity Materials . . . . . . . . . . . . . . . . . . . . . 3.4.1 Common Mode Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Measurement Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 20 20 21 23 25 26 26 27 28 30 31 31 32 32 34 35 36 37 38

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4 High Energy Electron Irradiation of Delafossite Metals . . . . . . . . . . . . 4.1 Principles of High Energy Electron Irradiation . . . . . . . . . . . . . . . . . . 4.1.1 Distribution of Introduced Defects . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Sample Temperature During Irradiation . . . . . . . . . . . . . . . . . 4.1.3 Electron Irradiation of Other Ultrapure Materials . . . . . . . . . 4.2 The SIRIUS Electron Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Accelerator Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Microstructure Design for Irradiation Experiments . . . . . . . . . . . . . . 4.3.1 Effects of Irradiation on Typical FIB Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Epoxy-Free FIB Mounting Method . . . . . . . . . . . . . . . . . . . . . 4.4 In-situ Measurements During Irradiation . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 In-situ Resistivity Change at 2.5 MeV Electron Energy . . . . 4.4.2 Dependence on Electron Energy . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Theoretical Cross-Section Models . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Defect Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Comparison to the Unitary Scattering Prediction . . . . . . . . . . 4.4.6 Conclusions of the Energy Dependent Study . . . . . . . . . . . . . 4.5 Annealing of Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Examples Within Other Materials . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Annealing During Irradiation . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Annealing at Temperatures to Room Temperature . . . . . . . . . 4.5.4 Annealing at Temperatures Above Room Temperature . . . . . 4.6 Effects of Introduced Defects on Transport Properties . . . . . . . . . . . . 4.6.1 Matthiessen’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Magnetoresistance and Hall Resistance . . . . . . . . . . . . . . . . . . 4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Ballistic Transport in Square Junctions of Delafossite Metals . . . . . . . 5.1 Introduction to Ballistic Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Length Scales in Ballistic Physics . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Fabrication of Ballistic Regime Devices . . . . . . . . . . . . . . . . . 5.1.3 Some Previous Ballistic Regime Experiments . . . . . . . . . . . . 5.1.4 Previous Results in Delafossite Metals . . . . . . . . . . . . . . . . . . 5.2 Square and Cross Junction Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Van der Pauw Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Anisotropy of Conductivity Tensor . . . . . . . . . . . . . . . . . . . . . 5.2.3 Junctions in the Ballistic Regime . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Landauer-Buttiker Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 PtCoO2 and PdCoO2 Square Junctions . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Fabrication of Delafossite Squares . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Temperature Dependence of the Bend Voltage . . . . . . . . . . . . 5.3.3 Low-Temperature Bend Magnetoresistance . . . . . . . . . . . . . .

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Contents

5.3.4 Linearity of Bend Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 PdCoO2 Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Hall Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Landauer-Büttiker Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Calculation of the Bend Voltages . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Calculation of the Hall Voltage . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Decay of Ballistic Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Determination of Mean Free Path . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Decay with Square Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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114 115 117 120 123 125 127 127 129 132 132

6 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Appendix A: Decay of Bend Resistance Anisotropy with Increased Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Chapter 1

Introduction

An extremely large number of materials have now either been discovered in nature or synthesised within a laboratory: the inorganic material database Pearson’s Crystal Data holds structural data for over 200,000 different materials as of April 2022 [1]. Much effort is dedicated to measuring and interpreting the properties of these materials. In particular, there has been longstanding interest in characterising and understanding their ability to conduct electricity. The different form and strength of the interactions within each material lead to a wide range of possibilities: insulators, semiconductors, semimetals and metals are all found in nature. One manner in which to characterise this conductivity is by the mean free path, the average distance travelled by a conduction electron before it undergoes a collision which relaxes its momentum. In the overwhelming majority of materials, scattering from defects such as vacancies, missing ions in the lattice, or interstitials, ions displaced from their equilibrium position, ensures this length scale is very short, often only a few interatomic distances. A tiny fraction of materials, however, have an extremely high conductivity. Within these materials, the mean free path can reach near macroscopic scales of tens or even hundreds of microns. For most of history, membership of this category was limited to the ultraclean elemental metals, such as copper and gold. The defect concentration in high purity crystals of these materials is extremely small, limiting the amount of scattering and leading to mean free paths in the range of tens of nanometres at room temperature and millimetres at temperatures of a few Kelvin. Due in part to these conductivity properties, this handful of materials has had a disproportionate impact upon technology, particularly in communications. In the 20th century, however, it was realised that many technological applications required finding materials in which the conductivity could be controlled more easily than in these elemental metals. The high sensitivity of the conductivity of semiconductors to factors such as the applied electric field, the light level and their defect concentration led to them becoming the focus of such studies. This was initially within elemental 3D semiconductors such as germanium. However, various issues, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. H. McGuinness, Probing Unconventional Transport Regimes in Delafossite Metals, Springer Theses, https://doi.org/10.1007/978-3-031-14244-4_1

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1 Introduction

including around achieving a uniform distribution of the impurity dopant atoms needed for conductivity and the high current densities which would be required for technical applications led to a focus on conduction at the surfaces and interfaces of semiconductors. At such interfaces, the conduction electrons can be confined, by a number of techniques, to a thin, essentially two-dimensional, layer, leading to the formation of a 2D ‘electron gas’ (2DEG). Initially, the 2DEG at such surfaces was of a relatively low conductivity, such as that formed by applying an electric field at a Si-SiO2 interface, as used inside the transistors which are ubiquitous in modern technology. Later, however, developments in fabrication techniques enabled the production of semiconductor heterostructures, where 2DEGs are naturally formed at the interface between two different semiconductors with dissimilar band gaps, as reviewed in Refs. [2, 3]. This breakthrough lead to the use of higher conductivity materials, most notably those based on gallium arsenide, GaAs. Between the early 1970s and the late 1980s, the mobility of 2DEGs within AlGaAs-GaAs semiconductor heterostructures increased by a factor of over 10,000, largely due to the introduction of modulation doping, where the impurity dopant atoms which provide additional charge carriers are spatially separated from the conducting layer, dramatically reducing the level of scattering [4]. The associated long mean free paths, typically tens of microns, led to them and other ultrapure heterostructures becoming the new focus of research within high conductivity materials. This in turn resulted in many breakthrough fundamental physics experiments [5], such as the discovery of the fractional quantum Hall effect [6] and the realisation of quantum confinement phenomena such as quantum dots [7, 8]. In addition, technology based upon these ultrapure materials, such as the high electron mobility transistor, is already implemented in a number of real-world applications, for example within satellite receivers and mobile phones [9]. In the last few decades, however, due to the discovery of the cuprate high temperature superconductors in the 1980s, significant research effort has been focused on the opposite end of the spectrum of conducting materials, the highly correlated ‘bad metals’, where strong interactions and higher disorder levels can lead to a roomtemperature resistivity which is three orders of magnitude larger than that of copper. By 2010, however, micron scale mean free paths were being achieved in a new highly conductive 2D material, graphene [10, 11], resulting in intense study of its electronic transport due to the wide range of potential future applications. Even more recently, the Weyl and Dirac semimetals have emerged as another class of ultrahigh mobility materials [12, 13], with evidence suggesting that the origins of this behaviour are, in contrast to the low defect density of elemental metals, novel mechanisms which limit backscattering from defects [14]. Clearly, significant interest remains in investigating and understanding the electrical conductivity of an ever wider range of ultraclean materials. This thesis is devoted to the study of two compounds from a class of low resistivity materials distinct from all those described above: the delafossite metals PdCoO2 and PtCoO2 . Delafossites are layered oxides and the large majority are insulators or wide gap semiconductors. Remarkably, however, a small fraction are metals with a very low resistivity: the resistivity of PdCoO2 and PtCoO2 is smaller at room

1 Introduction

3

temperature than that of every metal except silver, copper and gold. This exceptional conductivity is especially surprising given that they are oxides, historically regarded as a low-conductivity class of materials. These two compounds, and the other highconductivity metallic delafossites, were discussed in detail in a recent review [15]. As will be examined in more depth in Chap. 2, PdCoO2 and PtCoO2 were first synthesised over fifty years ago, but many key aspects of their physics have only recently been determined. One reason for this is that, in addition to materials synthesis, a range of sophisticated experimental techniques are needed to fully establish the basic physical properties of new classes of metal. For electrical transport, characterisation of properties such as the Fermi surface and the Fermi velocity is vital to establish key quantities such as the mean free path. In the last few decades, the resolution of such characterisation techniques has improved remarkably, enabling complex materials to be studied with a precision only dreamed of at the time the simpler elemental metals were studied. For instance, ever higher magnetic fields are available for de Haas-van Alphen measurements, where oscillations of the magnetic susceptibility in the field are used to determine the Fermi surface, improving their ability to characterise these surfaces. Of particular relevance to the materials studied in this thesis are the improvements to angle resolved photoemission spectroscopy (ARPES). This technique, based on the photoelectric effect and especially useful for materials with quasi-2D electronic structure, can be used to directly characterise both the Fermi surface and the dispersion relation, therefore enabling properties such as the Fermi velocity to be determined [16, 17]. Due to resolution improvements, from factors such as the development of low energy laser ARPES [18] and improvements to the electron analyser, ARPES has emerged from being a niche technique a few decades ago to a major contributor to the characterisation of a wide range of materials. In addition to these improvements in spectroscopy, there have been advances in microstructuring techniques, via which structures can be created on the scale of a few microns or even nanometres. The advantages of such techniques for low-resistivity materials are two-fold. Firstly, they enable extremely precise electrical transport measurements within crystals of novel materials, which are frequently too small to allow electric contacts to be added by hand. Secondly, they allow for transport studies within the regime where the characteristic size of the device, such as the width of the conducting channel, is similar to the mean free path within the material. This regime, only accessible in high-conductivity materials, is known as the ballistic regime and has been shown to produce distinct and characteristic electronic transport behaviour. Historically, most microstructuring has been performed via electron beam lithography, often by producing nanometre-scale masks for use in the etching and deposition processes which define nanostructures. In recent years, however, the direct use of a focused ion beam (FIB) has become more common. In this technique, a highly-focused beam of ions scans in a set pattern across a crystal to remove material and thus create bespoke structures. Such structures can be created on the scale of nanometers to millimetres, depending on the specifications of the FIB. The FIB is a versatile tool and has been in use for several decades within the semiconductor industry [19] and for the preparation of biological microscopy samples [20], but only

4

1 Introduction

recently have FIBs been used to create devices for the characterisation of modern quantum materials. This novel usage, reviewed in Ref. [21], has enabled a number of measurements to be performed, such as examining Josephson junction physics and determining the critical current within iron-based superconductors, which would not have been possible without FIB-based techniques. These pioneering studies inspired further work across other material families. The delafossite metals are a class of material particularly well-suited to the application of these modern characterisation and microstructuring techniques. These measurements have shown not only that the delafossite metals are ultraclean, with lowtemperature mean free paths of up to 20 µm [22], but also that they have properties which make them distinct from all other ultraclean materials. Within these delafossites, the conduction is quasi-2D due to the large anisotropy of their electronic structure: the conductivity in the ab-plane is often a thousand times larger than the out-of-plane, c-axis, conductivity. In this sense, their transport properties have similarities with the 2D transport seen in semiconductor heterostructures, but there are also important differences. Unlike the semiconductor devices, where special fabrication techniques yield devices with one or at most a few conducting layers, the delafossite metals are essentially natural heterostructures of highly conductive metal layers interspersed by correlated electron insulator layers. They therefore contain thousands of parallel conducting layers in very close spatial proximity, separated by approximately 6 Å. In addition, their carrier density is more comparable to that of the 3D elemental metals. High quality ARPES measurements, as shown in Fig. 1.1 for PtCoO2 , have demonstrated that their Fermi surface is broadly cylindrical and hexagonal in cross-section, unlike the circular Fermi surfaces found in the 2DEGs and graphene. Initial resis-

Fig. 1.1 a A cross-section of the Fermi surface of PtCoO2 as measured by ARPES. b A comparison of the experimental data (ARPES) with the Fermi surface calculated by density functional theory (GGA+SO+U). Reprinted with permission of AAAS from P. Kushwaha, V. Sunko, P.J.W. Moll, L. Bawden, J.M. Riley, N. Nandi, H. Rosner, M.P. Schmidt, F. Arnold, E. Hassinger, T.K. Kim, M. Hoesch, A.P. Mackenzie and P.D.C. King, Nearly Free Electrons in a 5d Delafossite Oxide c The Authors, some rights reserved; Metal, Science Advances, Vol. 1, No. 9, pp. e1500692, 2015.  exclusive licensee AAAS. Distributed under a CC BY-NC 4.0 License (http://creativecommons. org/licenses/by-nc/4.0/)

1 Introduction

5

tivity measurements, made possible by the use of FIB microstructuring, show that this property may lead to unconventional transport effects which are not observed in other materials [22–24]. Studies of this class of material therefore offer a unique opportunity to probe the effects of properties such as a lower Fermi surface symmetry and metallic carrier densities on transport in ultrapure quasi-2D materials. While the existing spectroscopic and transport studies have successfully characterised many properties of the delafossite metals, a number of key questions remain unanswered. One of the most fundamental is: what is the underlying reason for their extreme conductivity, which is almost unique among all known ternary compounds? The history of ultraclean materials points to two likely explanations. Firstly, as is true for the highly conductive elemental metals, this could stem from an ultrapurity of the crystal, with defects being nearly absent. Alternatively, similarly to the Weyl and Dirac semimetals, defects may be present but the backscattering from these defects may be suppressed. Answering this question is vital, not only for understanding their fundamental physics, but also in order to guide the quest to grow crystalline, ultraclean delafossite thin films using techniques such as molecular beam epitaxy, something that is being attempted in several laboratories across the world but which has not yet been achieved [25–28]. In addition, the properties of delafossite metals suggest that they are an ideal model system for exploring new regimes of electrical transport in ultrapure materials. The hexagonal Fermi surface is of a higher complexity and lower symmetry than the circular Fermi surfaces within 2DEGs, but is far more accessible to theoretical modelling than the 3D Fermi surfaces of many elemental metals. The fact that the delafossite metals are also amenable to FIB microstructuring techniques enables experiments in precisely defined geometries for the first time in materials with metallic densities and non-circular Fermi surfaces. It is not known how these properties affect some of the characteristic transport signatures of the ballistic regime, such as those observed in the transport within micron scale junctions of other ultrapure materials. This thesis is devoted to an exploration of both of these topics. Introducing additional defects to the delafossite metals via high-energy electron irradiation sheds light on the origins of their extreme conductivity. Probing the transport in a FIB-defined square junction, with a similar scale to the mean free path, highlights the impact of the hexagonal Fermi surface, both on the transport behaviour within the ballistic regime and the range over which this regime is observed. The thesis is structured as follows: Chapter 2 outlines the properties of PdCoO2 and PtCoO2 which are most relevant for understanding the studies described within later chapters. Chapter 3 gives a basic overview of the FIB and the effects of FIB irradiation on the target material. The techniques used to create and measure the resistivity of microstructures made from the ultrapure delafossite metals are also outlined. Chapter 4 describes a series of electron irradiation experiments used to introduce additional defects to the delafossite metals, with the aim of measuring the impact of these additional defects on the resistivity and therefore determining the intrinsic

6

1 Introduction

defect concentration and the reason behind their extremely long mean free paths. The effect of these defects on the other transport properties of the material, such as the magnetotransport, is also studied. The two sections of the chapter are in some senses distinct, and could have been presented as two separate chapters by splitting off Sect. 4.6. The linking theme of the study of irradiated samples seemed strong enough for them to be presented as one single long chapter, but the reader can treat the two parts separately if they so wish. Chapter 5 investigates the transport within square-shaped junctions of PdCoO2 and PtCoO2 which were structured using the FIB to have a side length approximately equal to the electron mean free path, placing them within the ballistic regime. The observed unconventional transport effects are used to explore the impact of the hexagonal Fermi surface both on the behaviour within the ballistic regime and the range over which this regime can be observed. Chapter 6 gives conclusions and an outlook for future work. The two main results chapters, Chaps. 4 and 5, are designed to be self-contained. It therefore seems more natural to introduce the relevant theory within those chapters so that its link with the experimental results is made clear. For this reason there is no self-standing chapter devoted to background theory.

References 1. Villars P, . Cenzual K (2022) Pearson’s crystal data: crystal structure database for inorganic compounds (on DVD), ASM international 2. Alferov ZI (2001) Nobel lecture: the double heterostructure concept and its applications in physics. Electron Technol Rev Mod Phys 73(3):767–782 3. Harris JJ, Pals JA, Woltjer R (1989) Electronic transport in low-dimensional structures. Rep Prog Phys 52(10):1217–1266 4. Pfeiffer L, West KW, Stormer HL, Baldwin KW (1989) Electron mobilities exceeding 107 cm2 /Vs in modulation-doped GaAs. Appl Phys Lett 55(18):1888–1890 5. Beenakker C, van Houten H (1991) Quantum transport in semiconductor nanostructures. Solid State Phys 44:1–228 6. Tsui DC, Stormer HL, Gossard AC (1982) Two-dimensional magnetotransport in the extreme quantum limit. Phys Rev Lett 48(22):1559–1562 7. Ashoori RC (1996) Electrons in artificial atoms. Nature 379:413–419 8. Hanson R, Kouwenhoven LP, Petta JR, Tarucha S, Vandersypen LMK (2007) Spins in fewelectron quantum dots. Rev Mod Phys 79(4):1217–1265 9. del Alamo JA (2011) The high-electron mobility transistor at 30: impressive accomplishments and exciting prospects. In:Proceedings of the 2011 international conference on compound semiconductor manufacturing technology 10. Dean CR, Young AF, Meric I, Lee C, Wang L, Sorgenfrei S, Watanabe K, Taniguchi T, Kim P, Shepard KL, Hone J (2010) Boron nitride substrates for high-quality graphene electronics. Nat Nanotechnol 5(10):722–726 11. Bolotin KI, Sikes KJ, Hone J, Stormer HL, Kim P (2008) Temperature-dependent transport in suspended graphene. Phys Rev Lett 101(9):096802 12. Wang S, Lin B-C, Wang A-Q, Yu D-P, Liao Z-M (2017) Quantum transport in Dirac and Weyl semimetals: a review. Adv Phys: X 2(3):518–544

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13. Yan B, Felser C (2017) Topological materials: Weyl semimetals. Annu Rev Condens Matter Phys 8(1):337–354 14. Liang T, Gibson Q, Ali MN, Liu M, Cava RJ, Ong NP (2015) Ultrahigh mobility and giant magnetoresistance in the Dirac semimetal Cd3 As2 . Nat Mater 14(3):280–284 15. Mackenzie AP (2017) The properties of ultrapure delafossite metals. Rep Prog Phys 80(3):032501 16. Cattelan M, Fox N (2018) A perspective on the application of spatially resolved ARPES for 2D materials. Nanomaterials 8(5):284 17. Lv B, Qian T, Ding H (2019) Angle-resolved photoemission spectroscopy and its application to topological materials. Nat Rev Phys 1(10):609–626 18. Zhou X, He S, Liu G, Zhao L, Yu L, Zhang W (2018) New developments in laser-based photoemission spectroscopy and its scientific applications: a key issues review. Rep Prog Phys 81(6):062101 19. Giannuzzi L (2006) Introduction to focused ion beams: instrumentation. Theory, techniques and practice, Springer, US 20. Narayan K, Subramaniam S (2015) Focused ion beams in biology. Nat Methods 12(11):1021– 1031 21. Moll PJW (2018) Focused ion beam microstructuring of quantum matter. Annu Rev Condens Matter Phys 9(1):147–162 22. Nandi N, Scaffidi T, Kushwaha P, Khim S, Barber ME, Sunko V, Mazzola F, King PDC, Rosner H, Moll PJW, König M, Moore JE, Hartnoll S, Mackenzie AP (2018) Unconventional magneto-transport in ultrapure PdCoO2 and PtCoO2 . npj Quantum Mater 3(66) 23. Moll PJW, Kushwaha P, Nandi N, Schmidt B, Mackenzie AP (2016) Evidence for hydrodynamic electron flow in PdCoO2 . Science 351(6277):1061–1064 24. Bachmann MD, Sharpe AL, Barnard AW, Putzke C, König M, Khim S, Goldhaber-Gordon D, Mackenzie AP, Moll PJW (2019) Super-geometric electron focusing on the hexagonal fermi surface of PdCoO2 . Nat Commun 10:5081 25. Harada T, Fujiwara K, Tsukazaki A (2018) Highly conductive PdCoO2 ultrathin films for transparent electrodes. APL Mater 6(4):046107 26. Brahlek M, Rimal G, Ok JM, Mukherjee D, Mazza AR, Lu Q, Lee HN, Ward TZ, Unocic RR, Eres G, Oh S (2019) Growth of metallic delafossite PdCoO2 by molecular beam epitaxy. Phys Rev Mater 3(9):093401 27. Sun J, Barone MR, Chang CS, Holtz ME, Paik H, Schubert J, Muller DA, Schlom DG (2019) Growth of PdCoO2 by ozone-assisted molecular-beam epitaxy. APL Mater 7(12):121112 28. Hagen DJ, Yoon J, Zhang H, Kalkofen B, Silinskas M, Börrnert F, Han H, Parkin SSP (2022) Atomic layer deposition of the conductive delafossite PtCoO2 . Adv Mater Interfaces 9(12):2200013

Chapter 2

The Ultrapure Delafossite Metals PdCoO2 and PtCoO2

As described in Chap. 1, the delafossite metals are a new class of ultrahigh purity metals which can be seen as a model system for the study of electrical transport within low resistivity materials. In this chapter, some of their attributes will be reviewed in more detail. The goal is not to give a complete history of this set of materials, which has been summarised within a recent review [1], but to describe the properties which are most relevant in motivating and enabling the experimental studies described within later chapters. In the most general terms, delafossites are materials with the chemical composition ABO2 where the A is Pt, Pd, Cu or Ag and B is a transition metal, such as Co or ¯ Cr. The most common delafossite crystal structure, which has the space group R 3m, is shown in Fig. 2.1. The A sites are arranged in triangularly coordinated planes. Between these planes lies a triangularly coordinated layer of B-site cations, each of which sits in the centre of an oxygen octahedron. In addition, each of the oxygen cations is linearly co-ordinated to an A-site cation. The triangular co-ordination can lead to frustrated magnetism when the B-site cation is magnetic. The large majority of delafossites, like other layered oxides, are either semiconductors or insulators, but, in 1971, the first ultrapure delafossite metals were reported. In a set of three studies by Shannon and collaborators [2–4] which detailed the properties of multiple delafossites, nine new delafossites were synthesised. Two of the single crystals grown, PdCoO2 and PtCoO2 , had remarkably small resistivities in-plane at room temperature, now established to be 3.1 μcm in PdCoO2 and 1.8 μcm in PtCoO2 [5]. These values are records for oxide materials and, remarkably, are even smaller than the resistivity of elemental Pt or Pd. The resistivity was also shown to be very anisotropic, with the c-axis resistivity being up to three orders of magnitude larger than that in the plane. The in-plane conductivity is also extremely high at low temperature. At 5 K, the resistivity is as small as 7.5 ncm in PdCoO2 and 20 ncm in PtCoO2 , corresponding to mean free paths of around 20 µm and 5 µm respectively. These are comparable to values found in graphene and the 2D electron gases within ultrapure semiconductor © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. H. McGuinness, Probing Unconventional Transport Regimes in Delafossite Metals, Springer Theses, https://doi.org/10.1007/978-3-031-14244-4_2

9

10 Fig. 2.1 The delafossite crystal structure with the ¯ space group R 3m

2 The Ultrapure Delafossite Metals PdCoO2 and PtCoO2

A BO2

c a

b

heterostructures studied in the 1980s and 1990s described in Chap. 1, in both cases only after years of refinement of the sample preparation techniques. Mean free paths this long are typically only significantly exceeded within carefully prepared elemental metals, and it is highly unexpected to find these conduction properties in an oxide compound, especially given that the growth method does not involve any purification steps. Although the low room temperature resistivity and the resistive anisotropy were established in the pioneering work of Shannon and colleagues, surprisingly little investigation of these metals was undertaken for around two decades. For example, higher purity synthesis of PdCoO2 was not reported until 2007 [6]. Three other metallic delafossites are known to have high conductivities: PdRhO2 , PdCrO2 and AgNiO2 . The former was only synthesised in single crystal form in 2017 [7]. It has greater spin-orbit coupling due to the larger Rh atom and a slightly higher room temperature resistivity of 10 μcm, but most of the measured properties are qualitatively similar to those of PdCoO2 and PtCoO2 . PdCrO2 , unlike the three other non-magnetic compounds, is a frustrated antiferromagnet with a Néel temperature around 37 K [8]. In AgNiO2 , the conduction is based on the states of the transition metal Ni, resulting in physics somewhat different to that in the Pd- and Pt-based compounds. Although these three materials are mentioned here for completeness, none has been studied within the experiments in this thesis, so they will not be discussed further here. More details about them can be found in the review article, Ref. [1].

2.1 Synthesis of Single Crystals

11

2.1 Synthesis of Single Crystals The ultrahigh purity of these delafossite metals is surprising given the methods by which they are grown. In contemporary syntheses, both PdCoO2 and PtCoO2 are grown from ground powders in an evacuated quartz tube via the following reaction: (Pd, Pt)Cl2 + 2CoO → (Pd, Pt)CoO2 + CoCl2 as detailed in Refs. [7, 9]. During the reaction, the crucible is heated to high temperatures of several hundred degrees and gradually cooled down in a multiple step process. After this process, clusters of single crystal platelets are formed within the tube. The crystal size is known to depend on the ratio of the reactants and the temperature profile [7]. No annealing or other additional purification steps are necessary: the crystals as grown have the extremely high conductivities described above. This property, unusual even within ultrapure materials, makes understanding the origin of their high conductivity, as explored in Chap. 4, even more vital. A scanning electron microscopy (SEM) image of a typical PtCoO2 crystal is shown in Fig. 2.2. The size of PtCoO2 crystals in the ab-plane is usually below 1 mm, with PdCoO2 crystals being slightly larger. There are often hexagonal terraces on the surface of the crystal, as seen in Fig. 2.2. Regions without this thickness variation, which are necessary to facilitate accurate transport measurements, are usually less than 500 µm square, providing an upper limit on the size of transport devices. The crystal thickness along the c-axis is typically 5 µm for PtCoO2 and 15 µm for PdCoO2 . These dimensions, especially combined with the low resistivity, create difficulty for traditional methods of measuring the resistivity by causing the resistance of a hand-mounted sample to be very small at low temperatures. Indeed, it was not until the development of the FIB-based microstructuring techniques described in Chap. 3 that the resistivity was accurately measured.

Fig. 2.2 A scanning electron microscope (SEM) of a typical PtCoO2 crystal. Most of the growth edges reflect the triangular lattice symmetry. On the surface, there is a single linear and two hexagonal shaped growth terraces

a

200 μm a

12

2 The Ultrapure Delafossite Metals PdCoO2 and PtCoO2

2.2 Fermi Surface and Electronic Structure Another key property that makes the ultrapure delafossite metals appealing for an in-depth study is the relative simplicity of their electronic structure. The high conductivity elemental metals have 3D, sometimes multiple band, Fermi surfaces which can cause difficulty for modelling. Most of the other ultrapure materials have circular Fermi surfaces. As will be shown here, the complexity of the Fermi surfaces of PdCoO2 and PtCoO2 sits between these two limits. This allows the delafossite metals to be used to test the effects of the electronic structure and the Fermi surface on transport whilst enabling comparison with relatively simple theoretical models, such as will be performed in Chap. 5. The delafossite formal valence has been known to be A1+ B3+ O2− 2 since the initial reported synthesis [2]. In PdCoO2 and PtCoO2 , Co is in the low spin 3d 6 state, meaning that the compounds are non-magnetic, in contrast to Cr in PdCrO2 . Early photoemission studies determined that the states at the Fermi energy in PdCoO2 and PtCoO2 were primarily Pd/Pt based [12–14]. Specific heat measurements in PdCoO2 suggested Pd 4d-5s hybridisation, as the coefficient of the T-linear term lay between the typical values for s and d electron materials [13]. Similarly, Pt 5d-6s hybridisation was proposed for PtCoO2 [14]. Later X-ray absorption spectroscopy in PdCoO2 showed a metallic Pd spectrum and a correlated electron insulator Co spectrum [15]. In a simplified picture, therefore, these delafossite metals are natural heterostructures of high conductivity metallic layers and strongly correlated insulating transition metal oxide layers. The quasi-2D nature of the conduction lends itself naturally to angle resolved photoemission spectroscopy (ARPES) measurements. Such studies in PdCoO2 and PtCoO2 have shown their bulk Fermi surfaces to be single band, half filled, and with a broadly hexagonal cross-section, as shown in Fig. 2.3 [9, 16]. The PtCoO2 Fermi sur-

1.5

b)

PdCoO2

1.5 PtCoO 2 1

0.5

0.5 -1

k y (Å )

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Fig. 2.3 The cross-section of the Fermi surfaces of a PdCoO2 and b PtCoO2 using the harmonic expansions determined from photoemission and quantum oscillation measurements [9–11]

2.2 Fermi Surface and Electronic Structure

13

Fig. 2.4 a The electronic structure of PtCoO2 in the direction -K as measured by ARPES. b Dots showing the peak position of fits to the momentum distribution curves along the -K and -M direction. Linear fitting to this data allows for the calculation of the Fermi velocity. Reprinted with permission of AAAS from P. Kushwaha, V. Sunko, P.J.W. Moll, L. Bawden, J.M. Riley, N. Nandi, H. Rosner, M.P. Schmidt, F. Arnold, E. Hassinger, T.K. Kim, M. Hoesch, A.P. Mackenzie and P.D.C. King, Nearly Free Electrons in a 5d Delafossite Oxide Metal, Science Advances, Vol. 1, No. c The Authors, some rights reserved; exclusive licensee AAAS. Distributed 9, pp. e1500692, 2015.  under a CC BY-NC 4.0 License (http://creativecommons.org/licenses/by-nc/4.0/)

face has sharper corners and more rounded facets than that of PdCoO2 . The faceting present on both Fermi surfaces is suggestive of a d-orbital contribution. However, despite this hexagonal form, nesting and consequent density wave formation has not been observed in either material, possibly as the nesting wavevector would be incommensurate [1]. The ARPES studies also enabled measurement of the dispersion of the conduction band, such as shown in Fig. 2.4 for PtCoO2 , which was linear, sharp and steep in both cases. The Fermi velocities are consequently very large, in the case of PtCoO2 within 20% of the free electron value [9]. This demonstrates the nearly free electron nature of the conduction, which stems from the s-orbital contribution to the conduction band. The conclusions from ARPES are supported by de Haas van Alphen (dHvA) measurements by Hicks et al. [10] for PdCoO2 and Arnold et al. [11] for PtCoO2 . In both materials, two high frequencies were found, corresponding to neck and belly orbits, alongside a beating frequency and a difference frequency attributed to magnetic interaction. These oscillations had a 1/ cos(θ ) angular dependence where θ is the angle of the magnetic field to the z-axis, confirming the broadly cylindrical nature of the Fermi surface. The small deviation from this dependence was decomposed into harmonics describing the tiny warping in the z-direction, further confirming the nearly 2D nature of the Fermi surface and justifying the use throughout this thesis of a 2D

14

2 The Ultrapure Delafossite Metals PdCoO2 and PtCoO2

approximation when modelling the transport properties [10]. Lifshitz-Kosevich fits to the temperature dependent oscillation amplitudes determined the cyclotron masses to be around 1.5 me and 1.05 me for PdCoO2 and PtCoO2 respectively, where m e is the free electron mass, in agreement, within experimental error, with the value from ARPES of 1.14 me for PtCoO2 [9]. Band structure calculations have been performed to complement these experimental probes. Early density functional theory (DFT) studies, such as that by Eyert et al. [17] concentrated on showing the dominance of Pd 4d or Pt 5d orbitals at the Fermi energy. Importantly, they also showed that including spin-orbit coupling is necessary to prevent a second band crossing the Fermi level in PtCoO2 [9, 18]. However, the small electron masses established experimentally are inconsistent with a localised d-orbital. Later DFT calculations by Ong [19] showed a small Pd 5s contribution at the Fermi level for PdCoO2 , providing evidence for the hybridisation. In PtCoO2 , similar results were found for Pt 6s, especially when the calculations were performed in a local orbital instead of plane wave basis [9]. It is this combination of d and s electron physics, leading to a highly faceted Fermi surface occupied by nearly free electrons, which enables the novel transport behaviour in the ballistic regime which will be studied in Chap. 5. For both materials, a sizeable Co on-site U of around 6 eV is required within an LDA+U treatment in order to provide good agreement with the observed cyclotron masses and Fermi surface warping [9–11]. This suggests that the conduction is sensitive to the correlation in the insulating layers, despite the fact that the in-plane conduction occurs in the metallic Pt/Pd layers. Without this addition, the contribution of Co 3d at the Fermi level is overestimated and the system becomes less two-dimensional. This shows that the combination of both the highly correlated insulating layers and the highly conductive metallic layers, not just the latter alone, is important in producing the unique properties of the delafossite metals.

2.3 Electrical Transport Properties The history of electronic transport studies in the delafossite metals emphasises the importance of careful measurement in low resistivity materials with a high resistivity anisotropy. For example, when the current is injected from the top surface, the anisotropy ensures that the current penetrates the full depth of the crystal only after travelling an extended distance in the ab-plane. If this distance is not reached before the voltage contacts, the measured resistance is enhanced. This will be discussed in more detail in Sect. 3.3.1. Effects such as these can have a significant impact on the reported material properties. For example, in the first published synthesis of PtCoO2 , a 3 μcm room temperature resistivity [2] was reported. The first experiments on larger, higher purity single crystals by Kushwaha et al. [9] lowered this to 2.1 μcm, before the latest study by Nandi et al. [5], in which focused ion beam microstructuring techniques were used to ensure a homogeneous, full thickness current profile, resulted in a record for an

2.3 Electrical Transport Properties

15

oxide of 1.8 μcm at room temperature (Fig. 2.5). This demonstrates that only the most careful studies prevent spurious additional contributions to the resistivity. The growth of high-purity PdCoO2 crystals was first reported in 2007 [6]. Despite the motivation of this study being partly to search for superconductivity, no superconductivity has been reported in the metallic delafossites down to 15 mK. The temperature dependence of the PdCoO2 resistivity, both in the ab-plane and along the c-axis, was first carefully measured by Hicks et al. [10], as shown in Fig. 2.6. The behaviour in PtCoO2 , although not reported in such detail, is qualitatively similar. The large resistivity anisotropy confirms the quasi-2D nature of the conduction. For the c-axis resistivity, Fig. 2.6b, Fermi liquid behaviour occurs with an a + Ac T 2 + cT 5 temperature dependence where AC , a and c are all constants. However, the ab-plane resistivity, ρab , Fig. 2.6a, agrees better with an activated function a + be−T /T0 , where a, b and T0 = 165 K are fitted constants, than the typical a)

b) PdCoO2

3

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ρab (μΩcm)

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Fig. 2.5 The temperature dependence of the in-plane resistivity for a PdCoO2 and b PtCoO2 between 2 K and 300 K, adapted from [5]

Fig. 2.6 The temperature dependence below 30 K of the resistivity of PdCoO2 a in the ab-plane b along the c-axis. Note the different overall scale of the resistivity. Reprinted figure with permission from C. W. Hicks, A. S. Gibbs, A. P. Mackenzie, H. Takatsu, Y. Maeno and E. A. Yelland, Quantum Oscillations and High Carrier Mobility in the Delafossite PdCoO2 , Physical Review Letters, Vol.109, No.11, pp.116401, 2012. Copyright 2022 by the American Physical Society

16

2 The Ultrapure Delafossite Metals PdCoO2 and PtCoO2

Fermi liquid and Bloch–Grüneisen T 2 + T 5 dependence. There is a small resistivity upturn at temperatures below 10 K, which varies between crystals and is more prominent in PtCoO2 . This has an undetermined origin, but it does not show the weakening in a magnetic field that would be expected for the Kondo effect [20]. The cause of the activated behaviour is still not fully understood, but one possible explanation is phonon drag. In most materials, the phonons are at equilibrium. Normal electron-phonon collisions are able to relax the electron momentum, and often dominate the resistivity. However, in some materials, such as the alkali metals [21], the electron flow drags the phonons out of equilibrium and the processes which return the phonon distribution to equilibrium are slow. In this situation, normal electron-phonon collisions only transfer momentum between electrons and phonons without relaxing the momentum of the combined distribution. Therefore, they do not contribute to the resistivity. The temperature dependence of the resistivity is therefore determined by Umklapp electron-phonon scattering. If, as for PdCoO2 , if the Fermi surface is not close to the Brillouin zone boundaries, there is an energy barrier for this scattering process related to the minimum wavevector, kU , as shown in the inset of Fig. 2.6a. This leads to a characteristic temperature TU = c M kU /k B where  is the reduced Planck constant, c M is the speed of sound in PdCoO2 and k B is Boltzmann’s constant. Below this temperature, Umklapp scattering is suppressed and the temperature dependence of the resistivity is exponential. The Seebeck coefficient of PdCoO2 also shows evidence of phonon drag [22]. As well as these effects at zero field, the dominance of Umklapp scattering may be important in contributing to unconventional magnetoresistance properties, as will be explored in the defect introduction studies described in Chap. 4. As described previously, the resistivity-derived mean free paths of PdCoO2 and PtCoO2 at low temperature are around 20 µm and 5 µm respectively. The origin of these long mean free paths will be studied in Chap. 4. These high values are, however, in contrast with the Dingle mean free paths, determined from dHvA measurements to be 0.6 µm [10] and 0.3 µm [11]. This suggests that small angle scattering, which reduces the Dingle mean free path but only weakly affects the resistivity, is more frequent than large angle scattering. Additionally, as stated by Arnold et al. [11], the hexagonal Fermi surface further limits the impact of small angle scattering on the resistivity. Scattering between states on the same facet of the hexagon does not significantly change the electron trajectory, which determines the resistivity, but would still broaden the Landau levels and therefore decay the quantum oscillations. This potential preservation of electron trajectories even in the presence of scattering also has consequences for the transport behaviour on the scale of the mean free path within the ballistic regime, as will be explored in Chap. 5. In summary, PtCoO2 and PdCoO2 are layered oxide metals which, as-grown, have both a very small resistivity in-plane and a large resistivity anisotropy, leading to quasi-2D transport. The origin of this extreme conductivity is not fully understood and will be explored further in Chap. 4. Their Fermi surfaces are broadly hexagonal in cross-section and are formed from a single hybridised band with contributions from both s and d electron physics. The impact of this non-circular Fermi surface for transport on the scale of the mean free path will be studied in Chap. 5. For ease

2.3 Electrical Transport Properties

17

Table 2.1 Some properties of PdCoO2 and PtCoO2 most relevant to the electronic transport PdCoO2 PtCoO2 Source Resistivity at 300K (μcm) Resistivity at 5K (μcm) Mean free path at 5K (µm) 3D carrier density (cm−3 ) Fermi wavevector (m−1 ) Fermi velocity (ms−1 )

3.05

1.82

[5]

0.008

0.030

[5]

20

5

[5]

2.45 × 1022

2.42 × 1022

[1]

0.96 × 1010

0.95 × 1010

[10, 11]

7.5 × 105

8.9 × 105

[9, 10]

of future reference, Table 2.1 details some of the properties of PdCoO2 and PtCoO2 which are most relevant for later discussions.

References 1. Mackenzie AP (2017) The properties of ultrapure delafossite metals. Rep Prog Phys 80(3):032501 2. Shannon RD, Rogers DB, Prewitt CT (1971) Chemistry of noble metal oxides. I. Syntheses and properties of AB O2 delafossite compounds. Inorg Chem 10(4):713–718 3. Shannon RD, Prewitt CT, Rogers DB (1971) Chemistry of noble metal oxides. II. Crystal structures of platinum cobalt dioxide, palladium cobalt dioxide, copper iron dioxide, and silver iron dioxide. Inorg Chem 10(4):719–723 4. Shannon RD, Rogers DB, Prewitt CT, Gillson JL (1971) Chemistry of noble metal oxides. III. Electrical transport properties and crystal chemistry of AB O2 compounds with the delafossite structure. Inorg Chem 10(4):723–727 5. Nandi N, Scaffidi T, Kushwaha P, Khim S, Barber ME, Sunko V, Mazzola F, King PDC, Rosner H, Moll PJW, M. König, Moore JE, Hartnoll S, Mackenzie AP (2018) Unconventional magneto-transport in ultrapure PdCoO2 and PtCoO2 . npj Quantum Mater 3(66) 6. Takatsu H, Yonezawa S, Mouri S, Nakatsuji S, Tanaka K, Maeno Y (2007) Roles of highfrequency optical phonons in the physical properties of the conductive Delafossite PdCoO2 . J Phys Soc Jpn 76(10):104701–104701 7. Kushwaha P, Borrmann H, Khim S, Rosner H, Moll PJW, Sokolov DA, Sunko V, Grin Y, Mackenzie AP (2017) Single crystal growth, structure, and electronic properties of metallic delafossite Pd Rh O2 . Cryst Growth Des 17(8):4144–4150 8. Takatsu H, Yoshizawa H, Yonezawa S, Maeno Y (2009) Critical behavior of the metallic triangular-lattice heisenberg antiferromagnet PdCr o2 . Phys Rev B 79(10):104424(1)– 104424(7) 9. Kushwaha P, Sunko V, Moll PJW, Bawden L, Riley JM, Nandi N, Rosner H, Schmidt MP, Arnold F, Hassinger E, Kim TK, Hoesch M, Mackenzie AP, King PDC (2015) Nearly free electrons in a 5d delafossite oxide metal. Sci Adv 1(9):e1500692 10. Hicks CW, Gibbs AS, Mackenzie AP, Takatsu H, Maeno Y, Yelland EA (2012) Quantum oscillations and high carrier mobility in the delafossite PdCoO2 . Phys Rev Lett 109(11):116401

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2 The Ultrapure Delafossite Metals PdCoO2 and PtCoO2

11. Arnold F, Naumann M, Rosner H, Kikugawa N, Graf D, Balicas L, Terashima T, Uji S, Takatsu H, Khim S, Mackenzie AP, Hassinger E (2020) Fermi surface of PtCoO2 from quantum oscillations and electronic structure calculations. Phys Rev B 101(19):195101 12. Higuchi T, Tsukamoto T, Tanaka M, Ishii H, Kanai K, Tezuka Y, Shin S, Takei H (1998) Photoemission study on PdCoO2 . J Electron Spectrosc Relat Phenom 92(1–3):71–75 13. Tanaka M, Hasegawa M, Higuchi T, Tsukamoto T, Tezuka Y, Shin S, Takei H (1998) Origin of the metallic conductivity in PdCoO2 with delafossite structure. Phys B 245:157–163 14. Higuchi T, Baba D, Yokoyama Y, Hasegawa M, Takei H, Fukushima A, Shin S, Tsukamoto T (2003) Electronic structure of delafossite-type oxide PtCoO2 by resonant-photoemission spectroscopy. Jpn J Appl Phys 42:5698–5699 15. Noh H-J, Jeong J, Jeong J, Sung H, Park KJ, Kim J-Y, Kim H-D, Kim SB, Kim K, Min BI (2009) Orbital character of the conduction band of delafossite PdCoO2 studied by polarizationdependent soft x-ray absorption spectroscopy. Phys Rev B 80(7):073104(1)–073104(4) 16. Noh H-J, Jeong J, Jeong J, Cho E-J, Kim SB, Kim K, Min BI, Kim H-D (2009) Anisotropic electric conductivity of delafossite PdCoO2 studied by angle-resolved photoemission spectroscopy. Phys Rev Lett 102(25):256404 17. Eyert V, Frésard R, Maignan A (2008) On the metallic conductivity of the delafossites PdCoO2 and PtCoO2 . Chem Mater 20(6):2370–2373 18. Ong KP, Singh DJ, Wu P (2010) Unusual transport and strongly anisotropic thermopower in PtCoO2 and PdCoO2 . Phys Rev Lett 104(17):176601(1)–176601(4), 2010 19. Ong KP, Zhang J, Tse JS, Wu P (2010) Origin of anisotropy and metallic behavior in delafossite PdCoO2 . Phys Rev B 81(11):115120(1)–115120(6) 20. Nandi N (2019) Can hydrodynamic electrons exist in a metal? A case study of the delafossite metals PdCoO2 and PtCoO2 . Ph.D. thesis, Technischen Universität Dresden, Dresden 21. Bass J, Pratt WP, Schroeder PA (1990) The temperature-dependent electrical resistivities of the alkali metals. Rev Mod Phys 62(3):645–744 22. Daou R, Frésard R, Hébert S, Maignan A (2015) Large anisotropic thermal conductivity of the intrinsically two-dimensional metallic oxide PdCoO2 . Phys Rev B 91(4):041113(1)– 041113(5)

Chapter 3

Creation and Measurement of Microstructures of Ultrapure Materials

As described in the previous chapter, transport measurements of the ultrapure delafossite metals are challenging. One method to mitigate some of the issues is to use a focused ion beam (FIB) to create devices with a geometry specifically designed to maximise the resistance whilst also compensating for issues such as current inhomogeneities. This method is used within the electron irradiation studies described in the next chapter. In addition, FIB microstructuring enables devices with micron-scale dimensions to be made from delafossite materials in order to examine unconventional regimes of transport, as will be explored in Chap. 5. Although the FIB has existed since the mid 1970s, the initial uses were primarily for mask and circuit repair in the semiconductor industry [1], for imaging in the biological sciences [2] and for creation of transmission electron microscopy (TEM) samples [3]. In recent years, the FIB has begun to be used to structure modern quantum materials into bespoke devices, enabling measurements not possible with traditional techniques, as reviewed by Moll [4]. Since FIB microstructuring was a key foundation of much of the work presented in this thesis, the relevant techniques will be described in depth in this chapter. With any relatively new microstructuring technique, a number of areas must be addressed. Firstly, the minimum feasible device size, which is typically limited by the resolution of the technique, must be determined. As least as important is the extent of any damage which the technique imparts upon the material. Less prominent, but important for practical use, is the design flexibility and the procedure and total time required to produce a device, which often determine the reasonable limits of experimental studies. All of these factors will be discussed within this chapter. At the end of the chapter, the full FIB microstructuring process for a standard delafossite metal sample will be detailed alongside a description of the experimental configuration. Some key factors to consider when making resistance measurements of a low resistivity material, such as current inhomogeneities and common mode voltage, will also be discussed. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. H. McGuinness, Probing Unconventional Transport Regimes in Delafossite Metals, Springer Theses, https://doi.org/10.1007/978-3-031-14244-4_3

19

20

3 Creation and Measurement of Microstructures of Ultrapure Materials

3.1 The Focused Ion Beam In order to provide background for later discussion of aspects such as the resolution and the damage caused by FIB irradiation, a basic understanding of the key components of a FIB system, including the possible ion sources, is required. In this section, this will be provided alongside a brief description of the important aspects of the two key processes used to create the FIB structures used within this thesis: FIB milling and FIB assisted material deposition.

3.1.1 Basic Design The key features of a FIB are a source of ions and an evacuated column with electrodes to accelerate the ions, a set of apertures used to define the ion beam current, electrostatic lenses for focusing the beam and octupole plates for beam scanning. A schematic of a FIB column is shown in Fig. 3.1. The target sample is placed on a moveable stage within an evacuated chamber which has at least one ion detector for imaging. Both the column and the chamber are evacuated to a low pressure, around 10−8 and 10−6 Torr respectively, to prevent interaction between the ions and other particles. When the ions contact the sample surface, energy is transferred to the lattice atoms. If this energy is sufficient to overcome the surface binding energy, atoms will be sputtered, or removed, in a pattern determined by the beam scan pattern. The

Fig. 3.1 A schematic of a typical dual beam FIB-SEM system. For clarity, the FIB column is shown vertically but, in most systems, the SEM column is vertical

Ion source

Collimator lens

Beam path

Extraction/suppression electrodes Aligning octupole Aperture

Beam blanker Scanning octupole Objective lens Microchannel plate (MCP) Gas injection system Sample

M

SE

5-axis stage

3.1 The Focused Ion Beam

21

number of atoms removed per incident ion, the sputter rate, is controlled by factors such as the ion species, the acceleration voltage and the incidence angle of the beam. An ion column aims to provide a steerable ion beam with the smallest possible focal spot for a given current. This is achieved with electrostatic, rather than magnetic, lenses due to the high ion mass. The beam current is primarily determined by the apertures, with values from 1 pA to around 1 µA possible, depending on the ion source. A typical acceleration voltage is around 30 kV, with lower voltages sometimes used to decrease the level of damage by reducing the ion penetration depth. Most modern FIB systems are paired with a scanning electron microscope (SEM) for higher-resolution imaging, as shown in Fig. 3.1. The target area is held at the coincidence point, where the focal points of the two beams meet. The sample stage is rotated such that the surface is orthogonal to the SEM beam during electron imaging and to the FIB beam during milling of material. Additional components, such as a gas injection system used for FIB assisted material deposition, may be installed. A number of detectors are typically installed for use with both the SEM and FIB. These are usually sensitive to either electrons which are backscattered from the surface or secondary electrons/ions, which are those emitted from the sample during irradiation with the ion or electron beam. As well as the sample topology, the contrast in secondary electron images is primarily related to differences in the conductivity, whereas the secondary ion emission tends to differentiate between materials.

3.1.2 Ion Sources An important component within a FIB system is the ion source. When designing a source, the aim is to increase the reduced brightness, the beam current per unit area at a given acceleration bias, whilst reducing the energy spread of the ions, which determines the resolution-limiting chromatic aberration [5]. Most FIB systems use a liquid metal ion source (LMIS), but, in recent years, other sources have also begun to be widely used. The LMIS, shown schematically in Fig. 3.2a, was introduced in 1979 following experiments in ion emission [7, 8]. A metal, nearly always gallium, is held in a reservoir above a sharp tip, around 10 µm in radius, made from tungsten. The gallium is melted by a heater, which wets the tip. A set of electrodes produce an electric field, resulting in the formation of a 5 nm radius Taylor cone at the tip due to the combined effect of the electrostatic forces, the surface tension and the droplet pressure. As the voltage increases, field ionisation causes ions to be emitted, with the typical acceleration voltage being 5–30 keV. An advantage of this source over a solid source is that it is self-sharpening, with the cone remaining sharp even after ions are emitted. The brightness of a LMIS is relatively high, but the energy spread is affected by Coulomb forces. The limitations of a LMIS are most significant as the beam current I is increased. As shown in Fig. 3.3, above around 5 nA, the beam diameter begins to increase rapidly, approximately as I 1.5 , due to spherical aberration. At a 100 nA current the diameter is over 1 µm, and, additionally, the beam has a substantial non-

22

3 Creation and Measurement of Microstructures of Ultrapure Materials LMIS

a)

GFIS

b)

Atom

Heater wires

Electron

Plasma chamber Ion

Ga reservoir Antenna Suppressor electrode Liquid Ga Taylor cone

Wb tip Extractor electrode Extractor electrode

Ion beam

Source electrode

Ion beam

Fig. 3.2 Schematics of a a liquid metal ion source b a gas field ionisation source, adapted from [6]

Fig. 3.3 The spot size of a 30 keV FIB beam as a function of current for a LMIS Ga ion source and a ICPS Xe ion source, based on data from [10]

Gaussian profile and beam tail [9]. This makes the use of the LMIS impractical at currents above around 50 nA. To fill the gap at high beam current, plasma sources were traditionally used. However, their lifetime was limited due to anode and cathode degradation. The breakthrough came in 2006 with the introduction of the inductively coupled plasma source (ICPS), as shown in Fig. 3.2b [9]. In this source, a gas, often xenon, is inductively coupled to a coil which forms a radio frequency antenna. A current driven through this coil, at a frequency designed to excite only the electrons, ionises some

3.1 The Focused Ion Beam

23

atoms and creates a plasma. A set of electrodes then accelerate the ions and emit them from the source. The ICPS has an energy spread comparable to the LMIS. At small currents, the diameter of the beam is larger than that from the LMIS but, as seen in Fig. 3.3, above around 30 nA it is smaller. The highest practical current is approximately 2 µA, where the spot size is of order 1 µm. Importantly, the rate of material removal, the sputter rate, is around 20–100 times greater than with a Ga LMIS, enabling larger structures to be created much more quickly [11, 12]. During the work described in this thesis, two dual beam FIB-SEM systems manufactured by FEI were used. A Helios NanoLab G3 CX FIB with a Ga LMIS source (GFIB) was used for the smallest microstructuring, typically on the scale of a few microns. For larger devices, with pattern features often a few hundred microns in scale, a Helios G4 PFIB UXe FIB with a Xe ICPS (PFIB) was also used, substantially decreasing the time needed for sample production.

3.1.3 Milling with the FIB To mill a sample with a FIB, a pattern is created which defines the area which the beam should scan over. The desired depth and beam current are also chosen, and the total time automatically calculated based on known sputter rates for materials such as silicon. The beam will then pass multiple times, typically in a serpentine motion, over each part of the pattern until the desired depth is achieved. Either serial milling, where one pattern trench is milled at a time, or parallel milling, where a single pass is made over each trench and the process is repeated until the depth is reached, can be chosen. Parallel milling is required if pattern features overlap to prevent material being redeposited into a previously milled trench. Factors such as the time for each pass and the width between each line of the scan can all be edited. The total time for patterning is dependent primarily on the current, with larger currents significantly reducing the time but also increasing the spot size of the beam and therefore decreasing the pattern resolution. During the milling processes, some of the sputtered material is deposited back onto the surface of the sample. The amount and form of this redeposited material varies depending on the geometry. As the aspect ratio, the ratio of the depth of a milled trench compared to its width, rises, the rate of reposition increases until it is equal to the milling rate. At aspect ratios larger than this value, typically around two, milling is no longer possible. To increase the visibility of the crystal sidewalls, which is important for thickness measurements, the aspect ratio should be kept as small as possible, ideally less than one. This redeposited material can also cause issues in the form of spurious pathways for electrical conduction. Redeposition is a general challenge for FIB milling, but in most cases the redeposited material has a high resistance, due to its small depth and amorphous nature. Redeposition of the ultrapure metallic delafossites, however,

24

3 Creation and Measurement of Microstructures of Ultrapure Materials

Fig. 3.4 A schematic of the FIB polishing process. Before polishing, the corner of the sample is rounded and the sidewall is not straight. There is also a substantial layer of redeposition. After polishing with the beam at a glancing angle, the corner is square and the redeposition layer has been removed

Fig. 3.5 SEM images of FIB-milled sidewalls of devices a without polishing b with polishing. On the unpolished sidewall, there is a large amount of redeposition on both the crystal and the epoxy and the corner is rounded. This may cause shorts and makes thickness measurements difficult. On the polished sidewall, the square sidewall and lack of redeposition enable accurate thickness measurements

remains highly conductive, and can easily cause significant shorts, so limiting and removing redeposition is especially important when milling these materials. An important step to prevent these shorts is sidewall polishing to remove redeposition. In this process, as shown in Fig. 3.4, the FIB beam is rotated to a glancing angle of 1–2° to the wall. The beam is then swept in a number of linear passes up to the final intended sidewall position. As well as removing redeposition, this step also ensures the sidewalls are vertical. Examples of unpolished and polished sidewalls are shown in Fig. 3.5. As will be shown in Sect. 3.2.3, the polishing step also reduces the size of the damage layer in the crystal.

3.1 The Focused Ion Beam

25

Fig. 3.6 A schematic of the IA-CVD process. The precursor-containing gas is continuously added during the ion beam scanning time

3.1.4 Deposition of Materials Using the FIB In contrast to milling, a FIB can be used to deposit materials in a process known as ion-assisted chemical vapour deposition (IA-CVD), first introduced in 1984 [13]. Whilst this process is highly flexible due to its maskless nature, the main limitation is the small range of the required precursor gases: at the moment, only those able to deposit Pt, W, SiO2 and C are typically available. During IA-CVD, as shown in Fig. 3.6, an organometallic gas is introduced to the FIB chamber from a nozzle positioned close to the desired region of deposition. This gas adsorbs onto the nearby surfaces and the FIB beam is then scanned over the desired deposition area. As this occurs, secondary electrons break the bonds in the precursor gas. In theory, this process should remove only the highly volatile organic materials, leaving the desired material deposited on the surface. The remaining gas is then pumped out of the chamber. If the beam current or the dwell time, the time spent on each pixel of the pattern, is too high, milling can occur rather than deposition. During the deposition described in this thesis, the maximum current used in pA was no larger than the intended deposition area in square microns multiplied by a factor of five (for the GFIB) and a factor of ten (for the PFIB), with a dwell time of around 50 ns. All these parameters were empirically determined to match the requirements for working with PtCoO2 and PdCoO2 within our FIB-SEM systems. It is important to be aware that FIB deposited metals are far from pure. Indeed, it has long been known that the composition of the deposition is an amorphous mix of different materials. For example, one study found the ‘platinum’ deposited was around 73% C and 10% Ga with only a 17% Pt content [14]. The resistivity is therefore around a factor of 50 larger than the pure metal for Pt and W wires [15]. The resistivity of W wires can be lowered by applying a high voltage in-situ to cause annealing, but the deficit to the pure metal remains a factor of 10 [15]. No improvement was reported for Pt wires after this procedure [15].

26

3 Creation and Measurement of Microstructures of Ultrapure Materials

In theory, IA-CVD can be used to provide electrical contacts to the surface of a crystal. However, the high resistivity results in a contact resistance of typically a few kiloohms, which is too high for work with low resistivity materials such as the delafossite metals. In the absence of deposition techniques for high conductivity materials, one solution is to use the deposited material solely as a platform for metal deposited by another method, such as sputtered gold. This is described and implemented in Chap. 4. A second challenge is providing physical contact between a substrate and the raised crystal surface. If the deposition is attempted in a straight line from the sample surface to the substrate, horizontal terraces usually form on the crystal edge which do not reach the substrate. This problem has two main solutions: either placing the sample in epoxy to provide a smooth ramp or building a ‘staircase’ of deposited steps to reach the top of the sample. The latter idea will be also be discussed in more detail in Chap. 4.

3.2 Interaction of FIB Ions with Materials When considering FIB structuring materials as described in the previous section, an important aspect to address is the level of damage caused when the FIB ions interact with a material. If the damaged region is a significant fraction of the sample size, then the properties measured may be changed by the FIB processing, which is clearly undesirable, especially for the highly sensitive resistivity measurements which will be described in Chaps. 4 and 5. To aid experimental studies, it is helpful to examine the processes which occur within the sample during FIB irradiation, the methods which can be used to estimate the length scales over which these processes occur and the approaches which mitigate the damage. All of these aspects will be considered in this section.

3.2.1 Collision Cascade When an ion impinges on the surface of a material, it may be either reflected, backscattered or transmitted inside. In the last of these possibilities, which is by far most likely, the ion is then slowed by both electronic and nuclear stopping processes. Inelastic collisions cause energy to be transferred from the ions to the electrons within the material through electronic stopping. The excitation of these electrons will then result in the emission of photons or even ionisation. If the ionised secondary electrons originate within a few nanometers of the surface, they can be emitted from the sample. These are the electrons primarily used in imaging. However, the majority of energy lost by the ions is transferred to the nuclei of the material. These nuclear stopping collisions are nearly elastic, with only a small amount of energy given up to bremsstrahlung and nuclear excitation. Above a mate-

3.2 Interaction of FIB Ions with Materials Fig. 3.7 A schematic of a sample collision cascade for Ga ions into a material, based on [16]. Most of the damage is caused by secondary nucleus-nucleus collisions rather than direct ion collisions

27 Incoming ion trajectory Primary collision

Emitted electron Sputtered ion

Secondary collision

Typically 10 - 30 nm Implanted ion Approximate damage limit

rial dependent energy transfer threshold, typically 10–50 eV, the nucleus will be displaced, forming an interstitial-vacancy pair. Some of the atoms close to the surface will overcome the surface binding energy and will therefore be sputtered from the material. Other nuclei will displace further nuclei in secondary collisions, forming a chain of collisions usually referred to as a collision cascade, as sketched in Fig. 3.7. The size of this collision cascade is the main determinant of the damage layer. Except in the very thinnest samples, the ion will be stopped and therefore implanted within the target material. As the displacement energy barrier varies between atomic species, in compounds sometimes certain elements have a substantially smaller binding energy and will therefore be more frequently sputtered. This process, known as preferential sputtering, can result in a surface layer with a higher concentration of some elements than others, such as in the Weyl semimetal NbAs, where a superconducting, Nb-rich surface layer is formed after FIB irradiation preferentially removes the As [17].

3.2.2 Mechanisms of Damage During the collision cascade, as well as the desired sputtering, damage occurs to the underlying material. This includes the development of an amorphous surface layer, ion implantation and sample heating and charging. Collision cascades from multiple ion impacts combine and form an amorphous layer, comprised both of displaced atoms from the target material and implanted ions. In addition, collisions for which the energy transfer is below the threshold for displacement cause the lattice to become heated via phonon emission [18]. This is usually only a significant issue for samples with low thermal conductivity, typically semiconductors or insulators, where good

28

3 Creation and Measurement of Microstructures of Ultrapure Materials

thermal anchoring is not possible [16]. Charging of the sample can also be a problem with these materials. Therefore, mitigating steps such as flooding the sample with low energy electrons from a flood gun or a SEM are often used [19]. As mentioned previously, in most cases the irradiation ion will be implanted into the sample. The fractional content of Ga/Xe near the surface depends on the sputtering yield of the material, but simulations suggest values as high as 50% are possible [16]. With reactive ions such as Ga, interactions can occur within the material which change the electrical and physical properties. For example, the doping in semiconducting materials can be varied by Ga ion implantation [20] which is sometimes exploited to produce microstructures [21]. One advantage of a Xe-ion PFIB over a Ga-ion GFIB is that the Xe ions are not likely to have a strong reaction with the sample atoms. In general, however, neither charging, heating or chemical interaction have been found to have a significant impact during FIB milling of delafossite metal samples. The development of an amorphous layer remains the most significant potential impact upon the conductivity of these materials.

3.2.3 Estimating the Scale of Damage Having established that the creation of an amorphous layer as being the most important damage mechanism for FIB structuring of delafossite metal materials, it is important to determine the scale of this layer. This can, in principle, be estimated using techniques such as molecular dynamic calculations [22]. However, the most common approach is to use Monte Carlo simulation, typically via a program called Stopping Range of Ions in Matter (SRIM) [23, 24]. SRIM uses the binary collision approximation, meaning that the ions propagate in linear paths between a series of independent collisions with the sample nuclei. Between these nuclei collisions, energy is only lost to the electrons. Within SRIM, the average range of the ions and also the size of the collision cascade can be predicted for a material with a given chemical composition and density and for a particular ion species, incident angle and energy. In addition, the sputter yield, the number of atoms removed per ion, can be estimated. Whilst there is considerably uncertainty in these values, SRIM can be useful especially for comparison between different ion species and energies. Typically simulation of a few hundred ions is needed to reach convergence on the output parameters. Figure 3.8 shows the ion trajectories and collision cascade stemming from 500 30 keV Ga ions incident perpendicular to a layer of PtCoO2 . The ions penetrate an average distance of around 10 nm into the sample, but the size of the collision cascade is significantly larger. Figure 3.9 shows the same quantities but for Xe ions instead. The range of both the ions, at around 8 nm, and the collision cascade is smaller than for Ga, due to the increased ion mass. This also ensures the sputter yield of the Xe ions, at around 15 atoms per ion, is larger than for Ga ions, at 11 atoms per ion, ensuring milling would be quicker with a PFIB than a GFIB with the same ion density.

3.2 Interaction of FIB Ions with Materials

29

Fig. 3.8 A SRIM simulation of the trajectories in a PtCoO2 layer of a 500 perpendicularly-incident 30 keV Ga ions and b the atoms displaced after this irradiation

Fig. 3.9 A SRIM simulation of the trajectories in a PtCoO2 layer of a 500 30 keV Xe ions and b the atoms displaced after this irradiation

Another way to gauge the depth of damage is to examine the depth at which phonons are created in the material. Figure 3.10 shows the rate of energy loss to phonon production as a function of depth for Ga and Xe ions and at both a polishing angle and a perpendicular one. The glancing polishing angle and Xe ions both exert damage over a smaller region, but, in all cases, the production is confined to around a 30 nm depth. These values are typical: most materials have an ion range between 10 nm and 100 nm, with the typical amorphous layer being about 20 nm [16, 25]. One limitation of a SRIM calculation is the fact that crystal structures are not included. Compounds are treated as amorphous solids with random atom placement. Crystalline effects such as channelling, where ions can penetrate more deeply by propagating along channels in the crystal where nuclear stopping is diminished, cannot be accounted for. The results for the level and range of Ga implantation are, however, generally broadly consistent with experimental values from atom probe tomography [26].

30

3 Creation and Measurement of Microstructures of Ultrapure Materials

Fig. 3.10 Rate of energy lost to phonon production as a function of depth into the material for 30 keV a Ga ions and b Xe ions at both a polishing and straight-on angle. The damage layer is smaller for the Xe ions as the ion range is lower

For transport measurements of a FIB-structured delafossite sample with a typical minimal dimension of 1 µm, an amorphous damage layer of around 30 nm is not likely to cause significant issues. Indeed, Shubnikhov-de Haas oscillations have been observed in a PdCoO2 crystal FIB-milled with 30 keV Ga ions to a width of 6 µm, demonstrating that the bulk of the crystal remains of a very high purity [27]. This conclusion is also strongly reinforced by the ballistic transport results which will be reported in Chap. 5, which are only possible if the mean free path in the bulk of the sample remains micron-scale.

3.2.4 Mitigating Damage If required, the level of damage can be mitigated by a number of measures. Reducing the acceleration voltage decreases the ion penetration distance and therefore the size of the collision cascade. This reduces the amorphous layer in silicon from around 20 nm at 30 keV to 2 nm at 5 keV [28, 29]. Using a small current is also desirable, as this reduces the beam spot size and therefore increases the resolution with which geometric features are defined. The level of material redeposition and the slope of the sidewalls are also reduced [30]. Decreasing the current, however, increases the structuring time, which is approximately linear in current for delafossite metals, so a balance is often required. With these steps, alongside other measures such as changing the beam scan direction, crystalline samples can be prepared for TEM studies with thicknesses below 10 nm [31]. In our transport measurements, the minimum microstructure dimension was around 3 μm and such methods were therefore unnecessary.

3.3 Device Production

31

3.3 Device Production As well as using FIB microstructuring, there are a number of other considerations when producing a delafossite metal device for resistivity measurements, which are not standard with higher resistivity materials. These, along with an overview of a typical device production process, will be detailed here in order to provide information useful for future studies.

3.3.1 Current Homogeneity One challenge in measuring the resistivity of highly anisotropic materials when topinjection current contacts are used is ensuring a homogenous current distribution throughout the sample thickness. To achieve this, the current must travel parallel to the c-axis. However, the ratio of the distance travelled in this direction compared to that within the ab-plane is small due to the large resistivity anisotropy. If the current path in the device is too small for the current to penetrate the whole depth, the conduction will only occur in the top portion of the crystal, producing a higher resistance. The depth contributing to the electrical transport, presumed to be the whole thickness of the crystal, will be overestimated. Therefore, the resistivity will be overestimated. Indeed, in early measurements of the ultrapure delafossite metals, resistivities higher than the now-accepted values were reported, likely at least in part due to this effect. One simple manner in which to combat anisotropies is to extend the path of the current before the voltage contacts are reached. This can be via a meander, a long snake shaped path, sculpted using the FIB in both current contacts. The ratio between the distance√travelled parallel to the c-axis, lc , compared to that in the ab-plane, lab , is lab /lc = ρc /ρab and in PtCoO2 , ρc /ρab reaches a maximum √ of around 1000, meaning that for the typical crystal depth of 3 µm, at least a 3 1000 = 100 µm meander would be required on both sides of the device to ensure current homogeneity. This is easily achieved in most delafossite crystals. In addition, thinning the crystal, via a wide scan of the FIB beam across the area intended for both the active device and the meanders before they are structured, can assist with current homogenisation by reducing the depth the current needs to penetrate. A disadvantage is the damage which occurs to the sample, but, as shown in Sect. 3.2.3, the damage layer should not be significant for a delafossite crystal thinned to 1 µm or above. This method was used with several samples measured within this thesis, none of which showed an anomalously high resistivity. Even within a thinned region, any sudden change in the thickness of the sample, even those of relatively trivial size such as a small terrace, were found to introduce significant inhomogeneity to the current. Therefore, it is important to ensure the entire region of the active device, including the current meanders, are free from any terraces or step edges. This constraint is typically the limiting factor on the size and shape of device any particular crystal can host.

32

3 Creation and Measurement of Microstructures of Ultrapure Materials

3.3.2 Handling of the Samples In addition to the requirements due to current inhomogeneity, the handling of delafossite metal crystals requires an unusual degree of care. The crystals, which grow into thin platelets, are extremely brittle. Movement via tweezers is therefore not possible without shattering the crystal. The most reliable alternative method is to use electrostatic forces. For larger samples, above around 200 µm in the plane, this is easily achieved by a small tip made from PTFE tape on the end of a toothpick. For even smaller samples, MiTiGen polymer MicroTools can be used. Both of these should ideally be charged by, for example, rubbing them against a piece of clothing before use. Due to the light sample mass, these electrostatic forces are sufficient to hold the sample to the tip or tool and allow for sample movement. If a larger force is required, the loop shaped tools can be dipped in a solvent in order to increase the adhesion.

3.3.3 Typical Device Production Process The final process to make a standard FIB sample from a delafossite crystal for electrical transport measurements is described below. This will presume the use of a GFIB throughout, but some larger scale microstructuring stages can also be completed with a PFIB where indicated. Some stages of the process are illustrated in Fig. 3.11. Within this method, epoxy is used to hold the crystal and provide a smooth ramp for sputtered gold contacts. In the work described in Chap. 4 of this thesis, an adapted, epoxy free, method was required for electrical irradiation experiments. This method will be described within that chapter.

Fig. 3.11 Selected stages from the device creation process

3.3 Device Production

33

1. Select a sample from the batch. For the most accurate measurements, there should be a region, free from steps on both sides of the crystal, which is large enough for the whole device and any homogeneity meanders. 2. Clean a sapphire substrate approximately 5 × 5 × 0.8 mm in size. This is typically done by placing it in isopropanol and then acetone within an ultrasonic cleaner for five minutes each, followed by a short period in an oxygen plasma. 3. Make a flat epoxy droplet from two-component Araldite 2, a 5 min rapid epoxy, using a tool with a small tip, such as an eyelash or a thin wire. The droplet should be flat across the whole crystal area and at least as deep as the crystal. As well as providing adhesion for the sample, this epoxy bubble will absorb some of the differential thermal contraction between the sapphire and the crystal and provide a base for sputtered electrical contacts. 4. Place the crystal into the droplet and wait a few minutes for the epoxy to dry. Then, cure the epoxy at 100°C for one hour to increase the hardness. 5. Mask off the edges of the substrate with Kapton tape to prevent electrical shorts from the sputtered gold (Fig. 3.11, step a). 6. Using a Lesker deposition system, initially clean the sample and substrate surfaces via argon etching for around 10 min. Afterwards, electron-beam deposit a 5 nm Ti sticking layer followed by a 200 nm layer of sputtered Au, both at an angle of 15° to ensure the sides of the epoxy droplet are covered (Fig. 3.11, steps b and c). 7. Remove the Kapton mask. Adhere the substrate onto a SEM stub and ground it by using Silver Conductive Adhesive 503, supplied by Electron Microscopy Services. This adhesive can later be removed using isopropanol or acetone. Place the stub into a dual beam FIB-SEM system (with either a GFIB or PFIB if no pattern element is smaller than around 20 µm) and focus the coincidence point to the sample surface. 8. Rotate the stage angle to 52° so that it is orthogonal to the FIB beam. Define the desired pattern, including the meanders for current homogenisation. For the active device region, increase the intended final dimensions by around 10% to allow for later polishing. 9. Use a large scan of the FIB beam to remove the top Au and Ti layer from the entire device, including the meanders. Usual currents are 15 or 60 nA, with a typical patterning time of around 30 min. The crystal can also be thinned over a longer period if a decreased crystal thickness is desired. 10. Set the pattern to mill, using the smallest feasible current, typically 9 or 15 nA. The patterning time is typically around one night, with the milling occurring between 1.5 and 2 times as quickly as the time predicted by the FEI software for Si (Fig. 3.11, step d). 11. Inside a GFIB, rotate the stage to 53.5° (1.5° glancing FIB beam) and polish the sidewalls of the active device to the final dimensions. A clear division between the sample and the epoxy should be visible after this process is completed. Typically 2–3 h. 12. Using a GFIB or PFIB, with the stage rotated to 52°, cut the Au and Ti layer in the surrounding sample to form contact pads. Typically 1 h (Fig. 3.11, step e).

34

3 Creation and Measurement of Microstructures of Ultrapure Materials

Fig. 3.12 a A SEM image of a square device before the final cuts were made in the Au to define the contact pads. b An optical microscope image of the same device, but after the square was reduced in size using a FIB

13. Remove the sample from the FIB chamber. Attach an annealed 50 µm thick Au wire to each contact pad using silver epoxy (Fig. 3.11, step f). 14. Mount the substrate inside a 28-pin Kyocera chip carrier and then attach the ends of the wires to the desired contact pads using silver epoxy. The total FIB time to make a single sample, such as that shown in Fig. 3.12, is therefore in the region of 12 h, for a sample of the typical dimensions and complexity as those studied in this thesis research, with this including around 5 h of user input. The whole process, from crystal to active device, can be completed over a couple of working days. In addition, the same sample can later be returned to the FIB for further microstructuring (Fig. 3.12b), which is performed whilst the sample remains mounted inside the chip carrier with all wires connected. Most of these dimensional changes take less than 4 additional hours of cutting and polishing time. This makes any thinning study involving changing the dimensions of the sample rapid, and generally limited by measurement, rather than microstructuring, time. This flexibility enabled the size dependent studies which will be described in Chap. 5. The minimum device size is typically constrained by measurement issues rather than the FIB resolution. Whilst channels a few hundred nanometres in width can be produced in delafossite metals using a GFIB, widths less than around 1 µm are susceptible to fatal damage due to differential thermal contraction at cryogenic temperatures. Encasing the sample in epoxy can prevent this damage, but with the significant disadvantage that the device geometry can no longer be changed.

3.4 Measurement of Low Resistivity Materials As well as careful FIB microstructuring, the resistivity measurement must be tuned to accommodate the high conductivity of delafossite samples. A number of key factors which must be considered during such measurements will be discussed here.

3.4 Measurement of Low Resistivity Materials

35

Fig. 3.13 Resistivity measurement schematics of a delafossite sample for a the standard method of producing a current for an ac measurement, with a shunt resistor. There is a large common-mode voltage. b A measurement with the active common-mode rejection current source. The commonmode contribution to the signal has been eliminated

3.4.1 Common Mode Voltage One of the challenges for resistivity measurements of high conductivity materials is the common mode voltage. Whilst this voltage is always present, issues occur when the sample resistance is small compared to the resistance of the contacts and wires leading to the sample. In this case, the common mode voltage is large compared to the voltage difference between the contacts and thus will begin to affect lock-in amplifier measurements of the voltage difference, as will be explained below. A typical manner in which ac currents are created is by using an ac voltage supplied by a lock-in amplifier and a shunt resistor as shown in Fig. 3.13a. However, some of the samples measured in this thesis had a low temperature resistance, R S , around 10 μ. The resistance of the contacts and the wires, R L , was typically around 5  on each side. This means that, using this experimental configuration, the common mode voltage is much larger than the voltage stemming from the sample. With a typical 5 mA measurement current, I , and two voltage contacts with voltages V1 and V2 , the common mode voltage VC M is VC M =

I (2R L + R S ) V1 + V2 = ≈ 25 mV 2 2

(3.1)

whereas the voltage difference V is V = V1 − V2 = I R S ≈ 50 nV which is a ratio of 5×105 smaller than the common mode voltage.

(3.2)

36

3 Creation and Measurement of Microstructures of Ultrapure Materials

The voltage measured at a lock-in amplifier, VL I , is VL I = G(V +

VC M ) rC M

(3.3)

where G is the gain from any amplification stages and rC M is the common mode rejection ratio. Most lock-in amplifiers, such as the Stanford Instruments SR830, can reject common mode voltage up to around 100 dB, which means rC M = 105 . Above this ratio, the common mode signal begins to substantially influence the measured voltage, in this case contributing a 25 × 10−3 /105 = 250 nV signal before amplification, far larger than the experimental signal. To compensate for this, a current source with common mode rejection can be used. When using such a device, the current is sourced from both sides with opposite sign, ensuring that the ground is effectively at the centre of the sample. In the ideal measurement, V1 ≈ −V2 and the common mode voltage is therefore near zero. In our setup, this is achieved by a bespoke dual-sided current source designed by Mark Barber and Alexander Steppke, as described in Ref. [32]. This source has an active feedback mechanism to sense any offset voltage at the sample. If the two sides of the sample and wiring have an identical resistance and an equal amplitude of current is fed from both sides, the ground is automatically in the sample centre. If one side is more resistive, however, VC M will be non-zero and the ground will move from the centre of the sample. The current source, therefore, senses the voltage at the sample via the voltage contacts and adjusts the current input accordingly, to ensure the ground is exactly at the midpoint of the two voltage contacts. Currents up to 9 mA are possible using this source whilst maintaining excellent common mode rejection.

3.4.2 Measurement Configuration The measurement configuration must also meet a number of requirements. With the low resistance of the samples, careful steps must be taken during measurement to decrease the noise level. In addition, during a thinning study where multiple measurements are required, a measurement system where the sample can be changed relatively quickly is desirable. Finally, in many cases, several simultaneous voltage measurements are required. A Quantum Design Physical Property Measurement System (PPMS) with a 9T magnet provides measurement capability down to at least 4 K within a magnetic field and with the ability for rapid sample changes. However, the inbuilt resistivity measurement system only allows for five simultaneous voltage measurements and noise usually limits the measurement to signal sizes above 20 nV. In addition, the PPMS puck is not compatible with the 28-pin leadless Kyocera chip carriers (LCCs) used for sample mounting. A bespoke PPMS insert, previously designed and constructed by Nabhanila Nandi [33], was therefore used for the resistivity measurements. Due to the size

3.5 Conclusions

37

Fig. 3.14 Images of a the lowest part of the custom PPMS insert b the LCC socket at the end of the insert. Images courtesy of E. Zhakina

of the LCCs, they could be removed from the insert and placed onto a SEM stub for further FIB structuring whilst remaining on the LCC. This decreases experimental time and also reduces the risk of sample damage. The insert, shown in Fig. 3.14, has 12 twisted pairs of measurement wires leading to a small breakout box in the lower section. The breakout box allows for reconfiguration of the wiring between the twisted pairs and the pins on the chip carrier, offering flexibility in voltage wiring. Therefore, up to 11 voltage pairs and one current pair can be simultaneously defined, each between any two pins on the LCC. In addition, careful design limits the possibility of wire vibration and loops, both of which can cause significant noise. The insert is also electrically isolated from the PPMS to prevent ground loops and limit noise. Thermometry close to the sample is provided by a Cernox temperature sensor, with a heater also available for PID control of the temperature, especially during field ramps. A difference between this temperature and the PPMS sample temperature existed during measurement, demonstrating the need for this additional measurement. The voltage was measured using a Synktek MCL1-540 lock-in amplifier. This amplifier is capable of measuring √ up to 10 voltage channels simultaneously. The noise floor, at around 1.8 nV/ Hz, is also advantageous for measurement of low resistivity samples. The whole system was controlled via bespoke Matlab programs, based on those originally written by Mark Barber. These programs enable remote access to control the PPMS temperature and field, as well as recording all desired data.

3.5 Conclusions In this chapter, key aspects of the application of focused ion beam microstructuring and low noise resistance measurement techniques to transport studies of delafossite metal samples have been described. The FIB has been shown to be a highly flexible tool which enables rapid creation of bespoke structures. These structures can be

38

3 Creation and Measurement of Microstructures of Ultrapure Materials

designed to, for example, enhance the resistance and therefore improve the signal to noise ratio, as will be exploited in Chap. 4. The FIB microstructuring process also allows for changes to the size of a device after measurement, which will be used in Chap. 5 to examine the impact the size of a structure has on the transport, whilst preventing uncertainty being introduced by comparing devices within different crystals. In addition, the ability of the FIB to deposit certain materials, as described in this chapter, is vital to the epoxy-free methods of sample mounting which will be utilised as described in Sect. 4.3.2 for electron irradiation studies. The damage imparted by the FIB during structuring has also been explored. By performing SRIM simulations, the typical scale of the amorphous damage layer induced by FIB irradiation in the delafossite metals has been shown to be a few tens of nanometers. This is insignificant to the bulk transport within the structures studied in the experiments described in this thesis, which have a minimum size of a few microns. Finally, the standard process to both produce and measure the resistance of FIB-structured samples from delafossite metal crystals was described, with the aim of providing valuable information and areas for consideration for current and future studies.

References 1. Reyntjens S, Puers R (2001) A review of focused ion beam applications in microsystem technology. J Micromechanics Microengineering 11(4):287–300 2. Narayan K, Subramaniam S (2015) Focused ion beams in biology. Nat Methods 12(11):1021– 1031 3. Basile DP, Boylan R, Baker B, Hayes K, Soza D (2011) FIBXTEM - focussed ion beam milling for TEM sample preparation. MRS Proc 254:23 4. Moll PJW (2018) Focused ion beam microstructuring of quantum matter. Annu Rev Condens Matter Phys 9(1):147–162 5. Gierak J (2009) Focused ion beam technology and ultimate applications. Semicond Sci Technol 24(4):043001–0430024 6. Smith NS, Notte JA, Steele AV (2014) Advances in source technology for focused ion beam instruments. MRS Bull 39(04):329–335 7. Clampitt R, Aitken KL, Jefferies DK (1975) Intense field-emission ion source of liquid metals. J Vac Sci Technol 12(6):1208 8. Seliger RL, Ward JW, Wang V, Kubena RL (1979) A high-intensity scanning ion probe with submicrometer spot size. Appl Phys Lett 34(5):310–312 9. Smith NS, Skoczylas WP, Kellogg SM, Kinion DE, Tesch PP, Sutherland O, Aanesland A, Boswell RW (2006) High brightness inductively coupled plasma source for high current focused ion beam applications. J Vac Sci Technol B: Microelectron Nanometer Struct 24(6):2902–2906 10. Bassim N, Scott K, Giannuzzi LA (2014) Recent advances in focused ion beam technology and applications. MRS Bull 39(April):317–325 11. Altmann F, Young RJ (2014) Site-specific metrology, inspection, and failure analysis of three-dimensional interconnects using focused ion beam technology. J Micro/Nanolithography MEMS MOEMS 13(1):011202(1)–011202(11) 12. Kellogg S, Schampers R, Zhang S, Graupera A, Miller T, Laur W, Dirriwachter A (2010) High throughput sample preparation and analysis using an inductively coupled plasma (ICP) focused ion beam source. Microsc Microanal 16(S2):222–223

References

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13. Gamo K, Takakura N, Samoto N, Shimizu R, Namba S (1984) Ion beam assisted deposition of metal organic films using focused ion beams. Jpn J Appl Phys 23(5):L293–L295 14. De Teresa JM, Córdoba R, Fernández-Pacheco A, Montero O, Strichovanec P, Ibarra MR (2009) Origin of the difference in the resistivity of as-grown focused-ion- and focused-electron-beaminduced Pt nanodeposits. J Nanomater 936863:2009 15. Reguer A, Bedu F, Tonneau D, Dallaporta H, Prestigiacomo M, Houel A, Sudraud P (2008) Structural and electrical studies of conductive nanowires prepared by focused ion beam induced deposition. J Vac Sci Technol B: Microelectron Nanometer Struct 26(1):175–180 16. Volkert CA, Minor AM (2007) Focused ion beam microscopy and micromachining. MRS Bull 32(05):389–399 17. Bachmann MD, Nair N, Flicker F, Ilan R, Meng T, Ghimire NJ, Bauer ED, Ronning F, Analytis JG, Moll PJW (2017) Inducing superconductivity in Weyl semimetal microstructures by selective ion sputtering. Sci Adv 3:e1602983 18. Giannuzzi L (2006) Introduction to focused ion beams: instrumentation. Theory, techniques and practice. Springer, US 19. Munroe P (2009) The application of focused ion beam microscopy in the material sciences. Mater Charact 60(1):2–13 20. Hirayama Y, Okamoto H (1985) Electrical properties of Ga ion beam implanted GaAs epilayer. Jpn J Appl Phys 24(2, 12):L965–L967 21. Hirayama Y, Saku T (1989) Conductance characteristics of ballistic one-dimensional channels controlled by a gate electrode. Appl Phys Lett 54(25):2556–2558 22. Averback RS, Ghaly M (1994) A model for surface damage in ion-irradiated solids. J Appl Phys 76(6):3908–3910 23. Ziegler JF (2004) SRIM-2003. Nucl Instrum Methods Phys Res 219–220(1–4):1027–1036 24. Ziegler JF, Biersack JP, Ziegler MD (2013) The stopping and range of ions in matter, SRIM2013 25. Giannuzzi L, Stevie F (1999) A review of focused ion beam milling techniques for tem specimen preparation. Micron 30(3):197–204 26. Thompson K, Lawrence D, Larson D, Olson J, Kelly T, Gorman B (2007) In situ site-specific specimen preparation for atom probe tomography. Ultramicroscopy 107(2–3):131–139 27. Moll PJW, Kushwaha P, Nandi N, Schmidt B, Mackenzie AP (2016) Evidence for hydrodynamic electron flow in PdCoO2 . Science 351(6277):1061–1064 28. Giannuzzi LA, Geurts R, Ringnalda J (2005) 2 keV Ga+ FIB milling for reducing amorphous damage in silicon. Microsc Microanal 11(S02):29–35 29. Giannuzzi L, Leer BV, Ringnalda J (2007) Evidence for a critical amorphization thickness limit of Ga+ ion bombardment in Si. Microsc Microanal 13(S02):1516–1517 30. Prenitzer B, Urbanik-Shannon C, Giannuzzi L, Brown S, Irwin R, Shofner T, Stevie F (2003) The correlation between ion beam/material interactions and practical FIB specimen preparation. Microsc Microanal 9(03):216–236 31. Lechner L, Biskupek J, Kaiser U (2011) FIB target preparation for 20 kV STEM - a method for obtaining ultra-thin lamellas. Microsc Microanal 17(S2):628–629 32. Barber ME (2017) Uniaxial stress technique and investigations into correlated electron systems, Ph.D. thesis, University of St Andrews 33. Nandi N (2019) Can hydrodynamic electrons exist in a metal? A case study of the delafossite metals PdCoO2 and PtCoO2 , Ph.D. thesis, Technischen Universität Dresden, Dresden, 2019

Chapter 4

High Energy Electron Irradiation of Delafossite Metals

As described in Chap. 1, there has been longstanding interest in materials with anomalously high conductivities and extremely long electron mean free paths. Historically, the high conductivity elemental metals such as copper and gold, and, more recently, the two-dimensional electron gases formed within semiconductor heterostructures were the only materials able to reach these limits. However, recent developments in material synthesis has allowed for the study of novel materials such as graphene and the Weyl and Dirac semimetals. In addition, the introduction and improvement of microstructuring techniques, such as the FIB methods described in the previous chapter, enable the electronic transport within these materials to be increasingly well characterised. There is often a difference between elemental metals and many modern materials, however, in terms of the underlying reason for their high conductivity at low temperature. The high conductivity of the elemental metals is generally considered to stem from an extremely high purity, albeit often aided by purifying techniques such as vacuum annealing and zone refining [1]. This property effectively means there are few defects to scatter from. In many modern materials, however, defects are often present but large-angle backscattering, the scattering most important to resistivity, is suppressed by a variety of mechanisms. Reaching micron-scale mean free paths in GaAs-AlGaAs heterostructures required multiple fabrication advances over many years, including the development of modulation doping to move the defects out of plane and suppress this backscattering [2]. Weyl and Dirac semimetals, such as Cd3 As2 [3] and WP2 [4] can have low temperature resistivities measured in ncm. For these materials, however, there is a large difference, up to a factor of 104 , between the scattering time measured from quantum oscillation measurements, which are sensitive even to small angle scattering, and the scattering time measured from resistivity [3]. This is highly suggestive of a mechanism which significantly suppresses backscattering from defects, rather than extreme purity. As well as the band structure in the bulk, this may be partly due to © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. H. McGuinness, Probing Unconventional Transport Regimes in Delafossite Metals, Springer Theses, https://doi.org/10.1007/978-3-031-14244-4_4

41

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4 High Energy Electron Irradiation of Delafossite Metals

the nature of the surface states, which are known to strongly suppress backscattering compared to a Fermi surface without spin texture [5]. For the high conductivity metallic delafossites described in Chap. 2, however, the picture is less clear. If the ultrapurity view is taken, the mean free path is up to 2 × 105 lattice constants. This would seem to be an unfeasibly small defect level for an oxide material, especially one grown without a method known to produce high purity crystals or any additional purification steps. Alternatively, it is possible that defects do exist, but, due to a so far unknown mechanism, backscattering is strongly suppressed. Calculations considering the orbital structure of the PtCoO2 Fermi surface suggest that orbital-momentum locking leads up to a factor of four reduction in scattering, compared to a single-orbital model for a Fermi surface of the same size and dispersion [6]. This is not sufficient, however, to explain the mean free paths orders of magnitude longer than those in other materials. There is a disparity between the lifetime extracted from resistivity and Shubnikov de Haas measurements, but this is a modest factor of around 10 for both PdCoO2 [7] and PtCoO2 [8, 9]. As yet, therefore, no experimental or theoretical probe has been able to distinguish between the two possibilities for delafossite metals. Surprisingly, introducing disorder can provide insight into the origin of the high conductivity. If, when scattering centres are added, the resistivity increases in an expected manner, it is unlikely that there is a significant suppression of the scattering from disorder. This would consequently imply that the as-grown crystal has an extraordinary purity. Alternatively, if there is little change in the resistivity as the level of disorder is increased, there may be a mechanism strongly reducing the scattering from both intrinsic growth defects and introduced defects. This chapter describes such a defect-introduction study carried out by high energy electron irradiation of PtCoO2 at cryogenic temperatures. Varying the energy of the irradiation electrons whilst measuring the rate of resistivity increase allows for the introduced defect concentration to be determined by comparison with theoretical models. Once this is known, comparing the observed change in resistivity to predicted values at such a defect concentration enables distinction between the two possibilities described above. In addition, introducing defects in this controlled manner also allows for precise measurement of their impact on the general transport properties of the delafossite metals, such as the magnetoresistance. This study will therefore provide insight into the origin and nature of the extreme conductivity of the metallic delafossites, which is needed for both understanding their fundamental physics and for future technological aims.

4.1 Principles of High Energy Electron Irradiation High energy electron irradiation is a mechanism which adds disorder in a particularly well controlled manner. In this procedure, electrons with a typical energy of a few MeV are used to irradiate a material and remove atoms, therefore creating defects.

4.1 Principles of High Energy Electron Irradiation 10

6

Pb ions

Xe-FIB Ga-FIB

Particle mass (MeV/c 2)

Fig. 4.1 A contour plot showing the maximum energy transferred to Pt atoms by the most common methods of irradiation as a function of the irradiation particle mass and energy, based on a figure in Ref. [10]

43

10

4

He H

10 2

10 0

10 -2 10 -2

SIRIUS electron

10 0

10 2

10 4

Particle energy (MeV)

Although irradiation with species such as protons or heavy ions has historically been more common, the benefit of electron irradiation is that the energy transferred to the atoms within the irradiated material is significantly reduced due to the small electronic mass, hence only point defects are created. In Fig. 4.1, the maximum energy transferred to a platinum atom as a function of the kinetic energy and mass of the irradiation species is shown as a contour plot. All of the electron irradiation described in this thesis was performed at the SIRIUS accelerator located within the Laboratory for Irradiated Solids at the École Polytechnique, France, with a maximum electron kinetic energy of 2.5 MeV. The maximum possible energy transfer to Pt atoms at this facility, around 100 eV, is several orders of magnitude smaller than for the more typical proton or heavy ion irradiation. Even the typical energy transfer during focused ion beam microstructuring, as described in Chap. 3, is significantly larger. The smaller energy transfer is crucial as it limits the type of defects created. At intermediate energies, typical for the focused ion beam techniques discussed previously, the first platinum atom displaced by the irradiation species has sufficient energy to displace other platinum atoms in secondary collisions. These atoms then displace further atoms, leading to a collision cascade as described in Sect. 3.2.1. At even higher energies and masses, the irradiation particle can transfer sufficient energy to displace multiple platinum atoms directly, producing columns of displaced atoms. Such columns can be useful, for example in vortex pinning experiments in superconductors, but in our work we aim to avoid them. At the energies achievable with electron irradiation, the energy transferred is insufficient for either of these processes. Generally, only a single platinum atom can be displaced per irradiation electron. This atom will then be displaced from the equilibrium lattice position, leaving behind a vacancy, a missing atom in the lattice. The displaced atom will then move to another position in the crystal, becoming an interstitial. These vacancy-interstitial defect pairs are known as Frenkel pairs. The vacancy-interstitial distance varies depending on factors such as the electron energy

44

4 High Energy Electron Irradiation of Delafossite Metals

and the material, but is generally small enough to allow each pair to be considered to be independent from other pairs.

4.1.1 Distribution of Introduced Defects There are a number of important factors to consider in any such irradiation study, such as the distribution of introduced defects within an irradiated material. For example, if the range of an irradiation electron within a material is small compared to the thickness, it would be expected that the defects would be concentrated in the region of crystal closest to the beam, which is undesirable. The range can be estimated using the ESTAR database available from the National Institute of Standards and Technology [11]. This database includes a material and energy dependent quantity known as the stopping power, which describes the slowing of an irradiation electron by interactions with both the electrons and the nuclei in the target material. This can then be used to calculate the range within the continuously slowing down approximation (CSDA) range, which is known to yield accurate estimates for the range of electrons in a material. Figure 4.2 shows the variation of this quantity for PtCoO2 over the electron energy range used in the experiments reported in this thesis. The range is larger than 0.4 mm at all energies, over four orders of magnitude greater than the typical penetration depth of the FIB ions described in Sect. 3.2.3 and a factor of 80 larger than the typical 5 µm crystal depth used in this experiment. The great majority of electrons, therefore, pass through the crystal. The electrons which do interact with the sample may be expected to scatter only once and at points throughout the full thickness. The technique is therefore ideal for achieving the desired homogenous distribution of defects, which should affect the resistivity throughout the whole crystal.

1.6 1.4

CSDA range (mm)

Fig. 4.2 The CSDA range of electrons in PtCoO2 over the range of experimental electron energies, calculated using the ESTAR tables of NIST [11]

1.2 1 0.8 0.6 0.4

1

1.5

2

Electron kinetic energy (MeV)

2.5

4.1 Principles of High Energy Electron Irradiation

45

4.1.2 Sample Temperature During Irradiation The temperature of the sample during irradiation is also important: it must be sufficiently low that there is no possibility that the vibration of the ionic lattice leads to recombination of Frenkel pairs during irradiation. The energy barrier for partial recombination can be small: in copper around 70% of defects introduced at 10 K have recombined by 40 K [12], and in platinum around 80% have recombined by 30 K [13]. The time per migration jump, t, is dependent on the energy barrier U , typically of the order of 0.1 eV, t=

1 fl e

− k UT

(4.1)

B

where T is the temperature, k B is the Boltzmann constant and fl is the vibration frequency of the lattice [14]. The vibration frequency can be approximated in the Einstein model by h fl = k B  E where h is Planck’s constant and  E is the Einstein temperature. For PdCoO2 ,  E ∼ 1100 K [7]. The temperature must therefore be low enough that no such jumps are made within the time of the experiment, te , T
W , the voltage is approximately zero due to the separation of the voltage contacts from the current path. In the ballistic regime, however, electron trajectories along the axis of the device led to a dip at zero field, as in the bend resistance. A typical decay of the magnitude of this feature as L is increased is shown in Fig. 5.12b. As seen in Fig. 5.12b, if the width of the junction is small enough, an exponential decay with two different regimes of l B occurs [64, 65]. Explanations for the origin of this effect differ, but most attribute the rapid, width dependent decay at small L to boundary scattering within the junction and the slow, more width independent, decay at larger L to scattering within the bulk. This bulk scattering would dominate in the Ohmic regime and thus the longer mean free path corresponds better to the wide channel, Ohmic regime value.

106

5 Ballistic Transport in Square Junctions of Delafossite Metals

Simulations support this explanation by suggesting that the short decay length is more sensitive to the boundary scattering specularity, whereas the long decay length is more sensitive to bulk impurity scattering [46]. Takagaki et al. attribute the slower decay to electrons injected with a forward trajectory which have less boundary interaction [65]. Both mean free paths were always shorter than the resistivity-derived value, however, likely due to small angle scattering affecting the former more than the latter. In Sect. 5.5, the results of related length scale studies of delafossites will be presented.

5.2.4 Landauer-Buttiker Theory The novel behaviour seen in the experiments described in the previous sections provided motivation for a new theoretical understanding of the ballistic phenomena. With the intrinsic non-locality of the ballistic regime, however, Ohmic methods of calculating the resistivity, which rely on a local relationship between the current density and electric field, were clearly no longer appropriate. A new approach was required. Landauer [66] had developed an alternative framework in which the current and voltage terminals are considered to be reservoirs of electrons with a certain chemical potential. If all the chemical potentials are equal, no current flows. By biasing the chemical potentials with respect to one another, however, electrons flow between terminals. The conductance, G, in a two terminal measurement at zero temperature is determined by the probability, T , that an electron at the Fermi energy is transmitted to the second terminal rather than being reflected back into the first, G=

2e2 NT h

(5.15)

where N is the number of modes at the Fermi level in the terminals. Each mode at the Fermi level therefore transmits a conductance quantum 2e2 / h with a probability T . This formula is independent of the length of the conductor. The number of modes in each contact can be large: Eq. 5.15 is used much more widely than the few mode limit. An important assumption in the derivation, however, is that inelastic scattering only takes place in the contacts, not in the device. The inverse of the conductance in Eq. 5.15, G −1 , can be split into two separate terms, h 1−T h + 2 2 2e N 2e N T = G C−1 + G −1 D

G −1 =

(5.16) (5.17)

which are effectively a contact resistance, G C−1 , independent of both T and the length of the channel, and a device resistance, G −1 D , which stems from the main channel.

5.2 Square and Cross Junction Devices

107

When there is a large amount of scattering in the conductor, T is small and therefore the device resistance dominates. In fact, Ohm’s law can be recovered in the limit where T is small [67]. In the other limit, with T = 1, whilst there is no resistance from the channel, there is still a finite, length-independent resistance stemming from the existence of contacts. This is a key point: even in the limit of an infinite mean free path, the conductance remains finite. The Landauer formula for two terminals was extended to include additional terminals by Büttiker. The Büttiker multiprobe formula [68] states that the net current, Ii , out of contact i at zero temperature is ⎤ ⎡  2e2 ⎣ Ii = N j Ti← j (μ F )μ j ⎦ (Ni − Ni Ri )μi − h j, j=i

(5.18)

where μ j is the chemical potential at contact j, N j is the number of modes at the Fermi level in contact j, Ti← j is the probability an electron at the Fermi energy emitted from contact j is transmitted to contact i and, similarly, Ri is the probability an electron emitted from contact i is reflected back into the same contact. In the following discussion, the arrows and μ F labels in the transmission coefficients will be suppressed for brevity: Ti j = Ti← j (μ F ). If the net current and the transmission coefficients for each contact are known, the set of simultaneous linear equations formed by applying Eq. 5.28 at each contact can be solved to determine the chemical potentials μ j . The transmission coefficients are not completely unconstrained, however. To ensure that no current flows when all the chemical potentials are equal, a sum rule is required, 

Tq p =



q

T pq

(5.19)

q

which essentially states that the sum of the transmission probabilities out of a contact must be equal to the sum of the transmission probabilities where that contact is the recipient. The second symmetry of the transmission coefficients, time reversal, is more subtle. In a coherent conductor, where there is only elastic scattering, a scattering matrix S with components si j can be defined where Ti j =

 m

n

tmn =

 m

|smn |2

(5.20)

n

where the transmission tmn and scattering matrix components smn are from mode n in contact j to mode m in contact i and the sums are over all modes at the Fermi surface in the relevant contact. As shown by Datta [67] from the Schrödinger equation, the scattering matrix obeys a reciprocity relation at infinitely small voltage biases,

108

5 Ballistic Transport in Square Junctions of Delafossite Metals

smn (B) = snm (−B)

(5.21)

in a magnetic field B. When the square magnitude of both sides of Eq. 5.21 is summed over all the modes in both the transmitting and receiving contacts, this leads to a reciprocity relation for the transmission coefficients,   |smn (B)|2 = |snm (−B)|2 (5.22) n

m

n

m

Ti j (B) = T ji (−B)

(5.23)

where the sums are over all the modes m and n in the terminals i and j respectively. Unlike the sum rule, which stems from fundamental current conservation, there is no fundamental reason why this symmetry should exist in incoherent systems, where inelastic scattering occurs within the device. In practice, however, within the linear regime of voltage and current, the symmetry in Eq. 5.23 is always obeyed [22, 67]. Büttiker [68, 69] investigated the effect of incoherence and found that inelastic scattering is effectively equivalent to including an additional voltage contact in a coherent system. In a Green’s functions picture, both contribute to the self energy term [67]. In either case, the transmission coefficients change, but the reciprocity relation, Eq. 5.23, remains valid. When quantum interference effects become important, current-voltage nonlinearities can occur even at small currents, leading to an observed violation of Eq. 5.23 [70, 71]. In Sect. 5.4 current conservation and time reversal symmetry will be shown to contribute in strongly constraining the form and symmetries of the voltage signal in delafossite metal junctions. The remaining step in determining the expected conductance is the calculation of the transmission coefficients Ti j . Every ballistic effect described above can be explained in a Landauer-Büttiker picture by a change to these transmission coefficients. The calculation method can be either classical or quantum mechanical. In the latter case, typically used when quantum interference effects are important, generally the Green’s functions are determined [72] and the Fischer-Lee relation [73] is then used to calculate the S-matrix and the transmission coefficients. In the classical case, the coefficients tend to be calculated by billiard ball simulations, in which the electrons are treated as noninteracting, solid particles which obey Newton’s laws and the Lorenz force.

5.3 PtCoO2 and PdCoO2 Square Junctions Every study of micron scale four terminal junctions within the ballistic regime thus far, including those described in the previous section, has been within a material with a circular Fermi surface. The effect of the hexagonal Fermi surface of PtCoO2 and PtCoO2 on the behaviour of such devices is therefore unknown. The previous studies

5.3 PtCoO2 and PdCoO2 Square Junctions

109

in long channels, discussed in Sect. 5.1.4, imply however that the Fermi surface may have a significant impact upon the behaviour and perhaps lead to novel phenomena. Therefore, inspired by the Montgomery method used to measure resistance anisotropy in the Ohmic regime, we developed a similar approach to establish whether anisotropy was present in the electrical transport of PtCoO2 junctions within the ballistic regime, as will be described in this section. This was made possible by the development of the FIB techniques for ultrapure materials described in Chap. 3, as will be described in more detail below.

5.3.1 Fabrication of Delafossite Squares To facilitate the study in delafossite metals, a series of square shaped PtCoO2 junctions with a side length, w, typically 40 µm initially, were fabricated. The typical contact width, c, was 4 µm. Unlike with the circular Fermi surface shown in Fig. 5.13a, the orientation of the hexagonal PtCoO2 Fermi surface relative to the square had to be considered. The squares were fabricated with one of two different Fermi surface orientations, shown in Figs. 5.13b and c. These were designed to vary the ease of transmission along the diagonal: the enhanced orientation should increase this transmission and the diminished one should decrease it. This can be seen in Fig. 5.13, where the enhanced orientation has two of the six main electron directions orientated along the diagonal, whereas the diminished orientation has no such property. It is therefore possible to imagine that the transport in the ballistic regime would be different between the orientations. For each crystal, the orientation of the Fermi surface was determined by visible hexagonal growth terraces or crystal edges, such as those marked in Fig. 5.14. In PtCoO2 , the Brillouin zone is known to be parallel to these edges, with the Fermi surface orientation therefore determined by a 90° rotation. After this orientation was known, standard FIB microstructuring techniques, as described in Chap. 3, were used to fabricate each square with the desired geometry. In all cases, a meander structure was added to each contact to homogenise the current, as discussed in Sect. 3.3.1,

Circular

Enhanced

I

I

Bend 1 -

1

2 FS

3

ΔV

-

4

- ΔV +

+

I

-

Bend 2

w

FS

3

+

ΔV

-

4

- ΔV +

I

+ 1

2

1

c

I

Bend 1 -

+

2

+

+

Bend 2

Bend 1 -

+

Diminished

+

+

I Bend 2

FS 3

ΔV

-

4

- ΔV +

Fig. 5.13 A schematic of the Fermi surface orientation relative to the square geometry for a circular Fermi surface and the enhanced and diminished orientations in PtCoO2 , alongside the contact configuration for the bend voltage measurements

110 Fig. 5.14 An SEM image of a PtCoO2 square microstructure, with a larger initial starting size of w = 95 µm. The crystal has been thinned in the darkened region to increase the resistance and therefore the signal to noise ratio

5 Ballistic Transport in Square Junctions of Delafossite Metals

Au layer removed

Current homogeneity meander

Cut in Au layer

120° crystal growth edge Thinned region

150 µm

as the current was top injected from gold contacts. In some squares, with as-grown thicknesses larger than 5 µm, a wide scan of the FIB beam over a large section of the crystal surface was used to reduce the thickness to less than 2 µm before the square was structured, in order to increase the signal to noise ratio. An example can be seen in Fig. 5.14. In total, one enhanced orientation, E1, and five diminished orientation squares, D1 to D5, were made from PtCoO2 crystals, with a single PdCoO2 diminished orientation square, PdD1, also being fabricated. In every square, two bend voltage measurements were taken, in configurations also indicated in Fig. 5.13. Both measurements, Bend 1, VB1 = V12,43 = V4 − V3 when the current flows from contact 1 to contact 2 and Bend 2, VB2 = V23,41 , were made using the custom resistivity PPMS insert described in Sect. 3.4.2. This also enabled measurement with a magnetic field of up to 9T parallel to the c-axis. In some cases, Hall measurements, V13,42 and V42,13 were also made. Typical AC currents of 9 mA at 123 Hz were supplied using the common mode rejecting current source described in Sect. 3.4.1, with the output voltage read using the low-noise Synktek lock-in amplifier. Due to the symmetry of the conductivity tensor, as discussed in Sect. 5.2.2, both bend voltages should be identical within the Ohmic regime. This equality provided a robust method to check for electrical shorts due to, for example, FIB related material redeposition: if such shorts were present, the bend voltages were significantly (at least 50%) different at room temperature. Further FIB processing was able to remove these shorts. Such a reliable method to check for electrical shorts does not exist for other FIB microstructures. An advantage of using a FIB-based microstructuring method, especially compared to lithography based approaches, is that transport measurements can be taken and the square size can then be reduced using the FIB. By repeating these two steps several times, different square sizes can be explored within the same section of crystal. Figure 5.15 shows a square with a side length of 95 µm and a square from exactly the same part of the original crystal but now cut down to a side length of 10 µm. Performing the measurement in this way reduces inconsistencies due to the natural variance in purity between crystals or even between sections of the same crystal.

5.3 PtCoO2 and PdCoO2 Square Junctions

a)

c-axis

50 µm

111

b)

c-axis

50 µm

Fig. 5.15 A PtCoO2 square device structured using a FIB at the same scale with a side length of a 95 µm and b 10 µm, after several intermediate FIB structuring and measurement stages

This fabrication method for PtCoO2 squares therefore allows for a number of physics issues to be explored, not only regarding the presence and magnitude of any anisotropy within the ballistic regime electrical transport but also to the length scale over which any such anisotropic effects can be observed. These factors will be studied in the next few sections.

5.3.2 Temperature Dependence of the Bend Voltage A simple initial measurement is to compare the temperature dependence of the two bend voltages as the square size is reduced. The results of such a measurement are shown in Fig. 5.16a for an enhanced orientation PtCoO2 square, Square E1, and Fig. 5.16b for a diminished orientation square, Square D1, for side lengths between 7 µm and 35 µm. In each case, the quantity plotted is VB1,B2 t/I where I is the measurement current and t is the square thickness, to compensate for any variance in these quantities between devices and measurements. Within the Ohmic regime, the two bend measurements are in the Van der Pauw configuration and are identical regardless of the Fermi surface orientation due to the Ohmic regime resistivity isotropy of the triangular lattice discussed in Sect. 5.2.2. The resistivity can therefore be calculated using the Van der Pauw equation, Eq. 5.5. The measurements were indeed nearly identical and, using this equation, gave a resistivity 1.8 ± 0.1 µcm at room temperature when averaging across all squares and measurements, in agreement with the accepted value of 1.82 µcm [74]. This suggests the FIB structured square devices accurately measure the resistivity in the Ohmic regime, with no evidence of significant electronic shorts. Indeed, the bend voltages remain near-identical at temperatures around 80 K and above in Fig. 5.16 across both square orientations and all square sizes. This is as expected in a Van der Pauw measurement of an isotropic material, where there is no dependence on the size of the square or the choice of bend voltage, and is good evidence that all devices are in the Ohmic regime at all temperatures above 80K.

112

5 Ballistic Transport in Square Junctions of Delafossite Metals

Below around 50 K, however, these properties become distinguishing. In the enhanced orientation square, Fig. 5.16a, the behaviour of the two bend voltages is similar at all square sizes and temperatures, with only a small difference, likely due to fabrication asymmetries. At low temperature, there is a strong size dependence: in the smaller square sizes, both voltages rapidly decrease with temperature, eventually crossing the x-axis and becoming negative. This is similar to the behaviour previously found in materials with a circular Fermi surface, and suggests that the mean free path is long enough that many electrons are following collimating trajectories and are transmitting along the square diagonal without scattering, resulting in a negative voltage. In the diminished orientation square, however, there is a strong divergence in behaviour between the two bend voltages. The Bend 2 voltage, VB2 , is qualitatively similar to the behaviour in the enhanced orientation square, albeit with a weaker decrease and with the zero-crossing beginning at a smaller square size. This suggests that, as expected, the enhanced Fermi surface orientation increases the collimation and hence the probability of transmission along the diagonal more than the diminished orientation, enhancing the negative voltage. The Bend 1 voltage, however, rapidly increases at low temperature, most strongly in the smallest squares. Such an increase has not previously been reported in any ballistic regime junction and suggests that the possible trajectories in the ballistic regime impede transport between the voltage contacts in this configuration. Regardless of the exact underlying trajectories, the transport within the square has effectively become strongly anisotropic along the two directions parallel to the square sides: at 5 K in the 7 µm square, the difference between the two voltages is approximately equal to the entire signal at 90 K. The data from PtCoO2 squares therefore shows a strong ballistic anisotropy effect.

a) 800

b) 800

Enhanced Square E1

Diminished Square D1

600

400 Square size Bend 1 Bend 2 35 µm 35 µm 20 µm 20 µm 15 µm 15 µm 10 µm 10 µm

200 0 -200

Vt/I (µΩµ m)

Vt/I (µΩµm)

600

400 Square size Bend 1 Bend 2 35 µm 35 µm 20 µm 20 µm 15 µm 15 µm 10 µm 10 µm 7 µm 7 µm

200 0 -200

0

20

40

60

Temperature (K)

80

100

0

20

40

60

80

100

Temperature (K)

Fig. 5.16 The temperature dependence of V B1,B2 t/I below 100 K for a E1 (an enhanced orientation square) and b D1 (a diminished orientation square). The 10 µm data for E1 are near-identical for the two measurements and is therefore difficult to resolve visually

5.3 PtCoO2 and PdCoO2 Square Junctions Square E1

Enhanced

b) 600

100

500

50

400 Vt/I (µΩµ m)

Vt/I (µΩµ m)

a) 150

113

0 -50 Square size Bend 1 Bend 2 40 µm 40 µm 20 µm 20 µm 15 µm 15 µm 10 µm 10 µm

-100 -150 -200 5K -250 -6

-4

-2

0

2

4

Square size Bend 1 Bend 2 95 µm 95 µm 75 µm 75 µm 50 µm 50 µm 40 µm 40 µm 20 µm 20 µm 15 µm 15 µm 10 µm 10 µm 7 µm 7 µm

Squares D1 & D2 Diminished

300 200 100 0 -100 5K

6

-200

-6

-4

-2

w/r c

0 w/rc

2

4

6

Fig. 5.17 The magnetic field dependence of V B1,B2 t/I at 5 K for a Square E1 and b Squares D1 and D2, diminished orientation squares with initial sizes of 40 µ and 95 µm respectively

5.3.3 Low-Temperature Bend Magnetoresistance To explore this anisotropy further, a magnetic field was applied parallel to the c-axis with the sample held at 5 K, in the same manner as the data for another material shown in Fig. 5.9. Figure 5.17 shows the voltage in both squares as a function of w/rc where w is the square size length and rc is the cyclotron radius. This quantity is proportional to the magnetic field but allows geometric resonances which occur at certain values of w/rc to be compared between different square sizes. In the enhanced orientation square, Fig. 5.17a, there is a trough at zero field, similar to the observations on other materials reviewed in Sect. 5.2.3. As is the case for the temperature dependence (Fig. 5.16), there is no significant difference between the two bend voltages. In the smallest squares, the voltage at low field is negative. As the field increases, the voltage rises rapidly, becoming positive and reaching a peak at w/rc ≈ 2. This corresponds to the first focusing field in transverse electron focusing. At this point, most electron trajectories are focussed into a contact next to the current injection contact, resulting in a peak in the voltage. A weaker higher harmonic at w/rc ≈ 4 can also be seen for the 10 µm and 15 µm square sizes. As the square size increases, the amplitude of the negative dip decreases. In these larger squares, fewer electrons transmit across the square diagonal without scattering, corresponding to a weaker ballistic signal compared to the Ohmic background. Figure 5.17b shows the behaviour for two diminished orientation squares, Square D1 between 7 µm and 40 µm and Square D2 between 50 µm and 95 µm. At the smallest square sizes, there is a very large difference between the two bend voltages. At zero field, the Bend 2 measurement is qualitatively similar to those in Square E1, albeit with a smaller magnitude of the dip due to the weaker collimation. The Bend 1 measurement, however, has a peak at zero field. At larger magnetic field values, above around w/rc = 2, the behaviour of both bend voltages becomes nearly identical. This suggests that guiding trajectories, as shown in Fig. 5.10d, dominate. A number of focusing peaks, both the initial peak at w/rc ≈ 2, and harmonics at even-integer

114

5 Ballistic Transport in Square Junctions of Delafossite Metals

Fig. 5.18 The magnetic field dependence of VB1,B2 t/I at 5 K for Square D2, which had a starting size of 95 µm

70 65

Square D2 Diminished

Bend 1

50 µm 95 µm

60 Vt/I (µΩµm )

40 µm

75 µm

55 95 µm

50

75 µm

45

50 µm 40 µm

40

Bend 2

5K

35

-6

-4

-2

0 w/rc

2

4

6

multiples of this value, can be seen in both voltages. In neither of the squares are there any signs of voltage fluctuations stemming from quantum interference effects, such as seen in Refs. [75, 76], suggesting that the behaviour has a semiclassical origin. The form of these features may be understandable, but their scale and the length scales at which they are observed are surprising. Nearly all of the previous studies concentrated on devices smaller than the mean free path. Here, even the smallest PtCoO2 square, at 7 µm, is larger than the typical 5 µm mean free path of our crystals and would therefore remains outside the ballistic regime in the most simplistic definition. Despite this, the size of the peak in the 7 µm square is around a factor of ten larger than the Ohmic background: it is an extremely strong effect. Even at a square size of 40 µm, there is still a large difference between the bend voltages in the diminished orientation, equating to 50% of the background signal. To highlight the behaviour at even larger square sizes, Fig. 5.18 shows the data from Fig. 5.17b only for Square D2, which had the larger initial size of 95 µm. The peak and the trough clearly exist up to a square size of at least 50 µm. Surprisingly, however, there remains a difference between the two bend voltage measurements even at a 95 µm square size, which is over 15 times the average mean free path in PtCoO2 . This extended nature of the ballistic signal has not been reported in other materials, and its possible origins will be considered in more detail in Sect. 5.5.

5.3.4 Linearity of Bend Resistance Despite the signatures of a strong ballistic effect demonstrated in the previous sections, a number of checks must be made regarding the origin and nature of this signal. For example, nonlinearities in the relationship between voltage and current are not uncommon within the ballistic regime for junctions formed from semiconductor heterostructures [45] and graphene [77]. These effects are often attributed to ‘hot’ electrons injected with an energy higher than the equilibrium value, which can con-

5.3 PtCoO2 and PdCoO2 Square Junctions 500

Vt/I (µΩµm)

Fig. 5.19 The magnetic field dependence of VB1,B2 t/I at 5 K for Square D1 with a 7 µm side length for 0.09, 0.9 and 9 mA currents

115 Current 9 mA 0.9 mA 0.09 mA

Square D1 Diminished 7µm 5K

400

300

200

100 -2

-1

0

1

2

Field (T)

sequently have different transport properties such as a shorter mean free path [78]. The energy scales in metallic delafossites are different due to the significantly larger carrier density, meaning that, at 5 K, k B T /E F is around 1 × 10−4 in PtCoO2 compared to 0.03 in GaAs-based heterostructures. Nevertheless, the presence or absence of any nonlinearity should be determined. Figure 5.19 shows the magnetic field dependence of VB2 t/I at 5 K for the 7 µm Square D1 at three different currents: 9 mA, 0.9 mA and 0.09 mA. At no point is there a significant change in the resistance between these three current values, suggesting that there is no change in the ballistic behaviour with current and the signal is linear over these two orders of magnitude in current. As well as demonstrating that there is no significant impact of electron heating on the signal, this linearity should ensure that the transmission coefficients should follow the reciprocal relation, Eq. 5.23, as was described in Sect. 5.2.4. This will be shown in Sect. 5.4 to contribute in constraining the experimental signal to have certain symmetries.

5.3.5 PdCoO2 Square Another important test is to observe the changes to the signal when the mean free path is varied, as the ballistic behaviour should be highly sensitive to this quantity. All of the square junction experiments described previously have been in PtCoO2 , where the mean free path is approximately 5 µm. The compound PdCoO2 has a longer mean free path, typically 20 µm, which should lead to a larger range of square sizes over which the ballistic effects can be observed. The behaviour within a PdCoO2 square is therefore important to measure in order to ascertain whether this expected extension is observed. Accordingly, a diminished orientation PdCoO2 square, Square PdD1, was fabricated with a large initial size of 75 µm. Figure 5.20 shows the temperature dependence and dependence on w/rc at 5 K of the two bend voltages within this square as the side length was gradually reduced from 75 µm to 15 µm. Due to an instrumental issue, there is no temperature dependence

116

5 Ballistic Transport in Square Junctions of Delafossite Metals

a) 1000

600

100

Vt/I (µΩµ m)

800

Vt/I (µΩµ m)

b)

Square size (w) Bend 1 Bend 2 75 µm 75 µm 50 µm 50 µm 30 µm 30 µm 20 µm 20 µm

400 200 0

40 60 Temperature (K)

0

5K

-200 20

50

-50

Diminished Square PdD1

0

Square size (w) Bend 1 Bend 2 75 µm 75 µm 50 µm 50 µm 30 µm 30 µm 20 µm 20 µm 15 µm 15 µm

Square PdD1 Diminished

-6

80

-4

-2

0

w/r

2

4

6

c

Fig. 5.20 a The temperature dependence of t V B1,B2 /I for a PdCoO2 square with the diminished orientation, Square PdD1, at square sizes between 75 µm and 20 µm. b) The dependence of t VB1,B2 /I on w/rc at 5 K in the same square at square sizes between 75 µm and 15 µm

data at 15 µm. It is clear that the ballistic effects begin to strengthen at a significantly larger square size than within PtCoO2 , as expected from the longer mean free path. The form of the ballistic signal and the difference in behaviour between the two bend measurements, however, is qualitatively similar to in PtCoO2 . There is a small peak feature at around w/rc = 0.3, present in the Bend 2 measurement in both materials but more prominent in PdCoO2 , stemming from an unknown geometric resonance. The voltage oscillations due to electron focusing are stronger in the PdCoO2 square, likely due to the longer mean free path and the large ratio of w/c. If these are followed to higher values of w/rc than in Fig. 5.20, it can be seen that, in the smallest squares, counterintuitively, there are a smaller number of oscillations.

a) 120

b)

100

Vt/I (µΩµ m)

80

Square PdD1 Dimin. Bend 2

Square size (w) 50 µm 30 µm 20 µm 15 µm

2rC

60 40 20 0

c

5K

-20 -3

-2

-1

0

1

2

3

c/rc

Fig. 5.21 The magnetoresistance data of the Bend 2 measurement of Square PdD1 with an x-axis renormalisation as c/rc where c is the contact width

5.3 PtCoO2 and PdCoO2 Square Junctions

117

To understand the reason behind this behaviour, Fig. 5.21a shows the Bend 2 magnetoresistance data plotted on a renormalised x-axis, c/rc where c is the contact width. In this plot, the oscillations cease around c/rc = 2, which is the value at which a full cyclotron orbit fits within the contact. Beyond this point, 2rc < c, as shown in Fig. 5.21b, all orbits will enter the adjacent contact, regardless of the magnetic field value. Therefore, there is no focusing effect. As focusing oscillations occur at multiples of the focusing field B F = 2w/rc , as the square size w decreases, the corresponding value of rc decreases, and therefore the peaks occur at higher field values. In the smallest squares, the field at which c/rc = 2 occurs before there are many multiples of B F . After this point, there is little dependence on the side length of the square, as seen in Fig. 5.21a. Overall, it is clear that the ballistic anisotropy at a larger square size is stronger in PdCoO2 than PtCoO2 , providing evidence of the required close dependence on the mean free path which is characteristic of the ballistic regime. The exact decay of the anisotropic signal will be examined in more detail in Sect. 5.5.

5.3.6 Hall Voltage As well as the bend voltage, the Hall voltage was investigated in some of the initial four terminal junction studies within other materials, resulting in a number of novel signatures of ballistic physics, as reviewed in Sect. 5.2.3. The impact upon the Hall voltage due to the hexagonal Fermi surface of PtCoO2 is therefore interesting to investigate, especially with regard to the symmetries of the signal. It is important to note that, unlike the two bend voltages, the Hall voltage is constrained by the Onsager relations. From the geometry of the Fermi surface with respect to the square diagonals in the enhanced orientation, as shown in Fig. 5.22, it may be expected that the two Hall measurements shown, V13,42 and V42,13 , would have very different behaviour, due to the enhanced transmission between contacts 1 and 3 indicated by the purple arrow. As will be shown below, however, the Onsager reciprocal relations ensure that V13,42 (B) = V42,13 (−B) and vice versa. Typically the Onsager reciprocal relations for electrical conductivity are expressed in terms of the components of the conductivity tensor, σi j

b) ΔV

-

FS

+ FS

3

B

1

I

- I

2

-

1

+

2

+

a) + - ΔV

Fig. 5.22 Schematics of the measurement configurations for two possible Hall voltage measurements a V13,42 and b V42,13 alongside the enhanced orientation of the PtCoO2 Fermi surface

4

3

B

4

118

5 Ballistic Transport in Square Junctions of Delafossite Metals

σi j (B) = σ ji (−B)

(5.24)

where B is the magnetic field [79]. The conductivity tensor, however, is ill-defined in the ballistic regime due to the lack of a local relationship between the current density and electric field. Despite this, as shown by Büttiker [68], a global resistance symmetry can be derived which does not depend on the local symmetry, Rab,cd (B) = Rcd,ab (−B) Vc (B) − Vd (B) Va (−B) − Vb (−B) = Iab Icd Vab,cd (B) = Vcd,ab (−B)

(5.25) (5.26) (5.27)

where Va is the voltage at contact a and Iab refers to a current from contact a to contact b. Therefore, V13,42 (B) = V42,13 (−B) and the anisotropy which exists between the two bend resistance measurements cannot be present in these two Hall voltage measurements. Although the Onsager relations expressed in Eq. 5.25 constrain the behaviour expected in the two measurement configurations, there are no constraints on the symmetry of each measurement individually in a magnetic field. This can be seen in the Hall voltage measurements at 5 K, shown in Fig. 5.23a and b for both square orientations. At high field, both measurements are linear in field and do not depend on the square size or Fermi surface orientation. The gradient at high field is within 15% of the value in the Ohmic regime. At low field, however, the behaviour diverges. The voltage in the enhanced orientation square, Fig. 5.23a, is antisymmetric in the field at all square sizes. At small square sizes, the voltage is enhanced at small field compared to the linear Ohmic regime behaviour. The Hall voltage in the diminished orientation square, Fig. 5.23b, shows very different behaviour. In particular, there is a finite ‘Hall’ voltage at zero field, suggesting that there is an asymmetry in the transmission to the voltage contacts even at zero field, likely stemming from the Fermi surface orientation. There is also a large enhancement of the Hall voltage over the Ohmic value at small fields. In addition, the overall behaviour is not antisymmetric, providing another demonstration of the anisotropy introduced by the non-circular Fermi surface. This is also in contrast to the bend voltage, which remains symmetric in the field in all Fermi surface orientations. These symmetries will be discussed further in Sect. 5.4. Figures 5.23b and c show the same voltage data as subfigures a and b, but with the x-axis rescaled as w/rc and vertically offset for clarity. In each trace, there is also a linear fit to the data at high field. Here, the oscillations in the data from the diminished orientation square can be seen to align, suggesting they stem from geometric resonances at certain values of w/rc where the transmission into one of the voltage contacts is enhanced. In particular, in both orientations, there is evidence at around w/rc = 2 of a plateau-like feature likely corresponding to the last Hall plateau seen with other materials (Fig. 5.11) and coinciding with the first focusing

5.3 PtCoO2 and PdCoO2 Square Junctions

119

peak in the bend voltage. On this plateau, guiding trajectories dominate and nearly every electron reaches a contact adjacent to the injection contact regardless of the exact field, resulting in a constant voltage. In neither orientation does evidence exist of a suppression of the Hall voltage at very low field, as is experienced in some other materials, which is not surprising given the observed sensitivity to the level of collimation in other materials. Despite this, it is clear that the Hall data demonstrate strongly ballistic behaviour overall and a lack of antisymmetry which has not been observed in other materials. The highly non-anti symmetrical Hall voltage in the diminished orientation squares, however, does not violate Büttiker’s extension of the Onsager relations, presented in Eq. 5.24. The Hall resistance for both possible voltage configurations is shown in Fig. 5.24a for Square D1 at 7 µm and 5 K. Both measurements are highly asymmetric in magnetic field, but the relation R13,42 (B) = R42,13 (−B) is still obeyed, as shown in subfigure b, where the field values for the second measurement are flipped, producing an identical set of data to the first configuration. This ‘textbook’ agreement with the Onsager-Büttiker relation is a powerful self consistency

a)

b)

1.5

1.5

Square size

0.5 0 -0.5 -1

0.5 0 -0.5 -1

Enhanced Square E1, 5K

-1.5 -4

-3

-2

-1

0

1

2

3

40 µm 20 µm 15 µm 10 µm 7 µm

1

VHt/I (mΩµm)

1

VHt/I (mΩµm)

Square size

40 µm 20 µm 15 µm

-1.5 -4

4

Diminished Square D1, 5K -3

-2

-1

Field (T)

c)

d)

1.5

Square E1, 5K

1 20 µm 0.5 40 µm 0

2

3

4

Square D1, 5K

7 µm

3 10 µm 2 15 µm 1 20 µm 0 40 µm Square size

Square size -4

1

5 4

VHt/I (mΩµm)

VHt/I (mΩµm)

15 µm

-6

0

Field (T)

-2

0

w/rc

2

4

6

-6

-4

-2

0

2

4

6

w/rc

Fig. 5.23 a, b The variation of V H t/I where VH = V13,42 as a function of magnetic field at a variety of square side lengths for a) an enhanced orientation square between −2.5 and 2.5 T b a diminished orientation square between −4.5 and 4.5 T. c,d) The data from a and b rescaled in x as w/rc and vertically offset for clarity alongside a linear fit to each curve at high field. Square E1 was fatally damaged before measurements could be completed at a smaller size

120

a)

5 Ballistic Transport in Square Junctions of Delafossite Metals 3

1 0 -1 -2 -3

-6

-4

-2

0

2

4

6

8

Field (T)

5K

V13,42 (B) V42,13 (-B)

1 0 -1 -2

Diminished Square D1, 7 µm -8

3 2

VHt/I (mΩµm)

VHt/I (m )

2

b)

5K

V13,42 (B) V42,13 (B)

-3

Diminished Square D1, 7 µm -8

-6

-4

-2

0

2

4

6

8

Field (T)

Fig. 5.24 a The variance of the Hall voltage with field for the two configurations for the 7 µm Square D1 at 5K. b The data from a but with the data for the second configuration flipped in field, to show the adherence to the Onsager relation V13,42 (B) = V42,13 (−B)

that validates both Büttiker’s work and the fidelity of the data presented here for our squares.

5.4 Landauer-Büttiker Analysis The experimental studies in delafossite metal squares described in the previous section have demonstrated a number of new phenomena not previously observed in ballistic junctions, including a dependence on the Fermi surface orientation and an asymmetrical bend resistance. Within these data, there are number of symmetries which do not seem to immediately follow from the geometry of the device and the Fermi surface. For example, the magnetoresistance is highly symmetric in all cases, despite the fact the electron injection into the square is asymmetric, whereas the Hall resistance is antisymmetric only when the square has the enhanced Fermi surface orientation. This implies that the fundamental laws, such as current conservation and time reversal symmetry, combined with geometric symmetries may be causing the experimental signal to have certain symmetries. Further insight into these symmetries and the physics underlying the data shown in Sects. 5.3.2 to 5.3.6 can be obtained by performing a Landauer-Büttiker analysis of the square junctions, using the method described in Sect. 5.2.4. Such analyses of the bend and Hall voltages within four terminal junctions are common, for example by Bandyopadhyay [80], but all assume four fold symmetry between the contacts. Therefore, these analyses are not usually applicable when the Fermi surface symmetry is non-circular. In the analysis here the symmetry will be reduced to two-fold to accommodate the hexagonal delafossite Fermi surface in the enhanced and diminished orientations.

5.4 Landauer-Büttiker Analysis

121

The Büttiker multiprobe formula (Eq. 5.28) can be used to show that the net current into a contact in a four point measurement at zero temperature is ⎤ ⎡ 4  2e2 N ⎣ Ii = Ti j (μ F )μ j ⎦ (1 − Ri )μi − h j=1, j=i

(5.28)

where μ j is the chemical potential at contact j, N is the assumed-equal number of modes in each contact, Ti j is the probability an electron emitted from contact j is transmitted to contact i and, similarly, Ri is the probability an electron emitted from contact i is reflected back into the same contact [68]. An important note is that N = k F c/π ≈ 12,000 with a typical c = 4 µm contact width in our PtCoO2 devices, which is very different to the few modes in the initial 2DEG devices. Despite this, strongly ballistic behaviour is still observed. The transmission, Ti j , and reflection, R j , coefficients are normalised such that Rj +



Ti j = 1

(5.29)

i,i= j

which essentially ensures that an electron emitted from a contact is either transmitted to another contact or reflected back into the emission contact. The relevant form of Eq. 5.28 for each of the four contacts can be combined to form a matrix equation, ⎡

⎤ I1 ⎢ I2 ⎥ 2e2 N ⎢ ⎥= ⎣ I3 ⎦ h I4

⎡

TA ⎢ −T21 ⎢ ⎣ −T31 −T41

−T12 TB −T32 −T42

−T13 −T23 TA −T43

⎤⎡ ⎤ −T14 μ1 ⎢ μ2 ⎥ −T24 ⎥ ⎥⎢ ⎥ −T34 ⎦ ⎣ μ3 ⎦ TB μ4

(5.30)

 where the normalisation, Eq. 5.29, ensures T A = 1 − R1 = j, j=1 T j1 and equivalent expressions for TB,C,D for contacts 2 to 4. In both of the experimental orientations, contacts 1 and 3 and, separately, contacts 2 and 4 are equivalent in terms of their orientation relative to the Fermi surface, as can be seen in Fig. 5.25, and so their transmission coefficients are identical. Therefore, the overall number of transmission coefficients in Eq. 5.30 can be reduced, namely, T21 = T43

(5.31a)

T41 = T23 T32 = T14

(5.31b) (5.31c)

T34 = T12 T31 = T13

(5.31d) (5.31e)

T42 = T24 T A = TC

(5.31f) (5.31g)

122

5 Ballistic Transport in Square Junctions of Delafossite Metals

a)

b) 2

R2

T12 T21

T32 T 42 T13 T R3

23

3

T43

T31

T24

1

R1

T41 T14

R4

T34

4

I

Bend 1 -

c)

+ 1

2 T23

+ I Bend 2

Bend 1 -

T41

3

4

- ΔV +

+ 1

2

+ I -

+ ΔV -

FS

I

Bend 2

FS

3

+ ΔV 4

- ΔV +

Fig. 5.25 a The transmission coefficients from each of the contacts in a square device. The colours denote coefficients made identical by the Fermi surface symmetry in both experimental orientations. The measurement configuration for bend voltage in b a diminished orientation device which is likely to enhance T23 and T41 and c an enhanced orientation device, likely to enhance T13 and T31

TB = TD

(5.31h)

at all values of magnetic field. These substitutions reduce Eq. 5.30 to ⎡ ⎤ TA I1 ⎢ I2 ⎥ 2eN ⎢ −T21 ⎢ ⎢ ⎥= ⎣ I3 ⎦ h ⎣ −T31 I4 −T41 ⎡

−T34 TB −T32 −T42

−T31 −T41 TA −T21

⎤⎡ ⎤ −T32 μ1 ⎢ μ2 ⎥ −T42 ⎥ ⎥⎢ ⎥ −T34 ⎦ ⎣ μ3 ⎦ TB μ4

(5.32)

where there are now eight distinct, field-dependent transmission coefficients. In addition to the symmetries of the transmission coefficients due to the geometry, there are also constraints due to the more general symmetries discussed in Sect. 5.2.4. Current conservation, Eq. 5.33, ensures each row and column sums to the same value, for example, T21 (B) + T41 (B) = T34 (B) + T32 (B)

(5.33)

when the first row and column are considered. Furthermore, time reversal symmetry, Eq. 5.23, leads to Ti j (B) = T ji (−B) for all values of i and j. As shown below, these transmission coefficient constraints will lead to symmetries in the experimental voltage. Although this is a zero temperature expression, a finite temperature calculation effectively involves an average over E = 3.5k B T at a temperature T , providing the thermal energy k B T is small compared to the Fermi energy E F [81]. Due to the large E F in a metallic delafossite compared to many semiconductor systems, at 5 K E/E F is around 4 × 10−4 . It would therefore be unlikely that there would be a significant difference in the transmission coefficients over this energy range due to a variation in, for example, the scattering or the cyclotron radius, justifying the use of a zero temperature expression.

5.4 Landauer-Büttiker Analysis

123

5.4.1 Calculation of the Bend Voltages Equation 5.32 describes a set of four linear equations which must be solved for each of the experimental measurement configurations. Generally, the approach is to set the net current at the voltage contacts to be zero due to their high resistance. The value at the current contacts is determined by the experimental configuration. These constraints are then used to determine the chemical potential at each contact and, consequently, the predicted experimental voltage difference. For the first bend voltage, the current flows between contacts 1 and 2 and the voltage difference is measured between contacts 3 and 4, I = [I, −I, 0, 0]. Without loss of generality, the chemical potential at one of the voltage contacts can be set to zero, in this case μ4 = 0, reducing the matrix dimensions to 3 × 3. Solving Eq. 5.32 for the chemical potentials with these constraints gives a bend resistance, VB1 (B), which follows μ4 − μ3 VB1 (B) = I eI h T41 T32 − T31 T42 = 2 2e N D(T41 + T21 )

(5.34) (5.35)

where D = T41 T34 + T21 T32 + T34 T31 + T32 T31 + T41 T42 + T21 T42 + 2T31 T42 . Using the symmetries expressed in Eqs. 5.33 and 5.23, it can be shown that D is symmetric in the magnetic field. The sign of the bend resistance is therefore determined by the proportion of electrons which transmit between contacts linked vertically in Fig. 5.25a, T41 T32 , compared with the fraction which transmit along the square diagonal, T31 T42 . If more pass along the diagonal, the bend voltage becomes negative. The above calculation determines the voltage in a single 2D layer. In a delafossite metal, there are effectively multiple 2D layers in parallel, with the number equal to N L = t/(d/3) where t is the sample thickness and d is the c-axis lattice constant. In order for a comparison with the experimental voltage, the voltage calculated from the Landauer-Büttiker calculation would need to be divided by N L , t = VB1

VB1 d 3t

(5.36)

which reduces the voltage significantly, but does not affect the sign. At a negative magnetic field, VB1 (−B) h T32 (B)T41 (B) − T31 (B)T42 (B) = 2 I 2e N D(T32 (B) + T34 (B)) VB1 (B) = I

(5.37) (5.38)

as T21 (B) + T41 (B) = T34 (B) + T32 (B) from Eq. 5.33. Despite the asymmetry that may be expected from the diminished orientation in Fig. 5.25c, where electrons are

Fig. 5.26 The variation of V t/I with magnetic field where V = VB1,B2 for Square D1 at 5K with w =10 µm, with and without symmetrisation of the data around zero field

5 Ballistic Transport in Square Junctions of Delafossite Metals 500

Bend 1 sym. Bend 1 non-sym. Bend 2 sym. Bend 2 non-sym.

5K, 10 µm Square D1 400 Diminished

Vt/I (µΩµm)

124

300 200 100 0 -100 -5

-4

-3

-2

-1

0

1

2

3

4

5

B (T)

injected at different angles with respect to the two sides of the square, in a device with perfect symmetry, the signal is predicted to be symmetric in magnetic field. This is seen within the experimental data. Figure 5.26 shows the magnetic field dependence of the bend voltages in the 10 µm Square D1 at 5 K, both with and without symmetrisation. There are only small asymmetries, likely due to slight differences in the fabricated contacts. For the second bend voltage, I = [0, I, −I, 0] and therefore, following the same procedure, μ1 − μ4 VB2 (B) = I eI h T34 T21 − T31 T42 = 2 2e N D(T41 + T21 )

(5.39) (5.40)

which is not the same as VB1 (B)/I unless T34 T21 = T32 T41 . For this bend voltage, the signal is negative when a larger proportion of electrons are transmitted between horizontally linked contacts (T34 T21 ) in Fig. 5.25a than along the square diagonals. Only in the case where transmission between horizontally and vertically linked contacts in Fig. 5.25a is equally likely, meaning T34 T21 = T32 T41 , such as with a circular Fermi surface, are the two bend voltages equal. The conclusion here is that the Bend 1 signal is negative if more electrons traverse between diagonally linked contacts rather than vertically linked ones in the illustration in Fig. 5.25. Similarly, the Bend 2 signal is negative if more traverse diagonally rather than horizontally. These two bend voltages are only the same if horizontal and vertical transmission are equally likely. For circular Fermi surfaces in the ballistic regime, the enhancement of diagonal transmission ensures a negative voltage. The probabilities of vertical and horizontal transmission are smaller but identical, meaning the two bend voltages remain the same. Similarly, for PtCoO2 and PdCoO2 , for the enhanced orientation, diagonal transmission is most likely, and horizontal and vertical transmission are lesser but equally

5.4 Landauer-Büttiker Analysis

125

likely, resulting in two identical, negative bend voltages. For the diminished orientation, however, the probability of vertical transmission is greatly enhanced, with diagonal transmission remaining more likely than horizontal transmission. These means the two bend voltages are no longer identical and only Bend 1 is negative.

5.4.2 Calculation of the Hall Voltage The Hall voltage can be calculated via the same method, wiht the measurement configurations now as in Fig. 5.27. In the first Hall configuration, Fig. 5.27a, the current now flows between contacts 1 and 3, I = [I, 0, −I, 0]. This leads to a voltage, VH 1 (B), which follows VH 1 (B) μ4 − μ2 = I eI h T41 − T21 . = 2 2e N D

(5.41) (5.42)

As expected, the Hall voltage is determined by the difference between the transmission probability to the contact immediately left of the emission contact, T41 , compared to the one immediately to the right, T21 . At negative field, VH 1 (−B) h = 2 I 2e N h = 2 2e N

1 (T41 (−B) − T21 (−B)) D 1 (T32 (B) − T34 (B)). D

(5.43) (5.44)

The Hall voltage is therefore antisymmetric in magnetic field only if the anticlockwise and clockwise transport coefficients can be related by inverting the field, T34 (B) = T32 (−B) and T32 (B) = T34 (−B). This is likely true in the enhanced ori-

b) Hall 2

+ T13

T41

B

1

FS

FS

3

T31

+

ΔV

+

+ - ΔV

- I

2

I

1 T23

-

2

-

a) Hall 1

4

3

B

4

Fig. 5.27 The contact configuration for a the first Hall measurement b the second Hall measurement, with the a diminished and b enhanced Fermi surface orientations indicated. The key transmission coefficients enhanced by these orientations are indicated. This does not imply an exclusive link between orientation and measurements: data was taken in both configurations for each orientation

126

5 Ballistic Transport in Square Junctions of Delafossite Metals

entation, where there is a line of symmetry of the Fermi surface along the diagonal of the square. It is not likely to be true, however, in the diminished orientation, when the same symmetry line is instead orientated parallel to an edge of the square. This behaviour is seen in the experimental data. Unlike the symmetry required of the bend voltage, the symmetries of the transmission coefficients do not ensure any antisymmetry in the magnetic field. For the second Hall measurement configuration, as shown in Fig. 5.27b, the current flows between contact 4 and 2, I = [0, −I, 0, I ]. This configuration is chosen to ensure the sign of the voltage in the Ohmic regime is consistent with the experimental measurement. μ1 − μ3 VH 2 (B) = I eI h T32 − T34 = 2 2e N D VH 1 (−B) = I

(5.45) (5.46) (5.47)

Considering this voltage at negative field, VH 2 (−B) h 1 = 2 (T32 (−B) − T34 (−B)) I 2e N D h 1 = 2 (T41 (B) − T21 (B)) 2e N D VH 1 (B) = I

(5.48) (5.49) (5.50)

Therefore, regardless of the Fermi surface orientation, VH 2 (−B) is always the same as VH 1 (B) in this Landauer-Büttiker model, as must be true in the linear regime due to the Büttiker extension to the Onsager relations expressed in Eq. 5.24. In conclusion, the Landauer-Büttiker analysis presented here provides insight as to the symmetries and anisotropies of the experimental data. In particular, it shows that the anisotropy of the bend resistance measurements in the diminished orientation can be explained by an anisotropy in the transmission probabilities parallel to each of the square sides. Similarly, the negative bend resistances are due to a high probability of transmission along the square diagonal. The symmetry of the bend resistance in the magnetic field found experimentally has been shown to be due to a combination of the fundamental symmetries of the transmission coefficients, due to current conservation and time reversal symmetry, and the geometric symmetry of the square with respect to the Fermi surface. Conversely, this analysis demonstrates that these symmetries do not lead to antisymmetry being required in the Hall resistance, in agreement with the data shown in Fig. 5.23. Antisymmetry only occurs when there is a line of symmetry of the Fermi surface parallel to the square diagonal, which, in our study, is only in the enhanced orientation. Time reversal symmetry and the Onsager relations, however, are always obeyed within this analysis, as they are within the experiment.

5.5 Decay of Ballistic Anisotropy

127

5.5 Decay of Ballistic Anisotropy

As well as the nature and symmetry of the novel ballistic phenomena, an important limit to determine in any ballistic regime device is the manner in which a given effect decays as the various length scales are changed. As shown in Sect. 5.3.3 and discussed again in Sect. 5.5, it is clear that the novel effects in PtCoO2 squares can be observed over a length scale far longer than the typical 5 µm mean free path. The exact form of the decay is therefore important to ascertain. Here two methods will be used to examine this decay: changing the square size and changing the mean free path.

5.5.1 Determination of Mean Free Path To compare the geometric and electronic transport length scales between different squares, it is important to accurately determine the mean free path within each device. The average transport mean free path, derived from the resistivity, is around 5 µm in PtCoO2 , but the value varies between each crystal due to differences in the concentration and form of growth defects and impurities. In the Ohmic regime, for a material with resistivity ρ, the transport mean free path, l, which is an average around the Fermi surface, is l=

m∗vF ne2 ρ

(5.51)

where m ∗ is the effective mass of the transport electrons, v F is the Fermi velocity, e is the electronic charge and n is the density of carriers [82]. The derivation of this equation, however, relies upon the existence of a local relationship between the current density and electric field that does not exist in the ballistic regime (see discussion in Sect. 5.1.1). This intrinsic non-locality also means that the resistivity is ill-defined. In fact, if Eq. 5.51 was applied to some of the zero field data within the square devices, the calculated mean free path would be negative. The desired mean free path is the value that would occur in the same crystal but within a bulk device, defined as being when the boundary scattering does not significantly contribute to the overall resistance and the transport behaviour is within the Ohmic regime. This property is nontrivial to determine when, due to the long lived ballistic effects, most of the square devices had an initial size around 30 µm, at which ballistic effects were already important. At high field, however, many of the ballistic effects are suppressed. There is no clear dependence on the square size, Fermi surface orientation or bend measurement configuration, suggesting that the behaviour is approximately Ohmic. This is supported by the fact that the cyclotron radius above 5 T is below 1.25 µm, less than half the width of the thinnest contacts and thus suggesting that the electrons are less able

128

5 Ballistic Transport in Square Junctions of Delafossite Metals

to sense the experimental geometry. If the behaviour at these field values is assumed to be Ohmic, the high field resistance and bulk sample magnetoresistance can be used to estimate the Ohmic contribution to the resistance of such a square at zero field. This would be the zero field resistance if the ballistic effects were negligible. Expressions from the Ohmic regime can then be used with this value to determine the resistivity and the mean free path. Therefore, to estimate the mean free path, the resistance at points between 6 and 9 T was measured. The magnetoresistance, known from a previous study in bulk devices [74], was used alongside this data to estimate the Ohmic resistance contribution at zero field. The resistivity was calculated via the Van der Pauw equation, Eq. 5.5, and the mean free path was then determined using the Ohmic expression, Eq. 5.51. The result of this procedure for each of the square devices is displayed in Table 5.2. In large PtCoO2 squares, such as 95 µm, the weakened nature of the ballistic effects means that a reasonable estimate of the bulk resistivity can be made directly by assuming the zero field behaviour is near to Ohmic, with a negligible ballistic contribution. The mean free path can then be determined by using the Van der Pauw equation, Eq. 5.5, and Eq. 5.51 directly with the zero field resistance. Results for this method for the two larger PtCoO2 squares are also shown in Table 5.2. In each case, the calculated mean free path agrees within uncertainty between methods, suggesting that Method 1 is a valid approach, yielding accurate estimates of the mean free path. To provide further confidence in this method, nearly all the mean free path values in the PtCoO2 squares are around the established 5 µm average, with only that of Square D5 being significantly shorter. This crystal was from a batch verified from other studies to have a consistently lower purity compared to the other PtCoO2 crystals, which were from the same high purity batch.

Table 5.2 Mean free path at 5K of the square devices as determined by a method using the high field resistance (Method 1) and by zero field data from square sizes larger than 95 µm (Method 2) where possible. Errors determined from the variance between multiple data sets from different square sizes Sample Method 1 l (5K) (µm) Method 2 l (5K) (µm) E1 D1 D2 D3 D4 D5 PdD1

4.8 ± 0.1 5.5 ± 0.8 6.1 ± 0.4 4.88 ± 0.04 5.6 ± 0.1 2.60 ± 0.05 17 ± 2

– – 6.1 ± 0.5 – – 3.1 ± 0.5 –

5.5 Decay of Ballistic Anisotropy

129

5.5.2 Decay with Square Size One method to examine the decay of the ballistic regime anisotropy is to change the square size. As stated previously, the FIB-based microstructuring method has an advantage in that the square size can be reduced after measurement and then a smaller size can be studied. In other junction studies, the depth of the trough in the magnetoresistance at zero field is often taken as an indicator of the size of the ballistic effects. In a similar vein, the difference between the bend voltages in the diminished orientation squares at zero field and 5 K, V t/I , will be used here. An example is shown in Fig. 5.28a. An advantage of this choice is that, within the Ohmic regime, V = 0. This is not the case for the trough alone, which remains finite in the Ohmic regime if the material has a bulk magnetoresistance, as is the case for the delafossite metals. Therefore, the separation of the ballistic contribution from the Ohmic contributions is automatic only when the difference V is used, ensuring this is a highly sensitive parameter for gauging the size of the ballistic signal. Figure 5.28b shows the variation of V t/I with the ratio, w/l, of the square side length, w, to the mean free path l calculated in the previous section, for the four diminished orientation squares with a similar purity. The decay rate is similar between different squares but there are clearly two distinct regimes. At the smallest square sizes, there is a rapid decay. Around w/l ∼ 5, however, the decay becomes much slower. This slower decay can be tracked to at least w/l = 15, which is far beyond the traditional limits of the ballistic regime of w/l = 1. There is a good agreement between the data and the exponential decay V t bw = Ae− l I

a) 600

b)

Square D1 7 µm 500 Diminished

(5.52)

103

Square Data Fit D1 D2 D3 D4

ΔVt I

300

ΔV t/I (µΩµm)

Vt/I (µΩµ m)

400

200 100

10

2

10

1

10

0

0 -100 5K -200

-6

-4

-2

0 w/rc

2

4

6

0

5

10

15

w/l

Fig. 5.28 The decay of the difference between the peak and trough, V t/I as a function of the ratio of square size and mean free path for a number of diminished orientation squares

130

5 Ballistic Transport in Square Junctions of Delafossite Metals

where A is a constant and b has two different constant values, b S and b F , in different ranges of w/l. Specifically, at w/l < 5, b F = 0.65 and at w/l > 5, the decay is a factor of three slower, with b S = 0.21. The origin of this double decay is uncertain. On a qualitative level, this double decay appears to agree with that of the peak measured within the transfer junction geometries described in Sect. 5.2.3 (with data shown in Fig. 5.12). Like the anisotropy in delafossite metals, this is a sensitive parameter to measure the scale of ballistic effects, as the peak height is zero in the Ohmic regime. In these structures, the rapid decay was attributed to boundary scattering, whereas the slower decay was considered to be due to scattering in the bulk from defects, which may be the cause of the double decay in the delafossite square junctions. It must be noted, however, that the previously reported decay lengths, even for the slower decay, were always smaller than the mean free path l, whereas here, the characteristic decay length, l/b, is 1.5l for the rapid decay and 4.8l for the slow decay. In PtCoO2 , there seems to be an enhancement to the range over which these ballistic effects occur. A possible reason for this extended range of the ballistic effects is the non-circular Fermi surface. All of the transfer junction studies reported have been performed on materials with circular Fermi surfaces. As shown in Fig. 5.29, if a collision between an electron on a circular Fermi surface and a defect occurs in the bulk of a device, there is a vanishingly small chance the direction of motion after the scattering is similar to the pre-collision direction. For a hexagonal Fermi surface, there are only six primary electron directions, and so the chance of the direction of motion remaining nearly the same post-collision is significantly increased. This property may increase the likelihood of maintaining the initial trajectory even in the presence of some scattering, enhancing the range over which ballistic effects can be observed. Although the previous discussion suggests that w/l is the parameter which controls the decay rate, all of the data in Fig. 5.28 is from samples of PtCoO2 which have a similar l. To further establish the reliability of the decay in w/l, it is helpful to make a comparison with a case where the mean free path is significantly different. In the measured data there are two possibilities: the lower purity PtCoO2 square, Square D5, and the PdCoO2 square, Square PdD1. Figure 5.30 shows the decay of the signal for both of these squares, alongside Squares D1 and D2. For both squares, there is an offset in the y-axis, which relates to the overall size of the signal compared to the data from the high purity PtCoO2 . The reason is unknown, but is possibly related to factors such as the ratio of contact and square size, which is different between each set of squares due to the large variance in square size.

Fig. 5.29 An illustration of the possible directions of motion post-scattering for a circular and a hexagonal Fermi surface Collision

With a circular Fermi surface

With a hexagonal Fermi surface

5.5 Decay of Ballistic Anisotropy 10

ΔV t/I (µΩµm)

Fig. 5.30 The decay of the difference between the two bend voltages in the diminished orientation, V t/I , as a function of the ratio of square size and mean free path for three PtCoO2 squares and a PdCoO2 square

131 3

Square Data Fit D1 D2 D5 PdD1

102

101

10

0

0

5

10

15

w/l

Despite the difference in signal size, for Square D5, the rate of the slow decay, fitted to the data with w/l > 5, is b S (Square D5) = 0.20, within 10% of the value for Square D2, suggesting this is a common decay rate and that w/l controls the rate. There is some evidence of a rapid decay at w/l < 5, but the small mean free path prevented fabrication of squares with a ratio of w/l sufficiently small to fully compare this decay to other squares. The smallest side length for Square D5 in Fig. 5.30 has w =10 µm and squares smaller than this value would have w/c < 3, where effects of the finite contact width begin to dominate the signal. For the PdCoO2 square, the rapid rate of decay at small w/l is very similar to that of the high purity PtCoO2 squares, with b F (Square D1) = 0.65 and b F (Square PdD1) = 0.61, despite the absolute values of w and l being around four times larger in PdCoO2 than PtCoO2 . The square sizes measured in Square PdD1 are not large enough to reach the slow decay regime, which would require values of w in the range of 150–200 µm, not feasible from the current possible crystal sizes when the size of the homogeneity meanders is also considered. These observations further demonstrate that the decay rate is determined by the ratio w/l, rather than the value of w alone, across both a range of mean free paths and different delafossite materials. This is additionally supported by measurements where this ratio was varied by increasing the temperature, and therefore decreasing l, rather than changing the square size, as described in Appendix A, where a very similar decay rate is seen as a function of w/l. In this appendix, the rate is also shown to be independent of the source of the scattering, unlike the Ohmic regime studies in Sect. 4.6 where different behaviour occurred with defect scattering than with phonon scattering. This is another example of the significant differences between the Ohmic and ballistic regimes in terms of their transport behaviour.

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5.6 Conclusions In conclusion, this chapter has explored the behaviour of micron scale squareshaped PtCoO2 and PdCoO2 junctions. These devices have shown strongly ballistic behaviour, in accordance with their micron scale mean free paths and studies of other ultrapure materials, but have also demonstrated a novel resistance anisotropy not observed in other materials. This anisotropy in the ballistic regime stems from their non-circular Fermi surfaces and is present in both bend and Hall resistance measurements. The level of anisotropy can be controlled by the size of the square and the orientation of the Fermi surface. An analysis using the Landauer-Büttiker equation shows that current conservation and time reversal symmetry, alongside geometric symmetries, produce several strong constraints for the symmetries of these measurements, all of which are reflected in the experimental data. The scale of the anisotropy is a sensitive parameter for the size of the ballistic effects. This parameter has been demonstrated to decay with a rate that depends only upon the ratio of the mean free path to the square side length. There are two distinct decay regimes distinguished by the size of this ratio. These studies have shown, for the first time, that ballistic behaviour in the delafossite metals persists far outside the point where the mean free path and square side length are equal, possibly due to the non-circular Fermi surface. This behaviour therefore highlights the importance of both including an accurate representation of the Fermi surface in theoretical modelling of ultrapure materials and considering the possible unconventional and long range nature of any observed ballistic effects when studying transport within these materials, especially in those with non-circular Fermi surfaces.

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Chapter 6

Conclusions and Outlook

In conclusion, this thesis has explored both the origins of the ultrahigh conductivity of the delafossite metals PdCoO2 and PtCoO2 and the impact their properties have on electronic transport on the scale of the mean free path. These materials are ideally placed to test many of the assumptions made regarding transport within ultraclean materials. They have metallic carrier densities, quasi-2D transport and a hexagonal Fermi surface: a unique set of properties within ultrapure materials. The origins of the low resistivity were explored by a defect introduction study performed using high energy electron irradiation. By measuring the dependence of the rate of change of the resistivity during irradiation on the electron energy and comparing this with predictions of theoretical models of atom displacement, the added concentration of defects introduced was calculated. The observed change in resistivity is as expected from unitary scattering models: there is no evidence of a significant suppression of scattering. The low resistivity instead stems from an extremely small defect density in the metallic layers: as small as 1 in 120,000 Pd atoms in PdCoO2 and 1 in 30,000 Pt atoms in PtCoO2 . This is a remarkably small value, especially for as-grown oxide materials synthesised without any purification steps, and the underlying reasons should be investigated in order to provide the insight needed for synthesis of further ultrapure materials. The calculation of the stability, equilibrium stoichiometry and defect energies of materials is at forefront of modern theoretical condensed matter physics [1], and the data presented here will hopefully aid the refinement of such calculations. Even in the absence of theory, the work strongly motivates follow up studies on materials in the delafossite structure to see whether it is possible to grow insulating or semiconducting delafossites with similarly low defect levels. The measurements and calculations necessary for this study also led to electron irradiation becoming a method which can be used to introduce a known quantity of homogeneously distributed point defects into delafossite metals. This ability is powerful and will allow for future experiments where the impact of these defects on transport in unconventional regimes, such as the ballistic regime, is studied. In our © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. H. McGuinness, Probing Unconventional Transport Regimes in Delafossite Metals, Springer Theses, https://doi.org/10.1007/978-3-031-14244-4_6

137

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6 Conclusions and Outlook

work, however, this enabled a detailed study of the effects of additional defect scattering on the electronic transport in the Ohmic regime. A violation of Matthiessen’s rule and unconventional effects in the magnetotransport demonstrate that defect scattering and phonon scattering have different impacts on the electronic transport, which can be qualitatively explained by the scattering time anisotropy around the Fermi surface varying between the two mechanisms. Again, the data strongly motivate theoretical work, this time a full quantum transport calculation including a realistic parameterisation of the Fermi surface and the full phonon density of states. In addition to this information gained on the origin of their unusual conductivity, this thesis has explored the use of the delafossite metals for examining transport within the ballistic regime, when the device geometry is on the scale of the mean free path. Measuring the transport within square shaped junctions has demonstrated a novel resistance anisotropy, which only exists within the ballistic regime and is attributable to the non-circular Fermi surface. This anisotropy is a highly sensitive gauge for the strength of the ballistic behaviour within a material and has shown that this behaviour remains present, in a diminished form, at a square size over 15 times larger than the mean free path. This study has therefore reiterated that the shape of the Fermi surface can have profound effects on the phenomena observed within the ballistic regime and must be considered within any model of such systems. In particular, the scale over which ballistic effects are observed may be enhanced by this property, meaning that the Fermi surface must be considered when looking for evidence of unconventional transport effects on larger scales, such as those of electron hydrodynamics. In general, the studies described within this thesis show that a relatively small deviation from a circular Fermi surface can have a profound impact upon the transport behaviour, in the Ohmic and particularly in the ballistic regimes, and therefore stress the importance of making models as comprehensive as possible when interpreting experiments which search for novel physics.

Reference 1. Freysoldt C, Grabowski B, Hickel T, Neugebauer J, Kresse G, Janotti A, Van de Walle CG (2014) First-principles calculations for point defects in solids. Rev Mod Phys 86(1):253–305

Appendix A

Decay of Bend Resistance Anisotropy with Increased Temperature

As described in the main text, the ratio w/l plays a key role in determining the observed properties of the square devices. In Sect. 5.5, two methods for varying this ratio were described: reducing w using a FIB or measuring crystals with a different l from growth. There is a third way to change w/l, namely by varying the temperature. Increasing the temperature from the base 5 K introduces phonon scattering in the bulk of the crystal, reducing l. As seen in Sect. 4.6, defect scattering affects the magnetoresistance of Ohmic regime PtCoO2 devices differently than phonon scattering. It is therefore important to determine if this is also true for the anisotropic effects, which have a different origin, in ballistic physics. Increasing the temperature introduces a temperature and field dependent Ohmic voltage offset to the bend voltages. As these voltages are identical in the Ohmic regime, the offset is identical and therefore their difference at zero field remains equal to the difference between the ballistic contributions to their voltage. This quantity can therefore still be use to gauge the strength of the ballistic anisotropy, even at elevated temperature. Figure A.1a shows the magnetic field dependence of the bend voltages within Square D1 at a number of temperatures between 5 K and 110 K when the side length was 7 µm. The Ohmic contribution to the voltage increases due to the phonon scattering, mostly clearly at high field. However, the ballistic effects at low field are also suppressed due to the increased scattering, with both the size of the peak and trough and their difference rapidly decreasing as the temperature increases. The decay of this difference, V at zero field (normalised using the sample thickness t and the current I ) is shown in Fig. A.1b as a function of w/l for w/l < 8. At w/l > 8, it is difficult to accurately determine the difference given the large Ohmic background. In the figure, this ratio is varied through either increasing the temperature with a constant 7 µm side length or reducing the square side length whilst maintaining the temperature at 5 K, as described in the main text. The mean free path at elevated temperature was estimated from the resistivity of a bulk sample with a similar mean free path at low temperature as that calculated for Square D1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. H. McGuinness, Probing Unconventional Transport Regimes in Delafossite Metals, Springer Theses, https://doi.org/10.1007/978-3-031-14244-4

139

Appendix A: Decay of Bend Resistance Anisotropy with Increased Temperature

a)

1400

T Bend Bend 2 1

1200

5K 20K 30K 40K 50K 60K 70K 80K 90K 110K

Vt/I (μΩμm)

1000 800 600 400

b)

10 3

Scattering varied by: Square size Temperature Fit to temp. data for w/l < 5

ΔVt/I (μΩμm)

140

10 2

200 0

Diminished Square D1, 7 μm

-200 -6

-4

-2

0 2 Field (T)

4

6

10

1

0

2

4

6

8

w/l

Fig. A.1 a The magnetic field dependence of the bend voltages in Square D1 at a 7 µm size between 5 K and 110 K. b The dependence of V t/I for the same square as a function of w/l where the latter ratio is varied by either changing the square size or the temperature

in Sect. 5.5.1. The mean free path cannot be determined directly from the data for Square D1 at low field as the strongly ballistic signal prevents Ohmic expressions being used. For further discussion of this point, see Sect. 5.5.1. Both methods produce a similar decay, with the rapid increase in the signal occurring at approximately the same value of w/l ∼ 5. The gradient of this fast decay for the temperature-dependent study is b F (D1, Tdep) = 0.84, within 25% of the value derived from the size dependence study in the same square, thus providing further evidence that this ratio determines the strength of the anisotropy. The similar decay also suggests that the origin of the scattering makes little difference to the effect it has on the ballistic anisotropy, at least in the region of rapid decay. Unlike the bulk sample magnetoresistance studied in Sect. 4.6, the scale of the anisotropy does not depend upon whether the scattering is due to either defects, as is the case at 5 K, or phonon scattering, which is dominant at higher temperatures. This highlights that there can be important differences between the response of Ohmic and ballistic physics to additional scattering and that careful consideration must always be made of the form and nature of the conduction in a material when creating transport models.