• Aucun résultat trouvé

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 figure2.5 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 chapter4.

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 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 ferro-electrics with possible designs of device architectures [36] and nanoelectronic components such as domain wall diodes [37] and switches [38].

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 figure2.7. The domain walls are characterised by the angle between the polarisation vectors.

Conduction was observed at 180 and 109domain walls of BiFeO3 [35], with no conduction at 71walls. Figure2.7 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, density functional theory (DFT) calculations

Figure 2.6:Illustration of 109, 71and 180domain walls in rhombohedral BiFeO3. From [39].

Figure 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 109and 180than at 71walls and the bulk. From [35].

2.4 Domain wall conduction

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 eV and 0.2 eV for 109and 180 walls. Second, the 109domain 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 eV and 0.5 eV for 71and 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 O2partial 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.

Figure 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. From [45]

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 head-to-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 figure 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]. 2 Conduction at charged domain walls was also observed at domain walls with head-to-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.

Tilting of 180 domain walls was also shown to enhance conduction at domain walls in highly curved domains in Pb(Zr0.2Ti0.8)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].

Figure 2.9:Domain wall conduction in Pb(Zr0.2Ti0.8)O3 grown on a SrTiO3 sub-strate. (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 subse-quent 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. From [56].

Conduction at uncharged 180 domain walls was also observed by the

2.4 Domain wall conduction

Paruch group in the relatively simpler tetragonal Pb(Zr0.2Ti0.8)O3thin films grown on a SrTiO3 substrate [56]. Currents localised at the domain walls, as shown in figure2.9(a).

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 currents are inhomogeneous and observed only at the domain walls, as can be seen by comparing the current and domain configuration maps in figure2.9b-c.

At tip voltages between -1.5 V 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 figure2.10leading to locally charged walls further attracting charge carriers.

Figure 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. Adapted from [28].

Domain wall conduction was also shown at the University of Geneva to be reversibly switchable in thin films of Pb(Zr0.2Ti0.8)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 minute annealing

step at 300C in ultra-high vacuum. Upon re-exposure to ambient condi-tions, 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 figure 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 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 chapter4. 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 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 chapter 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

2.4 Domain wall conduction

Figure 2.11: Schematic of the proposed mechanism of switchable domain wall conduction in Pb(Zr0.2Ti0.8)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. From [57].

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.

CHAPTER 3

Crackling noise and avalanches

This chapter introduces fundamental concepts of crackling noise. First, a sim-ple 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 im-plemented 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 crack-ling, universality and renormalisation group methods used in the context of crackling [63–65]. Insightful and detailed information on elastic models can be found in [66–71] while readers interested in plasticity models can consult [72]. Readers that are still interested in crackling after reading this chapter are also encouraged to consult [73].

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 figure3.1(a). 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 [74,75]. The induced signal is irregular, having spikes of a broad range of sizes and intervals with no activity.

Figure 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 [76]. (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. Adapted from [75]

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 φthrough 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 [77, 78].

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)

3.1 Crackling noise

and are coupled to their nearest neighbours through an interaction strength J. Disorder is introduced by means of a random field hi drawn at each site from a Gaussian distribution with standard deviation R. The resulting Hamiltonian is then

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 the system cansnap,crackleorpop, 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 calledsnappingand 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 calledpopping, 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 [79]. Crackling noise is very common and is seen in compressed porous materials [80], earthquakes [81,82], collective decision-making [83], solar flares [84], cell-front motion [85], the stock market [86], mass-extinctions [87], fluids in porous media [88], martensitic phase transitions [89] and ferroelastic and ferroelectric switching [90–93], 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)

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 [79]. Crackling noise is very common and is seen in compressed porous materials [80], earthquakes [81,82], collective decision-making [83], solar flares [84], cell-front motion [85], the stock market [86], mass-extinctions [87], fluids in porous media [88], martensitic phase transitions [89] and ferroelastic and ferroelectric switching [90–93], 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)