Scanning Probe Studies of Structural and Functional Properties of Ferroelectric Domains and Domain Walls (Springer Theses) 3030723887, 9783030723880

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Springer Theses Recognizing Outstanding Ph.D. Research

Philippe Tückmantel

Scanning Probe Studies of Structural and Functional Properties of Ferroelectric Domains and Domain Walls

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.

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Philippe Tückmantel

Scanning Probe Studies of Structural and Functional Properties of Ferroelectric Domains and Domain Walls Doctoral Thesis accepted by University of Geneva, Geneva, Switzerland

Author Dr. Philippe Tückmantel Department of Quantum Matter Physics University of Geneva Geneva, Switzerland

Supervisor Prof. Patrycja Paruch Department of Quantum Matter Physics University of Geneva Geneva, Switzerland

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-030-72388-0 ISBN 978-3-030-72389-7 (eBook) https://doi.org/10.1007/978-3-030-72389-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 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

In loving memory of my father, Joachim

Supervisor’s Foreword

Understanding the size scaling of the jerky, highly nonlinear dynamics of elastic interfaces in a complex disorder landscape is of significant fundamental interest for diverse physical systems ranging from earthquakes and fractures to periodic flux line lattices and flame front propagation. It also governs the switching of the order parameter—magnetisation, polarisation, or spontaneous strain—in ferroic materials, which is of key importance for applications in data storage and processing, nonlinear optics, actuation, and many others. Ferroelectric domain walls, essentially extended topological defects separating regions with different polarisation in their parent ferroelectric phase—and at which this polarisation is locally minimised—provide an interesting model system for such crackling investigations, where the complex effects of long-range elastic/strain and dipolar interactions and a rich, heterogeneous disorder landscape can be controlled via different growth conditions. In addition, they offer a unique possibility of studying switching dynamics under a wide range of fields, and could thus provide important insight into possible changes in the spatial correlations between creep and depinning, as recently reported in theoretical investigations. Recently, much research interest in the field of ferroelectrics has also focused on the functionalities of domain walls, and more broadly complex polarisation textures in these materials. These can present emergent properties absent from the parent phase, such as electrical conductance or topologically non-trivial reorientation of the polarisation. Identifying, understanding, and controlling these unusual, highly localised properties could lead to their integration as active device components in ferroelectrics-based nanoelectronics. For both by basic physical interest and the promise of novel/improved applications, piezoresponse force microscopy and related scanning probe microscopy techniques to image electric current, electrostatic surface potential, or magnetic fields have become the key tool, allowing unprecedented nanometre-scale resolution. During his Ph.D. research, Philippe Tückmantel addressed both of these subjects, investigating epitaxial thin films of ferroelectric Pb(Zr0.2 ,Ti0.8 )O3 to better understand the role of different defect landscapes on the distribution of nanoscale switching event sizes observed directly via piezoresponse force microscopy, and mapping the distinct piezoresponse signatures and functional responses at the crossings of individual twin domains in these films. In order to carry out the measurements, Philippe vii

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Supervisor’s Foreword

contributed to the development of a computer vision-based image correction algorithm allowing single-pixel fidelity in tracking scanning probe microscopy image series. The key results of this thesis are (i) analysis of the crackling dynamics of domain walls during polarisation switching in Pb(Zr,Ti)O3 thin films, distinguishing for the first time in ferroelectrics between events occurring in different dynamics regimes (creep versus depinning), and yielding unexpectedly high values of the scaling exponents when only event statistics during domain wall creep are analysed, consistently lowered by the presence of large, more snapping-like events above critical tip bias; (ii) identification of complex piezoresponse signals, possibly related to rotational polarisation textures, at twin domain crossings, which moreover show high susceptibility to electric fields and applied mechanical force. I hope that you, the reader, whether from the field of ferroelectrics and functional materials or fracture analysis and statistical physics will find fresh insight and connection between these different subjects in Philippe’s work, and that you walk away as fascinated and intrigued by the physics of ferroelectric domain walls as we continue to be. Geneva, Switzerland March 2021

Prof. Dr. Patrycja Paruch

Preface

While ferroelectric materials are already widely used in industrial applications as capacitors, transducers, actuators, and infrared detectors for their high bulk dielectric, piezoelectric, and pyroelectric constants, the domain walls separating regions of differing polarisation orientations have also proven to be of significant interest. As nanoscale objects that can be created, moved, erased, and imaged with nanometric precision, they offer the possibility of using ferroelectric materials as model systems for the study of fundamental aspects of interface physics. The results from investigations of crackling physics in particular can be extended to a wide variety of systems, from fractures in rocks and earthquakes to collective decision-making and solar flares. Throughout this thesis, the nanoscale properties of domain walls in thin films of Pb(Zr0.2 Ti0.8 )O3 are studied using scanning probe microscopy. The dynamics of polarisation reversal are first explored in the context of crackling noise. The characteristic exponent of the power law probability distribution of event sizes were extracted from measurements of multiple switching events under different applied electric fields. These power law distributions, which are a typical feature of crackling noise, show exponent values larger than typical elastic model predictions, themselves highly dependent on the dynamic regimes included in the statistical analysis. These findings suggest that the dynamic regimes of the systems investigated in crackling noise studies must be carefully determined beforehand. Domain walls have also been shown to exhibit properties absent from the domains themselves, such as enhanced electric conductivity, offering the prospect of reconfigurable nanoelectronic devices where the domain walls themselves are the active components. While a variety of mechanisms can result in enhanced conduction at the domain walls, locally increased defect concentration at the walls has been identified as a contributing mechanism in several ferroelectric materials. In this thesis, potential correlations are investigated in Pb(Zr0.2 Ti0.8 )O3 domain walls, between nanoscale geometrical distortions and variations in electric conductivity. Both the geometric and conductivity variations can be related to varying defect densities. The absence of strong correlations between geometrical distortions suggests that the defects causing the conduction mostly affect the domain wall tilting through the domain wall thickness, thus forming locally charged interfaces. ix

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Preface

Lastly, much recent research has focused on polarisation patterns in ferroelectric materials forming complex topological structures such as flux-closure domains, polar vortices, and polar skyrmions, which can also exhibit properties differing from the bulk material. Here, the crossings of ferroelastic twin domains were investigated. Scanning probe microscopy and nonlinear optical microscopy measurements suggest potential polarisation rotation structures at the crossings. Furthermore, the crossings themselves seem to exhibit higher sensitivity to mechanical pressure than the surrounding bulk medium. This work shows that nanoscale studies of ferroelectric domain walls can provide new insights into crackling systems, as well as revealing spatio-temporal correlations in the domain wall dynamics, which has implications for an extremely wide variety of physical systems showing crackling behaviour. Geneva, Switzerland

Dr. Philippe Tückmantel

Publications Related to This Thesis • I. Gaponenko, P. Tückmantel, J. Karthik, L.W. Martin, and P. Paruch, Towards reversible control of domain wall conduction in Pb(Zr 0.2 Ti0.8 )O3 thin films, Appl. Phys. Lett. 106, 162902 (2015) • I. Gaponenko*, P. Tückmantel*, G. Rapin, M. Chhikara, and P. Paruch, Computer vision distortion correction of scanning probe microscopy images, Scientific Reports 7, 669 (2017) • P. Tückmantel*, I. Gaponenko, N. Caballero, J.C. Agar, L.W. Martin, T. Giamarchi, P. Paruch, Local probe comparison of ferroelectric switching event statistics in the creep and depinning regimes in Pb(Zr 0.2 Ti0.8 )O3 thin films, submitted to Physical Review Letters

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Acknowledgements

Completing a Ph.D thesis is a work of endurance, and not just for the authors. Many more people are making it possible. This work would not have been possible without funding. For this, I would like to thank the Swiss National Science Foundation for funding the research presented here through grants 200021_178782 and 200021_153174, as well as the SNSF R’Equip grant 206021_150777 funding for equipment used throughout this thesis. This manuscript would also not exist if Patrycja Paruch had not welcomed me into her group, first for a summer internship, then again for my master’s thesis and yet again for my Ph.D. Her guidance and unending supply of enthusiasm and positivity have kept me going over the years. I am lucky to have had Salia Cherifi-Hertel, Marty Gregg and Thierry Giamarchi as jury members, and I am grateful for their careful reading of the manuscript, insightful questions and productive collaborations and discussions. I am also thankful to the past and present members of the Paruch group with whom I have had the pleasure to interact and discuss over the years: Cédric Blaser, Fedir Borodavka, Ralph Bulanadi, Nirvana Caballero, Manisha Chhikara, Seongwoo Cho, Kumara Cordero Edwards, Iaroslav Gaponenko, Jill Guyonnet, Loic Musy, Guillaume Rapin, Christian Weymann and Benedikt Ziegler. Warm thanks also go to our group dog Lewis for making his presence known during meetings in various and creative ways. I would like to also show my gratitude to all other people in the Quantum Matter Physics department and beyond that I was lucky to spend time with: Margherita Boselli, Mireille Conrad, Marios Hadjimichael, Céline Lichtensteiger and Marc Philippi. The productive (and not so productive) discussions I have had with all of you have been some of the highlights of my time at the University of Geneva. Special thanks go to Iaroslav Gaponenko, whose coding experience has been of great help. Iaroslav also introduced me (and the rest of the group) to the nerdy wonders of Python, trained me to tame the fickle and unpredictable animal that ultra-high vacuum AFM can be, and showed me that bludgeoning equipment with a hammer does, in some circumstances, solve performance issues. No equipment has been harmed in the making of this thesis though. xiii

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Acknowledgements

I would also like to show my gratitude to Nirvana Caballero for her enlightening theory input and for taking the time to discuss my lingering doubts on my understanding of aspects of disordered elastic systems theory. Guillaume Rapin needs to be thanked not only for his enthusiasm for all things made of cheese (and food-related in general), but also for finding a way through the labyrinth of Python module versions required to make particular legacy pieces of code work, thus making Chap. 6 possible. I am grateful to Nirvana Caballero and Céline Lichtensteiger for their careful reading of Chaps. 3 and 2, respectively, and for their insightful comments. I would like to thank Sandro D’Aleo, Marco Lopes and Sebastien Muller for their technical support throughout the thesis, and Nathalie Chaduiron, Fabienne Hartmeier and Dragana Pantelic for helping me through the sometimes circuitous labyrinths of administration. Closer to home, I wish to express my gratitude to my family for their support throughout the years. My father Joachim’s unlimited curiosity for, and knowledge of, all things from mushrooms and Sudoku solving programmes to intricate details of particle accelerator technology has been one of the main reasons I was drawn to physics in the first place. I am also greatly indebted to my mother Jutta for her support in the good and bad times, and to all members of my family in general for providing such a positive environment to grow up and live in. My friends Alex, Damien, Eric, Marvin, Michele, Ryan and Soltane also have my deepest thanks for all the good times spent together, and I am lucky to have them all in my life. I should also not forget to thank my cat Oliver shown in Fig. 1 for allowing me the honour and privilege of giving him pets, for providing endless laughs and for showing me hitherto unknown keyboard shortcuts. Last but not least, I would like to thank the love of my life, Jennifer. You have been an unwavering source of support, patience and strength, both in and out of the confines of the lab, and you make me wake up everyday with a smile. I cannot wait to begin the next chapter(s) of my life with you.

Fig. 1 Oliver performing proofreading duties

Contents

1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 Ferroelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Pb(Zr1−x Tix )O3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Emergence of Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Domain Wall Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 8 9 9 16

3 Crackling Noise and Avalanches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Crackling Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 An Experimentalist’s Guide to Renormalisation Group Theory . . . . 3.3 Avalanches in Elastic Systems in a Disordered Medium . . . . . . . . . . 3.4 Plasticity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Crackling Noise Experiments in Ferroelectric Materials . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 21 24 28 29 31

4 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Thin Film Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Off-Axis RF Magnetron Sputtering . . . . . . . . . . . . . . . . . . . . . 4.1.2 Pulsed Laser Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 X-Ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Atomic Force Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Basic Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Conductive AFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Piezoresponse Force Microscopy . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Distortion Correction Algorithm . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 34 35 35 38 38 41 41 49 52

5 Crackling at the Nanoscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Sample Characterisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Switching at a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Extraction of Switching Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 56 59 61 xv

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5.5 Power Law Fitting and Characterisation . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Maximum-Likelihood Estimator . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Estimating Size Cutoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Boxing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Characteristic Size Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Constant Driving Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Spatial Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 64 65 67 69 72 75 76 78

6 Correlations Between Domain Wall Currents and Distortions . . . . . . 6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Domain Wall Conduction in Pb(Zr0.2 Ti0.8 )O3 . . . . . . . . . . . . . . . . . . . 6.3 Preprocessing Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Domain Wall Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Domain Wall Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Topographical Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 80 82 82 84 84 85 90 91

7 Crossings of Ferroelastic Twin Domains . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Sample Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 PFM Measurements of Twin Domain Crossings . . . . . . . . . . . . . . . . . 7.2.1 Crossing 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Crossing 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Other Crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Imaging Artefacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 SHG Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Basic Description of SHG Microscopy . . . . . . . . . . . . . . . . . . 7.3.2 SHG Microscopy Measurements of Twin Domains . . . . . . . . 7.4 Sensitivity to Mechanical Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Effect of Increasing Mechanical Force . . . . . . . . . . . . . . . . . . 7.4.2 Stability of Mechanically Induced Polarisation Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93 93 96 97 101 103 105 107 107 107 109 109 111 112 113

8 Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Chapter 1

Introduction and Motivation

A wide variety of systems respond to external stimuli in abrupt, jerky events. The statistical distributions of parameters such as the sizes of these events and their released energies follow power laws with characteristic exponents. The first documented example of such a system is the so-called Barkhausen noise heard when wrapping a coil around a ferromagnetic material and slowly reversing its magnetisation. An incredibly diverse range of systems have since then been shown to exhibit crackling behaviour, from fractures in rocks and concrete to earthquakes, collective decision making, the evolution of stock market prices and solar flares. These systems might appear at first glance not to share any common characteristics. However, they can often be grouped into so-called universality classes sharing large-scale statistical properties in terms of the characteristic power law exponents of the distributions of crackling events. Importantly, these exponents do not depend on the details of the physics driving those systems at their constituting level. This concept, called universality, is relevant theoretically, as insights into a system within a universality class can be applied to other systems within the class. It is also relevant practically as universality means that technologies such as fracture testing using Barkhausen noise type of measurements can be performed on many materials without prior knowledge of the details of their microstructure, and without actually fracturing them. Furthermore, mine shafts, geological structures or civil-engineering works can be monitored using similar techniques in order to detect early signs of structural failure. In this context, ferroelectric materials, characterised by multiple switchable spontaneous electric polarisation directions have shown to be good model systems for the study of crackling phenomena in terms of the dynamics of polarisation reversal. In these materials, regions of uniform polarisation called domains can be created, moved, and erased using electric fields, while simultaneously recording the acoustic noise or electrical currents caused by the polarisation reversal process. This makes ferroelectrics practical materials in which crackling noise can be studied much more © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. Tückmantel, Scanning Probe Studies of Structural and Functional Properties of Ferroelectric Domains and Domain Walls, Springer Theses, https://doi.org/10.1007/978-3-030-72389-7_1

1

2

1 Introduction and Motivation

rapidly and safely than other systems such as earthquakes. Furthermore, the motion of the interface between domains, called domain walls, shows very rich physics. At low driving force, it exhibits slow thermally activated motion, called creep, followed by depinning then viscous flow regimes at higher forces, with recent theoretical work suggesting spatial as well as temporal correlations of the switching events in the creep regime. While past studies have allowed characteristic scaling exponents to be measured in ferroelectric single crystals, they lack the spatial information that would allow studies of such correlations. Ferroelectrics are also a tried and much tested model system for testing predictions of theories treating interfaces as elastic membranes embedded in disordered media promoting wandering. In ferroelectrics, interfaces between domains, called domain walls have been extensively used in this context. In addition, domain walls have also shown in the past decades to exhibit a range of properties that are absent from bulk ferroelectric materials, such as domain wall photovoltaicity and electrical conduction. While more complex mechanisms have been shown to give rise to such conduction, defects appear to play a crucial role even in the more simple ferroelectric materials. Combined with the ability to create, move, and erase domain walls in a controllable way, these properties have led to the idea that these interfaces themselves could form the active component of future nanoelectronic devices, which has led to domain walls gaining significant interest in the ferroelectrics community. At the same time, ferroelectric domains have been shown to exist in a broad range of forms from alternating stripe domains to structures showing polarisation rotations such as flux-closure domains, vortices, and polar skyrmions. The stabilisation of these domain patterns is usually given by an interplay between the electrostatic boundary conditions at the ferroelectric surfaces and the strain applied to the material via the substrate on which it is grown. Recently, strain gradients within the material have also been shown to stabilise rotations of the polarisation, potentially giving another tool to control polarisation patterns. This thesis focuses on the nanoscale study of ferroelectric domain and domain wall properties in thin films of Pb(Zr0.2 Ti0.8 )O3 , using atomic force microscopy techniques. Fundamental dynamic properties of domain walls are studied in terms of their crackling statistics. A potential direct correlation between geometrical distortions of the domain walls and their enhanced conductivity is investigated, as both are affected by the local disorder. Furthermore, the nanoscale polarisation patterns of junctions of twin domains where high multi-directional strain gradients occur are studied. The thesis is structured as follows. Fundamental concepts of ferroelectricity are presented in Chap. 2, with an emphasis on perovskite oxides and the technologically relevant Pb(Zr1.x Tix )O3 . The formation of domains is briefly discussed and a literature review of research on domain wall conduction is presented, including some of our past work predating the present doctoral research. In Chap. 3, crackling noise and avalanches are introduced. The basic idea behind renormalisation group theory applied in this context is shown, as well as how this approach allows universal behaviour in the system to be identified. Further details are given on two classes of models that have been very successful in describing avalanche

1 Introduction and Motivation

3

phenomena; elastic and plasticity models. This chapter ends with a literature review of recent crackling studies in ferroelectric materials. Chapter 4 introduces the reader to experimental concepts and techniques involved in this thesis. Thin film growth using off-axis radio-frequency magnetron sputtering and pulsed laser deposition is introduced, as well as characterisation of the resulting crystalline structure using x-ray diffraction. Atomic force microscopy (AFM) techniques are described in more detail. The key components for a working AFM are discussed, as well as two of the basic measurement modes upon which most AFM techniques are built. Piezoresponse force microscopy (PFM), which allows the nanoscale imaging of ferroelectric domains is presented and various ways of implementing this technique are shown. Finally, an algorithm used in this study to correct for common imaging artefacts due to the AFM instrumentation is introduced. This algorithm will prove immensely valuable for most of the following chapters. Once these introductory chapters are concluded, studies of crackling statistics in ferroelectrics are presented in Chap. 5. The size distributions of polarisation switching events are investigated at the nanoscale in two samples exhibiting different disorder landscapes established during growth. The characteristic power-law exponents of these size distributions are extracted, first within the regime of thermally activated (creep) polarisation reversal, then over the entire range of accessed dynamic behaviours, including the depinning regime. The measured exponents are found to be higher than theoretically expected, and to be higher in the creep than in the depinning regime. These results highlight the importance of careful consideration of the dynamic regime of the studied system when crackling experiments are performed. The measurement scheme used here allows for direct visualisation of the spatial distribution of switching events, which tentatively suggests that there is a spatial correlation between switching events in the creep regime, but that they are more uniformly distributed along the domain wall in the depinning regime. In Chap. 6, preliminary attempts at directly correlating the magnitude of domain wall currents and local distortions and meandering of the domain walls are presented. The measures of the domain wall distortions in terms of their local curvature and deviation from the average domain wall position are shown, and mapping of the local topographical curvature is used as a proxy for the tip-sample contact area. No strong direct correlation was observed as of the time of writing of this thesis. This could point to the presence of distinct domain wall-defect interactions in the plane of the film, orthogonal to the polarisation axis, and along the direction of the polarisation axis. The former in that case could govern the measured geometry of the domain wall, while the latter is key for the conduction properties. However, more detailed analysis and follow up work are planned. In Chap. 7, PFM is used to map the polarisation patterns at high resolution at and around junctions of ferroelastic twin domains in Pb(Zr0.2 Ti0.8 )O3 . The PFM signals are shown to be consistent with tail-to-tail in-plane polarisation components at the twin domains, while hints of polarisation rotations at the crossings are observed. Artefacts due the restricted geometry of the problem, which is a common feature of lateral PFM are discussed and related to the present measurements. Complementary measurements using second harmonic generation (SHG) appear to show consistent results with the PFM studies.

Chapter 2

Ferroelectricity

2.1 General Properties All crystals can be classified into one of the 32 crystal classes. Of these classes, 21 lack an inversion centre. Of these 21 classes, 20 are piezoelectric and display an electric polarity when subject to external stress (and vice-versa). Of the 20 piezoelectric classes, 10 are polar and display a spontaneous polarisation which is also temperature dependent, making crystals within these classes pyroelectric. To be ferroelectric, a material requires not only to be polar but also to exhibit at least two possible polarisation states in the absence of external electric field and it must be possible to switch between the spontaneous polarisation states repeatedly with an electric field of the proper magnitude and orientation. Because of these symmetry constraints, ferroelectric materials are also pyroelectric and piezoelectric as illustrated in Fig. 2.1, making them a technologically relevant class of materials. Thanks to their high piezoelectric, pyroelectric and dielectric constants, they are materials of choice for transducers and actuators, infrared detectors and capacitors. Ferroelectric materials exhibit a hysteresis of the polarisation upon cycling of an external electric field, an example of which is shown for a real sample in Fig. 2.2a. The polarisation at zero field (or bias) is called the remanent polarisation, Pr . When the voltage is increased and reaches the coercive voltage Vc , the polarisation of domains not oriented parallel to the field switches and aligns with the external field. If the field is further increased, the measured polarisation can increase as a result of dielectric charging of the material. During polarisation switching, a current can also be measured between the electrodes connected to the sample, as shown in Fig. 2.2b, due to changes in the surface bound charges caused by the reversal of the polarisation. Integration of these currents can be used to extract the polarisation of the material. As discussed in Chap. 3, polarisation reversal occurs in discrete events called jerks, which can be studied by carefully measuring these switching currents. At the microscopic scale, different mechanisms can give rise to the polarisation states. In order-disorder ferroelectrics such as polyvinylidene fluoride (PVDF), fer© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. Tückmantel, Scanning Probe Studies of Structural and Functional Properties of Ferroelectric Domains and Domain Walls, Springer Theses, https://doi.org/10.1007/978-3-030-72389-7_2

5

6 Fig. 2.1 Ferroelectric materials are also pyroelectric and piezoelectric

2 Ferroelectricity

Piezoelectric Pyroelectric Ferroelectric

roelectricity arises as an ordering of initially randomly oriented electric dipoles upon cooling under a critical temperature. In displacive ferroelectrics, the dipolar moments are caused by ionic displacements within the lattice spontaneously appearing below a critical temperature. Although perovskite ferroelectrics were first considered as displacive, the exact mechanism for ferroelectricity in some of the most common perovskite ferroelectrics such as BaTiO3 and PbTiO3 is under debate [2], and a mixed character is possible. Some of the most studied and technologically relevant ferroelectric materials have a perovskite structure. These materials have a chemical formula ABX3 , where A and B are cations and X is usually oxygen. The ideal perovskite lattice is cubic with the A atoms located at the corners of the cube, B in the centre and the oxygen (X) atoms are located at the centres of the faces of the cube and form an octahedron around the central atom. An ideal cubic realisation of such a structure is shown in Fig. 2.3a. The perovskite structure is highly robust to changes in the atoms in the A and B sites, leading to a wide variety of materials adopting this structure and rich physical properties [3], as well as the relative ease of preparing solid solutions of different materials within a perovskite structure. Furthermore, materials with a perovskite structure can show a variety of symmetry-lowering distortions such as distortions of the octahedral shape and symmetry, rotations of the octahedron or off-centering of the cations. These distortions can lead to significant changes in the material properties such as the colossal magnetoresistance in LaMnO3 [4], control of the metal-insulator transition in rare-earth nickelates [5] and the emergence of ferroelectricity in a wide variety of perovskite oxides [6]. The presence of these distortions can often be rationalised by the ionic radii r of the A and B cations and oxygen anions, which control the optimal packing of atoms within the √ lattice. Ideally, the lattice parameter a and the ionic radii should be related by a = 2(r A + r O ) = 2(r B + r O ), leading to a cubic lattice. This relation is often not satisfied, leading to deviations from the cubic structure which can be estimated from the Goldschmidt tolerance factor [7]. r A + rO t=√ 2(r B + r O )

(2.1)

Materials with t < 1 have an A atom that is small, leaving ample space between oxygen octahedra. These materials are often not ferroelectric and for 0.7 < t
1, the central cation has enough space to move and polar distortions are often favoured, giving rise to ferroelectricity, as in BaTiO3 and PbTiO3 where the tolerance factors are 1.07 and 1.03 respectively [8]. The stability of ferroelectricity in PbTiO3 is further assisted by the smaller size of the A cation, Pb, along with its hybridisation with the O ions. Both lead to the stabilisation of a tetragonal structure. The Ti ion also bonds with the closer oxygen ion, further pulling the Ti off-centre. Simultaneously, the Ti and oxygen octahedron are displaced in the same direction with respect to the Pb ion, albeit with a higher shift amplitude for the oxygen octahedron, shifting the centres of charge and further contributing to the dipole moment [9]. The resulting structure is shown in Fig. 2.3b, c for polarisations pointing up and down respectively. PbTiO3 is a widely used and studied material and is commonly grown in solid solutions by replacing some of the central Ti atoms with other elements such as Zr, where it forms Pb(Zr1−x Tix )O3 .

8

2 Ferroelectricity

Fig. 2.4 Phase diagram of Pb(Zr1−x Tix )O3 showing the high temperature cubic paraelectric phase (PC ) and at temperatures lower than ∼600 K, the ferroelectric phases with rhombohedral (FR ), monoclinic (FM ) and tetragonal (FT ) symmetries as the Ti percentage increases. The morphotropic phase boundary (MPB) separates the rhombohedral and tetragonal phases, while a “morphotropic phase” with monoclinic symmetry separates the rhombohedral and tetragonal phases at temperatures lower than ∼400 K. Reproduced with permission from [12]

2.2 Pb(Zr1−x Ti x )O3 Pb(Zr1−x Tix )O3 is commonly used in transducers, actuators and capacitors because of its high piezoelectric constants and relative permittivity. It is a solid-solution of ferroelectric PbTiO3 and antiferroelectric PbZrO3 and as such, can exhibit either character, depending on the relative content of Ti and Zr. The phase diagram of Pb(Zr1−x Tix )O3 , shown in Fig. 2.4 exhibits a composition-dependent phase transition at x ≈ 53%, called morphotropic phase boundary, resulting in an enhancement of the piezoelectric properties of the material as it goes through structural changes, from rhombohedral to tetragonal [10] above ∼300 K, and via a monoclinic phase below [11]. In the high PbTiO3 content region of the phase diagram, the material has relatively simple properties, with only the cubic to tetragonal structural transition, unlike BaTiO3 which undergoes several transitions [13]. The possible orientations of the polarisation tend to be simpler than many other materials like BiFeO3 whose rhombohedral symmetry at room temperature [14] leads to up to eight possible polarisation directions. Furthermore, the polarisation orientation can be controlled by growing thin films of materials epitaxially on substrates with a different lattice parameter, allowing strain to be applied to the material. In Pb(Zr1−x Tix )O3 , the polarisation can be forced to point out of the film plane through compressive strain. Using a substrate with a lattice parameter close to that of Pb(Zr1−x Tix )O3 such as DyScO3 allows the formation of narrow twin regions in films thicker than 10 nm, where the polarisation points along the in-plane crystallographic axes as a means to reduce the misfit strain [15, 16]. These factors make Pb(Zr1−x Tix )O3 a good system to study fundamental aspects of ferroelectricity and unusual properties that ferroelectric materials can exhibit at the nanoscale and that will be discussed in the next section.

2.3 Emergence of Domains

9

2.3 Emergence of Domains The polarisation within a ferroelectric material leads to bound charges at the surfaces, which generate a so-called depolarisation field E dep = − P0 which is antiparallel to the polarisation and tends to suppress it. This field can be larger than the coercive field and suppress the polarisation if left uncompensated, especially in thin films where a complete screening of the surface bound charges is more difficult. There are, however, ways in which this problem can be avoided [17], summarised in Fig. 2.5. The depolarisation field can be screened by external charges coming either from adsorbates such as the layer of water present on all surfaces at finite humidity, or through metallic electrodes that can be deposited on the surface. External adsorbates have a surprisingly high impact on the polarisation of thin films of ferroelectric material and can control the polarisation direction in the absence of externally applied electric field [18], change the surface chemistry [19], and affect the dynamics of polarisation switching when external electric fields are applied to the material [20–22]. The screening can also originate from free charges within the material accumulating at the surfaces. If the screening is not adequate, the polarisation itself can change by rotating and pointing in the plane of the film in order to increase the distance between the bound charges and decrease the depolarisation field or by forming domains of different polarisation orientations leading to a net absence of surface charges. Although at first glance the depolarisation field might seem to be an unwanted phenomenon, the interplay of the electrostatic boundary conditions with the strong effect of strain applied by the substrate can lead to very rich and complex configurations of domains where the polarisation can be oriented away from the crystalline axes, despite the energetic cost due to the lattice strain. Ferroelectric domains come in many forms, from Kittel [23–25] and Landau–Lifshitz [26–28] (or flux-closure) domains illustrated in the bottom of Fig. 2.5 to polar vortices in superlattices of PbTiO3 and SrTiO3 [29], bubble domains [30] and ferroelectric skyrmions [31]. Ferroelectric domains can also be patterned artificially, through the use of fixed electrodes connected to the sample or by using conducting atomic force microscope tips as scanning local electrodes to “write” polarisation patterns at the nanoscale, as will be discussed in Chap. 4.

2.4 Domain Wall Conduction The ability to study ferroelectric materials at ever smaller scale thanks to techniques such as transmission electron microscopy and scanning probe microscopy has allowed experimentalists to explore the nanoscale properties of the interface between domains, called domain walls. These interfaces have been shown to exhibit properties that are absent from the bulk material, such as superconductivity at twin domain walls in oxygen deficient WO3 [32] and photovoltaic effect at domain walls in BiFeO3 [33, 34]. While most of these emergent domain wall properties appear to be

10

2 Ferroelectricity

Fig. 2.5 Possible consequences of the depolarisation field. On the left, the field is screened by free charges provided either internally or externally. On the right, the polarisation of the material can change in order to reduce or avoid the depolarisation field. Reproduced with permission from [17]

specific to certain materials, the enhanced conduction at domain walls in ferroelectric thin films has been shown to occur in a wide variety of ferroelectric materials, with different mechanisms put forward to explain the phenomenon. It offers the tantalising possibility of reconfigurable devices where domain walls are the active components. A proof of principle for memristive applications was shown in the first report of domain wall conduction in BiFeO3 [35], which led to domain wall conduction being investigated in a wide range of ferroelectrics with possible designs of device architectures [36] and nanoelectronic components such as domain wall diodes [37] and switches [38].

2.4 Domain Wall Conduction

11

Fig. 2.6 Illustration of 109◦ , 71◦ and 180◦ domain walls in rhombohedral BiFeO3 . Reproduced with permission from [39]

Fig. 2.7 Conduction at domain walls in BiFeO3 . a Topography showing no features corresponding to the domain wall position. b, c in-plane and out-of-plane piezoresponse force microscopy images of the domains, with the interpretation of domain types shown in (c). d conductive atomic force microscopy image showing the higher conduction at 109◦ and 180◦ than at 71◦ walls and the bulk. Reproduced with permission from [35]

Conduction at domain walls in ferroelectric materials was first observed in rhombohedral BiFeO3 in which three types of domain walls exist (71◦ , 109◦ and 180◦ ), as shown in Fig. 2.6. The domain walls are characterised by the angle between the polarisation vectors. Conduction was observed at 180◦ and 109◦ domain walls of BiFeO3 [35], with no conduction at 71◦ walls. Figure 2.7 shows the sample topography (a), out-of-plane and in-plane piezoresponse force microscopy images used to infer the domain wall types (b, c) and the local conduction map (d) showing higher currents at 109◦ and 180◦ domain walls. Two explanations were put forward to explain this. First, den-

12

2 Ferroelectricity

sity functional theory (DFT) calculations suggested that the band gap was lower at the domain walls with a lowering of only 0.05 eV for 71◦ walls, compared to 0.1 and 0.2 eV for 109◦ and 180◦ walls. Second, the 109◦ domain wall implies that the polarisation component parallel to the domain boundary has a discontinuity, causing a potential step across the wall, which is screened by charge carriers accumulating at the domain wall. This first work triggered several further studies on this system. Bandgap lowerings were later measured by scanning tunnelling microscopy measurements (STM), showing significantly larger lowerings than predicted by the density functional theory calculations, yielding 0.2 and 0.5 eV for 71◦ and 109◦ domain walls, respectively [40]. Further DFT calculations [41] suggested that the ground state obtained in [35] was not the optimal one. The optimal state calculated in [41] showed no significant band gap lowering. Therefore, segregation of oxygen vacancies or off-stoichiometry at the domain wall was proposed as a possible explanation of the domain wall conductivity. These results are in principle not incompatible with the decreased band gap measured by STM, which could result from states within the band gap being provided by defects segregating at the domain walls. Domain wall conduction was later observed at 71◦ domain walls as well in thin films cooled down at a lower oxygen partial pressure after growth, suggesting a potential role of oxygen vacancies [42]. This idea was confirmed in a study of the domain wall conduction as a function of O2 partial pressure during cooling, showing a systematic increase of the domain wall currents in a series of films cooled after growth with successively lower oxygen partial pressure [43]. A hysteresis opening was observed between the currents in the increasing and decreasing voltage branches of current-voltage curves performed at the domain walls, suggesting that a dynamic mechanism for the conduction exists as well. This hysteresis and the corresponding currents were linked to microscopic reversible or irreversible changes in the polarisation [44], generating measurable transient switching currents. Domain wall conduction was further discovered in a wide variety of materials. In rare-earth manganites, it was first observed in oxygen-deficient YMnO3 [46], suggesting once more that defects such as oxygen vacancies have an impact on the domain wall conduction. Enhancements of the conduction were then observed in ErMnO3 [45] and HoMnO3 [47] where the domain walls can exhibit either headto-head or tail-to-tail polarisations, where the polarisation vectors at the domain wall point towards or away from each other respectively, as shown schematically in Fig. 2.8. These configurations are electrostatically unfavourable and tend to attract screening charge carriers within the film. The conduction was found to be suppressed in head-to-head and enhanced in tail-to-tail walls, suggesting a p-type conductivity in these materials which was confirmed by Hall-effect measurements at the domain walls [48]. These results have encouraged efforts to tune the domain wall conductivity through donor doping in ErMnO3 [49]. Conduction at charged domain walls was also observed at domain walls with headto-head and tail-to-tail polarisation components in BaTiO3 [50] and Cu-Cl boracite [51] respectively, highlighting the efficiency of these types of walls as efficient components in promoting higher domain wall currents.

2.4 Domain Wall Conduction

13

Fig. 2.8 Schematic example of a a tail-to-tail, b neutral and c head-to-head domain wall. At these boundaries, bound charges arise, which are positive in head-to-head and negative in tail-to-tail domain walls. The bound charges therefore tend to attract screening charge carriers, creating a conducting channel along the domain wall. Reproduced with permission from [45]

Fig. 2.9 Domain wall conduction in Pb(Zr0.2 Ti0.8 )O3 grown on a SrTiO3 substrate. a cross-sections of the domain wall currents with successive tip bias. b, c local current at −1.625 V and corresponding domain map acquired by piezoresponse force microscopy where the dark and bright contrasts correspond to out-of-plane down and up polarisations respectively. d, e Local current at −2.25 V and subsequent polarisation map showing additional contributions to the currents due to local polarisation reversal. f Average currents as a function of tip bias with hysteresis loop measured by PFM in inset. Reproduced with permission from [56]

Tilting of 180◦ domain walls was also shown to enhance conduction at domain walls in highly curved domains in Pb(Zr0.2 Ti0.8 )O3 [52], where the conductance was observed to be metallic. In LiNbO3 single crystals, conduction can be activated by super band gap illumination, injecting photocarriers into the material which accumulate at the charged domain wall [53]. Domain wall conduction in this material has also been shown to be tunable through control of the domain wall tilt [54] and to potentially lead to conductivities 13 orders of magnitude higher than in the bulk [55]. Conduction at uncharged 180◦ domain walls was also observed by the Paruch group in the relatively simpler tetragonal Pb(Zr0.2 Ti0.8 )O3 thin films grown on a SrTiO3 substrate [56]. Currents localised at the domain walls, as shown in Fig. 2.9a. Three regimes were established. Conductive currents with no sign of domain wall displacements were observed for tip voltages of up to −1.4 V with no sign of polarisation reversal in the PFM images and no hysteresis between the forward and backward sections of I-V ramps performed at the domain walls. The resulting

14

2 Ferroelectricity

Fig. 2.10 Map of the local dipole moment orientations and magnitudes acquired by transmission electron microscopy showing that the domain wall is not perfectly straight and has local steps where the domain wall is charged. Reproduced with permission from [28]

currents are inhomogeneous and observed only at the domain walls, as can be seen by comparing the current and domain configuration maps in Fig. 2.9b, c. At tip voltages between −1.5 and −2.7 V, current hysteresis openings are seen, indicating microscopic polarisation changes while, for even higher bias, currents are observed, shown in panel (d) corresponding to polarisation switching shown in panel (e). The corresponding current magnitudes are shown in panel (f).The domain wall currents were measured as a function of temperature in order to establish the dominant transport mechanism. The conduction within the domain walls was found to most closely match to Poole-Frenkel hopping in which the increased conductivity is provided by carriers jumping between trap states. The enhanced conductivity of the domain walls was thus attributed to a combination of defects segregating at the domain walls and providing states within the band gap, along with local steps in the domain wall observed by transmission electron microscopy [28] shown in Fig. 2.10 leading to locally charged walls further attracting charge carriers. Domain wall conduction was also shown at the University of Geneva to be reversibly switchable in thin films of Pb(Zr0.2 Ti0.8 )O3 grown on DyScO3 substrates [57]. The films showed no enhanced domain wall currents as grown. However, the conduction could be activated by a 30 min annealing step at 300 ◦ C in ultra-high vacuum. Upon re-exposure to ambient conditions, the conduction was switched off and this process was shown to be repeatable within the limits of alterations in the film stoichiometry. The proposed mechanism for this effect, shown in Fig. 2.11, was a form of defect engineering through the interplay of the high sensitivity of oxygen vacancy mobility to temperature [58] and the removal of surface adsorbates changing electrostatic boundary conditions. In the as-grown configuration, the polarisation of the film at the surface is screened by surface adsorbates shown in yellow. In films grown on SrTiO3 , the overall higher density of oxygen vacancies shown in shades of blue allows for a conducting pathway to form when the defects accumulate at domain walls. In films grown on DyScO3 , the defect density and their distribution does not allow formation of a conductive pathway along the domain wall through the film thickness. However, the annealing carried out on these films at least partially

2.4 Domain Wall Conduction

15

Fig. 2.11 Schematic of the proposed mechanism of switchable domain wall conduction in Pb(Zr0.2 Ti0.8 )O3 . a, b In the as-grown state, the film grown on SrTiO3 has a higher density of defects at the domain walls (shown in shades of blue) than the film grown on DyScO3 , allowing domain wall conduction to be observed in as-grown films on the former substrate, but not the latter. c The ultra-high vacuum thermal annealing at least partially removes the surface adsorbates changing the electrostatic boundary conditions and promoting a redistribution of defects, while the domain writing process further redistributes and injects defects into the material. Both these factors are thought to lead to an activation of the domain wall conduction. d When the film is exposed to ambient conditions, it recovers the surface adsorbates and the oxygen vacancies are redistributed in a way that does not allow for a conducting pathway to occur. Reproduced with permission from [57]

removes surface adsorbates and modifies the electrostatic boundary conditions, as witnessed by a global reversal of the polarisation throughout the sample. These changes can in principle lead to a redistribution of defects, further helped by the higher annealing temperature promoting a higher mobility of defects. The domain writing process with sharp atomic force microscopy tips also leads to injection and further redistribution of defects as discussed in Chap. 4. Both these factors are postulated to allow for the conduction to occur after annealing. When the samples are exposed to ambient conditions, they recover a layer of surface adsorbates (switching the polarisation again) and the domain wall conduction is lost. The emerging picture from this vast body of work is that multiple mechanisms can contribute to the enhanced conduction at domain walls in different systems and to varying degrees. While charged domain walls lead to generally higher domain wall conductivities, which have in some cases been shown to be metallic [52, 59], these are limited to specific materials and domain wall types, while conduction through defects often provides lower currents but appears to be a more general feature. As such, defects and their role in functional properties of materials are an interesting subject of study. First principles and density functional theory calculations suggest

16

2 Ferroelectricity

that defects such as oxygen and cationic vacancies have a lower formation energy at domain walls [60, 61], causing them to migrate and accumulate there. Furthermore, defects are also known to have a significant influence on both the static properties of domain walls, through the domain wall roughness, as well as on the dynamic properties of the walls, as will be discussed in more detail in Chap. 3. Defects also provide nucleation sites for new domains and pin the domain walls as they advance through the material, affecting the switching dynamics of materials and their functional properties [62]. Given the effect of defects both on the geometrical and functional properties of domain walls, it would be interesting to study whether the magnitude of domain wall currents can be correlated with the local wall curvature, which can be assumed to reflect the local pinning and defect strength and/or density.

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51. McQuaid RG et al (2017) Injection and controlled motion of conducting domain walls in improper ferroelectric Cu-Cl boracite. Nat Commun 8:1–7 52. Maksymovych P et al (2012) Tunable metallic conductance in ferroelectric nanodomains. Nano Lett 12:209–213 53. Schröder M et al (2012) Conducting domain walls in lithium niobate single crystals. Adv Funct Mater 22:3936–3944 54. Lu H et al (2019) Electrical tunability of domain wall conductivity in LiNbO3 thin films. Adv Mater 31:1–7 55. Werner CS et al (2017) Large and accessible conductivity of charged domain walls in lithium niobate. Sci Rep 7:1–8 56. Guyonnet J et al (2011) Conduction at domain walls in insulating Pb(Zr0.2 Ti0.8 )O3 thin films. Adv Mater 23:5377–5382 57. Gaponenko I et al (2015) Towards reversible control of domain wall conduction in Pb(Zr0.2 Ti0.8 )O3 thin films. Appl Phys Lett 106:162902 58. Gottschalk S et al (2008) Oxygen vacancy kinetics in ferroelectric PbZr0.4 Ti0.6 O3 . J Appl Phys 104:114106 59. Stolichnov I et al (2015) Bent ferroelectric domain walls as reconfigurable metallic-like channels. Nano Lett 15:8049–8055 60. He L, Vanderbilt D (2003) First-principles study of oxygen-vacancy pinning of domain walls in PbTiO3 . Phys Rev B 68:134103 61. Paillard C et al (2017) Vacancies and holes in bulk and at 180◦ domain walls in lead titanate. J Phys: Condens Matter 29:485707 62. Damjanovic D (1998) Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics. Rep Prog Phys 61:1267

Chapter 3

Crackling Noise and Avalanches

This chapter introduces fundamental concepts of crackling noise. First, a simple Ising model with disorder is discussed to give an intuitive understanding of the phenomenon. Then, examples of systems in which crackling noise is observed are provided and used as a springboard to discuss universality and give a phenomenological introduction to renormalisation group theory. Two types of very popular models used to describe crackling are then discussed: elastic interface in disordered media and plasticity models. The chapter ends with a brief overview of the literature of crackling in ferroelectric materials and the advantages and disadvantages of the measurement techniques implemented in these materials so far. For further information, some excellent reviews have been written on these topics. James Sethna, Karin Dahmen and Christopher Myers give an excellent introduction to the subject of crackling, universality and renormalisation group methods used in the context of crackling [1–3]. Insightful and detailed information on elastic models can be found in [4–9] while readers interested in plasticity models can consult [10]. Readers that are still interested in crackling after reading this chapter are also encouraged to consult [11].

3.1 Crackling Noise The story of crackling begins in 1919, when Heinrich Barkhausen conducted an experiment in which the existence of magnetic domains was first indirectly evidenced. In this experiment, a coil of wire is wrapped around an iron bar, which is connected to an earphone, and a magnet is placed close to the bar, as shown in Fig. 3.1a. As the magnet is brought closer and the magnetic field through the bar is increased, a crackling sound can be heard, similar to static noise on a phone line [12, 13]. The induced signal is irregular, having spikes of a broad range of sizes and intervals with no activity. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. Tückmantel, Scanning Probe Studies of Structural and Functional Properties of Ferroelectric Domains and Domain Walls, Springer Theses, https://doi.org/10.1007/978-3-030-72389-7_3

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Fig. 3.1 a Replica of the original Barkhausen noise measurement setup. The external magnet is visible on the right, while the iron bar is in the coil at the centre of the image and pointing out of the picture [14]. b Example of Barkhausen pulse signals in polycrystalline FeSi. The time-derivative of the magnetic flux show irregular pulses, of various magnitudes and separated by quiet intervals. Reproduced with permission from [13]

Barkhausen concluded that this sound, later called Barkhausen noise, was due to the switching of entire ferromagnetic domains to align with the external field, inducing changes in the magnetic flux φ through the pick-up coil. These flux changes cause voltage spikes through the coil and sound through the earphone. It was soon realised that the sound was due not to entire ferromagnetic domains abruptly switching, but to the motion of the domain walls driven by the external field. It was then understood that the motion of the domain walls happens in series of discrete jumps as the wall is pinned and depinned, rather than in a smooth continuous fashion [15, 16]. The jumps correspond to the domain wall going from one metastable state to another by an avalanche process, where an initial motion of the domain wall triggers further depinnings of the interface. This can be qualitatively understood with the help of a simple model where a cubic arrangement of ferromagnetic domains oriented either up (Si = 1) or down (Si = −1) and are coupled to their nearest neighbours through an interaction strength J. Disorder is introduced by means of a random field h i drawn at each site from a Gaussian distribution with standard deviation R. The resulting Hamiltonian is then

H=−

 i

(H (t) + h i )Si − J



Si S j

(3.1)

i, j

where H (t) is the externally applied magnetic field. This model is known as the random field Ising magnet model (RFIM). The competition between pinning by the disorder and nearest-neighbour interaction can lead to very different behaviour of the magnetic switching under external field. Three main regimes can be identified, where

3.1 Crackling Noise

21

the system can snap, crackle or pop, as a reference to the Rice Krispies mascots. If the pinning due to the disorder is much weaker than the nearest-neighbour interaction so that the local disorder hi is typically much smaller than the interaction energy, and if the external field is slowly increased, the switching will happen in a single large system-spanning avalanche. This is called snapping and is similar to chalk or a pencil snapping in one large-scale event when a threshold external force is reached. If on the other hand the pinning due to the disorder is much larger than the interaction energy, the local domains will flip almost independently and the switching will happen in very small jumps. This is called popping, in reference to the many popping sounds of similar amplitude that can be heard when popcorn is heated in the microwave. At a critical disorder Rc , the switching of the domains is a mixture of snapping and popping and switching events happen on a broad range of scales. This is referred to as crackling. In this regime, the individual switching event sizes and energies take on a power-law distribution over a broad range of scales. While Heinrich Barkhausen was the first to hear crackling in ferromagnetic materials, he was most certainly not the first to hear crackling, as this phenomenon can be observed in a wide variety of systems. In fact, crackling can be commonly observed for example when crumpling a piece of paper or wrapping [17]. Crackling noise is very common and is seen in compressed porous materials [18], earthquakes [19, 20], collective decision-making [21], solar flares [22], cell-front motion [23], the stock market [24], mass-extinctions [25], fluids in porous media [26], martensitic phase transitions [27] and ferroelastic and ferroelectric switching [28–31], to name but a few. This phenomenon is interesting for two reasons. First, the fact that events happen both on small and large scales and can be described by the same (power) laws over a large range of size suggests that a theoretical description requires detailed knowledge of neither the microscopic nor macroscopic details of the system. Second, some systems which at first sight seem very different and can be expected to be driven by very different physics share statistical properties, like the characteristic power-law exponents in their size and energy distributions. Both of these reasons are addressed by renormalisation group theory.

3.2 An Experimentalist’s Guide to Renormalisation Group Theory Renormalisation group theory was developed in the 1940s in the context of quantum field theory and used later in various fields such as cosmology, quantum mechanics and condensed matter physics. Renormalisation group theory provides an efficient tool to describe continuous phase transitions, in which the correlation length, the distance over which fluctuations in the microscopic degrees of freedom are correlated to each other, becomes infinite and the system is scale-invariant. The basic idea of renormalisation group applied to crackling noise is to start from a microscopic description of the system and iteratively shrink, or coarse-grain

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the system in such a way that microscopic degrees of freedom are removed but the large-scale properties are preserved. As the system is coarse-grained and the model parameters are renormalised, the system can be seen as flowing in the space of model parameters. In this space, which has as many dimensions as there are parameters in the model, fixed points can exist, at which coarse-graining yields the same model and the system is self-similar. The process of iterative shrinking (coarse-graining) of the system can be visualised in Fig. 3.2 in the example of the RFIM model of Eq. 3.1. The model is seen as the red line on the figure. Depending on the starting values of the disorder R, the system flows towards either snapping or popping. At Rc , the model flows towards a fixed point S ∗ , where avalanches occur on a broad range of scales and are power-law distributed. Power-law distributions result naturally from the coarse-graining process if one assumes that the distribution that is considered remains the same upon coarsegraining. The distribution of event sizes is typically of the form D(S) ∼ S −τ Fs , where τ is a universal exponent independent of the microscopic details of the model and Fs is a function controlling the large scale cutoff to the power-law behaviour. The further the parameters of a model are from their critical value corresponding to S ∗ in Fig. 3.2, the shorter the cutoff to the pure power-law behaviour. FS is modeldependent and not universal, and will depend on details of the system. Multiple models starting with different microscopic descriptions can converge towards the same fixed point, meaning that they share large scale properties which are independent of the microscopics of the particular systems under consideration. These models share critical exponents and are said to be part of the same universality class. Universality is a very powerful concept, which in principle allows the large-scale statistics of any system within its universality class to be predicted by using insight obtained from a single system within the universality class. Different systems that are expected to have very different physics at small scale can share a universality class. For example, the distribution of slip sizes in slowly compressed microcrystals has been shown to be part of the same universality class as the size distribution of avalanches in soft magnets [11]. Universality is useful, not only as an abstract theoretical tool but also for applications such as non-destructive stress testing. The Barkhausen noise of slowly compressed crystals depends on the applied stress, which allows the failure stress to be extracted in a non-destructive way from the noise spectrum. The same idea can also be applied to the monitoring of structures and mine shafts. In these applications, universality means that the details of the materials or rocks that are being monitored do not matter as long as they belong to the expected universality class. Although systems within a universality class share critical exponents, the cutoff functions differ from model to model. In principle, this allows different models belonging to the same universality class to be distinguished. Experimental data obtained with different values of some relevant parameter can be plotted in terms of the dependence of the cutoff function on that parameter. For the RFIM for example, the cutoff function for the distribution of event sizes D(S) takes the c ) where σ is related to the fractal dimension of the form D(S) ∼ S −τ Fs (S σ R−R R

3.2 An Experimentalist’s Guide to Renormalisation Group Theory

23

Fig. 3.2 Visualisation of renormalisation group flow in the space of model parameters in the context of the RFIM. The model flows towards a fixed point S ∗ in which the model is self-similar. Reproduced with permission from [1]

avalanches. D(S)/S −τ can then be plotted as a function of S σ (R − Rc )/R in order to collapse curves of different avalanches at different values of R, as shown in Fig. 3.3. This is not easy however. On the theoretical side, it requires some knowledge of the form of the scaling function. On the experimental side, the data needs to be acquired over a range large enough to see the effect of the cutoff function experimentally, which requires repeated measurements for different values of relevant parameters of the model (here R). Measuring power-law exponents is already not easy as measurements have to be performed over several decades and enough data needs to be gathered to be statistically relevant. Measuring exponents over a range of scales wide enough to see the cutoffs to the studied probability distributions is experimentally highly non-trivial. The RFIM model discussed above is only one of many models showing crackling behaviour. One class of models called elastic interface models has been very successful in describing a variety of systems such as soft magnets, slip faults, contact lines of fluids on rough surfaces and will be discussed in the next section.

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Fig. 3.3 Scaling collapse of avalanche sizes in the RFIM where r = (R − Rc )/R. Dint refers to the distribution of avalanche sizes integrated over the entire external field hysteresis loop in a soft magnet and τ¯ is the corresponding power-law exponent. σ is another universal exponent related to the fractal dimension of the avalanches. Reproduced with permission from [32]

Fig. 3.4 Description of the coordinate system describing the interface. The interface is shown in blue and is propagating along the x direction. The function u(z, t) describes the interface position

3.3 Avalanches in Elastic Systems in a Disordered Medium Elastic models have proven very powerful in describing the statics and dynamics of various systems, from domain walls in ferroelectric and ferromagnetic materials [5, 7, 9] to Abrikosov lattices [33], wetting fronts [34] and crack propagation [35]. Elastic models provide a general description of interface statics and dynamics by focusing on the boundary itself and including only a few key ingredients rather than the detailed microscale physics of the system. The interface is modelled as an elastic boundary of dimension d, with m transverse components along the direction of movement of the interface and living in a D-dimensional space, with D = d + m. Elastic models can describe a moving interface such as a ferroelectric or ferromagnetic domain wall as well as periodic structures such as Abrikosov vortex lattices in type-II superconductors. Here, the case of a d-dimensional interface living in D = d + 1 is discussed. The interface is described using a displacement function u(z, t) where z is the position along the interface and x is the transverse coordinate, describing the direction of propagation (Fig. 3.4). u(z) is a single-valued function, meaning that this description does not take overhangs or nucleation sites into account. In the case of real interfaces, where these overhangs do exist, they can be interpolated out. The main ingredients affecting the interface shape are the external force driving the overall movement of the interface, the interface elasticity which tends to favour a flat line configuration (or plane or hyperplane depending on the interface dimensionality), and the disorder, which pins

3.3 Avalanches in Elastic Systems in a Disordered Medium

25

the interface and promotes meandering. The system displays glassy physics as the energy landscape exhibits many local minima and the interface wanders through consecutive metastable states. The competition between these three a priori simple ingredients leads to the rich physics of avalanches and scale-invariance of event size and energy distributions. A simple description of the dynamics of the interface is the quenched Edwards–Wilkinson equation

η∂t u(z, t) = f (z, t) + F(u(z, t), z) + c∇ 2 u(z, t) + μ(z, t)

(3.2)

where η is a microscale friction coefficient, f (z, t) describes the applied external force, and F(u(z, t), x) is a random force which mimics the effect of the disorder. c∇ 2 u(z, t) describes a short-range interface elasticity with modulus c, while μ(z, t) describes thermal noise. It is assumed that the disorder is uncorrelated along the interface direction, whereas along the transverse coordinates, the disorder is usually described as being correlated in one of two ways, each belonging to a distinct universality class [36]. In the so-called random-bond case illustrated in Fig. 3.5a, c, it is assumed that the disorder affects the phases on both sides of the interface symmetrically. In the case of uniaxial ferroelectrics, random-bond disorder affects the depth of both energy wells corresponding to the two stable polarisation states in the same way. The pinning potential in random-bond disorder is correlated on a short range and only impurities located close to the interface contribute to its pinning. In the random-field case shown in Fig. 3.5b, d however, both sides of the interface are affected asymmetrically and one polarisation orientation is favoured over another. In random-field disorder, all the impurities within the region corresponding to the phase favoured by the external force contribute and the pinning potential has long-ranges correlations. At a temperature of 0 and below a critical external force f c , the interface is pinned with no observable motion. At f >> f c , the interface velocity is linear with the driving force and the defects act as a viscous drag. Closer to the critical force, however, where f > f c , the interface is depinned and the overall velocity exhibits a power-law increase v ∼ ( f − f c )β , where β is a universal exponent. Crossing through f c , the system undergoes a second-order dynamic phase transition with the velocity as an order parameter and the external field as a control parameter. At f = f c , motion of the interface occurs in a wide range of event sizes and with a power-law distribution P(S) ∼ S −τdep f s (S/Sc ), where f s is a cutoff function of the sizes, which decays sharply as S ≥ Sc and is a constant for S < Sc . Sc is a characteristic event size which depends on the external force, the dimensionality of the interface and its static roughness. In this regime, avalanches are spatially uncorrelated and no aftershocks are observed, as opposed to earthquakes, where following large events, the faults rearrange in a series of smaller earthquakes. At low temperature and below f c , the interface is in the so-called creep regime, where the motion of the boundary is thermally activated. This regime has been experimentally demonstrated by magnetooptic Kerr imaging in ferromagnetic Pt/Co/Pt films [37] and by piezoresponse force

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Fig. 3.5 a, b Schematic of how impurities affect the pinning in the random-bond and randomfield cases respectively. The defects contributing to the pinning are shown in red. After [9]. c, d In the case of a ferroelectric, the disorder affects the depths of the free energy minima as defined by the Ginzburg–Landau–Devonshire (GLD) theory, where the free energy F = 21 a P 2 + 41 b P 4 + 1 6 6 c P is minimised in terms of the polarisation order parameter P. The energy wells are affected symmetrically in the random-bond case and asymmetrically in the random-field case

microscopy in ferroelectric materials [38–40]. This regime is particularly rich and interesting. The overall interface velocity in the creep regime follows a stretched exponential behaviour v ∼ ex p(−βUc ( ffc )μ ) where β = 1/kb T , Uc is the characteristic height of the barriers the interface needs to cross and μ is an exponent dependent on the type of disorder and the dimensionality of the interface. The motion typically happens in two steps. In the first step, a portion of the interface of typical length lopt moves through thermal activation, triggering a rearrangement of the interface on a larger scale, through a fast avalanche process in the second step, as shown in Fig. 3.6. As a consequence of the glassy physics describing the interface, the optimal 1 thermal nucleus size lopt ∼ f − 2−ζ is inversely proportional to the external force f and ζ describes the size scaling of the domain wall roughness [5]. The size distribution −τ f s (Seve /Sc ) where the cutoff of these events also follows a power-law P(Seve ) ∼ Seve size increases as the driving force decreases.

3.3 Avalanches in Elastic Systems in a Disordered Medium

27

Fig. 3.6 In the creep regime, a thermal nucleus of typical size lopt moves first, triggering a larger avalanche. After [5]

This means that most events are smaller than lopt since the probability of events larger than Sc is cut off by f s . It is therefore the few larger events that drive the overall motion of the interface according to the creep law. Another interesting feature is the spatial localisation of events in the creep regime recently identified in theoretical studies [41]. In d = 1 and in the presence of randombond disorder, the avalanche events in the creep regime tend to cluster together in space and time (Fig. 3.7), similar to aftershocks in earthquakes, while these correlations are absent in the depinning regime. The distribution of cluster sizes follows a power-law with a crossover of exponents at S = Sc . Below Sc , the size exponent in the random bond case is that of equilibrium with τeq = 0.8, whereas above Sc , it follows the depinning characteristic exponent of τdep = 1.11 (Fig. 3.8).

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Fig. 3.7 In the creep regime, avalanches are predicted to cluster in space and time, with aftershocks as seen in earthquakes, while in the depinning regime where f > f c , the avalanches are uncorrelated and no aftershocks are predicted. Reproduced with permission from [41]

Fig. 3.8 The distribution of clusters of event sizes in the creep regime is predicted to show a crossover from the 1D equilibrium exponent τeq = 0.8 to a larger exponent τdep = 1.11 compatible with the 1D depinning exponent of 1.11. Reproduced with permission from [41]

3.4 Plasticity Models Plasticity models have also shown success in describing avalanche behaviour in rocks and granular materials, ferroelastic materials and earthquakes [10]. In this model, it is assumed that a material under strain has weak spots, which cause slips when the local stress at a position r exceeds the weak spot’s threshold value θt,r . When slips happen, the stress is redistributed to other weak spots, which will cause further slips if the local strain exceeds their thresholds and trigger avalanches. Slips will

3.4 Plasticity Models

29

continue to occur until the stress is lowered below an arrest threshold θa,r . This class of models can be used to describe weakening or hardening materials by defining different successive arrest threshold values. Decreasing and increasing threshold values model brittle materials and hardening materials respectively. This model can be described in the continuous limit as η∂t d(r, t) = F + σint (r, t) − f p (d, r, {d(r, t  < t)}) where d(r, t) is the accumulated slip, η is a friction coefficient, F the externally applied stress, and σint is the accumulated stress at point r and time t and is described as  t σint = −∞ dt  dr  J (r − r  , t − t  )[u(r  , t  ) − u(r, t)] with J the (positive) coupling between slips of weak spots. f p describes the pinning stress preventing slips as long as the accumulated stress is below threshold. f p is position and history dependent. The model has been extensively studied in the discrete version where the material is discretised into N sites. In mean-field theory, the coupling function J is replaced with a constant and the stress is redistributed equally over all weak spots, J (r ) = J/N . A renormalisation group treatment of this model predicts a stress-dependent power-law distribution of avalanche size P(S, F) ∼ S −τ G(S(F − Fc )1/σ ) where G(x) ∼ Aex p(−Bx), τ = 1.5 and σ = 2. While A and B are non-universal constants, τ and σ are universal [42]. In the case where avalanches are collected over a range of stresses from F = 0 to F = Fc , the avalanche size distribution is Pint (S, F) ∼ S −τ¯ G int (S/Smax ), with τ¯ = τ + σ = 2 [43, 44].

3.5 Crackling Noise Experiments in Ferroelectric Materials The investigation of crackling noise in ferroelectric materials was inspired by early work on optical microscopy measurements of the propagation of single ferroelastic needle domains in slowly strained LaAlO3 , where smooth motion of the needle tip was found to be superimposed with abrupt jumps of a wide range of amplitudes. The energy released by these events were found to be power-law distributed with a characteristic exponent = 1.8 ± 0.2 [45]. This study was followed by work [28] on the progression of many needles in single crystals of LaAlO3 and PbZrO3 . In this study, the samples were fixed at two extremities and pressed in the middle while the height of the samples was measured with a precision of ∼5 nm. The drops in sample height were measured under slowly increased strain. The maximum velocity of the drops was used as a proxy for the energy released during the jerky events, from which power-law exponents of the energy distributions were found to be = 1.6 ± 0.1. The relatively large error bars on these experiments mean that the measured exponent values are compatible both with that observed for single needles, and also with models of elastic interfaces in random media in the mean-field approximation, predicting = 1.5 [43, 44].

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The same group then studied the motion of 90◦ ferroelastic domains in ferroelectric BaTiO3 by acoustic emission measurements [29]. The domains were switched by slowly increasing the voltage between electrodes connected to the sample sides and a piezoelectric sensor was used to detect the crepitation of the moving domains. The acoustic emission data was acquired over six full hysteresis loops. The energy of the switching events was extracted by calculating the square of the integrated jerk signals and their distribution was shown to exhibit a power-law over 5 decades with no visible cutoffs and an average exponent of = 1.65 ± 0.15. Because the data was collected over a range of driving forces, the expected energy exponent is int = 1.67 [44] and the extracted exponents are compatible with this value. Amplitude exponents are extracted as the maximum voltage amplitude picked up by the receiver. A time exponent linked to the duration over which the jerk signals are above a given detection threshold is measured as well. From these, amplitude and duration exponents can be extracted and are found to be, within error bars, compatible with exponent equalities predicted theoretically. The authors also observe aftershocks following Omori’s law predicting the probability P of an aftershock at time t following a main jerk event as P ∼ t − p , with p = 1. Jerky motion of charged ferroelastic domain walls in BaTiO3 was also investigated both optically by birefringence imaging and electrically by acquiring the displacement currents while switching [31]. Both methods yielded energy exponents of 1.6. Crackling was also observed through the displacement currents in three PZT ceramics [30] (of proprietary composition). Energy exponents were obtained through the square of the time derivative of the displacement currents. The energy exponents obtained here were between 1.61 ± 0.04 and 1.73 ± 0.04. Measurements were performed at different temperatures below the Curie temperature and, while the most likely exponent was seen to be temperature independent, the authors observe hints of a potential temperature-dependent higher exponent increasing from 1.8 to 2 in the temperature range of 373–423 K, which they tentatively attribute to depinning from dislocations. While measurements through acoustic emission and displacement currents are very powerful and allow data to be acquired over a large range of energies and other exponents related to avalanche durations to be extracted, they are indirect measurements and fundamentally lack the ability to directly access single events. Furthermore, these measurements do not allow a focus on specific regimes of the domain wall motion such as the creep or depinning regimes and the extracted histograms most likely mix together jerks belonging to both regimes. The optical microscopy data discussed earlier could in principle allow the switching event sizes to be extracted directly, but the lateral resolution is too low to pick up small events typical of the creep regime. The ability to observe jerks directly with high resolution could provide additional information on the spatial correlations between jerks as predicted to occur in the creep regime [41]. It would also allow potential differences in contributions to the overall size distributions coming from different types of jerks to be distinguished. Such measurements could furthermore potentially discriminate between events occurring under weak collective pinning and close to strong pinning sites such as extended defects like twin domains. In this regard, techniques like atomic force microscopy could be a valuable tool.

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30. Tan CD et al (2019) Electrical studies of Barkhausen switching noise in ferroelectric PZT: critical exponents and temperature dependence. Phys Rev Mater 3:1–6 31. Casals B et al (2020) Avalanches from charged domain wall motion in BaTiO3 during ferroelectric switching. APL Mater 8 32. Sethna JP (2007) Crackling noise and avalanches: scaling, critical phenomena and the renormalization group. In: Bouchaud JP, Mezard M, Dalibard J (eds) Les houches summer school proceedings, vol 85. Elsevier, Amsterdam, pp 257–288 33. Blatter G et al (1994) Vortices in high-temperature superconductors. Rev Mod Phys 66:1125– 1388 34. Moulinet S, Guthmann C, Rolley E (2002) Roughness and dynamics of a contact line of a viscous fluid on a disordered substrate. Eur Phys J E 8:437–443 35. Bonamy D, Bouchaud E (2011) Failure of heterogeneous materials: a dynamic phase transition? Phys Rep 498:1–44 36. Chauve P, Giamarchi T, Le Doussal P (2000) Creep and depinning in disordered media. Phys Rev B 62:6241–6267 37. Lemerle S et al (1998) Domain wall creep in an Ising ultrathin magnetic film. Phys Rev Lett 80:849–852 38. Paruch P, Giamarchi T, Triscone JM (2005) Domain wall roughness in epitaxial ferroelectric PbZr0.2 Ti0.8 O3 thin films. Phys Rev Lett 94:3–6 39. Paruch P et al (2006) Nanoscale studies of domain wall motion in epitaxial ferroelectric thin films. J Appl Phys 100:051608 40. Tybell T et al (2002) Domain wall creep in epitaxial ferroelectric Pb(Zr0.2 Ti0.8 )O3 thin films. Phys Rev Lett 89:097601 41. Ferrero EE et al (2017) Spatiotemporal patterns in ultraslow domain wall creep dynamics. Phys Rev Lett 118:1–6 42. Dahmen KA, Ben-zion Y, Uhl JT (2009) Micromechanical model for deformation in solids with universal predictions for stress-strain curves and slip avalanches. Phys Rev Lett 175501:1–4 43. Leblanc M et al (2012) Distribution of maximum velocities in avalanches near the depinning transition. Phys Rev Lett 109:1–5 44. Leblanc M et al (2013) Universal fluctuations and extreme statistics of avalanches near the depinning transition. Phys Rev E 87:1–13 45. Harrison RJ, Salje EKH (2010) The noise of the needle: avalanches of a single progressing needle domain in LaAlO3 . Appl Phys Lett 97:021907

Chapter 4

Experimental Methods

In this chapter, the experimental techniques relevant to the present thesis are presented, starting from thin film growth and characterisation, to atomic force microscopy based measurements and post-processing techniques used to correct for instrument artefacts.

4.1 Thin Film Growth The drive to study high quality crystals of ferroelectric materials in thin film form historically stems in part from conflicting reports of the critical thickness at which ferroelectricity sets in in BaTiO3 [1–3], later attributed to variations in the crystalline quality and defect structure, which both affect the material’s ferroelectric behaviour. Also, applications of ferroelectrics for pyroelectric detectors, transducers and filters, for example, require high quality ferroelectric materials. Both of these factors pushed ferroelectrics to be grown as thin films. The study of ferroelectric materials in thin film form allows detailed investigations of the critical thickness at which ferroelectricity emerges [4], of the effect of electrostatic and mechanical boundary conditions in the form of electrodes [5–7] and adsorbates [8–10] at the film interfaces in the former and strain in the latter [11]. The interplay of strain (mainly due to the substrate) and electrostatic boundary conditions allows the favoured polarisation state to be controlled and to stabilise complex polarisation configurations (flux closure [12– 14], bubble domains [15], vortices [16], skyrmions [17]) The thin films used in this study were grown by off-axis RF magnetron sputtering and pulsed laser deposition. In this section, a brief description of these techniques will be given. For further information, the reader is referred to the chapter on growth and novel applications of epitaxial oxide thin films of reference [18].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. Tückmantel, Scanning Probe Studies of Structural and Functional Properties of Ferroelectric Domains and Domain Walls, Springer Theses, https://doi.org/10.1007/978-3-030-72389-7_4

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Fig. 4.1 Schematic of an off-axis magnetron sputtering system showing the plasma required to sputter the target, as well as the size difference between the electrodes allowing the Ar+ atoms to bombard the target despite the high frequency of the AC field used to generate the plasma

4.1.1 Off-Axis RF Magnetron Sputtering In sputtering, a target with the same stoichiometry as the desired material is bombarded with particles that knock atoms, ions and molecular complexes out onto a substrate. The substrate is heated to promote good crystalline growth of the incoming target material. The particles bombarding the target are often Ar+ ions obtained by forming a plasma in a vacuum chamber. While a relatively simple setup where a DC field is used to create the plasma works with metals, sputtering of insulating materials requires a more complex design. A DC field promotes charging of the target with Ar+ ions, which create an electric field that eventually extinguishes the plasma (Fig. 4.1). To solve this problem, an AC field is used between the target and the wall of the chamber instead. The substrate and chamber walls are grounded while the target forms the biased electrode. To prevent cyclic sputtering of the target and the substrate due to the alternating field direction, the frequency of the field is chosen in a range where the electrons can react to the AC field but not the Ar ions, typically ∼13 MHz. Because of the asymmetry in the electrode size, when the field is oriented towards the target, electrons accumulate close to it, generating a net negative bias, while

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when the electric field direction is reversed, the electrons are spread out over the chamber walls. This generates an average electric field pointing towards the target, which attracts the Ar+ towards it. A cylindrical magnet called a magnetron is also placed behind the target to further confine the electrons close to the target preventing resputtering of the film. When growing oxides, oxygen is typically injected into the chamber to prevent reduction of the film by the vacuum and is ionised by the plasma along with the Ar. As these oxygen ions would tend to also sputter the growing film if the substrate is placed directly facing the target, the substrate is usually placed at 90◦ to the target. This is called “off-axis”.

4.1.2 Pulsed Laser Deposition In pulsed laser deposition (PLD), the ablation of the target is performed by laser pulses which locally heat the target to temperatures high enough to evaporate all its constituting elements and form an ablation plume of target material that transfers to the heated substrate (Fig. 4.2). The target is typically rotated or scanned in order to prevent the formation of holes which can change the angle of the plume and affect the deposition profile. The kinetic energy of ions in the ablation plume can be high enough to cause resputtering of the film, for example in the case of Pb in materials like Pb(Zrx Ti1−x )O3 . This problem can be solved by changing the distance between the substrate and the target, the laser energy, or by using targets that are over-stoichiometric in Pb. Reflection high-energy electron diffraction (RHEED) is often used in PLD. In RHEED, an electron gun is pointed towards the sample surface at grazing incidence. The beam diffracts from the surface and forms an interference pattern that can be picked up using a photoluminescent detector. This system allows the growth of the thin film to be characterised and controlled in-situ and in real time. Both techniques described here can be used to epitaxially grow ferroelectric thin films of high crystalline quality.

4.2 X-Ray Diffraction X-ray diffraction is a commonly used technique to probe the crystalline structure of solid materials, in which a beam of x-rays diffracts from the regularly spaced atoms of the crystal and the resulting interference pattern is collected by a detector. Constructive interference occurs when the path difference of the X-ray beam between two consecutive atomic planes is an integer multiple of the beam wavelength. This is described using Bragg’s law 2dsin(θ ) = nλ, where θ is the incident beam angle with respect to the atomic plane, d the spacing between the planes, λ the wavelength of the incoming photons and n is an integer. X-ray light has a wavelength that is of

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Fig. 4.2 Basic schematic showing the principle of PLD

the same order of magnitude as the typical interatomic distances in crystals, making interference effects more visible (Fig. 4.3). X-rays are commonly produced using a vacuum tube made of a cathode and an anode, where the cathode is heated in order to eject electrons which are accelerated by the field applied between the electrodes, until they collide with the anode, emitting light in the X-ray range of wavelength. The tube, sample and detector are mounted on a motor-controlled diffractometer allowing accurate control over their relative angles (Fig. 4.4). Two main types of scans were performed in this thesis: ω-2θ and reciprocal space maps. In the former, the sample and detector both rotate, with the detector rotating at twice the angular speed of the sample. This means that the diffraction vector remains normal to the film surface throughout the scan and only its length changes. This type of scan allows the c-axis of the films to be determined from the angle of the film peak. Because the number of planes from which the X-ray beam diffracts is finite, the diffraction pattern shows finite-thickness oscillations which can be used to determine the thickness (Fig. 4.5). In reciprocal space maps ω-2θ scans are performed for a range of ω values. This allows the in-plane lattice parameters of the films to be extracted.

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Fig. 4.3 Illustration of Bragg’s law. The incoming x-ray light is scattered elastically from atoms of multiple crystalline planes. Constructive interference occurs when the path difference between light scattered from different planes (in thicker red) is an integer multiple of the x-ray wavelength

Fig. 4.4 Illustration of the basic setup and angles of the diffractometer

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Fig. 4.5 ω-2θ scan of a SrRuO3 thin film on SrTiO3 (001) substrate, around the (002) reflection of SrTiO3 . The substrate peak is visible on the right while the film peak and finite-thickness oscillations can be seen on the left

4.3 Atomic Force Microscopy Atomic force microscopy (AFM) is a nanoscale surface probing technique, which, since its invention and first proof of principle in 1986 [19], has become a standard part of the experimental condensed matter scientist’s tool kit. From being a tool to probe topographies at nanometric resolution, AFM has been improved upon and features were added to probe a wide variety of properties, from magnetic and ferroelectric structures to local conduction and surface potentials, local stiffness, friction and adhesion forces to name only a few. This technique can probe physical properties in various environments: in liquids, air or other gases and at pressures down to the 10−12 mbar range, and temperatures ranging from below 1 K to about 1000 K. In this section, we will cover the basic operation of an AFM, in its most typical setup. We will describe how the topography of a sample can be acquired in the so-called contact and intermittent contact modes. We will then describe how AFM can be used to measure local currents and image and write ferroelectric domains at the nanoscale.

4.3.1 Basic Operation In its most basic form, an AFM requires (Fig. 4.6): • • • •

A tip with nanoscale radius at the end of a flexible cantilever A way to measure the deflection of the cantilever An XYZ scanner with high resolution Control electronics

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Fig. 4.6 Basic setup of a typical AFM capable of scanning in contact and tapping mode

The AFM tip, an example of which is shown in Fig. 4.7, acts as a nanoscale force sensor. When it is brought close to the sample, interactions with the sample cause deformations of the cantilever. In principle, at long ranges, from a few nanometres to a few Angstroms, attractive Van Der Waals forces dominate while, at shorter ranges, the Pauli exclusion principle dominates and the cantilever feels a repulsive force. However, other forces depending on the particular material, tip and environment often significantly contribute to the deformation of the cantilever. Electrostatic and magnetic forces measurably affect the cantilever at distances of up to hundreds of nanometres, while surface contaminants or the water meniscus present on all objects in ambient environment can result in adhesive forces when the tip is in contact with the sample. The deformations due to these interactions can be picked up by a detection system using a laser beam that reflects from the cantilever into a position sensitive detector (PSD). The vertical and lateral bending of the cantilever is picked up by the corresponding deflections on the photodetector, DV and D H , respectively and provides information on forces acting out of the sample plane through DV and torques on the cantilever due to forces in the sample plane through D H . This laser beam system allows atomic-scale bending of the cantilever to be detected. The scanner moves the tip over the sample surface in a raster pattern in the sample plane and provides a 2D map of the tip interactions with the sample. Most of the time, the vertical extension of the scanner is controlled to keep the tip-sample interaction at a constant user-defined value by a feedback loop in the control electronics.

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Fig. 4.7 SEM images of a Co/Cr coated tip from Bruker. a The chip holding the cantilever with the tip marked by a white arrow. b Closeup on the tip

The variety of AFM modes can be divided into two categories: static and dynamic modes. In static modes, the tip is scanned in contact with the sample surface whereas in dynamic modes the cantilever oscillates, close enough to the surface for interactions to affect the cantilever oscillations. In the most basic implementation of the static mode called contact mode illustrated in Fig. 4.8a, only the topography is probed. The tip is pushed into contact with the surface until the vertical deflection DV reaches a setpoint value. The tip is then scanned over the sample while a feedback loop adjusts the column extension to keep DV at the setpoint value. The topography image is then formed by mapping the column extension as a function of position. Information on local friction force variations can be extracted by also measuring the lateral deflection as a function of position at the same time as the topography is extracted. In intermittent contact or tapping mode shown in Fig. 4.8b, the tip is fixed on a piezoelectric actuator, to which a driving AC voltage is applied in order to oscillate the tip close to its resonance frequency. A lock-in amplifier is used to acquire the amplitude of the oscillating vertical deflection signal. The interaction between tip and sample depends on their relative distance and therefore affects the tip oscillation amplitude. The corresponding amplitude of the oscillation in DV is used as a feedback signal to modulate the scanner’s vertical extension and acquire the sample topography. At the same time, the phase signal can be acquired and can give information on local variations of the sample stiffness. The ability to acquire various channels of information simultaneously is one of the strengths of AFM, which allows the topographical information to be correlated with other signals such as local conduction or electric polarisation in ferroelectric materials.

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Fig. 4.8 Principle of a contact and b tapping mode. In contact mode, the tip is held at a constant setpoint of vertical deflection by a feedback loop, whereas in tapping mode, it is held at a constant setpoint vertical deflection oscillation amplitude. When encountering a topographical feature, the vertical deflection (contact) or amplitude (tapping) change and the feedback loop adjusts the tip height to recover the setpoint value Fig. 4.9 Schematic of conductive AFM. A bias Vti p is applied to the tip-sample junction and the current flowing through is converted to a voltage Vamp by a transimpedance amplifier

4.3.2 Conductive AFM In conductive AFM (c-AFM), shown in Fig. 4.9 a metallic tip is used as a scanning top electrode in contact with the sample. A DC bias is applied to the tip while the sample is grounded (or vice-versa) and a transimpedance amplifier is used to convert the current passing through the circuit into a proportional output voltage. With the proper voltage/current calibration, this allows the local currents through the sample to be mapped with a lateral resolution of the order of 10 nm. Performing conductive AFM on thin films grown on insulating substrates, as in the present study, requires the presence of a back electrode between the film and the substrate. It is this backelectrode that is set to ground.

4.3.3 Piezoresponse Force Microscopy The first experiments using atomic force microscopy in order to image ferroelectric domains were performed in dynamic mode with the tip oscillating 1–100 nm away from the surface [20–22] and where the oscillating tip experiences a force gradient. This gradient affects the effective spring constant of the cantilever. The change in sign of the charges at the surface of the ferroelectric material as the tip scans across a domain wall leads to changes in the force gradient, which in turn affects the cantilever resonance frequency. These changes can be used to map the position of

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Fig. 4.10 Snapshot of the expected behaviour of a piezoelectric or ferroelectric material under a positive tip bias (downward-oriented electric field). Upon application of an AC field, up and down polarised materials show a 180◦ shift in the phase of the cantilever oscillation with respect to each other

the underlying ferroelectric domains. Measurements were also performed in contact mode with electrically insulating Si3 N4 tips, where a contrast in the lateral deflection was detected and later attributed to the tip acquiring an electric polarisation from the ferroelectric material, which, it was assumed, relaxes slower than the scanning time [21–23]. This tip polarisation was though to cause torques on the cantilever due to the electric field created by either the bound charge at the surface or the charge of the layer screening this bound charge. Other works [24] used metallic tips scanning in contact mode and a DC bias applied to the tip to cause surface deformation of the ferroelectric material due to the converse piezoelectric effect. The material undergoes a mechanical deformation in response to the application of an electric field through the relation Si = d ji E j , where S, d and E are the strain and piezoelectric tensors and the electric field respectively. These deformations were visible in the topography signal and subsequent scans using insulating tips were used to separate topographical from ferroelectric domain contributions. At around the same time, other studies [25–27] were performed using what is now called single-frequency piezoresponse force microscopy (PFM). In this technique, an oscillating electric field at frequency ω is applied to the material by a metallic tip in contact with the sample. The converse piezoelectric effect causes a periodic deformation of the material under the tip, which in turns leads to a periodic bending of the cantilever, also at frequency ω. A lock-in amplifier is used to demodulate the vertical deflection amplitude and phase at ω, where the amplitude signal is proportional to the local polarisation magnitude. Polarisation vectors oriented anti-parallel to each other yield a 180◦ phase shift with respect to each other upon application of the AC bias, as illustrated in Fig. 4.10, showing the polarisation-dependent deformation of a ferroelectric material upon application of a down-oriented electric field. Therefore, the phase signal from the lock-in amplifier gives information on the polarisation orientation of the material under the tip. When scanning across a ferroelectric domain wall in PFM, a 180◦ contrast is therefore expected in the PFM phase. A drop in the PFM amplitude is expected at

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Fig. 4.11 Left: Expected PFM phase and amplitude for a line scan crossing through an-up oriented domain in a down-oriented area. Right: PFM phase and amplitude image of a stripe shaped ferroelectric domain

the domain wall position. This is due to the cancelling contributions to the PFM amplitude at the domain wall itself, the typically 10-15 nm resolution of PFM being larger than the domain wall width, and to the null vertical polarisation at the domain wall position (Fig. 4.11). Typically, the cantilever is driven at frequencies of ∼1–100 kHz and the corresponding oscillation amplitudes are close to the AFM noise levels. To increase the signal to noise ratio, the drive frequency can be set close to the contact resonance frequency ∼300–500 kHz of the tip-sample system. The drawback of this technique is that the contact resonance frequency can vary across a sample, for example when scanning over topographical features, which can potentially lead to significant crosstalk between the topography and PFM phase and amplitude signals. One way to solve this problem is to track the resonance peak, as is done in dual frequency resonance tracking (DFRT) PFM.

4.3.3.1

Dual Frequency Resonance Tracking

In this technique, the resonance peak is tracked by applying a bias through the tip-sample system at two frequencies simultaneously, typically 5–15 kHz apart and centred around the contact resonance frequency. Two lock-in amplifiers track the cantilever oscillation amplitude and phase at each of the two drive frequencies and a feedback loop dynamically controls the central frequency by using the difference in amplitudes A2 − A1 as an error signal (Fig. 4.12). Driving the sample at resonance frequencies results in a significant improvement of the signal to noise ratio as can be seen in Fig. 4.13. This allows detection and imaging of finer features without introducing crosstalk with the topography. DFRT

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Fig. 4.12 Operating principle of DFRT PFM. a Two carrier frequencies are used to drive the tipsample system and two lock-in amplifiers track the amplitudes of the cantilever oscillations at the two frequencies. b A feedback loop shifts the central frequency in order to keep the oscillation amplitude difference at 0. Reproduced with permission from [28]

has become a standard for PFM imaging and several manufacturers now offer built-in DFRT implementations in their systems. Further improvements have been made, extending the idea of increasing the number of simultaneously applied frequencies to applying a whole frequency range. In band-excitation [29] for example, the tip bias is applied as a band of frequencies around the resonance peak, allowing full tracking of the resonance, determination of the Q-factor, giving information on mechanical properties of the sample.

4.3.3.2

Switching Spectroscopy PFM

Measurements of ferroelectric hysteresis loops can be brought down to the nanoscale with switching spectroscopy PFM [30]. In this technique, the AFM tip is held stationary on the sample surface, while a bias waveform shown in Fig. 4.14 is applied. The waveform consists of a superposition of a DC component used to locally affect the polarisation with an AC component used simultaneously to measure the PFM phase and amplitude. The DC waveform has a triangular envelope but is split in “on” and “off” sections illustrated in Fig. 4.14 in order to separate electrostatic

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Fig. 4.13 Line scans of a domain in single frequency (left) and DFRT (right). The PFM phases and amplitudes have a much higher signal to noise ration in DFRT Fig. 4.14 Illustration of the bias waveform applied in SSPFM. A “reading” AC bias used to acquire PFM phase and amplitude signals is superimposed with a DC bias with a triangular envelope. The PFM phase in the “off” steps allows the ferroelectric hysteresis loop to be constructed, while the amplitude typically shows a characteristic butterfly shape. An example of acquired signals is shown in Fig. 4.15

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Fig. 4.15 Example of an SSPFM measurement performed on one point, showing two hysteresis loops in the phase signal on top and butterfly loops in the amplitude at the bottom

and ferroelectric contributions to the acquired signals. The PFM phase in the “off” sections is used to reconstruct the ferroelectric hysteresis, while the amplitude shows a characteristic butterfly shape where the minima positions correspond to the coercive bias as shown in Fig. 4.15. These measurements are usually performed on a grid in order to extract ferroelectric hysteretic behaviour with spatial resolution [7, 31], down to single defects [32].

4.3.3.3

Ferroelectric Domain Lithography

The ability to use the AFM tip to locally switch the polarisation allows lithography of ferroelectric domains at the nanoscale by applying electric fields higher than the coercive field of the studied material. During this process, the tip can be moved in any desired pattern, allowing domains of various sizes and shapes to be written (Fig. 4.16). This aspect has been used extensively to study the switching dynamics of ferroelectric domains [33, 34], measure ferroelectric hysteresis loops at the nanoscale [30], as well as study fundamental aspects of disordered elastic systems and ferroelectric domains [35–37].

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Fig. 4.16 DFRT PFM (a) phase and (b) amplitude of an artificially written circular domains

It is worthy to note that the sharpness of AFM tips lead to locally high electric fields which can often lead to injection or redistribution of defects and damage to the film [38, 39]. While this is often an unwanted effect, it can in some cases be used to locally alter the defect landscape in a material.

4.3.3.4

Lateral and Vector PFM

Imaging of ferroelectric materials using PFM is not limited to domains where the polarisation points out of the sample plane. Because ferroelectric materials can have non-zero shear piezoelectric tensor components, if domains in the sample plane are present, the tip field can also cause torsion of the cantilever through the shear motion of the material shown in Fig. 4.17. The periodic torsion of the cantilever due to the AC tip field can be picked up by demodulating the amplitude and phase of the corresponding oscillation of the horizontal deflection D H . The measurement setup is the same as for vertical PFM, with the only difference being that the lateral deflection signal corresponding to torques of the cantilever are demodulated instead of the vertical deflection. In vector PFM [40], the vertical and lateral oscillation amplitudes are acquired simultaneously in order to image the vertical and lateral components of the sample polarisation vector. In general, the corresponding resonance modes would have very different frequencies and the tip is simultaneously biased at two different frequencies chosen to excite both modes. Interpretation of lateral PFM images is more difficult than in vertical PFM. First, the stiffness of the cantilever is much higher for rotations around the long axis than bending, which typically leads to a much lower signal in lateral PFM than in vertical

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Fig. 4.17 Non-zero d15 components of the piezoelectric tensor cause a torsion of the cantilever which depends on the polarisation orientation and for an AC tip bias, can be picked-up as a non-zero oscillation amplitude in the horizontal deflection

PFM. Second, any single lateral PFM image only acquires information on the in-plane polarisation component perpendicular to the long axis of the cantilever. This means that at least two images need to be obtained at different cantilever orientations and the polarisation directions need to be reconstructed from the set of images obtained. An example of this is displayed in Fig. 4.18. Third, in vertical PFM, the absolute orientation of the polarisation can easily be determined by using electric field pulses of known polarity to locally switch the polarisation, then comparing the PFM phase of the written domain of known polarisation orientation with that of the as-grown polarisation state. This is not easily done in lateral PFM. Though domains can in principle be written using the trailing lateral field of the tip [41], this is requires a more complex writing process [42]. Furthermore, lateral PFM signals are subject to artefacts linked to broken symmetries in the investigated materials. These artefacts are discussed in further detail in Sect. 7.2.4.

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Fig. 4.18 Example of an in-plane polarisation pattern and the corresponding expected lateral PFM phase and amplitude contrasts for different orientations of the cantilever with respect to the polarisation pattern. After [43]

4.3.4 Distortion Correction Algorithm Atomic force microscopy is often used to track the evolution of a system with time or through the variation of an external parameter. These types of measurements requiring sequential AFM images have been used to study a wide variety of systems such as collagen self-assembly [44], nano-encapsulation processes for drug delivery [45], effects of drugs on mechanics and structure of bacteria [46], magneto-electric couplings in multiferroic materials [47, 48], and electromechanical effects in graphene [49]. In ferroelectric materials, sequential measurements were used to track the evolution of ferroelectric domains [7, 50, 51] and to study the conduction at domain walls under increasing electric field [52, 53]. In these sequential measurements, significant further insight can sometimes be gained by directly correlating individual nanoscale features across time or changing external parameters, which requires tracking features across the sequentially acquired images. In this respect, atomic force microscopy suffers from drawbacks related to the piezoelectric actuators used to control the tip position with respect to the sample. AFM images can be formed from either the left-to-right (trace) or right-to-left (retrace) branches of the raster scanning pattern. Due to piezoelectric hysteresis effects, images formed from the trace and retrace sections will exhibit relative distortions which are usually maximal at the middle of the image. Moreover, fast movements of the tip require fast changes in the voltage applied to the scanner. In these conditions, the scanner moves fast for most of the distance, then much slower for the remainder. This second slow step is called scanner creep and leads to further distortion in AFM images. Furthermore, temperature changes or gradients are an additional source of distortions, called thermal drift. Hardware solutions exist to minimise these phenomena. Atomic force microscopes can be enclosed in thermally insulated or temperature-controlled casings to reduce thermal drift. In so-called closed-loop systems, the position of the column can

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Fig. 4.19 Distortion correction algorithm work flow a The reference and target images are loaded. b Matching points in the topography and their respective positions are found automatically. c–e The coordinate transform required to map the target topography onto the reference is computed and applied to all channels in the target scan. Reproduced with permission from [58]

be measured externally, typically through capacitive sensors, in order to correct for the aforementioned effects in real time. However, closed loop systems are sometimes difficult to implement or financially out of reach. In these cases, various software solutions have been developed. Real-time corrections can be performed with the drawback of requiring complex calibration routines [54, 55]. Relative corrections of distortions can also be performed after image acquisition [56, 57] using topography information to extract a mapping between subsequent images. An algorithm using a similar approach as the latter has been developed in the Paruch group [58]. The correction of relative distortion between two AFM scans within a series is performed in the following steps and illustrated in Fig. 4.19: 1. Within the series of AFM images, a reference scan is chosen by the user. 2. Matching points and their respective coordinates in the topography signals are found automatically between the reference (xr,i , yr,i ) and the target scan (xt,i , yt,i ). 3. The coordinates of the matching points are used to generate a polynomial transform that maps the target topography onto the reference. 4. The transform is applied to all scan channels in the target scan. The algorithm uses methods implemented in the OpenCV Python library for computer vision. The feature detection can be performed by the SIFT [59], SURF [60] or ORB [61] methods while the matching is performed by the FLANN [62] algorithm. The computation of the transformation and corrected target coordinate map are performed using functions implemented in the Scikit-image transformations and Scipy ndimage libraries, respectively. A demonstration of the algorithm is shown in Fig. 4.20. The first and last scans in a series of 21 PFM scans are used to demonstrate the viability of the algorithm. Figure 4.20a, d shows the topographies of the first and last scans respectively. Slight shifts and distortions are present between the two scans. A map of the pixel by pixel difference between the scans is shown in Fig. 4.20b. The first topography is used as a reference to correct the last scan in the series and the resulting corrected topography is shown in Fig. 4.20c. The white areas on the left and bottom indicate where the image had to be compressed in order to fit the topography and show that the last scan is not only shifted with respect to the reference but also slightly distorted. Figure 4.20e shows the difference between the corrected last scan and the reference. The

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Fig. 4.20 Use case of the distortion correction algorithm on the first and last scans in a series of 21 images taken consecutively. a Topography of the first scan, used as a reference. b Uncorrected topography of the last scan showing slight shifts and distortions with respect to the reference. b Map of height differences between the uncorrected last scan and reference showing inhomogeneous differences due to the relative distortions. e Height difference map between the last corrected scan and reference. The differences are now much smoother. c Corrected topography of the last scan showing that the image had to be compressed in order to match the reference. f Histograms of the maps shown in b in blue and e in red. The histogram of corrected height differences is much narrower indicating that the correction was effective

Fig. 4.21 Phase images corresponding to a the reference and d the last scan in the series of consecutive PFM images shown in Fig. 4.20. b The phase difference map between the last uncorrected scan and reference show the mismatch between the two images, while the differences between the corrected last scan and reference are much smoother. c Domain wall positions extracted by binarising the phase signals of the reference (blue) and uncorrected (red) PFM scans showing large shifts. f The domain wall positions of the corrected scan show an almost perfect overlap with the reference

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4 Experimental Methods

map of differences is now much smoother. This can also be seen in the histogram of differences between the reference and uncorrected scan in blue and corrected scan in green. The distribution in the latter is much narrower highlighting the effectiveness of the correction. The same warping was used to correct the PFM phase signal. Figure 4.21 shows the reference and uncorrected PFM phases, as well as differences between the reference and uncorrected and corrected phases. Figure 4.21c, d show the extracted domain wall position of the reference scan in blue. The uncorrected domain wall position shown in panel (c) shows strong shifts and distortions, while the corrected domain wall position is almost perfectly overlapping the domain wall position in the first scan in the series. This algorithm was used to correct for distortions between sequential AFM measurements presented in Chaps. 5 and 6.

References 1. Feuersanger A, Lublin P (1963) Electrical properties and structure of barium titanate films. J Electrochem Soc 110:C192 2. Feuersanger A, Hagenlocher A, Solomon A (1964) Preparation and properties of thin barium titanate films. J Electrochem Soc 111:1387–1391 3. Slack J, Burfoot J (1971) Electrical properties of flash evaporated thin films of InTe. J Phys C 4:898–909 4. Lichtensteiger C, Dawber M, Triscone J-M (2004) Ferroelectric size effects. In: Rabe K, Ahn C, Triscone J-M (eds) Physics of ferroelectrics - a modern perspective. Springer, Heidelberg, pp 305–336 5. Pintilie L et al (2008) The influence of the top-contact metal on the ferroelectric properties of epitaxial ferroelectric Pb(Zr0.2Ti0.8)O3 thin films. J Appl Phys 104 6. Lichtensteiger C et al (2014) Tuning of the depolarization field and nanodomain structure in ferroelectric thin films. Nano Lett 14:4205–4211 7. Lichtensteiger C et al (2016) Built-in voltage in thin ferroelectric PbTiO3 films: the effect of electrostatic boundary conditions. New J Phys 18:1–12 8. Wang RV et al (2009) Reversible chemical switching of a ferroelectric film. Phys Rev Lett 102:2–5 9. Garrity K et al (2013) Ferroelectric surface chemistry: first-principles study of the PbTiO3 surface. Phys Rev B - Condens Matter Mater Phys 88:1–11 10. Domingo N et al (2019) Surface charged species and electrochemistry of ferroelectric thin films. Nanoscale 11:17920–17930 11. Schlom DG et al (2007) Strain tuning of ferroelectric thin films. Ann Rev Mater Res 37:589–626 12. Jia CL et al (2011) Direct observation of continuous electric dipole rotation in flux-closure domains in ferroelectric Pb(Zr, Ti)O3. Science 331:1420–1423 13. McQuaid RG et al (2014) Exploring vertex interactions in ferroelectric flux-closure domains. Nano Lett 14:4230–4237 14. Tang YL et al (2015) Observation of a periodic array of flux-closure quadrants in strained ferroelectric PbTiO3 films. Science 348:547–551 15. Zhang Q et al. Nanoscale bubble domains and topological transitions in ultrathin ferroelectric films. Adv Mater 1702375 16. Yadav Y et al (2016) Observation of polar vortices in oxide superlattices. Nature 530:198–201 17. Das S et al (2019) Observation of room-temperature polar skyrmions. Nature 568:368–372

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18. Rabe KA et al (2004) Modern physics of ferroelectrics: essential background. In: Rabe K, Ahn CH, Triscone J-M (eds) Physics of ferroelectrics - a modern perspective. Springer, Heidelberg, pp 1–29 19. Binnig G, Quate CF, Gerber C (1986) Atomic force microscope. Phys Rev Lett 56:930–933 20. Saurenbach F, Terris BD (1990) Imaging of ferroelectric domain walls by force microscopy. Appl Phys Lett 56:1703–1705 21. Lüthi R et al (1993) Surface and domain structures of ferroelectric crystals studied with scanning force microscopy. J Appl Phys 74:7461–7471 22. Lüthi R et al (1994) Statics and dynamics of ferroelectric domains studied with scanning force microscopy. J Vac Sci Technol B 12:2451 23. Gruverman A et al (1995) Domain structure and polarization reversal in ferroelectrics studied by atomic force microscopy. J Vac Sci Technol B 13:1095–1099 24. Kolosov O et al (1995) Nanoscale visualization and control of ferroelectric domains by atomic force microscopy. Phys Rev Lett 74:4309–4312 25. Güthner P, Dransfeld K (1992) Local poling of ferroelectric polymers by scanning force microscopy. Appl Phys Lett 61:1137–1139 26. Franke K et al (1994) Modification and detection of domains on ferroelectric PZT films by scanning force microscopy. Surf Sci Lett 302:283–288 27. Gruverman A, Auciello O, Tokumoto H (1996) Scanning force microscopy for the study of domain structure in ferroelectric thin films. J Vac Sci Technol B 14:602 28. Rodriguez BJ et al (2007) Dual-frequency resonance-tracking atomic force microscopy. Nanotechnology 18 29. Jesse S et al (2007) The band excitation method in scanning probe microscopy for rapid mapping of energy dissipation on the nanoscale. Nanotechnology 18 30. Jesse S, Baddorf AP, Kalinin SV (2006) Switching spectroscopy piezoresponse force microscopy of ferroelectric materials. Appl Phys Lett 88:1–4 31. Jesse S, Lee HN, Kalinin SV (2006) Quantitative mapping of switching behavior in piezoresponse force microscopy. Rev Sci Instrum 7 32. Kalinin SV et al (2008) Probing the role of single defects on the thermodynamics of electricfield induced phase transitions. Phys Rev Lett 100:2–5 33. Gruverman A, Wu D, Scott JF (2008) Piezoresponse force microscopy studies of switching behavior of ferroelectric capacitors on a 100-ns time scale. Phys Rev Lett 100:3–6 34. Polomoff NA et al (2009) Ferroelectric domain switching dynamics with combined 20 nm and 10 ns resolution. J Mater Sci 44:5189–5196 35. Paruch P, Giamarchi T, Triscone JM (2005) Domain wall roughness in epitaxial ferroelectric PbZr0.2Ti0.8O3 thin films. Phys Rev Lett 94:3–6 36. Paruch P et al (2006) Nanoscale studies of domain wall motion in epitaxial ferroelectric thin films. J Appl Phys 100 37. Catalan G et al (2008) Fractal dimension and size scaling of domains in thin films of multiferroic BiFeO3. Phys Rev Lett 100:35–38 38. Ievlev AV et al (2018) Nanoscale electrochemical phenomena of polarization switching in ferroelectrics. ACS Appl Mater Interfaces 10:38217–38222 39. Kalinin SV et al (2011) The role of electrochemical phenomena in scanning probe microscopy of ferroelectric thin films. ACS Nano 5:5683–5691 40. Kalinin SV et al (2006) Vector piezoresponse force microscopy. Microsc Microana 12:206–220 41. Crassous A et al (2015) Polarization charge as a reconfigurable quasidopant in ferroelectric thin films. Nat Nanotechnol 10:614–618 42. Béa H et al (2011) Nanoscale polarization switching mechanisms in multiferroic BiFeO3 thin films. J Phys Condens Matter 23 43. Guyonnet J (2013) Growing up at the nanoscale: studies of ferroelectric domain wall functionalities, roughening, and dynamic properties by atomic force microscopy. PhD thesis 44. Gale M et al (1995) Sequential assembly of collagen revealed by atomic force microscopy. Biophys J 68:2124–2128

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45. Oliva M et al (2002) Sequential atomic force microscopy imaging of a spontaneous nanoencapsulation process. Int J Pharm 242:291–294 46. Wu Y, Sims RC, Zhou A (2014) AFM resolves effects of ethambutol on nanomechanics and nanostructures of single dividing mycobacteria in real-time. Phys Chem Chem Phys 16:19156– 19164 47. Heron JT, Schlom DG, Ramesh R (2014) Electric field control of magnetism using BiFeO3based heterostructures. Appl Phys Rev 1 48. Trivedi H et al (2018) Sequential piezoresponse force microscopy and the ‘small-data’ problem. NPJ Comput Mater 4:28 49. Chhikara M et al (2017) Effect of uniaxial strain on the optical Drude scattering in graphene. 2D Mater 4:025081 50. Gruverman A et al (2005) Direct studies of domain switching dynamics in thin film ferroelectric capacitors. Appl Phys Lett 87:1–4 51. Huey BD et al (2012) High speed SPM applied for direct nanoscale mapping of the influence of defects on ferroelectric switching dynamics. J Am Ceram Soc 95:1147–1162 52. Guyonnet J et al (2011) Conduction at domain walls in insulating Pb(Zr0.2Ti0.8)O3 thin films. Adv Mater 23:5377–5382 53. Gaponenko I et al (2015) Towards reversible control of domain wall conduction in Pb(Zr0.2Ti0.8)O3 thin films. Appl Phys Lett 106 54. Lapshin RV (2015) Drift-insensitive distributed calibration of probe microscope scanner in nanometer range: approach description. Appl Surf Sci 359:629–636 55. Lapshin RV (2016) Drift-insensitive distributed calibration of probe microscope scanner in nanometer range: virtual mode. Appl Surf Sci 378:530–539 56. Rahe P, Bechstein R, Kühnle A (2010) Vertical and lateral drift corrections of scanning probe microscopy images. J Vac Sci Technol B 28:31–38 57. D’Acunto M, Salvetti O (2011) Pattern recognition methods for thermal drift correction in atomic force microscopy imaging. Pattern Recognit Image Anal 21:9–19 58. Gaponenko I et al (2017) Computer vision distortion correction of scanning probe microscopy images. Sci Rep 7:1–8 59. Lowe D (1999) Object recognition from local scale-invariant features. In: 7th IEEE international conference on computer vision 60. Bay H et al (2008) Speeded-Up Robust Features (SURF). Comput Vis Image Underst 110:346– 359 61. Rublee E et al (2011) ORB: an efficient alternative to SIFT or SURF. In: International conference on computer vision 62. Muja M, Lowe D (2009) Fast approximate nearest neighbors with automatic algorithm configuration. In: 4th international conference on computer vision theory and application, pp 331–340

Chapter 5

Crackling at the Nanoscale

As discussed in Chap. 3, previous studies of crackling noise in ferroelectric materials lack spatial information and do not distinguish between the creep and depinning regimes of interface velocity. Theoretical work [1] suggests that in disordered elastic systems theory, which has been successful at describing the static and dynamic properties of ferroelectric domain walls, avalanches cluster in space and time in the creep regime, while in the depinning regime, individual avalanches appear to be uncorrelated. Experimental [2, 3] and theoretical [4] studies of propagating crack fronts, which are well modelled by elastic interfaces in disordered media with longrange interactions, have shown that close to but above the depinning force, global avalanches are formed by local disconnected clusters, whose aspect ratio depends on the roughness exponent of the crack. The size distributions of both the global and local avalanches follow power law behaviours, albeit with different exponents related by τcluster = 2τglobal − 1. These results point to the importance of distinguishing between the creep and depinning regimes when studying crackling and show that spatial information about individual jerky events provides further insight into the systems under study. In this chapter, crackling behaviour is studied in two ferroelectric thin films of Pb(Zr0.2 Ti0.8 )O3 using PFM to investigate the distribution of the jerky event sizes with nanoscale spatial resolution.

5.1 Sample Characterisation The two samples, henceforth labelled PZT-Nuc and PZT-Mot, consist of thin films of Pb(Zr0.2 Ti0.8 )O3 , epitaxially grown on (001)-oriented SrTiO3 substrates, with a SrRuO3 back electrode. PZT-Nuc and PZT-Mot were grown by off-axis RF magnetron sputtering and pulsed laser deposition, respectively. Both samples are 60–70 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. Tückmantel, Scanning Probe Studies of Structural and Functional Properties of Ferroelectric Domains and Domain Walls, Springer Theses, https://doi.org/10.1007/978-3-030-72389-7_5

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Fig. 5.1 Surface topographies of a PZT-Nuc and b PZT-Mot. The root-mean-square roughness in the two samples are comparable, at 0.3 nm and 0.4 nm respectively. The topographical patterns in PZT-Mot are a consequence of dislocations forming during the growth [5]

nm thick and show similar surface roughnesses of 0.3 and 0.4 nm for PZT-Nuc and PZT-Mot, respectively, but with quite different surface morphologies seen in Fig. 5.1. As shown in Fig. 5.2a, b, XRD 2θ/ω scans along the (001) crystalline axis give a similar c-axis parameter of 4.14 Å for PZT-Nuc and 4.18 Å for PZT-Mot. Reciprocal space maps shown in Fig. 5.2c, d show that PZT-Nuc has relaxed with an a-axis parameter of 3.95 Å, while PZT-Mot is still mostly strained to the substrate with an a-axis of 3.905 Å, although some shift of intensity of the peak towards higher a-axis values can be observed, suggesting partial relaxation. The surface morphology and X-ray diffraction data in PZT-Mot are consistent with dislocations forming during growth as a result of sample-substrate mismatch strain altering the local adatom binding energy. This results in a growth mode transition from step-flow to 3D nanoscale islands oriented along the crystalline axes as the sample grows thicker [5]. Wavelength dispersion spectroscopy on both samples and analysis of the Pb/Zr ratios suggest that PZT-Mot has a slightly higher Pb content than PZT-Nuc with ratios of 8.5 ± 2 and 5.6 ± 0.8, respectively. Both samples have an as-grown monodomain up polarisation oriented out of the plane of the film. The differences in morphology, relaxation of mismatch strain, and Pb stoichiometry can be expected to lead to very different defect landscapes. The resulting variations in domain wall pinning can be seen in Fig. 5.3 with noticeably different degrees of roughening of domains written under similar conditions in the two films, suggesting higher variations in defect densities in PZT-Nuc.

5.2 Measurements Measurements were performed in ultra-high vacuum at pressures of ∼10−11 mbar in an Omicron VT beam deflection AFM, thus avoiding fluctuations in the switching

5.2 Measurements

57

Fig. 5.2 a, b 2θ/ω scans along the c-axis, aligned on the (001) SrTiO3 peak for PZT-Mot and PZT-Nuc respectively. c, d Reciprocal space maps in PZT-Mot and PZT-Nuc taken around (-103) showing in-plane relaxation in PZT-Nuc, while PZT-Mot is mostly strained to the substrate

Fig. 5.3 PFM images of domain walls written with 10 V in PZT-Nuc (top) and 8 V in PZT-Mot (bottom), showing different roughness, indicative of higher variations in defect densities in the former

dynamics due to varying relative humidities observed in ambient conditions [6–8]. First, domains of opposite polarisation are written with a high positive scanning probe tip bias. Local polarisation switching events are then induced by scanning the film surface while applying a DC bias to the scanning probe lower than the bias used for domain writing. In order to gradually switch the polarisation and monitor the evolution of the domains, the measurements are performed stroboscopically. Switching scans with a defined DC bias, gradually incremented from scan to scan by

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Fig. 5.4 PFM phase images of the same written domain after various switching scans. a As-grown region (bright contrast) between two written domains (dark contrast), without subsequent switching scan. b–d The same region after scans at 3.25, 3.875 and 4.0 V. In this sample, when switching from as-grown up to down polarisations, the reversal occurs first by inward motion of the domain wall b, followed by mixed domain wall motion and nucleation of new domains c. The nucleated domains rapidly expand outwards and coalesce d

a fixed interval, are alternated with PFM imaging scans allowing the evolution of the domain structure to be imaged at nanoscale resolution.1 Snapshots of the resulting domain configurations in PZT-Nuc are shown in Fig. 5.4. The resulting switching event size distributions are acquired separately in the as-grown and written regions in both samples Relative distortions in successive scans due to creep and thermal drift of the AFM column are removed by applying the distortion correction algorithm described in Sect. 4.3.4. This allows individual regions throughout the entire measurement series to be correlated with high accuracy. Each switching scan induces a series of switching events, mapped as the differences between the PFM images preceding and succeeding the switching scans. The tip bias triggering polarisation reversal is extracted for each pixel by first binarising all the PFM phase images corrected for local distortions. Pixels with a value of 1 correspond to the polarisation orientation that is favoured by the tip bias, while pixels corresponding to the opposite orientation are assigned a value of 0. For each pixel, the tip bias at which the binarised phase changes from 0 to 1 is assigned as its local switching bias. Some pixels flicker between 0 and 1 during multiple PFM scans before remaining at a value of 1 until the end of the measurement series. For these pixels, the median value of the switching biases during the flickering portion of the measurement is assigned as that pixel’s switching scan. This procedure allows the maps of the local switching bias shown in Fig. 5.5 to be generated. The colour represents the tip bias triggering the local events. From these, 1 The

DC bias of the ith scan Vi is kept constant throughout the switching scan but is increased from one switching scan to the next by a fixed interval V of 125 mV in PZT-Nuc and 100 mV in PZT-Mot. The switching scan i + 1 is then performed with a DC bias of Vi + V .

5.2 Measurements

59

Fig. 5.5 Local polarisation switching maps extracted from alternating switching scans and PFM measurements under increasing tip bias. The left a, b and right c, d columns show the switching bias in PZT-Nuc and PZT-Mot respectively. The top a, c and bottom b, d rows show switching maps acquired with a negative and positive tip bias respectively, corresponding to switching from written down-polarised domains to up and from as-grown up to down

polarisation reversal event sizes can be extracted as the area of individual regions sharing the same switching bias. As will be discussed in more detail in Sect. 5.4, these local switching bias maps also allow domain nucleation, motion, and merging events to be distinguished.

5.3 Switching at a Glance In this section, the differences in polarisation switching between PZT-Nuc and PZTMot are discussed in detail. Qualitative differences in the switching mechanisms can be clearly seen between the two samples, as well as between switching from the asgrown (up to down) and the written (down to up) polarisation states under positive and negative tip bias Vti p , respectively. For PZT-Nuc, polarisation reversal proceeds predominantly by the nucleation and growth of new domains, with relatively little motion of the pre-existing walls of the initial domain configuration. This is particularly noticeable under negative tip bias, shown in Fig. 5.5a, where the two side walls of the down-oriented central stripe domain move inwards by only 20 nm between −1.0 and −1.5 V. Beyond this threshold, multiple new up-oriented domains nucleate in the central region and grow rapidly outwards, gradually merging until almost complete polarisation reversal at −4 V. Under positive tip bias, we observe significantly higher

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Fig. 5.6 a Normalized switching rates in PZT-Nuc and PZT-Mot samples, showing earlier onset of switching at negative bias, but more rapid full polarisation reversal at positive bias. Changes in switching rates corresponding to the onset of rapid domain nucleation and merging can also be observed. b First and second SSPFM hysteresis loops averaged over 25 locations. Changes of the positive coercive bias between the second and first loop are indicative of injection and redistribution of defects, known to occur in atomic force microscopy measurements on ferroelectric thin films [8–10]

activation thresholds for polarisation switching via domain wall motion, which sets in at 3 V. The motion contributes to somewhat larger 100 nm inward displacements of the existing domain walls. Subsequently, very rapid growth of new domains via point nucleation occurs beyond 3.75 V, leading to almost complete polarisation reversal. For PZT-Mot, meanwhile, polarisation reversal under both positive and negative tip bias proceeds almost exclusively via the motion of pre-existing domain walls seen

5.3 Switching at a Glance

61

in Fig. 5.5c, d. For positive tip bias, these walls, initially relatively flat, begin moving via small displacements of around 25 nm at close to ±1 V. For negative tip bias, the motion appears quite regular and generalised to the entire domain walls, while for positive tip bias the displacements remain extremely limited until approximately 2.3 V is reached, at which point large jumps can be observed, leading to complete polarisation reversal. Under negative tip bias, the domain walls also appear to roughen more noticeably while they move. The effect of domain writing history on the overall switching behaviour can also be seen from the evolution of the proportion of the switchable surface. This is calculated as the ratio of the switched area to the total area in which the polarisation is initially oriented opposite to the applied electric field, and shown in Fig. 5.6a. In both samples, onset of domain wall motion occurs at a higher bias when switching from the asgrown polarisation state with positive bias, and the switched surface proportion first increased slowly close to the onset of switching, then much more rapidly at higher bias. At negative bias, corresponding to switching written domains where additional defects caused by the writing are expected to provide additional pinning to the domain walls, the switching is much more gradual. In both samples, the switching rates clearly show a difference between a relatively limited regime of domain wall motion at lower positive bias, and then more rapid switching, whether by further domain wall motion or nucleation and growth of new domains at higher bias values. The influence of domain writing history is further observed in the differences between the first and second local hysteresis loops, taken with a stationary tip and averaged over 25 locations, acquired by switching spectroscopy PFM [11] and shown in Fig. 5.6b. Changes in the positive bias branch in the second loop seem to suggest that the high bias applied during the first loop affected the local switching thresholds in the films. Because of these differences in the overall switching dynamics, the event size distributions are extracted separately for positive and for negative tip bias in each sample.

5.4 Extraction of Switching Events For each measurement series, the respective contributions of domain nucleation, wall motion, and merging can be obtained by analysing individual switching events to determine their connectivity to other domains of the same orientation. The cumulative event maps of these processes during polarisation reversal in the two samples are shown in Fig. 5.7. In subsequent analysis, nucleation events are excluded from the size distributions as their sizes are expected to be determined by the critical nucleus [12], and we consider only domain wall motion and merging. The ability to discriminate between event types should allow the size distributions to be extracted separately for these two types of events, exploring potential variations in their power law scaling exponents. However, the present measurements do not yet provide sufficient statistics to warrant their separate treatment.

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Fig. 5.7 Maps showing different types of switching events—domain nucleation (red), domain wall motion (blue) and domain merging (green)—in the PZT-Nuc a, b and PZT-Mot c, d samples at negative and positive tip bias, respectively. The scalebars are 200 nm

To analyse the crackling statistics in the different samples and voltage polarities, the critical force separating the creep and depinning regimes is determined in each case. Initially, only events in the creep regime are included in the event size distributions in order to avoid exponent crossover between regimes, and for a valid comparison of the event size statistics in the two samples and opposite tip bias polarities. For each tip bias we determine the effective domain wall displacement by calculating the equivalent disc radius of the total area switched at each switching scan, as shown in Fig. 5.8. At low tip bias, the overall displacement remains small but starts to increase non-linearly when the driving force is increased and the displacement exceeds the noise threshold, most clearly visible on a logarithmic scale in the insets of Fig. 5.8. This displacement threshold defines the lower cutoff of the tip bias window defining the creep regime. Since the depinning transition is characterised by a sharp upturn of the velocity response to the applied field [13], a lower bound for the value of the depinning bias can be obtained from the intercept of the linear fit to the region of rapid displacement [14], as indicated by the star marker in Fig. 5.8b–d. This estimate of the critical force defines the upper tip bias cutoff. The analysis of the average domain wall displacements shows a qualitatively different behaviour for positive and negative bias. In the former case, where the polarisation is switched from the preferred as-grown Pup to the Pdown state, the velocity appears to follow a typical creep to depinning transition. With negative bias however, where the switching occurs in written domains whose defect landscape has

5.4 Extraction of Switching Events

63

Fig. 5.8 Average domain displacement at each switching scan with linear fits of the rapid increase at higher bias, whose intercept at zero (star marker) is used to extract a lower bound on the depinning bias. This bias is used as an upper cutoff to the events included in the power law fitting. Insets show the average domain wall displacement on a logarithmic scale, highlighting abrupt changes in displacement. The first abrupt change is used as a lower cutoff to the switching events included in the power law fitting. Lightly coloured areas show the tip bias range included in the power law fitting. a PZT-Nuc, Vti p < 0, b PZT-Nuc, Vti p > 0, c PZT-Mot, Vti p < 0, d PZT-Mot, Vti p > 0

been altered by the high writing bias, the velocity curve shows a dome-like feature, possibly related to different pinning hierarchies. The sudden drop in domain wall displacement in PZT-Mot for V < 0 between −1.5 and −2.0 V seen in Fig. 5.8c is likely a result of surface contaminants temporarily adhering to the tip and changing the effective field applied to the sample. In the case of PZT-Nuc for V < 0 shown in Fig. 5.8a, where the switching is dominated by multiple nucleation sites, each giving rise to a separate growing domain, the decrease of the switching after 4.0 V is likely caused by the lack of available unswitched areas, leading to low displacements. In this case, the same voltage window is used as in the as-grown case. The event sizes in the creep regime are extracted from these boundaries and their distribution can be analysed. These events are displayed in the maps of Fig. 5.9.

5.5 Power Law Fitting and Characterisation Establishing whether a set of data is distributed as a power law is not an easy task. Many heavy-tailed distributions exhibit a linear behaviour on a log-log plot and power law fitting must be performed with care. Fitting of power laws described

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Fig. 5.9 Map of switching events occurring in the creep regime of interface motion and included in the power law fitting. These events do not include nucleation and span only a fraction of all the local polarisation reversal events, whose contours are shown by the thin black lines

by p(x) = Ax −α is often performed by linearising the distribution as log( p(x)) = log(A) − αlog(x). A linear regression is performed, typically using a least-squares cost function and the slope of the regression is assigned as the power law exponent. Although this method is easily implemented, it suffers from multiple shortcomings, such as the estimation of the errors on the fitting parameters and assessing the goodness of fit. Indeed, the standard error on a least-squares fit assumes that the error on the independent variable follows a Gaussian distribution centred around the “true” value. However, even if the error on x is Gaussian, the error on log(x) is not, making error estimates invalid. Furthermore, the coefficient of determination R 2 typically used to estimate the goodness of fit tends to be high for a wide variety of heavytailed distributions, which will look linear on a log-log plot, giving a false sense of confidence that the data is generated by a power law distribution. Methods have been devised to fit power law distributions and estimate the characteristic exponents more reliably, notably by using the maximum-likelihood estimator.

5.5.1 Maximum-Likelihood Estimator The maximum likelihood estimator finds the value of α that maximises the probability that the data was drawn from a power law distribution. The power law probability distribution takes the form

5.5 Power Law Fitting and Characterisation

p(x) =

65

α − 1 x −α ( ) xmin xmin

where xmin is the minimum value of x for which the data is power law distributed and α is the power law exponent. The probability that the data was drawn from a distribution with parameters xmin and α is proportional to

p(x | α, xmin ) =

n  α−1 i=1

xmin

(

x xmin

)−α

where xi are the individual data points, which are assumed to be independently and identically distributed random variables. The calculation of this probability is made computationally less costly by taking the logarithm of p(x | α, xmin ) in order to convert the product into a sum. Maximising log( p(x | α, xmin )) and solving for α yields

αˆ = 1 + n

 n i=1

ln

xi

−1

xmin

l where αˆ is the estimated value of the power law exponent. The maximum likelihood estimator depends on the lower cutoff to the power law distribution xmin . This parameter usually needs to be estimated separately. The standard error on αˆ can be estimated from the fact that the MLE is asymp2 totically Gaussian with variance (α−1) and yields n αˆ − 1 σ= √ n

5.5.2 Estimating Size Cutoffs Experimentally, power law distributed data typically have a lower scale cutoff xmin due to physical restrictions such as lower limits of the system size or resolution and noise in the data acquisition. The choice of xmin when performing power law fits is very important. If xmin is too low, data that is not power law distributed will be included in the regression. Since for heavy-tail distributions like power laws, most of the data is on the lower end of x values, this can lead to significant deviations in

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Fig. 5.10 Fitted power law exponent values αˆ as a function of xmin for synthetic data following a power law above xmin and an exponential distribution below, with xmin = 10, α = 2.0, and N = 3000. The values of αˆ are averaged over 100 realisations of the synthetic data. The lowest values of xmin within the plateau in αˆ can be used to estimate the real xmin

the fitted value of the exponents. If xmin is chosen to be too large, valuable data is excluded from the fit, leading to higher errors in the regression parameters. A popular approach is to perform separate regressions for all values of xmin within the data range, each yielding a separate exponent α. ˆ When plotting αˆ against xmin , a plateau in αˆ should be reached for a range of xmin values. For xmin to the left and right sides of that plateau, αˆ deviates rapidly because on the left side, data not following a power law is included in the regression and on the right the sample size is reduced, as is illustrated in Fig. 5.10. xmin is then visually chosen to be the smallest value of x within the plateau. Although this technique can be effective for very large sample sizes, plateaus are not always unique or clearly visible and this method becomes somewhat subjective. Another method which has been shown to be more objective and reliable [15], especially for smaller data sets, involves choosing the value of xmin that minimises a measure of distance between the cumulative probability distribution functions (CDF) of the experimental data and the corresponding regression. The distance is measured between the CDFs rather than the probability distribution functions (PDFs) because the CDF is more robust to fluctuations due to finite numbers of data points. This is especially true on the large end of x, where the data is the scarcest. The most common distance metric used in this context is the Kolmogorov-Smirnov distance, defined as the maximum distance between the CDFs of the data and fit:

D K S = max | Cdata (x) − C f it (x) | x≥xmin

The power law fitting in this study was performed using the maximum-likelihood method with the Powerlaw Python package implementation [16], following the methods described in this section [15]. Power law fits are inspected individually in order to assess the quality of the fit and check for potential multiple values of xmin yielding

5.5 Power Law Fitting and Characterisation

67

Fig. 5.11 Typical characterisation of a power law fit on switching event sizes in our samples. a–c show the KS distance, standard error σ and exponent τ as a function of the lower size cutoff, for the entire Smin range. The triangles indicates the xmin value that minimises the KS distance. The KS distance can be seen to decrease sharply towards the minimal value of 243 nm2 before increasing sharply. d–f KS distance, σ and τ over restricted ranges showing a plateau in exponent values. The CDF g and PDF h of the fit and data show very good agreement

similar KS distances. Two different values of xmin could yield KS distances that are very close. In those cases, the lowest value of xmin is kept in order to include as much data as possible into the regression. An example of characterisation is shown in Fig. 5.11. The KS distance, standard error σ and the fitted power law exponents τ are plotted against xmin . This allows the stability of the fitted exponents and standard error to be checked. The CDFs and PDFs of the data and regression are also visually inspected. The maximum-likelihood estimation of the characteristic size exponent τ , combined with the KS method for finding xmin seem to give reliable regression exponents within the limits of the quantity of data and the range of event sizes that are accessible in this measurement setup.

5.5.3 Boxing Further complications can arise when the domain walls are mobile enough to allow separate switching events occurring at subsequent passes of the biased tip to connect into an apparent single event, as illustrated schematically in Fig. 5.12. The measurements on PZT-Mot with Vti p < 0 show single switching events spanning the whole length of the domain wall. However, we know based on past studies of domain growth under a stationary biased tip [17] that the actual switching events must necessarily be local, given the effective dwell time which can be estimated based on the speed of the scanning tip during writing (∼1–10 ms per pixel during a

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Fig. 5.12 a During switching scans, individual passes of the biased tip (dashed arrows) trigger local switching events. If the domain walls are mobile enough, depending on the interplay of the tip bias and the disorder, the individual switching events become interconnected and will appear as a single larger event on the subsequent imaging of the domain configuration b

line scan). That these switching events appear to coalesce into a single domain wall motion event is a consequence of scanning the entire area with a DC bias before performing a PFM imaging scan. However, this approach can lead to very large apparent event sizes and very low statistics. In order to extract meaningful size exponents in this case, the domain wall areas were split into boxes as shown in Fig. 5.13. Here, we illustrate the boxing procedure with the dataset from measurements on PZT-Nuc Vti p >0, as this allows a direct comparison of the exponent resulting from the boxing and from the standard power law fitting, providing validation of the technique. Event sizes were acquired for each box. To avoid correlations between adjacent boxes, the events for all the even and odd boxes were combined into two corresponding single distributions of events. Events occurring in even and odd numbered boxes were fitted separately. The box width is also an important parameter. Narrow boxes lead to a higher number of events when the data from all the boxes are combined, but also limit the possible event sizes within the boxes. Wide boxes mean fewer total events taken into account in the fitting. Therefore, fits were performed for box widths between 1 and 40 pixels (7.81–312 nm). To analyse such a high number of fits rapidly and consistently, and to take into account the case of close KS values mentioned earlier, for each box size, the possible xmin values were split into 10 intervals as shown in Fig. 5.14.

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Fig. 5.13 Example of a boxing scheme. The switching events taken into account are colour-coded according to the tip bias at which they occurred

10 fits were performed for each box width, in which xmin was chosen as the value minimising the KS distance within each of the 10 intervals. Figure 5.15 shows the resulting exponents, number of points and CDF of the data and corresponding fit for a box width of 10 pixels (78 nm). The thicker marker corresponds to the best overall fit. This procedure was repeated for all box sizes and for even and odd boxes. All the fit parameters (the optimal xmin within the interval, σ; the estimated error on τ , N) were recorded. Fits were eliminated by filtering out the ones with σ > 0.5, small box widths (typically 7–15 pixels or 55–120 nm) where the effective binning of the event sizes does not allow good fit, and high xmin values (typically around 150 nm2 . The final exponent attributed to the measurement corresponds to the average τ of the fits that passed the filtering process. Figure 5.16 shows the exponent values and error estimates for the fits before and after filtering. The filtered fits have an average exponent of τ = 1.90 ± 0.04, compatible with the exponents obtained without boxing on this series (τ = 1.98 ± 0.09), thus validating the boxing procedure in the instances where it is the only statistically viable approach.

5.6 Characteristic Size Exponents We find that for both samples and voltage polarities, the event size statistics follow the expected power law scaling over up to two decades of event sizes, as shown in Fig. 5.17a. For all measurement series, the lower cutoff to the event size was found to vary between 122 and 305 nm2 (2–5 pixels). The obtained exponent values vary

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Fig. 5.14 Kolmogorov-Smirnov (KS) distance, standard error and fitting exponent as a function of the size cutoff xmin . The dashed lines indicate the local fitting intervals

between τ = 1.98 and 2.87. For PZT-Nuc Vti p > 0, the size exponent of 1.98 is compatible with field-integrated mean-field models predicting τ = 2. This model also describes characteristic energy exponents measured in switching current measurements in Pb(Zr0.2 Ti0.8 )O3 in a parallel-plate capacitor geometry and acoustic measurements in BaTiO3 [18–20]. In PZT-Nuc Vti p < 0, included for completeness over a bias window equivalent to that of PZT-Nuc Vti p > 0, we observe a higher value of τ = 2.85. This reflects a higher prevalence of smaller-sized switching events, which can be a consequence of the restricted space for large events due to the high density of nucleation sites. An overall higher pinning caused by the high electric fields applied in the writing process can also be a contributing factor. Similarly in PZT-Mot, the values obtained for Vti p > 0 and Vti p < 0 give exponents of τ = 2.87 and 2.57, respectively.

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Fig. 5.15 Fitting exponents, number of points taken into account in the fits, and probability distribution function (PDF) for each of the fits. The data is shown with circular markers and the fits with lines. The colours correspond to the fitting intervals in Fig. 5.14. The average value of τ and standard deviations are shown by the black horizontal lines

Fig. 5.16 a Exponents for all boxing fits with no selection applied. The histogram of fitted exponents is shown on the right. b Exponents after filtering out the fits with σ > 0.5, box widths smaller than 55 nm (7 pixels) and xmin larger than 150 nm2 , with the corresponding histogram of exponent values on the right

Fig. 5.17 a Event size distributions and power law fit with no bias cutoffs, for PZT-Nuc Vti p > 0, Vti p < 0 and PZT-Mot, Vti p > 0. b Fitted exponents for box widths of 60–310 nm, clustering around τ = 2.57. The histogram of fitted exponents is displayed on the right

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Fig. 5.18 a Event size distributions and power law fit for PZT-Nuc Vti p > 0, Vti p < 0 and PZT-Mot, Vti p > 0 in the creep regime. The power law fitting on PZT-Nuc, Vti p < 0 was limited to events triggered by tip bias within the same bias window as for PZT-Nuc, Vti p > 0. The measurement series are shifted for clarity. b Fitted exponents for box widths of 60–310 nm, clustering around τ = 2.24. The histogram of fitted exponents is displayed on the right

Finally, since all previous studies of avalanche dynamics in ferroelectric and ferroelastic systems did not distinguish events occurring in different dynamics regimes [18–20], we also extracted for comparison the values of the scaling exponents in our switching studies when all nucleation, motion and merging events were included in the dataset and no cutoffs in the tip bias were applied. The resulting event size distributions and corresponding power law fits are shown in Fig. 5.18. We find that the exponents are lower when no bias cutoffs are applied, reflecting the overall larger event size occurring at tip biases higher than the upper cutoff values. This comparison highlights the importance of the consideration of different dynamics regimes, and could potentially be used as a means to distinguish between them.

5.7 Constant Driving Force To compare avalanche statistics in the creep and depinning regimes, we next focused on switching in PZT-Nuc under a scanning tip biased at two different constant voltages of −3.5 and −5.0 V. Figure 5.19 shows the resulting maps of the local time at which the polarisation at individual pixels reversed in units of the switching scan number. In the −3.5 V case, we observe only very slow switching dynamics. In particular, the pre-existing written domain walls appear to be strongly pinned in their initial positions, with only small displacements occurring in the first few switching scans where the domain walls are exploring more favourable configurations of the potential landscape in their immediate proximity, as shown in Fig. 5.19a. We also observe very few nucleation events, occurring stochastically throughout the measurement series. The newly nucleated domains expand slowly under the applied tip bias as can be seen from close ups of some of the domains in Fig. 5.19b. The stochastic nucleation of new domains throughout the measurement series, their ultraslow

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Fig. 5.19 Maps of switching time in measurements on written domain structures on PZT-Nuc at a, b −3.5 V and c −5.0 V. b shows close-ups of nucleated domains illustrating their slow growth. Time is expressed in units of the switching scan number, where each scan takes ∼15 min

outward growth with time and the rapidly saturating growth of the initially written domain walls qualitatively suggest that the growth is occurring in the creep regime.2 At a higher tip bias of −5 V, the polarisation reversal is much more rapid and polarisation reversal events are much larger, although some down-oriented domains remain by the end of the measurement as seen by the white areas in the middle of Fig. 5.19c. To study differences in the growth of nucleated domains at both tip biases, the size of individual domains expressed as their equivalent disc radius are extracted and shown in Fig. 5.20. In the Vti p = −3.5 V case, the domain wall radius increases 2 It

is tempting to compare the values of the tip bias in the constant bias measurements shown here and in the measurements shown earlier where the tip bias is increased incrementally. However, these measurements were performed with separate tips which can lead to changes in the effective field applied to the sample. Moreover, the domain writing process can affect the tip and adsorbates at the surface of the sample can accumulate at the tip, further affecting the effective tip field significantly. Direct comparisons of switching dynamics at a given voltage between measurement series are therefore difficult, though within a given measurement series, the effective tip field usually does not appear to change significantly.

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Fig. 5.20 Equivalent disc radii of nucleated domains and corresponding event maps where the switching scans are all performed at the same DC tip bias. a At a tip bias of −3.5 V, the domains grow logarithmically with time, consistent with creep motion. b At a tip bias of −5.0 V, the radius of nucleated domains grows linearly with time, indicative of a depinned domain wall motion. The inset shows the corresponding map of event types, where events in red, blue and green correspond to nucleation, motion and merging of domains respectively

logarithmically with time, while in the Vti p = −5.0 V case, the radii of nucleated domains expand linearly with time. We believe the logarithmic domain growth observed at low tip bias can be seen as a marker of ultraslow dynamics. As the tip bias is applied, the domain wall initially explores favourable surrounding configurations, allowing larger switching events to occur. However, when the tip bias is low compared to the characteristic barrier height, the switching event size rapidly decreases, and this more rapid growth is followed by a slower expansion due to thermal activation. The effects of individual events are also clearly seen in the dynamics. At high tip bias, where we can presume the applied field strength exceeds the characteristic barrier height, the growing domains expand at a constant rate into the surrounding region, and the effects of individual switching events cannot be distinguished in the dynamics. We note here that past studies of ferromagnetic domain walls in the framework of disordered elastic systems [21, 22] under a constant uniform field showed a constant, field-dependent velocity, and thus a linear expansion of the domain with time, in all the dynamic regimes of creep, depinning, and flow. However, these measurements were carried out by magneto-optic Kerr microscopy with optical resolution over very large segments of domain wall, and over relatively long times. In studies of ferroelectric domain walls, in contrast, logarithmic domain growth with time was observed under a static tip [17, 23, 24], but in this case related to the highly spatially inhomogeneous electric field. It is possible that in the scanning biased tip configuration, a long enough series of measurements, and significantly larger scan areas could allow a more uniform average domain wall dynamics to be observed even in the creep regime, as in the ferromagnetic studies. Such measurements are challenging, however, as they would require a prohibitively large number of PFM scans, leading to significant wear of the AFM tip. From the distribution of event sizes at −5.0 V, we obtain an exponent of τ = 2.10 ± 0.05 over 2.5 decades of event sizes in the depinning regime. In the creep regime however, we find much higher exponent values of τ = 3.36 ± 0.16,

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Fig. 5.21 Probability distributions of polarisation reversal event sizes at −3.5 V (a) and −5.0 V (b) and their corresponding power law fits. In the former, switching events are predominantly very small and their distribution is confined to low values, making the fit unreliable

emphasising the role of very small switching events, although we note that the fitting is more difficult, and our confidence in the exponent value is far lower in this case. The size distributions and corresponding power law fits are shown in Fig. 5.21.

5.8 Spatial Correlations As a further comparison between the creep and depinning regimes, the spatial as well as temporal structuring of switching events was studied as a function of increasing tip bias in PZT-Nuc. Strong spatial correlations below the depinning field have been predicted to occur, and almost no correlations beyond it [1]. The spatial distribution of events was analysed for the measurement on PZT-Nuc with Vti p > 0. Figure 5.22 shows the results on three domain walls. On the left the switching events corresponding to the inward domain wall motion (before nucleation and rapid merging). The estimated lower bound to the depinning force is 3.76 V. The panel on the right shows on the vertical axis, a horizontal line by line breakdown of the switching events. The events in the Nth line of the left panel correspond to the events in the Nth line of the right panel. The horizontal axis shows the bias at which the switching event occurred. The colour code corresponds to the horizontal extent of the switching event. The plots suggest spatial clustering of switching events deep in the creep regime, with a gradual broadening of the event clusters as the tip bias is increased. For tip bias greater than 3.0 V, switching events are observed throughout the interface length. However, significantly more statistics would be necessary to confirm this observation, which we hope to address in further investigations.

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Fig. 5.22 Left: Map of switching events along a moving domain wall in PZT-Nuc when the positive DC tip bias during switching scans is incrementally increased at each scan. Right: Map of the position of switched pixels along the vertical axis in the left panel as a function of the applied tip bias. The colour code represents the depth (along the horizontal axis) of the switching event at that scan line. The map shows spatial correlation of events at low bias which breaks down as the switching bias is increased

5.9 Conclusions We observe markedly different switching dynamics in Pb(Zr0.2 Ti0.8 )O3 films with different defect landscapes established during sample growth, and discriminate between creep and depinning regimes as a function of the applied tip bias. While the switching event statistics all show power law scaling summarised in Table 5.1, we find significant variations in the value of the scaling exponent τ . The exponents range from 1.98 ± 0.05 to 2.87 ± 0.12 in the creep regime when considering only switching events occurring at driving fields below the depinning threshold. Lower values ranging from 1.81 ± 0.05 to 2.56 ± 0.1 are found when the distribution of events is not restricted to the creep regime, and switching events occurring during the entire driving field window are included. The dynamics of individual growing domains was also investigated at a constant tip bias of −3.5 and −5.0 V. Analysis of the resulting domain wall nucleation and motion suggest that in these measurements, a tip bias of −3.5 V induces creep motion of the written domain walls and newly nucleated domains while at −5.0 V, the applied Table 5.1 Table of size exponents τ measured in two samples with different defect landscapes Measurement No tip bias cutoffs With tip bias cutoffs PZT-Nuc Vti p > 0 PZT-Nuc Vti p < 0 PZT-Mot Vti p > 0 PZT-Mot Vti p < 0 PZT-Nuc Vti p =-3.5V PZT-Nuc −5.0V

1.81 ± 0.05 2.51 ± 0.06 2.56 ± 0.1 2.23 ± 0.17 Vti p =3.36 ± 0.165 2.1 ± 0.05

1.98 ± 0.09 2.85 ± 0.1 2.87 ± 0.12 2.57 ± 0.2 N.A N.A

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force exceeds the critical barrier and allows depinning and more rapid dynamics. The characteristic size exponents were also fitted with the tip bias kept constant, in both of these measurements. While at −3.5V, the fitted exponent of 3.36 ± 0.16 is less reliable due to the overall range of accessed switching event sizes, the elevated value nonetheless highlights the importance of very small events at low driving forces. At −5.0 V, the fitted exponent is found to be 2.1 ± 0.05. Overall, the exponents extracted in this study are significantly higher than the ones predicted by an elastic model, which suggests that deviations from this model could potentially occur or that more statistics and a broader range of scales need to be accessed in order to measure the size exponents more accurately. Questions remain as to whether the polarisation reversal through a scanning tip rather than a field applied homogeneously throughout the probed volume of material could lead to deviations from the expected power law exponents. Although studies of switching dynamics in BiFeO3 suggest that the overall dynamics is the same in both cases [25], this question deserves more careful study and theoretical support. These results also show the importance of distinguishing between creep and depinning regimes in experimental measurements of characteristic exponents linked to crackling. Furthermore, larger switching event sizes need to be made accessible in order to probe a larger range of sizes and increase the confidence that the distribution of event sizes follows a power law. Assuming that the distribution of events does follow a given universality class, reaching event sizes large enough to access the upper cutoff to the power law distribution might further allow particular models within that universality class to be distinguished. Direct comparison of switching event sizes and currents could be allowed by acquiring the switching currents during the switching scans, which could allow independent extraction of size and energy exponents as defined in [18] and direct spatially-resolved comparison with the resulting domain configurations. Further information could also be acquired by depositing arrays of micrometerscaled electrodes on the surface of the films and performing similar measurements as in this work but with the driving force applied through these electrodes instead of through the tip. This setup could offer significant advantages. First, the electric fields applied through the sample would in this case be much more homogeneous than through a tip and resemble the field applied in theoretical models much more closely, and the time during which the field is applied would be controlled much more accurately. Second, applying the electric field through electrodes should in principle remove questions related to the effective tip field, ever-present in scanning probe measurements. This would also allow measurements to be performed on multiple electrodes and the acquired data to be pooled into a single distribution, significantly increasing the available statistics. Interestingly, the switching currents could be acquired simultaneously and potentially allow simultaneous extraction of the size exponent defined as the measured sizes of individual domains or defined as the integrated current above noise threshold as is done in [18]. Although this setup would have the drawback of lower spatial resolution as the PFM scanning would have to be performed through the electrode, it could provide a very powerful tool for direct measurement of critical exponents in ferroelectric with spatial resolution.

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References 1. Ferrero EE et al (2017) Spatiotemporal patterns in ultraslow domain wall creep dynamics. Phys Rev Lett 118:1–6 2. Måløy KJ et al (2006) Local waiting time fluctuations along a randomly pinned crack front. Phys Rev Lett 96:1–4 3. Tallakstad KT et al (2011) Local dynamics of a randomly pinned crack front during creep and forced propagation: an experimental study. Phys Rev E 83:1–15 4. Laurson L, Santucci S, Zapperi S (2010) Avalanches and clusters in planar crack front propagation. Phys Rev E 81:1–6 5. Pandya S et al (2016) Strain-induced growth instability and nanoscale surface patterning in perovskite thin films. Sci Rep 6:26075 6. Blaser C, Paruch P (2015) Subcritical switching dynamics and humidity effects in nanoscale studies of domain growth in ferroelectric thin films. New J Phys 17 7. Ievlev AV et al (2014) Humidity effects on tip-induced polarization switching in lithium niobate. Appl Phys Lett 104 8. Domingo N et al (2019) Surface charged species and electrochemistry of ferroelectric thin films. Nanoscale 11:17920–17930 9. Kalinin SV et al (2011) The role of electrochemical phenomena in scanning probe microscopy of ferroelectric thin films. ACS Nano 5:5683–5691 10. Ievlev AV et al (2018) Chemical phenomena of atomic force microscopy scanning. Anal Chem 90:3475–3481 11. Jesse S, Baddorf AP, Kalinin SV (2006) Switching spectroscopy piezoresponse force microscopy of ferroelectric materials. Appl Phys Lett 88:1–4 12. Miller RC, Weinreich G (1960) Mechanism for the sidewise motion of 180◦ domain walls. Phys Rev 117:1460–1466 13. Chauve P, Giamarchi T, Le Doussal P (2000) Creep and depinning in disordered media. Phys Rev B 62:6241–6267 14. Paruch P et al (2006) Nanoscale studies of domain wall motion in epitaxial ferroelectric thin films. J Appl Phys 100 15. Clauset A, Shalizi C, Newman M (2009) Power law distributions in empirical data. Soc Ind Appl Math Rev 51:661–703 16. Alstott J, Bullmore E, Plenz D (2014) Powerlaw: a python package for analysis of heavy-tailed distributions. Public Libr Open Sci One 9:1–18 17. Tybell T et al (2002) Domain wall creep in epitaxial ferroelectric Pb(Zr0.2Ti0.8)O3 thin films. Phys Rev Lett 89 18. Tan CD et al (2019) Electrical studies of Barkhausen switching noise in ferroelectric PZT: critical exponents and temperature dependence. Phys Rev Mater 3:1–6 19. Salje EKH et al (2019) Ferroelectric switching and scale invariant avalanches in BaTiO3. Phys Rev Mater 3:1–8 20. Casals B et al (2020) Avalanches from charged domain wall motion in BaTiO3 during ferroelectric switching. APL Mater 8 21. Lemerle S et al (1998) Domain wall creep in an ising ultrathin magnetic film. Phys Rev Lett 80:849–852 22. Ferré J et al (2013) Universal magnetic domain wall dynamics in the presence of weak disorder. Comptes Rendus Physique 14:651–666 23. Kleemann W (2007) Universal domain wall dynamics in disordered ferroic materials. Annu Rev Mater Res 37:415–448 24. Guyonnet J (2013) Growing up at the nanoscale: studies of ferroelectric domain wall functionalities, roughening, and dynamic properties by atomic force microscopy. PhD thesis 25. Steffes JJ et al (2019) Thickness scaling of ferroelectricity in BiFeO3 by tomographic atomic force microscopy. Proc Natl Acad Sci U S A 116:2413–2418

Chapter 6

Correlations Between Domain Wall Currents and Distortions

6.1 Motivation As discussed in Sect. 2.4, in Pb(Zr0.2 Ti0.8 )O3 , one mechanism for the appearance of conduction at ferroelectric domain wall involves the accumulation of charged defects such as oxygen vacancies at the domain boundaries, providing a conducting pathway. In materials where the domain wall conduction is provided by such defects, the currents at the walls are therefore expected to scale with the concentration of defects. Such a relationship has indeed been observed in studies of thin films of La-doped BiFeO3 , where oxygen vacancies introduced by lowering the oxygen partial pressure during the post-growth cooling phase were shown to modulate the magnitude of the domain wall currents [1] by increasing the Fermi energy at the walls [2]. Doping has also been shown to increase the conductivity of domain walls in Ti-doped ErMnO3 [3]. In the particular case of conduction at 180◦ domain walls in Pb(Zr0.2 Ti0.8 )O3 thin films, the conduction was attributed to domain wall tilting, leading to locally charged domain walls screened by charged defects such as oxygen vacancies. Independently, considering domain walls as elastic interfaces in disordered media, fluctuations in the density of defects lead to variations in the disorder potential energy landscape. These fluctuations act as pinning sites and promote wandering, resulting in a characteristic self-affine roughness at equilibrium. Bearing these both in mind, one might therefore ask whether the functional properties of the domain walls (in this case their increased electrical conductivity) can be linked directly to their geometric properties in terms of their distortions from a straight, elastically optimal configuration. Measurements carried out on freshly written nano-domains in Pb(Zr0.2 Ti0.8 )O3 thin films showed strongly conductive domain walls with respect to the ferroelectric phase itself, with a metallic conductance character. However, in this case the nanodomains had both a high curvature as seen from the top surface as well as tilts along the polarisation axis, either of which could influence the conduction [4]. Subsequent theoretical studies suggested that tilts in face play an important role [5], but did not © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. Tückmantel, Scanning Probe Studies of Structural and Functional Properties of Ferroelectric Domains and Domain Walls, Springer Theses, https://doi.org/10.1007/978-3-030-72389-7_6

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address the question of curvature. Thus, disentangling the effects of local domain wall curvature and tilt and how they affect the domain wall conduction could prove to be an interesting study. In this chapter we therefore present preliminary work exploring whether a connection can be established between the local domain wall deformations and magnitude of the currents in Pb(Zr0.2 Ti0.8 )O3 thin films.

6.2 Domain Wall Conduction in Pb(Zr0.2 Ti0.8 )O3 The measurements analysed here show conduction at domain walls for the first time in Pb(Zr0.2 Ti0.8 )O3 and were published in [6]. The thin film studied in this work is the same as PZT-Nuc studied in Chap. 5. In [6] down-polarised, stripe shaped domain patterns were written in the monodomain up-polarised film. PFM scans in which the domain configuration was imaged at high resolution were then alternated with conductive AFM scans in which the currents were acquired. Furthermore, I– V measurements were carried out on designated grids across the domain walls, at different temperatures between 120 and 330 K. In these measurements, the voltage was ramped up to a maximum value Vmax before being ramped down to 0 V. The temperature and voltage-dependent grid measurements were used to establish the dominant conduction mechanisms. Differences between the current magnitudes in the forwards and backwards voltage ramps allowed to establish whether a hysteresis in the current could be observed. Such a hysteresis would appear as a consequence of microstructural changes in the domain wall leading to displacement currents, even when such events occurred below the resolution of PFM measurements. The domain wall currents showed a diode-like behaviour with preferential conduction when a negative tip bias is applied as opposed to a positive bias. This asymmetry is attributed to the asymmetric electrodes consisting of a sharp Co/Cr coated AFM tip on top and a SrRuO3 back electrode at the bottom. The conduction at negative tip voltage shows three regimes. Between –0.5 and –1.4 V, currents are observed only at the domain walls. Neither motion of the interface, nor hysteresis in the currents are observed, suggesting that changes in the domain wall structure smaller than the PFM resolution do not play a dominant role. Between –1.5 and –2.7 V, currents are still limited at the domain wall with no visible changes in the polarisation configurations but at this higher bias range, a hysteresis in the forwards and backwards voltage ramps is observed. At higher bias, the currents increase significantly and are visible throughout the domain structure indicative of irreversible polarisation switching. Temperature dependent I–V measurement show a thermally activated conduction with the onset bias for conduction decreasing as the temperature increases. The conduction mechanism inferred from all these measurements suggests the dominant process to be Poole–Frenkel hopping of carriers between trap states, assisted by tunnelling from the tip to the film surface. The density of trap states is not high enough for conduction to occur within the domains. However, accumulation of charged defects such as oxygen vacancies at the domain walls [7] would allow for

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Fig. 6.1 Domain wall currents measured in c-AFM at a –0.625 V, b –1.0 V and c –1.375 V. The current magnitude fluctuates along the domain walls. Areas with higher currents highlighted with red circles in a can be seen to remain more highly conductive throughout the measurement series. Panel c shows the labelling of the domain walls as well as the polarisation orientations. The film is monodomain polarised up and the written domains are polarised down

conduction to occur. Charged defects could be further attracted to the walls by local steps in the domain wall observed by transmission electron microscopy [8], which would necessitate screening. Subsequent density functional theory calculations of domain wall conduction in PbTiO3 support these conclusions [9]. This scenario is consistent with the variations in the magnitude of the domain wall currents observed in the c-AFM scans shown in Fig. 6.1. The domain wall currents appear to be locally varying in intensity, while current hotspots, some of which are highlighted with red circles, are visible at the same location throughout the measurement series. In the present work, the alternating PFM and c-AFM scans are used in order to extract the domain wall positions and assess their local geometric distortions in order to establish whether the magnitude of such distortions can be directly correlated with the magnitude of the observed c-AFM currents.

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6.3 Preprocessing Steps In order to assess such a correlation, the positions of the domain wall in the successive PFM and c-AFM scans need to be matched and distortions due to AFM scanner artefacts need to be removed. To do this, the distortion correction algorithm presented in Sect. 4.3.4 is applied to the PFM/c-AFM scan series, with the first PFM scan used as a reference. Following this correction, the domain wall positions in the PFM scans are extracted by binarising the phase signal in the PFM scan prior to the cAFM measurement, directly giving the domain wall positions as the contours of the binarised contrast image. This results in known domain wall positions in the PFM scans that match the position of the corresponding domain wall currents in the c-AFM scans.

6.3.1 Domain Wall Curvature To assess the magnitude of the domain wall distortions, two criteria are used based on the local domain wall curvature and local deviation from the average domain wall position. To calculate the local curvature, segments of domain wall of fixed length ldw are fitted with a circle and the fitted radius is assigned to the domain wall position at the middle of the segment. The segment is then shifted by one pixel and the process is repeated. ldw is therefore an important parameter. As illustrated in Fig. 6.2 for domain walls 1 and 2 showing different roughness, the segment length affects the minimal size of the deviations in domain wall position that result in a significant curvature. If the interface length is too short, small-scale variations in the domain wall position due to instrument noise are fitted as circles with high curvatures. If the segment is too long, deviations that can be significant are ignored. In this work, interface lengths of 150 nm are used for the fitting. The choice of segment length is for now based on visual inspection and on knowledge of the typical noise in the domain wall positions due to noise in the PFM or imperfect corrections of the relative distortions in consecutive AFM measurements. This procedure yields a distribution of fitted radii shown in Fig. 6.3a that shows two peaks at ∼100 nm and 109 nm. The latter peak is attributed mostly to fitting artefacts due to changes in curvature directions within the segment. The corresponding points in the interfaces are shown in Fig. 6.3b and are excluded from the analysis of correlations between domain wall distortions and currents.

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Fig. 6.2 Configurations of domain walls a 1 and b 2, where the colour code corresponds to the local domain wall radius. Low radii corresponding to high curvatures appear in yellow. The domain wall radii are shown for fitting interface lengths ldw of 100, 150 and 200 nm. As this length increases, short regions yield larger fit radii, providing a means of filtering fluctuations due to noise in the domain wall positions. The domain wall positions are shown over three pixels in width to improve visibility

Fig. 6.3 a Histogram of the domain wall radii fitted with a sliding interface length of 100 nm. The histogram shows two peaks, at ∼100 and ∼109 nm. The latter peak is attributed to issues in fitting regions where the curvature changes direction. b Map of assigned peak in the domain wall radii histogram. Black and orange colours are assigned to the low and high radii peaks respectively

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Fig. 6.4 Domain wall displacements from the average position, for all four walls studied. For wall 1, the average was calculated over a section excluding the bottom half of the image, where large overhangs are observed

6.3.2 Domain Wall Displacement Another metric used to assess the distortion is the distance from the mean domain wall position illustrated in Fig. 6.4, motivated by the description of elastic interfaces in terms of their displacement functions. As can be seen by comparing Figs. 6.2 and 6.4, the domain wall displacement typically gives smaller fluctuations of the interface distortion than the domain wall curvature.

6.3.3 Topographical Curvature Another effect that can influence the local domain wall currents is the local tipsample contact area. The tip radius is typically of ∼30–50 nm. In principle, the better the radius of curvature of the sample matches that of the tip, the higher the contact area, and thus intuitively the measured currents should be larger. As can be seen in Fig. 6.5, this would correspond in principle to higher currents at locally convex surface curvature, and lower currents at locally concave surface curvature for a material with otherwise uniform conductive properties. To this effect, the local surface curvature was also extracted. The procedure is similar to that described in Sect. 6.3.1, extended to two spatial dimensions. A sphere is fitted on a square window of the sample topography of width wz and the extracted radius is assigned to the pixel

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Fig. 6.5 Illustration of the varying contact surface due to the topographical radius of curvature. Convex and concave sections are assigned positive and negative radii respectively

at the centre of the surface. The window is shifted by one pixel and the process is repeated. To distinguish between convex and concave curvatures, which lead to different contact areas for the same radius, a positive or negative sign is given to the radius of curvature if the centre of the circle lies above or below the sample surface, respectively, as shown in Fig. 6.5. A fitting window width of wz = 40 nm, corresponding to the typical radius of an AFM tip, is used to extract the curvature of the topography in Fig. 6.6a. This process yields the surface radius of curvature map shown in Fig. 6.6b. As can be seen from the topographical radii map and its corresponding distribution histogram shown in Fig. 6.7, the fitted radii are mostly much larger than the tip radius, peaking at ∼800 nm.

6.4 Preliminary Results The approach followed here is to extract the domain wall distortions, topographical curvature and domain wall currents at each point along the domain walls. However, because the distortion correction algorithm does not always perfectly correct distortions, a window around the wall has to be used in order to make sure that the domain wall currents and the correct topographical curvature are captured. To this effect, a window 5 pixels (40 nm) wide, with 2 pixels on each side of the domain wall position, and 1 pixel (8 nm) high was used. The dimensions of the window were chosen based on the observation that in this measurement series, most of the uncorrected

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Fig. 6.6 a AFM topography of the area shown in Fig. 6.1 and b corresponding map of topographical radii of curvature Fig. 6.7 Distribution of fitted topographical radii of curvature, showing a peak at ∼±800 nm

distortions occur on the horizontal axis and the typical distortions remaining after application of the distortion correction algorithm are of the order of 2 pixels. The full width at half maximum of the current peak is typically of ∼6–8 pixels (45–50 nm). In this selected window, the sum of the c-AFM currents is calculated in order to integrate through the conducting channel and increase the signal. This sum of currents is assigned as the current at the domain wall position in the window centre. The topographical curvature assigned to that same domain wall position is the maximum

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Fig. 6.8 Wall 1 position with colour coding corresponding, from left to right, to the domain wall curvature, displacement from average, topographical curvature and domain wall current

curvature over the 5 pixel window, based on the hypothesis that the highest curvature will lead to the highest contact area and measured current. The resulting maps of the local domain wall curvature, displacement from the average position, maximum topographical curvature and current sum are shown for each wall in Figs. 6.8, 6.9, 6.10 and 6.11. The currents shown in these figures were acquired at a tip bias of –1.375 V. At this tip bias, neither domain wall motion, nor hysteresis between the forwards and backwards branches of I–V ramps were observed and we therefore conclude that the currents are conduction rather than displacement currents. The domain wall positions were extracted from the PFM scan preceding the c-AFM scan at Vti p = –1.375 V. A visual inspection of Figs. 6.8, 6.9, 6.10 and 6.11 suggests an absence of strong direct correlation between the extracted metrics of domain wall distortion and tipsample contact and domain wall current magnitude. Wall 1, which is the roughest of the walls studied does have regions where both the local domain wall and topography curvatures match corresponding regions of higher domain wall current. However, other similarly highly curved regions in the domain wall and topography do not exhibit higher currents. This can further be seen by plotting the current as a function of the domain wall and topography curvatures and domain wall displacement. Panels (a, c, e) of Fig. 6.12 show these plots for all domain wall points, while panels (b, d, e) show the distribution of the curvatures and domain wall displacement. As can be seen by comparing each plot with its corresponding histogram, the regions of higher currents in the plot also

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Fig. 6.9 Wall 2 maps of curvature, displacement from average, topographical curvature and domain wall current

Fig. 6.10 Wall 3 maps of curvature, displacement from average, topographical curvature and domain wall current

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Fig. 6.11 Wall 4 maps of curvature, displacement from average, topographical curvature and domain wall current

correspond to the regions of higher prevalence of the curvatures and displacement. This suggests that the higher values of the currents observed for lower domain wall radius might simply be an effect of the larger statistical sample in this range of radii. The same observation can be made regarding the domain wall displacements and topographical radius of curvature. These correlations, or lack thereof, need to be analysed in further detail using specialised statistical techniques for skewed distributions such as calculating rankbased correlation coefficients, which will be the focus of further work. Another possibility that needs to be considered is whether the distortions in the domain wall positions are due to topographical features affecting the writing process, rather than varying concentrations of defects. Figure 6.13 shows the topography of the written structure, while the corresponding domain wall positions are shown by black lines. The large overhanging features observed in wall 1 seem to propagate within topographical valleys, while the top section of wall 4 seems to skirt along the edges of islands in the film surface. A more detailed analysis of such correlations is therefore required as well. It is also possible that the curvature in the sample surface plane is a poor proxy for the distortions of the wall along the polar axis, which would be energetically much more costly and require screening for instance by defects [10]. Such distortions of the domain wall along the polar axis have been imaged both at the scale of a few unit cells [8] in Pb(Zr0.2 Ti0.8 )O3 and at the order of 10–100 nm by Cherenkov SHG microscopy in LiNbO3 [11], and shown to induce higher conductivity.

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Fig. 6.12 Domain wall currents plotted as a function of a, c, e domain wall radius, topographical curvature and domain wall displacement. b, d, f Corresponding histograms of domain wall and topographical radii, and domain wall displacement respectively

Fig. 6.13 Overlay of atomic force microscopy topography and written domain wall position extracted from PFM phase images

Refinements of the preliminary analysis shown here are the subject of ongoing work.

6.5 Conclusion In this chapter, preliminary work is presented, establishing whether a direct correlation can be observed between the local distortions of the domain walls as seen by PFM and the amplitude of the domain wall currents measured by c-AFM. Maps of the local domain wall distortions were acquired, using both the local curvature and displacements from the average domain wall position. To investigate the contribution of the tip-sample contact area, the local curvature of the sample surface was acquired

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as well. At this stage of the analysis, no strong correlations between the domain wall currents, surface curvature and local distortions have been observed. However, many avenues remain to be explored. Important parameters such as the length of the section of domain wall used to fit the local curvature could be justified on more fundamental grounds. The section length used here is based on knowledge of typical errors on the domain wall position due to noise in the PFM phase signal and imperfect correction of distortions in subsequent scans. However, criteria based on typically expected sizes of the domain wall distortions could improve the domain wall curvature mapping. Similarly, the estimation of the tip-sample contact surface was made by extracting the surface curvature as a proxy for the contact surface. Though useful, this parameter can be refined in order for it to scale with the topographical curvature in the same way as the actual contact area. Furthermore, due to the high skewness of the extracted domain wall, surface radii of curvature, and domain wall displacement, more refined statistical analysis needs to be performed in order to confirm the presence or absence of a correlation between the domain wall currents and geometrical parameters thought to control them. If indeed no correlation can be observed, it is possible that the in-plane domain wall meanderings are not linked to fluctuations of the charged defects important for domain wall conduction. These could rather be due to other disorder contributions, such as pinning by topographical features and random-bond type uncharged defects, as indeed previously reported in Pb(Zr1−x Tix )O3 thin films [10] and (Pb,La)(Zr,Ti)O3 ceramics [12]. Instead, local distortions of the domain wall along the polar axis, invisible in PFM images, could be the dominant component, potentially suggesting that the in-plane and out-of-plane roughness of the domain wall are essentially independent. Additional information could be gained by applying clustering techniques such as basic k-means clustering or Gaussian mixture models in which the variance of the clusters does not have to be symmetric in all input features. These techniques could provide information on potential characteristic classes of behaviours. Principal component analysis could be applied in order to identify combinations of features best explaining the observed currents. Complex interplays between the domain wall distortions, contact surface contributions and domain wall currents could potentially be uncovered by using simple regression techniques such as decision trees, which have the advantage of being both robust to data skewness and interpretable. The avenues mentioned here are the subject of ongoing work.

References 1. Seidel J et al (2010) Domain wall conductivity in La-doped BiFeO3 . Phys Rev Lett 105:2010– 2012 2. Farokhipoor S, Noheda B (2011) Conduction through 71◦ domain walls in BiFeO3 thin films. Phys Rev Lett 107:3–6

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3. Holstad TS et al (2018) Electronic bulk and domain wall properties in B-site doped hexagonal ErMnO3 . Phys Rev B 97:1–7 4. Maksymovych P et al (2012) Tunable metallic conductance in ferroelectric nanodomains. Nano Lett 12:209–213 5. Eliseev EA et al (2012) Domain wall conduction in multiaxial ferroelectrics. Phys Rev B 85:1–11 6. Guyonnet J et al (2011) Conduction at domain walls in insulating Pb(Zr0.2 Ti0.8 )O3 thin films. Adv Mater 23:5377–5382 7. He L, Vanderbilt D (2003) First-principles study of oxygen-vacancy pinning of domain walls in PbTiO3 . Phys Rev B 68:134103 8. Jia CL et al (2011) Direct observation of continuous electric dipole rotation in flux-closure domains in ferroelectric Pb(Zr, Ti)O3 . Science 331:1420–1423 9. Paillard C et al (2017) Vacancies and holes in bulk and at 180◦ domain walls in lead titanate. J Phys: Condens Matter 29:485707 10. Paruch P, Giamarchi T, Triscone JM (2005) Domain wall roughness in epitaxial ferroelectric PbZr0.2 Ti0.8 O3 thin films. Phys Rev Lett 94:3–6 11. Kämpfe T et al (2014) Optical three-dimensional profiling of charged domain walls in ferroelectrics by Cherenkov second-harmonic generation. Phys Rev B 89:1–4 12. Pertsev NA et al (2011) Quasi-one-dimensional domain walls in ferroelectric ceramics: evidence from domain dynamics and wall roughness measurements. J Appl Phys 110:052001

Chapter 7

Crossings of Ferroelastic Twin Domains

Ferroelectric materials have been shown to exhibit a wide variety of domain configurations such as flux-closure [1–3] and bubble domains [4], vortices [5] and ferroelectric skyrmions [6]. These types of structures can potentially exhibit functional properties such as enhanced conduction, as observed at vortex cores in BiFeO3 [7]. Usually, these exotic configurations arise from an interplay between strain applied by the substrate and electrostatic boundary conditions. In this regard the regions around ferroelastic twin domains such as those illustrated in Fig. 7.1a can help stabilise polarisation rotations [8] due to the high strains and strain gradients observed at the twin domain boundaries. A local strain map around a ferroelastic domain wall in PbTiO3 is shown in Fig. 7.1b along with the corresponding polarisation orientations. The highest strains are located close to the substrate, under the twin domain. Phase field simulations suggest that in these regions, the flexoelectric effect can favour various polarisation structures [9], as shown in Fig. 7.1c, such as tail-to-tail polarisation configurations. High strains and strain gradients in multiple directions can be expected at crossings of such twin domains, which can potentially lead to complex polarisation textures with possible rotations and unusual functional properties. In this chapter, PFM and SHG are used to study these ferroelastic twin domain junctions in Pb(Zr0.2 Ti0.8 )O3 thin films in order to establish their polarisation structure and response to external stimuli.

7.1 Sample Characteristics The sample investigated here is a PLD grown thin film of Pb(Zr0.2 Ti0.8 )O3 with a thickness of ∼90 nm, grown epitaxially on a (001) pseudocubic oriented DyScO3 substrate, with a ∼30 nm thick SrRuO3 back electrode. The growth was performed at the Department of Materials Science and Engineering at the University of California, Berkeley. Finite thickness oscillations visible in the ω − 2θ scan shown in Fig. 7.2b © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. Tückmantel, Scanning Probe Studies of Structural and Functional Properties of Ferroelectric Domains and Domain Walls, Springer Theses, https://doi.org/10.1007/978-3-030-72389-7_7

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Fig. 7.1 a Schematic side-view of ferroelastic twin domains shown in red. b Strain map and polarisation orientations measured by TEM in [8] showing polarisation rotations around the twin domain. c Phase field simulations of possible polarisation configurations around ferroelastic domains. a, b reproduced with permission from [8]. c reproduced with permission from [9]

Fig. 7.2 a Schematic representation of the sample studied here, consisting of a 90 nm thick epitaxially grown layer of Pb(Zr0.2 Ti0.8 )O3 on a ∼30 nm thick back electrode layer of SrRuO3 on a DyScO3 substrate. b ω/2θ scan of the film around the (002) pc substrate peak. Faint finite size oscillations are visible suggesting good crystalline quality. c, d Reciprocal space maps taken around the (–103) pseudocubic peak along both in-plane lattice directions, showing that the film is mostly strained to the substrate. e AFM topography of the sample showing ferroelastic twin domains more clearly visible in the PFM amplitude image shown in f

suggest a good crystalline quality. Reciprocal space maps shown in Fig. 7.2c, d were acquired around the (–103) pseudocubic substrate peak, along both in-plane crystalline directions. The alignment in Q x between the substrate (top peak) and the film (bottom peak) shows that the film is mostly strained to the substrate. The tapping mode AFM scan shown in Fig. 7.2e presents a topography with visible straight lines at right angles and a low surface RMS roughness of 0.25 nm. A PFM

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Fig. 7.3 Vertical PFM images of the sample showing the ferroelastic domains, before poling with a positive voltage of 9 V on the large square and –9 V in the small square, shown with red lines. b PFM amplitude image of the resulting written domain. Some of the short ferroelastic domains have been erased by the poling process, with minimal changes in the ferroelastic domains around the written structure. c PFM phase image showing a contrast with the background where a positive tip bias was applied, indicating that the film is polarised up outside of the ferroelastic domains

scan whose amplitude signal is displayed in Fig. 7.2f shows that the straight lines visible in the topography correspond to ferroelastic twin domains crossing the sample at right angles. These domains are visible as long narrow regions with low vertical PFM amplitude due to the in-plane polarisation in the twin domains. Interestingly, the lines in the topography extend further along their long axis than the corresponding low amplitude regions in the PFM scans. This can be seen by comparing the length in the topography and PFM signals of the line crossing from bottom left to the upper right and through the centre the scan frame in Fig. 7.2. Moreover, there appears to be a hierarchy in the twin domains, more visible in Fig. 7.3a with domains several microns long connected at a 90◦ angle by shorter twin domains. The twin domains are typically separated by distance of ∼350 nm and embedded in a monodomain up-oriented polarisation configuration, as confirmed by comparing the as-grown polarisation state to a defined domain structure written by the application of a positive tip bias in a large square and negative bias in a smaller square. Vertical PFM scans of the resulting domain structure, illustrated in Fig. 7.3c show a phase contrast with the background where a positive tip bias was applied. The application of the bias seems to erase some of the shorter twin domains as can be seen by comparing the vertical PFM amplitudes before and after writing, shown in Fig. 7.3a, b. Regions scanned multiple times with the tip pressing on the surface with a force of ∼250 nN and with no DC tip bias do not show visible changes in the twin domains.

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7.2 PFM Measurements of Twin Domain Crossings A set of four crossings of ferroelastic twin domains (or a-domains) were studied with vector PFM in order to simultaneously investigate the horizontal and vertical polarisation components at and around the crossings. In all lateral PFM scans shown in this chapter, the cantilever is oriented along the vertical direction in the images. In this orientation, only in-plane polarisation components that are horizontal in the image lead to torsion of the cantilever and lateral PFM phase and amplitude signals. Because the signal of most interest is the lateral component of the polarisation, which covers only a small fraction of the scan area, PFM scans were performed in single frequency rather than in DFRT mode. In the latter case, the lateral resonance peak would not be present for most of the scan potentially leading the feedback loop on the amplitude signals drifting away from the resonance peaks. The excitation bias amplitudes for the vertical and lateral piezoresponse were set to 0.5 and 1 V respectively, while the frequencies were ∼300 kHz and ∼1.05 MHz, close to their respective contact resonance frequency. The force applied by the tip was ∼100 nN.1 The crossings studied here are all made of the longer ferroelastic domains extending for several micrometers along the length of the sample, as shown in Fig. 7.4. More detailed vector PFM scans shown in Fig. 7.5 seem to capture peculiar polarisation structures. As can be seen in Fig. 7.5a, c, some of the crossings exhibit drop shaped features with low vertical amplitude, about ∼50 nm in diameter and located slightly off the domain crossings, and with no corresponding contrast in the vertical phase signal. The lateral amplitude signal in Fig. 7.5b shows lines of minimum amplitude surrounded by narrow regions where the lateral amplitude is higher than the background. At the crossings, the lines of minimum amplitude are distorted away from a straight configuration. In the lateral phase shown in Fig. 7.5d, a 180◦ contrast can be seen at the twin domains. Away from them, the LPFM phase decays back to the background value. To study these crossings in more detail, high resolution vector PFM scans were acquired on the four crossings marked by numbers in Fig. 7.5. The crossings were imaged with the cantilever long axis at angles of ∼0◦ , ∼45◦ and ∼90◦ with respect to the vertical twin domains in Fig. 7.5. This allows the lateral polarisation components to be extracted in different directions and a more complete picture of the in-plane polarisation vector directions to be established. The acquired images on crossing 1 are shown for ∼0◦ , ∼45◦ and ∼90◦ in the panels Figs. 7.6, 7.7 and 7.8, respectively. The columns show the lateral PFM amplitude force was extracted by using Hooke’s law F = kx where x is the vertical deformation of the cantilever and k is the cantilever spring constant. The vertical displacement is obtained by x = S I where S is the setpoint vertical deflection value and I is the inverse optical lever sensitivity (InvOLS). The InvOLS is effectively the conversion factor between vertical deflection and vertical deformation of the cantilever. It is extracted by bringing the tip into contact with the surface and pushing the tip while measuring the vertical extension z of the AFM column and the vertical deflection V D . The InvOLS is obtained by a linear regression on a V D versus z plot. k was determined by the thermal noise method described in [10].

1 The

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Fig. 7.4 PFM map encompassing the studied crossings, highlighted with a red circle. All the selected crossings are of the longer ferroelastic domain category, observed to be more stable under out-of-plane polarisation switching

and phase, and vertical amplitude. The vertical phase is not shown, as no significant contrast is visible. The important features in the PFM images are described and the interpretation of the polarisation configurations is discussed for each angle. The overall interpreted polarisation pattern based on the PFM is then illustrated and potential artefacts in the imaging process are discussed.

7.2.1 Crossing 1 7.2.1.1

Polarisation Pattern: 0◦ Images

At 0◦ , consistent with the scan covering all the investigated crossings of Fig. 7.5, the lateral amplitude shows a vertical line of minimum amplitude highlighted in red in all images of Fig. 7.6 with values higher than the background on both sides of the line. The lateral phase shows a 180◦ contrast at the position of the red line. This could suggest the presence of a domain wall separating in-plane polarisation vectors with non-zero components along the horizontal direction of the image, schematically indicated by the red arrows as a tail-to-tail domain wall. In principle, one could argue that a head-to-head domain wall is also possible. Indeed, when assessing in-plane polarisation configurations, the absolute orientations are usually not known and only the relative angles between the in-plane polarisation components of domains are known. In our images however, the region of lateral amplitude higher

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Fig. 7.5 Vector PFM image of the investigated crossings. Crossings 1 and 2 are discussed in detail. Crossings 3 and 4 show similar characteristics to crossings 1 and 2 respectively. a, c Vertical PFM amplitude image showing the ferroelastic domains with drop shaped features located slightly offcentre of some of the crossings. The corresponding phase image shows no significant contrast. b, d Lateral PFM image showing higher amplitudes at the ferroelastic domain positions with distortions and pinching at the crossings. The vertical lines of minimum amplitude in b correspond to the boundary where the phase shifts by 180◦ in the phase image

than the background seems to extend further to the left of the red line than to the right, with a gradual amplitude decrease. This suggests that the twin domain plunges into the film to the left of the line and the gradual decrease of the amplitude is due to the decreasing tip electric field leading to a lower in-plane PFM signal. A similar effect is seen in the vertical amplitude, where the amplitude increases back to the background value over a longer distance on the left than on the right. From this, it can be assumed due to electrostatic consid-

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Fig. 7.6 Lateral amplitude and phase and vertical amplitude image of crossing 1 taken at an angle of 0◦ between the cantilever long axis and the domain symbolised by the red stripe on the left of the figure. The red lines indicate the region of low lateral amplitude and high phase contrast, with the red arrows indicating the horizontal in-plane polarisation components inferred from the images

Fig. 7.7 Lateral amplitude and phase and vertical amplitude image of crossing 1 taken at an angle of 90◦ between the cantilever long axis and the domain symbolised by the red stripe on the left of the figure. The green lines indicate the region of low lateral amplitude and high phase contrast, while the arrows indicate the in-plane polarisation components, inferred in a similar way as in the previous caption

erations that an in-plane polarisation component of the a-domain pointing to the left is more favourable. If the polarisation of the twin domain pointed to the right, tail-totail and head-to-head polarisations would appear at the left and right interfaces of the twin domain respectively, leading to electrostatically unfavourable charged domain walls.

7.2.1.2

Polarisation Pattern: 90◦ Images

A similar analysis can be performed on images where the structure is rotated counterclockwise by ∼90◦ with respect to the tip and displayed in Fig. 7.7. This time, the ferroelastic twin domain appears to be plunging into the thickness of the film on the right side. The amplitude increase to the left of the green line is faint however, possibly due the fact that the a-domain highlighted by the green line is not perfectly vertical on the image, leading to a smaller horizontal polarisation

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Fig. 7.8 Lateral amplitude and phase and vertical amplitude image of crossing 1 taken at an angle of 45◦ between the cantilever long axis and the domain symbolised by the red stripe on the left of the figure. The green and red lines indicate the regions of low lateral amplitude and large changes in phase contrast. The arrows of corresponding colour indicate the polarisation orientations inferred from the previous images. The phase contrast is self-consistent; all areas with an in-plane polarisation component pointing to the left and right have bright and dark phase contrasts respectively

component. Tip wear can also contribute to lower signal as the images taken at ∼90◦ and ∼45◦ are more noisy and the amplitude values are lower.2

7.2.1.3

Polarisation Pattern: 45◦ Images

To check whether these measurements are consistent, scans were performed at an angle of ∼45◦ with respect to the configuration shown in Fig. 7.6. The in plane polarisation orientations inferred from Figs. 7.6 and 7.7 are illustrated with the red and green arrows, while the lines of corresponding colour highlight the line where the lateral PFM phase shifts by 180◦ and where the lateral amplitude is at a minimum in Fig. 7.8. The assigned polarisation components from images taken at 0◦ and 90◦ are consistent in that all regions where the horizontal component of the assigned polarisation directions in Fig. 7.8 points to the left exhibit a bright phase contrast, while those pointing to the right consistently show a dark phase contrast. The interfaces between the dark and bright phase contrasts show significant deviations from straight lines and the red and green lines seem to pinch off close to the centre of the crossing. The lateral amplitude at the centre of the pinch is higher, suggesting that the polarisation might rotate from the configurations shown in red arrows to the one shown in green.

2 Blunting

of the tip with scanning is a known effect, well illustrated in [11], which decreases the lateral resolution and blurs small features. This is probably a significant effect as large scans needed to be performed when the angle was changed in order to find the investigated crossings again. Because images were taken for each angle at all crossings before changing to the next angle, all images taken at ∼45◦ and ∼90◦ show lower resolution. The phase signal, which is the most sensitive, still shows a 180◦ contrast at the green line corresponding to the position of the a-domain.

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Close to the centre of the pinch, in the region highlighted by a black circle, the vertical amplitude is lower than anywhere else in the crossing, suggesting that the out-of-plane polarisation component is overall at its lowest.

7.2.1.4

Polarisation Pattern: Interpretation

Based solely on the images shown above, the PFM data suggests a polarisation pattern as summarised schematically in Fig. 7.9. Figure 7.9a shows a schematic cut of an adomain through the film thickness showing the tail-to-tail polarisation component. The polarisation component at the back of the a-domain is illustrated with a non-zero vertical component as the vertical amplitude in Fig. 7.6 goes back to the background value within a shorter distance than the corresponding decay of the lateral amplitude. This picture is consistent with phase field simulations of ferroelastic domains in Pb(Zr0.2 Ti0.8 )O3 where the transverse flexoelectric coefficient is shown to control polarisation rotations at the edges of the twin domains, particularly at the filmsubstrate interface [9], as shown in Fig. 7.1c. This interpretation is also consistent with local tilting and lattice deformations in order to accommodate the out-of-plane lattice mismatch between the in-plane and out-of-plane polarised regions [12, 13]. Figure 7.9b shows the polarisation pattern interpretation at and around the crossing, as seen from above the film. The centre area is indicated by a black circle to highlight the fact the structure at the heart of the crossing is complex and not well understood. The exact polarisation configuration close to the centre is very difficult to determine through the PFM alone, as the acquired signals are an integral of the piezoresponse throughout the thickness of the film and the three-dimensional structure of the crossing itself is unknown. Although the electric field decays through the surface, the PFM measurements probe a significant thickness of the film. The depth up to which PFM signals are above the noise threshold can be roughly estimated from the width of the slow decrease of the lateral amplitude in Fig. 7.6, which was used to infer the direction in which the twin domain buries into the film. The width of the region where the lateral amplitude slowly decreases and is significantly above noise level is of ∼50–70 nm, suggesting that the measurement integrates the piezoresponse through at least ∼50–70 nm of the film thickness (since the twin domains usually go into the film at an angle of approximately 45◦ ). This means that the signals observed at the crossings could be superpositions of different polarisation configurations occurring at different depths.

7.2.2 Crossing 2 A similar overall structure is seen in the second crossing as labelled in Fig. 7.5, except that a drop shaped feature located slightly off the centre of the crossing is visible in the vertical amplitude signal. The feature is pointed away from the dominant polarisation directions of both twin domains. At the same location in the lateral

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Fig. 7.9 Schematic interpretation of the overall polarisation structure. a Side view of the polarisation pattern around a twin domain. b Top view of the crossing. The centre area is marked with a black circle to indicate low certainty on the detailed polarisation structure

signal, a “u” shaped bending of the line of minimal amplitude is visible, especially at ∼0◦ . At this angle, the lateral amplitude signal is decreased to the right of the “u” shaped structure (and the drop shaped feature). At approximately the same position in the image taken at ∼90◦ , the lateral amplitude is enhanced. It is therefore possible that the ferroelastic domain schematically represented in green in the markers at the left of Fig. 7.10 either interrupts the other one or lies above it. The drop shaped feature in the vertical amplitude and corresponding bends in the lateral amplitude are located approximately where the largest strain can be expected, based on the strain map of Fig. 7.1b. A more detailed understanding of the three dimensional structure of the crossing is therefore required to understand the polarisation pattern better. In addition, we note that previous optical microscopy studies of switching dynamics in single crystal of relaxor Pb(Mn1/3 Nb2/3 )O3 -PbTiO3 (PMN-PT) single crystals showed long and thin domains propagating from the intersections of a-domains, as shown in Fig. 7.11. The formation mechanism for those domains might be similar to the drop shaped features, which are also observed to protrude at 45◦ with respect to both a-domains. The high resolution of PFM can be a powerful tool to study such formations.

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Fig. 7.10 Detailed vector PFM images of twin domain crossing 2 at a ∼0◦ , b ∼90◦ and c ∼45◦ . The interpretations of the in-plane polarisation components and domain boundaries are indicated with red and green arrows and lines for the corresponding twin domains schematically represented in the boxes on the left of each panel

7.2.3 Other Crossings Crossings 2 and 4 show a similar structure to crossings 1 and 2 respectively, with a similar interpretation of the resulting polarisation structures. It is also worthy to note that, possibly as a consequence of the apparent hierarchy of twin domains, diverse types of structures seem to exist at the crossings. In some cases, a clear pinching of one of the ferroelastic domains can be seen, as in Fig. 7.12a, especially at crossings between long and short twin domains. The drop shaped features can be seen at other crossings, such as the one illustrated in Fig. 7.12b but are not systematically present. It would therefore be of interest to map more crossings in order to get a better representation of how different the polarisation patterns are at the different crossing types and whether the drop shaped features are found only at the junctions of the longer type of twin domains. Also, verifying that the drop shaped features systemat-

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Fig. 7.11 Formation of domains at 45◦ with respect to both a-domains in PMN-PT observed by optical microscopy. Reproduced with permission from [14]

Fig. 7.12 Multiple types of PFM signatures can be observed at the twin domain crossings. a The intersections of long and short twins tend to show a pinching of the shorter twin domain. b While the drop shaped features discussed in Sect. 7.2.2 are not always present at junctions of long twins, they can be observed throughout the sample

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ically point away from the polarisation vectors of both twin domains would confirm that this is a systematic feature of these structures. Before doing this however, it is vitally important to understand the origin of the contrasts observed in our images.

7.2.4 Imaging Artefacts Although PFM is a powerful tool to study ferroelectric domains at the nanoscale, it is not devoid of artefacts making the interpretation of images more challenging [15]. In essence, PFM detects the periodically modulated interaction between the tip and the sample caused by the applied AC tip bias. Several phenomena can therefore mimic a PFM response, such as surface charging and ionic migration [16, 17]. Various artefacts in PFM images can also arise as a result of local variations in sample properties.

7.2.4.1

Topograhical Features

Sample topography variations can have a strong effect on the contact resonance frequencies through local changes in the local tip-sample contact area [18–20] and should affect both the vertical and lateral PFM signals. This effect is more marked at the resonance frequency if no tracking of the resonance peak is performed, as illustrated in Fig. 7.13. In our case, PFM images were acquired close to but not at the resonance peak, where this effect should be less visible. Furthermore, vertical PFM images acquired at single frequency and in DFRT mode, in which the resonance is tracked, show very similar PFM signals. This suggests that variations in the contact resonance frequency should not be a major source of contrast in our images, at least assuming that the sensitivity to contact resonance changes is similar in lateral and vertical PFM signals. Another mechanism through which local topographical variations can affect the lateral PFM signal is by a breaking of the symmetry along the direction perpendicular to the cantilever long axis (horizontally on the images), where the cantilever is sensitive to in-plane polarisation components. In this case, even a fully out-of-plane polarised material on a highly enough tilted surface can give an apparent lateral PFM signal due to the lack of mutually counter-balancing responses on both sides of the cantilever [21]. The sample measured in this work presents topographical changes of the order of ∼0.4 nm away from the twin domains. Around the twins, the local sample height changes are comparable and of up to ∼0.6 nm. Therefore, if topographical changes were contributing significantly to the measured lateral PFM signals, sharp amplitude variations with maximal values comparable to the ones measured close to the twin domains would be seen away from the twin domains. As can be seen in Fig. 7.5b, d, no significant lateral PFM signals are observed away from the twin domains.

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Fig. 7.13 Effect of variations in tip-sample contact area on vertical PFM signals acquired at resonance. a On a smooth surface, line scans of the vertical PFM phase and amplitude signals (thick lines) closely match the expected signals shown in thin lines at the bottom of the panel. b When the tip is driven at the contact resonance frequency on a rough surface, strong variations from the ideal phase and amplitude signals can be observed as a result of variations in tip-sample contact area. Reproduced with permission from [19, 20]

7.2.4.2

Local Inhomogeneities

In lateral PFM, not only breaking of the topographical symmetry can lead to fluctuations in the amplitude and phase signals. In principle, any breaking of symmetry in the material properties affected by the measurement in the direction perpendicular to the cantilever long axis can cause a torsion of the cantilever [15, 22, 23]. Even at well-defined 180◦ domain walls between fully out-of-plane oriented domains, a lateral PFM amplitude increase can be seen at the domain wall as a result of the local breaking of symmetry close to the domain boundary leading to an effective in-plane response [24]. Local variations in dielectric constant, bow-waves due to surface adsorbates and tip apex asymmetry are other examples in which spurious lateral PFM contrast can in principle be observed [23]. Some effects discussed here, such as changes in the contact resonance frequency and changes in the mechanical properties of the sample can be identified by simultaneously acquiring the phase and amplitude response across the entire resonance peak, as is done in band-excitation methods [25]. However, local changes in material properties and their effect on the lateral PFM signals can be very challenging to disentangle from the ferroelectric contributions of in-plane domains. Therefore, they cannot be excluded at this stage. Ideally, the crossings of twin domains and the polarisation profile across individual twins should be studied with a separate but complementary method to PFM. Given the intrinsically nanoscale size of these structures, few methods can be used to effectively study them. One of these techniques is second harmonic generation (SHG) microscopy.

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7.3 SHG Measurements 7.3.1 Basic Description of SHG Microscopy Second harmonic generation is a non-linear optical process in which two photons of frequency ω interact with a non-linear material, emitting a photon of double the input frequency. The dominant process for this phenomenon occurs in materials with broken inversion symmetry, which makes SHG microscopy a good tool to probe ferroic order [26]. The setup used to acquire the SHG images, from measurements performed in the group of Dr. Cherifi-Hertel at IPCM-Strasbourg, is illustrated in Fig. 7.14. A laser beam with a wavelength centred around 800 nm is sent through a polariser and focused on the film surface. The outgoing SHG light passes through another polariser (referred to as the analyser) and to a detector. The beam is scanned across the surface of the film to produce SHG intensity maps. In this setup, only the polarisation components lying within the film plane are probed. The polarisation of the outgoing light can be described as P 2ω = χ (2) (E ω )2 , where χ (2) and E ω refer to the SHG susceptibility tensor and electric field produced by the incoming light. The susceptibility tensor is linked to the material symmetry and careful choice of the sample orientation and polariser and analyser angles allow χ (2) to be reconstructed. While the lateral resolution is in principle limited by the laser beam size, the SHG signal is sensitive to local field variations smaller than the lateral resolution [27]. Furthermore, SHG can allow imaging through the depth of the material [28, 29], thus providing a very good tool to probe three-dimensional polarisation patterns in ferroelectric materials.

7.3.2 SHG Microscopy Measurements of Twin Domains The SHG intensity was collected and mapped as a function of position for different polariser and analyser angle combinations. Polar plots shown in Fig. 7.15a, b, in which the SHG intensity is acquired for various analyser angles confirm that the polarisation within the twin domains points perpendicular to their long axis. Intensity maps show alternate dark and bright contrast lines that are in principle consistent with destructive interference at head-to-head or tail-to-tail domain walls. Intensity maps of the same area with various polariser and analyser configurations, displayed in Figs. 7.16 and 7.17 show enhanced SHG signals at the crossings. A possible drop shaped feature is highlighted in the inset of Fig. 7.16, as it lies slightly off the centre of the nearest visible twin domain crossing. This area shows enhanced SHG signal for all tested polariser and analyser combinations, in principle consistent with polarisation rotations in the sample plane. For some of the junctions, enhancements can be seen for all polariser and analyser combinations as well. Although these results are consistent with PFM measurements and might appear encouraging, it has to be pointed out that to interpret the SHG data with certainty,

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Fig. 7.14 Schematic setup of SHG measurements shown in this chapter. The laser light is sent through a polariser and is focused before interacting with the sample. The outgoing SHG light passes through another polariser before hitting the detector. Both polariser and analyser angles can be changed in order to probe different elements of the SHG susceptibility tensor. The beam can be scanned on the surface to produce an intensity map. Reproduced with permission from [29]

Fig. 7.15 a, b Polar plots indicating that the overall orientation of the polarisation in the ferroelastic domains lie perpendicular to their long axis. c SHG maps showing alternated high and dark contrast lines, in principle consistent with destructive interference at tail-to-tail domain walls

the twin domain structure needs to be modelled and the SHG signature of this structure needs to be simulated and compared with the acquired data. This is especially important with structures like the ones investigated here where various interfaces exist between ferroelastic and c-oriented domains which can produce their own SHG signal as they also break inversion symmetry. In this regard, a collaboration has been

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Fig. 7.16 SHG intensity map with corresponding polariser and analyser orientations shown on the left. The twin domain crossings show enhanced signals. The area highlighted with the dashed square could correspond to a drop shaped feature and shows enhanced SHG response as well. High SHG response at this feature is also observed with polariser and analyser perpendicular to each other and at 45◦ with the twin domains

started with the Department of Dielectrics of the Institute of Physics at the Czech Academy of Sciences in order to model the lattice structure at and around the crossing.

7.4 Sensitivity to Mechanical Force 7.4.1 Effect of Increasing Mechanical Force Given the complex structure of the crossings, incorporating high strains and strain gradients and the potential polarisation rotations observed by PFM, the response of the twin structures to external perturbation such as mechanical force and electric fields could provide further information. In particular, if the drop shaped features are a consequence of the large strain gradients observed around the twin domains, their size can be expected to be affected by mechanical force. To investigate their stability to such mechanical force, vector PFM measurements were performed with increasing force applied by the tip. The results are shown in Fig. 7.18. Up to 685 nN, the drop shaped feature increases in size. It keeps a domainlike structure in lateral PFM signals with 180◦ phase contrast and ring of lower amplitude. In the vertical amplitude, the drop shaped region remains an area of minimum amplitude whose surface increases in size as the force is increased. The twin domain that is vertical in the image appears to plunge into the film to the right, as indicated by the higher lateral amplitude to the right of the minimum amplitude line. For forces of about 1010 nN, a line of domain-like features appears in the

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Fig. 7.17 SHG intensity maps of the same area under different polariser and analyser angle combinations. At some twin domain crossings, the SHG signal is enhanced for all angle combinations

lateral PFM, that seems to lie above the twin domain. Around it, small domains in the lateral PFM image, above the twin domain, seem to start opening. In the vertical PFM signals, amplitude rings with corresponding 180◦ phase shifts can be observed suggesting a reversal of the vertical polarisation components in these areas. The drop shaped feature also seems to have a down-oriented vertical polarisation component. Overall, these images are consistent with a two-step process in which the uppolarised areas above the twin domain acquire an in-plane tilt due to the force applied by the tip. At the higher force range explored here, the out of plane polarisation components are reversed in the drop shaped feature first, then in other areas around it, suggesting that the centre of the crossing is indeed more responsive to external mechanical perturbation than any other structure in the sample. The vertical polarisation domains at the top of the scan frame form a line parallel to the horizontal twin. These domains are aligned with a twin domain that is visible as a faint decrease in the vertical amplitude at 1340 nN. Overall this suggests that the regions around the twin domains are the most susceptible to polarisation orientation changes due to mechanical pressure. In particular, the drop shape feature appears to be more sensi-

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Fig. 7.18 Vector PFM performed on a twin domain crossing with increasing mechanical force applied by the tip. The drop shaped feature increases in size in the lateral PFM, while starting at around 1010 nN, it seems to acquire a down-oriented vertical polarisation component. Around the twin domains present in the images, features consistent with in-plane domains appear at around 1010 nN, while at higher forces, out of plane down-oriented polarisation components are also visible close to the twin domains

tive than the areas directly above the twin domains plunging into the film, as its size increases already at lower force which could be an indication that this area is already highly strained.

7.4.2 Stability of Mechanically Induced Polarisation Changes Following the PFM scans with increasing tip mechanical force, the structure was scanned regularly at a constant low force of ∼200 nN in order to monitor the relaxation of the polarisation changes induced by the high mechanical forces applied previously. As can be seen in Fig. 7.19, the visible features in the lateral domain relax somewhat but the ones located along the twin domains, including the drop shaped feature seem to remain stable for at least up to four hours. The vertical components however seem to relax somewhat more, although features are still visible along the twin domains after four hours. The drop shaped feature seems to have relaxed back

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Fig. 7.19 Vector PFM images acquired with a constant low tip force of ∼200 nN for a period of four hours. Both in-plane and out of plane features preferentially relax when away from twin domains

to a region of low vertical amplitude, suggesting the initial configuration is indeed close to optimal for both elastic and electrostatic considerations.

7.5 Conclusions The crossings of twin ferroelastic domains in a Pb(Zr0.2 Ti0.8 )O3 thin film grown on a DyScO3 substrate were investigated using PFM and SHG microscopy. The sample shows a hierarchy of twin domains, with domains extending for several microns intersected by perpendicular, shorter, and more densely spaced twin domains. Various types of crossings seem to co-exist within the sample. Some junctions of long domains exhibit drop shaped features extending away from the directions in which both twin domains plunge into the film, while other crossings do not show such features. The junctions between long and short domains show a pinching of the shorter twin. Vertical and lateral PFM images suggest that a tail-to-tail polarisation component exists at some of the crossings of long twins, consistent with phase field predictions [9]. The junctions of long twins exhibit a complex structure suggesting

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possible polarisation rotations. The drop shaped features show a higher sensitivity to mechanical pressure than both the surrounding twin domains and as-grown regions. Potential artefacts cannot be excluded however, especially in the lateral PFM signals, which are the most prone to imaging artefacts due to local symmetry breaking. Band excitation PFM in which the response is acquired over a band of frequencies would allow the amplitude signals across the whole resonance to be reconstructed, which could allow artefacts due to changes in the contact resonance frequency to be identified. However, artefacts due to local symmetry breaking are much more challenging to assess. Furthermore, the interpretation of the resulting PFM images are complicated by the intrinsically three-dimensional nature of the structure, where variations through the thickness are expected as a consequence of the angle at which the twin domains propagate through the film. Encouragingly, SHG imaging of the twin structures does not exclude the interpretation of the polarisation orientations constructed from the PFM images. However, to obtain a full picture, a better understanding of the three-dimensional structure of the twin domain junctions is again required. In this regard, a collaboration has been opened with the Czech Academy of Sciences and work is ongoing. Complementary measurements of the same crossings in PFM and SHG would prove valuable in order to directly compare the same structures investigated with both techniques. Though difficult to perform, depth resolved SHG images could also prove extremely valuable for understanding the structure of the twin domain junctions. Combining the high lateral resolution of PFM and in-depth measurement capabilities of SHG could potentially allow unresolved questions as to the overall polarisation structures at the crossings to be answered. Meanwhile, a careful study of the response of the crossings with external bias might provide further insight complementary to the study of the effect of mechanical force.

References 1. Jia CL et al (2011) Direct observation of continuous electric dipole rotation in flux-closure domains in ferroelectric Pb(Zr, Ti)O3 . Science 331:1420–1423 2. McQuaid RG et al (2014) Exploring vertex interactions in ferroelectric flux-closure domains. Nano Lett 14:4230–4237 3. Tang YL et al (2015) Observation of a periodic array of flux-closure quadrants in strained ferroelectric PbTiO3 films. Science 348:547–551 4. Zhang Q et al, Nanoscale bubble domains and topological transitions in ultrathin ferroelectric films. Adv Mater 1702375:1702375 5. Yadav Y et al (2016) Observation of polar vortices in oxide superlattices. Nature 530:198–201 6. Das S et al (2019) Observation of room-temperature polar skyrmions. Nature 568:368–372 7. Balke N et al (2012) Enhanced electric conductivity at ferroelectric vortex cores in BiFeO3 . Nat Phys 8:81–88 8. Catalan G et al (2011) Flexoelectric rotation of polarization in ferroelectric thin films. Nat Mater 10:963–967 9. Cao Y, Chen L-Q, Kalinin SV (2017) Role of flexoelectric coupling in polarization rotations at the a-c domain walls in ferroelectric perovskites. Appl Phys Lett 110:202903

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10. Hutter JL, Bechhoefer J (1993) Calibration of atomic-force microscope tips. Rev Sci Instrum 64:1868–1873 11. Blaser C (2014) Probing nanoscale limits of polarization switching and controlling electronic properties in devices combining carbon nanotubes and ferroelectrics, PhD thesis 12. Pertsev NA, Zembilgotov AG (1995) Energetics and geometry of 90◦ domain structures in epitaxial ferroelectric and ferroelastic films. J Appl Phys 78:6170–6180 13. Utsugi S et al (2009) Domain structure of (100)/(001)-oriented epitaxial PbTiO3 thick films with various volume fraction of (001) orientation grown by metal organic chemical vapor deposition. Appl Phys Lett 94:2007–2010 14. Ushakov AD et al (2018) Direct observation of the domain kinetics during polarization reversal of tetragonal PMN-PT crystal. Appl Phys Lett 113:112902 15. Gruverman A, Alexe M, Meier D (2019) Piezoresponse force microscopy and nanoferroic phenomena. Nat Commun 10:1–9 16. Bark CW et al (2012) Switchable induced polarization in LaAlO3 /SrTiO3 heterostructures. Nano Lett 12:1765–1771 17. Borowiak AS et al (2014) Electromechanical response of amorphous LaAlO3 thin film probed by scanning probe microscopies. Appl Phys Lett 105:012906 18. Rabe U (2006) Atomic force acoustic microscopy. In: Bhushan B, Fuchs H (eds) Applied scanning probe methods 19. Rodriguez B et al (2010) Dynamic and spectroscopic modes and multivariate data analysis in piezoresponse force microscopy. In: Kalinin S, Gruverman A (eds) Scanning probe microscopy of functional materials. Springer, New York 20. Proksch R, Piezoresponse force microscopy with Asylum Research 21. Peter F et al (2005) Analysis of shape effects on the piezoresponse in ferroelectric nanograins with and without adsorbates. Appl Phys Lett 87:1–4 22. Ruediger A (2010) Symmetries in piezoresponse force microscopy. In: Kalinin S, Gruverman A (eds) Scanning probe microscopy of functional materials, vol 3. Springer, New York, pp 54–67 23. Peter F et al (2005) Contributions to in-plane piezoresponse on axially symmetrical samples. Rev Sci Instrum 76:1–3 24. Guyonnet J et al (2009) Shear effects in lateral piezoresponse force microscopy at 180◦ ferroelectric domain walls. Appl Phys Lett 95:132902 25. Jesse S et al (2014) Band excitation in scanning probe microscopy: recognition and functional imaging. Annu Rev Phys Chem 65:519–536 26. Denev SA et al (2011) Probing ferroelectrics using optical second harmonic generation. J Am Ceram Soc 94:2699–2727 27. Dong Z et al (2015) Second-harmonic generation from sub-5 nm gaps by directed self-assembly of nanoparticles onto template-stripped gold substrates. Nano Lett 15:5976–5981 28. Kämpfe T et al (2014) Optical three-dimensional profiling of charged domain walls in ferroelectrics by Cherenkov second-harmonic generation. Phys Rev B 89:1–4 29. Cherifi-Hertel S et al (2017) Non-Ising and chiral ferroelectric domain walls revealed by nonlinear optical microscopy. Nat Commun 8:1–9

Chapter 8

Conclusions and Perspectives

In this thesis, the structural and functional properties of the ferroelectric domain walls as well as the fundamental physics of polarisation switching dynamics in ferroelectrics were studied in thin films of Pb(Zr0.2 Ti0.8 )O3 with nanoscale scanning probe microscopy techniques. In the context of crackling phenomena, jerky switching events were triggered by a DC electric field applied by the tip during switching scans. These scans were alternated with PFM measurements mapping the domain configuration with nanoscale resolution. Detailed study of the connectivity of these jerky events with the surrounding domain configuration allowed the contributions due to domain nucleation, motion and merging to be separated. From subsequent increases in the overall switched area, the critical force separating the creep and depinning regimes was estimated. The characteristic event size exponent was measured first by including only events occurring in the switching regime, then by including all events within the probed voltage window, as has been done in previous studies. This work has shown that particular care needs to be taken to correctly identify the dynamic regime when studying crackling, as event size exponents were systematically higher in the creep regime than when all events are included. The overall values of the exponents were higher than theoretical predictions, potentially suggesting nanoscale deviations from elastic models. This is the first time that the distribution of jerky event size distributions were measured directly with nanoscale spatial resolution in ferroelectrics, allowing the characteristic size exponents to be measured, as well as spatial correlations within the creep regime to be studied. Hints of spatial clustering of events consistent with theoretical studies [1] were observed, although more statistics are required to confirm such correlations. Simple improvements to the measurement setup were suggested, where the polarisation switching bias is applied through patterned arrays of micrometre sized thin Pt electrodes. This strategy would in principle show numerous advantages as it would increase the size of the statistical sample and provide a much more spatially homo© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. Tückmantel, Scanning Probe Studies of Structural and Functional Properties of Ferroelectric Domains and Domain Walls, Springer Theses, https://doi.org/10.1007/978-3-030-72389-7_8

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geneous and well-controlled electric field, both in time and in space. Furthermore, these modifications to the measurement protocol would offer the possibility of simultaneously determining the characteristic size exponents by measuring the event sizes directly through the PFM imaging and indirectly, by studying the switching currents. The ability to distinguish between creep and depinning regimes by PFM, used in previous work [2] and in the present study, combined with high-resolution imaging of domains has shown that PFM studies can be a valuable tool in the investigation of crackling phenomena in ferroelectrics. Subsequently, the link between fundamental aspects of disordered elastic systems (in terms of their distortions) and functional properties of ferroelectric domain walls in Pb(Zr0.2 Ti0.8 )O3 (in terms of their enhanced conductivity) was studied. Measurements on written domain structures alternating PFM imaging of the ferroelectric domains and c-AFM scans of the domain wall currents were used to extract local distortions of the domain walls using two metrics: one based on the local curvature of the interfaces, and the other based on their displacements from their average position. The effect of the tip-sample contact area was included by extracting the local radius of curvature of the sample surface. At the present stage, this work has found no direct strong correlations between the functional properties of domain walls in terms of their local conductivity and local curvature and displacement from the average domain wall positions. Hints of overall higher currents with more highly curved sections of the domain walls might well be a consequence of the asymmetric distribution of extracted curvatures, showing a preponderance of highly curved domain wall sections. There are multiple future perspectives, however, as refinements on the extraction of the local domain wall curvature and tip-sample contact area can be made. Detailed analysis of the extracted correlations still needs to be performed and further insight could be gained by identifying typical classes of behaviour through clustering techniques. Complex interplays between the extracted geometrical metrics and domain wall currents can potentially be uncovered through simple robust and interpretable techniques such as decision trees. Furthermore, possible correlations between the shape of the ferroelectric domains as established during growth and local topographical curvature could imply that the local domain wall distortions are a consequence of pinning by topographical features, rather than by charged defects such as oxygen vacancies, which are thought to be an important component in the enhanced domain wall conductivity. In such a scenario, random bond type disorder, such as uncharged structural defects could provide pinning of the studied domain walls, leading to the characteristic roughening behaviour observed by PFM in the film plane, perpendicular to the polar axis. Along the polar axis, meanwhile, independent charged defect accumulation at local discontinuities (tail-to-tail and head-to-head steps) could promote the highly inhomogeneous conducting behaviour observed by c-AFM. These are all possibilities that will be explored in future work. Finally, the polarisation patterns at and around junctions of ferroelastic twin domain walls in Pb(Zr0.2 Ti0.8 )O3 were mapped at high resolution using vector PFM. The crossings were studied at three different angles in order to establish a more

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complete picture of the orientation of the local polarisation vectors. The PFM data suggests that an in-plane tail-to-tail component exists at the back of the ferroelastic domain walls, consistent with phase field simulations predicting such structures due to transverse flexoelectric fields [3]. The detailed structure of the crossing itself remains challenging to reconstruct with certainty due to the complex threedimensional structure of the crossing. The PFM data suggests potential in-plane rotation of the polarisation with tail-to-tail components at the centre. However, artefacts due to the restricted geometry of the studied structures, which are particularly challenging to identify in lateral PFM signals, cannot be excluded at this stage. The twin domain crossings were also studied using SHG microscopy, showing tentatively consistent behaviour with the PFM imaging. Collaborations initiated with theoreticians working on simulations of domain wall structures within the GLD approach should elucidate the equilibrium structure at twin domain crossings, which will be immensely helpful in the interpretation of both the PFM and SHG data. To summarise, fundamental nanoscale properties of domain walls and domain structures were investigated in Pb(Zr0.2 Ti0.8 )O3 by scanning probe microscopy techniques, showing that a wealth of information can be extracted using these measurements. Multiple avenues of research and open questions are left, encouraging collaborations with specialists of complementary techniques and theory colleagues. Various improvements and further studies were proposed to bring the results shown in this thesis further, which will of course be the subject of future work.

References 1. Ferrero EE et al (2017) Spatiotemporal patterns in ultraslow domain wall creep dynamics. Phys Rev Lett 118:1–6 2. Tybell T et al (2002) Domain wall creep in epitaxial ferroelectric Pb(Zr 0.2 Ti0.8 )O3 thin films. Phys Rev Lett 89:097601 3. Cao Y, Chen L-Q, Kalinin SV (2017) Role of flexoelectric coupling in polarization rotations at the a-c domain walls in ferroelectric perovskites. Appl Phys Lett 110:202903