Prof Hermann Kohlstedt

Prof Agnès Barthélémy

Dr Julie Grollier

Cécile Carrétéro

Éric Jacquet

Sophie Collin

Dr Stéphane Xavier

Dr Stéphanie Girod

Karin Garcia Hernandez

Dr Hiroyuki Yamada

Dr Flavio Y Bruno

Prof. Sylvain Saïghi

Dr Jean Tomas

Dr Gwendal

Dr Frédéric Nguyen

Dr Van Dau

Frédéric Petroff

Christine Marchand

Anne Dussart

I thank all of you for your contributions. Thank you, Julie, for introducing me to the field of neuromorphic computing and your motivating, never-ending ideas all around this topic. Your broad interest, ranging from fundamental physics and even neuroscience to industry-ready applications, encouraged myself to look for inspiration in very different directions. Finally, I learned a lot from your presentation skills, motivated by your determined striving to make your research accessible to very different publics. Thank you, Vincent, for the joint paper writing sessions. You definitively rekindled my admiration for precise and yet compelling formulations

Introduction and state of the art

From the beginning of the development of microcircuits on, computer architectures have been optimized to process information in a predetermined and deterministic way. Natural data that are present in a highly unorganized and noisy form therefore are a very tough task for these classical systems. The human brain, however, excels at finding a familiar face out of a big crowd or concentrating on the words of a speaker in the middle of a conversation. The wish to confer electronic systems with these and the various other accompanying capabilities therefore motivates many researchers to fathom the functioning of the brain in order to transpose its principles into artificial circuits. This endeavor constitutes the field of neuromorphic computing.

One important building block for the realization of these large-scale self-learning systems are tiny, nano-scale electronic devices called memristors. This work concentrates on a promising solid-state memristor type based on ferroelectric oxides and combines fundamental materials research [START_REF] Heber | Materials science: Enter the oxides[END_REF] with the aim of using these memristors for new, bio-inspired applications. This introductory chapter 1 presents the ideas and principles behind memristor-based neuromorphic circuits and then takes a look at the physics background of ferroelectric tunnel junctions -the base of this memristor [START_REF] Chanthbouala | Jonctions tunnel ferroélectriques ou magnétiques : nouveaux concepts de memristors[END_REF]. In the following chapter 2, we address its fundamental, static properties and try to expand the understanding of the electrical transport through the ferroelectric barrier. In order to realize larger neuromorphic systems, its components need to fulfill certain performance requirements. Chapter 3 shows that ferroelectric tunnel junctions can be fabricated into practical solid-state devices that exhibit fast switching in combination with good endurance and retention performances. Another critical issue for the integration of memristors into self-learning circuits is the understanding of their dynamic characteristics. In chapter 4, we present a study of the ferroelectric polarization reversal and demonstrate how this understanding can be used to realize autonomously learning components. After summarizing this work, we finally provide perspectives of fundamental physics-based questions around ferroelectric tunnel junctions and related systems as well as on the future of neuromorphic computing with memristors in chapter 5.

Memristor-based artificial neural networks

This section presents the context of this work. After taking a look at the current computing landscape, we show how inspirations from biological systems may lead to new, efficient architectures that can enable fruitful uses of the overwhelming amount of data we experience today. The work concentrates on artificial neural networks and, following a short introduction of their properties, we present the motivation of the idea to construct them with small analog devices, called memristors. This section then closes with an overview of the different memristor types and a state of the art of memristor-based artificial neural networks.

En route to neuromorphic chips

The current computer architecture is optimized to treat digital information following strict procedures. However, as more and more powerful devices find their way into the every-day life, these systems are confronted with natural data, such as images and sounds. Brain-inspired computing schemes may enable the efficient treatment of these inputs.

Limits of classical computer architectures

Nowadays, all commercial computers are based on the von Neumann architecture: information is stored in a memory unit and must be shifted to a central processing unit (cpu) in order to be used in logic operations [START_REF] Burks | Preliminary discussion of the logical design of an electronic computer instrument[END_REF]. The development of the latter over the years follows Moore's law, a trend that the semiconductor industry has been following since the 1970s [START_REF] Moore | Cramming More Components onto Integrated Circuits[END_REF]. In order to achieve this performance increase, the microprocessor designers put more and more transistors, the plodders of classical cpus, together. This was made possible by advances in the lithography processes that allowed to shrink the sizes of the complementary metal-oxide-semiconductor (cmos) structures. shows that this trend, for a long time, implicated an increase in the processor frequency. In the last ten years however, this evolution has stalled because of one major issue: the ever growing power consumption that -together with the shrinking sizes -poses severe problems for the dissipation of the accompanying heat. As a result, parts of the cpu have to be switched off temporarily (the so-called "dark silicon") in order to prevent a harmful overheating of the circuit [START_REF] Merritt | ARM CTO: power surge could create 'dark silicon[END_REF][START_REF] Venkatesh | Conservation Cores: Reducing the Energy of Mature Computations[END_REF][7][START_REF] Ball | Computer engineering: Feeling the heat[END_REF]. From 2005 on, the microprocessor designers try to evade this problem by integrating multiple, parallel cores on a same chip. However, this strategy merely is a partial solution and does not provide the same performance gain [START_REF] Moore | Multicore is bad news for supercomputers[END_REF]. Evolution of the number of transistors, the clock frequency, the power consumption, and the number of cores of a typical microprocessor over the years. The lines only serve as visual guides. [data from Ref. 10] Another fundamental problem, known under the term "von Neumann bottleneck", arises from the nature of this computer architecture: as the data have to be moved between the memory 1 and the processor and as a consequence of the limited speed of these operations, the system as a whole is slowed down [START_REF] Wulf | Hitting the Memory Wall: Implications of the Obvious[END_REF]. Finally, the structure miniaturization in cpus has led to very small dimensions nowadays: in the current generation of semiconductor devices, the node size has reached 14 nm. At these length scales, the influence of crystal and fabrication defects becomes higher and higher and emerges as one of the crucial problems for the deterministic computing principle.

Brain-inspired computing

While classical computers are unbeatable calculators for arithmetic operations with welldefined in-and outputs, their performance is far from that of the human brain 2 when it comes to the processing of noisy, "natural" data. Prime examples for these tasks are the classification of (parts of) images and voice recognition. With the strongly growing amount of data, collected by consumer devices and in all business areas as well as in medical applications, new paradigms to handle and make use of these are needed. In 2005 already, P. Dubey [START_REF] Dubey | Recognition, mining and synthesis moves computers to the era of tera[END_REF] from Intel predicted "recognition, mining and synthesis" as the great computational challenges of our time.

The interest to copy certain functionalities of the brain is not new. Even J. von Neumann looked for inspiration in the brain when he thought how large, complex systems could be made more tolerant to defects [START_REF] Neumann | Probabilistic logics and the synthesis of reliable organisms from unreliable components[END_REF]. Another important aspect is the energy consumption: consuming less than 25 W, the brain can fulfill tasks that are highly power-hungry to compute in the conventional way [START_REF]Principles of neural science[END_REF]. In the past, many efforts have therefore been made in order to reproduce these astonishing properties. The high complexity [START_REF] Mountcastle | Perceptual neuroscience: the cerebral cortex[END_REF] of the brain, however, often hid the essential ingredients of its working principles, leading to several waves of growing and caving interest [START_REF] Temam | The Rebirth of Neural Networks[END_REF]. The current fascination for neuromorphic computing strongly relies on the work by C. Mead & M. Ismail who proposed several realizations of brain-inspired chips with conventional hardware methods at the beginning of the 1990s [START_REF] Mead | Analog VLSI Implementation of Neural Systems[END_REF][START_REF] Mead | Neuromorphic electronic systems[END_REF].

Artificial neural networks

In a simple picture, the brain can be described as a network composed of neurons that are inter-connected by synapses: a neural network (nn). Here, after a short presentation of these two building blocks, we take a look at how artificial neural networks (anns) can be recreated in software simulations and implemented in hardware systems. Apart from the use for data processing, these systems may also provide insights into the functioning of the brain [START_REF] Gerstner | Theory and Simulation in Neuroscience[END_REF][START_REF] Sharp | Power-efficient simulation of detailed cortical microcircuits on SpiNNaker[END_REF]. shows the basic structure of a neural network. It is generally assumed to be organized in several layers that each contain a certain number of nodes. Information can flow between nodes of one layer to nodes of the next adjacent layer. 3 However, each connection is associated with a certain connection strength that describes how much of the information is transferred. As we will see in the following, this simple model is widely used to describe the functioning of the brain and is at the basis of many brain-inspired software as well as hardware projects. The nodes in figure 1.1.2 are the neurons which are connected by synapses. In the following, we take a closer look into these two elements.

Neurons

In the body, neurons are the base cells of the nervous system. They process and transmit information, e. g. from sensory inputs. The basic structure of a biological neuron, shown in figure 1.1.3 , can be divided in three parts. The first is the cell body (soma) that contains 3) In recurrent networks, information is also allowed to flow "backwards". Moreover, this organization in distinct layers clearly is a simplification.

the nucleus. From here, dendrites, the second part, ramify as thin structures and form a tree-like configuration. Finally, the axon forms an elongated structure that can measure more than 1 m. In a simple picture, signals from pre-neurons enter at the dendrites, are processed as local electrical excitations, and then travel along the axon before being transmitted to post-neurons. These electrical cell excitations are called action potentials or spikes and their typical amplitude of about 50 mV can be measured by voltage probes. In 1952, A. L. Hodgkin & A. F. Huxley [START_REF] Hodgkin | A quantitative description of membrane current and its application to conduction and excitation in nerve[END_REF] found that Na + and K + ions crossing the cell membrane at specific gates called channels are at the base of the electric excitations in neurons. They developed an electronics-based, mathematical description of their dynamics that is, however, often too complicated for practical uses. In 1907 already, L. Lapicque [START_REF] Lapicque | Recherches quantitatives sur l'excitation électrique des nerfs traitée comme une polarisation[END_REF] came up with a simpler description, the so-called leaky integrate-andfire (lif) model. Here, the neuron is assumed to accumulate incoming signals until a certain threshold is reached and then itself sends a spike to the following neurons. If, for a long time, no signals arrive, the information is assumed to decay, or "leak", over time. Owing to its simplicity, this model and its numerous variants are widely used in software and hardware implementations [START_REF] Gerstner | Spiking Neuron Models: Single Neurons, Populations, Plasticity[END_REF]. 4

Synapses

Synapses, located at the intersections of cells, are responsible for the signal transmission between different neurons. The information then flows from the axon of the pre-neuron to the dendritic system of the post-neuron. 5 In average, there are about 10 000 synaptic connections per neuron in the human brain and this high degree of connectivity results in a massively parallel architecture conjectured to be the core ingredient of the brain's capacities [START_REF]Principles of neural science[END_REF][START_REF] Azevedo | Equal numbers of neuronal and nonneuronal cells make the human brain an isometrically scaled-up primate brain[END_REF][START_REF] Hawkins | Why Neurons Have Thousands of Synapses, A Theory of Sequence Memory in Neocortex[END_REF].

In the body, two major types of synaptic information transmission exist: electrical and chemical. In an electrical synapse, the neurons are in direct contact so that electrical signals as well as ions can directly transit from one cell to the other. In a chemical synapse, shown in figure 1.1.4 , the neurons are separated by the so-called synaptic cleft. Here, the axon spike leads to the release of neurotransmitters that can bind to receptor proteins at the post-neuron surface. The transmission speed of electrical synapses is very high (of the order of 10 µs), whereas chemical synapses can induce delays of 0.1 to 10 ms [START_REF]Principles of neural science[END_REF]. Apart from transmitting transient information between neurons, synapses play a crucial role in the long-time dynamics of the brain [START_REF] Bliss | Long-lasting potentiation of synaptic transmission in the dentate area of the anaesthetized rabbit following stimulation of the perforant path[END_REF]. In the 1950s, D. O. Hebb [START_REF] Hebb | The organization of behavior: a neuropsychological theory[END_REF] postulated a neuropsychological theory according to which changes in the synaptic connection strengths are at the heart of the learning process. A stronger synaptic connection means a good signal transmission, and a weak connection damps the signal. All synaptic weights together form the system's memory. The fact that the synaptic connection strength changes is described by the term synaptic plasticity. However, the biological mechanisms at the origin of synaptic plasticity are still under discussion; among others, these may include a higher number of released neurotransmitters and a higher sensibility of the receptors [START_REF] Gaiarsa | Long-term plasticity at GABAergic and glycinergic synapses: mechanisms and functional significance[END_REF][START_REF] Lamprecht | Structural plasticity and memory[END_REF][START_REF] Citri | Synaptic Plasticity: Multiple Forms, Functions, and Mechanisms[END_REF]. As a function of the stability, two different types of plasticity can be distinguished: Long-term potentiation (ltp) describes long-lasting (of the order of minutes to days), stable changes of the synaptic weight, 6 and rapidly (of the order of tens of milliseconds to minutes) decaying changes are called short-term potentiation (stp). 7 In a very simple model, the synaptic weight is adapted as a function of the activity of the pre-neuron: the more often it fires a spike, the stronger the connection becomes. However, it is easy to see that this rule would eventually result in a general supersaturation. Another parameter that can influence plasticity is the firing rate [START_REF] Kirkwood | Common forms of synaptic plasticity in the hippocampus and neocortex in vitro[END_REF][START_REF] Sjöström | Rate, Timing, and Cooperativity Jointly Determine Cortical Synaptic Plasticity[END_REF]. Finally, experiments found evidence for another plasticity mechanism called spike-timing-dependent plasticity (stdp) [START_REF] Sjöström | Rate, Timing, and Cooperativity Jointly Determine Cortical Synaptic Plasticity[END_REF][START_REF] Markram | Regulation of Synaptic Efficacy by Coincidence of Postsynaptic APs and EPSPs[END_REF][START_REF] Bi | Synaptic Modifications in Cultured Hippocampal Neurons: Dependence on Spike Timing, Synaptic Strength, and Postsynaptic Cell Type[END_REF]. According to stdp, the synaptic connection is strengthened only if there exists a causality between the pre-and post-synaptic neuron activities. It therefore closely follows the Hebb's rule [START_REF] Hebb | The organization of behavior: a neuropsychological theory[END_REF] and has been identified as a crucial ingredient for self-learning anns. In section 4.2 from page 101 on, we will elaborate on stdp and its implications for learning processes.

Theory and implementations in software

One of the simplest forms of an nn is the perceptron. In 1958, F. Rosenblatt [START_REF] Rosenblatt | The perceptron: A probabilistic model for information storage and organization in the brain[END_REF], inspired by Hebb's work [START_REF] Hebb | The organization of behavior: a neuropsychological theory[END_REF] proposed a single-layer network that is able to learn to classify linearly separable inputs. Multi-layer neural networks (nns) were later found suitable for even more complicated classification tasks. In 1982, J. J. Hopfield [START_REF] Hopfield | Neural networks and physical systems with emergent collective computational abilities[END_REF] described an often-cited model of a recurrent neural networks (rnns) in which neurons of a same layer are interconnected and can therefore enter in self-sustained dynamics [START_REF] Little | The existence of persistent states in the brain[END_REF]. 8 While in first models, the signals were always assumed to be of continuous nature, biology also inspired nns with spike-based inter-neuron communication [START_REF] Hopfield | Pattern recognition computation using action potential timing for stimulus representation[END_REF].

From the first theoretical works on anns on, researchers thought about simulating them on classical computers to harness the brain's astonishing capabilities [START_REF] Rumelhart | Learning representations by back-propagating errors[END_REF][START_REF] Lippmann | An introduction to computing with neural nets[END_REF]. One of the standard tasks in which the software implementations proved to excel is pattern recognition in and classification of data such as in digital images [START_REF] Bishop | Neural networks for pattern recognition[END_REF].

Nowadays, so-called deep neural networks are widely used [START_REF] Schmidhuber | Deep learning in neural networks: An overview[END_REF][START_REF] Lecun | Deep learning[END_REF]. They reached a new milestone in speech recognition [START_REF] Collobert | A Unified Architecture for Natural Language Processing: Deep Neural Networks with Multitask Learning[END_REF] and set new records in image recognition and classification with very low error rates [START_REF] Ciresan | Multi-column deep neural networks for image classification[END_REF][START_REF] Krizhevsky | ImageNet Classification with Deep Convolutional Neural Networks[END_REF][START_REF] Taigman | DeepFace: Closing the Gap to Human-Level Performance in Face Verification[END_REF]. Deep learning, originally invented in the 1970s [START_REF] Ivakhnenko | Polynomial Theory of Complex Systems[END_REF], now is at the heart of all advanced speech recognition systems [START_REF] Hannun | Deep Speech: Scaling up end-to-end speech recognition[END_REF] and is soon Go-ing [START_REF] Silver | Mastering the game of Go with deep neural networks and tree search[END_REF] to respond to all kinds of questions people ask their mobile phones [START_REF] Newton | Bots are here, they're learning -and in 2016, they might eat the web, The Verge[END_REF]. Recently, researchers even used it to create images out of noise [START_REF] Mordvintsev | Inceptionism: Going Deeper into Neural Networks, Google Research Blog[END_REF].

The training of deep nns typically is a supervised process during which numerous selected input examples are presented together with their respective labels and the internal connection strengths are adapted accordingly. Not only does this phase require a big set of pre-classified inputs but it also demands immense computation resources when run on conventional architectures. So far, graphics cards have proven to be the most efficient hardware due to their inherent massive processing parallelization [START_REF] Krizhevsky | ImageNet Classification with Deep Convolutional Neural Networks[END_REF][START_REF] Lee | Facebook to open-source AI hardware design, Facebook Code[END_REF][START_REF] Hemsoth | Deep Learning Pioneer Pushing GPU Neural Network Limits, The Next Platform[END_REF]. However, even after this step, away from the serial computation on classical cpus, the training still consumes huge amounts of power: One sole graphics card needs more than 200 W and these are usually assembled in big clusters. Building a computer whose internal architecture approaches that of an ann promises a tremendous reduction in power consumption. A first step in this direction is the use of field-programmable gate arrays (fpgas) [START_REF] Cassidy | FPGA Based Silicon Spiking Neural Array[END_REF], but ultimately only fundamentally different, neuromorphic hardware may provide the brain's capabilities at manageable power costs. The necessity for numerous pre-classified inputs in the training process also poses a problem, but unsupervised learning may come to the rescue. A key element to its realization is the learning rule according to which the synaptic connection strengths evolve and a strong potential is attributed to stdp. However, the typical current implementations of deep nns rely on time-independent floating-point information flow and thus do not provide the possibility to use stdp. Closer to their biological examples, spike-based anns could realize unsupervised learning at low power consumption. 9

Implementations in classical hardware

Ideas of building bio-inspired computer circuits already existed in the beginning phase of the modern computer development [START_REF] Neumann | Probabilistic logics and the synthesis of reliable organisms from unreliable components[END_REF][START_REF] Mcculloch | A logical calculus of the ideas immanent in nervous activity[END_REF][START_REF] Widrow | Adaptive switching circuits[END_REF]. First demonstrations of a basic neuromorphic system on the macro-scale were achieved by B. Widrow & M. Hoff in 1960 [START_REF] Widrow | Adaptive switching circuits[END_REF] and a first realization of a small neuromorphic cmos circuit on the micro-scale 8) An older but comprehensive overview of the different nn topology types was, e. g., given by [START_REF] Jain | Artificial neural networks: a tutorial[END_REF]. 9) For further details, see section 4.2 from page 101.

was presented by M. Holler et al. in 1989 [61]. Shortly after, C. Mead presaged that using analog cmos hardware in very large scale systems would allow for the realization of potent anns [START_REF] Mead | Analog VLSI Implementation of Neural Systems[END_REF][START_REF] Mead | Neuromorphic electronic systems[END_REF].

Since then, the research around cognitive computing with large-scale integrated cmos circuits has seen a considerable rise in interest [START_REF] Modha | Cognitive Computing[END_REF]. Neuromorphic accelerators and classical cpus could soon be combined on a same die, working in parallel synergy so that tasks are given to the most efficient system [START_REF] Chen | BenchNN: On the Broad Potential Application Scope of Hardware Neural Network Accelerators[END_REF][START_REF] Kumar | Introducing Qualcomm Zeroth Processors: Brain-Inspired Computing, Qualcomm[END_REF]. Although several first large prototypes of cmos-based ann implementations have been presented in the last years [START_REF] Schemmel | A wafer-scale neuromorphic hardware system for large-scale neural modeling[END_REF][START_REF] Merolla | A million spiking-neuron integrated circuit with a scalable communication network and interface[END_REF][START_REF] Walter | Neuromorphic implementations of neurobiological learning algorithms for spiking neural networks[END_REF][START_REF] Indiveri | Memory and Information Processing in Neuromorphic Systems[END_REF] (also see figure 1.1.5 ), one crucial drawback of this pure-cmos ansatz persists: in order to achieve the rather complex neuron and synapse behavior needed, a combination of numerous circuit elements is necessary and results in large area requirements [START_REF] Indiveri | Neuromorphic silicon neuron circuits[END_REF]. Figure 1.1.6 shows a cmos synapse circuit that covers an area of 50 µm 2 as presented by J. Schemmel et al. in 2006 [71]. Since, optimization of the architecture 10 has yielded smaller structures [START_REF] Seo | A 45nm CMOS neuromorphic chip with a scalable architecture for learning in networks of spiking neurons[END_REF] that, however, remain relatively area-hungry. Recalling the fact that, in the brain as in all competitive anns, synapses outnumber neurons by several orders of magnitude, efforts to reduce the synapse size are the unambiguous priority. One possible solution to shrink their footprint was proposed by B. Widrow & M. Hoff in 1960 [START_REF] Widrow | Adaptive switching circuits[END_REF]: they thought of emulating synapses by relatively simple transistor-like devices called memistors. However, these three-terminal devices never met with much success. Then, in 2007, G. S. Snider published the idea to emulate synapses by new two-terminal nano-scale resistive-switching elements called memristors [START_REF] Snider | Self-organized computation with unreliable, memristive nanodevices[END_REF]. This proposal has since seen a huge interest and is at the origin of this work.

Memristors

Memristors are resistive-switching devices endowed with memory: "memory-resistors". First ideas to use them in the context of bio-inspired circuits already came up in 1976, when L. Chua & S. M. Kang linked its characteristics to the Hodgkin-Huxley neuron model [74]. However, it was only from 2007 on that the interest in neuromorphic hardware based on memristors aroused, when more concrete proposals were published by several researcher groups [START_REF] Snider | Self-organized computation with unreliable, memristive nanodevices[END_REF][START_REF] Snider | Spike-timing-dependent learning in memristive nanodevices[END_REF][START_REF] Pershin | Memristive model of amoeba learning[END_REF][START_REF] Linares-Barranco | Memristance can explain spike-time-dependent-plasticity in neural synapses[END_REF][78][START_REF] Jo | Nanoscale Memristor Device as Synapse in Neuromorphic Systems[END_REF][80].

Definition

The concept of the memristor was originally proposed by L. Chua as a result of symmetry considerations in theoretical electronics in 1971 [81]. As a resistor with memory, this new theoretical element completes the relations between the four fundamental electronic variables (figure 1.1.7 ). L. Chua also designed a new circuit symbol which has hence become the standard representation of the memristor. Five years later, L. Chua & S. M. Kang extended this work to "memristive systems", whose definition imposed somewhat looser theoretical restrictions [74]. In their general form, these are defined by the two following equations: ξ = f(ξ, x, t) and y = g(ξ, x, t) x , where x denotes the system input (either voltage v or current i), y denotes the output (either current i or voltage v, respectively) and the function g yields either the system's resistance R or its conductance G, respectively. The function f describes how the internal system state ξ evolves. In short, these equations define a non-linear resistor whose resistance depends on the history of input signals 11 and which may also explicitly depend on time t. 11) Which is why the system is said to have a memory.

From the first publication in 2008 on that used the term "memristor" to label such a device [82], a multitude of other physical systems have been denoted with the same name. Far from the original theory-based definition of a memristor, the term is now used for all devices of which the resistance can be stabilized at multiple different values. 12 In the scope of this work, although aware of the theory-motivated discussion around this matter [84], we will use the term "memristor" for all resistance-switching devices that display more than two stable resistance states.

Memristor types

Resistance switching properties have been observed in many different systems. While research in this field is strongly motivated by the quest for a new electric memory, from 2008 and the first "memristor"-titled publication [82] on, more and more work uses this new, appealing term. In this section, we give a short overview of various physical effects that can be at the base of resistance changes in micro-to nano-scale structures, without being exhaustive. For further information and more complete picture, the reader may turn to recent review articles [85][86][87].

This thesis concentrates on a memristor that is based on a thin ferroelectric tunnel barrier in which the resistance is modulated by the direction of the ferroelectric polarization. We will introduce this system separately in section 1.2 from page 15 on.

Memristors based on ion movements

Systems in which the resistance switching is based the movements of ions currently form the largest memristor class. In general, they consist of a thin layer of an insulating material between metallic electrodes. Also termed resistive random-access memory (rram), these devices have seen a tremendous interest in the recent years that is mainly based on the great projected potential for memory applications. Consequently, many extensive review articles cover the state of the art of these devices [88,89] and display the wide range of insulating materials in which these resistance-switching phenomena can be observed [90,[START_REF] Wei | ReRAM is Real[END_REF]. Although the detailed physics underlying the switching mechanism have not been elucidated yet, rram devices can be divided in two general classes, based on the type of the migrating ions.

In layer structures where an insulating transition-metal oxide (such as, e. g., HfO x , TaO x , TiO x , SrTiO 3 , ZnO x , or WO x ) is sandwiched between inert electrodes, the application of external voltages or currents can lead to non-volatile resistance changes [START_REF] Wong | Metal-Oxide RRAM[END_REF]. The physical origin of the switching is believed to be related to redox-reactions and the movement of oxygen-vacancy anions that can alter Schottky barriers at the electrode interfaces [START_REF] Sawa | Resistive switching in transition metal oxides[END_REF] (figure 1.1.8 -b) or lead to conductive filaments across the insulating layer (figure 1.1.8-a). Also mixed switching with both effects was observed [START_REF] Yang | Memristive switching mechanism for metal/oxide/metal nanodevices[END_REF][START_REF] Kubicek | Uncovering Two Competing Switching Mechanisms for Epitaxial and Ultra-Thin Strontium Titanate-Based Resistive Switching Bits[END_REF]. Furthermore, these systems can display either bipolar or unipolar resistance switching and are summarized under the terms valence-change memory (vcm) or thermochemical memory (tcm), respec- 12) In reaction to this, L. Chua meanwhile redefined the term to include all resistance-switching memory devices [83].

tively [88]. 13 In the latter case, the switching is temperature-mediated and caused by Joule heating. The other resistance-switching structure is based on one electrochemically active electrode (e. g. Ag or Cu) and one inert electrode in combination with an insulating barrier such as a solid electrolyte (e. g. GeTe or GeS 2 ) or a dielectric (e. g. SiO 2 or a-Si). Under high electric fields, cations from the active electrode can migrate through the insulator and form metallic filaments as shown in figure 1.1.8-a [START_REF] Terabe | Quantized conductance atomic switch[END_REF]. This type of memory is known under different names, such as electrochemical metallization (ecm) cell, programmable metallization cell (pmc), solid-state electrolyte memory, conductive-bridge random-access memory (cbram), and atomic switch [90].

While devices based on these structures have shown some convincing performance properties [START_REF] Govoreanu | 10x10 nm 2 Hf/HfO x crossbar resistive RAM with excellent performance, reliability and low-energy operation[END_REF][START_REF] Lee | A fast, high-endurance and scalable non-volatile memory device made from asymmetric Ta 2 O 5 -x/TaO 2 -x bilayer structures[END_REF] and are already considered for memory applications [START_REF] Wei | ReRAM is Real[END_REF], the physical mechanisms at the origin of the resistance switching are not fully understood yet [99]. For example, the ongoing research in this field has shown cation movement even in structures based on transition-metal oxides (HfO x , TiO x , and TaO x ) [100]. Furthermore, many systems only present resistive switching after an initial forming step for which high current densities are needed and that still need to be optimized [START_REF] Sawa | Resistive switching in transition metal oxides[END_REF]101,102].

Memristors based on phase-change materials

Phase-change materials (pcms) exhibit a strong contrast in their electrical and optical properties between their different structural phases. Under current pulses, the Joule effect can partly melt the memory cell and the following temporal temperature profile either leads to a rapidly melt-quenched amorphous or a more slowly crystallized phase [103,104]. Owing to the temperature-mediated effect, the switching is unipolar. These materials have been under investigation for a long time already [105] and have been proposed for neuromorphic and logic applications [106, 107] -apart from the now apparently realized "storage-class" memory [108, 109]

Spintronic memristors

Spintronic memristors are based on structures in which the magnetization of thin films can be influenced by spin-polarized currents. Magneto-resistive phenomena then provide a means for the electrical readout of the magnetization state [110,111]. After the theoretical concept [112], a first demonstration of a more-than-binary magnetic memory with a resistance-readout has been published in 2011 [113].

Other memristors

Another emerging memristor idea is based on metal-insulator phase transitions (mits) in Mott materials [114]. For example, VO 2 has shown to become highly conductive under the application of electric fields [115,116]. However, also the influence of oxygen ion migration and local Joule heating have been considered [117,118]. Despite many open questions, e. g. on the origin of the non-volatile resistance changes, these systems have already attracted a strong interest [119][120][121]. In Pr 1-x Ca x MnO 3 , the apparently purely electric process of charge injection and trapping has been identified as one possible mechanism [122].

Apart from the two-terminal devices present so far, there are also ideas to use threeterminal transistor-like structures [123] and memristive behavior was demonstrated in field-effect transistors (fets) combined with carbon nanotubes [START_REF] Zhao | Nanotube devices based crossbar architecture: toward neuromorphic computing[END_REF] or organic materials [START_REF] Alibart | An Organic Nanoparticle Transistor Behaving as a Biological Spiking Synapse[END_REF]. Finally, also classic flash-memory transistors were shown to be able to store several resistance states in a two-terminal configuration [START_REF] Ziegler | Memristive operation mode of floating gate transistors: A two-terminal MemFlash-cell[END_REF].

State of the art of memristor-based neural networks

After the proposition by K. Likharev to use "analog nano-devices" for neuromorphic circuits in 2003 [START_REF] Likharev | Neuromorphic CMOL circuits[END_REF], G. S. Snider four years later specifically highlighted the potential of memristors to imitate the function of synapses in anns [START_REF] Snider | Self-organized computation with unreliable, memristive nanodevices[END_REF]. He also proposed more concrete ideas on how to realize stdp-based learning circuits with memristors [START_REF] Snider | Spike-timing-dependent learning in memristive nanodevices[END_REF].

Several research groups picked up on the idea, and in 2010 proposed a simple circuit for associative memory [START_REF] Pavlov | Conditioned reflexes: An investigation of the physiological activity of the cerebral cortex[END_REF] with memristors [78]. Two years later, the demonstration of this basic principle set the experimental work in motion [129] -further fueled by promising simulations of learning in memristor-based spiking anns by D. Querlioz et al. [START_REF] Querlioz | Simulation of a memristor-based spiking neural network immune to device variations[END_REF].

In order to reach a high inter-neuron connectivity, there is agreement that the memristive synapses should be arranged in a crossbar architecture, such as the one shown in figure 1.1.9 ), without however demonstrating a successful operation yet [START_REF] Park | RRAM-based synapse for neuromorphic system with pattern recognition function[END_REF][START_REF] Park | Nanoscale RRAM-based synaptic electronics: toward a neuromorphic computing device[END_REF]. Nonetheless, there are first results on the training and operation of crossbarstructured memristor arrays in combination with external, laboratory-sized equipment: [START_REF] Alibart | Pattern classification by memristive crossbar circuits using ex situ and in situ training[END_REF]. In 2015, the same research group extended this operation to a basic pattern classification task -this time relying on online learning rules [START_REF] Prezioso | Training and operation of an integrated neuromorphic network based on metal-oxide memristors[END_REF]. Similar results were also reported by S. Eryilmaz et al. using pcm memristors [START_REF] Eryilmaz | Experimental demonstration of array-level learning with phase change synaptic devices[END_REF][START_REF] Eryilmaz | Brain-like associative learning using a nanoscale non-volatile phase change synaptic device array[END_REF]. ) notwithstanding, an autonomously functioning memristor-based ann has not been demonstrated yet. The stdp learning rule might once enable nonsupervised learning (see section 4.2 for details). Its implementation with memristors has been proposed by T. Serrano-Gotarredona et al. [START_REF] Serrano-Gotarredona | A Proposal for Hybrid Memristor-CMOS Spiking Neuromorphic Learning Systems[END_REF] and J. M. Cruz-Albrecht et al. already performed simulations of a suitable cmos circuit [START_REF] Cruz-Albrecht | A scalable neural chip with synaptic electronics using CMOS integrated memristors[END_REF]. One important goal of this work therefore is to make another step towards the realization of memristor-based artificial neural networks for autonomous, non-supervised learning.

The ferroelectric memristor

This section introduces the concept of ferroelectric tunnel junctions (ftjs), starting from the fundamental physics of ferroelectric materials. In the following, several models describing the dynamics of polarization reversal are presented. The last part of this section then deals with the electrical transport across ferroelectric ultra-thin films and how it is influenced by the polarization direction.

Ferroelectrics

Ferroelectrics are materials with a spontaneous electric polarization that can be switched by application of an external electric field. They belong to the broad class of dielectricsinsulating solids that exhibit an electric polarization under application of an electric field, breaking the symmetry of the material. Another source of symmetry breaking can stem from the crystal structure of the solid. In this case, the symmetry is broken permanently and defines which effects are permitted in the respective material. All dielectrics with a non-centrosymmetric structure (with one exception) display piezoelectricity: mechanical stress leads to the separation of opposite charges inside the solid and polarizes it. The other way around, through the inverse piezoelectric effect, an externally applied electric field induces a deformation of these materials.

A subset of piezoelectrics possesses a unique axis of symmetry which makes its members polar and leads to a spontaneous electric polarization. This polarization changes upon a variation in temperature: this is described as pyroelectricity. Finally, the spontaneous polarization of certain pyroelectrics can be switched under application of an external electric field and remains non-zero on removal of the field. These materials are called ferroelectrics. First experimental evidence of ferroelectricity was discovered in 1920 by Joseph Valasek during experiments on Rochelle salt [START_REF] Valasek | Piezo-Electric and Allied Phenomena in Rochelle Salt[END_REF]. As a summary, the hierarchy of the different dielectric classes is depicted in figure 1.2.1 The name "ferroelectricity" stems from the similarity between the hysteresis cycles of electric polarization as a function of electric field and those obtained on magnetic materials when measuring the magnetization of a ferromagnet as a function of magnetic field. Such a typical, macroscopic ferroelectric hysteresis cycle is shown in figure . Starting from the origin, the material is poled under application of an electric field E so that the polarization P increases. At high fields, its value goes into saturation which may be accompanied by a small remaining variation from relative permittivity. The extrapolation of this high-field regime to zero field yields the saturation (or spontaneous, see above) polarization P S . Once the electric field is reduced to zero, a ferroelectric material shows a finite permanent (or remnant) polarization P R . Then, under application of an opposite electric field, the polarization reduces until reaching zero at the coercive field E C and finally goes into saturation as under positive electric fields. As already mentioned above, the (macroscopic) polarization depicted in figure 1.2.2 stems from the non-centrosymmetric structure of the crystal. This leads to a non-symmetric distribution of charges in the unit cell and therefore the appearance of a dipole moment p. The macroscopic polarization of the solid P is then given by the sum over all unit cell moments p i normalized by the unit cell volume v i : P = ∑ i p i /v i . While charges are still compensated in the bulk solid, this will result in a net charge at the solid's surface. Thus, the polarization amplitude can also be understood in terms of surface charge and its unit is in the same as that of charge density, [P] = C m m -3 = C m -2 . In certain materials, such as in most perovskites, the polarization is of displacive origin as opposed to order-disorder governed ferroelectricity in BiFeO 3 (bfo) -a particular ferroelectric that will be discussed below. It should, however, be noted that a clear distinction between these two is not always possible [START_REF] Sicron | Nature of the ferroelectric phase transition in PbTiO 3[END_REF][START_REF] Stern | Character of Order-Disorder and Displacive Components in Barium Titanate[END_REF].

To complete the theoretical bases of ferroelectrics, the reader is also referred to the alternative description by R. Resta & D. Vanderbilt [START_REF] Resta | Theory of Polarization: A Modern Approach[END_REF]. Their theory is based on a more dynamic ansatz, directly related to the way the ferroelectric polarization of a solid is normally measured: the motion of electric charges.

Basics

This section starts by introducing Landau-Devonshire theory to describe the energetics of ferroelectric polarization. Afterwards, we present perovskites as an important system for applications as well as more fundamental research purposes. In the last part, the phenomena reining the properties of ultrathin ferroelectric films are presented.

Landau-Devonshire theory

In order to describe phase transitions in ferroelectrics, we give a brief look into Ginzburg-Landau-Devonshire (gld) theory here, as it provides a simple yet powerful model of its energetics as well as an insight into the role of temperature. Its origin lies in the work of Lev D. Landau. He proposed to use a phenomenological expression to describe a system's free energy and write it as a Taylor expansion in the relevant order parameter (in the case of a ferroelectric, its polarization P) [START_REF] Landau | On the theory of phase transitions. I[END_REF]. Years later, Devonshire developed a theory specifically for ferroelectrics using the same approach [START_REF] Devonshire | Theory of barium titanate -Part I[END_REF][START_REF] Devonshire | Theory of barium titanate -Part II[END_REF].

In the simple case of a uni-axial ferroelectric, the polarization has two stable states: P and -P. As, a priori, there is no energy difference between these two directions, the free energy must be an even function of P and the Taylor expansion of the intrinsic contributions can only contain even powers of P. It is then only the existence of an external electric field E that breaks this symmetry. Under these considerations, the expression of the Gibbs free energy density G takes the following form:

G = -E P + a 2 P 2 + b 4 P 4 + c 6 P 6 + ...
For the description of most phenomena, there is no need to take into account higher orders in P than six. Although, in general, all parameters depend on temperature T, in the framework of mean-field theory its dependence can be reduced to one parameter: a = a 0 (T -T 0 ). Here, T 0 corresponds to the Curie temperature T C that describes the onset of the phase transition.

Above T 0 , the material is paraelectric and the global minimum of the Gibbs free energy lies at P = 0. Second-order transitions can then be modeled by setting all parameters to positive values. Upon decreasing temperature below T 0 , the global minima correspond to the spontaneous polarization P S that continuously increases at lower temperatures as shown in figure 1.2.3 -a. For the case of first-order transitions, one assumes b < 0. This leads to the appearance of local energy minima with non-zero polarization even above T 0 , as shown in figure 1.2.3-b. Upon decreasing temperature, these minima become global for T < T C . However, the non-polar state at P = 0 stays locally stable until decreasing the temperature below T 0 < T C . At the Curie temperature the saturation polarization discontinuously jumps to zero.

In the presence of an external electric field E, the symmetry of the energy potential is broken and one of the before equally deep potential wells (figure 1.2.3-c) becomes the global minimum (see figure 1.2.3-d). Likewise, the energy barrier separating both polarization states now changes and asymmetrizes, too. (In first order, the energy barrier linearly depends on the applied field E.) Thus, the transition from the unfavorable high-energy configuration to the global minimum becomes more likely whereas the reverse transition is suppressed. Upon further increase, as depicted in figure 1.2.3-e, one polarization state eventually becomes unstable.

Initially, the theory only describes the free energy of a uniformly ordered medium. In order to treat more realistic, inhomogeneous configurations, Lev D. Landau and Vitaly L. Ginzburg introduced a term, that accounts for local changes in the order parameter [START_REF] Ginzburg | Polarization and Piezoelectric Effect in BaTiO 3 Near the Ferroelectric Transition Point[END_REF]. In the case of a ferroelectric material, this term depends on the gradient of the polarization direction so that the free energy expression (up to the sixth order in P) grows to:

G = -E P + a 2 P 2 + b 4 P 4 + c 6 P 6 + γ|∇P| 2 .
For a more complete formulation of the free energy in a given material, one can include many other contributions in this phenomenological model. This then enables the description of the interactions of e. g. ferroelastic, magnetic, and stress-induced couplings to the ferroelectric polarization. Finally, it should be underlined that Landau theory does not provide any means to estimate the values of its parameters. They have to be derived from experimental data or calculations based on ab-initio theory [START_REF] King-Smith | Theory of polarization of crystalline solids[END_REF].

Perovskites

The ferroelectric materials used in this work typically crystallize in the perovskite structure ABO 3 . In this stoichiometric formula, A and B designate electropositive elements forming cations that transfer electrons to the oxygen O.

In the case of an undistorted highly-symmetric cubic unit cell (see figure 1.2.4 -a), the A-cations occupy the cube corners whereas the B-cations are located in the cube center. The oxygen anions are face-centered so that they form an octahedron around the B-cation leading to its six-fold coordination. The A-cation then has twelve nearest neighbors. This structure is centrosymmetric and can therefore not display ferroelectricity. Its response to an external electric field is purely paraelectric.

A perovskite in which interactions are governed by purely ionic bonds is always centrosymmetric. It is only due to orbital hybridization between the oxygen atoms and the A-or B-site cations that a shift in the atomic positions can be stabilized. In many cases, the 2p-orbitals of the oxygen may hybridize with the ex-centered B-ion [START_REF] Cohen | Electronic structure studies of the differences in ferroelectric behavior of BaTiO 3 and PbTiO 3[END_REF]. If the A-cation has s 2 -type orbitals as it is the case for Pb or Bi, it may hybridize with the oxygen p-orbitals. This leads to anisotropic s-orbitals and can stabilize the ferroelectric displacement [START_REF] Hill | Why Are There so Few Magnetic Ferroelectrics?[END_REF]. Many perovskite crystals undergo a number of structural phase transitions as a function of temperature. The cubic unit cells then deforms and tetragonal, orthorhombic or rhombohedral distortions can be stabilized (see figure 1.2.4-b-d). For materials in the form of thin films another means to stabilize various crystal structures is the use of strain. By choosing the appropriate substrate and a certain crystal orientation of its surface, perovskites under compressive (figure 1.2.5 ) are nowadays commercially available, this approach is now widely used [START_REF] Schlom | Elastic strain engineering of ferroic oxides[END_REF][START_REF] Li | Effect of substrate constraint on the stability and evolution of ferroelectric domain structures in thin films[END_REF][START_REF] Zembilgotov | Ultrathin epitaxial ferroelectric films grown on compressive substrates: Competition between the surface and strain effects[END_REF][START_REF] Schlom | Strain Tuning of Ferroelectric Thin Films[END_REF][START_REF] Martin | Advanced synthesis techniques and routes to new single-phase multiferroics[END_REF][START_REF] Sando | Crafting the magnonic and spintronic response of BiFeO 3 films by epitaxial strain[END_REF] and will be further detailed using the example of the bfo thin films.

Size effects in ultrathin films

For a long time, ferroelectric polarization was believed to vanish in thin films and the presence of an intrinsic thickness limit was assumed [START_REF] Lines | Principles and Applications of Ferroelectrics and Related Materials[END_REF][161][START_REF] Ishikawa | Size Effect on the Phase Transition in PbTiO 3 Fine Particles[END_REF]. However, in 1998, the measurements of finite saturation values in thin films of a ferroelectric polymer [START_REF] Bune | Twodimensional ferroelectric films[END_REF] and, one year later, in thin films of perovskite Pb(Zr 0.2 Ti 0.8 )O 3 (pzt) down to 4 nm in thickness [START_REF] Tybell | Ferroelectricity in thin perovskite films[END_REF] disproved this assumption. The purely intrinsic effects treated by gld theory did not account for the extrinsic phenomena that become critical at these length scales and in experimental samples.

Moreover, in thin ferroelectric films, the electrostatic conditions at the interfaces become increasingly important. In typical capacitor structures where the ferroelectric film is sandwiched between two metallic electrodes, the surface charges from the ferroelectric polarization attract screening charges in the metal. In the case of an ideal metal, viz. a perfect conductor, these screening charges would be located in an infinitesimal thin plane directly at the interface, completely screening all polarization charges, as illustrated in figure 1.2.7 -a. In a realistic material, however, the screening charges will be distributed over a finite depth, the screening length λ eff (see figure 1.2.7-b). Using Thomas-Fermi theory, its value can be estimated to be of the order of interatomic distances [165, p. 342]. 14 As depicted in figure 1.2.7-d, this spread of the charges in the metals then leads to a non-zero electrostatic potential at the interfaces and results in a finite depolarization field E d [166]. As its name indicates, this electric field tends to destabilize the ferroelectric polarization and, in the case of very thin films, drives the formation of domains with opposite polarization polarity [START_REF] Mitsui | Domain Structure of Rochelle Salt and KH 2 PO 4[END_REF][START_REF] Batra | Thermodynamic stability of thin ferroelectric films[END_REF][START_REF] Streiffer | Observation of Nanoscale 180°Stripe Domains in Ferroelectric PbTiO 3 Thin Films[END_REF][START_REF] Chen | 180°Ferroelectric Stripe Nanodomains in BiFeO 3 Thin Films[END_REF].

From a macroscopical point of view, the influence of the depolarization field can be noticed as a decrease of the coercive field at small film thicknesses. After a prediction based on first-principles calculations of a metal-ferroelectric interface in a realistic perovskite structure based on BaTiO 3 (bto) [START_REF] Junquera | Critical thickness for ferroelectricity in perovskite ultrathin films[END_REF], this has first been experimentally observed by M. Dawber et al. [START_REF] Dawber | Depolarization corrections to the coercive field in thin-film ferroelectrics[END_REF]. Calculations using a phenomenological theory by N. A. Pertsev & H. Kohlstedt [START_REF] Pertsev | Elastic Stabilization of a Single-Domain Ferroelectric State in Nanoscale Capacitors and Tunnel Junctions[END_REF] predicted stable polarization in down to 1 nm thin films of Pb(Zr 0.5 Ti 0.5 )O 3 and bto between metallic electrodes [174]. Finally, surface modifications at the atomic level also can contribute to the stabilization of ferroelectric polarization in thin films, e. g. when small, interfacial ion displacements in the electrode contribute to the charge screening [START_REF] Junquera | Critical thickness for ferroelectricity in perovskite ultrathin films[END_REF]175,176].

Another, now successfully employed method to stabilize ferroelectricity in small dimensions is the use of substrates that exert epitaxial strain on the thin film [START_REF] Schlom | Elastic strain engineering of ferroic oxides[END_REF][START_REF] Li | Effect of substrate constraint on the stability and evolution of ferroelectric domain structures in thin films[END_REF][START_REF] Zembilgotov | Ultrathin epitaxial ferroelectric films grown on compressive substrates: Competition between the surface and strain effects[END_REF][START_REF] Schlom | Strain Tuning of Ferroelectric Thin Films[END_REF][START_REF] Martin | Advanced synthesis techniques and routes to new single-phase multiferroics[END_REF][START_REF] Sando | Crafting the magnonic and spintronic response of BiFeO 3 films by epitaxial strain[END_REF][177][178][179]. This allows the stabilization of certain polar crystal phases in thin films that are not stable 14) See section 1.2.2.1.2 on page 30 for estimations based on experimental results. in the bulk. As an example, a substrate with an in-plane lattice parameter smaller than that of the potentially ferroelectric material will result in a compressive strain. In a simple picture, the thin film's unit cell compensates this in-plane compression by expanding perpendicularly. 15 This deformation may then enhance the tetragonality of the material and stabilize a phase with (enhanced) out-of-plane polarization. Moreover, also the Curie temperature can be increased by substrate strain. As discussed above, for perovskites, the availability of substrates with a wide range of lattice parameters (see figure 1.2.6) nowadays opens many possibilities for this approach.

Dynamics of ferroelectric polarization reversal

While gld theory can describe the ferroelectric polarization in the bulk or on a very microscopic level (cf. section 1.2.1.1.1 on page 16), it is not easily suited for the reversal process in realistic samples. In those, due to the presence of defects, the influence of finite temperatures, and other factors, the switching of the ferroelectric polarization does not occur homogeneously. Local conditions, such as defects, may favor a reversal and therefore lead to the creation of small switched regionsnuclei. Once they exist, the neighboring polarization can be switched by the expansion of these domains -which is generally referred to as growth.

This section provides an overview of the theories of polarization reversal. Starting from universally valid considerations on domain wall (dw) motion as an important part of the reversal process, the second part will introduce more specific models that try to describe the polarization switching dynamics in ferroelectrics through nucleation and growth.

Domain walls at rest and in motion

In a homogeneously polarized ferroelectric thin film (figure 1.2.8 -a), an electric field of sufficient amplitude against the polarization direction will eventually lead to the creation of small reversed zones, the nuclei (figure 1.2.8-b). Their formation starts from one of the interfaces and then the nucleus grows forwards through the film, forming a needle-like structure (figure 1.2.8-c). Afterwards, the reversal can proceed by sideways growth (figure 1.2.8-d). While the forward growth of the needle structures is a very fast process, the lateral expansion of the domains generally is much slower [START_REF] Dawber | Physics of thin-film ferroelectric oxides[END_REF]. It is therefore this process that governs the reversal. To describe its dynamics, one often switches to the perspective of the interfaces that separate the two regions of different polarization direction -domain walls (dws) -and then examines their motion. The energies that determine the structure of dws in ferroelectrics are influenced by dipolar interactions, strain, atomic bonding, interface-related phenomena, and even magnetic interactions in multiferroics. Additionally, ferroelectric dws are very narrow compared to their magnetic counterparts. While the width of magnetic dws can be of the order of several 100 nm, in ferroelectrics, dws are down to one unit cell thin [START_REF] Scott | Ferroelectric memories[END_REF][START_REF] Jia | Direct Observation of Continuous Electric Dipole Rotation in Flux-Closure Domains in Ferroelectric Pb(Zr,Ti)O 3[END_REF][START_REF] Tagantsev | Domains in Ferroic Crystals and Thin Films[END_REF].

Apart from the interactions that are homogeneous throughout the film, local defects can attract or repel a dw. This phenomenon is called pinning by disorder. At rest, a dw will therefore not follow a straight line but bend in order to minimize its energy as a function of the interaction with the defects, as schematized in figure 1.2.9 . Owing to their small width, ferroelectric dws are very susceptible to disorder and can be pinned by small, locally confined defects.

As a function of the amplitude of the external force F, i. e. the applied electric field in the case of ferroelectrics, the dw velocity v passes through different regimes that are summarized in figure 1.2.10 . Under strong forces, the dw velocity will linearly grow with the applied force: v = ηF, where η is the friction coefficient [184,[START_REF] Yang | ac dynamics of ferroelectric domains from an investigation of the frequency dependence of hysteresis loops[END_REF]. For smaller forces, the influence of the defects becomes important. At zero temperature, there exists a threshold depinning force F dep so that only for F > F dep the dw can locally overcome the disorder energy barriers and therefore advance with an average velocity v ∝ (F

-F dep ) β .
Here, the exponent β depends on the kind of disorder [184,[START_REF] Yang | Direct Observation of Pinning and Bowing of a Single Ferroelectric Domain Wall[END_REF].

At finite temperature, however, energy barriers U B may be overcome by thermal activation so that the velocity follows an Arrhenius law:

v ∝ exp [-U B (F) /k B T].
Depending on the strength and density of the disorder, the expression of the energy barrier takes two different forms as a function of the applied force, leading to fundamentally different propagation regimes. In the case of sparse but strong defects, the barrier is lowered by the external force F as U B (F) ∼ (F dep -F) and the regime is called thermally assisted flux flow (taff) [START_REF] Anderson | Theory of Flux Creep in Hard Superconductors[END_REF][START_REF] Kes | Thermally assisted flux flow at small driving forces[END_REF][START_REF] Ferré | Universal magnetic domain wall dynamics in the presence of weak disorder[END_REF]. . 190, 191] However, in the presence of weak but dense disorder, the dw will interact with a manifold of defects at once, leading to collective pinning. In this case, the energy barrier diverges for small F and the dw motion enters the so-called creep regime [START_REF] Blatter | Vortices in high-temperature superconductors[END_REF]. Here, the dw velocity evolves as

v ∝ exp [-( U B/k B T)( F dep/F) µ ],
where the dynamical exponent µ depends on the type of the disorder and the dimensionsionality of the domain wall [START_REF] Lemerle | Domain Wall Creep in an Ising Ultrathin Magnetic Film[END_REF]. In the creep regime, the dynamics are critically ruled by the static properties of the dw (through the dynamical exponent µ). A common concept used to describe the energies and their influence on the dw is that of elastic interfaces in a disordered medium [START_REF] Giamarchi | Dynamics of Disordered Elastic Systems[END_REF][START_REF] Nattermann | Scaling approach to pinning: Charge density waves and giant flux creep in superconductors[END_REF]. 16 The static energy of such a system is governed by two competing energy contributions: by the elastic energy 17 that favors a short interface and by the disorder that deforms the energy landscape [START_REF] Ferré | Universal magnetic domain wall dynamics in the presence of weak disorder[END_REF][START_REF] Guyonnet | Growing up at the nanoscale: studies of ferroelectric domain wall functionalities, roughening, and dynamic properties by atomic force microscopy[END_REF]. The dw will then adapt its form to minimize the total energy, which is simply the sum of these two contributions. Under an external force, the dw is assumed to jump from one stable configuration to another, slowly enlarging the reversed domain. Due to this strong connection between the static and the dynamics properties of these systems, 18 researchers often try to characterize the dws at rest to conclude on the underlying disorder [START_REF] Lemerle | Domain Wall Creep in an Ising Ultrathin Magnetic Film[END_REF][START_REF] Paruch | Domain Wall Roughness in Epitaxial Ferroelectric PbZr 0.2 Ti 0.8 O 3 Thin Films[END_REF][START_REF] Catalan | Fractal Dimension and Size Scaling of Domains in Thin Films of Multiferroic BiFeO 3[END_REF][START_REF] Metaxas | Domain wall dynamics in ultrathin ferromagnetic film structures: disorder, coupling and periodic pinning[END_REF][202].

Before these detailed investigations, the ferroelectric reversal dynamics were studied measuring the electric field dependence of the mean switching time. Based on such experiments with bto single crystals, in 1954, W. J. Merz formulated an empirical law, now known as Merz's law [START_REF] Merz | Domain Formation and Domain Wall Motions in Ferroelectric BaTiO 3 Single Crystals[END_REF]. He observed that the mean switching time t s depended on the electrical field E as:

t s ∝ t 0 e -E a /E , (1.2.1)
where E a denotes an activation field. Merz explained this dependence in the framework of classical nucleation theory by considering the energies involved in the nucleation of "long flat dagger[s]" [START_REF] Merz | Domain Formation and Domain Wall Motions in Ferroelectric BaTiO 3 Single Crystals[END_REF]. Six years later, R. C. Miller & G. Weinreich developed another, similar theory considering that the dws advance through the appearance of new nuclei at their front [START_REF] Miller | Mechanism for the Sidewise Motion of 180°Domain Walls in Barium Titanate[END_REF]. Their model, correspondingly called the Miller-Weinreich model, could explain the experimentally found electric field dependence of the domain wall velocity in many ferroelectrics. However, in thin films, the theoretical critical nucleus size predicted by the theory largely exceeds the film thickness [START_REF] Tybell | Domain Wall Creep in Epitaxial Ferroelectric Pb(Zr 0 .2Ti 0 .8)O 3 Thin Films[END_REF]. One possible reason could be that the Miller-Weinreich model neglects the finite-width of dws. This would lower the surface energy of new nuclei and therefore allow for smaller critical sizes [START_REF] Shin | Nucleation and growth mechanism of ferroelectric domainwall motion[END_REF]. Furthermore, this finding motivated a revival of the research regarding the creep dw motion mechanism and remarkable agreements between the static and dynamic properties of ferroelectric dws have been found, underlining the importance of creep theory [START_REF] Guyonnet | Growing up at the nanoscale: studies of ferroelectric domain wall functionalities, roughening, and dynamic properties by atomic force microscopy[END_REF][START_REF] Tybell | Domain Wall Creep in Epitaxial Ferroelectric Pb(Zr 0 .2Ti 0 .8)O 3 Thin Films[END_REF]207,[START_REF] Xiao | Domain wall roughness and creep in nanoscale crystalline ferroelectric polymers[END_REF].

Switching models

This section provides an overview of different models developed to explain the dynamics of polarization reversal in ferroelectric thin films. 16) This concept is universal and finds application in many areas of physics such as the vortex motion in superconductors [START_REF] Blatter | Vortices in high-temperature superconductors[END_REF], charge density waves, or crack propagation in solids [START_REF] Brazovskii | Pinning and sliding of driven elastic systems: from domain walls to charge density waves[END_REF], and magnetic dw motion [START_REF] Lemerle | Domain Wall Creep in an Ising Ultrathin Magnetic Film[END_REF][START_REF] Chauve | Creep and depinning in disordered media[END_REF][START_REF] Metaxas | Creep and Flow Regimes of Magnetic Domain-Wall Motion in Ultrathin Pt/Co/Pt Films with Perpendicular Anisotropy[END_REF]]. 17) The elastic energy is given as E el ∝ (∇u(r)) 2 d d r, where u(r) is the local dw deformation and d its dimensionality. This expression is equivalent to the length of the dw, in the approximation of small bending radii. 18) The dynamic exponent µ of creep motion in a random potential is in fact a direct function of the dimensionality d of the interface and the equilibrium (i. e. static) roughness exponent ζ of the dw, that characterizes the disorder:

µ = d-2+2ζ /2-ζ [198].

Single nucleus

In the simplest case of nucleation of one sole nucleus at a certain time t n with subsequent radial growth of the reversed domain and constant dw velocity v, the reversed area A would evolve for times t > t n as:

A(t) = π v d (t -t n ) d
, where the dimension d = 2 for lateral dw motion. However, in a realistic sample the reversible area is limited so that even for t → ∞ the reversed domain size A will not exceed a maximal area A 0 . 19 Henceforth, we will therefore use the relative reversed area S(t) = A(t)/A 0 for the description of the switching progress. Moreover, the limited size has direct influence on the growth process, as soon as domains hit the borders. Taking into account this limitation and combining it with a time-dependent dw velocity v(t), Y. Kim et al. [START_REF] Kim | Non-Kolmogorov-Avrami-Ishibashi Switching Dynamics in Nanoscale Ferroelectric Capacitors[END_REF] could reproduce their experimental results by a single-domain reversal model with a good agreement.

Homogeneous switching

For the case of several nucleation centers, the Kolmogorov-Avrami-Ishibashi (kai) model is widely used [START_REF] Avrami | Kinetics of Phase Change. II Transformation-Time Relations for Random Distribution of Nuclei[END_REF][START_REF] Ishibashi | Note on Ferroelectric Domain Switching[END_REF]. Using the "extended volume" defined by Avrami, it takes into account the restriction that new nuclei may not form in the previously reversed volume. Under the assumption of an infinite ferroelectric and a homogeneous nucleation, the Kolmogorov-Avrami-Ishibashi (kai) model predicts a switching process where the fraction S of reversed domains evolves as:

S(t) ≈ 1 -exp - t τ n , (1.2.2)
and the characteristic time scale is given by τ, which typically is a function of the applied electric field. The parameter n represents the dimension of the process: in the case of a finite number of initial nuclei and no further nucleation, n = d, whereas in the case of a constant nucleation rate during the switching, n = d + 1, with the dimensionality of the dw d (see above). This model was employed to describe ferroelectric polarization reversal in different materials [START_REF] Li | Ultrafast polarization switching in thin-film ferroelectrics[END_REF][START_REF] So | Polarization switching kinetics of epitaxial Pb(Zr 0 .4Ti 0 .6)O 3 thin films[END_REF][START_REF] Chanthbouala | A ferroelectric memristor[END_REF]. In some cases, fits to the experimental switching data yielded physically reasonable values for n, indicating the crossover between different dimensionalities [START_REF] So | Polarization switching kinetics of epitaxial Pb(Zr 0 .4Ti 0 .6)O 3 thin films[END_REF][START_REF] Hu | Universal Ferroelectric Switching Dynamics of Vinylidene Fluoride-trifluoroethylene Copolymer Films[END_REF]. However, in other cases, non-physical values of n, even smaller than 1, make the interpretation in the framework of the kai model difficult. Two possible causes could be a strongly non-homogeneous nucleation rate or the neglect of finite size effects. Trying to solve the latter problem, V. Shur et al. [216] slightly modified the model to account for blocked dw motion at the sample edges.

Heterogeneous switching in disordered ferroelectrics

In contrast to the kai model, in which the dynamics are dominated by the domain growth, A. Tagantsev et al. [START_REF] Tagantsev | Non-Kolmogorov-Avrami switching kinetics in ferroelectric thin films[END_REF] proposed the nucleation-limited switching (nls) model. Here, the ferroelectric is assumed to be composed of many small regions that switch independently as soon as a nucleus forms. The model therefore neglects the time to reverse each region, leaving dw motion aside. Instead, the authors assume that the nucleation times τ have a widely spread distribution F(log τ) that governs the reversal progress. In their work, they could reproduce the experimental data by using a Lorentzian distribution with a large plateau around the mean value. Nonetheless, there is no direct physical motivation for this choice.

In 2007, J. Y. Jo et al. [START_REF] Jo | Domain Switching Kinetics in Disordered Ferroelectric Thin Films[END_REF] investigated the switching dynamics in polycrystalline pzt films and adapted the nucleation time distribution of the nls model to a pure Lorentzian. Apart from obtaining a very good agreement with their experimental results, they also provided a physical argument for this particular choice based on the distribution of dipole impurities in the ferroelectric.

Another switching model for highly disordered ferroelectrics was developed by S. Zhukov et al. [START_REF] Zhukov | Experimental and theoretical investigation on polarization reversal in unfatigued lead-zirconate-titanate ceramic[END_REF] -the inhomogeneous field mechanism (ifm). Based on the kai model and Merz's law, the authors assumed a distribution of the locally applied electric field and could extract this distribution from their experimental data on polarization reversal in pzt ceramics. However, one weak point of the ifm model is that it relies on several assumptions: apart from the two given above, the distribution also has to scale with the applied electric field and may not be too large [START_REF] Zhukov | Experimental and theoretical investigation on polarization reversal in unfatigued lead-zirconate-titanate ceramic[END_REF][START_REF] Genenko | Universal Polarization Switching Behavior of Disordered Ferroelectrics[END_REF].

In conclusion, the existence of this multitude of models to describe the switching dynamics in ferroelectrics shows that a fundamental understanding of the subject is still missing. Even though there is now much data for the polarization reversal in bulk or polycrystalline samples, the dynamics in thin and ultrathin ferroelectric films demand further investigations. Not only the lateral limits (such as the electrode size or film borders) but also the top and bottom interfaces may influence the switching. In chapter 4 on page 93, we take a closer look at the dynamics of polarization reversal in ferroelectric tunnel junctions.

Ferroelectric tunnel junctions

Ferroelectric tunnel junctions (ftjs) are based on a simple three-layer structure: an ultrathin ferroelectric layer that is sandwiched between two electrodes, as shown in figure 1.2.11 . The polarization of the ferroelectric thin film points to either one or the other electrode and can be switched by an electric field under application of a voltage across the barrier [START_REF] Garcia | Ferroelectric tunnel junctions for information storage and processing[END_REF]. In 1971, L. Esaki et al. [222] proposed this structure, which they called a "polar switch", and predicted a change in the tunnel resistance upon polarization reversal (cf. figure 1.2.12 ).

However, due to the belief that ferroelectricity could not exist in films with thicknesses allowing for tunneling transport, as mentioned above, this concept sank into oblivion for about 30 years. Meanwhile, in an intermediate step, ferroelectric diodes incorporating thicker ferroelectric films showed resistance changes with off/on ratios of up to 10 2 [START_REF] Blom | Ferroelectric Schottky Diode[END_REF][START_REF] Jiang | A Resistive Memory in Semiconducting BiFeO 3 Thin-Film Capacitors[END_REF]. The high resistances owing to the thick highly resistive ferroelectric layers, however, limit their benefit for practical applications as the very low current densities are detrimental for the information read-out. In the 2000s, epitaxial deposition methods, growth conditions, and characterization techniques had finally evolved to realize high-quality ultrathin ferroelectric films [START_REF] Bune | Twodimensional ferroelectric films[END_REF]175,[START_REF] Nagarajan | Size effects in ultrathin epitaxial ferroelectric heterostructures[END_REF][START_REF] Kim | Critical thickness of ultrathin ferroelectric BaTiO 3 films[END_REF][START_REF] Béa | Ferroelectricity Down to at Least 2 nm in Multiferroic BiFeO 3 Epitaxial Thin Films[END_REF]. This revived the interest in theoretical investigations on the subject [START_REF] Kohlstedt | Theoretical current-voltage characteristics of ferroelectric tunnel junctions[END_REF][START_REF] Zhuravlev | Giant Electroresistance in Ferroelectric Tunnel Junctions[END_REF][START_REF] Tsymbal | Tunneling Across a Ferroelectric[END_REF] and, shortly after, first experimental demonstrations of resistance-switching ferroelectric tunnel junctions could be made [START_REF] Gajek | Tunnel junctions with multiferroic barriers[END_REF][START_REF] Garcia | Giant tunnel electroresistance for non-destructive readout of ferroelectric states[END_REF][START_REF] Maksymovych | Polarization Control of Electron Tunneling into Ferroelectric Surfaces[END_REF][START_REF] Gruverman | Tunneling Electroresistance Effect in Ferroelectric Tunnel Junctions at the Nanoscale[END_REF][START_REF] Crassous | Giant tunnel electroresistance with PbTiO 3 ferroelectric tunnel barriers[END_REF].

In the following, I will first discuss the different mechanisms that contribute to the electrical transport in ferroelectric tunnel junctions (ftjs), then present the effects at the origin of the resistance change, and finally conclude this introductory section by an overview of the state of the art.

Concepts

This section contains the concepts at the base of the electrical characteristics of ftjs. After reviewing the three main electrical transport mechanisms contributing to the current across the ferroelectric barrier in ftjs, we discuss the different phenomena that may cause resistance changes in these structures.

Electrical transport mechanisms through ultrathin insulators

When a voltage V is applied across the ftj, several conduction mechanisms can contribute to the current across the ferroelectric barrier: direct tunneling (dt), Fowler-Nordheim tunneling (fnt), and thermionic injection (ti). They are presented in the following paragraphs.

Direct tunneling

As ferroelectric materials are insulating by definition, an energy barrier prevents charge carriers from freely flowing through them. However, for very thin films with thicknesses of some nanometers quantum-mechanical tunneling becomes possible [START_REF] Frenkel | On the Electrical Resistance of Contacts between Solid Conductors[END_REF][START_REF] Simmons | Generalized Formula for the Electric Tunnel Effect between Similar Electrodes Separated by a Thin Insulating Film[END_REF]. , commonly termed trapezoidal. Based on the work of W. F. Brinkman et al. [START_REF] Brinkman | Tunneling Conductance of Asymmetrical Barriers[END_REF] and using the Wentzel-Kramers-Brillouin (wkb) approximation, an applied voltage V results in the following tunnel current density:

E F eV ϕ B,1 ϕ B,2
j DT (V) = C exp α(V) ϕ B,2 - eV 2 3 /2 -ϕ B,1 + eV 2 3 /2 α(V) 2 ϕ B,2 - eV 2 1 /2 -ϕ B,1 + eV 2 1 /2 2 × sinh 3 eV 4 α(V) ϕ B,2 - eV 2 1 /2 -ϕ B,1 + eV 2 1 /2

Fowler-Nordheim tunneling

At high voltages, another regime dominates the conduction: Fowler-Nordheim tunneling (fnt). When the voltage exceeds the barrier height at the interface of the electrode it is applied to, the effective tunneling barrier width is modified. This is because part of the energy barrier profile in the ferroelectric then lies beneath the Fermi energy of the electrode, as schematized in figure 1.2.14 . The basics of this situation have been described as (cold) field emission of electrons from a metal into vacuum [START_REF] Fowler | Electron Emission in Intense Electric Fields[END_REF][START_REF] Dyke | Field Emission[END_REF]. A simplified and widely used expression for the current density in the fnt regime under an electric field E is given by

j FNT (E) = C e 3 16 π 2 h m * ϕ B E 2 exp - 4 2 m e m * 3 h e ϕ 3 /2 B E , (1.2.4)
with the potential barrier ϕ B and the effective mass of the tunneling charge carriers m * [START_REF] Pantel | Electroresistance effects in ferroelectric tunnel barriers[END_REF]. However, this basic expression can, in general, not reproduce experimental data quantitatively, which is why the prefactor C is often left free, taking values far from unity.

The general form of the expression for the current density has been calculated by R. G. Forbes [START_REF] Forbes | Physics of generalized Fowler-Nordheim-type equations[END_REF]. The author accounted for many different effects that can influence the electrical transport in the fnt regime. While R. H. Fowler & L. Nordheim [START_REF] Fowler | Electron Emission in Intense Electric Fields[END_REF] already included the reduction of the energy barrier due to image forces (Schottky-Nordheim barrier) [START_REF] Deane | The formal derivation of an exact series expansion for the principal Schottky-Nordheim barrier function v, using the Gauss hypergeometric differential equation[END_REF] in their original formula, these improvements are often neglected [START_REF] Forbes | Extraction of emission parameters for large-area field emitters, using a technically complete Fowler-Nordheim-type equation[END_REF]. Among others, also the exact value of the electric field might be locally modified, so that research concerning a complete theoretical description is still ongoing [START_REF] Forbes | Extraction of emission parameters for large-area field emitters, using a technically complete Fowler-Nordheim-type equation[END_REF][START_REF] Forbes | Reformulation of the standard theory of Fowler-Nordheim tunnelling and cold field electron emission[END_REF][START_REF] Forbes | Improved approach to Fowler-Nordheim plot analysis[END_REF]. In the context of this work, however, these details remain negligible, so that a model taking into account a small reduction of the barrier of image forces is sufficient [START_REF] Forbes | Use of Millikan-Lauritsen plots, rather than Fowler-Nordheim plots, to analyze field emission current-voltage data[END_REF]. This can be accounted for by slight variations of the prefactor C from 1. 20 Thermionic injection [START_REF] Mountcastle | Perceptual neuroscience: the cerebral cortex[END_REF] -Schema of thermionic injection (TI) at an energy barrier of height ϕ B under an applied voltage V and at finite temperature T > 0 K.

E F eV ϕ B TI Figure 1.2.
At non-zero temperatures, charge carriers have a finite possibility to overcome the interface barrier by thermal excitation. This process is described by thermionic injection (ti) 21 and becomes important for relatively high temperatures resp. applied voltages (approximately V 3k B T/e). In contrast to the simplified expressions used for dt and fnt above, in the framework of thermionic injection (ti), image force lowering is already taken into account (see figure 1.2.15 ). This leads to the following expression of the current 20) See section 2.4.1.1 on page 53 for details. 21) Also named "thermionic emission" (depending of the point of view) density under application of an electrical field E,

j TI (E) = A ** T 2 exp   - 1 k B T   ϕ B - e 3 E 4 π ε 0 ε ifl     ,
with the potential barrier ϕ B and the relative permittivity of the ferroelectric responsible for image force lowering ε ifl [START_REF] Pantel | Electroresistance effects in ferroelectric tunnel barriers[END_REF]. The effective Richardson constant A ** is material dependent but its value normally is close to that of the unmodified Richardson constant A * = 4πm e k 2 B e/h 3 ≈ 1.20 × 10 6 A m -2 K -2 [START_REF] Crowell | The Richardson constant for thermionic emission in Schottky barrier diodes[END_REF].

Resistance switching mechanisms in ferroelectric tunnel junctions

Many experiments on the metal/ferroelectric/metal structure presented above have shown that the resistance of these devices changes after application of an external voltage.

In some cases, the voltages had to be applied for several seconds, in other cases, short pulses on timescales of the order of nanoseconds were sufficient. For structures incorporating very thin ferroelectric films, allowing for tunneling transport, these resistance changes are summarized under the term tunnelling electroresistance (ter) [START_REF] Kohlstedt | Theoretical current-voltage characteristics of ferroelectric tunnel junctions[END_REF][START_REF] Zhuravlev | Giant Electroresistance in Ferroelectric Tunnel Junctions[END_REF] that is defined as the relative conductance between the resistances for both polarization directions (here defined as pointing to the left or the right), 22

TER = R -R R × 100 % .
Usually, R > R so that TER > 0 %. As the large resistance changes occurring in ftjs result in very high values when given in percent, the ratio between the high (R off ) and the low (R on ) resistance states is often used. 23 In the following, we present several possible mechanisms contributing to these resistance changes. Figure 1.2.16 summarizes the most important ones supposed to play a role in ftjs.

Electrostatic effects

For the following, we assume that the ferroelectric thin film is in a tetragonal phase so that its polarization P points either toward the metallic electrode 1 (M1) or electrode 2 (M2). As mentioned above (see section 1.2.1.1.3 on page 20), the surface charges of the polarization σ P = | P| in the ferroelectric layer will be screened in the metallic electrodes where they attract or repel free charge carriers, leading to a zone of accumulation or depletion, respectively (see figure 1.2.17 -a 1 , b 1 ). Using the Thomas-Fermi model of screening [START_REF] Ashcroft | Solid state physics[END_REF], the screening potential in the metal (1 or 2, respectively) as a function of the distance x 22) The definition closely follows the one used for the tunnel magnetoresistance (tmr) in magnetic tunnel junctions [START_REF] Jullière | Tunneling between ferromagnetic films[END_REF]. 23) For simplicity, we also use the term "off/on ratio" to denote R off /R on in this work. from the respective ferroelectric interface (1, 2) is given by

φ 1,2 (x) =    ∓σ S λ 1 ε 0 ε 1 e x/λ 1 for x ≤ 0 , ±σ S λ 2 ε 0 ε 2 e (d-x)/λ 2 for x ≥ d , (1.2.5)
where λ 1,2 are the Thomas-Fermi screening lengths in the electrodes, ε 1,2 are the ionic permittivities of the electrodes, and ∓σ S is the screening charge surface density [START_REF] Kim | Polarization Relaxation Induced by a Depolarization Field in Ultrathin Ferroelectric BaTiO 3 Capacitors[END_REF].

Under the assumption of a short-circuited ftj, φ 1 (∞) = φ 2 (∞) and the screening charge density amplitudes σ S are equal in both electrodes (as already used in equation 1.2.5). Making use of the condition of potential continuity at the interfaces, we can then give an expression for their value [START_REF] Pantel | Electroresistance effects in ferroelectric tunnel barriers[END_REF]:

σ S = P d ε stat λ 1 ε 1 + λ 2 ε 2 + d ,
where ε stat is the static relative permittivity of the barrier. Furthermore, the depolarization field inside the ferroelectric therewith can be given as:

E d = - P -σ S ε 0 ε stat .
Figure 1.2.17(a 2 , b 2 ) shows the resulting potential profile for both polarization directions.

In order to determine the electric transport through the ftj, one needs to calculate the energy barrier profiles E B (x). In the case of a trapezoidal barrier, exploited here, the profile is determined by the interfacial barrier heights that are composed of the initial barriers ϕ B,1 and ϕ B,2 and the changes induced by the screening charges -e φ 1,2 in the

M1 FE M2 0 d P x σ(x) λ 2 λ 1 (a 2 ) E d x φ(x) (a 3 ) x E B (x) E F ϕ B ϕ B,1 ϕ B,2 (b) (b 1 )
M1 FE M2 metallic electrodes depending on the polarization:

0 d P x σ(x) λ 2 λ 1 (b 2 ) E d x φ(x) (b 3 ) x E B (x) E F ϕ B ϕ B,1 ϕ B,2
ϕ B,i = ϕ B,i ± e φ i , ϕ B,i = ϕ B,i ∓ e φ i .
In order to estimate the value of the initial barriers at the metal-ferroelectric (or metalsemiconductor) interface, Anderson's rule [START_REF] Anderson | Germanium-Gallium Arsenide Heterojunctions[END_REF] can give a first, simple guess. Accordingly, the barriers at the interface of metal i are defined by the difference of the metal work function (wf) and the electron affinity χ of the ferroelectric:

ϕ B,i = WF i -χ , (1.2.7)
as schematized in figure 1.2.18 . It should, however, be noted that this simple picture ignores the influence of chemical bonding and a multitude of other effects that can be important at interfaces. As an example, image forces can significantly decrease the interface barrier. Therefore, only experimental measurements can yield trustworthy values and reveal the uncertainties of the model [START_REF] Robertson | Schottky barrier heights of tantalum oxide, barium strontium titanate, lead titanate, and strontium bismuth tantalate[END_REF]. For particular interfaces, results have been obtained by X-ray photoemission spectroscopy (xps) [e. g. [START_REF] Schafranek | Barrier height at (Ba,Sr)TiO 3 /Pt interfaces studied by photoemission[END_REF][START_REF] Wu | Direct spectroscopic evidence of charge reversal at the Pb(Zr 0.2 Ti 0.8 )O 3 /La 0.7 Sr 0.3 MnO 3 heterointerface[END_REF][START_REF] Chen | Polarization dependence of Schottky barrier heights at interfaces of ferroelectrics determined by photoelectron spectroscopy[END_REF][START_REF] Rault | Structure chimique et électronique des interfaces métal/ferroélectrique en fonction de la polarisation ferroélectrique[END_REF].

When the electrical transport through the ferroelectric barrier is dominated by tunneling mechanisms, the barrier height plays a crucial role. In the dt regime, for example, at low metal semiconductor vacuum level bias, the current density depends on the mean barrier height ϕ B and the thickness d as -a shows that dt dominates the conduction at small voltage whereas fnt becomes prevailing at higher bias. ti does not significantly contribute to the total current density for the parameters used in the calculation.

E F WF E C χ E V ϕ B Figure 1.2.
j ∝ √ ϕ B d exp [-d √ ϕ B ] V . ( 1
P = 3 µC cm -2 , ε stat = 60, ε ifl = 10, m * = 1, A ** = 1 × 10 6 A m -2 K -2 , λ 1 = 0.08 nm, λ 2 = 0.055 nm, ε M 1 = 8, ε M 2 = 2, and ϕ B,1 = ϕ B,2 = 1 eV. [taken from Ref. 221]
This changes when the barrier thickness is increased, as can be seen in figure 1.2.19-b: while dt still dominates the conduction for very thin barriers, ti takes over for larger thicknesses. The resulting tunnelling electroresistance (ter) in figure 1.2.19-c also demonstrates this transition: in the ti regime, very large resistance changes can be obtained at negative voltages, whereas under positive bias, the ter becomes even negative. As interesting as this could be for low-energy switching, 24 on the downside, the comparatively large thicknesses that are required for ti result in impractical high resistances. Therefore, most experimental realizations of ftjs present transport characteristics resembling the dt (at low voltage amplitudes) and fnt (at higher voltage amplitudes) regimes. Nonetheless, thanks to the optimization of growth and materials combination parameters, high resistance changes can be achieved at relatively low absolute resistance (resp. resistance-area product) values.

This purely electrostatic resistance switching mechanism is supposed to be the main cause for the high off/on resistance ratios measured in ftjs today. And there are strong indications for this, as explained in section 1.2.2.2 on page 36.

Atomistic interface effects

A break in the symmetry of the metal/ferroelectric/metal structure is a necessity for a resistance change that depends on the ferroelectric polarization direction. In the mechanism explained in the section above, the different electric screening of the polarization surface charges in the two electrodes is at the origin of a change in the tunneling barrier height. Taking a more detailed look at the interfaces between the ferroelectric and the electrodes, effects on the atomic level may also result in asymmetric changes, as sketched in figure 1.2.16-b.

H. Kohlstedt et al. [START_REF] Kohlstedt | Theoretical current-voltage characteristics of ferroelectric tunnel junctions[END_REF] suggested that the ion displacements in the ferroelectric layer could alter the interfacial barrier heights, resulting in an asymmetric trapezoidal profile. High-resolution imaging of the interfaces in a SrRuO 3 (sro)/BaTiO 3 /SrRuO 3 layer stack revealed different atomic arrangements depending on the deposition conditions for each interface [257]. Theoretical ab-initio calculations of the complex band structure in the barrier show that small changes in the bond lengths at the interfaces can modify the decay lengths of evanescent wave functions in the barrier [START_REF] Velev | Effect of Ferroelectricity on Electron Transport in Pt/BaTiO 3 /Pt Tunnel Junctions[END_REF][START_REF] Hinsche | Strong influence of complex band structure on tunneling electroresistance: A combined model and ab-initio study[END_REF][START_REF] Wortmann | Influence of the electronic structure on tunneling through ferroelectric insulators: Application to BaTiO 3 and PbTiO 3[END_REF].

Also the screening may be affected by minute ion displacements at the interfaces. This has been suggested based on results from ab-initio calculations on SrRuO 3 (sro) electrodes [176] and experimentally shown using X-ray measurements of the oxygen positions at a PbTiO 3 (pto)/SrTiO 3 (sto) interface [START_REF] Fong | Direct structural determination in ultrathin ferroelectric films by analysis of synchrotron x-ray scattering measurements[END_REF]. D. I. Bilc et al. [START_REF] Bilc | Electroresistance Effect in Ferroelectric Tunnel Junctions with Symmetric Electrodes[END_REF] underlined the importance of the precise atomic structure at the interfaces: they predicted a ter effect in supposedly symmetric junctions due to atomic interface differences.

Piezoelectric strain effects

As all ferroelectric materials are also piezoelectric, under the application of an electric field, they will exhibit the inverse piezoelectric effect. This can result in a change in the thickness of the ferroelectric layer and therefore in a change in the tunneling barrier width, as indicated in figure 1.2.16-c. Depending on the direction of the ferroelectric polarization, a given electric field leads to either compressive or tensile strain, i. e. to an increase or a decrease of the ferroelectric film thickness.

As the barrier width not only influences the screening in the electrodes but also directly enters the expression of the tunneling conductance, an over-exponential dependence of the resistance is expected. Nonetheless, based on the fact that the absolute changes of the film thickness are rather small, 25 H. Kohlstedt et al. [START_REF] Kohlstedt | Theoretical current-voltage characteristics of ferroelectric tunnel junctions[END_REF] have predicted resistance variations of up to 70 %. Ab-initio based calculations on symmetric ftjs by D. I. Bilc et al. [START_REF] Bilc | Electroresistance Effect in Ferroelectric Tunnel Junctions with Symmetric Electrodes[END_REF] later suggested that the inverse piezoelectric effect could lead to resistance changes up to a factor of 2.

Unfortunately, at high applied voltage V > V C , i. e., when the polarization is aligned to the electric field, the resistance contrast vanishes. However, as for small electric fields E, the strain is proportional to E and therefore the barrier thickness does not change at E = 0, a minimal voltage must be applied to detect a resistance difference.

Conversely, it should be noted that mechanical stress can also be applied externally to the ftj. In this case, the piezoelectric effect changes the effective polarization or structural parameters that lead to a resistance variation [START_REF] Zheng | Giant piezoelectric resistance in ferroelectric tunnel junctions[END_REF][START_REF] Luo | Tunable Tunneling Electroresistance in Ferroelectric Tunnel Junctions by Mechanical Loads[END_REF]. A possible application of this effect is the use for mechanical sensors.

Moving oxygen vacancies

In ferroelectric films with thicknesses of 20 nm and above tunneling through the film is strongly suppressed. These structures therefore do not fall into the category of ftjs. Nonetheless, they can display resistance changes after the application of external voltages as evidenced in the results on bfo by X. Ou et al. [START_REF] Ou | Forming-free resistive switching in multiferroic BiFeO 3 thin films with enhanced nanoscale shunts[END_REF]. The origin of this resistive switching phenomenon is believed to reside in the presence of oxygen vacancies in the ferroelectric films -a widely known problem in the growth of these layers. In the presence of an electric field, the charged vacancies move though the film, forming conducting filaments [266] or changing an interfacial Schottky barrier [START_REF] Ou | Forming-free resistive switching in multiferroic BiFeO 3 thin films with enhanced nanoscale shunts[END_REF]. This effect, by the way, is also suspected to be at the origin of the first results of resistive switching on ftj-like structures by J. Rodríguez Contreras et al. [267] [268]. It should be noted that the term ter normally does not cover resistance changes that are induced by ion (resp. vacancy) movements.

Other effects

The link between ferroelectricity and magnetism through interfacial or bulk effects has emerged as a new field of physics and nowadays constitutes the main part of the subject of multiferroics [START_REF] Garcia | Ferroelectric tunnel junctions for information storage and processing[END_REF][START_REF] Gajek | Tunnel junctions with multiferroic barriers[END_REF][START_REF] Tsymbal | Ferroelectric and multiferroic tunnel junctions[END_REF][START_REF] Pantel | Reversible electrical switching of spin polarization in multiferroic tunnel junctions[END_REF].

In ultrathin ferroelectric films with many small domains dw densities can be high. As electrical conduction at dws has been identified as a possible transport mechanism in ultrathin films with mixed domain states, this could contribute in addition to the tunneling mechanisms described before. Using locally resolved conductance maps, J. Seidel et al. [START_REF] Seidel | Conduction at domain walls in oxide multiferroics[END_REF] found increased current densities at 180 • dws in rhombohedral bfo films of 100 nm thickness. Even though dws in ferroelectric thin films are only some unit cells thick [183, pp. 274ff.], their high conductivity, that can be up to six orders of magnitude higher than in the bulk, makes them interesting for applications. Since then, apart from bfo [START_REF] Seidel | Conduction at domain walls in oxide multiferroics[END_REF][START_REF] Farokhipoor | Conduction through 71°Domain Walls in BiFeO 3 Thin Films[END_REF][START_REF] Maksymovych | Dynamic Conductivity of Ferroelectric Domain Walls in BiFeO 3[END_REF], conducting walls have also been evidenced in thin films of pzt [274], 25) Typical values of the piezoelectric coefficient are of the order of 50 pm V -1 . In clamped thin films of a nanometers, this coefficient is supposed to be even smaller.

LiNbO 3 [START_REF] Schröder | Conducting Domain Walls in Lithium Niobate Single Crystals[END_REF], and bto [START_REF] Sluka | Free-electron gas at charged domain walls in insulating BaTiO 3[END_REF]. The exact origin of their high conductivity has not been identified yet, but a high local oxygen vacancy concentration or free screening charge carriers from defects have been evoked as possible explanations together with structural changes that affect both the electrostatic potential and the electronic structure [START_REF] Vasudevan | Domain Wall Conduction and Polarization-Mediated Transport in Ferroelectrics[END_REF][START_REF] Matzen | Domains and domain walls in multiferroics[END_REF].

For ftjs, however, there exists no demonstration of a contribution of dw conduction.

In summary, the main resistance change mechanism in ftjs is due to the change of the tunneling barrier height by screening effects in the electrodes. This enables a purely electronic readout of the polarization state and, thanks to the high non-linearity of the polarization reversal, the readout can be be performed without perturbing the resistance state. For memory applications these two points, the non-destructive and resistive readout, represent important advantages over conventional ferroelectric random-access memory (feram) that requires a new write step after each information readout [START_REF] Ishiwara | Ferroelectric Random Access Memories: Fundamentals and Applications[END_REF]. Furthermore, the capacitive readout used in conventional feram limits the scalability of the device whereas in ftjs the off/on does not change with their lateral size. All these advantages of ftjs also contribute to the reason why companies and researchers have been very quick to file patents regarding the aspect of a use as memory element [280-286].

State of the art

This section provides an overview of the experimental results obtained on ferroelectric tunnel junctions at the start of this thesis.

First experimental demonstrations

First resistance switching measurements on ftj-like structures using Pb(Zr 0.52 Ti 0.48 )O 3 thin films were obtained by J. Rodríguez Contreras et al. [START_REF] Contreras | Resistive switching in metal-ferroelectric-metal junctions[END_REF] in 2003: they observed resistance changes of about 400 % at room temperature. Five years later, the same group presented a method to distinguish the origin of the resistance switching in these junctions by simultaneously measuring the ferroelectric properties and monitoring the resistance change of the junction. Only if the polarization reversal and the resistance change coincide, they can be linked. Based on this insight, the researchers finally concluded that their prior result did not originate in ferroelectric polarization switching but was more likely related to ion (resp. vacancy) movements [START_REF] Kohlstedt | Method to distinguish ferroelectric from nonferroelectric origin in case of resistive switching in ferroelectric capacitors[END_REF].

In 2007, M. Gajek et al. published first measurements of ter in tunnel junctions with ferroelectric barriers of La 0.1 Bi 0.9 MnO 3 [START_REF] Gajek | Tunnel junctions with multiferroic barriers[END_REF]. Two years later, three independent research groups demonstrated the connection between the ter effect and ferroelectricity [START_REF] Garcia | Giant tunnel electroresistance for non-destructive readout of ferroelectric states[END_REF][START_REF] Maksymovych | Polarization Control of Electron Tunneling into Ferroelectric Surfaces[END_REF][START_REF] Gruverman | Tunneling Electroresistance Effect in Ferroelectric Tunnel Junctions at the Nanoscale[END_REF]. Using locally resolved imaging of the ferroelectric polarization via piezoresponse force microscopy (pfm) in parallel with local resistance measurements via conductive-tip atomic force microscopy (afm), they could establish a direct link between the polarization direction and the local resistance on bare ferroelectric ultrathin films of bto and pzt. -a-f shows the direct link between the local polarization direction of a bto thin film and the local resistance. Furthermore, the resistances of the virgin (unpoled) and the positively and negatively poled states scale exponentially with the ferroelectric film thickness, as expected for tunneling transport, as shown in figure 1.2.20-j. Consequently, the ter also increases exponentially which is thus in good agreement with the electrostatic barrier model (figure 1.2.20-k, also refer to equation 1.2.8). A. Gruverman et al. [START_REF] Gruverman | Tunneling Electroresistance Effect in Ferroelectric Tunnel Junctions at the Nanoscale[END_REF] also measured local current versus voltage curves that were compatible with tunneling through the ferroelectric barrier and obtained a first estimate of the polarization-direction dependent barrier heights. P. Maksymovych et al. [START_REF] Maksymovych | Polarization Control of Electron Tunneling into Ferroelectric Surfaces[END_REF] used c-afm on 30-nm thick pzt films to simultaneously measure the local piezoresponse and the current as a function of the applied voltage across the ferroelectric. In this way, they could evidence a resistance change of a factor of 500, that was directly correlated to the ferroelectric polarization reversal. Owing to the relatively thick film and the high electric fields produced by the sharp atomic force microscopy (afm)-tip, the electrical transport was mainly dominated by Fowler-Nordheim tunneling in these structures, as illustrated in figure 1.2.21 -a. 26

Solid-state binary memories

After these c-afm-based experiments on bare ferroelectric surfaces, in 2011, D. Pantel et al. [START_REF] Pantel | Room-Temperature Ferroelectric Resistive Switching in Ultrathin Pb(Zr 0 .2Ti 0 .8)O 3 Films[END_REF] fabricated first ftj-structures with patterned electrodes of submicron size. Composed of 9-nm-thick pzt films sandwiched between lsmo bottom and Cu topelectrodes, these devices showed off/on ratios of more than 1500 with correlated coercive 26) Owing to the large thickness of the film, electrons are injected into the ferroelectric by Fowler-Nordheim tunneling but the electrical transport inside the pzt film occurs via hopping or other mechanisms [287]. Nonetheless, limiting the electron injection, fnt is the characteristic contribution to the current across the ferroelectric. At the beginning of 2012, A. Chanthbouala et al. [START_REF] Chanthbouala | Solid-state memories based on ferroelectric tunnel junctions[END_REF] deposited ultrathin strained bto films on La 0.67 Sr 0.33 MnO 3 / /NdGaO 3 with Au/Co top electrodes. These were contacted with a conducting afm tip so that a voltage could be applied. The characterization of the electrical transport showed that it was governed by direct tunneling. Using c-afm and pfm, the group evidenced a concurrent change in the electrical conduction and the pfm signals, a strong sign of the polarization-controlled resistance switching, as shown in figure 1.2.21-c. The authors underline the potential of these solid-state ftjs as memory devices [START_REF] Chanthbouala | Solid-state memories based on ferroelectric tunnel junctions[END_REF].

Polarization control of electronic states

Several groups demonstrated that the electric field originating from the ferroelectric surface charges can be harnessed to induce changes in the electronic properties of the electrodes. This provides an additional path to enhance the resistive switching performance of ftjs. Besides, the ter effect represents a highly sensitive tool to probe field-induced interfacial phase transitions.

Z. Wen et al. [START_REF] Wen | Ferroelectric-field-effect-enhanced electroresistance in metal/ferroelectric/semiconductor tunnel junctions[END_REF] fabricated ftjs in which a 3.4-nm-thick bto film is sandwiched between a metallic Pt electrode and a semiconducting Nb:SrTiO 3 (sto) (Nb doping from 0.1 to 1 %) electrode. They measured very large off/on ratios of more than 10 4 that could be linked to the polarization reversal occurring at similar voltages as confirmed by pfm measurements. These huge resistance changes originate from the electron depletion in the semiconducting electrode when the ferroelectric polarization is pointing away from the interface, as schematized in figure 1.2.22 -a. The resulting increase in the barrier height and width leads to a significant increase of the off resistance. Changing the doping level of the sto allowed for a tuning of the resistance contrast and confirmed the ferroelectric field-effect based mechanism. For this, they inserted an ultrathin (0.8 nm) layer of lcmo at one barrier interface in an epitaxially grown bto-based ftj with lsmo electrodes, shown in figure 1.2.22-b. They chose this La x Ca 1-x MnO 3 composition with x = 0.5, i. e. right on the frontier between a ferromagnetic metallic phase (for x < 0.5) and an antiferromagnetic insulating phase (for x > 0.5) at low temperature. In the ftj, the electric field from the adjacent ferroelectric polarization charges leads to an accumulation or depletion of charge carriers (in this case holes) at the lcmo interface depending on its orientation. This is equivalent to a slight shift in the lcmo composition (x < 0.5 or x > 0.5, respectively) and can trigger a phase transition in the ultrathin film. The group then compared the resistance switching characteristics of these junctions with reference measurements on lsmo/bto/lsmo structures at low temperatures. They found the observed 100-fold increase in ter in the system with an interfacial lcmo layer compatible with a polarization-induced mit.

Additional ab-initio calculations further support this mechanism. L. Jiang et al. [START_REF] Jiang | Tunneling Electroresistance Induced by Interfacial Phase Transitions in Ultrathin Oxide Heterostructures[END_REF] later demonstrated a similar effect using pzt with La x Sr 1-x MnO 3 electrodes. Changing the composition of the electrodes, the junctions exhibited a change in ter from positive to negative values as the polarization switching either triggered an mit or the inverse effect.

The polarization-mediated control of the interfacial electronic states of the electrodes in ftjs also permits the modulation of their magnetic properties [START_REF] Garcia | Ferroelectric Control of Spin Polarization[END_REF]. These artificial multiferroic structures demonstrate both tunneling magnetoresistance (tmr) as well as ter and -most importantly -exhibit a coupling between both order parameters. In view of highly efficient magnetization reversal via an electric field, this has spurred much interest in the last years [START_REF] Pantel | Reversible electrical switching of spin polarization in multiferroic tunnel junctions[END_REF][START_REF] Yin | Enhanced tunnelling electroresistance effect due to a ferroelectrically induced phase transition at a magnetic complex oxide interface[END_REF][START_REF] Garcia | Ferroelectric Control of Spin Polarization[END_REF][START_REF] Hambe | Crossing an Interface: Ferroelectric Control of Tunnel Currents in Magnetic Complex Oxide Heterostructures[END_REF][START_REF] Valencia | Interfaceinduced room-temperature multiferroicity in BaTiO 3[END_REF][START_REF] Yin | Coexistence of tunneling magnetoresistance and electroresistance at room temperature in La 0 .7Sr 0 .3MnO 3 /(Ba, Sr)TiO 3 /La 0 .7Sr 0 .3MnO 3 multiferroic tunnel junctions[END_REF].

Memristive behavior

As seen in section 1.2.1.2, the polarization in ferroelectric films reverses not as a whole but through the nucleation and growth of new domains. This opens new possibilities when these films are incorporated in ftj structures. A. Chanthbouala et al. [START_REF] Chanthbouala | A ferroelectric memristor[END_REF] first exploited the partial switching with bto ultrathin films in Au/Co/BaTiO 3 /La 0.67 Sr 0.33 MnO 3 / / NdGaO 3 ftjs (same structure as in [START_REF] Chanthbouala | Solid-state memories based on ferroelectric tunnel junctions[END_REF]). By applying short voltage pulses of 20 ns and varying amplitude, they succeeded in stabilizing multiple resistance levels (see figure 1.2.23

1 R = S R off + 1 -S R on . (1.2.9)
The presence of intermediate resistance states already qualifies ftjs as memristors. Over and above that, the authors demonstrated that by successive series of constant-amplitude voltage pulses, the resistance could be tuned very finely (figure 1.2.24 ). At any given time, the resistance state is a function of the history of voltage pulses applied. In a certain way, the ftj therefore possesses an intrinsic memory functionality. In this thesis, this property will be investigated in detail in chapter 4 on page 93 and therein proves essential for the use of ftjs as artificial synapses in hardware neural networks.

Fundamental properties of BiFeO 3 -based ferroelectric tunnel junctions

This chapter exhibits the fundamental properties of ferroelectric tunnel junctions (ftjs) based on a new promising material: BiFeO 3 . After introducing the properties of this multifaceted material and the structure of these new ftjs, their resistance switching properties and the link to the ferroelectric polarization configuration are demonstrated. As these are not solely defined by the ferroelectric layer but also by the composition of the structure as a whole, the next part then focuses on the influence of the top and bottom electrodes. Its material choice can have important effects on the static transport properties as well as on the resistance switching. Finally, the impact of the lateral size of the ftjs on the ferroelectric polarization reversal is investigated in the last part of this chapter.

Introduction to BiFeO 3

In the recent years, BiFeO 3 (bfo) has attracted a strong interest as it is the only roomtemperature multiferroic and the material with the highest ferroelectric saturation polarization. Multiferroics exhibit two or more ferroic orders -such as ferromagnetism, ferroelectricity, and ferroelasticity -simultaneously [179,[START_REF] Eerenstein | Multiferroic and magnetoelectric materials[END_REF][START_REF] Cheong | Multiferroics: a magnetic twist for ferroelectricity[END_REF]. 1 bfo combines ferroelectricity and antiferromagnetism with other salient structural and optical properties. Moreover, this perovskite exhibits a coupling between these properties which could be exploited in many innovating applications as well as for more fundamental research purposes [START_REF] Wang | Epitaxial BiFeO 3 Multiferroic Thin Film Heterostructures[END_REF][START_REF] Sando | BiFeO 3 epitaxial thin films and devices: past, present and future[END_REF].

-The chapter image is a derivative of "Grey foliose lichens of Parmelia type on rock" (wikipedia.org/

Properties of BiFeO 3 thin films

In bulk, bfo crystallizes in a rhombohedral structure (figure 2.1.2 -d) with a Curie temperature of T C = 1100 K and a saturation polarization close to 100 µC cm -2 [START_REF] Lebeugle | Very large spontaneous electric polarization in BiFeO 3 single crystals at room temperature and its evolution under cycling fields[END_REF]. Below a Néel temperature of T N = 640 K, it becomes antiferromagnetic, presenting a particular cycloidal order of the magnetic moments.

As discussed in sections 1.2.1.1.2 and 1.2.1.1.3, the deposition of perovskites on substrates with particular lattice parameters can help to stabilize new material phases by epitaxial strain. In the case of bfo, this leads to a rich phase diagram as a function of strain and temperature that is represented in figure 2 Biaxial strain (>+0.6 % or <-1.6 %) produces a change from the cycloidal to a G-type antiferromagnetic order without affecting the structure. Under compressive strain up to -4.2 %, the tetragonality, here expressed as the c/a ratio, increases. However, unlike the case of classical ferroelectrics such as BaTiO 3 where compressive biaxial strain induces an increase of the saturation polarization and of T C [177], in bfo antiferrodistortive rotations of the oxygen octahedra compete and result in a decrease of T C with compressive strain [302]. Between -3 % and -4.2 % strain, thin films of bfo crystallize in a mixed phase of the rhombohedral (R) and tetragonal-like (T) polymorphs. Stronger compression (more than -4.2 % strain) then stabilizes the so-called t-phase. Despite a pronounced tetragonality in which the ion displacement is orientated along the [001] axis, this phase still possesses a monoclinic structure (figure 2.1.2-b,c). Super-tetragonal monoclinic bfo films of Cm symmetry with giant tetragonality, i. e. with high c/a ratios of more than 1.25 (a ≈ 0.377 nm, c ≈ 0.465 nm [START_REF] Béa | Evidence for Room-Temperature Multiferroicity in a Compound with a Giant Axial Ratio[END_REF]), have been obtained when deposited on substrates exerting more than -5 % compressive strain [START_REF] Beekman | Phase Transitions, Phase Coexistence, and Piezoelectric Switching Behavior in Highly Strained BiFeO 3 Films[END_REF]. A purely tetragonal phase can be achieved in very thin, strained films of 4 nm: while exhibiting the same c/a ratio, these films show no sign of oxygen-octahedra tilting and possess a P4mm symmetry (figure 2.1.2-a) [START_REF] Pailloux | Atomic structure and microstructures of supertetragonal multiferroic BiFeO 3 thin films[END_REF]. This t-phase bfo is expected to have a large polarization of up to 150 µC/cm 2 [178, [START_REF] Ricinschi | A mechanism for the 150 µC cm 2 polarization of BiFeO 3 films based on first-principles calculations and new structural data[END_REF][START_REF] Dupé | Competing phases in BiFeO 3 thin films under compressive epitaxial strain[END_REF][START_REF] Hatt | Strain-induced isosymmetric phase transition in BiFeO 3[END_REF], which was partially confirmed by recent experiments [START_REF] Béa | Evidence for Room-Temperature Multiferroicity in a Compound with a Giant Axial Ratio[END_REF][START_REF] Zhang | Microscopic Origin of the Giant Ferroelectric Polarization in Tetragonal-like BiFeO 3[END_REF]. Nonetheless, the exact value is still under discussion [START_REF] Sando | BiFeO 3 epitaxial thin films and devices: past, present and future[END_REF]. As higher polarization promises an enhanced tunnelling electroresistance (ter) effect with high off/on ratios (see section 1.2.2.1.2 on page 30), we were interested in using t-phase bfo as tunnel barrier in a new type of ftj.

.1.1 Figure 2.1.1 . Cycloid 1 Cycloid 2 G-AF G-AF C-AF 2D-AF Bulk Ferroelectric T-phase T+R R-phase -8 -7 -6 -5 -4 -3 -2 -1 0

Fabrication and structural characterization of BiFeO 3 -based ferroelectric tunnel junctions

The main results of this section have been published in "Giant Electroresistance of Supertetragonal BiFeO 3 -Based Ferroelectric Tunnel Junctions", ACS Nano 7, 5385 (2013) [START_REF] Yamada | Giant Electroresistance of Super-tetragonal BiFeO 3 -Based Ferroelectric Tunnel Junctions[END_REF].

In order to stabilize this bfo phase with giant tetragonal distortion, YAlO 3 (yao) was chosen as substrate. With a bulk in-plane parameter of a = 0.371 nm in pseudocubic notation, in close vicinity to that of the t-phase bfo, yao is expected to exert more than 6.5 % strain on the bfo film (in comparison to the bulk bfo phase, see figure 1.2.6). Ultrathin films of 5 %-Mn-doped bfo were grown on (001)YAlO 3 substrates by pulsed laser deposition using a Nd:YAG laser. Mn doping was used in order to reduce the leakage [START_REF] Singh | Room temperature ferroelectric properties of Mn-substituted BiFeO 3 thin films deposited on Pt electrodes using chemical solution deposition[END_REF]. Ca 0.96 Ce 0.04 MnO 3 (ccmo) thin films, that are well lattice-matched with yao and therefore allow for an epitaxial growth of the bfo on top, were used as bottom electrodes. As pure CaMnO 3 (cmo) is insulating, the films were doped with 4 % of Ce by to obtain a metallic electrode [START_REF] Xiang | Strain-Mediated Phase Control and Electrolyte-Gating of Electron-Doped Manganites[END_REF].

The thicknesses of the bfo and ccmo layers were estimated by X-ray reflectivity (xrr) to be 4.6 nm and 21 nm, respectively (figure 2.1.3 -a. From this, the evolution of the c/a ratio close to the bfo/ccmo interface can be extracted, as shown in figure 2.1.4-b. In the ccmo layer, the c/a ratio approaches unity (1.02 ± 0.03) and slightly increases towards the bfo interface. In the bfo film, after 3 unit cells, the c/a ratio reaches very high values of about 1.25 ± 0.03, indicating that the film is in the t-phase with a very high tetragonality. Furthermore, the analysis of the haadf images allows to extract the displacements of the B-site ions (Fe or Mn) relative to the A-site ions (Bi or Ca) as schematized in figure 2.1.4-d. While the ccmo layer does not show any substantial shift in the ion positions, the Fe displacement reaches values of up to 40 ± 15 pm in the bfo film (figure 2.1.4-b). The sign of the displacement indicates that the bfo polarization is poiting toward ccmo. Using the Born effective ion charges from ab-initio calculations [START_REF] Neaton | First-principles study of spontaneous polarization in multiferroic BiFeO 3[END_REF], one can obtain an estimation of the saturation polarization P S corresponding to these displacements [START_REF] Jia | Unit-cell scale mapping of ferroelectricity and tetragonality in epitaxial ultrathin ferroelectric films[END_REF]. This analysis suggests values of around P S ≈ 100 µC cm -2 for the strained t-phase bfo used here [START_REF] Marinova | Depth Profiling Charge Accumulation from a Ferroelectric into a Doped Mott Insulator[END_REF]. Finally, the electron charges screening the ferroelectric polarization of bfo could be visualized (see figure 2.1.4-e) by the shifts of the Mn 3d (L 3 ) edges from electron energy loss spectroscopy (eels).

Solid-state ftjs with diameters ranging from 180 to 660 nm were defined by a single step e-beam lithography process and lift-off of top Pt (10 nm)/Co (10 nm) electrodes grown by sputtering [START_REF] Chanthbouala | Solid-state memories based on ferroelectric tunnel junctions[END_REF] as shown in figure 2.1.5 -a. These Pt/Co/BiFeO 3 /Ca 0.96 Ce 0.04 MnO 3

2) Laboratoire de Physique des Solides, CNRS UMR 8502, Université Paris Sud XI, Orsay, France.

3) NION UltraSTEM200 Cs, equipped with a cold-field emission electron source operated at 100 kV. Details available in [START_REF] Marinova | Depth Profiling Charge Accumulation from a Ferroelectric into a Doped Mott Insulator[END_REF]. junctions were connected electrically by a c-afm tip, see figure 2.1.5-b, to allow for electrical measurements as well as pfm imaging [START_REF] Gruverman | Direct studies of domain switching dynamics in thin film ferroelectric capacitors[END_REF][START_REF] Yang | Nanoscale Observation of Time-Dependent Domain Wall Pinning as the Origin of Polarization Fatigue[END_REF]. Throughout this work, they will be referred to as a-type samples. The next section details the measurement setup.

Setup for combined electrical transport and piezoresponse force microscopy experiments

The ftj top electrodes are contacted by a commercial Si afm tip coated with Cr/Pt (Budget Sensors) while the bottom electrode is fixed to ground level, as shown in figure 2.2.1 pfm imaging of the ferroelectric polarization configuration is possible through the top electrode of the junctions [START_REF] Gruverman | Direct studies of domain switching dynamics in thin film ferroelectric capacitors[END_REF]. pfm experiments were conducted with SR830 lock-in detection. A TTi TG1010 external source was used to apply a sinusoidal alternating current (ac) excitation of 0.6 V peak to peak and a frequency of 35 kHz with a dc offset of 0.2 V. The tip was grounded for pfm experiments.

Polarization and resistance switching characteristics

The main results of this section have been published in "Giant Electroresistance of Supertetragonal BiFeO 3 -Based Ferroelectric Tunnel Junctions", ACS Nano 7, 5385 (2013) [START_REF] Yamada | Giant Electroresistance of Super-tetragonal BiFeO 3 -Based Ferroelectric Tunnel Junctions[END_REF].

This section presents the basic characteristics of the ferroelectric polarization switching and the observed resistance contrast in the bfo-based a-type ftjs presented in section 2.1.2 on page 45. In the underlying experiments, individual ftjs are switched by applying 100-ns write voltage pulses at the top electrodes through the conductive afm tip while the bottom electrodes are connected to ground potential, using the setup presented above. We use pfm to image the ferroelectric domain configurations within the junctions. shows a typical hysteresis curve of resistance measured at low dc bias vs. write voltage pulses (V write ) obtained on a 180-nm-wide capacitor. The device resistance is initially 2 to 3 × 10 9 Ω in the virgin state. Applying negative voltage pulses with increasing amplitude virtually does not change the resistance state of the device until a threshold voltage of V write = -2.5 to -3 V (cycle 1). At this voltage, the resistance drops by five orders of magnitude to 2 to 3 × 10 4 Ω (on state). When V write is then swept up from -3 V to 3 V the resistance is stable up to a threshold voltage of 1.5 V where it starts increasing. The resistance progressively increases further when increasing V write to 3 V and saturates around 3 to 4 × 10 8 Ω in the off state. When sweeping again V write from 3 V to -3 V, the device resistance goes back to the on state at a slightly lower threshold V write of -2 V (cycle 2). Afterwards, it switches back to the off state in the same manner when V write again varies from -3 V to 3 V (cycle 3). The resulting off/on ratio is typically over 10 000 within a write pulse window of ±3 V. Importantly, intermediate resistance states can be stabilized by limiting the maximum V write to 2.5 V (cycle 4) or 1.5 V (cycle 5). This "analog", 4 multistate switching is at the basis of the memristive behavior of ftjs as mentioned in section 1.2.2.2.4 on page 40.

Resistance switching

Polarization switching

In order to understand the resistance variations observed in figure 2.3.1, we used pfm to visualize the ferroelectric domain structure of bfo through the top Pt/Co electrode. Practically, we first switch the ftj to a certain resistance through the application of voltage pulses and subsequently perform pfm. -a shows the piezoresponse phase (top) and amplitude (bottom) signals of a virgin 180-nm junction with a resistance of 1 to 2 × 10 9 Ω (figure 2.3.2-g). The phase is completely bright and the amplitude is strong and mainly homogeneous throughout the whole junction area. This corresponds to a homogeneous "down" polarization pointing towards ccmo (as deduced from the sign of the bias needed for ferroelectric switching). pfm shows identical virgin states for all ftjs with various diameters between 180 nm and 660 nm, with polarization pointing downwards; they all are in a high resistance state with a constant resistance-area product of about 3 to 4 × 10 7 Ω µm 2 . The reason for this systematic polarization orientation in the virgin state might lie in an internal electric field which favors one direction of polarization in these bfo films. One possible source could be the low oxygen pressure during the growth of bfo which might induce a compensation of surface charges by oxygen vacancies [START_REF] Kim | Direct observation of ferroelectric field effect and vacancy-controlled screening at the BiFeO 3 /LaxSr 1-x MnO 3 interface[END_REF] and also influence the polarization of defect dipoles [START_REF] Lee | Flexoelectric Control of Defect Formation in Ferroelectric Epitaxial Thin Films[END_REF]. Other contributions to the internal field are discussed in the following section 2.4 on page 52.

After switching the 180-nm junction to the on state (2 to 3 × 10 4 Ω) with negative voltage pulses, the pfm phase homogeneously shifts by 180°, except for a few regions on the right side of the device (figure 2.3.2-b). The pfm amplitude (bottom panel) is large only on the left side, consistent with a saturated state with polarization pointing up in this part of the junction. On the right side, the amplitude is lower, indicating that this half of the device is in a mixed state with incomplete switching; because the phase is globally dark, it consists of a majority of domains with upward polarization, and pinned nanodomains with downward polarization. Overall, most of the bfo polarization is pointing towards Co in the on state. Analysis of the pfm images yields a relative fraction of down domains (hereinafter designated as S) of only about 1 %.

When the junction resistance is slightly increased (5 × 10 4 Ω) by positive voltage pulses, bright domains nucleate/propagate around the preexisting bright nanodomains on the right side (figure 2.3.2-c, top panel). Accordingly, the amplitude is strongly decreased in this mixed area (figure 2.3.2-c, bottom panel) due to the higher density of domain walls at which the pfm amplitude vanishes and the averaging over several small domains. As the resistance reaches 1 × 10 5 Ω, the bright domains expand (figure 2.3.2-d, top panel) and the amplitude starts to kick in on the right side while decreasing on the left side.

For a resistance of 9 × 10 5 Ω, the down domains in the right area coalesce, as can be concluded from the now homogeneously bright phase (figure 2.3.2-e, top panel). On the left side, bright domains nucleate within the dark domain area. The amplitude of the two opposite domains is similar and only decreases at an intermediate region that resembles a broad domain wall (figure 2.3.2-e, bottom panel). As discussed in section 1.2.1.2.1 on page 22, ferroelectric dws are very narrow, with widths of the order of a nanometer, so that the broad decrease in amplitude observed here is most likely the result of an extended area with mixed polarization 5 and the limited resolution, partly due to the convolution with the afm-tip size.

After further application of positive voltage pulses, the ftj reaches the off state (2 × 10 8 Ω) in which the phase is completely bright with some noise in the left area and the amplitude 5) Which might be interpreted as a super-rough interface [START_REF] Rodriguez | Controlling Polarization Dynamics in a Liquid Environment: From Localized to Macroscopic Switching in Ferroelectrics[END_REF]. is strong except for the same area of the left side (figure 2.3.2-f). These observations suggest that nanodomains (smaller than the tip radius) with upward polarization persist in the off state. Once created (figure 2.3.1, cycle 1), these pinned nanodomains with upward polarization cannot be erased and lead to a lower resistance state in the off state than in the virgin state. Furthermore, they act as nuclei and diminish the negative threshold voltage needed to switch the capacitor to the on state in subsequent cycles (figure 2.3.1, cycle 2).

Connection between resistance and polarization switching

We investigated ten 180-nm junctions in the same way, combining pfm and electrical measurements in various resistance states. The fraction of down domains S is estimated for each pfm image. -a shows a plot of the resistance of these devices which increases as a function of S. The results can be interpreted using a simple model where down and up ferroelectric domains conduct in parallel, as schematized in figure 2.3.3-b and described by equation 1.2.9. The black line in figure 2.3.3-a is calculated with an on resistance of R on = 2.5 × 10 4 Ω (when the polarization points towards Co) and an off resistance of R off = 2.5 × 10 9 Ω (when it points towards ccmo). The experimental data of the ten junctions agree very well with this simple parallel conduction model that does not rely on any further variables. This clearly suggests that resistive switching in these Co/BiFeO 3 /Ca 0.96 Ce 0.04 MnO 3 tunnel junctions results from the nucleation and propagation of up and down ferroelectric domains. Furthermore, this also indicates that in ultrathin t-phase bfo films the conduction of ferroelectric domain walls (see section 1.2.2.1.2 on page 35) can be neglected over the tunneling conduction through domains. Otherwise, the mixed domain states (e. g. figure 2.3.2-c), with a presumably high concentration of dws, should exhibit the lowest resistances and the homogeneously polarized configurations -down or up -should both display higher resistances. This is clearly not the case in these ftjs, where we observe a very large resistance difference depending on the polarization direction. Although we cannot exclude second order contributions from dw conduction, the resistance change clearly is governed by the ferroelectric domain configuration. Conversely, this direct relation between the two variables also allows to use simple resistance measurements to probe the ferroelectric domain configuration. In chapter 4, we make use of this to examine the domain reversal dynamics in fully patterned ftjs under voltage pulses.

Influence of the electrode material

As explained in section 1.2.2.1.2 on page 30, the resistance change in ftjs is mostly interpreted as the result of a modification of the tunnel barrier profile or width upon polarization reversal of the ferroelectric barrier [214, 232-234, 290, 321-323]. In order to explain the resistance contrast, simple electrostatic models consider different screening lengths for the top and bottom electrodes [START_REF] Kohlstedt | Theoretical current-voltage characteristics of ferroelectric tunnel junctions[END_REF][START_REF] Zhuravlev | Giant Electroresistance in Ferroelectric Tunnel Junctions[END_REF][START_REF] Pantel | Electroresistance effects in ferroelectric tunnel barriers[END_REF]. M. Stengel et al. [START_REF] Stengel | Enhancement of ferroelectricity at metal-oxide interfaces[END_REF], however, have performed first-principle calculations that indicate the importance of microscopic atomic bond strengths where ionic relaxations govern the dielectric properties at the metal/ferroelectric interface. Additionally, recent experimental results also indicate that the atomic arrangement at the electrode/ferroelectric interface critically influences the electronic band profile [START_REF] Yamada | Strong Surface-Termination Effect on Electroresistance in Ferroelectric Tunnel Junctions[END_REF].

Various material combinations have been used to realize ftjs, giving rise to very different ter amplitudes (see section 1.2.2.2 on page 36). As one example, fully epitaxial junctions [START_REF] Yin | Enhanced tunnelling electroresistance effect due to a ferroelectrically induced phase transition at a magnetic complex oxide interface[END_REF][START_REF] Hambe | Crossing an Interface: Ferroelectric Control of Tunnel Currents in Magnetic Complex Oxide Heterostructures[END_REF] are realized with ferroelectric tunnel barriers made of highly-strained perovskite oxide layers (e.g. bto, pzt, bfo) and electrodes of strongly-correlated oxides (e. g. Nb:SrTiO 3 , (La,Sr)MnO 3 , (La,Ca)MnO 3 , (Ca,Ce)MnO 3 ). The ter in such systems is convoluted with strong electric-field-induced modifications of the interfacial electronic properties making the understanding of the interplay between tunneling conduction and polarization challenging, as evoked in section 1.2.2.2.3 on page 38. In order to partially reduce this overall complexity, in a first step, we used ordinary metals as electrodes in ftjs in the hope that this would facilitate the interpretation of ter to simpler schemes where only electrostatic modulations of the potential barrier profile are expected. In the second part of this section, we then exchanged the bottom electrode to investigate its effect. Unless noted otherwise, the experiments were conducted using the experimental setup presented in section 2.2 on page 48.

Top electrode

The main results of this section have been published in "Engineering ferroelectric tunnel junctions through potential profile shaping", APL Mater. 3, 061101 (2015) [START_REF] Boyn | Engineering ferroelectric tunnel junctions through potential profile shaping[END_REF].

We investigated the transport properties of ftjs where we only varied the top electrode (te) material. On the basis of the Co/BiFeO 3 /Ca 0.96 Ce 0.04 MnO 3 structure, presented above, we fabricated samples with different metals (W, Co, Ni, or Ir) forming the te, as schematized in figure 2.4.1 . bfo films of 4.6 nm were grown by pulsed laser deposition (same parameters for all samples) on yao substrates in order to stabilize the supertetragonal phase (T-bfo). We finally defined solid-state ftjs by e-beam lithography and lift-off of sputter-deposited tes of W, Co, Ni, and Ir with submicron diameters. The metallic electrodes are 10 nm thick and capped by 10 nm of Pt. Again, this allows us to conduct electrical measurements as well as image the ferroelectric polarization configuration through the electrode by pfm.

All measurements are conducted at room temperature. Current densities were determined from the junction areas measured by afm. -a shows the resistance-area (ra) product in the virgin, high-resistance state averaged over about ten junctions of each electrode metal. We note a clear increase of the ra with the work function of the te material [START_REF] Lide | CRC handbook of chemistry and physics: a ready-reference book of chemical and physical data[END_REF]. In addition, pfm imaging revealed homogeneous downward-pointing polarization in the virgin state of all junctions, as shown in figure 2.4.4 , the current-density-voltage characteristics J(V) of the virgin state are shown for representative junctions of each te. All junctions exhibit highly non-linear conduction with a strong asymmetry regarding the sign of the applied bias: at the same voltage amplitude, the current is much higher at negative than at positive bias. This asymmetry is strongest in the case of the W electrode and the least pronounced for the Ir electrode. Using a combination [328] of direct tunneling and Fowler-Nordheim tunneling,

Influence on electrical transport

J(V) = J DT (V) + p FNT × J FNT (V/d) ,
we could very well describe the experimental results by fitting the J(V) data. J DT (V) and J FNT (V/d) are the standard expressions for the two tunneling mechanisms as given in equations (1.2.3) and (1.2.4) on pages 28 and 29, respectively. Furthermore, d corresponds to the thickness of the bfo thin film that was set to d = 4.6 nm (as determined by xrr; see, e. g., figure 2.1.3-a on page 46) and p FNT is the relative amplitude of the Fowler-Nordheim tunneling contribution. 6 The junction area was fixed according to the dimensions measured by afm.

Fitting the experimental J(V) curves with all other parameters free, we noticed that a highly trapezoidal barrier with a very low barrier height at the bfo/ccmo interface was necessary to reproduce the results. We then made the hypothesis that the bfo/ccmo barrier height ϕ B,1 does not depend on the te and fitted the J(V) with a constant ϕ B,1 = 0.1 eV, leaving the top electrode/bfo interface barrier height ϕ B,2 , the effective electron mass m * , and the relative amplitude of the Fowler-Nordheim tunneling contribution p FNT free. The results are shown in figure 2.4.3 (solid lines). 6) p FNT therefore corresponds to the parameter C from equation 1.2.4. The large variations in current density with the te are essentially due to an increase of ϕ B,2 from W over Co and Ni to Ir, while the effective mass shows little variations with typical values of around m * = 0.2. We note that fitting with a slightly different barrier height of ϕ B,1 = 0.2 eV and/or considering an effective mass of m * = 0.2 has only minor influence on the results and gives rise to very similar conclusions. Furthermore, p FNT decreases as ϕ B,2 increases, suggesting that the contribution from Fowler-Nordheim tunneling vanishes as the average barrier height becomes bigger. The contribution of Fowler-Nordheim tunneling at negative voltage is confirmed by the strong drop of the ra value with voltage for junctions with W and Co electrodes whereas Ni and Ir electrodes lead to a more ohmic behavior (figure 2.4.2-a).

We can understand the changes of the interfacial barrier height ϕ B,2 by the intrinsic physical properties of the different metals: the increasing work functions of W over Co and Ni to Ir give rise to increasing barrier heights that limit Fowler-Nordheim tunneling and increase the resistance of the ftj (figure 2.4.2-b). This observation therefore follows the trend of Anderson's rule and, using his formula (equation 1.2.7 on page 32), we can obtain an estimate for the value of the bfo electron affinity in our system. Apart from the results on Ir which yield values of χ BFO/Ir ≈ 2.7 eV, the data on the three other samples give a consistent electron affinity of around χ BFO/(W,Co,Ni) ≈ 3.2 eV. This is very close to the value of rhombohedral bfo of 3.3 eV [START_REF] Clark | Band gap and Schottky barrier heights of multiferroic BiFeO 3[END_REF]. In spite of this surprisingly good agreement, one has to keep in mind that the metal work functions are, in general, influenced by interface effects that we cannot quantify here. The value of the estimated electron affinity of bfo is therefore to be taken with caution. We observe that the initial off-to-on switching threshold voltage is different depending on the te material and its amplitude seems to increase as a function of the te work function. This would be in agreement with an increase in the built-in electric field E bi = (ϕ B,2ϕ B,1 )/(d e) with the te work function [166,[START_REF] Gerra | Ferroelectricity in Asymmetric Metal-Ferroelectric-Metal Heterostructures: A Combined First-Principles-Phenomenological Approach[END_REF][START_REF] Liu | Effect of a built-in electric field in asymmetric ferroelectric tunnel junctions[END_REF]. As this electric field favors the downward polarization state, the application of higher electric fields is required to switch the polarization toward the te when ϕ B,2 -and therefore also E bi -increases.

Influence on resistance switching

Bottom electrodes of LaNiO 3

The main results of this section have been published in "Millionfold Resistance Change in Ferroelectric Tunnel Junctions Based on Nickelate Electrodes", Adv. Electron. Mater. 2, 1500245 (2016) [START_REF] Bruno | Millionfold Resistance Change in Ferroelectric Tunnel Junctions Based on Nickelate Electrodes[END_REF].

In order to stabilize the bfo t-phase, high in-plane strains are necessary which severely limit the choice of suitable substrates. Moreover, also the bottom electrode has to be chosen accordingly to keep the lattice parameters of the substrate. As an alternative to the ccmo/ /yao structure, we fabricated samples with bottom electrodes of LaNiO 3 (lno) on LaAlO 3 (lao) substrates and characterized them using their static and dynamic properties.

Fabrication and structural characterization

The heterostructures of BiFeO 3 /LaNiO 3 were grown by pulsed laser deposition (pld) on lao (001) substrates using a Nd:YAG laser. As for the samples with ccmo electrodes, the epitaxial growth of the films was controlled in situ by reflection high energy electron diffraction (rheed) and ex situ by afm as well as xrd. The thickness of bfo was varied from 0 to 13 u.c., as afterwards estimated by xrr. The thickness of lno, determined from rheed oscillations and verified by xrr, was kept constant at 10 nm.

xrd patterns of bfo (5 to 13 u.c.) samples, shown in figure 2.4.5 resulting in a large c/a ratio of 1.2 in the bfo film. We note that -at similar thicknessesthe c/a ratio in the bfo here is slightly lower than that of the bfo grown on yao substrates (cf. section 2.1.2 on page 45), which is consistent with the larger in-plane parameter of lao compared with yao. By combining stem-haadf with eels maps for La, Ni, and Fe (figure 2.4.5-c), the interface terminations were identified as LaO-NiO 2 -LaO-AlO 2 and BiO-FeO 2 -LaO-NiO 2 at the lno/lao and bfo/lno interfaces, respectively. Ruddlesden-Popper faults (that correspond to a missing plane of NiO 2 ) are occasionally observed within the lno films (as for example in figure 2.4.5-b), but do not modify the bfo/lno interface sequence. ftjs were structured from these BiFeO 3 /LaNiO 3 / /LaAlO 3 heterostructures in two different geometries. First, matrices of sub-micrometer devices (with diameters ranging from 180 to 660 nm) were fabricated following the same procedure as detailed in section 2.1.2 on page 45. In these a-type ftjs (Pt/Co/BiFeO 3 (9 u.c.)/LaNiO 3 ), the top electrodes are contacted by a conducting afm tip which allows us the comparison between transport properties and ferroelectric domain configuration by pfm of the same device. Second, fully patterned sub-micrometer ftjs (with diameters of 300 to 400 nm) were defined by the complex 5-step lithography process described in section 3.1.1 on page 69. These b-type ftjs with the same electrode materials (Pt/Co/BiFeO 3 (10 u.c.)/LaNiO 3 ) are connected by standard radio-frequency (rf) probes and their electronic transport properties can be probed as a function of temperature in a cryogenic system.

Polarization switching

Before the deposition of the top electrodes, we investigated the physical properties of the bare ultrathin films of bfo by afm. Applying a voltage of 2.2 V to the lno electrode induces a sharp 180°switching of the ferroelectric polarization (dark phase signal in figure 2.4.6-b). We note that the pfm amplitude signal is equally strong for both directions of polarization and only vanishes at the domain wall between the two (figure 2.4.6-c). In order to gain insights into the electronic transport mechanisms occurring in these films, we recorded c-afm resistance maps on different bfo thicknesses under a fixed lno potential of 1 V. The resulting average resistances of four different thicknesses of bfo (0, 2, 6, and 10 u.c.) are displayed in figure 2.4.6-d. As indicated by the error bars, corresponding to the standard deviation, all the films are very homogeneous and show no sign of electrical pinholes. Furthermore, the exponential increase of the resistance as a function of the thickness of the bfo films suggests that the electronic transport is dominated by tunneling through the ferroelectric. In summary, these scanning probe experiments confirm the high quality of the ultrathin bfo films on LaNiO 3 / /LaAlO 3 : they exhibit a well-defined pristine ferroelectric polarization that is switchable by the application of moderate voltages and their local resistance properties are homogeneous and characteristic of tunnel transport.

We then investigated ferroelectric polarization switching under application of voltage pulses in a-type ftjs. Transport measurements on a large set of a-type ftjs with different diameters ranging from 80 to 660 nm indicate that the pristine state with homogeneous downward polarization exhibits a constant resistance-area product of about 2.5 × 10 7 Ω µm 2 , as shown in figure 2.4.6-h. These homogeneous and reproducible transport characteristics of the ftjs reinforce the above-mentioned hypothesis that electrical transport is governed by tunneling through the ultrathin bfo films. We applied 100-ns voltage pulses through the c-afm tip to switch the polarization of the three ftjs in the diagonal line of figure 2.4.6-f. The resulting dark phase signal shows that the ferroelectric polarization of these solid-state ftjs is switchable -as demonstrated for the bare bfo films in figure 2 Polarization switching to this on state is accompanied by a resistance decrease of more than five orders of magnitude (figure 2.4.7-b). Upon application of a voltage pulse of opposite polarity (1 V), the polarization partially switches back to the downward state, leading to a slight increase of the resistance. The resulting intermediate state consists of well-defined up and down domains with equally high amplitudes (Int. in figure 2.4.7-a). We then apply a voltage pulse of 2 V, after which the phase of the off state resembles that of the virgin state (figure 2.4.7-a). However, the pfm amplitude in the bottom part of the ftj is much lower. As in the on state this exact area corresponds to the part with the highest amplitude, this observation suggests the possible creation of defects during the initial virgin-to-on switching. Furthermore, the partly depressed pfm amplitude in the off state is accompanied by a resistance drop of more than two orders of magnitude compared to the virgin state (Irrev. in figure 2.4.7-b). Thus, these ftjs show a clear bipolar switching of the ferroelectric polarization with a correspondingly large ter. However, the application of large electric fields (about 0.4 V/nm based on a simple estimation) seems to create defects in the domains with upward polarization that limit the reversibility of the resistive switching process.

As a consequence, we investigated another, similar ftj under a reduced amplitude window of pulse voltages. Figure 2.4.7-c reveals that, initially, the polarization is also pointing downward. A voltage pulse of -1.7 V then leads to the partial switching of the polarization: upward domains appear at the border of the junction as well as in a small central area. Even though only a minor fraction of the ftj is switched (about 20 % based on the pfm phase image), the virgin-to-on switching results in large ter of 10 4 (figure 2.4.7-d). In a similar way as previously explained in the upper panel of figure 2.4.7, successive voltage pulses of 1 V and 2 V result in the progressive switching of the polarization to an intermediate state and the off state, respectively. However, in contrast to figure 2.4.7-a, the pfm signal in the off state now resembles that of the virgin state, regarding both the phase and amplitude signals. These pfm observations (figure 2.4.7-c) are confirmed by the resistance comparison of the off and virgin states which display a reduced irreversibility (figure 2.4.7-d). Additional switching experiments show no further degradation of the resistance levels of the on and off states. We therefore conclude that moderate voltage pulses can limit the formation of defects during the switching of ftjs.

Temperature dependence of the resistances

In order to further characterize the electronic-transport mechanism and the influence of these defects, we performed temperature-dependent investigations on b-type ftjs. -a shows hysteresis cycles at room temperature and at 80 K on one of the ftjs (junction #1) for which we obtained the highest ter values. As it was the case for a-type ftjs, the resistance measured at 290 K is initially high (3 × 10 9 Ω) and negative voltage pulses induce the switching to the on state (3 × 10 3 Ω). Positive voltage pulses then lead to a progressive increase of the junction resistance until a saturated off state is reached (5 × 10 8 Ω). The irreversible resistance decrease from the virgin to the off state is minimized here, leading to very large off/on ratio (or ter) of more than 10 5 at room temperature. As shown in figure 2.4.8-a, the resistance evolution of junction #1 as a function of the pulse voltages is similar at 80 K. Resistance switching occurs in comparable write voltage windows, whereas the resistance levels of on and off states are slightly lower and higher, respectively, which results in an increase of the off/on ratio to about 3 × 10 6 at 80 K. 4) display a weak temperature variation with a slight increase by a factor of 2.5 from 290 to 80 K. This temperature dependence is in agreement with a direct tunneling transport through the ultrathin bfo film [START_REF] Stratton | Volt-current characteristics for tunneling through insulating films[END_REF][START_REF] Simmons | Generalized Thermal J-V Characteristic for the Electric Tunnel Effect[END_REF]. The on states of junctions #1 and #2 also show very little resistance variations from 290 to 80 K (less than 40 %). If we consider the simple electrostatic models presented in section 1.2.2.1.2 on page 30 that explain the ter by a variation of the average tunnel barrier height, the on state is defined by a lower barrier height than the virgin state. According to R. Stratton and J. G. Simmons, the temperature dependence of the tunnel resistance should then be higher in the on state than in the virgin state. The apparent discrepancy with our experimental results could thus reflect the temperature variation of physical parameters (polarization, screening) that increase the ter as the temperature decreases. Indeed, as the variation of the barrier height depends linearly on the ferroelectric polarization (cf. section 1.2.2.1.2 on page 30), already a small increase of bfo polarization from 290 to 80 K would be sufficient to explain the temperature variations of the virgin and on state resistances. Besides, the screening length asymmetry between the BiFeO 3 /LaNiO 3 and Co/BiFeO 3 interfaces could vary with temperature, which would also lead to an increase of the ter. In addition, figure 2.4.8-b shows typical temperature dependencies of the resistances of the patterned lno bottom electrodes in junctions #1 and #2. We observe a clear metallic behavior with a resistance decrease by a factor of 2.5 from 290 to 80 K. It is therefore possible that the enhanced metallic character of lno at 80 K influences the screening properties at the bfo/lno interface and increases the ter.

Thus, the virgin (with homogeneous downward polarization) and on states of the investigated ftjs display a weak temperature dependence that can be interpreted within a direct tunneling model. The off states of junctions #1 and #2, however, show a more pronounced temperature variation (figure 2.4.8-b) with a resistance increase by a factor of about 17 from 290 to 80 K. This relatively strong temperature dependence could indicate that thermally activated processes -such as defect-mediated inelastic tunneling -contribute to the transport of the off states. As we pointed out for a-type ftjs, the reduced resistance of the off state in comparison to the virgin state is characterized by areas with lower pfm amplitude (figure 2.4.7-a) that could indicate the presence of defects (such as oxygen vacancies). These defects may induce inelastic tunneling processes through chains of localized states in the barrier [START_REF] Yin | Enhanced tunnelling electroresistance effect due to a ferroelectrically induced phase transition at a magnetic complex oxide interface[END_REF][START_REF] Wen | Temperature-dependent tunneling electroresistance in Pt/BaTiO 3 /SrRuO 3 ferroelectric tunnel junctions[END_REF], as formalized by L. Glazman & K. Matveev [START_REF] Glazman | Inelastic Tunneling Through Thin Amorphous Films[END_REF]. Using their model and considering the physical parameters of the barrier, the experimental off-state temperature dependence can be reproduced considering a hopping transport through 2 to 3 states, as shown in figure 2.4.9 ). [START_REF] Glazman | Inelastic Tunneling Through Thin Amorphous Films[END_REF]. n = 0 then corresponds to direct tunneling (dt) [START_REF] Stratton | Volt-current characteristics for tunneling through insulating films[END_REF][START_REF] Simmons | Generalized Thermal J-V Characteristic for the Electric Tunnel Effect[END_REF]. Each data set is normalized to its respective value at T = 80 K. For the calculations shown here, we assumed a value of 1 nm for the radius of the localized states.

In previous investigations by Z. Wen

Conclusion

The experiments with bfo-based ftjs combining different materials as tes shed light on the electronic band profile of the system. Current-voltage transport measurements suggest direct tunneling through a highly-trapezoidal ferroelectric barrier with relatively high and very low barrier heights at the te/bfo and bfo/ccmo interfaces, respectively. The te/bfo barrier height increases with the te work function, leading to higher average tunnel barriers and therefore higher off state resistances. Together with the pronounced off/on resistance ratio for tes with higher work functions, this provides a way to optimize the static characteristics of ftjs. However, our results also show that the higher resistance contrast comes at the cost of deteriorated switching characteristics. Thus, the quest for giant off/on ratios seems to be associated with the increase of the built-in field in these work-function-designed ftjs and imposes a trade-off between static and dynamic properties.

The bfo-based ftjs with lno bottom electrodes show record ter values, reaching higher values than ftjs with ccmo electrodes. On one hand, this is surprising as the measured c/a ratio is smaller for the ftjs with lno bottom electrodes due to the wider in-plane lattice constant of the lao substrate in comparison to the yao substrate used for the standard ftjs. The c/a ratio is linked to the saturation polarization of the bfo thin film and we expect lower values in the case of lao. The resulting ter should thus be reduced. On the other hand, however, lno has a considerably lower resistivity than ccmo which allows to reduce the on-state resistance by more than an order of magnitude. Finally, it seems that this second effect prevails, resulting in a higher ter values for the ftjs with lno bottom electrodes. However, these ftjs also appear to be more prone to the degradation by defects. Apart from the slightly reduced substrate strain, this might also be linked to the different resistivities of the bottom electrodes. The more resistive ccmo layer could act as a voltage divider under high voltage pulse amplitudes and thus protect the bfo from too high electric fields. In the case of the lower resistance of the lno electrode, this protection might be less effective and result in a higher electric fields that create defects. In the future, more investigations are needed to control the interplay between the electric field required to switch the polarization and the associated formation as well as the movement of these defects.

In conclusion, the high ter values measured on bfo-based ftjs with lno bottom electrodes come at the expense of a less reliable switching. However, reducing the on resistance might still be an advantageous route for faster read-operations when the ftj is used as a memristive element for conventional memory or in neuromorphic applications. Besides that, high off/on ratios are crucial in order to realize practically large memristor crossbar arrays (section 1.1.4 on page 12). Complex strategies involving metal-insulator phase transitions or field-induced modulations of carrier concentrations at interfaces should be adopted to push ftj performance (e. g. such as those mentioned in section 1.2.2.2.3 on page 38). The large ter observed in ftjs combining bfo and lno opens exciting opportunities to locally control sharp mits via the ferroelectric field-effect with other rare-earth nickelates [START_REF] Catalan | Progress in perovskite nickelate research[END_REF].

In summary, these results show the large influence of the electrodes on the static and dynamic properties of ftjs. Further future exploration of different material combinations and the investigation of atomic-scale properties at the electrode interfaces will lead to even better understanding and the realization of other auspicious ftjs.

Effects of the lateral junction size

In order to investigate the influence of the lateral ftj size, we performed experiments on atype samples on which differently sized top electrodes have been deposited. Figure 2.5.1 Initially, all ftjs are in the high-resistive state. We then applied 100-ns voltage pulses to the top electrodes and after each measured the dc resistance at a low voltage of V read = -0.2 V. Under increasing pulse amplitudes from 0 to -3.5 V in steps of -0.5 V, the resistances of all ftjs decrease abruptly at pulses of about -2.5 to -3.5 V. These threshold voltages show no clear trend with the ftj diameter. Further application of negative voltage pulses has only little influence on the resistance. From this on state, the junctions are then switched back to the off state under increasing voltage pulses of up to 3.5 V. The resistances are comparable to their initial values for each ftj diameter. By performing a second hysteresis cycle that closely follows the shape of the first one, we can exclude any non-reversible effects. The measurements on a second sample of each ftj diameter show the good reproducibility of the switching cycle.

The off resistance decreases with increasing ftj diameter. Figure 2.5.2 shows the same hysteresis measurements for which we scaled all resistances with the respective area of each ftj. The resulting ra values in the off state of all top-electrode diameters are then very similar (about 5 × 10 7 Ω µm 2 ), confirming, once more, the homogeneous electron transport through the ferroelectric thin film. pfm phase images, presented in figure 2.5.3 -a, further reveal a uniform polarization configuration in the off state which points toward the ccmo bottom electrode (bright domains). As the pfm amplitude signal is high throughout the ftj area, this also underlines that the electrical conduction in the off state of all ftjs is governed by homogeneous tunneling through the bfo barrier. (m²) . With increasing size, the domain configurations display several small, dispersed nuclei throughout the ferroelectric bfo barrier. In the case of the largest ftj with a diameter of 1200 nm, the reversed nuclei with upward-pointing polarization are too small to be clearly noticed in the pfm phase signal (figure 2.5.3-b 5 ). However, the reduced amplitude that is mainly located in the center of the ftj indicates their presence.

In order to extract the fraction of upward-pointing domains 8 in the on states of the different ftjs the pfm phase signal can generally be used. However, its information is only reliable when the accompanying pfm amplitude is sufficiently high. The larger junctions unfortunately impose higher pfm scan speeds that reduce the amplitude. For this reason, the values of 1 -S for the larger junctions are characterized by very high uncertainties. Nonetheless, in first approximation, the absolute area that the domains with up polarization occupy is similar for the on states of all ftjs -independently of the top electrode diameter. The parallel-conduction model then suggests that their resistance values should also be similar, 9 which corresponds closely to what we observed in the resistance hysteresis measurements (cf. figure 2.5.1-b): all samples display on state resistances around R on ≈ 5 × 10 4 Ω. We therefore think that the off-to-on switching is again limited by the bottom-electrode resistance that acts as a voltage divider (also see section 2.4.2 on page 57). Owing to the strong non-linearity in the current-versus-voltage (i-v) characteristics of the tunnel barriers, it is however difficult to conclude its exact value. Moreover, as the electrical contact to the bottom electrode is made by silver paste at the sample border and the ccmo is unpatterned covering the whole substrate, the exact geometry is unknown. In consequence, we cannot give an estimation of its resistance here. Nonetheless, as the geometry of the latter is unchanged for the different ftj sizes, we expect similar values in all cases, which is underlined by the fact that all junctions display very close on state resistances.

Looking further into the pfm images of the ferroelectric domain configurations in the on state, we can make a general observation: the polarization seems to switch first at the borders of the ftjs. This is obvious for the smaller junctions (with diameters of 180 to 630 nm), where the pfm phase clearly indicates a dark contrast at the edges. Nonetheless, in the case of the largest ftj in which only a very small fraction is reversed, we can also identify a small domain at the junction border in the lower right part of the phase image in figure 2.5.3-b 5 . In order to understand these particular domain configurations, we simulated the electric field distribution inside the ferroelectric barrier. We can also estimate the lateral extend of the edge zones with enhanced electric fields. Their width is found comparable to (or even smaller than) the ferroelectric barrier thickness. Considering higher values for the dielectric constant of bfo tends to shrink these zones so that the higher fields are concentrated to even smaller areas. Consequently, as the extend of the higher fields at the top-electrode edges is very small in comparison to the ftj diameters, we do not expect any major difference in the ferroelectric domain switching dynamics of the different junction sizes due to these edge effects.

In conclusion, the switching experiments with differently sized ftjs underline the important role of the resistance of the ccmo bottom electrodes. The tunnel barrier resistance cannot fall significantly below this series resistance which translates to the fact that only a limited fraction of the polarization in the ftj is reversed.

Performances of the ferroelectric memristor

After the investigation of the fundamental properties of the bfo-based ftj in the previous chapter, we now turn towards its switching performance. After the presentation of a new sample design approaching the structures needed for electronic applications, we perform measurements that provide an insight into the switching dynamics of the ferroelectric polarization. In the following, we test the switching endurance properties and finally measure the stability of the polarization through a resistance-retention analysis.

Fully patterned ferroelectric tunnel junctions

The main results of this section have been published in "High-performance ferroelectric memory based on fully patterned tunnel junctions", Appl. Phys. Lett. 104, 052909 (2014) [START_REF] Boyn | High-performance ferroelectric memory based on fully patterned tunnel junctions[END_REF].

The fully patterned b-type ftjs, presented in this section can be contacted by standard rf probes. This allows us to conduct more reliable and longer electrical experiments than on the a-type ftjs used in the previous chapter that were limited by afm tip drift. However, the new structure does not allow pfm imaging anymore.

Sample structure and fabrication

The growth of BiFeO 3 (4.6 nm)/Ca 0.96 Ce 0.04 MnO 3 (20 nm) on yao substrates is done by pld as described previously in section 2.1.2 on page 45. A Mn-doped (5 %) target is again used for the growth of bfo. The fabrication process of the ftjs presented here (figure 3 ) consists of five major steps. During the first step, Pt (90 nm)/Co (5 nm) nanopillars (of 300, 400, and 500 nm in diameter) are defined by conventional e-beam lithography and lift-off on top of the bfo/ccmo/ /yao multilayer (figure 3.1.1-c). The 100-µm-wide bottom -The chapter image is a derivative of "Shrubby Sunburst Lichen" (flickr.com/photos/7147684@N 3/ 98285 549) by Jason Hollinger, used under CC-BY 2.0.

electrode is then patterned by etching the bfo/ccmo layers. In order to planarize and electrically insulate the top from the bottom electrodes of the pillar, a thin photoresist layer is deposited and polymerized. Subsequently, the top of the pillar is uncovered from the photoresist by reactive ion etching. Finally, macroscopic contact pads (shown in figure 3.1.2 ) are defined by optical lithography, followed by Ni (20 nm)/Au (200 nm) deposition and lift-off. The ftj is contacted from the left by a standard rf probe. This configuration allows to minimize reflections before the pulse arrives at the sample. The substrate being transparent, the adhesive tape that keeps the sample in place can be seen shining through.

Characterization statistics

Among the 50 ftjs on the sample, 10 % are either short-circuited or unswitchable. The remaining ftjs show resistance switching between a high resistive off state and a low resistive on state (see example in figure 3.1.3 -a). The off/on ratio is higher than 100 for 86 % and higher than 1000 for 64 % (figure 3.1.3-b). The virgin resistance of these ftjs is high with a mean value of 2 × 10 8 Ω measured at a dc voltage of -0.2 V. The high resistive junctions can be switched to a low resistive on state by successive application of 100-ns voltage pulses down to -2 V. Their on state resistance averages at 1 × 10 5 Ω with a low standard deviation of 3 × 10 4 Ω. Upon successive application of 100-ns voltage pulses with increasing amplitude up to 2.5 V, the resistance increases to finally recover the virgin value in the off state of 2 × 10 8 Ω with a standard deviation of 1.2 × 10 8 Ω.

These statistics and the consistency with our previous results on the a-type samples (see section 2.1.2 on page 45) show the reliability of the ferroelectric switching and the good control of the ftj fabrication process.

Endurance

Endurance describes the ability of an ftj to be switched many times without observing a substantial degradation of its properties. Due to the large potential of ferroelectric capacitors for memory applications, big efforts have already been undertaken in order to understand and minimize the switching endurance limitations of ferroelectrics [339]. All physical phenomena that contribute to the degradation of the switching performance are summarized under the term "polarization fatigue". The variety of different fatigue models that have been proposed so far shows the importance and complexity of the problem.

State of the art

Several studies have identified the apparition of charged defects as a possible source of fatigue. Especially oxygen vacancies are suspected to form under the large electric fields applied for polarization switching. Created during the switching cycles, these defects can act as pinning sites, hindering dw motion and preventing a full polarization reversal. Figure 3.2.1 illustrates this process in a capacitor structure based on 200-nm-thick pzt: charged defects accumulate near one of the electrodes and block dw propagation so that, in the severely fatigued state, the switchable polarization is strongly diminished [START_REF] Yang | Nanoscale Observation of Time-Dependent Domain Wall Pinning as the Origin of Polarization Fatigue[END_REF]. Another proposed fatigue mechanism is charge injection that inhibits nucleation of reversed domains. Also the formation of a non-switching "dead" layer near one of the electrodes could contribute to a degradation of the switching process. Finally, some studies even evidenced changes in the local atomic structure. For a review of fatiguerelated phenomena in thick (>100 nm) ferroelectric films see A. K. For capacitive ferroelectric memories such as ferams, based on ferroelectric films of more than 100 nm thickness, high endurance performances have nowadays been demonstrated: more than 10 14 switching cycles can be reached [343,345,346], which is largely sufficient for many applications [347]. For ftjs with thin ferroelectric films, however, only a few 10 3 cycles have been achieved [START_REF] Chanthbouala | Solid-state memories based on ferroelectric tunnel junctions[END_REF][START_REF] Wen | Ferroelectric-field-effect-enhanced electroresistance in metal/ferroelectric/semiconductor tunnel junctions[END_REF]. 1 These first results have so far only been obtained on bto-based ftjs and investigations on bfo are still missing. In view of its possible use in applications, the cycling potential of bfo-based ftjs thus still has to be demonstrated.

Measurements on standard junctions

The main results of this section have been published in "Giant Electroresistance of Supertetragonal BiFeO 3 -Based Ferroelectric Tunnel Junctions", ACS Nano 7, 5385 (2013) [START_REF] Yamada | Giant Electroresistance of Super-tetragonal BiFeO 3 -Based Ferroelectric Tunnel Junctions[END_REF].

We performed preliminary cycling measurements on standard a-type Pt/Co/BiFeO 3 / Ca 0.96 Ce 0.04 MnO 3 ftjs where the top electrode was contacted by an afm tip. shows fatigue measurements over 2000 cycles on a single device with a diameter of 300 nm. Each cycle consists in applying a pulse of 2.5 V, reading the resistance twice, applying a pulse of -3 V, and reading the resistance twice. Resistance reading is performed at a small dc voltage of -0.2 V. The resistance ratio between the two states stays stable at about 1000, with only slight oscillations of the extreme resistance levels. These endurance results are comparable to those obtained on bto-based ftjs in similar endurance testing conditions based on a scanning probe setup [START_REF] Chanthbouala | Solid-state memories based on ferroelectric tunnel junctions[END_REF]. However, the bfo-based ftjs display a higher resistance contrast during cycling. These experiments are limited by the afm-tip drift due to which the electrical contact is not stable.

Measurements on fully lithographed junctions

The main results of this section have been published in "High-performance ferroelectric memory based on fully patterned tunnel junctions", Appl. Phys. Lett. 104, 052909 (2014) [START_REF] Boyn | High-performance ferroelectric memory based on fully patterned tunnel junctions[END_REF].

Thanks to the fixed electrical contact to the fully lithographed b-type ftjs we were able to use them in much longer endurance measurements. The results are given after a short presentation of the employed measurement protocol.

Measurement technique

During preliminary measurements, we observed that the fully patterned ftjs exhibit significantly improved endurance performance. As the resistance reading operation at dc voltage takes about 2 s, verifying the resistance levels after each pulse is impractical for long experiments. 2 We therefore developed a special experimental procedure to assure stable switching by regular resistance measurements under cycling by voltage pulses, that is shown in figure 3.2.3 . The resistance is measured after a cycling sequence i during which an increasing number n i = 1, 2, 4, 8, . . . of positive and negative voltage pulses is applied. The length of these write pulses is 100 ns and an amplitude of V p = 1.5 V (-1.3 V) is used to set the high (low) resistance state. These pulses do not result in complete switching of the polarization and thus the resistance contrast in this endurance measurement is smaller than the off/on ratio obtained by higher pulse amplitudes. Nonetheless, we found the resistance switching to be most reliable under these slightly weaker pulses. After the cycling sequence, we verify both resistance levels under a dc voltage of V read = -0.2 V. However, we noticed that the ftj is sometimes blocked in the high resistance state where the application of the negative voltage pulse does not lead to resistance switching. In the case of a blocked junction, we discard the preceding train of cycles and automatically apply negative pulses of increasing amplitude (up to V p = -3 V in steps of ∆V = -0.5 V) until the resistance dropped below a defined limit of 10 6 Ω for the on state resistance.

Apply pulse cycles

n i times Verify resistances read read t t V V R ON R OFF i = i+1 n i +1 = 2 n i x i = i+1 n i+1 = 1 Deblocking procedure read t V R ON while R ON > R min R ON ≤ R min V j V j+1 = V j + ΔV i = 0 n i = 1
After this, we proceeded with the above mentioned cycling sequence restarting at n i = 1. This procedure therefore ensures that only cycling sequences without blocked highresistance states are counted for the endurance -at the expense of underestimating the final number of successful endurance cycles. shows a typical example of the endurance measurements. We reached up to 10 6 cycles with a resistance contrast of about two orders of magnitude. As mentioned above, this narrowed resistance difference is due to the reduced pulse voltages which only partly switch the polarization of bfo. This result presents the highest endurance in ftjs demonstrated so far and therefore is an important step towards the realization of memories based on ultrathin ferroelectric layers. During this experiment, the resistance is blocked and unblocked 1039 times, i. e., in average about every 4000 cycles, using the above procedure. Figure 3.2.5 -a shows the accumulated number of blocked states as a function of the cycle count. The data indicate that the blocking frequency increases during cycling and plotting it on a logarithmic scale suggests an exponential evolution. 3 This could be as sign of increased dw pinning by defects (see [START_REF] Yang | Nanoscale Observation of Time-Dependent Domain Wall Pinning as the Origin of Polarization Fatigue[END_REF] and figure 3.2.1). If the voltage pulses applied during cycling create or accumulate defects, an increasing probability for pinning would be expected. After cycling, switching is, indeed, less deterministic, which indicates more and more frequent pinning of domain walls, probably due to creation or rearrangement of defects, such as oxygen vacancies. for deblocking during the endurance measurement. We notice that in most cases a pulse of -1.5 V is sufficient and higher amplitudes are required much less often. A further analysis is shown in figure 3.2.5-c for which we calculated the relative fraction for each minimal deblocking pulse amplitude as a function of the cycling count. From this we can confirm that pulses of -1.5 V are sufficient for the majority of blocked states -as already observed in figure 3.2.5-b. But, what is more, we observe that this minimum deblocking amplitude does not increase during cycling. This indicates that, even if the pinning probability seems to increase, the average pinning strength stays roughly constant. Strong fluctuations of the deblocking amplitudes towards higher cycle numbers confirm that switching becomes less deterministic. Nonetheless, after cycling, the ftjs can still be switched to the initial off and on resistance states. This indicates that the switchable polarization of the t-phase bfo is not affectedin contrast to what was observed by S.-H. Baek et al. [342] and X. Zou et al. [348] in thicker (400 nm, respectively 40 nm) rhombohedral bfo films. Furthermore, even though defects seem to hinder dw motion, they do not contribute to increased leakage currents [348]. Finally, the switching to the low resistance state requires slightly higher voltage pulse amplitudes than before cycling as previously observed by S.-H. Baek et al. [342]. This could be related to an increased internal field by the movement of oxygen vacancies [START_REF] Yang | Nanoscale Observation of Time-Dependent Domain Wall Pinning as the Origin of Polarization Fatigue[END_REF]341,343,348], which destabilizes the on state. A possible means of slowing down the migration of oxygen vacancies and therefore enhancing this result is doping the bfo by La or Nb to reduce their mobility as demonstrated by D. Lee et al. [343].

Results

-0 . 5 V -1 . 0 V -3 . 0 V -2 . 5 V -2 . 0 V -1 . 5 V
In regard to the use of memristors in neuromorphic hardware their endurance performance is of high importance: as anns constantly adapt to new inputs and continue to learn, the conductances of its memristive synapses are in an ever-ongoing evolution. Fatigue phenomena would impact this essential property of bio-inspired anns. The good endurance performance measured with these bfo-based ftjs therefore is one of the key ingredients for their use in hardware anns.

Retention

Once the ftj is set to a certain resistance level, most applications rely on reading the same state any time later. The research and the following development process on ferroelectric memories, such as ferams, had to cope with polarization relaxation [START_REF] Scott | Ferroelectric memories[END_REF][START_REF] Ishiwara | Ferroelectric Random Access Memories: Fundamentals and Applications[END_REF], corresponding to an information loss in these memories. Depending on the exact memory type, the requirements regarding the retention properties vary. For data storage applications, stability must be assured for at least ten years [347]. Here, after a short overview of the possible mechanisms behind polarization relaxation, we examine the retention properties of standard and fully-patterned bfo-based ftjs.

State of the art

The polarization relaxation process in devices based on ferroelectric thin films is mostly driven by internal electrical fields that exist even in the absence of externally applied voltages. These fields can stem from several origins. As mentioned in section 2.4.1.2 on page 56, unequal work functions of the electrodes result in a potential drop across the ferroelectric that corresponds to a built-in electric field favoring one polarization direction and thus destabilizing the other [166]. As another source, A. Gruverman & M. Tanaka found that in presence of an interfacial oxygen vacancy layer electrons can be trapped and their accumulation then also results in an internal bias [START_REF] Gruverman | Polarization retention in SrBi 2 Ta 2 O 9 thin films investigated at nanoscale[END_REF].

In realistic thin film devices, screening of the polarization charges at the interfaces is never complete. 4 The remaining depolarization field drives the polarization into a multidomain state, eventually resulting in a zero net polarization. Additionally, to enable ter in ftjs, the ferroelectric/electrode interfaces necessarily have different polarization screening properties. This compulsorily entails incomplete screening at, at least, one interface. A. K. Tagantsev et al. evidenced another contribution to insufficient polarization screening: the presence of a non-ferroelectric (i. e., non-switching) layer at the interfaces can reduce interfacial screening and increase the depolarization field [START_REF] Tagantsev | Identification of passive layer in ferroelectric thin films from their switching parameters[END_REF].

Measurements on standard junctions

The main results of this section have been published in "Giant Electroresistance of Supertetragonal BiFeO 3 -Based Ferroelectric Tunnel Junctions", ACS Nano 7, 5385 (2013) [START_REF] Yamada | Giant Electroresistance of Super-tetragonal BiFeO 3 -Based Ferroelectric Tunnel Junctions[END_REF].

Retention experiments were performed on an array of four a-type ftjs with a diameter of 180 nm. All the devices were switched to different initial resistance states and imaged with pfm (figure 3.3.1 After an idle period of about three days (68 hours), the same devices were imaged by pfm again and measured electrically (figure 3.3.1-e-h). Note that the apparent distortion of the ftj top electrodes from a regular circular shape is only due to image drifts during pfm 4) Caused by finite screening lengths, as mentioned in section 1.2.1.1.3 on page 20.

imaging, not to effective geometric changes.

Measurements on fully lithographed junctions

The main results of this section have been published in "High-performance ferroelectric memory based on fully patterned tunnel junctions", Appl. Phys. Lett. 104, 052909 (2014) [START_REF] Boyn | High-performance ferroelectric memory based on fully patterned tunnel junctions[END_REF].

The fully lithographed ftjs (b-type samples) allow for monitoring the evolution of the resistance over longer time spans. The measurements presented here therefore enable deeper insights in possible relaxation processes.

Measurement technique

In order to evaluate the retention, we used the following measurement scheme for a representative junction. After initializing the ftj to the on-state resistance, a 100-ns write pulse of variable amplitude is applied. This sets the resistance state for each retention measurement. After each write pulse, the resistance read at a small dc voltage V read = -0.1 V is recorded in increasing time intervals up to 25 h of total measurement time.

Results

The evolution of the resistance for the differently initialized states is shown in figure 3.3.2 . The on and off states, i. e., when the polarization of the bfo is in a nearly saturated configuration, show very good retention. By extrapolation using a power-law dependence, we estimate that the off/on resistance ratio after 10 years would still be more than two orders of magnitude (≈375) on these 300-to 500-nm-wide ftjs. Comparably good retention properties have so far only been obtained on 30-µm-wide bto-based ftjs [START_REF] Wen | Ferroelectric-field-effect-enhanced electroresistance in metal/ferroelectric/semiconductor tunnel junctions[END_REF] so that our devices represent a 100-fold improvement for scalability. The intermediate resistance states (initialized by write pulses of 0.8 to 1.2 V) are less stable and tend to relax towards higher resistances. The strongest relaxation is observed after application of a pulse of 1.1 V, which initially set the resistance to 3 × 10 6 Ω, relaxing to 1 × 10 8 Ω after 25 h. The resistance fluctuations recorded by intermediate measurements indicate particularly unstable polarization configurations for resistance values in the range of 5 × 10 6 to 1 × 10 8 Ω.

.0 V 0.9 V 1.6 V 0.8 V 1.4 V 0.6 V 1.2 V 0.0 V 1.1 V Write pulse
The domain configuration in ferroelectric thin films is governed by two energies: the contribution from the depolarizing field and the domain wall energy -in addition to any configuration-independent electric fields. On the one hand, as the thickness of the ferroelectric layer decreases, the depolarizing field increases, favoring a multidomain polarization, as explained in section 1.2.1.1.3 on page 20. On the other hand, if energy is needed for the creation of dws, the system will tend to lower their length, which is minimized in a homogeneously polarized monodomain state. As the nearly saturated on and off states are more stable than intermediate, multidomain states, it seems that dw energy is significant in our system and outweighs the depolarization energy.

Additionally, the voltage threshold to destabilize the on state is lower than the voltage threshold of the off state (figure 2.3.1 on page 49). This indicates an energetically more favorable downwards polarization state which could result from internal electric fields in the ftj. A possible source of these internal fields has been discussed in section 2.4.1.2 on page 56: the difference in the barrier heights at the ferroelectric/electrode interfaces. The relaxation of intermediate states towards the off state is in agreement with this scenario of a built-in electric field.

Comparing these results on b-type ftjs with those on a-type ftjs, we notice two differences: the relaxation of intermediate and low resistance states towards higher resistance is less pronounced in a-type ftjs and their nearly saturated off state even exhibits a resistance decrease over time. These unequal relaxation tendencies are most likely linked to the different boundary conditions of the bfo film around the top electrode. In a-type ftjs, the surrounding bfo surface is free and might therefore be influenced by adsorbates [START_REF] Kalinin | Local potential and polarization screening on ferroelectric surfaces[END_REF][START_REF] Shin | Atomistic Screening Mechanism of Ferroelectric Surfaces: An In Situ Study of the Polar Phase in Ultrathin BaTiO 3 Films Exposed to H 2 O[END_REF] whereas it is covered by resist in b-type ftjs.

In order to further enhance the retention properties of the intermediate states, the asymmetry of the ftj could be reduced by using other electrode materials. However, in this way, the stability gain would probably come at the expense of a reduced off/on ratio. H. Lu et al. [START_REF] Lu | Enhancement of Ferroelectric Polarization Stability by Interface Engineering[END_REF] proposed a similar route by atomically engineering the interface terminations to enhance polarization stability.

When memristors are used in anns to store the synaptic connection strengths between neurons, spontaneous conduction changes due to relaxation could present a memory loss of the system. The good retention properties in the on and off states of these ftjs promise a stable operation once the ann has adapted to a new input pattern. During the learning process, in which the individual memristors are in intermediate conductance states, a weak relaxation process might even be advantageous. The slow drift towards higher resistances observed here might actually help to suppress noise, to free unneeded as well as redundant memory, and finally to stabilize the ann dynamics.

Switching speed in single-shot transmission measurements

Taking advantage of the ability to establish a stable electrical contact in combination with an improved circuit for rf measurements, we performed single-shot experiments on b-type ftjs. These allow us to gain direct insights into the resistive switching and -through the direct connection to the ferroelectric domain configuration -into the domain dynamics of the ultrathin ferroelectric.

Measurement setup and circuit characterization

In order to measure the transmitted ftj current during the application of a voltage pulse, a special setup was designed. The standard rf probes used below are not suited for transmission measurements. Therefore, the electrical contact to the ftj is made by Al wirebonding and the sample is placed in an Au-covered cavity equipped with SubMiniature version A (sma) connectors. shows a schematic of the measurement circuit. The main components are the same as used before: electrical dc measurements are conducted using a Yokogawa GS610 voltage source in series with a Keithley 6487 picoammeter and an Agilent 81150A pulse generator is used to apply voltage pulses to the ftjs. The output of the pulse generator is coupled to an attenuator (-3 dB) that helps to suppress reflections. A bias-tee (Mini-Circuits ZFBT-6GW+) combines the dc and rf signals. The top electrode of the ftj is connected to its output. An rf tee connector is placed right before and very close (< 2 cm) to the sample. As the ftj resistance is much higher than the impedance of the rf components 5 (50 Ω), the ftj side of the rf tee connector is effectively equivalent to an open circuit which leads to 50 Ω impedance matching at the tee. In this way, reflections should be very limited and we can record the voltage pulse as it occurs directly at the ftj using an oscilloscope with 50 Ω impedance (Agilent DSO9254A). The transmitted signal is split (Mini-Circuits ZFRSC-42) so that one part of the current I dir is directly recorded by the oscilloscope. The second splitter output is fed into a current amplifier (model Femto HCA-400M-5K-C, amplification 5 kV/A, bandwidth 500 MHz) 5) Including the coaxial sma and bnc cables. and this amplified signal I amp is recorded by the oscilloscope in parallel. By this method, we are able to monitor both the low current amplitudes of the ftj in the off state as well as the significantly higher currents that flow during the pulse in case the ftj is in the on state.

There are two different measurement modes: during pulsed single-shot measurements, the oscilloscope input recording the voltage pulse is set to an impedance of 50 Ω. For dc resistance measurements, the coupling of this input is changed to a purely capacitive mode in order to prevent the current from taking this path which presents a significant lower parallel resistance than the ftj. Ground is then made through the oscilloscope whose low impedance virtually has no influence on the measured current amplitude. shows typical current and voltage traces of a non-switching 1-µs voltage pulse with rise-and fall-times of 1.5 ns. The height of the pulse was set to -1.8 V which corresponds to the measured value at the beginning of the recorded trace. 6 However, figure 3.4.2-a reveals a small amplitude decrease of about 0.1 V over the full pulse length. This is due to the capacitance in the bias-tee which limits the maximal pulse length to 1 µs and is also the cause of the small voltage offset (∼ 0.1 V) at the end of the pulse. The direct current signal I dir (t) exhibits a strong noise floor of about 20 µA in amplitude and is at the limit of the oscilloscope resolution. The amplifier allows us to resolve current 6) The signals are corrected taking the attenuation and amplification factors into account.

changes of about 5 µA in the signal I amp (t) but saturates at ∼ 80 µA as it can be seen in figure 3.4.2-b,c. Combining the two signals I comb (t), we obtain a good sensitivity at low amplitudes while still being able to observe higher currents.

A closer look, shown in figure 3.4.2-b,c, reveals strong current peaks at the beginning and the end of the voltage pulse. We are able to reproduce these using a simple equivalent circuit model, shown in figure 3.4.3 -a, where the ftj itself is in parallel to a capacitance C and the overall current is limited by a series resistance R s . Fixing R ftj = R meas -R s , where R meas corresponds to the measured dc resistance, fits to over 300 singleshot measurements yield R s = 4.9 ± 0.2 kΩ and C p = 450 ± 21 fF. 7 Assuming a ccmo resistivity of ρ ccmo = 2 mΩcm [START_REF] Yamada | Ferroelectric control of a Mott insulator[END_REF], we can estimate the bottom electrode resistance to about 5 kΩ. This value agrees very well with that of the series resistance R s in the single-shot measurements. The main contribution to the parasitic capacitance comes from the geometry of the electrodes and contact pads, as sketched in figure 3.4.3-b. They form large parallel plates where they intersect below and above the ftj. The area of this capacitance is about A ≈ 30 µm × 100 µm, separated by 350 nm of resist (SU8 2002) with a relative permittivity of about ε SU8 ≈ 4 [355]. The corresponding capacitance of this configuration amounts to 300 fF -a value that very well matches the one extracted by the single-shot current peak fits. In comparison, the capacitance of the ftj itself is of the order of 50 fF or smaller 8 and thus has only a minor contribution. The temporal resolution of the measurements is limited to about τ rc ≈ 2.2 ns. For faster electrical characterizations, the geometry has to be adapted. On one hand, the ccmo bottom electrode could be made larger to reduce the series resistance. On the other hand, the low-resistive Au top contact pad could be redesigned in order to reduce the parallel-plate configuration. Finally, a larger distance between the two could also help in reducing τ rc .

Although the simple equivalent-circuit model can reproduce the data to a very good agreement, in the following we use a non-switching pulse as a reference. By subtracting the non-switching transient current traces from the data, we suppress all signals that 7) We only excluded obviously erroneous fits perturbed by resistance switching close to the beginning of the pulse. 8) Under the assumption of a relative permittivity ε bfo = 100 similar to the one measured on single-crystal rhombohedral bfo thin films [START_REF] Yakovlev | Interfacial effects on the electrical properties of multiferroic BiFeO 3 /Pt/Si thin film heterostructures[END_REF][START_REF] Lim | Enhanced dielectric properties in single crystal-like BiFeO 3 thin films grown by flux-mediated epitaxy[END_REF].

are not related to polarization reversal [START_REF] Scott | Switching kinetics of lead zirconate titanate submicron thin-film memories[END_REF]. This yields the combined current without reference I comb,∆ (t) which is completely free of any parasitic signals, such as the capacitance-charging peaks, (see figure 3.4.2-b,c) and is used for the following analysis.

Waiting times

For the following measurements, the ftj was initialized to the off or on state before each single-shot recording, respectively. We then investigated the transient current signal during voltage pulses that induced resistance changes in the ftj. shows the transient current responses to voltage pulses of different amplitudes. The ftj is initialized to the off state before each pulse, with dc resistances of R before ≈ 10 8 Ω. After the application of negative voltage pulses the dc resistance drops, reaching values from R after ≈ 10 5 to 10 6 Ω. As a rough general trend, the resistance change increases for larger pulse amplitudes. 9 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 The transient current traces clearly show the resistance switching. At the start of the voltage pulse, the current is very low, indicating that the ftj is still in the off state. After a certain delay, hereafter called waiting time t wait , the current abruptly increases. As the observed timescale of this increase is of the order of the RC constant τ rc ≈ 2.2 ns of the circuit, we are not able to determine the real speed of this transition here. Within the temporal resolution of our setup, the shape of the transition does not depend on the voltage pulse amplitude. After this abrupt increase, the transient current exhibits a slow amplitude decrease on a timescale of about 500 ns. The fact that this decrease shows an exponential shape for all applied voltage pulse amplitudes indicates that it is most likely linked to capacitive effects. The above-mentioned decrease of the voltage pulse amplitude over the pulse length -which occurs on the same timescale -is most certainly at its origin. The more pronounced decrease in the current signal can be explained by the highly non-linear nature of the electrical conduction mechanism, explained in section 2.4.1.1 on page 53: a small change in the applied voltage results in an exponential change of the current.

OFF to ON state switching

-3 0 0 -2 0 0 -1 0 0 0 A m p l i t u d e -2 . 0 5 V -2 . 1 5 V -2 . 3 0 V -2 .
The current amplitude after the switching event is not linked to the final resistance state, however. It increases nearly linearly as a function of the applied voltage which indicates that the current is not limited by the tunneling barrier which would most likely exhibit Fowler-Nordheim tunneling under these high electric fields (see section 1.2.2.1.1 on page 28). Indeed, the series resistance of the ccmo electrodes seems to prevail here and thus does not allow us to conclude on the resistance changes in the ftj itself once the current rises.

We extract the waiting times from pulsed measurements and plot them against the respective voltage pulse amplitudes V p (20 pulses per V p ) in figure 3.4.5 -a. The data show a clear trend: higher voltage amplitudes lead to shorter waiting times. Moreover, the distributions of the values are much broader under lower amplitudes. For |V p | ≤ 2.1 V, the resistance did not change during some pulses. This indicates waiting times longer than the pulse length of 1000 ns so that the mean values are skewed towards lower values. In the opposite case of high amplitudes, we observe that the waiting times do not decrease any further but remain at about 3 to 4 ns. We suppose that this again is most likely linked to the RC constant of the circuit which does not allow any further interpretation at these amplitudes.

-2 . 0 -2 . 5 -3 . 0 -3 . 5 1 1 0 1 0 0 1 0 0 0 W a i t i n g t i m e s M e a n w a i t i n g t i m e s W a i t i n g t i m e ( n s )

V o l t a g e p u l s e a m p l i t u d e ( V )

( a ) -1 . 4 -1 . 6 -1 . 8 -2 . 0 -2 . 2
M e a n w a i t i n g t i m e s F i t s ( M e r z ' s l a w )

( b )

1 / E ( n m / V ) In figure 3.4.5-b, the waiting times are plotted on a logarithmic scale against the reciprocal value of the applied electric field. 10 The waiting times data fall on a single straight line in this representation which is the same as predicted by Merz's law (see section 1.2.1.2.1 on page 22). Fitting the data with equation 1.2.1 under exclusion of the points under maximal and minimal voltage pulse amplitudes (see above), we obtain an activation field of E a = -1.80 ± 0.11 V/nm and a time constant of t 0 = 1.1 ± 0.5 × 10 -11 s. Although the activation field is very large in comparison to the critical fields in bulk ferroelectric, the value obtained here is in the same range as those measured in other ultrathin film samples [START_REF] Kim | Non-Kolmogorov-Avrami-Ishibashi Switching Dynamics in Nanoscale Ferroelectric Capacitors[END_REF][START_REF] Chanthbouala | A ferroelectric memristor[END_REF][START_REF] Pertsev | Coercive field of ultrathin Pb(Zr 0 .52Ti 0 .48)O 3 epitaxial films[END_REF]. N. A. Pertsev et al. [START_REF] Pertsev | Coercive field of ultrathin Pb(Zr 0 .52Ti 0 .48)O 3 epitaxial films[END_REF] evoked the large epitaxial strains of the ferroelectrics in these studies as one possible explanation for the large increase. 11 The highly non-linear i-v relation unfortunately does not allow us to determine the domain configuration, i. e., the fraction of down domains, from the transient current.

Being limited to low voltages in the dc transport characterization (cf. section 2.4.1.1 on page 53), we cannot extrapolate to the high voltage amplitudes applied during the pulses within an acceptable uncertainty range. Moreover, at the high field strengths during the voltage pulse, the resistance of the tunnel junction itself becomes comparable or even negligible in comparison to the series resistance of the ccmo bottom electrode (cf. section 2.4.2 on page 57). In this regime, the current is therefore no longer governed by the ferroelectric polarization configuration.

Nonetheless, the current trace reveals some information about the switching process. The very fast transition from the off to the on state stands in contrast to the long waiting time that precedes the switching. Interpreting these two phases in the framework of the classical nucleation-and-growth theory, we identify the waiting time with the time necessary for a nucleus to form. The fast transition is then associated to the polarization reversal by dw motion. The different timescales on which the nucleation and the growth occur clearly point out that it is the nucleation that limits the reversal dynamics in these ferroelectric ultrathin bfo films. In section 2.4.1.1 on page 53 we take a closer look into the subject combining pfm-based imaging of the domain configuration with electrical transport measurements.

ON to OFF state switching

For the characterization of the on-to-off transient currents, we initially set the ftj to the on state corresponding to dc resistances of R before ≈ 10 5 Ω. -a then shows a typical transient current trace under the application of a rectangular voltage pulse of V p = 1.5 V. The comparison of the dc resistance measurements before and after the pulse reveals that the ftj has switched from the on to the off state. The low amplitude of the transient current signal indicates that the transition occurs at the very beginning of the applied pulse so that no waiting time is observable -in contrast to what we observed during the off-to-on switching (cf. figure 3.4.4). This is true for all voltage pulses with amplitudes in the studied range from 0.5 to 2 V that induced considerable resistance increases. 10) The electric field E is directly calculated from the respective pulse voltage as E = V p /d, where d corresponds to the bfo film thickness of 4.6 nm 11) Also compare to the values obtained in section 4.1.3.1 on page 99. In order to cope with these supposedly very fast dynamics, we adapted the shape of the voltage pulses and applied waveforms with long rise times. Figure 3.4.6-b shows the typical transient current under such a triangular voltage pulse that is defined by the slope r v . We observe that the current initially increases with the applied voltage until it rapidly decreases at a certain moment. This decrease occurs during the rising edge of the triangular voltage pulse and indicates the onset of the resistance switching from the off to the on state. provides an overview of transient current traces that were collected under the application of voltage pulses with different slopes ranging from r v = 4.4 to 41 V/µs. They all exhibit the characteristics described above, so that we are able to identify a waiting time t wait (r v ) in each data set. We observe that higher slopes lead to shorter waiting times. In the case of the higher slopes shown in figure 3.4.7-c,d another finite current amplitude peak occurs around the maximum of the voltage pulse. This peak is most likely caused by Fowler-Nordheim tunneling in the off state. The very large electric fields applied here can lead to these high current densities. -a shows the waiting times extracted from measurements with different triangular pulses (as in figure 3.4.7) plotted against the respective voltage pulse slope. Here, we find confirmation of the trend observed above that higher slopes are linked to shorter waiting times. Before trying to dive into the interpretation of these data, we can also take a different perspective and extract another parameter from the measurements: figure 3.4.8-b shows the applied voltage V th at which the current amplitude starts to decrease and the on-to-off switching begins. We note that this threshold voltage V th varies only weakly as a function of the voltage pulse slope.

In order to understand the dependence of the waiting time with the voltage pulse slope, we start from Merz's law (equation 1. V o l t a g e p u l s e ( V )

r V = 3 1 V / µs ( c ) t ( n s ) C u r r e n t ( µA ) r V = 4 1 V / µs ( d ) t ( n s ) -1 0 1 2 3
V o l t a g e p u l s e ( V ) time t s can then be assumed to follow a Dirac distribution and thus simply reads as:

ϕ(t) = δ t -t s (V p ) = δ t -t 0 exp E a d V p ,
where V p corresponds to the applied voltage that, divided by the thickness d, gives the strength of the applied electric field. In the case of the triangular pulses, where the switching occurs during the rising edge, we can assume a time-dependent pulse voltage V p (t) = r v t. In this way, the switching time can be calculated as the mean value:

t s (r v ) = t ϕ(t) dt = t δ t -t 0 exp E a d V p (t) dt = t δ t -t 0 exp E a d r v t dt .
In order to express the root of the argument of the Dirac function above in terms of t, we have to resort to the Lambert W function [START_REF] Corless | On the LambertW function[END_REF]. With this we obtain an expression for the switching time as a function of the voltage slope r v :

t s (r v ) = t δ t - E a d r v W E a d r v t 0 dt = E a d r v W E a d r v t 0 . (3.4.1)
Although equation 3.4.1 cannot be written in terms of simple analytic functions, we can use it to numerically fit the data in figure 3.4.8-a. Fixing the attempt time to the value obtained for off-to-on switching in figure 3.4.5-b, t 0 ≡ 10 -11 s, we can well reproduce the data with an activation field of E a = 1.83 ± 0.02 V/nm. This value is very close to the one for off-to-on switching. However, the data can also be fitted under fixed values of t 0 from 10 -15 to 10 -10 s where the activation field takes values from 3.9 to 1.3 V/nm as can be seen in figure 3.4.8-a. Due to this large uncertainty, we cannot determine the parameters to a precision that would allow for a reliable comparison. The good match between the experimental data and the transformed Merz's law however suggests that it might also describe the waiting time during on-to-off switching.

In the exemplary transient current traces in figure 3.4.7, we observe that the current change occurs on a significant slower timescale than in the off-to-on switching. While during the latter the current increase takes place of the order of the RC constant (∼3 ns), the current decrease during on-to-off switching is visible for about 20 to 200 ns in the voltage-slope measurement. By taking a closer look back at the on-to-off switching under rectangular pulses (figure 3.4.6-a), we can identify a similar current decrease during the first 100 ns, despite the absence of any measurable waiting time.

These apparently large differences between off-to-on and on-to-off switching might originate from the parallel-conduction mechanism that governs the connection between the ferroelectric polarization configuration and the electrical resistance (cf. section 2.3.3 on page 51). As shown in figure 2.3.3-a, the same percentage change in the domain configuration results in significantly different resistance changes depending on whether the initial state is close to the on or the off state. For example, if half of the ftj area switches during a certain time ∆t, in on-to-off switching the resistance only increases by a factor of 2 whereas, in off-to-on switching, the resistance decreases by several orders of magnitude. As a consequence, the current decreases much slower during onto-off switching than it increases during off-to-on switching. The asymmetry imposed by the parallel-conduction mechanism could therefore explain the seemingly different timescales in the transient current signals for the two switching directions. Due to the tunnel current through the ultrathin ferroelectric bfo in our samples, the ferroelectric displacement current is hidden and can therefore not be used to determine the switched polarization [START_REF] Scott | Switching kinetics of lead zirconate titanate submicron thin-film memories[END_REF][START_REF] Grigoriev | Ultrafast electrical measurements of polarization dynamics in ferroelectric thin-film capacitors[END_REF].

In conclusion, we could directly visualize the ferroelectric switching dynamics in ftjs by these single-shot experiments. Under rectangular pulses, the off-to-on switching exhibits clear waiting times which we interpret as an indication that the polarization reversal is impeded by a first nucleation event. Using long voltage slopes, we could reveal that on-to-off switching follows very similar dynamics. Under pulse amplitudes of more than about 3 V the waiting times decrease below the RC-limited temporal resolution of the measurement setup, indicating very fast switching in under 2 ns. Finally, here again, the highly non-linear i-v characteristics of the ftjs prevent us from investigating the evolution of the ferroelectric domain configuration during the pulse. In order to learn more about the dynamics of the polarization reversal, we therefore performed cumulative reversal measurements that are presented in section 4.1 from page 93 on.

Even if biological systems typically evolve on much slower timescales (cf. section 1.1.2.1.2 on page 5), fast operation speeds are of interest for several reasons. First, prior to the in-situ use of an ann, accelerated training could allow for ready-to-go systems from the start on. Second, the inputs to neuromorphic hardware can be presented much faster than what typical biological systems experience. On the one hand, this could enable faster data treatment and speed up reaction times of the ann. On the other, these high speeds could open up the use of anns for new applications. 12 The highly non-linear dynamics of ftjs furthermore allow to vary their operation speed over a very large range and thus to set the priority either on power consumption or speed.

From ferroelectric domain dynamics to spike-timing-dependent plasticity

This chapter connects the dynamics of the ferroelectric polarization reversal in ultrathin bfo films to the implementation of spike-timing-dependent plasticity as a very important learning rule for unsupervised learning in artificial neural networks (anns) based on ftjs.

Ferroelectric domain dynamics in ultrathin films

The main results of this section have been submitted as a paper (under review).

As shown in section 1.2.1.2 on page 21, many different models have been developed to describe polarization reversal in ferroelectrics. However, investigations into the polarization dynamics of ultrathin films, such as the tunneling barrier in ftjs, are still missing.

Here, we combine pfm imaging with electrical measurements to tackle this issue.

Investigating the reversal by piezoresponse force microscopy and electrical measurements

We investigate the evolution of ferroelectric domains with voltage pulses by pfm. Starting from the on state, we apply voltage pulses of 1 V in amplitude and 100 ns in length to the ftj. Pfm imaging after an increasing number n = 1, 10, 100, . . . of cumulative pulses then provides insights into the nucleation and growth of the polarization reversal process. -a shows the corresponding pfm phase and amplitude images. In the on state, the majority of the bfo film is polarized upward. However, small areas with downward polarization remain visible in the bottom part of the phase image (figure 4.1.1-a, ON ) -The chapter image is a derivative of "Caribou moss" (commons.wikimedia.org/wiki/File:Caribou_ where also the amplitude signal is lower. After the application of one voltage pulse, this region indicates additional small ( 10 nm) domains with downward polarization 1 . It is only after 100 3 to 1000 4 extra pulses -or about 10 -5 to 10 -4 s of accumulated pulse time -that these domains start to coalesce as shown by the increased pfm amplitude. The amplitude signal in this lower region is finally maximized after 10 6 additional pulses 6 , corresponding to about 10 ms of accumulated pulse time. These images reveal that, during the on-to-off switching process, the polarization reversal is dominated by the reversal of small, independent domains rather than by the expansion of existing ones. 1 From each of the pfm phase images, we extract the relative fraction of down (bright) domains S. The evolution of S as a function of the cumulative pulse length is plotted in figure 4.1.1-b (black squares). It reveals that the reversed area continuously increases and finally saturates at a cumulated pulse duration of about 1 ms. We can now make use of the direct connection between the ferroelectric domain configuration and the electrical resistance in these ftjs. The established model of parallel conduction (cf. section 2.3.3 on page 51) allows for extracting the relative domain population from purely electrical transport measurements. Again starting from the on state, we apply cumulative pulses of the same amplitude (1 V) as for the pfm imaging. Thanks to the fast electrical resistance measurements, we can record the resistance progress in much more detail. The evolution of the normalized reversed area S that we calculated from the resistance values of all intermediate states is also shown in figure 4.1.1-b. We notice that these results are in very good agreement with the values that were directly extracted from the pfm images. This means that we can reliably probe the ferroelectric reversal dynamics by purely electrical measurements. Beyond that, this enables us to perform these investigations in much more detail: presents the evolution of the relative reversed area as a function of cumulative pulse time for different voltage pulse amplitudes from 0.5 to 1.5 V. Higher amplitudes lead to significantly faster switching: the reversal 2 takes about 4 × 10 -3 s under pulses of 0.6 V, whereas at 1.2 V it occurs after only about 5 × 10 -8 s.

) Pulse 1.5 V 1.4 V 1.3 V 1.2 V 1.1 V 1.0 V 0.9 V 0.8 V 0.7 V 0.6 V 0.5 V

Molecular dynamics simulations

B. Xu and L. Bellaiche carried out molecular dynamics (md) simulations of the ferroelectric polarization reversal in t-phase bfo at the University of Arkansas 3 in the framework of a collaboration.

They simulated the switching process under constant electric fields using a recently developed scheme that combines ab-initio calculations with molecular dynamics (md) simulations [START_REF] Wang | Atomistic Molecular Dynamic Simulations of Multiferroics[END_REF]. Therein, the total energy is provided by a first-principles-based effective Hamiltonian method [START_REF] Albrecht | Ferromagnetism in multiferroic BiFeO 3 films: A first-principles-based study[END_REF][START_REF] Kornev | Finite-Temperature Properties of Multiferroic BiFeO 3[END_REF] and inserted into an md simulation that simultaneously treats the dynamics of ions and magnetic degrees of freedom. A periodic 48×48×6 supercell (69 120 atoms) of the defect-free, super-tetragonal bfo (P4mm space group) is adopted and the electric field is applied along the pseudocubic [001] direction (opposite to the initial polarization direction). The temperature is kept at 10 K so as to minimize thermal fluctuations and the time step is set to 0.5 fs. -a shows polarization patterns at different times under a constant electric field of 2.5 V/nm. These images illustrate the inhomogeneous switching of polarization in bulk-like super-tetragonal bfo, similar to what is observed by pfm imaging in figure 4.1.1-a. This inhomogeneous 2) Here we refer to the time at which half of the switchable polarization is reversed. 3) Computational Condensed Matter Physics Group, Department of Physics and Institute for Nanoscience and Engineering, University of Arkansas, Fayetteville, Arkansas, USA.

reversal must therefore be intrinsic in nature and does not require the presence of defects in super-tetragonal bfo. Such findings contrast with the case of bulk-like bfo (in the rhombohedral phase) for which a homogeneous switching was predicted [START_REF] Bhattacharjee | Ultrafast Switching of the Electric Polarization and Magnetic Chirality in BiFeO 3 by an Electric Field[END_REF] and linked to large oxygen octahedra tilts that provide a specific homogeneous path for polarization reversal. Here, the reversed area is defined as any polarization that has a magnitude larger than 7 µC cm -2 (it reaches 100 µC cm -2 when being fully switched) in the [001] direction. 4 Simulation results of polarization switching under different electric fields are summarized in figure 4.1.3-c. They exhibit similar trends as the experimental data: larger fields lead to faster switching and a less largely spread reversal process.

Model of polarization reversal in ultrathin ferroelectric films

By combining the results from the experiments (above and in section 3.4.2 on page 85) and from the md simulations, we can gain important insights into the polarization reversal dynamics of these ultrathin bfo films. In the first part of this section, we present a model that describes this process under constant voltage pulse amplitudes. The second part then provides a generalization to arbitrary applied signals.

Nucleation-limited switching

Apart from the highly non-linear dependence of the switching time as a function of the applied voltage, the experimental and the simulation data also show that the reversal is "spread" over longer logarithmic timescales for lower voltages (cf. As suggested by J. Y. Jo et al., we assume a logarithmic Lorentzian shape for the distribution g(log τ) of the nucleation times,

g(log τ) = Γ π 1 (log τ -log τ 0 ) 2 + Γ 2 , (4.1.2)
characterized by its mean value log τ 0 and its half-width at half-maximum Γ. In general, these two parameters both depend on the applied voltage V (or the electric field respectively): τ 0 (V) and Γ(V). After inserting the distribution 4.1.2 into equation 4.1.1 and integrating, we obtain the time dependence of the relative reversed area under an applied voltage V:

S(t) ≈ 1 π arctan log t -log τ 0 (V) Γ(V) + 1 2 . ( 4.1.3)
Using this expression, we can now fit the switching data from the electrical measurements under different voltage pulse amplitudes. The resulting curves can very well describe the experimental data, as shown by the black lines in figure 4.1.4 -a. 6 Figure 4.1.4-b shows the underlying logarithmic Lorentzian distributions of the waiting times for several electric field strengths. We observe that higher fields lead to faster switching and to waiting times that are distributed in a narrower window around the mean value. At lower fields, the waiting time distribution becomes significantly larger and is shifted towards higher time values. The fits on the md simulation data (black lines in figure 4 tors [START_REF] Tagantsev | Non-Kolmogorov-Avrami switching kinetics in ferroelectric thin films[END_REF][START_REF] Jo | Domain Switching Kinetics in Disordered Ferroelectric Thin Films[END_REF]. However, despite the similar results in these two different systems, the origins of the observed dynamics are most likely very different. In the thicker ferroelectric layers, that are polycrystalline, the small nearly independently switching domains are probably separated by grain boundaries 7 and defects may play an important role. In contrast, the ultrathin bfo films used here are grown epitaxially (as verified by various techniques, cf. section 2.1.2 on page 45) and the md simulations indicate an intrinsic origin of the nls mechanism. In the experiments, interfacial effects may furthermore contribute to the large distribution of waiting times [START_REF] Gerra | Surface-Stimulated Nucleation of Reverse Domains in Ferroelectrics[END_REF].

) Pulse 1.5 V 1.4 V 1.3 V 1.2 V 1.1 V 1.0 V 0.9 V 0.8 V 0.7 V 0.6 V 0.5 V (a)
The dependence of the mean switching times τ 0 on the electric field E is shown in figure 4.1.4-c. Due to the idealized nature of bfo in the md simulations (no interface, no defects, no tunnel current), the timescale for polarization switching is much shorter than in experiments while the electric field is larger. Nonetheless, the data from measurements and simulations fall on a unique straight line when plotted on a logarithmic scale against the reciprocal electric field. Fitting this dependence using Merz's law (see equation 1.2.1 on page 24) yields an activation field of E a = 3.23 ± 0.12 V/nm and a time constant of t 0 = 1.1 ± 0.6 × 10 -13 s. The first -although surpassing the bulk critical field by orders of magnitude -is in concordance with the values obtained on other epitaxial ultrathin films [START_REF] Kim | Non-Kolmogorov-Avrami-Ishibashi Switching Dynamics in Nanoscale Ferroelectric Capacitors[END_REF][START_REF] Chanthbouala | A ferroelectric memristor[END_REF][START_REF] Pertsev | Coercive field of ultrathin Pb(Zr 0 .52Ti 0 .48)O 3 epitaxial films[END_REF] and comparable to the value that we obtained in section 3.4.2 on page 85. One possible explanation for the large discrepancy could lay in the large epitaxial strains of the ferroelectrics in these studies [START_REF] Pertsev | Coercive field of ultrathin Pb(Zr 0 .52Ti 0 .48)O 3 epitaxial films[END_REF]. The time constant t 0 corresponds to a frequency f 0 = 8.8 ± 0.5 THz which is of the order of the phonon frequencies in bfo

(∼1 THz) [367].
The value of the time constant found by the fits on the transmission measurement results in section 3.4.2 on page 85 is about two orders of magnitude higher. This can be explained by the fact that the transmission measurements are sensitive to large current changes that -owing to the parallel conduction mechanism -only occur towards the end of the reversal curve. This is especially true for the more reliable fits obtained on off-to-on switching single-shot measurements. We conclude that the cumulative measurement method allows for a much more detailed insight into the domain dynamics.

Generalization to arbitrary voltage signals

The nls model as expressed by equation 4.1.3 describes the full reversal of the ferroelectric domains starting from a uniform state and under constant voltage amplitude. In order to describe the ferroelectric switching dynamics under the influence of arbitrary voltage waveforms, the waveform is numerically divided in short segments of width ∆t (figure 4.1.5 -a). For sufficiently small ∆t, we can then assume a constant applied voltage V i . As the reversal dynamics depends on the present domain configuration and therefore on the value S i-1 before the segment i, we compute the time offset t offset i corresponding to a voltage V i and can only then apply equation 4.1.3 to calculate the value

S i = S(t offset i + ∆t, V i ) (figure 4.1.5-b).
Owing to the highly non-linear reversal dynamics 7) Based on the grain size of 200 nm given by J. Y. Jo et al. [START_REF] Jo | Domain Switching Kinetics in Disordered Ferroelectric Thin Films[END_REF], we can estimate the number of grains in their capacitor structure to about 10 000. The reversal of individual zones is therefore not detectable.

of ftjs where high voltage amplitudes govern the reversal process, the value of ∆t has to be chosen small enough not to smear out these maximal amplitudes. For the results presented in section 4.3.2 on page 110, we chose ∆t ≤ 20 ns.

S i = S(t offset i + ∆t, V i ).

Basics of spike-timing-dependent plasticity

Spike-timing-dependent plasticity (stdp) is a rule that determines how the weight of a synapse in a neural network evolves, i. e., governing the synaptic plasticity. On a larger network scale, this local rule governs the learning process of the system, which is why stdp is also often referred to as a learning rule. As a special form of Hebbian learning [START_REF] Hebb | The organization of behavior: a neuropsychological theory[END_REF], it is based on the principle of causality. Interestingly, this learning rule was proposed based on theoretical considerations [368, 369] and it was only later that its existence has been evidenced in biological systems [START_REF] Markram | Regulation of Synaptic Efficacy by Coincidence of Postsynaptic APs and EPSPs[END_REF][START_REF] Bi | Synaptic Modifications in Cultured Hippocampal Neurons: Dependence on Spike Timing, Synaptic Strength, and Postsynaptic Cell Type[END_REF]370].

In the spike-based communication between neurons, stdp stresses the importance of the relative timing of pre-and post-neuron spikes. One often adapted form of stdp can be resumed as follows: If the pre-neuron spikes and the post-neuron spikes shortly after, this second activity is -with a high probability -caused by the pre-neuron spike. Due to this causal link, the weight of the synapse connecting the two neurons is then strengthened. In contrast, if the pre-neuron spikes shortly after the post-neuron, no causal relationship between the activities can exist. In this case, the synaptic weight is weakened.

Although the idea of synaptic weight adjustment according to the causality of neuron activity was already proposed by Hebb [START_REF] Hebb | The organization of behavior: a neuropsychological theory[END_REF], he did not include the weakening of synaptic connection strengths in his theory. In classical Hebbian learning, the synaptic weights must therefore be normalized on a regular basis in order to prevent an overall average weight increase. This external maintenance process is no longer necessary with stdp as the neural network automatically balances the weights during learning [371].

In the following, we present the most important neuroscientific observations, an overview of the importance of stdp for learning in artificial neural networks (anns) and the stateof-the-art of memristor-based implementations. For a more detailed, biology focused history and the background of stdp, the reader may be referred to the review article by J. Sjöström & W. Gerstner [START_REF] Sjöström | Spike-Timing Dependent Plasticity[END_REF].

Observations in biology

Due to the weak electrical signals in biological systems, experimental evidence of the existence of stdp in neurons was published only after the theoretical predictions [START_REF] Markram | Regulation of Synaptic Efficacy by Coincidence of Postsynaptic APs and EPSPs[END_REF][START_REF] Bi | Synaptic Modifications in Cultured Hippocampal Neurons: Dependence on Spike Timing, Synaptic Strength, and Postsynaptic Cell Type[END_REF]370]. As one of the first, G.-Q. Bi & M.-M. Poo [START_REF] Bi | Synaptic Modifications in Cultured Hippocampal Neurons: Dependence on Spike Timing, Synaptic Strength, and Postsynaptic Cell Type[END_REF] recorded modifications in rat neurons. Their result, shown in figure 4.2.1 , nowadays is often referred to as the exemplary form of the stdp learning rule. Positive time delays correspond to pre-neuron spikes preceding the post-neuron activity and negative time delays correspond to pre-neuron spikes occurring after post-neuron spikes. In the first case, G.-Q. Bi & M.-M. Poo measured an increase of the connection strength (synaptic potentiation), whereas in the second case, they observed a decrease (synaptic depression). For short delays, the changes are very large and their amplitude decreases for increasing time delays (negative and positive). If the spikes occur with a time lag exceeding 80 ms, no modification is observed. This special form of stdp can be approximated by two decreasing exponential functions, a description that is often used for mathematical and simulation purposes. provides an overview of these results. Apart from the purely asymmetric form mentioned above and its inverse version, shown in figure 4.2.2-a,c, stdp can also lead to symmetric weight changes that only depend on the absolute time delay between pre-and post-neuron activity and not on their temporal order, shown in figure 4.2.2-d,e. Figure 4.2.2-b finally shows a mixed form, where arbitrary time delays more likely result in a synapse weakening. This large variety of stdp forms in biological synapses indicates that a unique rule may most likely not fit for all tasks so that a more complex system, such as the brain is based on the interplay of many. The observation of stdp in many different organisms -from insects, over frogs and rats, to humans -suggests that this learning rule is of fundamental importance for the processing of sensory inputs. For more information on the neuroscientific background and the mechanisms involved, the reader is referred to a review of the results on stdp in biological systems by Y. Dan & M.-M. Poo [START_REF] Dan | Spike Timing-Dependent Plasticity of Neural Circuits[END_REF].

Spike-timing-dependent plasticity in artificial neural networks

Many advances concerning stdp have been made by researchers in the field of computational neuroscience. The simulation of neural networks allows for the total control of all parameters regulating the learning and communication processes and therefore to examine the role of each independently and in detail. Many results that are given hereafter have been obtained by S. Thorpe et al. at the Centre de Recherche "Cerveau et Cognition" in Toulouse, France. 8 Traditionally, simulations of artificial neural networks are based on the concept of continuous information flow between neuron nodes, as mentioned in section 1.1.2. However, in 2001, S. Thorpe et al. underlined the importance of single spikes for the communication between different neurons: the popular rate-based concepts cannot account for the response speed to sensory inputs observed in biological systems [START_REF] Thorpe | Spike-based strategies for rapid processing[END_REF]. A visual input can provoke fast reactions on the order of 10 ms, for which rate-based communication would require excessive amounts of energy. Timing-sensitive information transmission would allow single spikes to transmit the same information in a much energy-effective way [START_REF] Rullen | Rate Coding Versus Temporal Order Coding: What the Retinal Ganglion Cells Tell the Visual Cortex[END_REF]. The importance of single spikes then must reflect in the learning rule through which the synaptic strengths in the neural network evolve. As already seen above, stdp is highly sensitive to single spikes.

Performing simulations of a spiking neural network with stdp, S. Song et al. observed that this learning rule can lead to a self-regulating system that does not require any outer reset or normalization procedures [START_REF] Song | Competitive Hebbian learning through spike-timing-dependent synaptic plasticity[END_REF]. Furthermore, stdp intrinsically introduces competition by strengthening synapses that take part in causal firing events, while at the same time suppressing non-causal, noise-induced activity. E. M. Izhikevich later found even complex neural-network dynamics emerge under stdp-based learning -such as the spontaneous self-organization into neuron ensembles and global synchronization phenomena [START_REF] Izhikevich | Polychronization: Computation with Spikes[END_REF].

T. Masquelier & S. J. Thorpe demonstrated the capability of stdp for unsupervised learning [START_REF] Masquelier | Unsupervised Learning of Visual Features through Spike Timing Dependent Plasticity[END_REF]. In their study, they presented a wide range of digital photograph data to a multiple layer spiking neural network. After a certain training period, the network had learned to classify the images itself -without any external control or feedback. The strength of stdp to intrinsically prioritize the most prominent features allows for a very fast recognition where already the first spikes carry the main information and later spikes refine the categorization [START_REF] Guyonneau | Neurons Tune to the Earliest Spikes Through STDP[END_REF]. illustrates how stdp leads to a weight concentration on synapses connecting early spiking pre-neurons.

The early spiking neurons generally respond to the salient features of an input signal (such as, e. g., an image [START_REF] Thorpe | Spike-based strategies for rapid processing[END_REF]). With stdp, one output neuron 9 will automatically specialize on one specific input pattern while signals from another input pattern will excite another output neuron. This learning process runs without any external control and is therefore termed unsupervised learning. The emergence of unsupervised learning under stdp is one of its major interests. 8) More information under cerco.ups-tlse.fr. 9) Or also an ensemble of several neurons. -s) Simple diagram of a single integrate-and-fire neuron with a threshold of 2 connected to five pre-neurons via synapses. At the start, all synapses are initialized to the same weight. (The order of the synapses in relation to the incoming spikes is chosen for readability reasons but does not limit the conclusions drawn here.) The panels show the incoming spikes as well as the evolution of the synaptic weights and of the internal state of the neuron. Before each spike wave, the neuron is reset to zero (b,h,n). (This could also be achieved by neuron leakage.) After only three spike waves, stdp has reinforced the synapses that are connected to early firing pre-neurons and has suppressed late arriving signals. [inspired by Ref. 381] R. P. N. Rao & T. J. Sejnowski even suggested that a neural network could learn to anticipate previously learned inputs with stdp [382]. This fast recognition has also been shown in a computational study for which T. Masquelier et al. used an stdp-based neural network to recognize a repeating spike pattern hidden in an equally dense (or "loud") spike-train background corresponding to a sound signal [START_REF] Masquelier | Spike Timing Dependent Plasticity Finds the Start of Repeating Patterns in Continuous Spike Trains[END_REF]. shows the working principle, that could for example explain the very fast human learning of repetitive audio patterns [START_REF] Agus | Rapid Formation of Robust Auditory Memories: Insights from Noise[END_REF].

A similar simulation has demonstrated unsupervised learning in connection to an eventbased camera [START_REF] Bichler | Extraction of temporally correlated features from dynamic vision sensors with spike-timing-dependent plasticity[END_REF]. stdp seems to be particularly suited to this kind of task, a problem whose solution is extremely computational expensive when using conventional methods. In biological systems, predicting inputs and filtering out signals of a noisy background are typical tasks for audition which is why, already in 1996, W. Gerstner et al. suggested that stdp could be involved [369]. Furthermore, other groups proposed stdp-based input (a) Layout of the simulated ann. All input neurons are linked to a single output neuron and the weights w i of the connecting synapses evolve according to a classical stdp learning curve. In addition to the inputs I i , a common oscillating signal i(t) is applied to all inputs. (b) Input spike trains from a small selection out of 2000 input neurons. The zoomed regions show spikes that correspond to the slightly noisy pattern ( ) and the random activity of other input neurons. The oscillating signal modulates the overall activity. (c) Evolution of the output neuron potential. It generally follows the oscillating input modulation but only spikes ( ) when the pattern is present. [adapted from Ref. [START_REF] Masquelier | Oscillations, Phase-of-Firing Coding, and Spike Timing-Dependent Plasticity: An Efficient Learning Scheme[END_REF][START_REF] Masquelier | The Timing of Vision -How Neural Processing Links to Different Temporal Dynamics[END_REF] processing of different senses such as olfaction [START_REF] Hopfield | Pattern recognition computation using action potential timing for stimulus representation[END_REF] and vision [START_REF] Rullen | Rate Coding Versus Temporal Order Coding: What the Retinal Ganglion Cells Tell the Visual Cortex[END_REF]. 10

Implementation with memristors

This section partially covers a review article published as "Plasticity in memristive devices for spiking neural networks", Front. Neurosci. 9, 51 (2015) [START_REF] Saïghi | Plasticity in memristive devices for spiking neural networks[END_REF].

In 2008, G. Snider proposed a mechanism that would allow the implementation of stdpbased learning in memristor networks [START_REF] Snider | Spike-timing-dependent learning in memristive nanodevices[END_REF]. 11 Thinking in terms of classical, synchronous communication between the different parts of a circuit, they came up with a repeating time-division scheme where rectangular pulses from the pre-and post-neurons would be emitted towards the memristor-based artificial synapse in a certain time slot. In this vein, the voltage across the memristor V mem (t) would correspond to the difference: -a. Through decreasingly large pulse-width modulation of the post-neuron signal, the memristor voltage would then depend on the exact relative timing of pre-and post-neuron signals. On the downside, this implementation of stdp requires clocked, synchronous communication and relies on the relatively complex pulse sequences emitted by cmos neurons. In 2010, S. H. Jo et al. published the first demonstration of stdp with memristors based on Ag atoms in a Si layer [START_REF] Jo | Nanoscale Memristor Device as Synapse in Neuromorphic Systems[END_REF]. For their measurements, they used a scheme similar to the one proposed by G. Snider [START_REF] Snider | Spike-timing-dependent learning in memristive nanodevices[END_REF].

V mem (t) = V pre (t) -V post (t), as sketched in figure 4.2.5
In 2009 already, B. Linares-Barranco & T. Serrano-Gotarredona had presented an alternative method to implement stdp with memristive synapses [START_REF] Linares-Barranco | Memristance can explain spike-time-dependent-plasticity in neural synapses[END_REF]. Their method relies 11) Already in 2007, G. S. Snider had published an ansatz going in the same direction, however, without mentioning stdp at the time [START_REF] Snider | Self-organized computation with unreliable, memristive nanodevices[END_REF].

(a) on specially designed pre-and post-neuron pulse waveforms (see figure 4.2.5-b), resembling the action potentials measured in biological neurons. Similar to the proposal by G. Snider, the voltage across the memristor is determined by the overlap of both waveforms. In contrast, however, due to the asynchronous communication B. Linares-Barranco & T. Serrano-Gotarredona assumed, pre-and post-neuron waveforms can be emitted at any time, t pre and t post , respectively. Then, the delay ∆t = t post -t pre by which the pre-and post-neuron waveforms arrive at the memristive synapse will determine the time-dependent memristor voltage:

V pre t -V post t V mem t ∆t V th (b) V pre t -V post t V mem t ∆t V th
V mem (t, ∆t) = V pre (t) -V post (t, ∆t)
. By choosing appropriate waveforms for pre-and post-neurons, this memristor voltage V mem (t, ∆t) can be tuned in a way that short, negative delays ∆t < 0 result in memristor voltages that decrease its conductivity (corresponding to a depression of the synaptic weight), whereas short, positive delays ∆t > 0 result in memristor voltages that increase its conductivity (corresponding to a potentiation of the synaptic weight). 12 In 2011, the same group showed how by tuning the exact shape of the neuron spike waveform one could obtain very different stdp learning curves [START_REF] Zamarreño-Ramos | On spike-timing-dependent-plasticity, memristive devices, and building a self-learning visual cortex[END_REF]. In order to calculate the conductance change of an idealized memristor-based synapse, they used a simple model of its dynamics under external voltages. One important property of this model is a pronounced non-linearity. C. Zamarreño-Ramos et al. even assumed a threshold voltage amplitude, so that pulses with amplitudes below this value do not lead to any conductance changes in the memristor. shows their results on the stdp responses for several neuron spike waveforms. Using a spike inspired by the action potential measured in biological synapses leads to an stdp curve (see figure 4.2.6-a) that resembles the one measured by G.-Q. Bi & M.-M. Poo [START_REF] Bi | Synaptic Modifications in Cultured Hippocampal Neurons: Dependence on Spike Timing, Synaptic Strength, and Postsynaptic Cell Type[END_REF] (figure 4.2.1). A similar response is predicted under a geometrical variant as shown in figure 4.2.6-b. By using particular spike shapes, the stdp curve can finally be designed to take many differentbiologically observed as well as completely artificial -forms, as shown in figure 4.2.6-c,d. 12) Please note: As already mentioned in respect to the results presented in figure 4.2.2, the time delay ∆t is sometimes inversely defined as ∆t = t pre -t post . Throughout this thesis, we will consistently use the following definition: ∆t = t post -t pre , so that positive delays ∆t > 0 indicate a causal relationship between the pre-and post-neuron spikes whereas negative delays ∆t < 0 indicate a non-causal (i. e., purely coincidental) relationship. Q. Lai et al. were the first to use this method for measurements on memristors with transistor structures incorporating RbAg 4 I 5 in combination with a polymer and controlling the channel conductivity [START_REF] Lai | Ionic/Electronic Hybrid Materials Integrated in a Synaptic Transistor with Signal Processing and Learning Functions[END_REF]. In the following years, many other groups have published stdp experiments on various kinds of memristors -from pcms, over organic materials, to the wide range of resistance switching materials that rely on the movement of ions. provides a temporal overview of these publications as of November 2015. This timeline further indicates the method used to implement stdp and the timescale on which the resistance changes occur in the respective system.

In unipolar memristors, such as those based on pcms and certain kinds of conductivefilament governed devices, the pre-and post-neuron pulses are designed so that their overlap results in the appropriate voltage amplitudes inducing conductance increase respectively decrease. Another possibility to use unipolar memristors is to combine two parallel, inversely connected devices, creating a bipolar memristor in the end [START_REF] Suri | Physical aspects of low power synapses based on phase change memory devices[END_REF]. Finally, M. Suri et al. suggested that even binary resistive memories could act as artificial synapses in a stochastic computing scheme [START_REF] Suri | Bio-Inspired Stochastic Computing Using Binary CBRAM Synapses[END_REF]. In this case, a probabilistic stdp mechanism would be used. D. Querlioz et al. have shown that spiking anns with stdp based on cmos neurons and memristive synapses can be very immune to device variations [START_REF] Querlioz | Simulation of a memristor-based spiking neural network immune to device variations[END_REF]. In their simulations, the ann performed well at recognition tasks even under 20 % variability of the memristor parameters. Overview articles containing more information on ideas to implement spiking cmos neurons for stdp with memristors have been published by T. Serrano-Gotarredona et al. [START_REF] Serrano-Gotarredona | STDP and STDP variations with memristors for spiking neuromorphic learning systems[END_REF] and S. Saïghi et al. [START_REF] Saïghi | Plasticity in memristive devices for spiking neural networks[END_REF].

Spike-timing-dependent plasticity in ferroelectric tunnel junctions

The main results of this section have been submitted as a paper (under review).

In order to implement stdp using ftjs, we make use of the overlapping-waveform mechanism introduced by B. Linares-Barranco & T. Serrano-Gotarredona [START_REF] Linares-Barranco | Memristance can explain spike-time-dependent-plasticity in neural synapses[END_REF] (see above) that presents important advantages: it allows for testing a wide variety of different stdp forms by changing the shape of the pre-and post-neuron waveforms. And furthermore, this way of implementing stdp does not require any computational steps in the neurons and even none in the artificial synapse. The weight (here: conductance) change is solely determined by the overlapping waveforms at the synaptic memristor, the ftj.

Shape of the neuron pulses

The dynamics of the ferroelectric polarization reversal in ftjs are a highly non-linear function of the applied voltage, as we observed in sections 3.4.2 and 4.1 (pages 85 and 93). As a direct consequence, for a given voltage pulse length τ p , there exists a threshold voltage amplitude V th (τ p ) so that pulses of smaller amplitudes do not induce any domain reversal, i. e., do not lead to any resistance change. This can also be observed in the hysteresis cycles where no resistance switching occurs at small pulse voltage amplitudes (cf. figure 2.3.1 on page 49). The existence of these thresholds allows for a straight-forward implementation of stdp in ftjs [START_REF] Linares-Barranco | Memristance can explain spike-time-dependent-plasticity in neural synapses[END_REF].

We emulate the spikes from pre-and post-neurons by the waveforms shown in figure 4 : rectangular 100-ns voltage pulses followed by smooth slopes of opposite polarity [START_REF] Lecerf | Excitatory and Inhibitory Memristive Synapses for Spiking Neural Networks[END_REF]. Importantly, the voltage never exceeds V ± th (100 ns) so that a single spike cannot induce a change in resistance, as shown in figure 4.3.2 When both pre-and post-neuron spikes reach the memristor with a short delay ∆t, their voltages superpose. Figure 4.3.2-b shows the case of ∆t > 0 in which the post-neuron spikes shortly after the pre-neuron. Here, the resulting voltage drop at the memristor V mem = V pre -V post momentarily exceeds the negative threshold voltage V - th . In the opposite case, in which the pre-neuron spikes shortly after the post-neuron (∆t < 0), V mem momentarily exceeds the positive threshold voltage V + th , as shown in figure 4.3.2-c. Owing to the shape of the spike waveforms and more specifically to their long slopes, the voltage amplitude at the memristor depends on the exact value of ∆t: it is maximal for small delays and falls off as the delay increases. th , leading to a decrease of the memristor conductance. (Please note that for the sake of clarity, the post-neuron spike voltages are displayed as -V post , i. e., relative to the memristor polarity.)

V pre t -V post t V mem t V + th V - th (a) |∆t| > τ p (b) ∆t > 0 (c) ∆t < 0

Spike-timing-dependent plasticity measurements

The measurements were conducted on fully patterned 13 ftjs with diameters of 400 to 500 nm using similar equipment as mentioned above. The pulses were applied through an Agilent 33250A arbitrary waveform generator (awg) and a Keithley 6487 source-measure unit (smu) was employed for dc resistance measurements. The two signals were combined by a bias-tee of the same type as used before. To allow for a simpler setup, we directly generated the combined waveform for a given delay ∆t in the awg. In this way, we can dispense with using a transmission circuit wiring. 14 shows the stdp curve obtained by applying the pre-and post-neuron spikes presented above. Negative delays (∆t < 0) lead to conductance decreases, whereas positive delays (∆t > 0) lead to conductance increases. Furthermore, the amplitude of the conductance change decreases as the pre-and post-neuron spikes are further apart so that only closely timed spikes produce a conductance change whereas long delays leave the device unchanged. The general shape of this stdp curve resembles the one measured on biological synapses by G.-Q. Bi & M.-M. Poo [START_REF] Bi | Synaptic Modifications in Cultured Hippocampal Neurons: Dependence on Spike Timing, Synaptic Strength, and Postsynaptic Cell Type[END_REF] (shown in figure 4.2.1). It presents the classical case of an stdp rule that strengthens causally linked neurons and weakens connections with non-causal activity. We can now make use of the direct relationship between S and R in these ftjs (section 2.3.3 on page 51) to describe the switching under single voltage pulses of different amplitudes and even the complicated waveforms that occur during stdp measurements. shows the stdp measurement results and the corresponding fits for three different junctions using the same technique (parallel fitting of hysteresis and stdp measurement data). The observed variability between the ftjs is of the order of the intrinsic variability of each device. This underlines the reliable stdp behavior between different junctions. The parameter set given by τ 0 (V) and Γ(V) constitutes a full characterization of the device and now allows us to predict the ftj's conductance variations under new, unexplored excitations. Two possible modifications of the standard pre-and post-neuron stdp waveforms are shown in the top panels of figure 4.3.6 . We can now use the individual ftj characterizations obtained before to predict the conductance variations under these new spike shapes (lines in figure 4.3.6). As expected [START_REF] Zamarreño-Ramos | On spike-timing-dependent-plasticity, memristive devices, and building a self-learning visual cortex[END_REF], they result in two new stdp curves: the one shown in figure 4.3.6-a is biologically realistic (cf. figure 4.2.2-e on page 102), whereas figure 4.3.6-b is an example of how to design and implement new learning curves with ftjs artificially. We then measure the conductance changes under the modified pre-and post-neuron waveforms in two different ftjs and figure 4.3.6 shows these experimental data alongside the predictions. The good agreement between the anticipated stdp curves and the measured data confirms that the dynamic response of these ftjs to new signals can be reliable predicted using the nls model. This constitutes an important step for the use of ftjs as artificial synapses in anns. The combination of the two models connecting the ferroelectric domain dynamics and the ftj resistance provides the basis for the realization of large scale systems. Only solid, physics-based memristor models, such as the one presented here, allow for the reliable simulation of the networks that is needed in their design process. Finally, this enables the realization of predictable anns endowed with unsupervised learning.

Outlook: combination with CMOS neurons

In a next step towards a neuromorphic computer, the ftj-based artificial synapses have to be combined with cmos neurons. This has been the goal of the Memristive hardware artificial neural networks (mhann) project. 15 S. Saïghi et al. at the University of Bordeaux have designed a cmos chip adapted to be used with ftjs [START_REF] Lecerf | Silicon neuron dedicated to memristive spiking neural networks[END_REF][START_REF] Lecerf | Développement d'un Réseau de Neurones Impulsionnels sur Silicium à Synapses Memristives[END_REF]. The chip consists of spiking lif neurons. Integration of incoming spikes is achieved by a capacitance from which a small leakage current can drain away through a connected resistance. When a neuron spikes, the accumulated charges on the capacitor are reset. A current conveyor ensures impedance matching for the spiked neuron communication. Spikes are generated by a circuit that allows slight adaptions of the shape based on the prototypical waveform presented in figure 4.3.1. -a shows the packaged circuit that can be mounted on a test board connecting it to ftj-based synapses.

Artificial neural networks require a very high inter-neuron connectivity with up to 10 000 synapses per neuron. In order to achieve this in hardware implementations with memristive synapses, the circuit architecture plays an important role. Lately, the crossbar structure 16 has emerged as a highly promising candidate to place artificial synapses as densely as possible on a wafer [START_REF] Likharev | Neuromorphic CMOL circuits[END_REF]412,[START_REF] Afifi | Implementation of biologically plausible spiking neural network models on the memristor crossbar-based CMOS/nano circuits[END_REF]. This is achieved by arranging the devices in a matrix (or array) form at the crossing points of conducting lines (or "bars"): metallic columns beneath and rows on top serve as electrodes, as shown in figure 4.4.2 . Here, the electrical bottom (top) contacts are shared by all units of a column (row) which significantly reduces their space footprint. Nonetheless, each device can still be addressed individually by applying a voltage to its row and grounding its column, leaving the other electrode lines floating. Furthermore, using the same read, write, and addressing circuits for all devices, further space savings are achieved. 15) More information under www.agence-nationale-recherche.fr/?Project=ANR-11-NANO-2. Carried out by partners at the IMS laboratory at the University of Bordeaux, the French National Institute for computer science and applied mathematics (Inria Saclay), and teams at Thales Research and Technology as well as the Unité Mixte de Physique CNRS/Thales. 16) In the following the term "crossbar architecture" is used synonymously. One important issue faced in realizing resistive crossbars is the sneak-path problem. Current may flow through unselected cells making a deterministic programming difficult. Of the various solutions that have been proposed [347,[START_REF] Zidan | Memristor-based memory: The sneak paths problem and solutions[END_REF][START_REF] Xu | Overcoming the challenges of crossbar resistive memory architectures[END_REF], highly non-linear i-v curves and switching dynamics of the memory (or memristor) element are a promising route. ftjs are endowed with those two characteristics intrinsically which is a big advantage for their high-density implementation. First products for classical memory applications were presented recently -"3D RAM" by Crossbar Inc. [START_REF] Jo | 3D-stackable crossbar resistive memory based on Field Assisted Superlinear Threshold (FAST) selector[END_REF]417] and "3D XPoint" by Micron and Intel [418] -after about ten years of intensifying research [START_REF] Zhao | Nanotube devices based crossbar architecture: toward neuromorphic computing[END_REF][START_REF] Kim | A Functional Hybrid Memristor Crossbar-Array/CMOS System for Data Storage and Neuromorphic Applications[END_REF][419][START_REF] Jo | High-Density Crossbar Arrays Based on a Si Memristive System[END_REF][START_REF] Kau | A stackable cross point Phase Change Memory[END_REF][START_REF] Chang | Oxide Double-Layer Nanocrossbar for Ultrahigh-Density Bipolar Resistive Memory[END_REF][START_REF] Kim | 32 × 32 Crossbar Array Resistive Memory Composed of a Stacked Schottky Diode and Unipolar Resistive Memory[END_REF]. First characterization measurements revealed reliable switching with only little contributions from sneak paths [START_REF] Yépez | Study of ferroelectric memristive nanodevices for neuromorphic applications[END_REF]. Nonetheless, the resistance of the ccmo bottom electrodes revealed to be high in comparison to the on state resistance. This effectively can act as a voltage divider so that higher pulse amplitudes are necessary for switching. In a new design, the deposition of a metallic layer on top of the free-lying parts of the ccmo lines might solve this problem.

Conclusions and outlook

This final chapter briefly summarizes the main outcomes of this thesis and provides some perspectives on possible future paths.

Conclusions

In this work, we investigated the static and dynamic properties of ftjs based on ultrathin films of BiFeO 3 (bfo) in view of their use as memristive synapses in neuromorphic hardware applications. These solid-state devices display non-volatile resistance switching under nanosecond voltage pulses with off/on ratios of up to 10 4 . Combining imaging of the spatial polarization distribution by pfm with investigations of the static electrical transport properties, we could establish the direct link between the ferroelectric domain configuration and the conduction mechanism in these BiFeO 3 -based junctions: mixed domain states correspond to intermediate resistance values that can be finely tuned by adapting the voltage pulse amplitude, length, and number. It is this analog switching that qualifies ftjs as memristors and makes them interesting for bio-inspired computing circuits such as anns.

On this basis, we started to further investigate the physics of ftjs. Varying the topelectrode material, we found that the tunnel-barrier profile crucially depends on the electrode work function. In principle, this allows to engineer the static and dynamic properties of an ftj by the choice of the electrode composition. However, maximizing the off/on ratio also leads to a strong asymmetry and a built-in field that destabilizes the ferroelectric polarization and impairs the switching process. As a result, we found that Co provides the best compromise between a high off/on ratio and a reliable switching.

Changing the bottom electrode from ccmo to LaNiO 3 allowed us to considerably lower -The chapter image is a derivative of "Unidentified lichen, Skeleton Coast, Namibia" (flickr.com/photos/ 42244964@N 3/39978 6673) by Frank Vassen, used under CC-BY 2.0.

the on-state resistance and therefore led to record off/on-ratio values. The reduced electrode resistance though probably also led to such high electric fields in the ferroelectric barrier that defects could form. Once they are present, they seem to impede the polarization switching and also lower the resistance in the off state by providing a conduction path through hopping. By analyzing the switching in ftjs of different sizes, we could affirm that electrical conduction is dominated by tunneling through the domains. Furthermore, these experiments also highlighted the critical role of the ccmo bottom-electrode resistance in the off-to-on switching process.

The fabrication of fully patterned ftjs with good yield characteristics is a crucial step to their integration in applications. In addition, we could demonstrate a high cycling endurance for these fully patterned devices which is important for the use of ftjs as memristive synapses. Measuring the resistance stability over time revealed a good retention in the extreme on and off states that are associated to nearly homogeneous ferroelectric domain configurations. Intermediate states, however, displayed a relaxation towards the off-state resistance that could be the consequence of a high energy cost for domain walls in the ferroelectric barrier. Furthermore, the direction of the presumed built-in field matches with the relaxation trend.

Experiments on a much faster time-scale provided first insights into the polarization reversal dynamics: using time-resolved single-shot measurements, we found that the ferroelectric switching can be preceded by a waiting time. We suppose that this initial inhibition period is linked to the time needed for a first reversed nucleus to appear. Under sufficiently large voltages, however, the waiting time disappears and the switching speed is only limited by the measurement setup, suggesting very fast switching of ferroelectric memristors in less than 3 ns. Ultimately, the strongly non-linear tunneling transport makes a detailed investigation of the whole switching process by transmission measurements difficult. This is why, in the following, we resorted to cumulative switching measurements in which the resistance is read after the application of several voltage pulses. Imaging of the evolution of the ferroelectric domain configuration revealed that the reversal is dominated by the nucleation of many small, independent domains. Exploiting the direct link between the ferroelectric domain configuration and the electrical transport properties in ftjs, we then investigated the influence of the applied voltage on the dynamics. These data could be well described by a nucleation-limited switching model -in very good accordance with the observations from pfm imaging.

Finally, we demonstrated that the resistance of these ftjs can evolve according to stdp, a learning rule that is conjectured to be at the heart of unsupervised learning in anns. Our measurements yielded reproducible results in a same ftj as well as in comparison to other ftjs. Combining the afore-established model of the ferroelectric reversal dynamics with the connection between the domain configuration and the resistance, we could fit our data and then reliably predict the stdp curves under new, unexploited excitations. This constitutes a very important step towards the realization of large-scale artificial neural networks.

Outlook

The present work shows the strengths of ftj-based memristors and of their potential use in bio-inspired neuromorphic hardware. In this section, we therefore provide a short outlook on the next possible steps, both concerning fundamental physics and in regard to their applications.

Ferroelectric tunnel junctions

Although the basic principles of the resistance-switching mechanism and the ferroelectric domain dynamics in ftjs are understood, systems with ultrathin ferroelectric layers still provide a huge playground for research, just as in view of potential applications.

The electric field generated by polarization charges at a ferroelectric/insulator interface could be used to drive metal-insulator phase transitions and thus to realize Mott transistors with very high off/on ratios [425]. Although first results on VO 2 were auspicious [116], the observed electronic phase transition was later linked to oxygen-ion migration [117]. The fast and reliable switching characteristics of bfo films on ccmo electrodes opens a new promising possibility that will be investigated in the French anr project ferromon [START_REF] Garcia | Projet FERROMON (Contrôle ferroélectrique d'un isolant de Mott aux échelles nanométrique/nanoseconde)[END_REF]. ccmo stands out due to its very rich phase diagram that can be explored by electrostatic doping. The additional contribution of an electric field could therefore lead to sharp metal-insulator phase transitions and even change the ferro-/antiferromagnetic order. Complementary advanced spectroscopy investigations of the interface cross section may provide new insights into the physical mechanisms contribute to the fundamental understanding of the interplay between ferroelectricity and electronic as well as magnetic properties at BiFeO 3 /Ca 0.96 Ce 0.04 MnO 3 interfaces.

In the case of ftjs with LaNiO 3 electrodes further investigations might be necessary to elucidate the mechanisms behind the defect formation. Then, the substantially lower resistances of LaNiO 3 electrodes will enable even higher off/on ratios and -all the more important -will lead to higher on state currents that are required for fast read operations [347].

A crucial next step after the demonstration of fully patterned ftjs will be their integration on Si to make them compatible with conventional cmos techniques. In order to grow these epitaxial ultrathin perovskite films and to preserve their ferroelectric character, oxide buffer layers, such as SrTiO 3 , are typically used. Some groups have already demonstrated the ferroelectric character of BaTiO 3 [START_REF] Abel | A strong electro-optically active lead-free ferroelectric integrated on silicon[END_REF][START_REF] Abel | Controlling tetragonality and crystalline orientation in BaTiO 3 nano-layers grown on Si[END_REF] as well as of bfo on Si [START_REF] Coll | Nanocrystalline Ferroelectric BiFeO 3 Thin Films by Low-Temperature Atomic Layer Deposition[END_REF] The intrinsic nucleation-limited switching that we discovered in the super-tetragonal bfo films allows for a fine control of the polarization configuration and therefore of the resistance state in an ftj. In magnetic multilayer systems, this is enabled by the controlled movement of individual domain walls [113,[START_REF] Grollier | Switching a spin valve back and forth by current-induced domain wall motion[END_REF]. While first results demonstrate the controlled nucleation and manipulation of single domain walls in ferroelectric thin films [START_REF] Whyte | Sequential injection of domain walls into ferroelectrics at different bias voltages: Paving the way for "domain wall memristors[END_REF][START_REF] Mcgilly | Controlling domain wall motion in ferroelectric thin films[END_REF][START_REF] Mcgilly | Velocity Control of 180°Domain Walls in Ferroelectric Thin Films by Electrode Modification[END_REF], more research in this direction might once combine the deterministic control of the polarization configuration with the high off/on ratios obtained in ftjs. By tuning the energy of ferroelectric domain walls in bfo, the reversal might be changed to favor propagation over nucleation. This then opens many possibilities to investigate the different dynamic properties of polarization switching.

Neuromorphic hardware with memristors

After several demonstrations of supervised learning in small memristor crossbars and stdp in individual devices, the next important step will be the realization of an autonomously learning memristor-based ann. This will not only require the fabrication of sufficiently large, fully functional crossbars but also the design and implementation of specially designed neuron circuits -that will certainly continue to rely on conventional cmos technology for some time. Several recent research projects 1 aim at finalizing first demonstrators and even commercial concepts of neuromorphic circuits with memristors have surfaced [START_REF] Nugent | AHaH Computing-From Metastable Switches to Attractors to Machine Learning[END_REF].

The learning rules that govern the evolution of the memristive synaptic weights may profit from new insights into biological processes. Specialized rules might enable much higher performances for certain tasks. In future anns, stdp might be extended to a triplet rule [START_REF] Pfister | Triplets of Spikes in a Model of Spike Timing-Dependent Plasticity[END_REF] and also the spike rates seem to have a certain importance for efficient learning in the brain [START_REF] Sjöström | Rate, Timing, and Cooperativity Jointly Determine Cortical Synaptic Plasticity[END_REF][START_REF] Senn | An Algorithm for Modifying Neurotransmitter Release Probability Based on Pre-and Postsynaptic Spike Timing[END_REF]. Finally, memristors might be used to realize electronic dynamical systems, such as reservoir computing circuits, that are conjectured to excel at real-time applications (e. g., speech recognition tasks) and could also profit from stdp [START_REF] Schrauwen | An overview of reservoir computing: theory, applications and implementations[END_REF][START_REF] Paugam-Moisy | Delay learning and polychronization for reservoir computing[END_REF].

At the current pace of research and development, it seems certain to say that neuromorphic hardware will eventually be used to solve tasks that appear insurmountable with current technology. Some of these applications hold a promising potential to improve, e. g., our knowledge on medicine. Apart from these undebatably beneficial use cases, others might still require fundamental discussions and considerations [START_REF] Russell | Research priorities for robust and beneficial artificial intelligence[END_REF]. ).

Voltage

v Current i Flux ϕ Charge q d ϕ = v d t d q = i d t Capacitor dq = C dv Resistor dv = R di Inductor dϕ = L di Memristor dϕ = M dq Figure A.

A.2.1.1.3 État actuel des réseaux de neurones basés sur des memristors

Déjà avant l'essor des memristors, il y avait des idées de les utiliser pour la réalisation de circuits neuromorphiques [START_REF] Snider | Self-organized computation with unreliable, memristive nanodevices[END_REF][START_REF] Likharev | Electronics Below 10 nm[END_REF]. Depuis les simulations prometteuses d'un réseau de neurones avec des synapses memristives en 2011 [START_REF] Querlioz | Simulation of a memristor-based spiking neural network immune to device variations[END_REF] et une première démonstration expérimentale -bien que encore très basique -en 2012 [129], plusieurs groupes de recherche ont publié des avancées dans ce domaine. Afin d'obtenir des hautes densités de memristors, ceux-ci sont généralement arrangés dans des structures matricielles (anglais : « crossbar »). F. Alibart et al. ont utilisé des memristors de TiO 2 pour réaliser un réseau contrôlé de l'extérieur [START_REF] Alibart | Pattern classification by memristive crossbar circuits using ex situ and in situ training[END_REF] et en 2015 M. Prezioso et al. ont démontré un apprentissage supervisé avec un dispositif similaire [START_REF] Prezioso | Training and operation of an integrated neuromorphic network based on metal-oxide memristors[END_REF]. Bien qu'il y ait également d'autres publications utilisant des types de memristors différents [START_REF] Eryilmaz | Experimental demonstration of array-level learning with phase change synaptic devices[END_REF][START_REF] Eryilmaz | Brain-like associative learning using a nanoscale non-volatile phase change synaptic device array[END_REF], tous ces résultats se limitent pour l'instant aux réseaux supervisés. Afin de réaliser un système avec un apprentissage autonome et non-supervisé, la stdp jouera probablement un rôle important. Une motivation centrale de ce travail est donc d'avancer vers l'implémentation de réseaux de neurones basés sur des synapses memristives en combinaison avec des circuits de neurones réalisés en technologies classiques.

A.2.1.2 Memristor basé sur des jonctions tunnel ferroélectriques

Le memristor étudié dans cette thèse s'appuie sur des dispositifs ferroélectriques. Les matériaux ferroélectriques possèdent une polarisation électrique spontanée qui peut être renversée par un champ électrique externe. La source microscopique de cette polarisation est un positionnement non-centrosymétrique des ions dans la maille cristalline. Tous les matériaux ferroélectriques sont également piézoélectriques et présentent donc un changement de leur polarisation en fonction d'une contrainte mécanique. En fonction du potentiel de barrière et de l'amplitude du champ électrique, il y a trois mécanismes qui peuvent contribuer au transport d'électrons à travers la couche vi ferroélectrique. La conduction par effet tunnel direct (dt) règne le transport pour des tensions modérées [START_REF] Frenkel | On the Electrical Resistance of Contacts between Solid Conductors[END_REF][START_REF] Simmons | Generalized Formula for the Electric Tunnel Effect between Similar Electrodes Separated by a Thin Insulating Film[END_REF][START_REF] Brinkman | Tunneling Conductance of Asymmetrical Barriers[END_REF]. Sous fort champ électrique dont l'amplitude excède la hauteur de barrière à l'interface, la largeur de barrière effective est réduite. Ceci est décrit par l'effet tunnel Fowler-Nordheim (fnt) [START_REF] Fowler | Electron Emission in Intense Electric Fields[END_REF][START_REF] Dyke | Field Emission[END_REF][START_REF] Pantel | Electroresistance effects in ferroelectric tunnel barriers[END_REF]. À des températures finies, des porteurs de charge thermiquement activés peuvent également contribuer. Le processus d'injection thermique (ti) devient important pour des températures ou des tensions élevées [START_REF] Pantel | Electroresistance effects in ferroelectric tunnel barriers[END_REF].

La figure A.2.2

Le changement de résistance observé après inversion de la polarisation de la couche ferroélectrique est appelé électrorésistance tunnel (ter) [START_REF] Kohlstedt | Theoretical current-voltage characteristics of ferroelectric tunnel junctions[END_REF][START_REF] Zhuravlev | Giant Electroresistance in Ferroelectric Tunnel Junctions[END_REF]. Plusieurs mécanismes peuvent en être la source ou y contribuer conjointement. En raison de la piézoélectricité de la barrière tunnel, son épaisseur peut être influencé par un champ électrique externe, ce qui impliquerait un changement substantiel de la transparence de la barrière pour des porteurs de charge [START_REF] Kohlstedt | Theoretical current-voltage characteristics of ferroelectric tunnel junctions[END_REF][START_REF] Bilc | Electroresistance Effect in Ferroelectric Tunnel Junctions with Symmetric Electrodes[END_REF]. Dans des fines couches sous fortes contraintes par contre, cet effet est fortement limité. Le mécanisme principal de la ter est basé sur la modification des hauteurs de barrières électrostatiques aux interfaces de la couche ferroélectrique avec les électrodes. Dans une structure réelle, l'écrantage des charges de polarisation dans les électrodes est imparfait ce qui modifie la hauteur de barrière à l'interface. Le renversement de la polarisation change le signe de cette modification et dans une ftj aux propriétés d'interface asymétriques, ceci entraîne une modification du profile de barrière. Ce mécanisme est présenté dans la figure A.2.4 et peut être simplifié à la modification de la hauteur de barrière moyenne. Étant donné la grande sensibilité du transport tunnel à la barrière, cet effet présente la contribution principale à la ter et permet donc une lecture et une écriture entièrement électrique du sens de la polarisation [START_REF] Garcia | Ferroelectric tunnel junctions for information storage and processing[END_REF][START_REF] Pantel | Electroresistance effects in ferroelectric tunnel barriers[END_REF].

Un premier résultat expérimental du ter a été publié en 2007 par M. Gajek et al. [START_REF] Gajek | Tunnel junctions with multiferroic barriers[END_REF]. Trois années plus tard, trois équipes de chercheurs indépendantes ont pu démontrer le lien direct entre cet effet et la ferroélectricité en combinant des mesures de transport électrique et de l'imagerie de la configuration de polarisation locale dans des fines couches de BaTiO 3 (bto) et Pb(Zr 0.2 Ti 0.8 )O 3 (pzt) [START_REF] Garcia | Giant tunnel electroresistance for non-destructive readout of ferroelectric states[END_REF][START_REF] Maksymovych | Polarization Control of Electron Tunneling into Ferroelectric Surfaces[END_REF][START_REF] Gruverman | Tunneling Electroresistance Effect in Ferroelectric Tunnel Junctions at the Nanoscale[END_REF].

Des premiers dispositifs à l'état solide, c'est-à-dire avec des électrodes déposées sur la couche ferroélectrique, ont été fabriqués avec du pzt par D. Pantel et al. en 2011 avec des ratios off/on de plus de 1500 [START_REF] Pantel | Room-Temperature Ferroelectric Resistive Switching in Ultrathin Pb(Zr 0 .2Ti 0 .8)O 3 Films[END_REF]. Peu arpès, en utilisant le bto, A. Chanthbouala et al. ont également démontré des ftjs qui pourraient être utilisées en tant que mémoire binaire [START_REF] Chanthbouala | Solid-state memories based on ferroelectric tunnel junctions[END_REF]. Le même groupe a aussi réalisé le premier memristor basé sur ces structures en exploitant la possibilité de ne renverser qu'une partie de la polarisation ferroélectrique [START_REF] Chanthbouala | A ferroelectric memristor[END_REF]. La stabilisation de configurations mixtes de polarisations up et down résulte en des résistances entre les valeurs extrêmes des états off et on. En outre, la valeur de résistance peut être contrôlée finement par des trains de pulses de tensions et -à un moment donné -est donc une fonction de l'histoire des pulses appliqués à ses bornes. C'est cette propriété de « mémoire » qui rend les ftjs intéressantes pour des applications bio-inspirées.

M1

FE M2

0 d P x σ(x) λ 2 λ 1 (a 2 ) E d x φ(x) (a 3 ) x E B (x) E F ϕ B ϕ B,1 ϕ B,2 (b)
(b 1 ) M1 FE M2 0 d P x σ(x) λ 2 λ 1 (b 2 ) E d x φ(x) (b 3 ) x E B (x) E F ϕ B ϕ B,1 ϕ B,2

A.2.2 Caractéristiques fondamentales de jonctions tunnel ferroélectriques basées sur BiFeO 3

Les jonctions tunnel ferroélectriques étudiées dans ce travail sont basées sur le BiFeO montre des cycles d'hystérèse résistive typiques. L'état initial correspond à une grande résistance de 2 à 3 × 10 9 Ω. Sous des pulses de tension d'environ -2,5 V et de 100 ns de durée la résistance décroit à une valeur de 2 à 3 × 10 4 Ω et la ftj se trouve dans l'état on. Sous des pulses de tension positive, la ftj peut atteindre l'état off avec une résistance de 3 à 4 × 10 8 Ω. En limitant l'amplitude de pulse, des états de résistances intermédiaires se laissent stabiliser. -a montre que l'état initial de la ftj, qui possède une résistance élevée, est associé à une polarisation uniforme pointant vers l'électrode inférieure (appelé down). L'état on, la plus basse résistance, est caractérisé par une polarisation quasi-uniforme vers l'électrode supérieure (appelé up, voir figure A.2.6-b). Seul le signal d'amplitude indique la présence de petits domaines down non-renversés. Sous l'application de pulses de tension positive, la proportion de domaines down augmente progressivement (figure A.2.6-c-e) jusqu'à ce que la ftj arrive dans l'état off (figure A.2.6-f). La configuration de la polarisation est proche de celle de l'état initial. La présence de quelques nano-domaines up non-reversés peut expliquer la résistance légèrement inférieure à celle de l'état initial.

Les images pfm permettent d'extraire la fraction qu'occupent les domaines down, cidessous appelée S. La figure A.2.7 -a montre la résistance électrique en fonction de S pour différents états mesurés dans dix jonctions. La dépendance observée se laisse expliquer par un simple modèle de conduction en parallèle (illustré dans la figure A.2.7-b) à travers des zones occupées par des domaines down, associées à une grande résistance, et des zones de domaines up, caractérisées par une résistance substantiellement plus petite.

A.2.2.2 Influence des électrodes supérieures

Des modèles simples lient l'effet de ter aux longueurs d'écrantage des électrodes [START_REF] Kohlstedt | Theoretical current-voltage characteristics of ferroelectric tunnel junctions[END_REF][START_REF] Zhuravlev | Giant Electroresistance in Ferroelectric Tunnel Junctions[END_REF][START_REF] Pantel | Electroresistance effects in ferroelectric tunnel barriers[END_REF]. Afin d'apprendre plus sur cette influence, nous avons fabriqué des ftjs avec des électrodes supérieures de W, Co, Ni et de Ir. L'interface inférieure reste inchangée.

La résistance des états initiaux, ou plus précisément le produit résistance-surface (ra), montre une claire dépendance avec la valeur du travail de sortie de chaque matériau ). Afin d'éclaircir cela, nous avons effectué des mesures de courant en fonction de la tension appliquée (i-v). En utilisant un modèle combinant les effets tunnel par direct tunneling et par Fowler-Nordheim tunneling, nous pouvons ensuite déterminer les hauteurs de barrière d'interface avec l'électrode supérieure. Les résultats montrent que les hauteurs de barrière augmentent avec le travail de sortie des électrodes, en bon accord avec la règle d'Anderson [START_REF] Anderson | Germanium-Gallium Arsenide Heterojunctions[END_REF]. Ces expériences montrent que le choix du matériau d'électrode est important. Il doit être fait en pesant les avantages de très grands ratios off/on et des désavantages d'une commutation répétée réduite. Les jonctions avec une électrode de Co se présentent ainsi comme le meilleur compromis pour nos besoins.

A.2.2.3 Influence de l'électrode inférieure

Nous avons également étudié des ftjs avec des électrodes inférieures basées sur un autre matériau, en remplaçant le ccmo par du LaNiO 3 (lno). La figure A.2.10 -a montre des cycles d'hystérèse résistive mesurés à 290 K et à 80 K. La commutation à température ambiante entraîne un grand changement de résistance et résulte donc dans un ratio off/on très élevé de 10 5 . A basse température, les valeurs de résistance dans les deux états, off et on, sont encore plus extrêmes, de manière à augmenter le ratio off/on à 3 × 10 6 . Pourtant, les tensions de seuil pour la commutation restent quasiment inchangées. La figure A.2.11-b montre que la résistance dans les états initiaux et on ne dépend que faiblement de la température -en bon accord avec une conduction par effet tunnel direct [START_REF] Stratton | Volt-current characteristics for tunneling through insulating films[END_REF][START_REF] Simmons | Generalized Thermal J-V Characteristic for the Electric Tunnel Effect[END_REF]. L'électrode inférieure présente une dépendance métallique.

Cependant, la résistance dans l'état off est caractérisée par une plus forte dépendance en température (figure A.2.11-b) et augmente d'un facteur d'à peu près 17 de 290 à 80 K. Ceci pourrait être dû à la contribution d'effets thermiquement activés au transport électrique. La présence de zones irréversibles observées par pfm (cf. figure A.2.10-a) pourrait indiquer la présence de défauts qui présenteraient un chemin de transport par effet tunnel inélastique à travers des états localisés dans la barrière [START_REF] Yin | Enhanced tunnelling electroresistance effect due to a ferroelectrically induced phase transition at a magnetic complex oxide interface[END_REF][START_REF] Wen | Temperature-dependent tunneling electroresistance in Pt/BaTiO 3 /SrRuO 3 ferroelectric tunnel junctions[END_REF][START_REF] Glazman | Inelastic Tunneling Through Thin Amorphous Films[END_REF]. Malgré l'influence de ces défauts, ces ftjs disposent de très grands ratios off/on qui, dans la moyenne, se situent autour de 10 4 . Ceci est le résultat de la faible résistance dans l'état on, fortement due à la grande conductivité de l'électrode inférieure de lno. Dans le cas des jonctions basées sur ccmo, c'est la résistivité de ce matériau qui limite de plus petites résistances dans l'état on. Néanmoins, cette résistance joue éventuellement un xiii rôle protecteur en limitant la tension appliquée aux bornes de la couche ferroélectrique. Ceci expliquerait la bonne performance en terme de reproductibilité des ftjs basées sur des électrodes inférieures de ccmo. De futures études pourraient tenter de les combiner avec des basses résistances dans l'état on, ce qui serait, par exemple, bénéfique pour la vitesse de lecture dans des systèmes de mémoire et neuromorphiques. Finalement, les grandes valeurs de ter observées dans ces ftjs combinant bfo et lno ouvrent des portes vers un contrôle précis d'une transition métal-isolant par effet de champ avec d'autres nickelates de terres rares [START_REF] Catalan | Progress in perovskite nickelate research[END_REF].

A.2.3 Performances de jonctions tunnel ferroélectriques basées sur BiFeO 3

Après l'étude des propriétés fondamentales des jonctions tunnel ferroélectriques basées sur bfo, nous effectuons des expériences autour de leurs performances en termes d'endurance, de rétention et de la vitesse de la commutation résistive.

Pour cela, un nouveau type d'échantillon (type b) a été fabriqué : basé sur le même empilement de couches que les échantillons utilisés ci-dessus (type a), leurs pistes de contact électrique leur permettent d'être contactés par des pointes radio-fréquence standard. Cette configuration assure des mesures plus stables et donne ainsi la possibilité d'effectuer des expériences plus longues. La caractérisation des 50 ftjs de l'échantillon révèle un très bon rendement (90 %) et des dispersions raisonnables des niveaux de résistances initiaux, on et off [START_REF] Boyn | High-performance ferroelectric memory based on fully patterned tunnel junctions[END_REF].

A.2.3.1 Endurance

Sous répétition de cycles d'hystérèse les matériaux ferroélectriques peuvent montrer des signes d'usure. Ce phénomène est appelé fatigue et peut se manifester, par exemple, par une polarisation commutable réduite. Étant donné l'intérêt des ferroélectriques en couches épaisses 3 pour des mémoires binaires [347], beaucoup d'efforts ont déjà été consacré à cette problématique [START_REF] Tagantsev | Polarization fatigue in ferroelectric films: Basic experimental findings, phenomenological scenarios, and microscopic features[END_REF]341]. Pour ces systèmes, plus de 10 14 cycles ont pu être démontrés [343,345,346]. Néanmoins, des expériences sur des couches très fines se sont limitées à quelques 10 3 cycles [START_REF] Chanthbouala | Solid-state memories based on ferroelectric tunnel junctions[END_REF][START_REF] Wen | Ferroelectric-field-effect-enhanced electroresistance in metal/ferroelectric/semiconductor tunnel junctions[END_REF].

Les mesures d'endurance sur les échantillons de type b montrent de bonnes performances jusqu'à 4 × 10 6 cycles (figure A.2.12 ). Vers la fin de l'expérience, la jonction reste de plus en plus souvent bloquée dans l'état de haute résistance et le basculement vers l'état on devient moins fiable. Du fait que cet état reste néanmoins accessible, nous concluons que cette fatigue n'est pas due à une réduction de la polarisation de saturation. Une source éventuelle du comportement observé pourrait être le mouvement de défauts, comme par exemple des lacunes d'oxygène [START_REF] Yang | Nanoscale Observation of Time-Dependent Domain Wall Pinning as the Origin of Polarization Fatigue[END_REF]341,343,348]. Afin de limiter cet effet, le dopage de la couche ferroélectrique par des matériaux tels que La ou Nb pourrait être envisagé [343]. montre que les niveaux extrêmes, c'est-à-dire les états off et on, sont très stables. L'extrapolation de l'hypothèse d'une dépendance suivant une loi de puissance suggère une rétention jusqu'à dix ans. Les états intermédiaires en revanche présentent tous une relaxation vers des plus hautes résistances. Leur instabilité permet des réflexions concernant les énergies associées. Le champ de dépolarisation favorise une configuration de domaines mixte et ne peut donc pas être à la source du comportement observé. Si, par contre, la création de parois de domaines nécessite une énergie, ceci favorisera un état uniforme. Nous concluons que dans nos ftjs, c'est cette dernière contribution énergétique qui importe.

La relaxation systématique vers les états de plus hautes résistances indique l'existence d'un champ interne [166,[START_REF] Gruverman | Polarization retention in SrBi 2 Ta 2 O 9 thin films investigated at nanoscale[END_REF]. Une voie vers une jonction plus stable pourrait donc consister à essayer d'influencer ce champ interne. Comme constaté auparavant, le choix des matériaux d'électrodes peut être crucial pour cela.

A.2.3.3 Mesures de courants transitoires

Afin d'étudier la dynamique du renversement de la polarisation, nous avons conçu un circuit expérimental qui nous permet d'enregistrer l'évolution temporelle du courant transitoire pendant chaque pulse de tension. Les principaux composants de ce montage sont un amplificateur de courant et un oscilloscope. Ces mesures de single-shot ont relevé l'existence d'une capacité non-négligeable de 450 ± 21 fF. Afin de s'affranchir des pics de courant dûs au chargement de celle-ci, nous utilisons des courbes de référence sans commutation. Elles sont soustraites des courbes montrées ci-dessous. contient quelques exemples du courant transitoire pendant la commutation de l'état off à l'état on. Au début, la jonction se trouve toujours dans l'état de haute résistance et le courant transmis et donc très faible. Après un temps d'attente t wait , le courant augmente quasi-instantanément. 4 La légère diminution consécutive de l'amplitude peut être attribuée à la haute non-linéarité du courant et à une faible réduction de l'amplitude du pulse de tension liée au circuit. Son amplitude ne dépend pas de la résistance finale mais est limitée par une résistance en série avec la jonction tunnel même. Comme nous l'avons évoqué ci-dessus par rapport aux mesures à courant continu, ceci est dû à la faible conductance de ccmo (l'électrode inférieure). -a montre les temps d'attente de la commutation de l'état off à l'état on extraites des évolutions temporelles des courants transitoires : plus la tension est élevée, plus la commutation est rapide. En traçant ces données en fonction de la valeur inversée du champ électrique 5 nous obtenons une dépendance linéaire qui suit la loi de Merz [START_REF] Merz | Domain Formation and Domain Wall Motions in Ferroelectric BaTiO 3 Single Crystals[END_REF]. L'ajustement de courbe donne un champ d'activation de E a = -1,80 ± 0,11 V/nm et un temps de relaxation de t 0 = 1,1 ± 0,5 × 10 -11 s. Bien qu'élevée, la valeur de E a est du même ordre que celles mesurées dans d'autres couches minces [START_REF] Kim | Non-Kolmogorov-Avrami-Ishibashi Switching Dynamics in Nanoscale Ferroelectric Capacitors[END_REF][START_REF] Chanthbouala | A ferroelectric memristor[END_REF][START_REF] Pertsev | Coercive field of ultrathin Pb(Zr 0 .52Ti 0 .48)O 3 epitaxial films[END_REF] et peut être influencée par la contrainte du substrat [START_REF] Pertsev | Coercive field of ultrathin Pb(Zr 0 .52Ti 0 .48)O 3 epitaxial films[END_REF].

.0 V 0.9 V 1.6 V 0.8 V 1.4 V 0.6 V 1.2 V 0.0 V 1.1 V Write pulse

La figure A.2.14

La figure A.2.15

Les longs temps d'attente indiquent le délai de formation d'un nucléus de polarisation opposée. La montée de courant suivante très rapide pourrait donc être liée au mouvement de parois de domaines. Les échelles de temps très différentes sur lesquelles ces deux 4) Le temps de montée du signal est en-dessous de la résolution temporelle du circuit qui est de l'ordre de 3 ns. mécanismes agissent montrent que le renversement est limitée par le processus de nucléation. Dans la section suivante, nous étudions cette dynamique par des études combinant l'imagerie par pfm et des mesures de transport sous pulses cumulés.

A.2.4 Dynamique de renversement de la polarisation ferroélectrique et règle d'apprentissage STDP

Dans un réseau de neurones réalisé avec des synapses memristives, la dynamique du système est intrinsèquement liée à la physique de chaque memristor. Ici, nous étudions donc le processus de la dynamique de renversement de la polarisation ferroélectrique dans des couches ultraminces en connexion avec l'implémentation d'une règle d'apprentissage, appelée spike-timing-dependent plasticity (stdp), dans des ftjs. Pour étudier la dynamique de renversement de la polarisation ferroélectrique dans les couches de bfo, nous avons combiné l'imagerie par pfm et les mesures de transport. La figure A.2.16 -a montre l'évolution de la configuration de domaines après l'application de pulses cumulés. En commençant dans l'état on qui est caractérisé par une majorité de domaines up, la polarisation se renverse progressivement jusqu'à ce qu'elle présente une configuration down quasi-uniforme dans l'état off. Les états intermédiaires sont caractérisés par une amplitude de pfm très faible ce qui indique la présence de petits domaines d'une taille en-dessous de la résolution. Cette nature très hétérogène du renversement est également confirmée par des simulations de dynamique moléculaire effectué par B. Xu et L. Bellaiche 6 En faisant varier la tension appliquée, nous obtenons les évolutions de la configuration ferroélectriques montrées dans la figure A.2.17 Les images pfm indiquent que ce processus passe par la nucléation de beaucoup de petits domaines plutôt que la croissance de domaines existants par le mouvement de parois. Les modèles de dynamique classiques, tels que modèle de Kolmogorov-Avrami-Ishibashi [START_REF] Avrami | Kinetics of Phase Change. II Transformation-Time Relations for Random Distribution of Nuclei[END_REF][START_REF] Ishibashi | Note on Ferroelectric Domain Switching[END_REF], ne peuvent pas décrire cela. En prenant en compte la nature très hétérogène du renversement, nous nous basons donc sur le modèle du renversement limité par nucléation de J. Y. Jo et al. [START_REF] Jo | Domain Switching Kinetics in Disordered Ferroelectric Thin Films[END_REF]. Leur description suppose que la couche ferroélectrique est partagée en de petites régions qui se renversent indépendamment et quasi-instantanément suite à un événement de nucléation. Les valeurs logarithmées des temps de nucléation suivent une distribution Lorentzienne. Finalement, ce modèle amène à une expression analytique de l'évolution de la fraction de domaine S en fonction du temps d'application t et de l'amplitude V d'une tension externe :

V 1.4 V 1.3 V 1.2 V 1.1 V 1.0 V 0.
S(t) ≈ 1 π arctan log t -log τ 0 (V) Γ(V) + 1 2 . (A.2.1)
Un ajustement de courbe suivant cette expression montre un très bon accord avec les données expérimentales (figure A.2.17-a). Plus la tension appliquée est élevée, plus les distributions de temps de renversement sont centrées à des temps plus courts et plus leur largeur est réduite (figure A.2.17-b). La figure A.2.17-c montre que les valeurs moyennes suivent la loi de Merz avec un champ d'activation de E a = 3,23 ± 0,12 V/nm et une constante de temps de t 0 = 1,1 ± 0,6 × 10 -13 s. Cette valeur de champ est comparable à celle obtenue par les mesures de transmission. La constante de temps correspond à une fréquence de f 0 = 8,8 ± 0,5 THz, proche des fréquences de phonons dans le bfo (∼1 THz) [367]. En comparaison avec la valeur obtenue des mesures de courant transmis, ces expériences de pulses cumulés donnent une échelle de temps plus rapide. Ceci peut être expliqué par la sensibilité du courant transmis aux grandes variations qui -dû xix au mécanisme de consécution en parallèle -se trouvent à la fin du renversement. En conséquence, les mesures de pulses cumulés nous permettent une étude plus détaillée de la dynamique de renversement de la polarisation ferroélectrique.

A.2.4.2 Spike-timing-dependent plasticity

La règle de la spike-timing-dependent plasticity (stdp) détermine comment le poids d'une synapse dans un réseau de neurones évolue. Cette plasticité synaptique locale règne sur l'apprentissage globale du système. C'est pour cela que la stdp fait partie des règles d'apprentissage. S'intégrant dans la ligne des idées de D. O. Hebb [START_REF] Hebb | The organization of behavior: a neuropsychological theory[END_REF], la stdp a été proposée sur la base de considérations théoriques [368, 369] avant d'être mise en évidence dans des systèmes biologiques [START_REF] Markram | Regulation of Synaptic Efficacy by Coincidence of Postsynaptic APs and EPSPs[END_REF][START_REF] Bi | Synaptic Modifications in Cultured Hippocampal Neurons: Dependence on Spike Timing, Synaptic Strength, and Postsynaptic Cell Type[END_REF]370]. Comme la stdp règle à la fois la potentialisation et la dépression synaptique, elle peut conduire à un système stable en contrebalançant les poids sans nécessité de nivellement extérieure [371,[START_REF] Song | Competitive Hebbian learning through spike-timing-dependent synaptic plasticity[END_REF]. Il a été montré que, grâce à son lien avec la causalité des activités neuronales, la stdp était un des piliers pour la réalisation de l'apprentissage non-supervisé dans les réseaux de neurones impulsionels [START_REF] Thorpe | Spike-based strategies for rapid processing[END_REF][START_REF] Masquelier | Unsupervised Learning of Visual Features through Spike Timing Dependent Plasticity[END_REF][START_REF] Guyonneau | Neurons Tune to the Earliest Spikes Through STDP[END_REF][START_REF] Bichler | Extraction of temporally correlated features from dynamic vision sensors with spike-timing-dependent plasticity[END_REF].

Le grand intérêt dédié aux réseaux memristifs a motivé des idées à réaliser la stdp avec ces composants résistifs. Parmi les différents concepts qui existent aujourd'hui [START_REF] Snider | Spike-timing-dependent learning in memristive nanodevices[END_REF][START_REF] Linares-Barranco | Memristance can explain spike-time-dependent-plasticity in neural synapses[END_REF][START_REF] Zamarreño-Ramos | On spike-timing-dependent-plasticity, memristive devices, and building a self-learning visual cortex[END_REF], nous avons choisi de nous baser sur celui de B. Linares-Barranco & T. Serrano-Gotarredona [START_REF] Linares-Barranco | Memristance can explain spike-time-dependent-plasticity in neural synapses[END_REF]. Leur proposition d'implémentation utilise le recouvrement entre Cela constitue une avancée importante vers l'usage de ftjs en tant que synapses artificielles dans des réseaux de neurones. Seuls des modèles de memristors fiables permettent d'effectuer des simulations nécessaires pour la conception de systèmes à grande échelle et, finalement, la réalisation de réseaux de neurones dotés d'un apprentissage non-supervisé.

A.2.5 Conclusions

Dans ce travail, nous avons étudié les propriétés statiques et dynamiques de jonctions tunnel ferroélectriques (ftjs) basées sur des couches ultraminces de BiFeO 3 en vue der leur usage en tant que synapses memristives pour des applications neuromorphiques. Ces dispositifs nanométriques présentent une commutation résistive très rapide en combinaison avec des contrastes de résistance off/on de 10 4 . Des configurations de polarisation ferroélectrique mixtes permettent de stabiliser des états intermédiairesune caractéristique essentielle pour leur usage dans des circuits analogues de calcul bio-inspirée comme des réseaux de neurones par exemple.

La variation de l'électrode supérieure a révélé que le profil de la barrière tunnel dépendait fortement du choix des matériaux et de leur travail de sortie. En combinaison avec une étude sur l'influence de l'électrode inférieure, ceci a montré que des bonnes performances xxii n'étaient atteignable qu'en faisant un compromis entre les propriétés statiques et une commutation reproductible. Des expériences d'impulsion unique de tension montrent un retournement de polarisation ultra-rapide. Nous avons approfondi l'étude de cette dynamique par des mesures d'impulsions cumulées. La combinaison de l'analyse du transport avec de l'imagerie par microscopie à force piézoélectrique nous a permis d'établir un modèle dynamique du memristor. Suite à la démonstration de la spiketiming-dependent plasticity, une règle d'apprentissage importante, nous avons donc pu prédire le comportement de ces synapses artificielles. Ceci représente une avance majeure vers la réalisation de réseaux de neurones sur puce dotés d'un auto-apprentissage non-supervisé. Abstract: Classical computer architectures are optimized to process pre-formatted information in a deterministic way and therefore struggle to treat unorganized natural data (images, sounds, etc.). As these become more and more important, the brain inspires new, neuromorphic computer circuits such as neural networks. Their energy-efficient hardware implementations will greatly benefit from nanodevices, called memristors, whose small size could enable the high synaptic connectivity degree observed in the brain.

In this work, we concentrate on memristors based on ferroelectric tunnel junctions that are composed of an ultrathin ferroelectric film between two metallic electrodes. We show that the polarization reversal in BiFeO3 films can induce resistance contrasts as high as 10 4 and how mixed domain states are connected to intermediate resistance levels.

Changing the electrode materials provides insights into their influence on the electrostatic barrier and dynamic properties of these memristors. Single-shot switching experiments reveal very fast polarization switching which we further investigate in cumulative measurements. Their analysis in combination with piezoresponse force microscopy finally allows us to establish a model describing the memristor dynamics under arbitrary voltage signals. After the demonstration of an important learning rule for neural networks, called spike-timing-dependent plasticity, we successfully predict new, previously unexplored learning curves. This constitutes an important step towards the realization of unsupervised self-learning hardware neural networks.

FR EN

Figure 1 . 1 . 1 Figure 1 . 1 . 1

 111111 Figure 1.1.1

Figure 1 . 1 . 1 -

 111 Figure 1.1.1 -Microprocessor trends. Evolution of the number of transistors, the clock frequency, the power consumption, and the number of cores of a typical microprocessor over the years. The lines only serve as visual guides. [data from Ref. 10]

Figure 1 . 1 . 2 Figure 1

 1121 Figure 1.1.2 Figure 1.1.2

Figure 1 . 1 . 2 -

 112 Figure 1.1.2 -Schematic representation of a simple feedforward neural network. The network shown here consists of three layers of which each contains certain number of nodes. All nodes in a given layer are connected to all nodes of the adjacent layers; their connection strength (or weight) w y,x can vary. (For the sake of simplicity, not all weights are shown in the figure.)

Figure 1 . 1 . 3

 113 Figure 1.1.3

Figure 1 . 1 . 3 -

 113 Figure 1.1.3 -Schematic representation of a biological neuron. Signals typically travel from the dendrites through the neuron soma and along the axon before being transmitted to another cell. [derived from Ref. 21]

Figure 1 . 1 . 4

 114 Figure 1.1.4

Figure 1 . 1 . 4 -

 114 Figure 1.1.4 -Schematic representation of a biological, chemical synapse. The signal from the pre-neuron is transferred by neurotransmitters across the synaptic cleft that bind to receptors at the pre-neuron surface. [derived from Ref. 21]

Figure 1 . 1 . 5 -

 115 Figure 1.1.5 -Photograph of a digital ANN implemented in CMOS. The core die contains a 1024×256 synapse array and 256 neurons and performs simple image recognition tasks with a (by a factor of more than 10 5 ) lower power consumption than that of an ann simulation on classical hardware. [taken from Ref. 65]

Figure 1 . 1 . 5

 115 Figure 1.1.5

Figure 1 . 1 . 6

 116 Figure 1.1.6

Figure 1 . 1 . 6 -

 116 Figure 1.1.6 -Circuit of CMOS synapse. The design allows for learning operation based on stdp at the individual synapse level. [taken from Ref. 71]

Figure 1 .

 1 Figure 1.1.7

Figure 1 . 1 . 7 -

 117 Figure 1.1.7 -Relations between the four fundamental electronic variables (voltage, electric current, magnetic flux, and electric charge) that are connected by the four ideal circuit elements (capacitor, resistor, inductor, and memristor). This schema also shows the new memristor circuit symbol ( ). [after Ref. 82]

Figure 1 . 1 . 8

 118 Figure 1.1.8

Figure 1 . 1 . 8 -

 118 Figure 1.1.8 -Resistive switching mechanisms in memristors based on ion movements. (a) Formation and rupture of a conducting filament. This conducting path can for example be formed by metal cations or oxygen vacancies. (b) Increase and decrease of an interfacial conducting zone due to the accumulation of oxygen vacancies or charge carriers. [taken from Ref. 93]

Figure 1 . 1 . 9 . 10 Figure 1

 119101 Figure 1.1.9 . Using rram memristors, K.-H. Kim et al. fabricated a memristive crossbar on top of a conventional cmos circuit and therewith demonstrated multi-level data storage in 2012 [131]. While this architecture would ultimately allow for a very short and direct connection to the peripheral circuits underneath [132], the fabrication process is not technically mature yet. S. Park et al. went another way, combining an rram-memristor crossbar with a separate cmos neuron chip (figure 1.1.10Figure 1.1.10

Figure 1 . 1 . 9 -

 119 Figure 1.1.9 -Memristor crossbar for the emulation of artificial synapses. In the example shown here, an rrambased memristor (incorporating thin films of TiO 2 /Al 2 O 3 ) is located at each crossing-point between a row and a column electrode line. [taken from Ref. 133]

Figure 1 . 1 . 10 - 11 Figure 1 . 1 . 11

 1110111111 Figure 1.1.10 -Memristor crossbar in combination with CMOS circuits. These hybrid anns are a first step towards fully integrated memristor-cmos systems. [taken from Ref. 134]

Figure 1 . 1 . 11 -

 1111 Figure 1.1.11 -Publication statistics on neuromorphic computing over the years. (a) Evolution of the number of publications that contain a word beginning with "memrist" in their title. (b) Evolution of the number of publications that contain the word "neuromorphic" in their title. [Source: Web of Knowledge search, November 2015]

Figure 1 Figure 1 . 2 . 1 -

 1121 Figure 1.2.1

Figure 1 . 2 . 2 -

 122 Figure 1.2.2 -Macroscopic ferroelectric hysteresis cycle: evolution of the overall polarization under an electric field sweep.

Figure 1 . 2 . 3

 123 Figure 1.2.3

E 2 Figure 1 . 2 . 3 -

 2123 Figure 1.2.3 -Landau theory energy potential landscapes. (a) Second-order transition as a function of temperature. (b) First-order transition as a function of temperature. (c) Gibbs free energy potential without electric field. (d) Asymmetric potential landscape under influence of a non-zero electric field. (e) Destabilization of one polarization state by a high electric field. Dotted lines in (a,b) correspond to a paraelectric state.

Figure 1 . 2 . 4

 124 Figure 1.2.4

Figure 1 . 2 . 4 -

 124 Figure 1.2.4 -Unit cell distortions of the perovskite structure. (a) Ideal cubic perovskite structure with the positions of the A-( ) and B-site ( ) cations as well as the oxygen anion ( ). (b) Tetragonal distortion enabling a polarization along the [001] direction. (c) Orthorombic distortion enabling a polarization along the [101] direction. (d) Rhombohedral distortion enabling a polarization along the [111] direction. [inspired by Ref. 151]

Figure 1 . 2 . 5 -

 125 Figure 1.2.5 -Schema of the effect of in-plane strain on perovskite thin films. (a) Unstrained perovskite structure. (b) Biaxial compressive strain exerted by growth on a substrate with smaller in-plane lattice constants. (c) Biaxial tensile strain exerted by growth on a substrate with larger in-plane lattice constants. [adapted from Ref. 154]

Figure 1 . 2 . 5 - 6 Figure 1 . 2 . 6

 1256126 Figure 1.2.5 -b) or tensile (figure 1.2.5-c) strain can be grown in phases that are otherwise thermally unstable. As monocrystalline substrates with a large variability in lattice constants (see figure 1.2.6 Figure 1.2.6

Figure 1

 1 Figure 1.2.7

Figure 1 . 2 . 7 -

 127 Figure 1.2.7 -Surface charges and potential distribution in a ferroelectric tunnel junction. (a) Schematic charge distribution σ(x) at the interfaces of a ferroelectric layer and two adjacent perfect metals. The metals completely screen the polarization charges. (b) Schematic charge distribution σ(x) at the interfaces of a ferroelectric layer and two adjacent realistic metals in which the screening charges are spread over a finite screening length λ. Furthermore, not all polarization surface charges are compensated in the metal. (c) Electrostatic potential profile ϕ(x) with perfect metal interfaces. (d) Electrostatic potential profile ϕ(x) with realistic metal interfaces. The incomplete charge compensation at the interfaces gives rise to an internal depolarization field E d in the ferroelectric layer.

Figure 1 . 2 . 8

 128 Figure 1.2.8

Figure 1 . 2 . 8 -

 128 Figure 1.2.8 -Schematic polarization reversal process. (a) Initial uniform polarization antiparallel to the applied electric field. (b) Nucleation at the interfaces. (c) Forward growth of the nuclei. (d) Sideways growth. (e) Final uniform polarization parallel to the applied electric field. [taken from Ref. 180]

Figure 1 . 2 . 9

 129 Figure 1.2.9

Figure 1 .

 1 Figure 1.2.10

Figure 1 . 2 . 9 -

 129 Figure 1.2.9 -Pinning of an elastic domain wall by defects. The defects change the potential landscape of the dw and lead to a profile that can strongly differ from the straight dw in a perfect defect-free medium.

FFigure 1 . 2 . 10 -

 1210 Figure 1.2.10 -Variation of the DW velocity with the applied force. At finite temperature T > 0 K, the velocity passes through different regimes as a function of the amplitude of the external driving force (e. g., the applied electric field): from creep or thermally assisted flux flow (taff) over depinning to the flow regime under strong forces. [inspired byRef. 190,[START_REF] Giamarchi | Dynamics of Disordered Elastic Systems[END_REF]

Figure 1 . 2 . 11

 1211 Figure 1.2.11

Figure 1 . 2 . 12

 1212 Figure 1.2.12

Figure 1 . 2 . 11 -

 1211 Figure 1.2.11 -Basic, schematic structure of a ferroelectric tunnel junction. The polarization points either toward the top ( ) or toward the bottom ( ) electrode.

Figure 1 . 2 . 12 -

 1212 Figure 1.2.12 -Principle of the polar switch proposed by L. Esaki et al. [222]. (a) Layer structure of the device. (b) Schematic current versus voltage curve showing the resistance switching at certain negative and positive threshold voltages. [adapted from Ref. 222]

Figure 1 .

 1 Figure 1.2.14

FNTFigure 1 . 2 . 14 -

 1214 Figure 1.2.14 -Schema of Fowler-Nordheim tunneling (FNT) through an energy barrier profile of height ϕ B under an applied voltage V > ϕ B /e.

Figure 1 . 2 . 15

 1215 Figure 1.2.15

Figure 1 . 2 . 16

 1216 Figure 1.2.16

Figure 1 . 2 . 17

 1217 Figure 1.2.17

Figure 1 . 2 . 16 -

 1216 Figure 1.2.16 -Overview of several possible mechanisms contributing to resistance changes in FTJs. (a) Electrostatic effect in an ftj with a ferroelectric (fe) between metallic electrodes (M1 and M2). (b) Interface effect. (c) Strain effect: the reverse piezoelectric effect results in a change of the ferroelectric (fe) thickness. [adapted from Ref. 230]

Figure 1 . 2 . 17 - 1 , b 1 ) 2 , b 2 )

 12171122 Figure 1.2.17 -Electrostatic resistance change effect in asymmetric FTJs (i. e. with different screening lengths, λ 1 < λ 2 ). (a) Low resistance (on) state. (b) High resistance (off) state. (a 1 , b 1 ) Charge distributions of the bound charges in the ferroelectric layer and the free charge carriers in the metallic electrodes. (a 2 , b 2 ) Electrostatic potential profiles: due to the incomplete screening of the polarization charges, a finite depolarization field E d remains. (a 3 , b 3 ) Energy barrier profiles: the different screening lengths cause different changes ( and ) in the interfacial energy barriers upon polarization reversal. As a consequence, the intermediate barrier height changes (between ϕ B and ϕ B ), leading to a change in the tunneling resistance.

Figure 1 . 2 . 18

 1218 Figure 1.2.18

.2. 8 )

 8 This over-exponential dependence explains why already small changes in the barrier height can result in very large changes in the tunneling resistance. Taking into account the three different transport mechanisms presented above, D. Pantel & M. Alexe have calculated the respective contributions to the total current density flowing through an ftj structure for the two polarization directions [241]. For an intermediate barrier thickness of 3.2 nm, figure 1.2.19

Figure 1 . 2 . 19

 1219 Figure 1.2.19

Figure 1 . 2 . 19 -

 1219 Figure 1.2.19 -Theoretical tunneling current densities through an FTJ with contributions from direct tunneling (DT), Fowler-Nordheim tunneling (FNT), and thermionic injection (TI) for both polarization directions of the ferroelectric barrier. (a) Individual contributions from the three conduction mechanisms. (b) Thickness dependence of the total current density. (c) Thickness and voltage dependence map of the electroresistance (ER). The dashed lines indicate the regions in which the different mechanisms dominate the current. The parameters used for the calculations are P = 3 µC cm -2 , ε stat = 60, ε ifl = 10, m * = 1, A ** = 1 × 10 6 A m -2 K -2 , λ 1 = 0.08 nm, λ 2 = 0.055 nm,

Figure 1 . 2 . 20

 1220

Figure 1 . 2 . 20

 1220 Figure 1.2.20

Figure 1 . 2 . 20 -

 1220 Figure 1.2.20 -Demonstration of the direct link between the ferroelectric polarization direction and the tunneling resistance in ultrathin strained BTO films on La 0.67 Sr 0.33 MnO 3 (LSMO)// NdGaO 3 (NGO). (a-c) pfm phase images of the bto surface revealing the direction of the local polarization. (d-f) Subsequent locally resolved resistance maps of the same areas. (g-i) Resistance profiles from the above maps. (j) Thickness dependence of the resistance of virgin ( ), negatively poled ( ), and positively poled ( ) regions showing an exponential increase with bto thickness. (k) Thickness dependence of ter, that also exhibits an exponential increase with thickness. [taken from Ref. 232]

Figure 1 . 2 . 21

 1221 Figure 1.2.21

Figure 1 . 2 . 21 -

 1221 Figure 1.2.21 -Demonstrations of resistive switching controlled by ferroelectric polarization. (a) Simultaneous measurements of the local current and the pfm signal of a 30-nm pzt film. The electrical transport is dominated by Fowler-Nordheim tunneling [233]. (b) Measurements of the current density and corresponding resistance in parallel to the pfm hysteresis as a function of applied voltage on a 9-nm thick pzt-based ftj with a Cu electrode (area: 0.6 µm 2 ). The electrical transport is dominated by thermionic injection [288]. (c) pfm phase and amplitude as well as resistance hysteresis measurements driven by 100-µs voltage pulses on ftjs with 2-nm bto films and Co electrodes (area: 0.2 µm 2 ). Open and filled symbols correspond to different measurements to show reproducibility. The electrical transport is dominated by direct tunneling [289]. The top panels show a sketch of the respective sample structure and the measurement configuration. All three devices use lsmo bottom electrodes. [adapted from Ref. 221]

Figure 1 . 2 . 22

 1222 Figure 1.2.22

Figure 1 . 2 . 22 -

 1222 Figure 1.2.22 -Polarization-mediated control of electronic states in FTJ electrodes. (a) Sketches of the structure and the electron band diagrams of an ftj in which an ultrathin bto film is sandwiched between a metallic Pt electrode and a semiconducting Nb:SrTiO 3 electrode [290]. When the polarization points towards the Nb:SrTiO 3 electrode, the screening charges that accumulate inside the semiconductor result in a decrease of the conduction band and a lowering of the barrier height.The device therefore is in the on state (upper panel). When the polarization points towards the Pt electrode, the negative polarization charges at the interface to the Nb:SrTiO 3 electrode repel charge carriers, leading to a depleted zone and an additional Schottky barrier. The resulting increase of the tunneling barrier width increases the resistance substantially so that the device is in the off state (lower panel). (b) Sketches of the structure of an ftj in which an ultrathin bto film is sandwiched between a La 0.67 Sr 0.33 MnO 3 (lsmo) and a La 0.5 Ca 0.5 MnO 3 (lcmo) electrode[START_REF] Yin | Enhanced tunnelling electroresistance effect due to a ferroelectrically induced phase transition at a magnetic complex oxide interface[END_REF]. When the polarization points towards the La 0.5 Ca 0.5 MnO 3 (lcmo) interface, screening leads to a hole depletion. This makes lcmo metallic, decreasing the ftj resistance (upper panel). In the opposite case, when the polarization points away from the lcmo interface, holes are accumulated so that the lcmo becomes insulating, increasing the ftj resistance (lower panel). [sketches adapted from Ref. 221]

Figure 1

 1 Figure 1.2.23

Figure 1 . 2 . 23 -

 1223 Figure 1.2.23 -Intermediate resistance states in BTO-based FTJs. (a) Resistance hysteresis curves as a function of 20-ns write voltage pulse amplitude. By limiting the applied maximum voltage amplitude, intermediate resistance states can be stabilized. The resistance was measured under direct current (dc) voltage of 0.1 V. (b) Variation of the ftj resistance as a function of the fraction of down domains extracted from pfm images for each state. The black line corresponds to the model of parallel conduction. [adapted from Ref. 214]

Figure 1 . 2 . 24

 1224 Figure 1.2.24

Figure 1 . 2 . 24 -

 1224 Figure 1.2.24 -Evolution of the resistance of a BTO-based FTJ under the application of successive 20-ns write voltage pulses. (a) Resistance evolution. (b) Corresponding write voltage pulse amplitudes. [adapted from Ref. 214]

Figure 2 . 1 . 2

 212 Figure 2.1.2

FerroelectricFigure 2 . 1 . 1 -

 211 Figure 2.1.1 -Temperature-strain phase diagram of BFO thin films indicating the transition temperatures for different magnetic and ferroelectric phases. This diagram combines theoretical (lines) and experimental (symbols) studies. Hypothetical phase transition temperatures in unexplored regions of the diagram are represented by dashed lines and transitions between different magnetic orders are assumed vertical for simplicity. [adapted from Ref. 300]

Figure 2 . 1 . 2 -

 212 Figure 2.1.2 -Overview of the different crystal structures that BFO forms in strained thin films. (a) Tetragonal bfo, only in very thin, strongly compressive films; (b) the highly distorted T-like monoclinic Cm phase, under strong compressive strain; (c) the M A monoclinic phase, under moderate compressive strain; (d) the bulk-like rhombohedral phase that forms on (111)-oriented substrates; (e) the M B monoclinic phase under moderate tensile strain; and (f) the orthorhombic phase that can be stabilized by moderate tensile strain. The primitive pseudocubic perovskite unit cell (light grey) is repeatedly indicated as a reference for each structure. [adapted from Ref. 300]

Figure 2 . 1 . 3 -

 213 Figure 2.1.3 -a). ω-2θ X-ray diffraction (xrd) scans (figure 2.1.3-b) and reciprocal space maps (figure 2.1.3-c) reveal that the bfo film is in

Figure 2 . 1 . 3 - 4 Figure 2 . 1 . 4

 2134214 Figure 2.1.3 -Structural characterization of ultrathin BFO films on YAO substrates with CCMO bottom electrodes. (a) X-ray reflectivity (xrr) measurement with fit yielding bfo and ccmo thicknesses of 4.6 nm and 21 nm, respectively. (b) X-ray diffraction (xrd) pattern with labeling of the main diffraction peaks. (c) Reciprocal space map (rsm) showing the in-plane perfect lattice matching of the ccmo and bfo films with the yao substrate (x coordinate in-plane, y coordinate out-of-plane).[published inRef. 310]

Figure 2 .

 2 Figure 2.1.5

Figure 2 . 1 . 4 -

 214 Figure 2.1.4 -HAADF image analysis of a BiFeO 3 /Ca 0.96 Ce 0.04 MnO 3 /Ca 0.98 Ce 0.02 MnO 3 /CaMnO 3 // YAlO 3 heterostructure. (a) haadf image of the structure in the {110} pseudocubic (pc) projection. (b) Evolution of the (averaged) displacements of the B-site ions (Fe or Mn) relative to the A-site ions (Bi or Ca), of the local c/a ratio, and of the underlying averaged haadf intensity near the bfo/ccmo interface. (c) Haadf base image. (d) Atomic model of bulk tetragonal bfo. The Fe ion displacement is measured along the [001] direction in the {110} pc projection. (e) Excess charge per unit cell throughout the heterostructure extracted from electron energy loss spectroscopy (eels) measurements. [published in Ref. 313]

Figure 2 . 1 . 5 -

 215 Figure 2.1.5 -Structure of A-type FTJs. (a) afm topography image of the sample surface. Several top-electrode dots are visible. (b) Cross-sectional view of one ftj. The top-electrode diameters range from about 180 to 660 nm. A c-afm tip is used to electrically connect each device. [adapted from Ref. 289]

Figure 2 . 2 . 1 -Figure 2 . 2 . 1 -Figure 2 . 2 . 1 -

 221221221 Figure 2.2.1-a. This allows for a combination of electrical transport measurements and imaging of the ferroelectric polarization configuration by pfm. All the measurements are performed with a multimode Nanoscope IV (Digital Instruments) setup at room temperature and under nitrogen flow.

Figure 2 . 3 . 1

 231 Figure 2.3.1

Figure 2 . 3 . 1

 231 Figure 2.3.1

  Figure 2.3.2

Figure 2 .

 2 Figure 2.3.2

Figure 2 . 3 . 2 -

 232 Figure 2.3.2 -PFM imaging of different resistance states of a Pt/Co/BiFeO 3 /Ca 0.96 Ce 0.04 MnO 3 FTJ of 180 nm diameter. (a) Virgin state. (b) on state. (c-e) Successive intermediate states with increasing resistances. (f) off state. The top panel shows the pfm phase and the bottom panel shows the corresponding amplitude signals. Beneath are indicated the fractions of down domains S extracted from the images. (g) Resistances of the different states. Measurement error bars are of the order of the data point size. [published in Ref. 310]

  Figure 2.3.3

Figure 2 .

 2 Figure 2.3.3

Figure 2 . 3 . 3 -

 233 Figure 2.3.3 -Connection between the ferroelectric domain configuration imaged by PFM and the electric resistance level for different states of an FTJ. (a) Electric resistance versus the fraction of down domains extracted from pfm images for several junctions. The black line corresponds the parallel conduction model described in (b). (b) Schematic of the parallel conduction model. [published in Ref. 310]

Figure 2 .

 2 Figure 2.4.1

Figure 2 . 4 . 1 -

 241 Figure 2.4.1 -Schematic structures of the Pt (10 nm)/x (10 nm)/BiFeO 3 /Ca 0.96 Ce 0.04 MnO 3 //YAlO 3 FTJs where x corresponds to one of the following metals: (a) W, (b) Co, (c) Ni, (d) Ir.

Figure 2 .

 2 Figure 2.4.2

Figure 2 .

 2 Figure 2.4.2

Figure 2 .

 2 Figure 2.4.4 -e-h. The observed resistance variations are thus solely reflecting

Figure 2 . 4 . 2 -

 242 Figure 2.4.2 -Electrical characteristics of the virgin states of FTJs with top electrodes of W, Co, Ni, and Ir. (a) Resistance-area product as a function of the respective work function at an applied voltage of -0.2 V (open symbols ) and -0.5 V (filled symbols ). (Lines are only a guide to the eye.) (b) Barrier heights at the top electrode interface as determined by fits of the currentvoltage measurements as a function of the respective work function. (The line connects the mean values of each top-electrode material.) [published in Ref. 326]

Figure 2 .

 2 Figure 2.4.3

Figure 2 . 4 . 3 -

 243 Figure 2.4.3 -Current-density-voltage characteristics of the virgin states of representative BFO/CCMO FTJs with different top electrodes. Points indicate measured values; lines correspond to the respective fits using the model of combined direct tunneling and Fowler-Nordheim tunneling. (a) Comparative graph including data for all four top electrodes. The inset shows the schematic energy barrier profile used for the fits. (b-d) Measurements and corresponding fits for Co, Ni, and Ir electrodes, respectively. Fitted values of the barrier height at the top interface ϕ B,2 , the effective electron mass m * , and the relative amplitude of the Fowler-Nordheim tunneling contribution p FNT are indicated for each electrode. [published in Ref. 326]

Figure 2 .

 2 Figure 2.4.4-a-d shows typical hysteresis cycles obtained for ftjs containing the different tes. Each data point corresponds to the dc resistance measured at V read = -0.2 V after applying a voltage pulse of amplitude V write and duration of 100 ns. Starting in the virgin state which corresponds to the high-resistive off state (polarization pointing toward ccmo; open symbols in figure 2.4.4-a-d), negative voltage pulses lead to switching to the low-resistive on state (polarization pointing toward the te). These assumptions are confirmed by pfm images in the on and off states for the different tes (figure 2.4.4-e-h).

Figure 2 . 4 . 4 -

 244 Figure 2.4.4 -Switching characteristics of FTJs with different top electrodes. (a-d) Representative hysteresis cycles of resistance-area product versus write pulse voltage for the respective top electrode materials. The ftjs are initially in the high-resistive virgin state and then switch to the low resistance on state under application of negative voltage pulses with increasing amplitude (open symbols ). The application of positive voltage pulses gradually increases the resistance to the off state from which the low-resistive on state can again be reached under application of negative voltage pulses (filled symbols ). Voltage pulses of increasing amplitude are spaced by 0.1 V. (e-h) pfm phase (color) and afm topography data (3d profile) of representative ftjs for each top electrode in the off and on states. [published in Ref. 326]

Figure 2 . 4 . 5 -

 245 Figure 2.4.5 -a, revealed that the ultrathin films of bfo purely are in the t-phase and exhibit a large out-of-plane lattice parameter of c = 0.464 nm. M. Marinova et al. from the Laboratoire de Physique des Solides 7 used aberration-corrected stem to examine cross-sections of BiFeO 3 (10 u.c.)/ LaNiO 3 / /LaAlO 3 (001) structures.Figure 2.4.5-b shows a haadf image of the heterostructure and reveals that both lno and bfo are coherently grown on the lao substrates,

  Figure 2.4.5-b shows a haadf image of the heterostructure and reveals that both lno and bfo are coherently grown on the lao substrates, 7) Laboratoire de Physique des Solides, CNRS UMR 8502, Université Paris Sud XI, Orsay, France.

Figure 2 . 4 . 5 -

 245 Figure 2.4.5 -Structural characterization of BiFeO 3 /LaNiO 3 //LaAlO 3 samples. (a) ω-2θ xrd patterns on BiFeO 3 (n u.c.)/LaNiO 3 (10 nm)/ /LaAlO 3 (n = 5, 9, 13) samples showing the presence of pure t-phase bfo with giant tetragonality and the bottom (001)-oriented lno electrode. (b) High-resolution stem-haadf image of a cross-sectional BiFeO 3 (10 u.c.)/LaNiO 3 / /LaAlO 3 sample. (c) eels maps of La M 5 , Ni L 3 , and Fe L 3,2 edges collected on a different area of the same cross-sectional sample. Ruddlesden-Popper faults are indicated by arrows in (b) and (c). The scale bars in (b) and (c) are 1 nm. [published in Ref. 332]

  Figure 2.4.6

Figure 2 . 4 . 6 -

 246 Figure 2.4.6-a shows the topography maps revealing flat surfaces with terraces and one-unit-cell steps. We then used pfm to characterize the local polarization state of bfo. pfm maps performed on a bfo (10 u.c.) sample are shown

Figure 2 .

 2 4.6-e-g shows pfm images of a typical matrix of nine ftjs with diameters of 300 nm. In the virgin state, all ftjs are in a homogeneous, downwardpointing (toward lno) polarization configuration. This is shown by the bright phase signal in figure 2.4.6-f and the well-defined amplitude in figure2.4.6-g for six out of nine ftjs (top left and bottom right corners) that did not experience any voltage pulses. The polarization state of the ultrathin films is thus neither altered by the modification of the surface boundary conditions (Co/BiFeO 3 interface instead of a free bfo surface) nor by the nanofabrication process.

  .4.6-b.

Figure 2

 2

Figure 2 .

 2 Figure 2.4.7shows a more detailed investigation of the polarization and resistance switching of two a-type ftjs. The first ftj is initially in the typical homogeneous state with the polarization pointing downward (figure 2.4.7-a). After a single 100-ns voltage pulse of -2 V applied to the top electrode, the ferroelectric polarization is completely switched upward (toward Co), as indicated by the dark phase contrast in figure 2.4.7-a.

Figure 2 . 4 . 6 -

 246 Figure 2.4.6 -Polarization switching and electronic tunneling transport in FTJs with LaNiO 3 bottom electrodes. (a) Topography, (b) out-of-plane pfm phase, and (c) amplitude on a 6×6 µm 2 scan after writing a 3×3 µm 2 domain with upward polarization (V lno = 2.2 V) on a BiFeO 3 (10 u.c.)/ LaNiO 3 (10 nm)/ /LaAlO 3 sample. (d) Average resistances deduced from c-afm resistance maps performed on samples with bfo thicknesses of 0, 2, 6, and 10 u.c. (e) Topography, (f) out-of-plane pfm phase and (g) amplitude on a 3×3 matrix of a-type ftjs (3×3 µm 2 scan). The "tic-tac-toe" is achieved by applying voltage pulses of -2 V to the top electrode of the three diagonal devices (upward polarization with dark phase). The diameter of the ftjs is 300 nm. (h) Distribution of the resistancearea (ra) products of a-type ftjs in the virgin state as a function of the ftj diameter (averaged over 30 ftjs at V Co = -0.2 V). [published in Ref. 332]

Figure 2 . 4 . 7 -

 247 Figure 2.4.7 -Switching hystereses of FTJs with LaNiO 3 bottom electrodes. (a) pfm phase and amplitude imaging for an a-type ftj in the virgin, on, intermediate (Int.), and off resistance states (diameter of 300 nm). (b) Corresponding resistance states of the ftj from a) as a function of voltage pulses. A large negative voltage pulse (-2 V) results in a complete switching of the polarization. (c) pfm phase and amplitude imaging in the virgin, on, intermediate (Int.) and off resistance states for another a-type ftj (diameter of 420 nm). (d) Corresponding resistance states of the ftj from (c) as a function of voltage pulses. A reduction of the irreversibility (Irrev.) of the off versus virgin states in (d) compared to (b) results from the smaller negative voltage pulse applied (-1.7 V). [published in Ref. 332]

Figure 2 .

 2 4.8

Figure 2 .

 2 Figure 2.4.8

Figure 2 . 4 . 8 -

 248 Figure 2.4.8 -Resistances of B-type FTJs with LaNiO 3 bottom electrodes investigated as a function of temperature. (a) Hysteresis cycles of the resistance of junction #1 as a function of write voltage pulses measured at 290 to 80 K. (b) Temperature dependence of the virgin state of two junctions (#3 and #4), of the on, off, and bottom electrode states of junctions #1 and #2 from 290 to 80 K. (c) Temperature dependence of the ter of junctions #1 and #2 calculated from (b). [published in Ref. 332]

Figure 2 . 4 . 9

 249 Figure 2.4.9

Figure 2 . 4 . 9 -

 249 Figure 2.4.9 -Temperature dependence of the normalized conductance of FTJs with LNO bottom electrodes. The points correspond to the same experimental data as in figure 2.4.8-b. The lines are the calculated temperature dependencies under consideration of inelastic tunneling through chains of n localized states in the barrier using the model by L. Glazman & K. Matveev[START_REF] Glazman | Inelastic Tunneling Through Thin Amorphous Films[END_REF]. n = 0 then corresponds to direct tunneling (dt)[START_REF] Stratton | Volt-current characteristics for tunneling through insulating films[END_REF][START_REF] Simmons | Generalized Thermal J-V Characteristic for the Electric Tunnel Effect[END_REF]. Each data set is normalized to its respective value at T = 80 K. For the calculations shown here, we assumed a value of 1 nm for the radius of the localized states.

  et al., Pt/BaTiO 3 (4.8 nm)/SrRuO 3 tunnel junctions have shown strong resistance variations of two orders of magnitude in the off and on states from 50 to 290 K that the authors interpreted by these inelastic processes. Furthermore, Y. W. Yin et al. observed a strong decrease of the ter in La 0.67 Sr 0.33 MnO 3 / BaTiO 3 (3 nm)/La 0.5 Ca 0.5 MnO 3 junctions originating from a decrease of the off-state resistance by two orders of magnitude from 50 to 290 K. The ter variation is weaker here, suggesting a lower defect concentration. Despite the possible nuisance of defects that weaken the off-state resistance at room temperature, the ter observed in these bfo-based ftjs with lno bottom electrodes reaches very large values. While the ter obtained in junction #1 reaches more than 10 5 (figure 2.4.8-c), the off-state of the majority of junctions exhibits lower resistance values, as shown by the representative junction #2 (figure 2.4.8-b). Their resulting ter at room temperature is about 10 4 (figure 2.4.8-c).

Figure 2

 2 Figure 2.5.1-a shows the topography of five top electrodes with diameters of 180, 290, 400, 630, and 1200 nm as verified by afm. These images confirm the circular shape as well as smooth edges of the structured top electrodes of Pt (10 nm)/Co (10 nm).

Figure 2 . 5 . 1 -

 251 Figure 2.5.1 -Resistance hysteresis measurements for different FTJ sizes. (a) Topography images of the top electrodes recorded by afm for each ftj size. The lateral scalebar corresponds to 500 nm, whereas the height data is not to scale. (b) Resistance hysteresis measurements read at low dc voltage of V read = -0.2 V after the application of 100-ns voltage pulses with increasing/decreasing amplitudes. Open symbols denote the first hysteresis loop, filled symbols a second one, respectively. For each ftj size, the measurements on two different ftjs are shown.

Figure 2 . 5 . 2

 252 Figure 2.5.2

Figure 2 . 5 . 3

 253 Figure 2.5.3

Figure 2 . 5 . 2 -

 252 Figure 2.5.2 -Resistance-area (RA) values of hysteresis measurements for different FTJ sizes. The data are based on the resistance values shown in figure 2.5.1-b and the top electrode sizes correspond to those determined by afm topography images.

Figure 2 . 5 . 3 -

 253 Figure 2.5.3 -Ferroelectric domain configurations for different FTJ sizes. (a) pfm phase (left) and amplitude (right) images of the off state polarization configurations. (b) pfm phase (left) and amplitude (right) images of the on state polarization configurations. Scalebars for each ftj size correspond to 100 nm.

Figure 2 .

 2 Figure 2.5.3-b shows the pfm images of the on state for each ftj size. The pfm phase signals indicate that the polarization only switches partially so that the upward-pointing

Figure 2 . 5 . 4 Figure 2 . 5 . 4 -

 254254 Figure 2.5.4 -Electric field distribution in the ferroelectric barrier of an FTJ. The field strength values are calculated using a finite-elements method and the partial differential equation toolbox of Matlab to solve the Poisson's equation of the two-dimensional problem under Dirichlet boundary conditions.For this, we considered ε bfo /ε resist = 10 and the electrodes were assumed to be perfectly metallic. The chosen potential difference V between the top and bottom electrodes leads to a naively calculated electric field of V/d = 10 a.u. with the bfo thickness d. The x-y-axes ratio of the image equals 1 so that the geometry is not distorted. The white arrows indicate the direction of the local electric field while their length is proportional to its strength. (The image presented here shows a part of the calculated geometry that was cropped to the region of interest around the ftj.)

. 1 . 1

 11 Figure 3.1.1

Figure 3 . 1 . 2

 312 Figure 3.1.2

Figure 3 . 1 . 1 -Figure 3 . 1 . 2 -

 311312 Figure 3.1.1 -Structure of the fully patterned (B-type) ferroelectric tunnel junctions. (a) Optical microscope image of the chip after patterning showing the contact pads of 5×10 ftjs. The total size of the sample is about 5×10 mm 2 . (b) 3D representation of a zoomed area containing a few ftjs. The three parallel bars are the ground-signal-ground contact pads. (c) 3D sketch of one ftj showing the layer structure. (Not to scale.) [published in Ref. 338]

Figure 3 . 1 . 3

 313 Figure 3.1.3

Figure 3 . 1 . 3 -

 313 Figure 3.1.3 -Characterization statistics of the fully patterned B-type ferroelectric tunnel junctions. (a) Typical resistance hysteresis of an ftj as a function of pulse voltage. The junction is initially in the high resistive virgin state and switches to the low resistive on state upon application of negative voltage pulses with increasing amplitude and of 100 ns length. The application of positive voltage pulses gradually increases the resistance to the off state which restores the resistance of the virgin state. A step of 0.5 V is chosen for the voltage pulses. (b) Distributions of the virgin, on, and off states for the 45 working ftjs. The inset shows the distribution of the off/on ratio. Resistance was measured under dc voltage of -0.2 V. [published in Ref. 338]

Figure 3 . 2 . 1

 321 Figure 3.2.1

Figure 3 . 2 . 1 -

 321 Figure 3.2.1 -Schematic diagrams of a capacitor showing the time-dependent switching process at differently fatigued states. The arrows indicate the polarization direction of the domains. Charged defects ( ) lead to domain wall pinning and therefore hinder the polarization reversal. (Structure studied: Au/PbZr 0.4 Ti 0.6 O 3 (pzt, 200 nm)/SrRuO 3 ) [adapted from Ref. 317]

Figure 3 . 2 . 2

 322 Figure 3.2.2

Figure 3 . 2 . 2 -

 322 Figure 3.2.2 -Endurance measurement on an A-type Pt/Co/BiFeO 3 /Ca 0.96 Ce 0.04 MnO 3 FTJ contacted by an AFM tip. After each application of a write pulse, the resistance is read at a small dc voltage of -0.2 V. [published in Ref. 310]

Figure 3 . 2 . 3

 323 Figure 3.2.3

Figure 3 . 2 . 3 -

 323 Figure 3.2.3 -Schematic flowchart of the endurance measurement procedure.

Figure 3 . 2 . 4 Figure 3 . 2 . 4

 324324 Figure 3.2.4

Figure 3 . 2 . 4 -

 324 Figure 3.2.4 -Endurance measurement on a fully patterned Pt/Co/BiFeO 3 /Ca 0.96 Ce 0.04 MnO 3 FTJ. (a) Resistance as a function of cycle number. (b) Distribution of the on and off state resistances. Resistance was measured under a dc voltage of -0.2 V. [published in Ref. 338]

Figure 3 . 2 . 5

 325 Figure 3.2.5

Figure 3 .

 3 Figure 3.2.5-b provides a more detailed look into the blocked states by showing the evolution of the number of deblocked states for each voltage pulse amplitude used in the deblocking procedure (see figure 3.2.3) -in other words, the minimal pulse needed

  l e n u m b e r N u m b e r o f b l o c k e d s t a t e s l e n u m b e r R e l a t i v e f r a c t i o n

Figure 3 . 2 . 5 -

 325 Figure 3.2.5 -Characteristics of the blocked resistance states during endurance testing. (a) Accumulated number of blocked states as a function of cycle number. The inset shows the same data on a logarithmic scale. (b) Accumulated number of deblocked states for each deblocking voltage pulse amplitude as a function of cycle number. (c) Relative fraction of each deblocking voltage pulse amplitude as a function of cycle number. These values were extracted by using a moving window of 61 data points.

Figure 3 . 3 . 1 -

 331 Figure 3.3.1 -a-d). ftj #1 was switched close to the on state (figure 3.3.1-a). ftjs #2 and #3 were switched to two different intermediate resistance states (figure 3.3.1-b,c). ftj #4 was first switched to the on state and then close to the off state (figure 3.3.1-d).After an idle period of about three days (68 hours), the same devices were imaged by pfm again and measured electrically (figure3.3.1-e-h). Note that the apparent distortion of the

Figure 3 .

 3 3.1-e shows the pfm signal of ftj #1 where small bright domains have nucleated in the bottom part of the capacitor (top panel). The amplitude of this region indicated the presence of small reversed domains already in the initial state (bottom panel of figure 3.3.1-a) and exhibits a further reduction after the idle period (bottom panel of figure 3.3.1-e). The pfm images of ftjs #2 (figure 3.3.1-f), #3 (figure 3.3.1-g) and #4 (figure 3.3.1-h) show little to no change to the respective initial domain configuration, both in phase and amplitude.

  Figure 3.3.1-i displays the resistance level of the four devices, with relatively small resistance variations over three days. It should be noted that the large ac signal (compared to the coercive threshold voltages) required for pfm experiments sometimes slightly destabilizes the ferroelectric domain configurations. Overall, these bfo-based ftjs thus show good retention properties, even in intermediate resistance states.

Figure 3 . 3 . 1 -

 331 Figure 3.3.1 -Retention measurement on four Co/BiFeO 3 /Ca 0.96 Ce 0.04 MnO 3 FTJs with diameters of 180 nm in different resistance states contacted by AFM. (a-d) pfm images of the four junctions shortly after writing the respective resistance state. (e-h) pfm images of the same four junctions after 68 h. Due to image drifts during pfm imaging, the top electrodes appear slightly distorted in respect to their regular circular shapes. (i) Resistances of the four ftjs shortly after writing and after 68 h. [published in Ref. 310]

Figure 3 . 3 . 2

 332 Figure 3.3.2

Figure 3 . 3 . 2 -

 332 Figure 3.3.2 -Retention measurement on a fully patterned Pt/Co/BiFeO 3 /Ca 0.96 Ce 0.04 MnO 3 FTJ for different initial states that are set by the application of voltage pulses of the indicated amplitude. The dashed lines show an extrapolation to ten years under the assumption of a power-law dependence. Resistance was measured under a dc voltage of -0.1 V. [published in Ref. 338]

Figure 3 .

 3 Figure 3.4.1

Figure 3 .

 3 Figure 3.4.1

Figure 3 . 4 . 1 -

 341 Figure 3.4.1 -Circuit scheme used for single-shot transmission measurements. All rf connections are made via coaxial cables which, for the sake of clarity, are omitted here. The oscilloscope is triggered by the trigger signal of the pulse generator.

Figure 3 . 4 . 2 Figure 3 . 4 . 2

 342342 Figure 3.4.2

Figure 3 . 4 . 2 -

 342 Figure 3.4.2 -Single-shot measurement signals. (a) Voltage and current signal traces under application of a voltage pulse of V p = -1.8 V in amplitude and of τ p = 1 µs in length. The voltage pulse does not change the ftj resistance (R before = 1.3 × 10 8 Ω, R after = 1.2 × 10 8 Ω) so that the recorded current peaks are only due to parasitic circuit effects. (b) Magnification of the pulse start. (c) Magnification of the pulse end. Apart from the voltage pulse at the ftj ( ), the oscilloscope records the raw current I dir ( ) and the amplified current I amp ( ). The combined current signal I comb ( ) is used to fit ( ) the capacitance model presented in figure 3.4.3. After subtraction of a non-switching reference pulse, we obtain the combined current without reference I comb,∆ ( ).

Figure 3 . 4 . 3

 343 Figure 3.4.3

Figure 3 . 4 . 3 -

 343 Figure 3.4.3 -Parasitic capacitance model. (a) Equivalent circuit scheme of the ftj device. The ftj with resistance R ftj is in parallel to a capacitance C. The overall current is limited by a series resistance R s . (b) Simplified sketch of the electrode geometry in the ftj devices (proportions not to scale). The part that supposedly presents the main capacitance contribution is highlighted.

Figure 3 . 4 . 4 Figure 3 . 4 . 4

 344344 Figure 3.4.4

Figure 3 . 4 . 4 -

 344 Figure 3.4.4 -Single-shot measurements of OFF-to-ON switching. Typical transient current recordings under different voltage pulse amplitudes between -2.05 to -3.00 V. The peaks at the beginning (t = 0 ns) and the end (t = 1000 ns) of the voltage pulses are artifacts due to small fluctuations in the non-switching reference measurements.

Figure 3 .

 3 Figure 3.4.5

Figure 3 . 4 . 5 -

 345 Figure 3.4.5 -Waiting times for OFF-to-ON switching. The values are extracted from single-shot measurements. (a) Waiting times as a function of voltage pulse amplitude. (b) Mean waiting times as a function of the reciprocal electric field E. Error bars correspond to the standard deviation of the waiting times at a given voltage pulse amplitude. The data ( ) are fitted ( ) corresponding to Merz's law, under exclusion of some points at very high or low electric fields ( ) as detailed in the text. This yields an activation field of E a = -1.80 ± 0.11 V/nm and an attempt time constant of t 0 = 1.1 ± 0.5 × 10 -11 s.

Figure 3 . 4 . 6 Figure 3 . 4 . 6

 346346 Figure 3.4.6

Figure 3 . 4 . 6 -

 346 Figure 3.4.6 -Typical transient current for ON-to-OFF switching. (a) In the case of rectangular pulses, no measurable current flows during the voltage pulse (R before = 3 × 10 5 Ω, R after = 1 × 10 8 Ω).(b) In the case of a slowly increasing voltage (here in form of a triangular pulse with very long rise times resulting in a slope of r v = 16 V/µs), a finite current initially flows and increases with the increasing voltage. At a certain moment, the current amplitude decreases, indicating the onset of the resistance switching (R before = 1 × 10 5 Ω, R after = 5 × 10 8 Ω).

Figure 3 . 4 . 7 Figure 3

 3473 Figure 3.4.7 Figure 3.4.7

Figure 3 . 4 . 8

 348 Figure 3.4.8

Figure 3 . 4 . 8

 348 Figure 3.4.8

  2.1) by which we could successfully describe the off-to-on switching data (see figure 3.4.5-b). The probability density ϕ of the switching

Figure 3 . 4 . 7 -Figure 3 . 4 . 8 -

 347348 Figure 3.4.7 -Typical transient current traces for ON-to-OFF switching under triangular pulses with different slopes r v : (a) r v = 4.4 V/µs, (b) r v = 14 V/µs, (c) r v = 31 V/µs, and (d) r v = 41 V/µs. The switching can be linked to a certain waiting time (marked with a dotted line) after which the current decreases despite the increasing voltage. (Note the different time scales in the graphs.)

Figure 4 . 1 . 1

 411 Figure 4.1.1

Figure 4 . 1 . 1

 411 Figure 4.1.1

Figure 4 . 1 . 1 -

 411 Figure 4.1.1 -Cumulative switching of an FTJ. The results shown here were obtained on a junction with a diameter of 180 nm. (a) pfm phase and amplitude images of the on state as well as after application of multiple cumulative write voltage pulses of 1 V. The scale bar corresponds to 50 nm. (b) Evolution of the relative fraction of down domains S as extracted from the pfm phase images ( ) and from purely electric resistance using the model of parallel conduction ( ).

Figure 4 . 1 . 2 Figure 4 . 1 . 2

 412412 Figure 4.1.2

Figure 4 . 1 . 2 -

 412 Figure 4.1.2 -Polarization reversal probed by electrical measurements. Evolution of the normalized reversed area S extracted from electrical resistance measurements as a function of cumulative application time of voltage pulses for different amplitudes. The data were obtained on an ftj of 180 nm in diameter.

Figure 4 . 1 . 3

 413

Figure 4 . 1 . 3

 413 Figure 4.1.3

Figure 4 . 1 . 3 -

 413 Figure 4.1.3 -Polarization reversal in BFO predicted by molecular dynamics simulations. (a) Evolution of the polarization configuration along the [001] direction under an electric field of 2.5 V/nm. (b) Corresponding normalized switched relative area as function of time. (c) Normalized switched relative area as function of time under different electric fields. The fits are done using equation 4.1.3 based on the nucleation-limited switching model presented below.

  figure 4.1.2 and figure 4.1.3-c, respectively). In the standard kai model (see section 1.2.1.2.2 on page 25) this spread is exclusively determined by the dimensionality n of the reversal dynamics.

Figure 4 . 1 . 4

 414 Figure 4.1.4

  .1.3-c) reveal the same trends: for lower fields, the switching-time distributions are larger and centered at higher mean values, as displayed in the inset of figure 4.1.4-b.

Figure 4 . 1 . 4 -

 414 Figure 4.1.4 -Modeling the ferroelectric reversal dynamics. The measurements were obtained on an ftj with a diameter of 180 nm. (a) Evolution of the normalized reversed area extracted from electrical resistance measurements as a function of cumulative application time of voltage pulses for different amplitudes. The black lines show the respective fits using the reversal model from equation 4.1.3. (b) Distributions of waiting times for several electric fields corresponding to the fits in (a). The inset shows the fit results obtained on the md simulation data in figure 4.1.3-c. (c) Mean switching times τ 0 as determined by fits on the experimental and simulated data plotted against the respective reciprocal electric field (data points). The line shows the fit to Merz's law.

Figure 4 . 1 . 5

 415 Figure 4.1.5

Figure 4 . 1 . 5 -

 415 Figure 4.1.5 -Method for the model generalization to arbitrary voltage signals. (a) The voltage waveform is divided into small segments of a width ∆t during which the voltage V i is assumed to be constant. (Schematic only, ∆t not to scale.) (b) Based on the reversed area S i-1 before each segment i, we use equation 4.1.3 to find the time offset t offset i corresponding to a voltage V i . After this, we apply equation 4.1.3 again to calculate the new value S i = S(t offset

Figure 4 . 2 . 1

 421 Figure 4.2.1

Figure 4 . 2 . 1 -

 421 Figure 4.2.1 -Measurement of STDP in hippocampal rat neurons. Change in the excitatory post-synaptic current (EPSC) as a function of the relative timing between the onset of the excitatory post-synaptic potential (EPSP) and the peak of the post-synaptic action potential. The measurements have been obtained after repeated stimulation by pre-and post-neuron spikes of each relative timing and the data show the respective averaged changes. [adapted from Ref. 36]

  Figure 4.2.2

Figure 4 .

 4 Figure 4.2.2

  pre -t post (ms)

Figure 4 . 2 . 2 -

 422 Figure 4.2.2 -Schematic forms of STDP measured on several types of biological synapses. Depending on the relative timing of pre-and post-neuron spikes, the synaptic weight is either strengthened (long-term potentiation ) or weakened (long-term depression ). Note the inverse definition of the relative timing used here. The time values are only indications of the approximate time scales. [adapted from Ref. 374]

Figure 4 . 2 . 3

 423

Figure 4 . 2 . 3

 423 Figure 4.2.3

Figure 4 . 2 . 3 -

 423 Figure 4.2.3 -Synaptic weight modification via STDP tends to prefer first spikes. (a) Simplified stdp curve used in the illustration. (b-s) Simple diagram of a single integrate-and-fire neuron with a threshold of 2 connected to five pre-neurons via synapses. At the start, all synapses are initialized to the same weight.(The order of the synapses in relation to the incoming spikes is chosen for readability reasons but does not limit the conclusions drawn here.) The panels show the incoming spikes as well as the evolution of the synaptic weights and of the internal state of the neuron. Before each spike wave, the neuron is reset to zero (b,h,n). (This could also be achieved by neuron leakage.) After only three spike waves, stdp has reinforced the synapses that are connected to early firing pre-neurons and has suppressed late arriving signals. [inspired byRef. 381]

Figure 4 . 2 . 4

 424

Figure 4 . 2 . 4

 424 Figure 4.2.4

Figure 4 . 2 . 4 -

 424 Figure 4.2.4 -Pattern detection with STDP.(a) Layout of the simulated ann. All input neurons are linked to a single output neuron and the weights w i of the connecting synapses evolve according to a classical stdp learning curve. In addition to the inputs I i , a common oscillating signal i(t) is applied to all inputs. (b) Input spike trains from a small selection out of 2000 input neurons. The zoomed regions show spikes that correspond to the slightly noisy pattern ( ) and the random activity of other input neurons. The oscillating signal modulates the overall activity. (c) Evolution of the output neuron potential. It generally follows the oscillating input modulation but only spikes ( ) when the pattern is present. [adapted from Ref.[START_REF] Masquelier | Oscillations, Phase-of-Firing Coding, and Spike Timing-Dependent Plasticity: An Efficient Learning Scheme[END_REF][START_REF] Masquelier | The Timing of Vision -How Neural Processing Links to Different Temporal Dynamics[END_REF]

Figure 4 . 2 . 5

 425 Figure 4.2.5

Figure 4 . 2 . 5 -

 425 Figure 4.2.5 -Implementation of STDP with memristors. (a) Classical method based on clocked, synchronous communication proposed by G. Snider [75]. (b) Asynchronous, analog implementation based on specially designed pre-and post-neuron waveforms proposed by B. Linares-Barranco & T. Serrano-Gotarredona [77]. [adapted from Ref. 75, 77]

Figure 4 . 2 . 6

 426

Figure 4 . 2 . 6

 426 Figure 4.2.6

Figure 4 . 2 . 6 -

 426 Figure 4.2.6 -Different neuron spike waveforms and the corresponding simulated STDP curves. Taking identical pre-and post-neuron spikes, the simulations are based on a fictional memristor model with clear threshold voltages ±V th . Any |V mem | < V th is assumed to not induce any changes in the memristor conductance. [data from Ref. 389]

Figure 4

 4

Figure 4

 4 Figure 4.2.7

. 3 . 1

 31 Figure 4.3.1

Figure 4

 4 Figure 4.3.2

Figure 4 . 3 . 1 -

 431 Figure 4.3.1 -Shapes of the neuron spikes for STDP with FTJ-based memristors. (a) Pre-neuron waveform. (b) Post-neuron waveform. The voltage sign is displayed relative to the polarity used before which defines that the voltage is applied to the top electrode of the ftj.

Figure 4 . 3 . 2 -

 432 Figure 4.3.2 -Overlapping of pre-and post-neuron spikes with different time delays. The upper (middle) row displays the pre-(post-)neuron spike voltage V pre (-V post ) with a certain delay ∆t and the bottom row shows the resulting voltage drop at the memristorV mem = V pre -V post . (a)In the case of a delay longer than the length of the pre-or post-neuron spikes |∆t| > τ p , the two do not overlap. As a consequence, the voltage at the memristor stays below the thresholds and the conductance does not change. (b) The pre-neuron spikes first and the post-neuron spikes arrives by a delay ∆t > 0 later. The resulting voltage at the memristor momentarily exceeds the negative threshold voltage V - th , leading to an increase of the memristor conductance. (c) The post-neuron spikes first and the pre-neuron spikes arrives by a delay ∆t < 0 later. The resulting voltage at the memristor momentarily exceeds the positive threshold voltage V + th , leading to a decrease of the memristor conductance. (Please note that for the sake of clarity, the post-neuron spike voltages are displayed as -V post , i. e., relative to the memristor polarity.)

Figure 4 . 3 . 3

 433 Figure 4.3.3

Figure 4 . 3 . 3

 433 Figure 4.3.3

Figure 4 . 3 . 3 -

 433 Figure 4.3.3 -Measurement of STDP in an FTJ. Modulation of the device conductance ∆G as a function of the time difference between pre-and post-synaptic spikes ∆t. The differently colored data sets correspond to seven measurements collected on the same device (Sample S1) showing the reproducibility of the effect.

Figure 4 . 3 . 4

 434

Figure 4

 4 Figure 4.3.4 -a shows nested hysteresis loops of resistance as a function of voltage pulse amplitude. The stdp curve of the same device is displayed in figure 4.3.4-b. Combining the nls model and the technique presented in section 4.1.3.2 on page 99 with the parallel resistance model, we simultaneously fit the resistance-versus-voltage cycles and the stdp curve. Both resistance variations can be accurately replicated (lines in figure 4.3.4) using the same set of parameters τ 0 (V) and Γ(V).

Figure 4 . 3 . 4 -

 434 Figure 4.3.4 -Parallel fitting of hysteresis and STDP measurements results. (a) Multiple resistance hysteresis loops as a function of the voltage amplitude of 100-ns pulses. The dc resistance measured at -0.2 V is mainly determined by the maximum pulse amplitude of each cycle (Sample S2). (b) stdp curve obtained using the spikes presented in figure 4.3.1 (Sample S2). Lines in (a,b) indicate the fit result.

Figure 4 . 3 . 5

 435 Figure 4.3.5

Figure 4 . 3 . 5

 435 Figure 4.3.5

Figure 4 . 3 . 5 -

 435 Figure 4.3.5 -Standard STDP measurements on three different samples. (a) Sample S1. (b) Sample S2. (c) Sample S3.

Figure 4 . 3 . 6

 436 Figure 4.3.6

Figure 4 . 3 . 6 -

 436 Figure 4.3.6 -Prediction and implementation of different forms of STDP with FTJs. (a) Symmetrical stdp curve under identical pre-and post-neuron spikes (Sample S2). (b) Approximately rectangular stdp curve obtained when the slope in the neuron spikes is replaced by a constant plateau (Sample S3). The respective spike waveforms are shown in the top panels. Using the model parameters obtained in figure 4.3.4, the stdp curves of (a,b) can be predicted ( ).

Figure 4 . 4 . 1

 441

Figure 4 . 4 . 1

 441 Figure 4.4.1

Figure 4 . 4 . 2

 442 Figure 4.4.2

Figure 4 . 4 . 1 -Figure 4 . 4 . 2 -

 441442 Figure 4.4.1 -Combining spiking CMOS neurons with FTJ-based memristors. (a) Photograph of the spiking neuron chip. The die is packaged in a pin grid array with electrical contact made by wirebonding. (b) Photograph of the test circuit board on which the neuron chip as well as the memristor crossbar array are to be placed. (c) Optical microscopy image of the ftj-based 10×81 memristor crossbar array. The full width of the image corresponds to about 8 mm. (d) Photograph of the ftj-based memristor crossbar array from (c) with probing needles used for simple pulsed characterizations.

Figure 4 .

 4 Figure 4.4.1-c,d shows the ftj-based 10×81 memristor crossbar array we fabricated and tailored to be used with the cmos neuron chip. First characterization measurements revealed reliable switching with only little contributions from sneak paths[START_REF] Yépez | Study of ferroelectric memristive nanodevices for neuromorphic applications[END_REF]. Nonetheless, the resistance of the ccmo bottom electrodes revealed to be high in comparison to the

Figure A. 2 Figure A. 2 . 2 -

 222 Figure A.2.2 montre un cycle d'hystérèse ferroélectrique macroscopique. Sous l'application d'un champ électrique externe, la polarisation moyenne augmente jusqu'à ce qu'elle

Figure A. 2 . 3 .Figure A. 2 . 3 -

 2323 Figure A.2.3. La polarisation ferroélectrique pointe vers une des deux électrodes et peut être renversée par l'application d'une tension électrique aux bornes des électrodes. En 1971 déjà, L. Esaki et al. avaient prédit que la commutation du sens de la polarisation pourrait entraîner un changement de la résistance tunnel à travers cet empilement [222], avec un état à haute résistance (appelé état off) et un autre à basse résistance (appelé état on). Néanmoins, il fallut attendre la maturation des techniques de croissance et de caractérisation pour pouvoir réaliser des fines couches permettant le transport tunnel à travers la barrière ferroélectrique[START_REF] Gajek | Tunnel junctions with multiferroic barriers[END_REF][START_REF] Garcia | Giant tunnel electroresistance for non-destructive readout of ferroelectric states[END_REF][START_REF] Maksymovych | Polarization Control of Electron Tunneling into Ferroelectric Surfaces[END_REF][START_REF] Gruverman | Tunneling Electroresistance Effect in Ferroelectric Tunnel Junctions at the Nanoscale[END_REF][START_REF] Crassous | Giant tunnel electroresistance with PbTiO 3 ferroelectric tunnel barriers[END_REF].

Figure A. 2 . 4

 24 Figure A.2.4

  vii

Figure A. 2 . 4 - 1 , b 1 ) 2 , b 2 ) 3 , b 3 )

 24112233 Figure A.2.4 -Effets électrostatiques dans des FTJs asymétriques (c.à.d. avec des longueurs d'écrantage différents, λ 1 < λ 2 ). (a) État de basse résistance (on). (b) État de haute résistance (off). (a 1 , b 1 ) Distribution des charges dans la couche ferroélectrique et des charges dû aux porteurs de charge libres dans les électrodes métalliques. (a 2 , b 2 ) Profils des potentiels électrostatiques : en raison de l'écrantage incomplet des charges de polarisation, un champ de dépolarisation E d non-nul demeure. (a 3 , b 3 ) Profile des barrières énergétiques : les longueurs d'écrantage différentes induisent des changements de barrières aux interfaces différents ( et ) en cas d'un renversement de la polarisation. En conséquence, la hauteur de barrière moyenne change (entre ϕ B et ϕ B ), ce qui induit un changement de la résistance tunnel.

Figure A. 2 . 5 - 6 Figure

 256 Figure A.2.5 -Cycles d'hystérèse typiques d'une FTJ Pt/Co/BiFeO 3 /Ca 0.96 Ce 0.04 MnO 3 avec un diamètre de 180 nm. Le cycle 1 montre le basculement initial de l'état de haute résistance vers l'état de basse résistance on. Le cycle 2 montre une commutation reproductible entre les états on et off. Les cycles 3 à 6 correspondent à des cycles mineurs, où des états de résistance intermédiaires ont été stabilisés en limitant les tensions d'impulsion maximales.

Figure

  Figure A.2.7

  ix

Figure A. 2 . 6 -

 26 Figure A.2.6 -Images PFM de différents états de résistance d'une FTJ Pt/Co/BiFeO 3 / Ca 0.96 Ce 0.04 MnO 3 d'un diamètre de 180 nm. (a) État initial. (b) État on. (c-e) États intermédiaires successives avec des résistances croissantes. (f) État off. Les images supérieures montrent les signaux de phase et les images en dessous montrent des signaux d'amplitude correspondants. Les valeurs de fractions de domaines down S ont été extraites des images. (g) Résistances électriques des différents états. Les barres d'erreurs sont de l'ordre de la taille de points de données. [publié dans Ref. 310]

Figure A. 2 . 7 -

 27 Figure A.2.7 -Lien entre la configuration des domaines ferroélectriques imagée par PFM et le niveau de résistance électrique pour différents états d'une FTJ. (a) Résistance électrique en fonction de la fraction de domaines down S extraite des images pfm. Les résultats ont été obtenus sur plusieurs jonctions. La ligne noire correspond au modèle de conduction en parallèle décrit dans (b). (b) Schéma du modèle de conduction en parallèle. [publié dans Ref. 310]

  Figure A.2.8

Figure A. 2 . 8 -

 28 Figure A.2.8 -Caractéristiques électriques des états initiaux de jonctions avec des électrodes supérieures de W, Co, Ni ou Ir. (a) Produit résistance-surface en fonction du travail de sortie mesuré sous une tension de -0,2 V (symboles ouverts ) et de -0,5 V (symboles fermés ). (Les lignes sont de simples guides d'oeil.) (b) Hauteurs de barrière à l'interface avec l'électrode supérieure déterminées par ajustements de courbe des données couranttensions en fonction du travail de sortie respectif. (La ligne lie les valeurs moyennées pour chaque matériau d'électrode supérieure.) [publié dans Ref. 326]

Figure A. 2 . 9 -

 29 Figure A.2.9 -a-d montre des cycles d'hystérèse de ftjs de chaque matériau d'électrode supérieure. Le ratio off/on augmente avec la valeur du travail de sortie. Par contre, la commutation devient également plus instable. Seule les jonctions avec une électrode de Co et W font preuve d'une commutation stable et reproductible. Dans le cas de Ni, nous observons même une augmentation de la résistance sous des pulses de tension négative ce qui pourrait indiquer la formation de défauts.

Figure A. 2 . 10 -Figure A. 2 . 9 -

 21029 Figure A.2.10-b,d montre des cycles d'hystérèse résistive accompagnés d'images pfm qui montrent les configurations de la polarisation pour les différents états. L'état initial est caractérisé par une haute résistance et une polarisation down, comme dans le cas de ccmo. L'application d'un pulse de -2 V provoque un grand changement de résistance qui chute de 4 × 10 7 Ω à moins de 4 × 10 2 Ω. Par contre, dans la suite il n'est plus possible d'atteindre la résistance de l'état initial par des pulses de tension positive. L'état off montre des régions de polarisation

Figure A. 2 . 11

 211 Figure A.2.11

Figure A. 2 . 10 -Figure A. 2 . 11 -

 210211 Figure A.2.10 -Cycles d'hystérèse de FTJs de type A basées sur les électrodes inférieures de LaNiO 3 . (a) Images pfm d'une jonction d'un diamètre de 300 nm dans différents états. (b) Niveaux de résistance correspondants en fonction des amplitudes de pulses de tension. (c) Images pfm d'une autre jonction d'un diamètre de 420 nm dans différents états. (d) Niveaux de résistance correspondants en fonction des amplitudes de pulses de tension. La réduction de l'irréversibilité par rapport à celle observée dans (b) est due à une plus faible tension pour la commutation vers l'état on. [publié dans Ref. 332]

Figure A. 2 . 12

 212 Figure A.2.12

3 )Figure A. 2 . 12 -

 3212 Figure A.2.12 -Mesures d'endurance d'une FTJ de type B. (a) Résistance en fonction du nombre de cycles. (b) Distribution des résistances on et off. Les mesures de résistance ont été effectuées sous une tension de -0,2 V. [publié dans Ref. 338]

Figure A. 2 . 13

 213 Figure A.2.13

  xv

Figure A. 2 . 13 -

 213 Figure A.2.13 -Mesures de rétention d'une FTJ de type B pour des états différents initialisés par l'application de pulses de tension d'amplitudes indiquées. La ligne interrompue montre l'extrapolation à dix ans dans l'hypothèse d'une dépendance suivant une loi de puissance. Les mesures de résistance ont été effectuées sous une tension de -0,1 V. [publié dans Ref. 338]

Figure A. 2 . 14

 214 Figure A.2.14

Figure A. 2 . 15

 215 Figure A.2.15

5 )Figure A. 2 . 14 -

 5214 Figure A.2.14 -Évolution temporelle du courant transitoire pendant la commutation de l'état OFF à l'état ON.Les courbes montrent des enregistrements typiques pour des pulses de tension entre -2,05 à -3,00 V. Les pics au début (t = 0 ns) et à la fin (t = 1000 ns) des pulses de tensions sont des artefacts liés au bruit dans les mesures de référence.

2 MFigure A. 2 . 15 -

 2215 Figure A.2.15 -Temps d'attente de la commutation de l'état OFF à l'état ON extraits des évolutions temporelles des courants transitoires. (a) Temps d'attente en fonction de l'amplitude du pulse de tension. (b) Temps d'attente moyens ( ) en fonction de l'inverse du champ électrique E. Les barres d'erreurs correspondent à l'écart type de chaque amplitude de tension. L'ajustement d'une courbe ( ) suivant la loi de Merz donne un champ d'activation de E a = -1,80 ± 0,11 V/nm et un temps de relaxation de t 0 = 1,1 ± 0,5 × 10 -11 s. Certains points à haute et basse valeurs de champs ont été exclus de cet ajustement à cause des limitations physiques du circuit électrique.

xvii A. 2 . 4 . 1

 241 Dynamique de renversement de la polarisation ferroélectrique dans des couches ultraminces

Figure A. 2 . 16

 216 Figure A.2.16

  dans le cadre d'une collaboration. Ceci indique que ce mode de renversement est intrinsèque à la phase t du bfo et ne nécessite pas la présence de défauts. De chaque image, nous avons extrait la fraction relative de domaines down S (cf. figure A.2.16-b). Des mesures purement électriques -considérablement plus rapides que l'imagerie pfm -donnent également la valeur de S en utilisant le modèle de conduction en parallèle. Le très bon accord entre les deux types d'expériences (figure A.2.16-b) nous permet désormais d'étudier la dynamique par des mesures purement électriques.

Figure A. 2 . 16 -

 216 Figure A.2.16 -Commutation cumulative d'une FTJ. Les résultats montrés ont été obtenus avec une jonction d'un diamètre de 180 nm. (a) Images pfm de l'état on et des états intermédiaires suite à l'application de plusieurs pulses de tension de 1 V. La barre d'échelle correspond à 50 nm. (b) Évolution de la fraction relative de domaines down S extraite des images pfm ( ) et de mesures purement électriques en utilisant le modèle de conduction en parallèle ( ).

Figure A. 2 . 17

 217 Figure A.2.17

.

  Nous y observons que des tensions plus 6) Computational Condensed Matter Physics Group, Department of Physics and Institute for Nanoscience and Engineering, University of Arkansas, Fayetteville, Arkansas, USA. xviii hautes entraînent des renversements beaucoup plus rapides. Pour des analyses plus profondes, un modèle est nécessaire.

Figure A. 2 . 17 -

 217 Figure A.2.17 -Modèle du renversement de la polarisation ferroélectrique. Les mesures ont été effectuées avec une ftj d'un diamètre de 180 nm.(a) Évolution de la fraction relative de domaines down extraite de mesures de résistance as en fonction du temps cumulé de l'application de pulses de tension pour des amplitudes différentes. Les lignes noires montrent le modèle ajusté selon l'équation A.2.1. (b) Distributions des temps d'attente pour plusieurs valeurs de champ électrique extraites des mesures en (a). L'insert comporte les résultats obtenus pour les données de simulations de dynamique moléculaire. (c) Temps de renversement moyens τ 0 extraits des données expérimentales et des simulations en fonction de la valeur inversée du champ électrique (points). La ligne montre la loi de Merz ajustée aux données.

A. 2 . 4 . 2 . 1

 2421 Bases théoriquesSelon la stdp, le poids synaptique évolue en fonction du temps relatif entre les spikes des neurones post-et pré-synaptiques. La règle peut être résumée comme suit : si le neurone pré-synaptique émet un spike et peu de temps après le neurone post-synaptique émet également un spike, ces activités sont -avec une grande probabilité -liées et le poids de la synapse est renforcé par conséquent. Si, par contre, le neurone pré-synaptique émet un spike juste après le neurone post-synaptique, aucun lien causal n'est possible et le poids synaptique est donc affaiblit. La figure A.2.18

Figure A. 2 . 18 montre

 218 Figure A.2.18montre une des premières mesures du changement du poids synaptique dans un système biologique dont la forme est l'exemple typique d'une courbe de stdp[START_REF] Dan | Spike Timing-Dependent Plasticity of Neural Circuits[END_REF].

Figure A. 2 . 18 -

 218 Figure A.2.18 -Mesure de STDP dans des neurones hippocampiques d'un rat.Changement du poids synaptique en fonction du temps relatif entre des excitations de neurones pré-et post-synaptiques. Les valeurs montrées présentent des moyennes de plusieurs mesures. [adapté deRef. 36]

Figure A. 2 . 19 -

 219 Figure A.2.19 -Superpositions de spikes pré-et post-synaptiques avec des délais relatifs différents. L'axe supérieure (du milieu) montre la tension d'impulsion du neurone pré-(post-)synaptique V pre (-V post ) avec un délai ∆t et l'axe du bas montre la tension résultante aux bornes du memristorV mem = V pre -V post . (a)Si le délai excède la durée des spikes |∆t| > τ p , il n'y a pas de recouvrement. En conséquence, la tension au memristor reste en dessous du seuil et sa conductance reste inchangée. (b) Lorsque le spike post-neuronal arrive avec un délai ∆t > 0 après celui du neurone pré-synaptique, la tension au memristor excède momentanément la tension de seuil négative V - th et induit donc une augmentation de la conductance. (c) Lorsque le spike pré-neuronal arrive avec un délai ∆t < 0 après celui du neurone post-synaptique, la tension au memristor excède momentanément la tension de seuil positive V + th ce qui conduit à une réduction de la conductance du memristor.

Figure A. 2 . 19 .A. 2 . 4 . 2 . 2

 2192422 Figure A.2.19

Figure A. 2 . 20 -

 220 Figure A.2.20 -a) montrent que cette caractéristique existe dans nos ftjs. En utilisant les formes de pulses semblables à celles de la figure A.2.19, nous obtenons une courbe de stdp (figure A.2.20-b) similaire à celle mesurée dans un système biologique. Maintenant nous faisons usage du modèle de la dynamique du renversement de la polarisation et le combinons avec le modèle de conductance en parallèle pour décrire les résultats expérimentaux. Après l'ajustement des distributions des temps de renversement (équation A.2.1), le modèle reproduit les données avec un très bon accord. Cette description nous permet de prédire le comportement memristif sous des nouvelles formes d'impulsion. La figure A.2.21

Figure A. 2 . 21 montre

 221 Figure A.2.21 montre deux exemples de pulses pré-et post-synaptiques arbitraires. Les courbes de stdp calculées à partir du modèle reflètent ces changements. Les mesures que nous avons effectuées sont en très bon accord avec les prédictions.

Figure A. 2 . 20 -

 220 Figure A.2.20 -Application du modèle de la dynamique du renversement de la polarisation à l'hystérèse résistive et à la courbe de STDP mesurée. (a) Cycles d'hystérèse résistifs mineurs en fonction de l'amplitude de pulses de 100 ns. La résistance mesurée à -0,2 V est principalement déterminée par l'amplitude maximale de chaque sous-cycle (ftj S2). (b) Courbe de stdp expérimentale obtenue en appliquant le principe illustré dans la figure A.2.19 (ftj S2). Les lignes dans (a,b) indiquent le résultat d'ajustement du modèle de la dynamique du renversement de la polarisation.

Figure A. 2 . 21 -

 221 Figure A.2.21 -Prédiction et implémentation de différentes formes de STDP avec des FTJs. (a) Courbe de stdp symétrique (ftj S2). (b) Courbe de stdp approximativement rectangulaire (ftj S3). Les formes des impulsions pré-et post-synaptiques sont illustrées ci-dessus. En utilisant les paramètres du modèle obtenus sur les données de la figure A.2.20, nous pouvons prédire ces courbes de stdp ( ).

xxiii

  Université Paris-Saclay Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France Titre : Jonctions tunnel ferroélectriques : memristors pour le calcul neuromorphique Mots clés : memristor, neuromorphique, ferroélectrique, jonction tunnel, nanoélectronique Résumé : Les architectures d'ordinateur classiques sont optimisées pour le traitement déterministe d'informations pré-formatées et ont donc des difficultés avec des données naturelles bruitées (images, sons, etc.). Comme celles-ci deviennent nombreuses, de nouveaux circuits neuromorphiques (inspirés par le cerveau) tels que les réseaux de neurones émergent. Des nanodispositifs, appelés memristors, pourraient permettre leur implémentation sur puce avec une haute efficacité énergétique et en s'approchant de la haute connectivité synaptique du cerveau. Dans ce travail, nous étudions des memristors basés sur des jonctions tunnel ferroélectriques qui sont composées d'une couche ferroélectrique ultramince entre deux électrodes métalliques. Nous montrons que le renversement de la polarisation de BiFeO3 induit des changements de résistance de quatre ordres de grandeurs et établissons un lien direct entre les états de domaines mixtes et les niveaux de résistance intermédiaires. En alternant les matériaux des électrodes, nous révélons leur influence sur la barrière électrostatique et les propriétés dynamiques des memristors. Des expériences d'impulsion unique de tension montrent un retournement de polarisation ultra-rapide. Nous approfondissons l'étude de cette dynamique par des mesures d'impulsions cumulées. La combinaison de leur analyse avec de l'imagerie par microscopie à force piézoélectrique nous permet d'établir un modèle dynamique du memristor. Suite à la démonstration de la spike-timing-dependent plasticity, une règle d'apprentissage importante, nous pouvons prédire le comportement de ces synapses artificielles. Ceci représente une avance majeure vers la réalisation de réseaux de neurones sur puce dotés d'un auto-apprentissage non-supervisé. Title: Ferroelectric tunnel junctions: memristors for neuromorphic computing Keywords: memristor, neuromorphic, ferroelectric, tunnel junction, nanoelectronics

Thin films Substrates Lattice constant (Å)

		(La,Sr) 2 CuO 4 Bi 2 Sr 2 CuO 6 Bi 4 Ti 3 O 12 YBa 2 Cu 3 O 7 EuTiO 3 SrBi 2 Ta 2 O 9 BiMnO 3 BiFeO 3 BaTiO 3 PMN-PT Pr(Zr,Ti)O 3 (Ba,Sr)TiO 3	BaSnO 3 BiScO 3
	3.6	LuAlO 3 YAlO 3 NdAlO 3 LaAlO 3 3.7 3.8	SrTiO 3 NdGaO 3 LaGaO 3 DyScO 3 GdScO 3 NdScO 3 TbScO 3 SmScO 3 PrScO 3 EuScO 3 3.9 4.0	4.1	LaLuO 3 4.2
			NSAT LSAT SAGT	KTaO 3
	Figure 1.2.6 -Overview of the pseudocubic in-plane lattice constants of various perovskite and perovskite-related oxides. [Abbreviations used in the figure: (LaAlO 3 ) 0.29 -(SrAl 0.5 Ta 0.5 O 3 ) 0.71 (lsat), (NdAlO 3 ) 0.39 -(SrAl 0.5 Ta 0.5 O 3 ) 0.61 (nsat), Sr 1.04 Al 0.12 Ga 0.35 Ta 0.50 O 3 (sagt), Pb(Mg 0.33 Nb 0.67 )O 3 -PbTiO 3 (pmn-pt).] [adapted from Ref. 154]

DT Figure 1.2.13 -Schema of direct tunneling (DT) through

		a trapezoidal energy barrier pro-file with barrier heights ϕ B,1 and ϕ B,2 at the in-terfaces under an applied voltage V.
		As mentioned in section 1.2.1.1.3 on page 20, the ferroelectric polarization charges of the
		thin barrier at the interfaces to the electrodes will be screened by free electrons in the
		metals, changing the interfacial energy barrier. If, in the case of asymmetric interfaces,
		the screening lengths are different, the associated interfacial barrier heights ϕ B,1 and
		ϕ B,2 are modified unequally. This results in an asymmetric barrier profile, as depicted
	Figure 1.2.13	in figure 1.2.13

18 -Simple scheme of the band alignment in a metal- semiconductor heterojunction using Anderson's rule: The

  energy barrier for electrons ϕ B is defined by the difference of the metal work-function WF and the semiconductor electron affinity χ[START_REF] Anderson | Germanium-Gallium Arsenide Heterojunctions[END_REF].

Hysteresis loops for a typical Pt/ Co/BiFeO 3 /Ca 0.96 Ce 0.04 MnO 3 FTJ with a diame- ter of 180 nm. Cycle

	1 0 9
		Figure 2.3.1 -
	Ω)
	R (	1 2
		3
		4
		5
		6
	V w r i t e ( V )

1

shows the first switching from the high-resistive initial state to the lowresistive on state. Cycle 2 demonstrates the reproducible switching between on of off state. Cycles 3 to 6 correspond to minor loops, stabilizing intermediate resistance levels by limiting the maximum applied pulse voltages.

  about 10 5 cycles by pinning of charged dws that formed during polarization switching by 180°. Later, D.Lee et al. [343] demonstrated fatigue-free switching up to more than 10 10 cycles in 300-nm-thick La-doped bfo. Together with the work of Z. Cheng et al.[344] this shows that doping by Nb and La can significantly reduce the mobility of oxygen vacancies and therefore provides a way to suppress dw pinning and enhance the cycling performance.

Tagantsev et al. [340] and X. J. Lou [341]. Due to the large interest in bfo, several studies have already investigated its cycling performance. In rhombohedral bfo films (400 nm thick), S.-H. Baek et al. [342] observed fatigue after

  10) L. I. Zhang et al. indeed measured stdp in the visionary cells of frogs [370].

  and, recently, R. Guo et al. as well as H. Yamada et al. published first results of functional BaTiO 3 -based ftjs [430, 431].

1 )

 1 Such as, among others, the French anr project mira. 'ensemble de ces poids synaptiques d'un réseau de neurones constitue la mémoire du système. En 1949, D. O. Hebb postulait que l'apprentissage dans le cerveau était basé sur la modification des forces de connexions synaptiques[START_REF] Hebb | The organization of behavior: a neuropsychological theory[END_REF]. Ce processus de changement des poids est résumé sous le terme de la plasticité synaptique et la règle selon laquelle les poids évoluent est nommée « loi d'apprentissage ». En plus de l'influence éventuelle des taux de sollicitation d'une synapse, une autre loi appelée spike-timing-dependent plasticity (stdp) est basée sur les temps relatifs entre potentiels d'action[START_REF] Markram | Regulation of Synaptic Efficacy by Coincidence of Postsynaptic APs and EPSPs[END_REF][START_REF] Bi | Synaptic Modifications in Cultured Hippocampal Neurons: Dependence on Spike Timing, Synaptic Strength, and Postsynaptic Cell Type[END_REF]. Elle est supposée être à la base de l'apprentissage non-supervisé et nous montrons leur réalisation avec des memristors ferroélectriques.Aujourd'hui, des simulations de réseaux de neurones sur des ordinateurs classiques sont déjà utilisées pour des applications comme la reconnaissance et la classification d'images tout comme la reconnaissance vocale. Néanmoins, ces logiciels demandent des capacités de calculs très importantes, notamment pour les phases d'entrainement, et les ordinateurs sur lesquels ils sont exécutés consomment énormément d'énergie. En concevant des circuits électroniques avec des architectures directement inspirées de la structure des réseaux de neurones, ces problèmes pourraient être résolus. Afin d'atteindre le haut degré de connectivité inter-neuronale mentionné ci-dessus, les synapses doivent être imitées par des composants minimisés sans toutefois perdre leur fonctionnalité. Un candidat très prometteur pour cela est le memristor, un élément électronique qui peut être réalisé à l'échelle nanométrique.

	hdd ram	hard disk drive random-access memory
	I i-v if ifm rf rheed reflection high energy electron diffraction radio-frequency current measured as a function of applied voltage integrate-and-fire neuron model rnn recurrent neural network inhomogeneous field mechanism model for domain reversal rram resistive random-access memory (class of resistive memories, sometimes syn-
	K kai	afm rsm	onym for memristor) Kolmogorov-Avrami-Ishibashi model for domain reversal atomic force microscopy reciprocal space map
	ann awg B bfo L lao lcmo S sagt lif sma bnc bto C c-afm LaAlO 3 artificial neural network arbitrary waveform generator Sr 1.04 Al 0.12 Ga 0.35 Ta 0.50 O 3 La 0.5 Ca 0.5 MnO 3 coaxial cable connector type SubMiniature version A leaky integrate-and-fire neuron model BiFeO 3 coaxial cable connector type Bayonet Neill-Concelman BaTiO 3 conductive-tip atomic force microscopy lno smu source-measure unit LaNiO 3 lsat sro SrRuO 3 (LaAlO 3 ) 0.29 -(SrAl 0.5 Ta 0.5 O 3 ) 0.71 lsmo ssd solid-state disk La 0.67 Sr 0.33 MnO 3 ltp long-term potentiation stdp spike-timing-dependent plasticity
	stem cbram conductive-bridge random-access memory (synonym for ecm) scanning transmission electron microscopy ccmo molecular dynamics sto SrTiO 3 Ca 0.96 Ce 0.04 MnO 3 cmo CaMnO 3 cmos mhann French National Research Agency (anr) project: Memristive hardware artificial M md stp short-term potentiation neural networks complementary metal-oxide-semiconductor cpu mit T taff thermally assisted flux flow (regime for dw motion) metal-insulator phase transition central processing unit tcm thermochemical memory (class of unipolar resistive memories based on the
	D dc N ngo dt nls te dw nn ter nsat ti	NdGaO 3 movement of anions, one type of rram) direct current nucleation-limited switching model for domain reversal top electrode direct tunneling neural network tunnelling electroresistance domain wall (NdAlO 3 ) 0.39 -(SrAl 0.5 Ta 0.5 O 3 ) 0.61 thermionic injection
	E ecm O off tmr eels on V vcm F fe P pc pcm W wf feram ferroelectric random-access memory electrochemical metallization (class of resistive memories based on the move-tunneling magnetoresistance high-resistance state of a resistive memory device ment of metallic cations, one type of rram) low-resistance state of a resistive memory device valence-change memory (class of bipolar resistive memories based on the electron energy loss spectroscopy movement of anions, one type of rram) pseudocubic ferroelectric phase-change material (also: phase-change memory) work function fet pfm piezoresponse force microscopy wkb Wentzel-Kramers-Brillouin approximation field-effect transistor fnt pld pulsed laser deposition X xps X-ray photoemission spectroscopy Fowler-Nordheim tunneling fpga pmc programmable metallization cell (synonym for ecm) xrd X-ray diffraction field-programmable gate array ftj ferroelectric tunnel junction pto PbTiO 3 pmn-pt Pb(Mg 0.33 Nb 0.67 )O 3 -PbTiO 3 xrr X-ray reflectivity
	G gld pzt Y yao	Ginzburg-Landau-Devonshire theory Pb(Zr 0.2 Ti 0.8 )O 3 YAlO 3
	H haadf high-angle annular dark-field imagery using scanning electron microscopy R ra resistance-area

LA.2.1.1.2 Memristors Le memristor a été postulé par L. Chua en 1971 sur la base de réflexions portant sur la théorie de l'électronique [81]. Dans cette conception idéalisée du memristor, il vient compléter les relations entre les quatre variables électroniques fondamentales en liant la charge électrique au flux magnétique (figure A.2.1

Figure A.2.1

2.1 -Relations entre les quatre variables

  Suite à la première publication qui revendiquait la démonstration expérimentale d'un memristor en 2008[82], beaucoup d'autres groupes de recherche ont présenté des phénomènes dits memristifs dans des systèmes différents. Bien que conscients de la discussion autour du terme « memristor » dans sa conception théorique [84], dans cette thèse, nous utilisons ce mot pour dénommer tout composant résistif qui peut atteindre plus que deux niveaux de résistance.Les différents memristors se classent dans plusieurs catégories selon le phénomène physique sur lequel la commutation résistive est basée [85-87]. Une grande partie des memristors actuels est basée sur le déplacement d'ions dans une fine couche isolante[88, 89]. En fonction de la charge de ces ions, deux sous-types sont différenciés : Dans des matériaux comme HfO x , TaO x , TiO x , SrTiO 3 , ZnO x , ou WO x la commutation résistive est attribuée aux déplacements de lacunes d'oxygène anioniques qui provoquent une modification d'une barrière Schottky à l'interface avec les électrodes ou qui forment des filaments conducteurs. Dans l'autre variante de ce type de memristor, ce sont des cations métalliques issus des électrodes réactives qui migrent dans une couche isolante. D'autres classes de memristors se basent sur des matériaux à changement de phase [103] ou des phénomènes de spintronique[110, 113]. Finalement, il y a aussi des idées d'utiliser des matériaux à transitions métal-isolant afin de réaliser des memristors[121].

	électroniques fondamentales (tension, courant électrique, flux magnétique, et charge électrique) qui sont liées par les quatre éléments de circuits idéaux (capacité, résistance, inductance, et memristor). Le schéma montre également le nouveau symbole de circuits du memristor ( ). [d'après Ref. 82]

  3 (bfo). Ce matériau attire beaucoup d'attention à cause du fait qu'il est le seul multiferroïque à température ambiante et qu'il présente une polarisation ferroélectrique record[START_REF] Wang | Epitaxial BiFeO 3 Multiferroic Thin Film Heterostructures[END_REF][START_REF] Sando | BiFeO 3 epitaxial thin films and devices: past, present and future[END_REF]. En fonction de la température et de la valeur d'une contrainte extérieure, une large gamme de phases cristallines (avec des propriétés ferroélectriques et magnétiques différentes) peut être stabilisée. Avec l'objectif de maximiser les contrastes de résistance entre les états off et on, nous nous sommes intéressés à la fabrication de fines couches de la phase super-tetragonale de bfo qui présente une très forte polarisation ferroélectrique. Pour cela, des couches de bfo avec des épaisseurs de 4,6 nm sont déposées par ablation laser pulsée (pld) sur des substrats de YAlO 3 (yao)[START_REF] Yamada | Giant Electroresistance of Super-tetragonal BiFeO 3 -Based Ferroelectric Tunnel Junctions[END_REF]. La contrainte compressive du substrat permet de stabiliser le bfo dans la phase t[START_REF] Ricinschi | A mechanism for the 150 µC cm 2 polarization of BiFeO 3 films based on first-principles calculations and new structural data[END_REF][START_REF] Dupé | Competing phases in BiFeO 3 thin films under compressive epitaxial strain[END_REF][START_REF] Hatt | Strain-induced isosymmetric phase transition in BiFeO 3[END_REF]. L'analyse de la structure atomique de ces couches montre une tétragonalité élevée et confirme la présence de la phase t. être contactées par la pointe métallisée d'un microscope à force atomique (afm) afin d'effectuer des mesures ou des excitations électriques. En plus, la faible épaisseur de l'électrode supérieure permet toujours l'imagerie de la configuration de domaines ferroélectriques dans la couche de bfo par microscopie en mode piezoréponse (pfm).

	A.2.2.1 Lien entre ferroélectricité et commutation résistive
	La figure A.2.5	Figure A.2.5

L'intégration d'une couche de Ca 0.96 Ce 0.04 MnO 3 (ccmo) d'une épaisseur de 20 nm entre le substrat et la couche ferroélectrique définit une électrode inférieure. Le dépôt par pulvérisation cathodique de fines couches de Pt/Co et leur structuration par lithographie électronique définit des électrodes supérieures de diamètres de 180 à 660 nm. Ensuite, ces viii ftjs peuvent

[START_REF] Moore | Cramming More Components onto Integrated Circuits[END_REF] An extensive list of the different neuron models was compiled by A. Cassidy et al.[START_REF] Cassidy | Cognitive computing building block: A versatile and efficient digital neuron model for neurosynaptic cores[END_REF].[START_REF] Merritt | ARM CTO: power surge could create 'dark silicon[END_REF] In general, synaptic inter-neuron connections have been observed between many different parts of the neuron structure.

[START_REF] Venkatesh | Conservation Cores: Reducing the Energy of Mature Computations[END_REF] The same applies for weakening of the synaptic weight, called depression. 7) These timescales are only indicators and vary depending on the respective system.

[START_REF] Rupp | 40 Years of Microprocessor Trend Data[END_REF] This includes, inter alia, the organization of artificial synapses in the crossbar geometry (see section 4.4 on page 113).

[START_REF] Mountcastle | Perceptual neuroscience: the cerebral cortex[END_REF] Not necessarily under conservation of the unit cell volume.

[START_REF] Gerstner | Theory and Simulation in Neuroscience[END_REF] This might be due to structured electrodes or even structured ferroelectric thin films.

[START_REF] Gerstner | Spiking Neuron Models: Single Neurons, Populations, Plasticity[END_REF] If the field applied to switch the ferroelectric always corresponds to the high-resistive, low-current configuration, the switching energy could be minimized.

[START_REF] Moore | Cramming More Components onto Integrated Circuits[END_REF] In contrast to the binary, digital switching used in conventional logic elements.

[START_REF] Ball | Computer engineering: Feeling the heat[END_REF] This corresponds to 1 -S as introduced in section 2.3.2 on page 49.[START_REF] Moore | Multicore is bad news for supercomputers[END_REF] The contribution from the downward-pointing domains can be neglected due to their high ra values.

[START_REF] Heber | Materials science: Enter the oxides[END_REF] The measurements by A. Chanthbouala et al.[START_REF] Chanthbouala | Solid-state memories based on ferroelectric tunnel junctions[END_REF] are also limited by the fact that the electrical contact through a c-afm tip does not allow stable conditions for a longer period.

2) The experiment would take more than 46 days for 10 6 cycles. Using the procedure described here, we could carry out this experiment in about nine hours measurement time.

3) It should be noted that here the frequency is limited to 1 blocked state per cycle by definition, so that the exponential increase cannot hold for arbitrarily high cycle numbers.

[START_REF] Moore | Multicore is bad news for supercomputers[END_REF] Exemptions to this tendency can be observed in figure3.4.4 at voltage pulse amplitudes of -2.05 V and -2.30 V for which the transient current is higher than under the next larger voltage amplitude, respectively. The dc resistance changes show the same result.

[START_REF] Dubey | Recognition, mining and synthesis moves computers to the era of tera[END_REF] For example, they could be used for the automatic pre-filtering of particle detector data -a task that typically has to cope with large amounts of complexly linked data and very high bandwidths.

[START_REF] Heber | Materials science: Enter the oxides[END_REF] This is similar to the switching process observed under the application of pulses with increasing amplitudes (cf. figure 2.3.2 on page 51).

[START_REF] Moore | Cramming More Components onto Integrated Circuits[END_REF] This definition is in agreement with the experimental procedure: we determine the polarization configuration after the application of the voltage pulse, which is to say, in an equilibrium state in which the polarization magnitude can be assumed to be locally relaxed to its remnant value.

[START_REF] Neumann | Probabilistic logics and the synthesis of reliable organisms from unreliable components[END_REF] For details refer to section 3.1 on page 69.[START_REF]Principles of neural science[END_REF] As the one used for the measurements in section 3.4.1 on page 82.

Acknowledgments

However, a possible change in the nucleation process -from the growth of inherent, latent nuclei at small amplitudes (corresponding to n = 2) to a continuous nucleation throughout the reversal process at higher amplitudes (corresponding to n = 3) -could only explain a change of about one order of magnitude in this spread. Moreover, the pfm images taken at different intermediate stages of the switching process show the apparition of many small domains and that the reversal is determined by the nucleation of new rather than by the expansion of existing domains. This is confirmed by the evolution of the spacial polarization configuration extracted from the simulations (as shown in figure 4. 1.3-a). The kai model is thus not suited to describe the dynamics in these t-phase bfo films.

Based on the heterogeneous polarization switching observed by pfm and in the simulations, we here employ the nucleation-limited switching (nls) model by J. Y. Jo et al. [START_REF] Jo | Domain Switching Kinetics in Disordered Ferroelectric Thin Films[END_REF] (cf. section 1.2.1.2.2 on page 25). This description assumes that the ferroelectric is split into many small regions that switch independently from each other. Each individual region i is dominated by dw motion which is described by a kai process with a specific nucleation time τ i . This individual reversal occurs on a very fast timescale compared to the width of the distribution of the nucleation (or "waiting") times in the whole ferroelectric, in good agreement to the observations in section 3.4.2 on page 85. It is thus the distribution of these waiting times g(log τ i ) that determines the overall reversal process in the ferroelectric film. Assuming a continuous distribution function g(log τ), we obtain a general expression for the time-dependence of the reversed area fraction:

where n corresponds to the dimensionality of the domain reversal as defined in the framework of the kai model. 5 Writing this whole expression in terms of x = log t and χ = log τ, we obtain

g(χ) dχ .

Under the reasonable assumption that n 1, the double exponential dependence of the integration variable χ therein can be approximated by a Heaviside step function centered at x, yielding

Using the condition that the distribution g(χ) is normalized, the expression thus simplifies to:

Date

Source Material Timescale Method 2010/04/14 [START_REF] Jo | Nanoscale Memristor Device as Synapse in Neuromorphic Systems[END_REF] Ag in Si ms external 2010/06/11 [START_REF] Lai | Ionic/Electronic Hybrid Materials Integrated in a Synaptic Transistor with Signal Processing and Learning Functions[END_REF] polymer-layer + RbAg 4 I 5 ms overlap 2011/06/24 [START_REF] Seo | Analog memory and spike-timing-dependent plasticity characteristics of a nanoscale titanium oxide bilayer resistive switching device[END_REF] TiO x ms external 2011/08/01 [START_REF] Yu | An Electronic Synapse Device Based on Metal Oxide Resistive Switching Memory for Neuromorphic Computation[END_REF] HfO x /AlO x 10 ns (1 µs) overlap 2012/02/07 [START_REF] Krzysteczko | The Memristive Magnetic Tunnel Junction as a Nanoscopic Synapse-Neuron System[END_REF] MgO 1 to 10 s overlap 2012/02/08 [START_REF] Alibart | A Memristive Nanoparticle/Organic Hybrid Synapstor for Neuroinspired Computing[END_REF] nanoparticle-organic (NOMFET) PCM (Ge 2 Sb 2 Te 5 ) 10 ns overlap 2013/05/01 [START_REF] Subramaniam | Spike-Timing-Dependent Plasticity Using Biologically Realistic Action Potentials and Low-Temperature Materials[END_REF] HfO 2 ms overlap 2013/10/31 [START_REF] Shi | A correlated nickelate synaptic transistor[END_REF] SmNiO 3 ms external 2014/05/09 [START_REF] Li | Activity-Dependent Synaptic Plasticity of a Chalcogenide Electronic Synapse for Neuromorphic Systems[END_REF] PCM (AgInSbTe) 10 µs overlap 2014/06/18 [START_REF] Mandal | Novel synaptic memory device for neuromorphic computing[END_REF] Mn:HfO 2 ms external 2015/05/01 [402] WO x ms interaction 2015/05/08 [START_REF] Wang | Characterization and Modeling of Nonfilamentary Ta/TaO x /TiO 2 /Ti Analog Synaptic Device[END_REF] TaO x /TiO 2 ms overlap 2015/05/20 [START_REF] Prezioso | Self-Adaptive Spike-Time-Dependent Plasticity of Metal-Oxide Memristors[END_REF] TiO 2 ms overlap 2015/10/26 [START_REF] Tian | Graphene dynamic synapse with modulatable plasticity[END_REF] AlO x + graphene channel ms overlap 2015/10/29 [START_REF] Kim | Carbon Nanotube Synaptic Transistor Network for Pattern Recognition[END_REF] SIO 2 + CNT channel ms overlap

A.2 Résumé in French

Les ordinateurs classiques ont été optimisés pour le traitement déterministe de données structurées. En conséquence, des données appelées « naturelles » qui sont caractérisées par leur manque de structure et qui peuvent être accompagnées de bruit posent des difficultés sévères à ces architectures classiques. Le cerveau cependant excelle dans des tâches telles que la reconnaissance d'un visage connu dans une foule ou la concentration sur une voix dans une conversation. La volonté d'équiper des systèmes électroniques artificiels de ces performances impressionnantes pousse de nombreux chercheurs à essayer, d'une part, de comprendre le fonctionnement du cerveau et, d'autre part, de concevoir des circuits fondés sur ses principes de base. Ces efforts forment le domaine de recherche du « calcul neuromorphique » auquel cette thèse contribue.

Un ingrédient important pour la réalisation de systèmes auto-apprenants à grande échelle est un composant électronique nanométrique appelé memristor. Ce travail se concentre sur un type de memristor prometteur basé sur des oxydes ferroélectriques en couches ultraminces et combine l'étude fondamentale des propriétés des memristors avec l'objectif de les utiliser pour des nouvelles applications bio-inspirées.

A.2.1 Introduction

A.2.1.1 Réseaux de neurones basés sur des memristors

Parmi les architectures de calcul bio-inspirées, les réseaux de neurones occupent une place clé. Déjà utilisés à grande échelle dans des simulations, leur implémentation dans des circuits pourrait aider à manier les énormes quantités de données qui nous submergent aujourd'hui. Les synapses lient différents neurones et sont donc essentielles pour la transmission des signaux. L'énorme degré de connectivité neuronale 2 est supposé être l'une des bases fondamentales de la performance du cerveau. À chaque synapse est associé un poids qui définit l'atténuation des signaux transmis : un fort poids résulte en une bonne transmission tandis qu'une synapse avec un faible poids transmet un signal atténué.

2) Il y a en moyenne de l'ordre de 10 000 connexions synaptiques par neurone.

iii