7.4 Sensitivity to mechanical force
7.4.2 Stability of mechanically induced polarisation changes 112
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 figure7.19, 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 to a region of low vertical amplitude, suggesting the initial configuration is indeed close to optimal for both elastic and electrostatic considerations.
The crossings of twin ferroelastic domains in a Pb(Zr0.2Ti0.8)O3 thin film grown on a DyScO3 substrate were investigated using PFM and SHG mi-croscopy. 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 . The junctions of long twins exhibit a com-plex structure suggesting possible polarisation rotations. The drop shaped features show a higher sensitivity to mechanical pressure than both the
Figure 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 1010nN, while at higher forces, out of plane down-oriented polarisation components are also visible close to the twin domains.
Figure 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.
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 inves-tigated with both techniques. Though difficult to perform, depth resolved SHG images could also prove extremely valuable for understanding the struc-ture 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.
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(Zr0.2Ti0.8)O3with 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  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 mi-crometre sized thin Pt electrodes. This strategy would in principle show numerous advantages as it would increase the size of the statistical sam-ple and provide a much more spatially homogeneous 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  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 fer-roelectric domain walls in Pb(Zr0.2Ti0.8)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 struc-tural 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(Zr0.2Ti0.8)O3were mapped at high resolution using vector PFM. The crossings were studied at three different angles in order to establish a more 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 . The detailed structure of the crossing itself remains challenging to reconstruct with certainty due to the complex three-dimensional 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(Zr0.2Ti0.8)O3by scanning probe microscopy techniques, showing that a wealth of information can be ex-tracted using these measurements. Multiple avenues of research and open questions are left, encouraging collaborations with specialists of complemen-tary 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.
List of publications
A list of publications by the author during this thesis at the University of Geneva can be found below.
• I. Gaponenko, P. Tückmantel, J. Karthik, L.W. Martin, and P.
Paruch, Towards reversible control of domain wall conduction in Pb(Zr0.2Ti0.8)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. Mar-tin, T. Giamarchi, P. Paruch, Local probe comparison of ferroelec-tric switching event statistics in the creep and depinning regimes in Pb(Zr0.2Ti0.8)O3 thin films, submitted to Physical Review Letters
A PhD thesis is a work of endurance and many more people than the author are making it possible.
First of all, I would like to thank Patrycja Paruch for welcoming me in her group, first for a summer internship, then again for my master’s thesis and yet again for my PhD. 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 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, Man-isha Chhikara, Seongwoo Cho, Kumara Cordero Edwards, Iaroslav Gapo-nenko, 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.
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 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 chapter 6 possible.
I am grateful to Nirvana Caballero and Céline Lichtensteiger for their careful reading of chapters 3 and 2 respectively and their very 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 some-times 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 figureA.1, for allowing me the honour and privilege of giving him pets, providing endless laughs and for showing me hitherto unknown keyboard shortcuts.
Figure A.1:The actual author of this thesis, hard at work.
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.
 J. Guyonnet,“Growing up at the nanoscale: studies of ferroelectric domain wall functionalities, roughening, and dynamic properties by atomic force microscopy”, PhD thesis (2013).
 A. Bussmann-Holder, “The polarizability model for ferroelectricity in perovskite oxides”,Journal of Physics: Condensed Matter24, 273202 (2012).
 R. H. Mitchell,Perovskites Modern and Ancient (Almaz Press, On-tario, 2002).
 P. Fazekas,Series in Modern Condensed Matter Physics — Vol. 5 Lecture notes on electron correlation and magnetism(World Scientific, Singapore, 1999).
 S. Catalano et al., “Rare-earth nickelates RNiO3 : thin films and heterostructures”, Reports on Progress in Physics81, 046501 (2018).
 K. A. Rabe et al., “Modern physics of ferroelectrics: essential back-ground”, inPhysics of ferroelectrics - a modern perspective, edited by K. Rabe, C. H. Ahn, and J.-M. Triscone (Springer, Heidelberg, 2004), pp. 1–29.
 V. M. Goldschmidt, “Die Gesetze der Krystallochemie”, Naturwis-senschaften 14, 477–485 (1926).
 P. Goudochnikov and A. J. Bell, “Correlations between transition tem-perature, tolerance factor and cohesive energy in 2+:4+ perovskites”, Journal of Physics: Condensed Matter19, 176201 (2007).
 Ronald E. Cohen, “Origin of ferroelectricity in perovskite oxides”, Nature 358, 136–138 (1992).
 B. Jaffe, W. R. Cook, and H. Jaffe,Piezoelectric ceramics (Academic Press, London, 1971).
 B. Noheda et al., “A monoclinic ferroelectric phase in the Pb(Zr1-xTix)O3 solid solution”, Applied Physics Letters 74, 2059–2061 (1999).
 B. Noheda et al., “Tetragonal-to-monoclinic phase transition in a ferroelectric perovskite: The structure of PbZr0.52Ti0.48O3”,Physical Review B61, 8687–8695 (2000).
 G. H. Kwei et al., “Structures of the ferroelectric phases of barium titanate”,Journal of Physical Chemistry97, 2368–2377 (1993).
 F. Kubel and H. Schmid, “Structure of a ferroelectric and ferroelastic monodomain crystal of the perovskite BiFeO3”,Acta Crystallograph-ica Section B46, 698–702 (1990).
 A. H. G. Vlooswijk et al., “Smallest 90° domains in epitaxial ferro-electric films”,Applied Physics Letters91, 112901 (2007).
 Q. Y. Qiu, V. Nagarajan, and S. P. Alpay, “Film thickness versus mis-fit strain phase diagrams for epitaxial PbTiO3 ultrathin ferroelectric films”,Physical Review B78, 1–13 (2008).
 C. Lichtensteiger et al., “Ferroelectricity in ultrathin film capacitors”, inOxide ultrathin films: science and technology, edited by G. Pacchioni and S. Valeri, 2012 (Wiley, 2012) Chap. 12, pp. 265–230.
 R. V. Wang et al., “Reversible chemical switching of a ferroelectric film”,Physical Review Letters102, 2–5 (2009).
 K. Garrity et al., “Ferroelectric surface chemistry: First-principles study of the PbTiO3 surface”,Physical Review B - Condensed Matter and Materials Physics88, 1–11 (2013).
 K.-W. Park et al., “Humidity effect of domain wall roughening behav-ior in ferroelectric copolymer thin films”,Nanotechnology25, 355703 (2014).
 A. V. Ievlev et al., “Humidity effects on tip-induced polarization switching in lithium niobate”,Applied Physics Letters104, 092908 (2014).
 C. Blaser and P. Paruch, “Subcritical switching dynamics and humid-ity effects in nanoscale studies of domain growth in ferroelectric thin films”,New Journal of Physics17, 013002 (2015).
 S. K. Streiffer et al., “Observation of Nanoscale 180° Stripe Domains in Ferroelectric PbTiO3 Thin Films”,Physical Review Letters 89, 1–4 (2002).
 D. D. Fong et al., “Ferroelectricity in ultrathin perovskite films”, Science 304, 1650–1653 (2004).
 G. Catalan et al., “Fractal dimension and size scaling of domains in thin films of multiferroic BiFeO3”,Physical Review Letters100, 35–38 (2008).
 A. Schilling et al., “Domains in ferroelectric nanodots”,Nano Letters 9, 3359–3364 (2009).
 L. J. McGilly, A. Schilling, and J. M. Gregg, “Domain bundle bound-aries in single crystal BaTiO3 lamellae: Searching for naturally form-ing dipole flux-closure/quadrupole chains”,Nano Letters10, 4200–
 C. L. Jia et al., “Direct observation of continuous electric dipole rotation in flux-closure domains in ferroelectric Pb(Zr, Ti)O3”,Science 331, 1420–1423 (2011).
 Y. Yadav et al., “Observation of polar vortices in oxide superlattices”, Nature 530, 198–201 (2016).
 Q. Zhang et al., “Nanoscale Bubble Domains and Topological Transi-tions in Ultrathin Ferroelectric Films”,Advanced Materials1702375, 1702375.
 S. Das et al., “Observation of room-temperature polar skyrmions”,
 S. Das et al., “Observation of room-temperature polar skyrmions”,