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 .
This study was followed by work  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∼5nm. 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 [104,105].
The same group then studied the motion of 90◦ ferroelastic domains in ferroelectric BaTiO3by acoustic emission measurements . 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 crepita-tion 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 isint= 1.67  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 timetfollowing 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 . Both methods yielded energy exponents of 1.6.
Crackling was also observed through the displacement currents in three PZT ceramics  (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
3.5 Crackling noise experiments in ferroelectric materials
temperature-dependent higher exponent increasing from 1.8 to 2 in the temperature range of 373-423K, 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 . 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.
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 [107–109], 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 , of the effect of electrostatic and mechanical boundary conditions in the form of electrodes [111–113] and adsorbates [18, 19, 114] at the film interfaces in the former and strain in the latter . 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 [28, 116, 117], bubble domains , vortices , skyrmions ) 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 .
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.
Figure 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.
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 when the electric field direction is reversed, the electrons are spread out over the chamber
4.1 Thin film growth
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. 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(ZrxTi1−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.
Figure 4.2:Basic schematic showing the principle of PLD.
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 ferroelec-tric thin films of high crystalline quality.