domain: phys spi

text:
 M Laurent Garrigues

 M Mauro Taborelli

 M Giovanni Santin

CEA  M Julien Hillairet

Examinateur  M Christophe

Mme Mélanie

PAYAN  M Denis

Etudiant en dernière année d'école d'ingénieurs, votre offre de stage a retenu mon attention… » C'est par ces quelques mots, aussi banals soient-ils, que mon aventure au DPHY a débuté, en quittant ma douce campagne marseillaise pour Toulouse. Et c'est par ces quelques paragraphes, plus inspirés je l'espère, qu'est venu le temps de la clôturer, en remerciant celles et ceux qui ont contribué à sa réussite.

Commençons par le gros morceau, pour mon encadrant et directeur de thèse, Christophe. Au début du stage, j'avais peur de ne pas faire assez de physique… Eh bien peuchère, j'ai été servi ! Trois ans plus tard, lors de ma soutenance, tu diras même avoir apprécié que nous ayons vraiment discuté de physique pendant la thèse, et c'est un sentiment que je partage entièrement. Mais audelà des nombreux moments passés à se casser la tête ensemble sur la physique des isolants, ce sont tes qualités humaines exceptionnelles qui ont été déterminantes pour la réussite de ma thèse. C'est grâce à ton accompagnement, ta grande disponibilité, ta pédagogie à toute épreuve, en me guidant tout en me laissant la liberté d'explorer, qu'il existe désormais un Dr. Gibaru. Et même lorsque que le monde s'est arrêté de tourner, tu as tout de même continué à assurer un suivi irréprochable, à distance comme en face à face. Je te remercie sincèrement de m'avoir fait confiance pour mener à bien ce projet, ainsi que pour ton écoute et ton soutien infaillibles tout au long de la thèse. Je ne sais pas si mon chemin m'emmènera vers le bureau d'à côté ou vers un autre pays, mais quoi qu'il en soit, j'espère que nous pourrons continuer à collaborer ensemble, et que nous aurons l'occasion de nous revoir au bout du monde tout en dilapidant le budget conférence de nos instituts respectifs.

Je remercie sincèrement Mohamed pour m'avoir ouvert les portes de sa salle de manips, de laquelle je ressortais à chaque fois avec plus de questions que de réponses… Et c'est pour ça que ce volet de ma thèse, imprévu au départ, a été extrêmement enrichissant. J'ai beaucoup apprécié ton intervention lors de ma soutenance, même si c'était risqué de me laisser seul avec une manip en sachant que mes compétences expérimentales se résumaient à casser de la verrerie en TP de chimie ! Si ces expériences se sont bien déroulées, c'est aussi grâce à Sarah, qui nous a dépannés quand DEESSE se comportait en déesse et que l'alchimie ne passait pas avec ALCHIMIE.

En étant au centre d'une collaboration entre l'ONERA le CNES et le CEA, pour reprendre mon amorce de lettre de motivation, j'adresse mes remerciements à ces trois institutions pour avoir financé mes travaux de thèse, ainsi qu'à mes correspondants respectifs. Merci à Mélanie et Damien, mes encadrants du CEA, pour nos échanges et l'aide précieuse que vous m'avez apportée autour des développements de MicroElec. Malgré que nous n'ayons eu que peu l'occasion de nous voir, j'espère que nous pourrons continuer notre collaboration dans GEANT4. Côté CNES, bien que les choses aient été plus compliquées, je remercie Jérôme pour avoir lancé le sujet, Christophe pour s'être occupé de la période de transition, et Denis et Nicolas pour avoir assuré le suivi en fin de thèse. Si j'écris aujourd'hui ces mots en qualité de docteur, c'est bien entendu grâce à Mauro Taborelli et Isabel Montero, qui ont accepté d'être rapporteurs de ma thèse, ainsi qu'à Laurent Garrigues, Giovanni Santin et Julien Hillairet qui en ont été les examinateurs. Je vous suis reconnaissant d'avoir apporté avec bienveillance des remarques et discussions très pertinentes vis-à-vis de mes travaux de thèse. Je souhaite également remercier M. Taborelli pour la rigueur de son travail de révision compte tenu de la longueur de ce manuscrit, ainsi que M. Hillairet pour nous avoir dépannés quelques jours avant la soutenance. J'ai été chaleureusement accueilli pour cette thèse dans l'unité CSE, composée entre autres d'une bande de gai-lurons qui partent massivement en pause-café lorsque le logiciel maison au nom en S se met inexorablement à planter. Ce rituel est souvent animé par les histoires de Pierre, très sympathique chef d'unité regorgeant d'anecdotes insolites se passant dans une petite vallée béarnaise. Dans les restos et pique-nique CSE, on retrouve entre autres, Sébastien, Ludivine, Gaël, Julien, David, Thierry, et le duo Marc et Rémi (ta présentation à Seattle restera entre nous).

C'est en réalité dans tout le DPHY qu'il règne une ambiance exceptionnelle qui favorise les rencontres. J'ai eu peur lorsque j'ai entendu parler d'un triathlon du département pour la première fois. Mais il s'avère que la version DPHY de ce sport est bien plus agréable, tout comme les barbecues et repas de Noël organisés par Virginie et Angélica. Il me faut également mentionner Antoine, propriétaire de la plus belle Yaris diesel de France, pour tes interprétations libres et notes bleues en cours de guitare, et Pablo, pour ton aide sur les fonctions diélectriques en début de thèse et les matchs sanguinaires de baby-foot en fin de thèse. Pour ces derniers, la relève est assurée avec Antoine F, Nour qui m'a fait découvrir le baby assis, et Vincent l'éternel débutant en guitare. Merci aussi aux soeurs Pujol et à Nathalie pour m'avoir aidé à surmonter les bourbiers administratifs, et au fameux duo Claude et Stéphane, pour les lancements du programme spatial stéphanois et les douches d'azote liquide bienvenues en temps de canicule. Il m'est impossible de tous vous nommer un par un, mais je tiens à remercier l'ensemble des permanents pour entretenir cet esprit de partage, et car mes interactions avec chacun d'entre vous ont toujours été très plaisantes.

Cette ambiance agréable est en partie portée par les autres doctorants du département. Je dois forcément commencer par Le Bureau des Légendes, alias Le Bureau F215, où le café se prend toujours à 15h15. En face de mon bureau, on retrouvait Guillerme et sa fidèle casquette, homme recherché pour la somme de 3 Mégadollars, et à qui je lègue mon contrat de publicité pour Senseo. Puisque Midi Libre prédit mieux les orages géomagnétiques que Sérénade, il faudra peutêtre songer à te reconvertir…! Il y a ensuite mes autres camarades de promotion. Gaëtan, je suis certain que les compétences acquises après la formation boomer te serviront pour ton tour d'Europe à vélo. Lucas, j'espère que nos aventures rôlistes t'inspireront pour guider tes péripéties à vélo ou en bateau avec Juliet. J'aurai passé beaucoup de temps lors des pauses café avec Julie à déprimer ensemble sur les Fauteuils des Doctorants de Troisième Année™ et à nous demander pourquoi nous sommes rentrés de Namibie ou de Madagascar, lorsque mon PC était à 100% et que j'étais à 0%. Merci d'avoir sauvé (presque) toutes mes plantes de la canicule, et ainsi de m'avoir évité de transformer à nouveau ma voiture en camion Truffaut. Enfin, il y a les doctorants et alternants des autres promotions. Je pense notamment à Maxime le cycliste fou, Carla, le faux Quentin, Lucas, Agnès, Lauriane, Adrian, Paul, Nathan, le trio Gwen-Maria-Rabia droguées au thé et ayant la lourde tâche de faire honneur au bureau F215, et tous les autres arrivés dernièrement. Bon courage pour la suite, car vous allez en baver ! Pour terminer avec le DPHY, il y a ceux qui l'ont quitté. Avec, bien sûr, le groupe des San Fermin ! Juan (Comment allez-vous mon père…), Raphael (J'espère que les cabarets aixois te plaisent !), Neil (Mon premier co-bureau et dompteur de toros), je retiens nos légendaires roadtrips à Pamplona, dans les Pyrénées et en Aveyron (On doit toujours se le faire ce voyage à Marseille !). Je n'oublie pas non plus Mathias pour l'apéro dans son terrain de golf et ses Peugeot de collection, Manon et son interprétation parfaite de l'orc poli lors des sessions D&D, Martin le troisième membre du bureau F215 en notre temps et tennisman d'élite, ainsi que les autres anciens doctorants et post-docs : Eudes, Guillaume, Abdess, Pauline, Hector, Adrien… Last but not least, Loanne, je te remercie pour les (trop) nombreux cafés, mon entraînement de baseball avant d'aller à Seattle, et l'organisation des soirées Jeux+Sushi, mais je ne te remercie pas d'avoir tué tous les méchants que j'ai invoqués en tant que MJ tyrannique de notre campagne JDR.

Pour réussir une thèse, il faut aussi savoir sortir la tête du guidon. C'est pourquoi je suis très reconnaissant envers mes amies Chloé et Gabriela, pour tous les bons moments passés ensembles, et l'énergie que vous m'avez donnée pour entretenir ma santé mentale. Je suis très heureux que nos amitiés aient pu durer si longtemps, et j'espère qu'elles resteront autant solides pendant de nombreuses années à venir. Les activités artistiques que j'ai effectuées en parallèle m'ont également beaucoup apporté. Ainsi, je remercie Bhairavi de m'avoir offert l'opportunité de monter sur les planches et d'incarner les rôles de Sir Wilfrid Robarts et Semyon Semyonovich, en compagnie de mes camarades de Stagefright. Il y a aussi Michel, qui m'a évité d'abandonner la guitare après 2 mois à l'aide de ses cours. C'est là qu'Antoine, Guillerme et moi avons créé les célèbres Blues Stompers, le groupe qui ne sait jouer qu'un seul morceau (et pas bien)… Le mot de la fin sera pour remercier chaleureusement ma famille : mon père Daniel, ma mère Solange, mon beau-père Jean-Louis et ma belle-mère Odile. Merci infiniment de m'avoir accompagné tout au long de cette thèse, et particulièrement pendant les confinements successifs. Avoir dû faire une partie de ma thèse à distance dans un climat de panique générale était une lourde épreuve que je n'aurais jamais pu surmonter sans votre soutien, et je vous remercie de m'avoir offert un splendide pot de thèse pour célébrer ce doctorat de la meilleure manière qui soit. Merci pour tout, Papa et Mama, pour votre amour, pour avoir toujours cru en moi et pour les sacrifices que vous avez fait tout au long de ma vie afin de me permettre de poursuive mes études et d'en arriver à être docteur. Les mots ne peuvent transcrire toute la reconnaissance et l'amour que j'ai et que j'aurai éternellement pour vous.

Table of contents

Effects of the space environment and radiation on electronics and spacecraft systems

Introduction

Dielectric materials are widely used in spacecrafts and satellites, and are part of numerous essential components for the operation of the spacecraft. Indeed, dielectrics can be found in CMOS components, radiofrequency devices such as waveguides, solar panels for the power supply of the spacecraft, plasma thrusters for electrical propulsion, thermal coatings on the surfaces of the spacecraft, wirings… Therefore, several insulators such as metal oxides or polymers have been selected in space applications for their mechanical, thermal and electrical properties. However, the systems and devices on board of spacecrafts are exposed to the space environment, a very harsh medium that comprises of a variety of incoming radiation. Dielectric materials on board of the spacecraft are not spared, especially those used on the surfaces of the spacecraft, which are directly exposed to the space radiation. However, the behavior of insulators under irradiation is responsible of critical issues, due to their tendency to accumulate electrostatic charge under irradiation.

Therefore, one of the main concerns in spacecraft design and operation is to prevent the risk of electrostatic discharges and perturbations created by the radiative space environment. In result, the study of the transport of radiations through dielectric materials is capital in order to understand and prevent these risks. Most of the charging effects involved in dielectrics are directly linked to their electron emission properties. Indeed, electron emission happens when a material under electron irradiation is emitting additional electrons in vacuum. In such a case, dielectric materials can charge positively. On the other hand, the implantation of electrons from the space environment in the dielectric layer charges it negatively. As a result, the potentials reached by a dielectric material are directly dependent on its electron emission properties. The multipactor effect, which is the formation of an electron cloud in RF devices, is also guided by the electron emission of the materials of the device.

Consequently, evaluating the total electron emission yield (TEEY) of insulating materials is critical for these applications. However, it is especially difficult to perform experimental TEEY measurements on dielectrics, due to the modification of the TEEY caused by charging effects. Indeed, during experimental measurements, the TEEY has been observed to increase or decrease with time depending on the global evolution of the charge.

simulate the evolution of the secondary electron emission of insulators depending on the global charge.

Finally, several experimental studies have been conducted in conditions where the external charge effects have been removed or reduced, which are more representative of the conditions of use of some dielectrics in spacecrafts. These studies have shown that the TEEY can still be reduced with time, due to internal charge effects.

However, the dielectric materials on board of satellites are subjected to strong gradients of temperature, typically from -200°C to +200°C, which can significantly modify the transport of electrons. Hence, most studies, which were made in the controlled environment of an electron microscope, are not representative of the space environment. Most importantly, while experimental studies did demonstrate that internal charging effects could strongly modify the TEEY of insulators, they were unable to provide a definite explanation for the origin of these effects. Indeed, it is very difficult to investigate the transport of electrons through experimental TEEY measurements, and a simulation of the transport of electrons in dielectrics would be required instead. Nevertheless, the simulation studies on the TEEY of insulators have only focused on the global charge effects, and no simulations were made in conditions where only internal charge effects can modify the TEEY. In consequence, there is a lack of physical explanations on the dependence of the electron emission yield of insulators on the internal charge, and it is unsure which physical mechanisms are involved.

Consequently, the aim of this study is to develop a low energy electron transport model in dielectric materials for space applications. The objective of this model is to simulate the electron emission properties of dielectric samples, and to model the effect of the internal charge on the TEEY. Hence, our goal is to provide physical explanations for the misunderstood experimental observations made on dielectrics. Improving the comprehension of charge transport and electron emission in dielectric materials is of interest in many applications. Indeed, this misunderstanding prevents the precise evaluation of the electrostatic discharges and multipactor risks associated with dielectrics in space applications. The modification of the TEEY of insulators under charging is very disruptive in electron microscopy, where the image contrast can be severely modified if the electron emission is not steady. The mitigation of electron cloud effects is also a major concern in particle accelerators and fusion reactors, since it can be the source of large power losses.

In Chapter 1, we will describe the radiative space environment and its effects on satellite systems. We will highlight the importance of secondary electron emission, and present the experimental procedure used to measure the electron emission yield. This will allow us to mention the issues met when characterizing the electron emission of dielectric materials, due to the charge buildup created in the insulator by the electrons. We will also highlight that some effects of charging on the electron emission are still not thoroughly understood, and why a low energy electron transport model for dielectrics is needed to correctly evaluate their electron emission properties in space applications.

In order to compute the electron emission of insulators, we need to identify which electronmatter interactions shall be modelled, and this will be the focus of Chapter 2. Some of these interactions, such as the inelastic scattering, are common to all material types, whereas other interactions, such as trapping and phonon collisions, are specific to insulators.

We know that the kinetic energy of most secondary electrons is of a few eVs only. Therefore, the interaction models should extend down to this range, since we have to simulate the transport of any electrons that are susceptible to escape the material. However, most publicly available Monte-Carlo models have a low energy limit of a few keV or a few hundred of eV, which is why we need to develop our own interaction and transport models. In this regard, we can start from the work made previously in MicroElec for the TEEY of metals and semiconductors. In Chapter 3, we will present the improvements brought to MicroElec to extend it to the simulation of the TEEY of 16 materials. The model will be able to simulate the elastic, inelastic, and surface interactions of low energy electrons. The simulation results will also be validated with experimental data, to verify that our simulation of the transport of low energy electrons is accurate and that the Monte-Carlo model can correctly simulate the TEEY.

In Chapter 4, we will derive an analytical secondary electron emission yield model that is based on the physics of low energy electron transport. Indeed, the Monte-Carlo model developed in Chapter 3 can be used to extract data on the penetration depth, the ionizing dose-depth profile, and the transmission rate of electrons. Gathering data on these parameters is especially critical, due to the low availability of experimental data for electrons below 10 keV. We will then combine these parameters into a single analytical expression for the secondary electron emission yield. The aim will be to provide an expression which can be used as an input parameter in systems simulation packages, and to demonstrate that the TEEY can be obtained analytically by following a more physical approach, on the contrary of most TEEY models which rely on arbitrary parameters.

In Chapter 5, we will present a new Monte-Carlo code based on the model developed in Chapter 3, which will be dedicated to the simulation of charge transport in dielectric materials. In this regard, we will have to model the transport of the thermalized electrons, and of the holes left in the material after the generation of secondary electrons. The electric field generated by the charge density influences the drift of these very low energy particles, so it needs to be computed and taken into account as well. Finally, for all particles, the trapping, detrapping and recombination processes need to be implemented. Since silicon dioxide is by far the most studied material, several of the parameters that will be needed by our model should be more available for this material. Consequently, the charging simulation model will be developed for SiO2 only. However, the model we will create will be as general as possible, so that it can be extended to other insulators if the simulation parameters can be found.

The Monte-Carlo model developed in Chapter 5 should allow us to understand the physical processes behind insulator charging and its effect on the TEEY. Nevertheless, experimental measurements on SiO2 samples are needed in order to quantitatively validate the simulations. In Chapter 6 we will present the experimental measurements made during this PhD thesis on SiO2 thin film samples. We will conduct measurements of the TEEY depending on the incident energy, and time-resolved measurements of the TEEY at a single incident energy. Afterwards, we will explain the experimental observations, using the wide range of data we can extract from the simulations. We will clarify why the TEEY is decreasing with the positive charge buildup, and highlight several experimental artifacts and bias that can appear when measuring the TEEY of dielectric materials. We will then show how the charge buildup can falsify the TEEY obtained on an insulator, and what steps can be made during the experiment to avoid this falsification.

We will also study the evolution of the TEEY until its stabilization, and the differences in the two TEEY measurement facilities used in this work. Finally, we will move away from TEEY studies in a controlled environment, and explore how the electron emission properties of a dielectric can vary in the conditions of the space environment.

Chapter 1: Context and aim of the study

Dielectric materials are widely used in spacecrafts and satellites, and are part of numerous essential components for the operation of the spacecraft. Indeed, dielectrics can be found in CMOS components, radiofrequency devices such as waveguides, solar panels for the power supply of the spacecraft, plasma thrusters for electrical propulsion, thermal coatings on the surfaces of the spacecraft, wirings… Therefore, several insulators such as metal oxides or polymers have been selected in space applications for their mechanical, thermal and electrical properties. However, the systems and devices on board of spacecrafts are exposed to the space environment, a very harsh medium that comprises of a variety of incoming radiation. Electrons, protons, heavy ions and photons continuously hit the spacecraft and perturb its operation. Dielectric materials on board of the spacecraft are not spared, especially those used on the surfaces of the spacecraft, which are directly exposed to the space radiation. However, the behavior of insulators under irradiation is responsible of critical issues, due to their tendency to accumulate electrostatic charge under irradiation. The subsequent potential difference between the dielectric and the metallic surfaces can create sudden electrical discharges, which cause serious damage to the devices. Nevertheless, even under the protection of radiation shielding, insulators inside the internal components of the spacecraft can still accumulate charge and cause several issues.

The question of dielectric materials charging under irradiation is also not specific to the space technology community. For instance, in scanning electron microscopy, samples are hit by an incident electron beam to generate an image of the target. However, the charging of insulating targets under electron irradiation can severely affect the image contrast [1][2][3]. The electronic components of particle accelerators, such as the LHC at CERN, are also exposed to incident radiation, which can hinder the operation of the devices. Finally, the generation of parasitic electron clouds, which is amplified by insulator materials, is a major source of power loss in particle accelerator beam lines, fusion reactors, or space telecommunication RF devices.

Therefore, one of the main concerns in spacecraft design and operation, is to prevent the risk of electrostatic discharges and perturbations created by the radiative space environment. In result, the study of the transport of radiations through dielectric materials is capital in order to understand and prevent these risks.

In this chapter, we will first describe the unwanted effects generated by radiation in the electronics and systems of a spacecraft, and demonstrate the necessity of studying the transport of low energy electrons through matter. Indeed, low energy electrons are responsible of the secondary electron emission process, which is the production of electrons by a material under electron irradiation. This phenomenon is responsible of the charging of dielectric materials and the generation of electron clouds, hence a precise knowledge of the electron emission of a material is required to limit these risks. In this regard, we will present the experimental procedure used to measure the electron emission. This will allow us to mention the issues met when characterizing the electron emission of dielectric materials, due to the charge buildup created in the insulator by the electrons. We will also highlight that some effects of charging on the electron emission are still not thoroughly understood, and why a low energy electron transport model for dielectrics is needed to correctly evaluate their electron emission properties in space applications.

1.1 Effects of the space environment and radiation on electronics and spacecraft systems

A spacecraft in a Geostationary Earth Orbit (GEO) or Low Earth Orbit (LEO) is exposed to several sources of incoming radiation. First, the sun continuously ejects a flux of photons from gamma rays to radio and a flux of low energy protons and electrons as part of the solar wind.

The sun also generates highly energetic protons (keV -500 MeV) and ions (1-10 MeV/n) during solar flares. The spacecraft can receive galactic cosmic rays made of protons and ions up to 300 MeV/n as well. Finally, the electrons and protons trapped in the Van Allen belts by Earth's magnetic field constitute a strongly hostile region and a source of continuous radiation for the satellites orbiting around Earth. According to the incident radiation flux model GREEN [4] from ONERA, the flux of electrons received by a spacecraft in either a LEO or GEO orbit is greater than the flux of protons. As shown in Figure 1-1, the majority of incident radiation received by spacecrafts is electrons from 100 eV to 100 keV. Consequently, electrons will be mostly responsible of the charging of the dielectric surfaces of the satellite that are directly exposed to the space environment, such as solar panels or insulating coatings. Spacecrafts are equipped with shielding materials, which are designed to slow down the incident radiation and reduce the flux received by the internal components. These shieldings are mostly made of aluminum, due to weight constraints. However, the most energetic radiation can still go through the outer layers of the satellite, hit the internal electronics and generate parasitic effects. For example, an electron of 1 MeV is able to go through a 1 mm thick layer of aluminum [5,6]. Moreover, the high energy particles going through the shielding can also create a cascade of lower energy electrons, which can then reach the inside of the spacecraft and cause additional disturbances. We shall now list some of the issues and unwanted effects occuring in electronics exposed to incident radiation.

Total Ionizing dose

The effect of the total ionizing dose received by an electronic component exposed to high-energy electrons and protons is a cumulative degradation of its properties. When going through the insulator layer of CMOS or MOSFET components, the incident radiation can transfer its energy to the electrons of the material. Electron-hole pairs are then created, which can either recombine immediately or separate. Given that the mobility of electrons is much greater than the mobility of holes in the insulator, the electrons can get quickly swept away by an electric field. Holes on the other hand can get deeply trapped in the insulator and migrate by hopping between traps.

Given that the density of traps is greater at the insulator/semi-conductor interface, a large number of holes can get trapped in this region, which creates a charge density in the dielectric layer. This progressively generates a parasitic electrostatic potential, which can disrupt the current/voltage characteristics of the device. The change of these characteristics can have many unwanted effects, such as the apparition of a leakage current or a modification of the threshold voltage, which can hinder the operation of the component. In most recent electronics, the insulating layer only has a thickness of a few nanometers. Thus, it is required to study the transport of electrons and holes at the nanometric scale to quantify the charge left in the insulator, and the cumulative degradation of the device.

Displacement damage

High energy particles going through a material can transfer their energy to the atomic nuclei as well, and eject them from their position in the crystal lattice. The ejected atoms can also transfer their energy to other atoms of the lattice and create a displacement cascade. Some of these displacements can heal and the atoms can recover a regular position in the lattice. On the other hand, other defects created by the incident radiation may remain in the material and stabilize themselves as defect clusters, in an irregular position in the lattice. These clusters modify the electronic structure of the material, by creating additional energy levels in the band gap of semiconductors and insulators. Such levels are also susceptible to act as traps that can capture the charge carriers, and act as a recombination center. In semiconductors, these levels can also serve as an intermediate for the thermal generation of electron-hole pairs, as they can facilitate the emission of an electron into the conduction band. This is especially problematic in materials with a small bandgap or in regions of the material with a high electric field. Indeed, spontaneous thermal emission of electrons in the conduction band is possible in these conditions, which results in a parasitic current generated in the device. Such conditions are found in pixel arrays of sensors onboard of satellites, where the dark current can be amplified by the displacement damage caused by the space radiation. The quantification of this degradation necessitates an estimation of the Non Ionizing Energy Loss (NIEL) transferred by the particle to the atoms of the lattice [7], the simulation of the transport of the defects [8] and the modification of the electronic structure [9].

Single event effects

Single event effects in electronic components are caused by the passage of an energetic particle through a sensitive volume. These can create soft errors, such as Single Event Upsets (SEU), which is the change of state of bits in a memory [10], or Single Event Transients, a sudden voltage spike in the device. Destructive hard errors can also happen, such as Single Event Latchups, an excess of current in the device which can potentially cause overheating of the component.

Single event effects are mostly caused by high energy protons and ions. However, recent electronics have been shown to be also sensible to SEU triggered by electrons [11]. Indeed, these components have nanometric sensitive volumes, where a SEU may be triggered by an electron of a few keV [12]. To evaluate the risk of SEU in highly integrated electronics, it is needed to accurately model the dose deposited in a nanometric sensitive volume, in order to see if it exceeds the threshold level for the triggering of a SEU.

In the case of heavy ions, these particles create electron-hole pairs in a column that spreads around the trajectory of the particle, called trace. To accurately model the width of the trace and the dose deposited by the incident ion, it is necessary to explicitly simulate the transport of low energy secondary electrons created by the heavy ion [13,14]. Therefore, low energy electron transport models are needed to correctly evaluate the triggering thresholds of SEU in electronics.

Since all electronics exposed to radiation may suffer from single event effects, displacement damage or total ionizing dose effects, these are also a concern outside of the space environment, for instance in particle accelerator or nuclear electronics.

The multipactor/e-cloud effect

Low energy electrons (eV -keV) hitting a material can create a cascade of other low energy electrons inside of the material. These secondary electrons are then able to escape the material, through the secondary electron emission process. Macroscopically, the material is generating more electrons under electron irradiation. However, in RF cavities, the secondary electrons generated by one material of the device may then impact another material and generate additional electrons, up to the point that an electron avalanche is created in the device. This phenomenon is known as the multipactor effect, which hinders the functioning of RF devices and can create an electrostatic discharge if the quantity of electrons is too important. The multipactor effect can be easily understood in the example of a parallel plate waveguide, illustrated in Figure 1-2. The energy of most secondary electrons exiting a material is below 50 eV, which is not enough to create secondary electrons in another material. However, the electromagnetic waves circulating in the cavity carry an electric field that can accelerate the secondary electrons emitted by a plate towards the opposite plate. If the electrons have acquired enough energy, the process of secondary emission can happen, and the quantity of electrons in the device increases. The new electrons may then be accelerated towards the opposite plate and generate even more electrons, resulting in an avalanche of electrons that grows exponentially. Hence, we can understand that the multipactor effect can only happen under certain conditions. Indeed, the quantity of secondary electrons emitted by a material depends on the incident electron energy, and the properties of the material itself. For multipacting to occur, the material needs to be generating more secondary electrons than the quantity of incident electrons received.

In most materials, this condition is met for incident electrons having an energy of a few hundred of eVs. If the electron energy is too low or too high, the material is absorbing more electrons than it is generating electrons, and the multipactor is prevented. The amplitude of the electric field directly modifies the energy gained by the electrons in between the parallel plates, and can thus create or cancel the multipactor. The generation of electrons also needs to be synchronous with the electric field, which oscillates with time since it is generated by a wave. If the sign of the electric field changes when electrons are emitted from a plate, the field may instead accelerate these electrons back to where they were generated, therefore preventing multipacting. In consequence, the conditions of triggering of the multipactor effect are dependent on the power and frequency used in the device. The quantification of the electron emission of the materials used in the component is also mandatory, in order to compute the multipactor thresholds and triggering conditions.

However, the multipactor effect can also be worsened by dielectrics [15,16]. Indeed, dielectrics are strongly emissive over a large range of incident energies, which amplifies the production of electrons and the risk of a multipactor discharge. Moreover, as they charge under electron irradiation, the emission properties of dielectrics get modified, which complicates the estimation of the multipactor risk if the charge state of the material is unknown.

The multipactor and the generation of an electron cloud are also an issue in several other applications. In particle accelerators, electron clouds can be generated in vacuum cavities, where they can be accelerated by the pulsating electric field generated by the particle beam line. Notably, in the LHC, electron clouds are a major source of power loss and heat loads [17]. The multipactor effect is also a concern for fusion applications [18].

Surface and internal charging effects

Low energy particles from the space environment can charge the surfaces of the spacecraft, like solar panels and thermal coatings. The flux of incident electrons generates a negative charge on the satellite's surface, which opposes itself to the positive charge generated by photoemission and secondary electron emission, until the material reaches an equilibrium potential. However, the potential reached by dielectric materials may be different from the equilibrium potential of the metallic surfaces [19]. If this potential difference becomes too important, electrostatic discharges [20] or flash-overs [21] can occur, which can damage the power supply system and thus affect the satellite functions and jeopardize the mission. The potential reached by a dielectric material is a direct function of the transport of electrons and holes inside of the material, which influences the Radiation Induced Conductivity. This potential is also dependent on the electron emission and photoemission properties of the material. Consequently, charge transport models in space-used dielectrics are required to evaluate the charging of satellite surfaces. As for the multipactor effect, it is necessary to quantify the electron emission of the materials of the spacecraft, to correctly model the potential reached by a given surface.

Description and measurement of the secondary electron emission

In secondary electron emission, the emitted electrons have been stripped from the atoms of the material following the interactions of incident electrons with these atoms. Having materials that are emitting more electrons than they are receiving is a necessary condition for the multipactor or e-cloud effects. Therefore, it is especially critical to have an accurate knowledge of the electron emission properties of the materials used in the devices.

General description of the electron emission yield

The emission of secondary electrons is quantified by the Total Electron Emission Yield (TEEY).

It is defined as the ratio of the total number of electrons exiting the material compared to the number of incident electrons:

𝑇𝐸𝐸𝑌 = 𝑁𝑏 𝑒𝑥𝑖𝑡𝑖𝑛𝑔 𝑒 - 𝑁𝑏 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑒 - Equation 1-1
For most typical materials, the TEEY follows a standard behavior depending on the energy of the incident electrons, which is illustrated in Figure 123. First, the TEEY increases linearly at low energy. Indeed, the more energetic the electrons are, the more secondary electrons they can set into motion from the atoms of the material. The TEEY can increase over one, which means that from a macroscopic point of view, the material is generating additional electrons. This is the necessary condition for the apparition of the multipactor effect. The point where the TEEY curve goes over 1 is called the first crossover point, with the corresponding energy of incident electrons noted 𝐸 𝐶1 . Depending on the materials, 𝐸 𝐶1 generally ranges from a few tens to a few hundreds of eVs. The TEEY then increases to its maximum value 𝑇𝐸𝐸𝑌 𝑀𝑎𝑥 at the energy 𝐸 𝑀𝑎𝑥 . For most common metals and semiconductors, 𝑇𝐸𝐸𝑌 𝑀𝑎𝑥 can range from 1 to 2.5 [22], while it can go above 5 for some insulators. 𝐸 𝑀𝑎𝑥 is generally around a few hundreds of eVs, up to 1 keV for heavier metals such as gold or some insulators such as polycrystalline diamond [23]. Then, the TEEY decreases with increasing electron energy. Indeed, while the more energetic electrons will still produce more and more electrons, the penetration depth of the incident electrons also increases with the energy [5,6]. As a result, the secondary electrons are also produced deeper into the material [24,25], so that their probability of reaching the surface is reduced. In this sense, the energy of the maximum TEEY is the optimum of a compromise between the quantity of secondary electrons produced, and the depth at which the secondary electrons are generated in the material. In consequence, the TEEY diminishes for energies beyond 𝐸 𝑀𝑎𝑥 , and becomes lower than 1 beyond the energy 𝐸 𝐶2 , the second crossover point. When characterizing the TEEY of different materials, 𝑇𝐸𝐸𝑌 𝑀𝑎𝑥 , 𝐸 𝑀𝑎𝑥 , 𝐸 𝐶1 and 𝐸 𝐶2 are often used as points of comparison.

The energetic distribution of the electrons exiting the material also follows a standard behavior, shown in Figure 1234. This distribution can be separated in three regions depending on the energy of the electrons, which define three populations of electrons. The first region (I) is the distribution of true secondary electrons, which were put into motion in the electronic cascades created by the incident electrons. Their energy is centered around a few eVs, and it is generally admitted that the distribution of true secondaries is capped at 50 eV. This criterion is often used in experimental measurements to obtain the Secondary Electron emission Yield (SEY), which only quantifies the emission of true secondary electrons. However, this criterion may not be valid for low energy electrons. Indeed, the inelastically backscattered electrons constitute the second region of the histogram (II). This continuum is mostly made of incident electrons that have lost a part of their energy before exiting the material, or secondary electrons that were excited with a higher energy. As a result, some low energy incident electrons may lose a significant amount of energy and exit the material with an energy lower than 50 eV, but these can be counted as secondary electrons when using the experimental criterion. However, due to the uncertainty principle in quantum mechanics, we cannot know is a given electron is the incident electron that has lost some energy, or a secondary that was generated with an energy larger than 50 eV. Finally, a peak is observed around the energy of incident electrons 𝐸 0 (III). These are the elastically backscattered electrons, which have entered and exited the material without losing energy, or were reflected by the material's surface. In opposition to the SEY, the Backscattered Electron emission Yield (BEY) is defined as the ratio of elastically and inelastically backscattered electrons, including both regions (II) and (III). While the term "SEY" is sometimes used with the meaning of "total electron emission yield", in this study we will consider that the SEY only includes the contribution of secondary electrons. The sum of the SEY and BEY thus gives the TEEY.

The description of Figure 1-4 is a general overview of the energy distribution of secondary and backscattered electrons. Most materials will exhibit a spectrum that follows this standard curve. However, multiple small peaks may appear on the histogram at explicit energies. These are attributed to specific transition energies of electrons between the valence and conduction band, or the energy lost by the incident electrons after interacting with a given number of plasmon oscillations.

Factors of modifications of the electron emission yield

Several parameters can modify the value and shape of the TEEY, such as surface roughness, surface contamination, or the angle of incidence. In computer simulations, the target is often assumed to be a perfectly pure and flat sample. While some simulation codes attempt to model the effect of roughness [26,27] or surface contamination [28], these factors can be a source of discrepancy between the ideal materials from the simulations, and the real materials used in the experiment.

Effect of the angle of incidence

The incidence angle of electrons is defined relatively from the surface normal. At 0° the electrons have a normal incidence perpendicular to the material surface, and at 90° the electrons have a trajectory that is parallel to the surface. In Figure 1-5, the experimental TEEY of an etched sample of copper under various angles of incidence is presented. The TEEY was measured from 0° to 60° by R. Pacaud [29] in the DEESSE facility at ONERA. We can immediately observe that the TEEY increases with the incident angle, from 1.3 at 0° up to 1.8 at 60°. The first crossover point is also lowered from 170 eV at normal incidence to 70 eV at 60° incidence. This increase of the TEEY is explained by the fact that the incident electrons with a high incidence angle have a reduced penetration depth in the material. Consequently, they can generate more electrons close to the surface, which have an increased escape probability. Since more secondary electrons can easily escape the material, the TEEY increases for high incidence angles.

We can also see that the effect of the incidence angle is less marked for lower energy electrons. This is because low energy electrons under normal incidence are already captured in the first ten nanometers of the material's surface [5]. For very low energy electrons (below 50 eV), their probability of being elastically reflected by the surface increases with the incident angle, which prevents them from generating secondary electrons. For this reason, we can also observe an increase of the BEY of low energy electrons with the incidence angle.

Higher energy electrons (> 1 keV) have a larger penetration depth. They also produce more electrons than lower energy electrons but also at a greater depth. In result, when the incidence angle is increased, the greater number of secondary electrons generated by the higher energy electrons can more easily escape the material, resulting in a large increase of the TEEY. This is also why we observe a shift of the maximal energy of the TEEY, from 700 eV at 0° to 950 eV at 60°.

Effects of the surface chemistry and roughness

Technical materials used in spacecraft devices or particle accelerators have been exposed to air beforehand, and have a surface contamination layer made of several molecules including hydrocarbons, oxides, water or other species adsorbed on the surface. For some metals, such as aluminum or copper, a native oxide layer also appears immediately when the material is exposed to air. The surface contamination layer is only a few nanometers thick but it can still strongly modify the TEEY. Consequently, unless we want to get the TEEY of a technical material, experimental samples must be decontaminated before the measurement to obtain the TEEY of the pure material. However, to decontaminate the samples properly, they should be cleaned under vacuum conditions and not exposed to air between the decontamination and the TEEY measurement. For these reasons, TEEY measurements are also made in vacuum chambers to limit the deposit of contamination on the sample.

In Figure 123456, the TEEY of a copper sample measured as received and after decontamination are compared. The data was obtained by Plaçais et al. [30]. We can see a strong increase of the TEEY on the contaminated sample compared to the sample decontaminated by baking and erosion, with a maximum TEEY of 2.23 before decontamination and 1.1 after decontamination. The first crossover point is also shifted from 30 eV to 190 eV after decontamination. Therefore, contaminated surfaces increase the electron cloud production and the risk of multipacting.

Similar observations were made on other materials, such as silver [31]. The TEEY of contaminated materials can evolve during time due to the conditioning of the surface. This phenomenon is caused by the transformation of the contamination layer into a graphite layer by the incident electrons. Contrary to surface contamination, the graphite layer created by conditioning reduces the TEEY of the surfaces. Hence, this process is exploited in particle accelerators to reduce the formation of electron clouds [32]. However, this modification of the surface chemistry can only happen with a large electron fluence, much larger than what is received during standard TEEY measurements. For instance, the maximal dose received by the samples in standard TEEY measurements made in the DEESSE facility at ONERA is 0.5 nC/mm². On the other hand, the TEEY of a technical copper surface only starts to decrease due to conditioning for a dose of 10 µC/mm² [33] or 0.1 mC/mm² [34]. Consequently, if the samples are properly decontaminated, the TEEY data should not be modified by conditioning, and we should not observe a variation of the TEEY with time in the case of metallic or semiconducting targets.

The experimental and technical materials also have a surface roughness, which can either increase or decrease the TEEY depending on the roughness structures. Indeed, the electrons emitted from the bottom of the roughness asperities may hit the side walls and get recollected, thus reducing the SEY. However, the presence of roughness may induce an opposing effect, as the incidence angle of the primary beam can be significantly increased if they hit the sidewalls of the roughness patterns for instance. Thus, in many fields, materials with a specific surface state or roughness are developed to mitigate the TEEY. A widely used option is to engineer a specific surface roughness pattern which can trap the secondary electrons emitted by the surface 60°0°Contaminated Decontaminated [26,27,[35][36][37][38][39]. Several roughness patterns can be used to reduce the TEEY, such as rectangular and triangular grooves [35], rectangular [27] or trapezoidal [37,40] checkerboard patterns, or sawtooth grooves [36]. The efficiency of these roughness structures is shown in Figure 1-7, in the example of a silver surface. The TEEY results were obtained through Monte-Carlo simulations made during this thesis and published in [40]. We can see that the presence of these patterns on a surface will hinder the propagation of the secondary electrons emitted from the bottom of the structures.

The TEEY can thus be reduced by purely geometrical effects independent on the material. The efficiency of the different roughness patterns depends on various parameters, such as the aspect ratio of the structures, the proportion of tilted surfaces or the openness of the recollection valleys. The feasibility and choice of a given structure depends on the fabrication processes available. Some authors have proposed different means of creating specific roughness patterns, for instance chemically or by laser engraving. Efforts are also made to create such roughness patterns through electrodeposition [41].

Effect of magnetic fields

In a magnetic field 𝐵, the trajectory of a secondary electron emitted from the surface with an emission angle α and a kinetic energy 𝐸 = Where 𝜔 0 = 𝑒𝐵/𝑚, and 𝑅 𝐿 is the Larmor radius for an electron traveling perpendicularly to the field orientation. The presence of a magnetic field can significantly modify the TEEY of a rough surface through multiple effects. For a secondary electron emitted from the bottom of the structures, the presence of a magnetic field can modify its escape probability, depending on the intensity of the field and the position the secondary electron has been emitted from. Wang et al. [36] suggest that the trajectory followed by the secondary electrons in the magnetic field will increase the number of collisions with the side walls, lowering the escape probability and the TEEY. This can especially happen on a sawtooth surface, where electrons with a non-zero incident angle can collide with the vertical walls. In the case of a magnetic field with an orientation parallel to the flat surface normal, the secondary electrons emitted from a side wall can be recollected by the same wall depending on the radius of the helix.

There is also the possibility that this trajectory will reduce the probability of a collision with the side walls. This can especially happen with higher values of the magnetic field, when the helix radius becomes smaller than the width of the structures. In such a case, the secondary electrons can easily escape from the bottom of the structures or the tilted side walls. For extremely high values of the magnetic field on the order of the Tesla, the secondary electrons should travel almost vertically and the effect of surface roughness could be overridden for electrons with a normal incidence.

The increase or decrease of the TEEY caused by the magnetic field is largely dependent on the roughness of the surface. Fil et al. [18] have shown that the TEEY of a technical surface could be modified by 45% in the presence of a magnetic field, but the TEEY of a polished surface was only reduced by 5%. In particle accelerators and fusion reactors, strong magnetic fields are used to contain the plasma or particle beam. Hence, the modification of the TEEY of a rough surface under a magnetic field must be considered in these applications.

Experimental measurement procedure of the TEEY

Over the years, a lot of data has been collected on the electron emission yields of several materials, including metals, semiconductors, insulators and compound materials. Most of this experimental data has been gathered in Joy's database [42,43]. However, the data for electrons below 100 eV is scarce, and there are strong dispersions in the TEEY data for a given material. These can be explained by the differences in experimental setup (intensity of the vacuum or work function of the electron collector), or the surface state of the material (chemistry, roughness). For these reasons, two experimental TEEY measurement installations were designed at ONERA: DEESSE (Dispositif d'étude de l'Emission Electronique Secondaire Sous Electrons, Facility for emission of secondary electron under electron bombardment) and ALCHIMIE (AnaLyse CHImique et Mesure de l'émIssion Electronique, Chemical analysis and measurement of electronic emission). The equipments available in DEESSE and ALCHIMIE will be given in Chapter 6, where we will also detail the experimental TEEY measurements obtained with these facilities on insulating samples. An extensive description of the facilities can be found in the thesis of T. Gineste [31,44].

The general TEEY measurement procedure is based on two measurements of the current flowing through the sample. According to the current conservation law, the incident current 𝐼 0 , the emitted current 𝐼 𝐸 and the current flowing through the sample 𝐼 𝑆 follow the expression

𝐼 0 = 𝐼 𝐸 + 𝐼 𝑆 Equation 1-3
First, the sample holder is biased positively, for instance to a potential of +27V in DEESSE and ALCHIMIE. Due to the external electric field induced by the bias, practically all low energy electrons are recollected by the surface. The measured current 𝐼 𝑆 + is very close to the incident current (𝐼 0 ≅ 𝐼 𝑆 + ). In a second step, the sample holder is biased negatively (for example -9 V in DEESSE and ALCHIMIE), to force all secondary electrons to exit the sample. Indeed, as we will show in the next section, the charge buildup in insulators may prevent the escape of secondary electrons by electrostatic effects. The current 𝐼 𝑆 -measured in this case can be used to determine the emitted current using Equation 1-3 and the incident current 𝐼 0 obtained from the previous step, as 𝐼 𝐸 = 𝐼 0 -𝐼 𝑆 -. Finally, the TEEY is given by the ratio of the emitted current over the incident current, which is determined using both measurements 𝐼 𝑆 + and 𝐼 𝑆 -:

𝑇𝐸𝐸𝑌 = 𝐼 𝐸 𝐼 0 = 𝐼 0 -𝐼 𝑆 𝐼 0 = 𝐼 𝑆 + -𝐼 𝑆 - 𝐼 𝑆 + Equation 1-4
For metals and semiconductors, it is possible to compute the TEEY by sending a continuous current on the sample. For insulators however, a pulsed measurement method must be used to limit the effects of charging. In this case, the sample is irradiated during a given time (width of the pulse) followed by a relaxation period before sending another pulse of electrons. To compute the TEEY, a first series of pulses is sent with the sample biased positively measure the incident current 𝐼 0 , then a second series of pulses is sent with the sample biased negatively to measure the sample current 𝐼 𝑆 . In both cases, the current is averaged over the series of pulses. The TEEY is then computed using the Equation 1-4.

Issues related to the electron emission yield of insulators

Even in the absence of surface contamination, a significant variation of the TEEY with time can be observed on dielectric materials, on the scale of the microsecond or nanosecond depending on the measurement parameters. This is due to the creation of charges in the material, following the emission or absorption of electrons. Consequently, it can be very challenging to perform TEEY measurements on insulators due to the strong modifications of TEEY created by the charge buildup, and special care must be taken to limit the influence of charging. Despite this, some effects of charging on the TEEY are still not thoroughly understood, as we will show in this section.

According to the conventional theory [45][46][47], the charge buildup in the dielectric material is guided by the electron emission yield. Indeed, when a secondary electron is set into motion by a primary particle, a deficit of electron is now present where the secondary electron was initially localized. Due to the conservation of charges, this deficit is modeled as a positively charged hole, which can also move in the material. If the TEEY is greater than one, the material is emitting electrons, more holes than electrons are left inside of the sample, and the insulator is charging positively. On the opposite, when the TEEY is lower than one, more electrons than holes are implanted in the material, which is absorbing electrons and charging negatively.

External charge effects

Due to the charge buildup, the surface potential 𝑉 𝑠 of the sample is modified, depending on the total charge inside of the material. The difference of potential between the surface of the insulator and the other surfaces, including the electron gun generally set to the ground, creates an electric field outside of the material. This electric field can significantly modify the trajectories and energies of low energy electrons in vacuum. In this section, we will only focus on these external charging effects, which are due to the global charge in the material and affect the electrons outside of the material. The effects of the internal electric field will be detailed in the next subsection.

From the conventional TEEY curve, three situations can arise depending on the TEEY and the energy of electrons, illustrated on Figure 12345678. When the energy of incident electrons 𝐸 0 is below the first crossover point (I), the TEEY is lower than one and the material is charging negatively. For most dielectrics at room temperature, the first crossover point is about a few tens of eVs. The incident electrons are not energetic enough to generate a significant number of secondary electrons, and are implanted close to the surface. Because the surface potential 𝑉 𝑠 is negative, the incident electrons are slowed down by the electric field in vacuum and arrive at the surface with an effective landing energy 𝐸 𝐿 = 𝐸 𝑖𝑛𝑖𝑡 -𝑒𝑉 𝑠 , where 𝑒 = 1. 6 10 -19 𝐶. Since this energy is lower than their initial energy, this reduces the TEEY according to the conventional curve. The insulator continues to charge negatively, which amplifies the slowing down of electrons and the reduction of the TEEY. The charge buildup continues until the electrostatic potential energy 𝑒𝑉 𝑠 becomes greater or equal to the energy of the incident electrons. When this limit condition is reached, the electrons cannot overcome the electric field, which is strong enough to reflect them from the surface and make them directly hit the electron collector instead. Since the electrons cannot hit the surface, the global charge is not evolving anymore and a steady state is reached. In TEEY measurements, we observe a reduction of the TEEY until the electrons get reflected by the electric field. At this point, the TEEY suddenly increases to 1 and stabilizes, though we obviously do not measure the TEEY of the material anymore since the electrons are not penetrating in the material. For electron energies 𝐸 𝑖𝑛𝑖𝑡 between the two crossover points (II), the TEEY is greater than one and the material is charging positively. Since more electrons are exiting the material than entering it, a net positive charge is created inside of the material due to the remaining holes. In this case, the surface potential becomes positive, and the electrons in vacuum are accelerated in the direction of the surface by the electric field. The incident electrons arrive at the surface with a landing energy 𝐸 𝐿 = 𝐸 𝑖𝑛𝑖𝑡 + 𝑒𝑉 𝑠 that is increased compared to their initial energy. As the material continues to charge positively, the effective energy increases more and more towards the second crossover point. Thus we could suppose from an empirical point of view that if 𝐸 𝑖𝑛𝑖𝑡 is greater than the energy of the maximum TEEY (𝐸 𝑀𝑎𝑥 ), the TEEY should decrease as the material is charging. On the other hand, if 𝐸 𝑖𝑛𝑖𝑡 < 𝐸 𝑀𝑎𝑥 , the TEEY would first go through a phase of increase until 𝐸 𝑖𝑛𝑖𝑡 = 𝐸 𝑀𝑎𝑥 . However, in reality, we only observe a decrease of the TEEY. Indeed, the secondary electrons that have escaped the material in vacuum are also accelerated back to the surface by the electric field. Thus, if their energy is lower than 𝑒𝑉 𝑠 , the secondary electrons cannot overcome the electric field and are forced back onto the surface, where they get recollected by the material. Given that most true secondary electrons have energies of 10 eV and below, a surface potential of only a few volts is enough to force the recollection of a significant part of the secondary electrons. What's more, given the energies of electrons between the two crossover points (a few hundreds of eV), an energy differential of a few eV should not cause a significant shift on the TEEY curve. Hence, the recollection of secondary electrons is the main factor that leads to a strong decrease of the TEEY. When the TEEY reaches 1, no net charge is created in the material, so that the external electric field is not increased anymore. At this point, the TEEY and the total charge have reached an equilibrium. If the incident electron energy is beyond the second crossover point (III), the TEEY is lower than one and the material is charging negatively. For most dielectrics, the second crossover point is above 2 keV. As in the case (I), the surface potential is negative, the electrons are slowed by the electric field and arrive at the surface with an energy 𝐸 𝐿 = 𝐸 𝑖𝑛𝑖𝑡 -𝑒𝑉 𝑠 . However, contrary to situation (I), the energy of incident electrons is much higher, so that they are not electrostatically reflected if the field becomes too strong. Instead, the landing energy of the electrons progressively decreases to 𝐸 𝐶2 and the TEEY converges to one. From this macroscopic point of view, a surface potential of a few hundred volts or more may be necessary to slow the electrons down the energy of 𝐸 𝐶2 . The TEEY and charge buildup reach a steady state, as in situation (II). Since the macroscopic effects of insulator charging on the TEEY are well known, it is possible to modify the experimental parameters used during TEEY measurements to limit the influence of charging. First, the electron collector can be biased to a strong potential (a few hundred volts). This will generate an extracting electric field that is opposed to the attracting field created by the positive global charge in situation (II), and prevent the recollection of secondary electrons.

Another method, which is the one used in the DEESSE and ALCHIMIE facilities at ONERA DPHY, is to bias the sample holder to a given potential, which will force the surface potential of the dielectric sample at a certain value. With this method, the sample's surface potential can be set to a negative value of a few volts, which will prevent the recollection of secondary electrons when the material is charging positively. Even if the negative surface potential will slow down the incident electrons by a few eV, this should not create a significant shift of the TEEY, especially for electrons above a few tens of eV. Moreover, the shift of incident energy is controlled, and we can take it into account when plotting the TEEY curves. Therefore, using this method will allow us to measure the TEEY of dielectric materials while removing the perturbations of external charge effects.

However, the sample holder biasing method has a few limitations. First, the negative bias should be kept to small potentials of a few volts or tens of volts, in order to avoid a significant difference in the landing energy, which will create a shift of the TEEY. Nevertheless, this shift can be compensated by sending more energetic electrons in the first place. The main limitation of the method is its ability to compensate the positive surface potential created by the charge buildup in thick or bulk dielectric samples. Indeed, the capacitance C of a material is given by the formula

𝐶 = 𝜖 0 𝜖 𝑟 𝑆 𝐷 Equation 1-5
Where 𝜖 0 is the vacuum permittivity, 𝜖 𝑟 the relative permittivity of the material, 𝑆 is the area of the sample's surface and 𝐷 is the sample's thickness. In consequence, thicker materials have a small capacitance. In addition, the change in the surface potential ∆𝑉 𝑠 can be linked to the total charge buildup in the material ∆𝑄 by the relation [48]:

∆𝑉 𝑠 = ∆𝑄 𝐶 = ∆𝑄 𝐷 𝜖 0 𝜖 𝑟 𝑆 Equation 1-6
As a result, one can see that for the same charge, a thicker material will have a stronger variation in surface potential. For instance, thin films of silicon dioxide of 20nm thickness will only charge up to a potential of a few volts during the TEEY measurements. However, samples of mica, Al2O3 or Teflon with a thickness on the order of the micrometer or millimeter can charge up to hundreds or even thousands of volts when irradiated by electrons of several keV [49]. Consequently, it can be very difficult to limit the effects of charging for thicker samples. For this reason, most dielectric samples used in TEEY studies conducted in DEESSE are thin films of nanometric thickness, so that the external charging effects can be removed efficiently. This depends obviously on the transport properties of electrons in the target material and its capacity to retain charges. For instance, Belhaj et al. [48] were still able to use the sample holder biasing method on MgO bulk samples having a thickness of 2 mm, when studying the TEEY of 200 eV electrons during short pulses.

Internal charge effects

Several TEEY experimental measurements have been made using the two external charge removal methods detailed previously [23,48,[50][51][52]. Nonetheless, a decrease of the TEEY during time has still been observed for various materials for energies between the two crossover points. These materials include Teflon [53], MgO [48], Al2O3 [50] and diamond [23], among others. Due to the absence of external charging and electric field effects, this decrease cannot be attributed to the recollection mechanisms highlighted in the previous section, in the case of situation (II). It has also been observed that the final value of the TEEY could be more or less lowered depending on the current density, the irradiation time or the temperature. Indeed, the charge buildup inside of the material can significantly disrupt the path of electrons in the material. First, an electric field also appears inside of the material, because the charge density in the material is not constant. Since the secondary electrons generated in the first few nanometers close to the surface have the highest escape probability, the remaining holes create a positive charge. The incident electrons are then implanted deeper in the material, creating a negative charge. Depending on its sign and intensity, the internal field can accelerate the electrons towards the surface, which increases their escape probability and the TEEY, or accelerate them deeper in the material, which will have the opposite effect. Simulation studies [54][55][56] have shown that the transport of electrons in SiO2 starts to be significantly affected for internal fields of 0.5-1MV/cm, but it is unknown whether such fields were reached during the TEEY measurements mentioned above. Some experimental works [48] have proposed an explanation based on the recombination of secondary electrons with the holes created by the previous cascade. Nevertheless, this hypothesis has not been confirmed. Indeed, the TEEY obtained experimentally is the global result of the several physical processes involved in the transport of electrons. Hence, it is not possible to determine experimentally the contribution of each internal physical process to the decrease of the TEEY. Instead, simulation models for the transport of low energy electrons could be more suited to study the impact of charging on the electron emission yield. In this regard, many computational models for the charge buildup and its effect on the TEEY have been conceived [57][58][59][60][61][62][63][64][65][66]. However, to our knowledge, no simulation of the effect of positive charge buildup on the TEEY were made in conditions where the sample holder was negatively biased to prevent the recollection of secondary electrons. Hence, the simulations results in the case of positive charging mainly show the effects of the external electric field on the TEEY.

In conclusion, the internal charge effects are still difficult to investigate experimentally and these results are not thoroughly understood. This is why, in this thesis work, we will particularly focus on developing a simulation model and conducting experimental measurements, to study and explain the effect of internal charge on the transport of electrons and on the TEEY.

Double-hump electron emission yield curves

Multiple studies have reported experimental TEEY measurements on such samples that do not follow the standard behavior of TEEY from Figure 1-3 [67][68][69]. The measured TEEY exhibit a double-hump shape illustrated in Figure 1-12, with the apparition of a TEEY local minimum between the two humps. This behavior has mainly been reported for SiO2 thin film samples, although it has also been observed on other space-used dielectric materials [70]. A few explanations have been proposed for these observations. Ye et al. [71] have shown that the variation of the surface irradiated by the electron beam could create double hump TEEY curves as an experimental artefact. However, these observations were only made in the case of metallic samples with surface roughness structures of millimetric dimensions. For dielectric materials, Hoffman and Dennison [70] proposed an explanation linked to external charging effects. However, other works [67][68][69] have also observed multiple hump TEEY curves on SiO2 thin films of various thicknesses grown on a Si substrate. In these studies, the sample holder was negatively biased to avoid the recollection of secondary electrons, so that this behavior cannot be attributed to external field effects only.

Yi et al. [69] and Yu et al. [68] proposed that the local TEEY minimum can be removed by compensating the holes created in the SiO2 layer by electrons tunneling from the Si layer. This is in agreement with other works that showed a link between the hole density and the TEEY [48,72]. They also proposed that the second TEEY maximum is due to enha+nced compensation of the holes by electrons tunneling from the Si layer when the penetration depth of electrons is equal to the thickness of the SiO2 thin film layer. On the other hand, Rigoudy et al. [67] suggested that a TEEY minimum instead appears under such conditions, where a conductive channel evacuates the secondary electrons from the SiO2 layer under the effect of radiation induced conductivity. However, the local TEEY minimum appears in their measurements at the same incident energy for various SiO2 thicknesses. Notably, in these three studies, the TEEY local minimum was removed by a change of experimental parameters, such as increasing the positive bias of the collector [67] or the negative bias of the sample holder [69], or decreasing the incident current [68]. As a result, the apparition of the double hump TEEY curves could be linked to the conditions of measurement and could also be an experimental artefact.

Computing the TEEY with Monte Carlo simulations

In order to understand the physical processes behind the emission of secondary electrons and the internal charging effects, which are not easily accessible by experimental means, we can use Monte-Carlo simulation codes to model the transport of electrons in matter and the generation of secondary electrons. These codes combine physical models of the interactions of particles with matter, with random numbers to model the non-deteministic behavior of these particles. Many Monte-Carlo codes are publicly available to the scientific community, such as PENELOPE, Casino, FLUKA, among others. We can also cite the GEANT4 toolkit, which includes several physics packages for the transport of particles through matter. However, most interaction models have been developed for high energy particles, and transport models for low energy electrons are lacking. For instance, PENELOPE and Casino have a low energy limit of 50 eV for electrons. The MicroElec physics module of GEANT4, developed by CEA and ONERA, extends down to 16 eV, but is only valid for electrons in silicon. As a result, these simulation packages are unable to compute the secondary electron emission yield.

On the other hand, several Monte-Carlo codes have been developed specifically for the transport of low energy electrons down to a few eVs [26,27,39,[73][74][75][76][77][78][79]. These codes are able to simulate the secondary electron emission of various materials under electron irradiation, including metals and semi-conductors [77,79] or insulators (without charging) [80][81][82], or the emission of electrons under ion irradiation [83]. Most codes are focused on metals and semi-conductors, since the TEEY of dielectrics is heavily dependent on the charge buildup, which needs to be dynamically simulated. Despite this, some Monte-Carlo codes [57][58][59][60][61][62][63] are able to model the transport of charges in insulators and its effect on the TEEY, notably external effects due to the global charge. Most studies focus on Silicon Dioxide, which is the material with the most reference data. Nevertheless, none of these codes is publicly available to the community, and the observations made on the TEEY in the case of positive charging are mainly due to external effects.

In recent years, the OSMOSEE code [76] has been developed at ONERA, which is able to simulate the transport of electrons in aluminum down to a few eVs, and can model the electron emission yield of aluminum. Further developments at ONERA during the thesis of J. Pierron [25,27,77,84], in collaboration with CEA and CNES, have extended OSMOSEE and MicroElec's physics to simulate the electron emission of Al, Si and Ag, but these updates were not released in GEANT4. Most importantly to our study, both of these codes were designed for metals and semi-conductors, hence they are not able to simulate the charge transport in insulators. However, we should be able to use these simulation codes as a base and extend it to the simulation of the TEEY of insulators.

Conclusion of Chapter 1

In this first chapter, we have highlighted how the secondary electron emission process can disrupt the operation of several devices, such as spacecraft electronics or particle accelerators. In this regard, dielectric materials have proved to be especially problematic, for several reasons. First, these materials have a larger total electron emission yield compared to metals, which increases the risk of multipactor discharges in RF components. The electron cloud formation is also enhanced, which leads to a significant loss of power in particle accelerators. Second, insulator materials can charge depending on the electron emission yield, generating a potential gradient inside of the material. The charge can build up to the point that electrostatic discharges suddenly occur. These can severely damage the electronic components, for instance by creating destructive electrical arcs on the surface of solar panels. Therefore, evaluating the TEEY of insulating materials is critical for these applications. However, it is especially difficult to perform experimental TEEY measurements on dielectrics, due to the modification of the TEEY caused by charging effects. This is also problematic in scanning electron microscopy. Indeed, the image contrast depends on the TEEY of the target, which can be modified by the charge buildup if it is an insulator. On the one hand, the external charging effects are well known. It is understood how the external electric field generated by the global charge modifies the energy of the incident and secondary electrons, why this changes the TEEY, and how the effect of this electric field can be removed. On the other hand, the comprehension of internal charging effects is still lacking. It is not thoroughly understood why the TEEY decreases with time even when the external electric field has no influence on the TEEY, or why multiple-hump TEEY curves have been observed on dielectrics but not on metals.

For these reasons, the main objective of this thesis work will be to develop our own Monte-Carlo model for the simulation of the TEEY of insulators, in order to understand the effects of internal charging on the transport of electrons in dielectric materials and their TEEY. We will also perform experimental measurements on insulating samples, to study the TEEY and obtain reference data for our simulation. Before constructing our model however, we need to identify which electron-matter interactions need to be taken into account. We have also established that the charge buildup can also modify the TEEY in insulators through internal charge effects. Hence, a main concern of this work will be to study the transport of charge carriers and their physical interactions. The description of all these interactions will be the focus of Chapter 2, along with the Monte-Carlo procedure used to simulate the transport of electrons and the secondary emission process.

Chapter 2: Presentation of the low energy electron-matter interactions and the Monte-Carlo simulation procedure

We have established that the generation of secondary electrons in all materials and the charge buildup in dielectrics are due to the interactions of incident low energy electrons with matter. However, to build a simulation of the effects of internal charging on the TEEY of dielectric materials, we need to introduce the electron-matter interactions we will model in the next chapters. The simulation procedure used in this work, namely the Monte-Carlo simulation, will first be presented. This will allow us to highlight the key quantities we will need to know, in order to describe the electron-matter interactions. Some of these interactions, as we will show, are common to all material types, and are sufficient to model the TEEY of materials without charging effects. However, we know that the internal charge buildup affects the TEEY of insulators and we want to know which physical mechanisms are involved. For this reason, we will also present the transport of the charge carriers in insulators, and the specific interactions involved. Finally, we will mention some other quantities than the TEEY, which are relevant to the study of the transport of low energy electrons and which can be accessed with Monte-Carlo codes.

Fundamental quantities of interactions and their use in the Monte-Carlo simulation procedure

This section aims to describe the operation process of Monte-Carlo codes from a general point of view. The fundamental quantities used to desctibe the interactions of particles are the mean free paths and cross sections. These two quantities, which are necessary to Monte-Carlo codes, will also be defined in this section.

Interaction cross sections

The cross section (also abbreviated as XS) is an indication of the probability of occurrence for a given interaction. The cross section is defined having the dimension of a surface, and can be understood using the analogy of a flux of particles 𝐹 𝑖 directed towards a single scattering center, which will then be measured by a detector located beyond the scattering center [1]. During the interaction, some of the particles can be scattered by the diffusion center into a solid angle 𝑑Ω measured by the particle detector. This analogy is shown in Figure 2-1. The number of particles 𝑑𝑛 scattered by the center and hitting the detector is thus dependent on the incident flux 𝜙 𝑖 and the solid angle measured by the detector 𝑑Ω. When extrapolating this image to a solid, a multiple number of scattering centers 𝑁 𝑐 is now involved. If we suppose that each particle is only interacting with one center, for instance in the case of a flux of particles crossing a very thin film, the number of particles hitting the detector is thus given by [1]:

𝑑𝑛 = 𝑁 𝑐 𝜙 𝑖 𝑑𝜎 𝑑Ω 𝑑Ω Equation 2-1
In this equation, the differential cross section 𝑑𝜎 𝑑Ω appears, which expresses the probability of a particle of being scattered in the solid angle 𝑑Ω. The total cross section 𝜎 is obtained by integration of the differential cross section in Equation 2-1:

𝜎 = ∫ 𝑑𝜎 𝑑Ω 𝑑Ω Equation 2-2
By combining both equations, we can understand that the number of particles hitting the detector is directly proportional to the total cross section 𝜎. It is equal to the number of particles hitting an area the size of 𝜎, and not simply the number of particles hitting the area occupied by the scattering center. In result, the cross section associated with an interaction can be quite different from the area of the scattering center.

Interaction Mean free paths and the computation of the interaction lengths

The mean free path 𝜆 associated to an interaction is the average distance that can be traveled by a particle between two successive interactions. If we consider a flux of particles passing through a sample of thickness 𝑑𝑥 and area 𝐴, with a volume density 𝑁 of scattering centers, the 2.1 -Fundamental quantities of interactions and their use in the Monte -Carlo simulation procedure probability of a particle to be stopped by one of the diffusion centers is expressed by the ratio between the area occupied by the scattering centers multiplied by the cross section, divided by the total area of the sample:

𝑃 = 𝑁𝜎𝐴 𝐴 𝑑𝑥 = 𝑁𝜎𝑑𝑥 Equation 2-3
The variation of the intensity of the flux after the crossing of the sample is thus equal to the initial intensity multiplied by the interaction probability as

𝑑𝐼(𝑥) = -𝐼(𝑥) 𝜎𝑁 𝑑𝑥 Equation 2-4
This equation is a linear differential equation that can be integrated. The solution of this equation is given in the form of 𝐼(𝑥) = 𝐼 0 𝑒 -𝑁𝜎𝑥 . We can then use this solution to evaluate the probability 𝑑𝑃(𝑥) of a particle to be stopped between 𝑥 and 𝑥 + 𝑑𝑥 as:

𝑑𝑃(𝑥) = 𝐼(𝑥) -𝐼(𝑥 + 𝑑𝑥) 𝐼 0 = 𝑁𝜎𝑒 -𝑁𝜎𝑥 𝑑𝑥 Equation 2-5
We can then deduce the average distance 〈𝑥〉 between two interactions, by applying the definition of the expected value of a random variable to the Equation 2-5:

〈𝑥〉 = ∫ 𝑥 𝑑𝑃(𝑥) +∞ 0 = ∫ 𝑥 𝑁𝜎 𝑒 -𝑁𝜎 𝑥 𝑑𝑥 +∞ 0 = 1 𝜎𝑁 Equation 2-6
Which gives the value of the mean free path 𝜆 = 1 𝜎𝑁 ⁄ .

Monte-Carlo programs use a random selection process to choose the interactions made by the particles, taking into account the probabilities of each interaction. This selection is made at the beginning of each simulation step, where the program surveys the mean free paths 𝜆 𝑖 for each interaction depending on the properties of the tracked particle. Using a random number 𝑅 𝑖 from 0 to 1, an effective interaction length 𝑙 𝑖 is computed, using

𝑙 𝑖 = -𝜆 𝑖 ln(𝑅 𝑖 ) Equation 2-7
The program then selects the physical process with the shortest interaction length 𝑙 𝑖 , which will be the distance travelled by the particle during the simulation step. At the end of the step, the effective interaction lengths of the physical processes that were not selected are modified, to take into account the distance that was travelled by the particle. The procedure is then repeated with a new random sampling for each simulation step. On average over a great number of simulated events, the most probable interactions will be the ones that have a higher frequency of occurrence. However, the random sampling introduces a statistical dispersion to take into account the non-deteministic transport.

Common electron-matter interactions for all material types

An electron penetrating in a solid can undergo multiple interactions and collisions, some of which are common to all material types and can occur for electrons traveling in metals, semiconductors and insulators. These are given in Figure 2-2. The primary electron entering the solid can first interact with the nuclei of the atoms of the material in an elastic interaction, which results in a change of direction of the electron. The incident electron can also transfer a part of its energy to an electron of a material in an inelastic interaction, which can result in the generation of a secondary electron. The new electrons set into motion may in turn excite other electrons of the solid, creating an electronic cascade in the material. The phenomenon of secondary electron emission happens when the secondary electrons cross the surface and escape from the material. Given the large number of electrons set into motion in the secondary electron cascades, one can easily understand how a material can be emitting more electrons than it had received. However, the emission of secondary electrons into vacuum is not systematic. Depending on its energy and angle of incidence, an electron reaching the surface can be transmitted with a change of direction, or reflected by the surface. If a primary electron remains close enough to the surface, it may be able to escape the solid, which results in a backscattered electron. The peak of elastically backscattered electrons we have observed in the energy distribution of electrons exiting a material are the primary electrons which were either reflected by the surface potential barrier before entering the material, or have entered the material, made one or a few elastic interactions with a negligible energy loss, and then escaped the material. If the primary particle has lost some energy through inelastic interactions before escaping, it will be measured as an inelastically backscattered electron.

The description of these interactions is mandatory in order to compute the electron emission of any material, so we will have to include them in our low energy electron transport code for dielectrics. On the other hand, with only a modeling of the interactions we present in this subsection, we should already be able to compute the TEEY of metals and semi-conductors, which we will do in Chapter 3.

Elastic interaction

The elastic interaction is the interaction of an incoming electron with the coulombian potential created by an atom of the material, precisely by the nuclei and the strongly bound core shell electrons. There is a very large difference in mass between the electron and the nucleus+core electrons ensemble, hence the energy lost by the incident electron is very small (below a few meV). However, the electron can be significantly deflected from its initial trajectory, depending on its energy. Indeed, when the electron energy falls below a few hundred of eVs, the probability of the electron being strongly scattered increases. The electron may even be scattered in an opposite direction, with a scattering angle greater than 90°. Moreover, as the electron energy decreases towards the eV, the elastic interaction mean free path also decreases down to the order of the interatomic distances (a few Angstrom). As a result, low energy electrons, especially below 100 eV, are strongly scattered in random directions by the elastic interactions after only a few Angstroms. Hence, their motion is equivalent to a Brownian motion. This particularity of the transport of low energy electrons means that they cannot be treated the same way as higher energy electrons (> keV), or heavier particles such as protons.

To describe the elastic interaction of an electron with an atom, one not only needs to take into account the electric field generated by the nucleus but also the field generated by the neighboring atoms of the lattice. This description and the computation of the interaction cross sections can be made using the Partial Wave Analysis (PWA) method proposed by Mott [2]. This method is based on quantum mechanics, and the computation of the PWA cross sections is strongly dependent on the approximations used for the potential created by the atoms. Nevertheless, several works have used the PWA method to compute the total and differential elastic cross sections. Notably, in this work, we have chosen to use the ELSEPA [3] (ELastic Scattering of Electrons and Positrons by Atoms) code for the computation of elastic cross sections. This code is a database of elastic cross sections computations for electrons from 10 eV up to 1 GeV. The code covers elements from Z = 1 to Z = 103, which allows the user to simulate the elastic interaction in the corresponding monoatomic materials. ELSEPA can also be used to compute the elastic cross sections of molecules, if the user can provide additional parameters such as the molecule geometry. In this case, the cross sections can be used for the transport of electrons in compound materials.

Nevertheless, the validity of the partial wave method analysis method below 50 eV is questioned by several works [4][5][6]. Akkerman et al. [4] and Valentin et al. [5] have substituted the ELSEPA cross sections for electrons in Si below 50 eV by ab-initio cross sections. In SiO2, Schreiber & Fitting [6] have shown that the values of the cross sections given by the PWA below 100 eV become unphysical, with a mean free path that falls below the interatomic distance. This is because, in insulators, the collective oscillations of the lattice (phonons) must be taken into account for electrons below 100 eV.

Inelastic interaction

In addition to the nuclei of the atoms, an electron may interact directly with the other electrons of the material. An inelastic interaction happens when the incident electron transfers a part of its energy to one of these electrons, which can then be set into motion. Hence, the production of secondary electrons and the generation of the electron cascade occurs during the inelastic interactions. However, to correctly model the electron cascade, we need to know the energy lost by the primary particle, where the secondary electron is coming from, and what part of this energy is actually transferred to the electron put into motion. In this regard, we must study the electronic structure of the material, which contains information on the two distinct populations of electrons from the solid: the strongly bound core shell electrons, and the weakly bound valence or conduction electrons.

Description of the electronic structure of a solid

The electronic structure of an isolated atom is made of several discrete energy levels occupied by the electrons. According to Bohr and Rutherford [7], these levels are defined by four integer numbers. The principal quantum number 𝑛 describes the electron shell the level belongs to, with the deepest shell having a number of 1, and the furthest shell has the biggest number. The atomic shells are often referred to by a letter, starting from K for 𝑛 = 1, L for 𝑛 = 2, M for 𝑛 = 3, and so on. The azimuthal quantum number 𝑙 describes the subshell of the electron within the 𝑛 shell, and ranges from 0 to 𝑛 -1. The magnetic quantum number 𝑚 corresponds to the shape of the orbital within the subshell, and is contained between -𝑙 and +𝑙. Finally, the spin quantum number 𝑠 gives the orientation of the spin (+1/2 or -1/2). According to Pauli's exclusion principle, a single orbital (𝑛, 𝑙, 𝑚) can only contain two electrons, having a spin of -1/2 and +1/2.

It is important to note that the deepest shells are closer to the atomic nucleus, which means that their binding energy is greater than the furthest shells. For instance, the binding energy of the K shell of germanium (Z=32) is 11 keV according to the EADL atomic data library [8], whereas the furthest shell (N3) only has a binding energy of 6 eV. Heavier elements have more energy levels, thus the core shells get even closer to the nucleus, which increases their binding energy.

However, the outermost shells still have a binding energy on the order of the eV. For example, the binding energy of the K shell of carbon (Z=6) is 300 eV, which is much lower than for germanium. On the other hand, the energy of the furthest shell of carbon (L3) is 9 eV and similar to the one of germanium, even if germanium has more shells (12) than carbon (4).

The electronic structure of a solid is quite different from the structure of isolated atoms, as shown in Figure 2-3. Indeed, when two atoms are brought together and form a chemical bound, the outermost energy levels are shared and split between the two atoms. In the case of a solid, a very large number of atoms are gathered in a volume (on the order of 10 23 at/cm 3 ), and only separated by a few Angstroms. Due to this proximity, the external shells fuse into several continuums of energy levels, which are known as electronic bands. Some of these levels can be accessible (valence and conduction bands) or inaccessible (energy gaps between the bands). The core shells however are still strongly bound to the individual atoms, and the overlap of the energy levels of the deeper shells is limited. Hence, they can still be treated as discrete energy levels. At a temperature of 0 K, electrons fill the electronic structure starting from the core shells up until the Fermi level 𝐸 𝐹 , which is the highest energy level occupied at absolute zero. The properties of a material are heavily dependent on the location of the Fermi level and the band structure. Materials can be separated into three categories, shown in Figure 2-4. First, in metals, the Fermi level is located within the conduction band. As a result, at room temperature, the conduction band is partially filled with weakly bound electrons. These electrons can easily be excited to an unoccupied level of the conduction band by thermal activation, and therefore put into motion in the solid.

In semi-conductors and insulators however, the Fermi level is placed between the valence band and the conduction band, in the middle of an energy gap. Hence, for these materials, the conduction band is completely empty at 0 K and the valence band is completely filled. The width of this gap is what differentiates a semi-conductor from an insulator. Indeed, semiconductors generally have an energy gap of a few eV at most (around 1 eV for Si and Ge), whereas the energy gap of insulators is greater than 4 eV (8 eV for SiO2 and Al2O3). The energy gap in a semiconductor is small enough that the electrons of the valence band can jump this gap, by gaining energy through thermal excitation. It is also possible to introduce additional charge carriers in a semiconductor by doping, which is done by implanting other atoms that have a different valence from the intrinsic atoms of the material. The effect of doping is also to create additional energy levels in the band gap, which can facilitate the jump of electrons into the conduction band. In insulators however, the band gap is much too large to be jumped by the electrons of the valence band through thermal excitation. This transition is only possible if the insulator is subjected to a very large electric field, which can enable the valence electrons to tunnel through the energy gap. During an inelastic interaction, the primary electron transfers a part of its energy to a bound electron of the material and promotes it into an unoccupied level of the conduction band, above the Fermi level. The promoted electron becomes a secondary electron that can now move into the material and even exit it. Therefore, we can separate the sources of secondary electrons into two populations: strongly bound electrons from the core shells, and weakly bound electrons, located in the partially filled conduction band of metals, or at the top of the valence band of semiconductors and insulators.

Inelastic interactions with core shells electrons

As mentioned earlier, core electrons are situated within the deepest shells of the atoms of the material, and are therefore strongly bound to the nucleus. To promote a core shell electron into the conduction band, the binding energy of the energy level of the electron must be overcome. Given that the binding energy of core shells ranges from a few hundred to a few thousand of eV, the excitation of a core electron results in a large energy loss for the primary. However, all of this energy is not directly transferred to the secondary particle, since most of it is spent to promote the electron into the conduction band. Indeed, for an energy transfer 𝑇 lost by the primary electron, a secondary electron from a core shell with a binding energy of 𝐸 𝑏 will be generated with an energy

𝑄 = 𝑇 -𝐸 𝑏 -𝐸 𝑔𝑎𝑝 Equation 2-8
In semiconductors and insulators, the energy of the gap 𝐸 𝑔𝑎𝑝 must also be overcomed by the secondary electron, whereas for metals 𝐸 𝑔𝑎𝑝 = 0.

Contrary to the weakly bound electrons in the energy bands, the core electrons still belong to a given atom of the lattice. When an electron is promoted from a core shell into the conduction band through ionization, a vacancy is left at the place of this electron. However, this leaves the atom in an unstable condition. Indeed, the most stable electronic configuration for an atom is when electrons occupy the deeper levels and the shallowest levels are vacant. Therefore, some de-excitation and reorganization processes can occur in the ionized atom. An electron positioned in a higher energy level 𝐸 𝑛 can dissipate energy by the emission of an X-ray photon, in order to compensate a vacancy in a lower energy level 𝐸 𝑣 . The emitted photon has an energy ℎ𝜈 = 𝐸 𝑛 -𝐸 𝑣 . The core vacancy can also be compensated through an Auger process, where the dissipation energy ℎ𝜈 is instead transferred from an electron A at the level 𝐸 𝑛 to another electron B at an above level 𝐸 𝑚 > 𝐸 𝑛 . The electron A then compensates the vacancy, whereas the electron B is ejected from the atom, which corresponds to the production of an Auger electron.

The ionization of core shells can be modeled with the dielectric function theory [4,5,9], which is the approach we have followed in this work. This method, which can also model the inelastic interactions with weakly bound electrons, will be presented in Chapter 3.

Inelastic interactions with weakly bound electrons and collective interactions

Due to the large energy transfers required to excite a core electron, most of the energy losses of low energy electrons below a few keV will be through inelastic interactions with weakly bound electrons and collective oscillations. Indeed, the primary electron can also transfer a part of its energy to an electron below the Fermi level, in the valence band or in the partially filled conduction band of a metal, and promote it into an unoccupied state in the conduction band. As a first approximation, the secondary electron can be assumed to be generated from the Fermi level, which means that it is generated with an energy 𝑄 = 𝑇 -𝐸 𝑔𝑎𝑝 , where 𝑇 is the energy lost by the primary electron. However, almost all weakly bound electrons are located at an energy level 𝐸 𝑖𝑛𝑖𝑡 that is below the Fermi level 𝐸 𝐹 in a metal, or below the top of the valence band in a semiconductor or an insulator. Therefore, depending on the initial energy of the secondary electron in the band, its energy after the inelastic interaction is given by

𝑄 = 𝑇 -𝐸 𝑖𝑛𝑖𝑡 -𝐸 𝑔𝑎𝑝 Equation 2-9
In the materials studied in this work, the energy of weakly bound electrons 𝐸 𝑖𝑛𝑖𝑡 is on the order of a few eV up to a ten of eV. In a metal, if the energy of the incident electron is greater than the difference between 𝐸 𝐹 and the bottom of the conduction band, all weakly bound electrons of the conduction band may be excited into a vacant state. This is also the case in semiconductors and insulators if the primary electron energy is greater than sum of the energy of the gap and the width of the valence band 𝐸 𝑉𝐵 . However, if the energy of the primary is not high enough, only a part of the weakly bound electrons may be excited.

Instead of transferring energy to individual electrons, the primary electron may instead create a perturbation that displaces multiple weakly bound electrons from the valence or conduction band. Through coulombian attraction, these electrons are moved back towards their position at rest, which generates a collective oscillation of electrons in the volume of the material [10]. This oscillation is called a plasmon, and characterized by its frequency 𝜔 𝑝 . When a primary electron is interacting with a volume plasmon, an energy ℏ𝜔 𝑝 is first transferred to the plasmon. The plasmon then decays after a given lifetime by transferring its energy to a weakly bound electron, which is excited into a secondary electron. Since the energy of the plasmon is quantized, the interaction of electrons with volume plasmons corresponds to a very marked peak on the energy loss spectrums, which is often the most probable energy transfer for low energy electrons. Hence, weakly bound electrons, whether by direct impact or plasmon relaxation, are the principal source of secondary electrons. These inelastic interactions will be modeled with the dielectric function theory in Chapter 3, as for core shell electrons. However, the dielectric theory cannot model surface plasmons, which are another type of collective oscillations. Indeed, a volume plasmon oscillates longitudinally along the incident electron's propagation axis, whereas a surface plasmon is a transverse oscillation that can only happen in the first few nanometers of the solid's surface. Therefore, the probability of exciting a surface plasmon instead of a volume plasmon depends on the distance between the electron and the material's surface [1]. Although it is easy to evaluate this distance in the case of a perfectly flat surface, this becomes more complicated in the case of rough structured surfaces. The interaction models we will develop for the electron emission of metals and semiconductors in Chapter 3 need to be compatible with the simulation of 3D complex geometries. Indeed, we have shown in Chapter 1 that surface roughness could be a way of reducing the TEEY, which can be studied with Monte-Carlo transport through 3D geometries [11,12]. Hence, to ensure the portability of our models, we have not decided to model surface plasmons, and assumed that all plasmons are volume plasmons.

Interactions with the materials's surface

In vacuum, electrons are not subjected to a coulombian force, and their energy can be expressed relatively to the energy level of vacuum. Therefore, the potential energy 𝐸 𝑝 of the electron is null and its total energy 𝐸 is only made of the kinetic energy 𝐸 𝑐 . At the vacuum/material boundary however, the periodic potential of the crystal is disturbed by the discontinuity induced by the interface. This results in the generation of a potential barrier that the electron must pass.

When the electron enters the material, its needs to overcome the surface potential barrier, which is defined as the difference between the bottom of the conduction band and the energy level of vacuum. The electron's energy is now expressed relatively to the bottom of the conduction band.

Due to this change of reference, an electron penetrating in the material will lose potential energy; this loss is transformed into a gain of kinetic energy. An electron exiting the material will, on the contrary, gain potential energy and lose kinetic energy. This change of energy is illustrated in Figure 2345.

Figure 2-5: Energy changes for an electron penetrating in a material

A secondary electron created in the material needs to have a greater energy than the surface barrier to cross it. This threshold is defined as the work function for a metal, or the electron affinity for an insulator. These energies correspond to the minimal energy required to eject an electron from the solid. Hence, the surface potential barrier greatly limits the number of low energy secondaries emitted.

Indeed, the work function of most metals and semiconductors is about 4 to 5 eV, which prevents electrons below this energy from escaping. In some insulators however, the electron affinity can be as low as 0.9 eV in SiO2 and Al2O3. Therefore, even very low energy secondaries of only a few eV can still escape the surface of such insulators.

The surface processes can be simulated with different models of potential barrier crossing, such as a square barrier [13] or an exponential barrier [14]. Other models, such as image force barrier models [15], take into account the perturbation generated by the incident electron as it approaches the surface. However, these models are more complicated and computationally expensive. Hence, we have chosen to retain the exponential barrier model in the case of our study.

Physical interactions of electrons and charge carriers in insulating materials

In addition to the elastic, inelastic and surface interactions, several processes need to be taken into account to correctly model the charge buildup in the insulator and its effect on the transport of electrons. Indeed, due to the large band gap, the electron-phonon processes become predominant for the transport of electrons in insulators, whereas they can be neglected in metals and semiconductors for the computation of the TEEY.

Due to charge conservation, an electron-hole pair is generated during an inelastic interaction, with the hole being left at the place of the secondary electron generated. Since we know that the internal charge buildup is modifying the TEEY, the transport of these holes must be simulated. We need to take into account the transport of thermalized electrons as well, which can move in the conduction band but are unable to cross the material surface due to their low energy. We can then differentiate the transport of ballistic electrons, which are primary and secondary electrons with an energy above 1 eV, with the drift charge carriers, which are holes and thermalized electrons. The transport of charge carriers, called drift, is very different from the transport of primary and secondary electrons we have treated so far, since it is enabled by thermal excitation and electric fields. Both drift charge carriers and ballistic electrons of only a few eV can get immobilized on traps of various natures. However, this immobilization is not definitive, and a trapped particle can escape from the trap through detrapping. Finally, electrons and holes may recombine with each other, either by the intermediate of a trap, or by geminate recombination. All the processes that will be modeled in our charge transport model for insulators are shown in Figure 23456.

Electron-phonon interactions

From the different band structures shown in Figure 2-4, we can understand how there is no threshold for the inelastic interaction in metals, since for any given energy transfer, we can always find an electron to excite above the Fermi level. In insulators and semiconductors however, an incident electron falling below the energy of the band gap is unable to lose energy through inelastic interactions. In semiconductors, this happens when electrons fall below 1 or 2 eV, but we do not need to simulate the transport of these electrons for the computation of the TEEY since their energy is lower than the electron affinity (4-5 eV). In insulators however, the energy gap is much larger (8-9 eV in SiO2 and Al2O3), but we still need to track the electrons down to the energy of the surface potential barrier, which can be below 1 eV (0.9 eV for SiO2 and Al2O3).

In the absence of energy loss models below the band gap, any electron falling below 9 eV will be able to escape the material, which leads to unrealistic TEEYs. Therefore, the electron phonon interactions, which are negligible in metals and semiconductors, must be considered in insulators as they are the main source of energy losses for electrons below the band gap.

Phonons are quantized vibrations of the lattice, which are generated when atoms collectively oscillate around their equilibrium position. These oscillations occur natively under the effect of temperature, which creates a thermal agitation in the solid. These collective oscillations of atoms can be propagated in the lattice under the form of waves. Instead of treating this phenomenon as a vibrational wave, we can use the particle-wave duality and treat them as a quasi-particle that is propagating in the lattice, which is known as a phonon. The quantization of these vibrations means that phonons correspond to distinct vibration modes, which are characterized by their vibrational frequency 𝜔. The population of phonons of vibrational mode 𝜔 is given by the Bose-Einstein statistic, which follows the relation

𝑁 𝑝ℎ = 1 𝑒 ℏ𝜔/𝑘 𝑏 𝑇 -1 Equation 2-10
Where 𝑘 𝑏 is Boltzmann's constant. We can see from this relation that the population of phonons tends to 0 when the temperature converges towards 0 K. Indeed, when the temperature increases, the oscillations of the atoms around their equilibrium position are amplified by the thermal agitation, which increases the number of phonons generated in the material. ⃗ ) ≠ 0), the vibration mode can be excited by a particle having a null wave vector, that is to say a photon. Hence, phonons that verify this condition are known as optical phonons. Indeed, optical phonons correspond to the vibration modes created when the positive and negative ions of the lattice vibrate and generate an oscillating dipole moment. This is why they can be created by the passage of a photon, and why optical phonon branches are only found in ionic or semi-ionic crystals. Acoustic phonons, on the other hand, correspond to the oscillations of neighboring atoms, and cannot be excited by a photon. However, the displacement of atoms also modifies the local electron density, which means that electrons may interact with these oscillations. Vibration modes can secondly be separated according to their orientation relatively to the propagation direction of the phonon. If the atoms oscillate along the propagation axis of the wave, it is known as a longitudinal phonon. Transverse phonons, on the other hand, create oscillations that are perpendicular to their propagation axis.

Electrons can in fact interact with both acoustic and optical modes, and either absorb and gain the energy of a phonon, or lose energy and create a phonon when traveling through the lattice.

In the case of SiO2, the energies of the vibrational modes of phonons range from a few meV for acoustic phonons, up to 0.15 eV for longitudinal optical phonons. Under the band gap energy, the emission of optical phonons will thus be the main energy loss for electrons, and must be modeled with electron-phonon interaction models to get accurate TEEYs. However, given that electrons only lose a small amount of energy when emitting a phonon, the escape probability of low energy secondaries is greater than in metals and semiconductors, which leads to insulators having a greater TEEY. The interactions of electrons with acoustic phonons cause a negligible loss of energy, but the scattering angle associated with the emission or absorption of acoustic phonons can be very large, and is quasi isotropic. Hence, for low energy electrons in insulators, acoustic phonons play a similar role to the elastic interaction, and are responsible of the random walk motion of electrons. As mentioned previously, the modeling of elastic scattering with the PWA method becomes invalid below 100 eV in insulators, which is why an acoustic phonon scattering model is often used in place in Monte-Carlo simulations of electron transport in dielectrics, for instance in SiO2 [6] or alkali halides [16].

Drift transport of electrons and holes

Ballistic electrons traveling in an insulator will then lose energy, first through inelastic interactions and second by optical phonon emission until they are thermalized in the conduction band. The electrons then reach a steady state energy on the order of 3/2 kT, which is about 40 meV at 300 K, and enter the drift transport regime. This regime is also followed by the holes created in the valence band during the inelastic interactions. Drift particles are continuously scattered by phonon collisions in random directions, and periodically gain or lose energy by absorbing and emitting phonons. Hence their transport is thermally enabled, and their energy depends on the temperature and the phonon population.

The drift transport of a charge distribution is often characterized by its drift velocity 𝑣 𝐷 ⃗⃗⃗⃗ , which must be differentiated from the thermal velocity 𝑣 𝑡ℎ ⃗⃗⃗⃗⃗ of the charge carriers. Indeed, the thermal velocity is the actual velocity of the charge carrier between two phonon scattering events and in the absence of an electric field, which is linked to its thermal energy. The drift velocity, on the other hand, is a macroscopic quantity, which is computed according to the definition of Figure 2345678. It is defined as the average displacement of the distribution of charge carriers after a given time. When there is no electric field (𝐹) in the material, the distribution spreads in every direction, since the charge carriers are deviated in a random direction after each scattering event with phonons. Hence, the drift velocity is equal to zero in this situation, since the average displacement of the distribution is null, but the individual charge carriers are absolutely not immobile. When an electric field is applied in the material, the charge carriers can be accelerated in between two scattering events, in the direction of the field for holes, or in the opposite direction for electrons. Even if the drift particles are still randomly scattered by phonons, their average displacement now follows the axis of the electric field. Hence, the distribution continues to spread due to thermal agitation, but globally moves in the direction of the electric field. The evolution of the drift velocity depends on the intensity of the electric field, and the transport properties of the charge carriers. The transport we have just described in Figure 2-8 is known as a Gaussian transport, where the charge distribution moves according to the electric field, and the drift velocity is a linear function of the field. The term µ connecting the velocity to the field is called the mobility, and depends on the effective mass of the carrier and the mean time of flight between two phonon collisions.

In the example of electrons drifting in SiO2, their mobility is 20 cm²/V/s at room temperature, and the drift velocity scales linearly with the electric field until 0.5 MV/cm [17,18]. Indeed, even if the field increases the energy of electrons between two collisions, the emission of optical phonons prevents the electrons from retaining this energy. However, if the field increases past 0.5 MV/cm, the energy losses by optical phonon emission are no longer sufficient, and the average energy of electrons increases. At this point, known as optical runaway the electrons are also mostly traveling along the axis of the electric field, and the spatial dispersion of the trajectories is severely reduced [17]. The acoustic phonon collisions are now the mechanism that prevents the average energy of electrons from increasing, until acoustic phonon runaway happens at 3-4 MV/cm [19]. For even higher electric fields of 8 MV/cm, the electrons have now acquired enough energy to be able to generate secondary electrons through impact ionization [20], until dielectric breakdown of the material occurs past 10 MV/cm.

The transport of holes in SiO2, on the other hand, is very different from electrons and cannot be modeled by Gaussian transport. Indeed, the transport of holes is instead dispersive [21,22]. This results in a transport that is strongly decorrelated from the electric field, with a mobility of only 10 -5 cm²/V/s, and can be modeled with the Continuous Time Random Walk Theory [23,24] instead. In other materials it is possible that the transport of electrons is dispersive instead, or the both charge carrier transports are Gaussian or dispersive. The transport properties of the charge carriers are largely dependent on the material. This is due to trapping mechanisms we will detail in the following subsection. Indeed, in some insulators, the drift particles are very mobile and can be evacuated easily. In other materials with traps that have a high density and/or a large depth, the transport of charges can be severely limited instead.

Trapping of charge carriers

Several types of traps exist in insulators, which introduce energy levels in the band gap. These traps, which are potential wells, can be due to the imperfections and impurities of the material, or the disorder in amorphous materials. Traps are able to immobilize electrons, holes, or both, for a given period. The traps can be separated in shallow and deep traps, which differ from the mechanisms involved to immobilize a particle. Traps are generally modeled in a macroscopic manner by a capture cross section 𝜎, which gives information on the attractiveness of the trap, and a trap density 𝑁. These can be combined into a capture mean free path 𝜆 = 1 𝜎𝑁 ⁄ .

Capture in shallow traps and polaronic transport

We can identify two different sources of shallow traps, which are both intrinsic to the material and not due to the presence of impurities. First, in polycrystalline and amorphous materials, the disorder and ruptures in the atomic bonds generates local modifications of the coulombian potential, which can immobilize electrons and holes due to the perturbation of the wave function of the charge carrier. This results in the generation of a continuum of localized states below the conduction band and above the valence band, which are known as Anderson states [25].

Second, depending on the electron-phonon or hole-phonon coupling strength in the material, a charge carrier may become self-trapped by forming a small polaron [26]. A polaron is a quasiparticle, made of an electron or a hole, and the strain field that surrounds the drift charge carrier, which can be interpreted as phonons surrounding the moving electron/hole. The effective mass of the carrier increases because the electron/hole is effectively dragging the lattice atoms along its path. This phenomenon is stronger in ionic crystals, due to the strong coulombian interaction between the charge carriers and the ions of the lattice, and much weaker in covalent crystals made of neutral atoms. The intensity of the electron-phonon coupling can be evaluated by the following coupling constant [10]:

1 2 𝛼 = 𝐶 ℏ𝜔 𝐿𝑂 Equation 2-11
Where 𝐶 is a deformation energy, and ℏ𝜔 𝐿𝑂 is the longitudinal optical phonon energy. In essence,

1 2
𝛼 can be interpreted as the number of phonons surrounding the electron [10].

Depending on the strength of the coupling, electrons and holes may create either a small or a large polaron. A large polaron is simply a mobile charge carrier that has an increased effective mass, whereas a small polaron can generate a deformation of the ions of the lattice and get selflocalized in between the ions, in the potential well generated by this deformation. Due to the degenerate valence band edge, the holes are more likely than electrons to become selfimmobilized by forming a small polaron [10]. This is why holes in SiO2 are much less mobile than electrons, because contrary to electrons they immediately become a small polaron after creation [22]. The hole polarons can get released from their trapping site by the thermal agitation, which means that their immobilization time in polaron traps is very short (10 -12 𝑠). However, the density of such traps is very large, on the order of the atomic density (10 22 𝑐𝑚 -3 ), since the polaron may self-localize in any interatomic site. If the temperature is not high enough to active this release, the polaron can instead tunnel between trapping sites, depending on the distance and difference in energy between the traps [27,28]. The density of localized states is also very large in amorphous materials (10 20 -10 21 𝑐𝑚 -3 ) due to the strong amount of disorder. Both localized states and polaron sites are very close to the conduction band or valence band edges, hence the depth of these traps ranges from a few hundredths to a few tenths of eV. Hence, the trapped charges only need a very small amount of energy to escape the trap. This energy can be provided by thermal excitation or by an electric field. However, the mean free path between two trapping events is low, due to the high trap density. Therefore, the charge carriers may move between traps through two transport mechanisms. They can either hop between traps by detrapping into the valence or conduction band and drifting until they are captured by another trap, or directly tunnel from one trap into another [28].

Capture in deep traps

Many types of defects and impurities are introduced in the fabrication process. These create energy levels in the middle of the band gap of the material, in which electrons or holes can fall, as shown in Figure 2-10. These traps are located several eV below the conduction band or valence band edge, and are an extrinsic property of the material. The nature and concentration of the deep traps is highly variable. For instance, in silicon dioxide [29], electron and hole traps can be generated by oxygen vacancies, sodium growth in SiO2, H2O molecules incorporated in the dielectric film, or other implanted ions. The SiO2/Si interface is also a highly disordered region with many broken bonds and implanted ions, such as W or Na + ions. This results in a strong density of traps in this area. The capture cross sections of all these traps are highly variable [29], depending on whether the trap is coulombic attractive (10 -13 -10 -15 cm²), neutral (10 -15 -10 -18 cm²) or coulombic repulsive (< 10 -18 cm²). This attractiveness of the trap depends on its natural charge state. For instance, some hole traps in SiO2 are initially neutral [29], and become positively charged when a hole is captured.

The potential wells created in the band gap by the deep traps have a depth of about 1 to 4 eV. However, their density is much lower than for shallow traps, ranging from 10 14 to 10 18 𝑐𝑚 -3 . Finally, the concentration of the traps is also dependent on the fabrication process. For instance the water related trap concentration in SiO2 can vary from 10 15 cm -3 for a dry oxide to 10 19 cm -3 for a wet oxide [30]. Deep traps are thus able to fix particles for a very long time, and the trapped charge carriers may only escape with the help of an increased temperature, or locally high electric fields. For some traps, the capture cross section has been observed to be field dependent. For instance, a reduction of the capture cross section of Coulomb-attractive electron traps was observed in SiO2 in fields greater than 0.5 MV/cm by Ning [31]. This was attributed to the increased energy of the electrons in the field, which makes them less attracted by the traps.

Detrapping of charge carriers

The charge carriers immobilized in a trap can be detrapped under the effect of thermal agitation.

The escape frequency 𝑊(𝐸 𝑖 ) for a trap level of depth 𝐸 𝑖 is commonly modeled by an Arrhenius law in the form of:

𝑊(𝐸 𝑖 ) = 𝑊 0 𝑒𝑥𝑝 (- 𝐸 𝑖 𝑘 𝐵 𝑇 ) Equation 2-12
With the frequency factor 𝑊 0 which describes the intrinsic mobility of the charge carrier in the trap. For shallow traps, thermal excitation is enough to allow the charge carriers to escape and hop between traps, but this is not the case of deep traps. Nevertheless, other detrapping enhancements, given in Figure 2-11, may increase the probability of the charge carrier from escaping a deeper trap. The potential barrier of the trap can first be lowered by an electric field 𝐹 due to the Poole-Frenkel (PF) effect [32,33], which is also commonly found in charge transport models for semiconductors. The lowering of the barrier reduces the effective depth of the trap, which increases the probability of escape by thermal excitation. This lowering can be quantified by the PF lowering factor, given by

∆𝐸 𝑖 = √ 𝑒 3 𝐹 𝜋𝜖 0 𝜖 𝑟 ⁄ Equation 2-13
From this expression, we can see that the potential barrier of the trap progressively lowers when the electric field increases, under a law in √𝐹. It is also possible to take into account the effect of Poole Frenkel lowering on the effective mobility, by introducing a dependency in √𝐹 and a lowering factor 𝛽 that can be determined experimentally [34]. It is possible however that the lowering factor deviates slightly from the Poole-Frenkel theory.

Detrapping can also be enhanced by the Phonon-Assisted Tunelling (PAT) effect. This phenomenon is made of two steps. First, the trapped charge carrier can absorb a phonon and get excited to a higher level in the trap, but still without escaping the potential well. However, the charge is now faced with a shallower trap barrier, where the probability of tunneling through this barrier is more favorable. Straight tunneling of the charge carrier through the potential well can only happen in shallow traps, unless the electric field is very strong (several MV/cm). This is why the charge carrier first needs to absorb a phonon, in order to increase its escape probability. The models used for the description of the Poole Frenkel and Phonon Assisted Tunneling enhancements used in this work are based on the work of Lemière et al. [35], and will be presented in more detail in Chapter 5.

Recombination mechanisms of electron-hole pairs

When a trap captures a charge carrier, the global charge of the trap is modified, and it can thus become more attractive to a particle of the opposite sign. For instance, the neutral hole traps become positively charged when they have captured a hole, which makes them strongly attractive for electrons. If a hole or an electron falls into a trap which is already occupied by the opposite particle, the two particles recombine and disappear, and the trap is freed. This is known as trap assisted recombination. As the material is irradiated, holes or electrons fill more and more traps, and the drift carriers have a higher probability of recombining in a trapping site. Hence, the recombination of electron-hole pairs reduces the quantity of free charges in dielectrics, and the trap assisted recombination we have just described must be modeled in our simulation. Moreover, since the density of trapped charges is reduced, the electric field generated by this density is also decreased, which lowers the detrapping enhancements and the energy gained by the charges in the field.

Figure 2-12: Recombination mechanisms in insulators -trap assisted recombination and geminate recombination

There is actually another type of recombination that can happen between a free electron and a free hole, which is known as geminate recombination. In this case, the free electron generated in an inelastic interaction has spontaneously been recaptured by the hole left in place. Onsager [36] has shown that the generation of an electron hole pair was conditioned by the electric field and the temperature. This was also noted by Hughes, who observed an increase of the electronhole pair yield with the electric field, starting from 0.1 MV/cm [18]. Indeed, the probability for the electron to escape geminate recombination can be expressed as [34,37] :

𝑃(𝑇, 𝐹) = 𝑃 0 exp ( 𝑟 𝑐 𝑟 0 𝑒𝑥𝑝(-𝜉) -1 𝜉 ) 𝑟 𝑐 = 𝑒 2 4𝜋𝜖 𝑟 𝜖 0 𝑘 𝑏 𝑇 𝜉 = 𝑒𝐹𝑟 0 𝑘 𝑏 𝑇 Equation 2-14
In this equation, 𝑟 𝑐 is equivalent to the capture radius that the electron must overcome to avoid recombination, and 𝑟 0 is the thermalization radius, which is the separation distance reached by the electron at the start of the diffusion. If this distance is too low compared to the capture radius, the electron cannot escape the attraction of the hole, and the pair recombine immediately. For instance, an electron in SiO2 at room temperature must be separated by at least 10 nm from its hole, in order for the two particles to diffuse apart instead of recombining immediately [18].

A major issue for this model however, is that there is no expression for the thermalization radius 𝑟 0 or the native yield 𝑃 0 , which must be determined arbitrarily or by fitting with experimental data. Given the importance of these parameters in the production yield, this can be a strong source of uncertainties in the simulation. Hence, we have chosen not to include geminate recombination in our simulation, and we will assume that the electron can always escape its parent atom. As we will show in Chapter 6, this approximation still allows us to correctly model the effects of internal charging on the TEEY, so it is possible that the effect of geminate recombination is weaker than other physical processes, and does not significantly influence the TEEY. It is also possible that the electric field reached in the sample during TEEY measurements is not strong enough to create a significant rise of the production yield. Indeed, geminate recombination must be taken into account in other applications, such as the modeling of radiation induced conductivity in polymers [34,38]. These materials can charge up to several kV, whereas the thin film dielectric samples used in this study do not charge past a couple of volts.

Other data on the transport of electrons accessible through Monte-Carlo codes

Since Monte-Carlo codes model the whole transportation of electrons through the material, several interesting quantities can be extracted from the code to quantify this transport. For instance, we can get information on the number and type of interactions, the energy and location of the secondary electrons generated, the charge distribution in the material… In this section, we focus on the characteristic parameters of the transport of electrons inside of the material, which will influence the TEEY.

Penetration depth and transmission rate

When electrons penetrate in a material, they lose energy by inelastic interactions with electrons of the target atoms, and are slowed down by doing successive interactions. Electrons can also be highly scattered by the target atoms' nuclei. As a result, each particle has an individual trajectory, as seen in Figure 2-14 (i) and Figure 2-13 [39]. iii.

The transmission rate and range are deduced from the distribution of depths.

The true range 𝑆 of a given trajectory is the total distance traveled until the electron comes at rest, i.e. the sum of all step lengths 𝑆 𝑖 traveled by the particle between each interaction, as illustrated on Figure 2-13. This parameter can be sampled for a large number of electrons to get the average true range 𝑆 ̅ , which is an indication of the total distance traveled by the electrons in average. Within the Continuous-Slowing-Down Approximation (CSDA), 𝑆 ̅ can be evaluated thanks to the following integral:

    E dQ dx dQ E S 0 1 Equation 2-15
This is the integral of the reciprocal of the stopping power (dQ/dx) over energy from a final to an initial value: It corresponds to the parameter 𝑆 ̅ from the definition of Figure 2-13. The CSDA range is the total distance that is effectively travelled by electrons. However, 𝑆 ̅ does not give information on the final position of the particles or in which direction they have traveled. But, the depth reached by the electrons depends, in addition to the slowing down induced by inelastic scattering, to the deviations generated by the inelastic and elastic interactions. In this regard, 𝑆 ̅ can be interpreted as a penetration depth that can only be reached by a theoretical electron with a strictly linear trajectory. As this is never the case for electrons, which are deflected by elastic interactions, this parameter is an unreachable limit for the actual range R. Moreover, 𝑆 ̅ is not accessible experimentally as the electrons do not behave in a deterministic way but follow statistical laws for each step.

In consequence, for many applications, the paths of the particles are expressed as a projection on the incident direction of the impinging particle, generally in the depth of the material. Thus, a more convenient method is to sample the final positions of electrons along this direction. This can be done following the method on Figure 2-13 to get the projected range 𝑅 for a given trajectory. For instance, sampling the distribution of the final depths 𝑅 reached by a large number of electron trajectories, as done on Figure 2-14 (ii), allows us to compute its average 𝑅 ̅ , which corresponds to the average of the depths reached by the electrons in a semi-infinite material.

In Chapter 4, we will study the penetration depth of an electron, which will be defined as the final depth reached by the electron when it comes at rest. The transmission rate through a thickness 𝑑 is thus defined as the proportion of electrons with a final position deeper than 𝑑. For electrons, that are highly scattered, two other parameters can be found in the literature to describe their trajectories. The extrapolated range and the practical range are commonly evaluated.

The extrapolated range (R0) is commonly defined following the method shown on Figure 2-14 (iii), taken as the point of intersection between the tangent at the steepest section of the transmission probability curve (P=0.5) and the depth axis (X-axis) [40,41]. In the following, the range value given by this point will be called 𝑅 0 . Similarly, the practical range can be obtained from the depth-dose profile in place of the transmission curve. Both parameters are evaluated in slightly different ways, but have often been used interchangeably [4] as they remain similar. They differ in their definition from a simple average over the penetration depths from each individual trajectory, but are more representative of the penetration distance of individual electrons. Hence they are commonly used to define the shielding thickness necessary to protect an equipment from radiation.

In this manuscript, we will refer to the extrapolated range using its definition from the transmission rate. At high energy (E>keV), the extrapolated ranges of electrons 𝑅 0 (𝐸) in cm are demonstrated to be inversely proportional to the density 𝜌 (g/cm 3 ). For this reason, the range in cm is commonly multiplied by the density to get a value in g/cm² as 𝑟(𝐸) = 𝑅 0 (𝐸) * 𝜌, which is then independent on the nature of the target material above a few keV. In Chapter 4, where we will study the extrapolated range of electrons, we will only use the range 𝑟(𝐸) in g/cm².

Ionizing dose

We have seen that when electrons penetrate in a solid, they transfer a part of their energy through inelastic interactions and the creation of electron-hole pairs. The ionizing dose is defined as the sum of energies deposed during ionization by an incident particle and by unit of mass, which is thus the average energy transferred to the electrons of the medium consecutively to inelastic electron/electron interactions. The number of secondary electrons set into motion in the medium is thus proportional to the ionizing dose. However, a part of this energy deposed in the material is dissipated in the form of backscattered electrons and secondary electrons escaping into vacuum. The remaining energy corresponds to the incident and secondary electrons that are captured by the material. Some examples of ionizing dose-depth profiles are given for low energy electrons in Al in Figure 2-15. When the electron energy increases from 25 eV to 100 eV, the dose deposed close to the surface increases, and so does the range of the electron, which is given by the end of the dosedepth curve. The dose deposed near the surface is directly linked to the TEEY, since the secondary electrons that are able to escape are the ones that were created near the surface. Hence, the TEEY at 100 eV is greater than at 25 eV. When the electron energy increases to 500 eV, more dose is deposed in total in the material, but the dose starts to be deposed much deeper. As we can see in the case of 2 keV electrons, the total dose is greater than for all other electron energies, due to the large number of inelastic interactions. However, the range of the electron is also much larger, and most of this dose is deposed in the depth of the material. The dose deposed near the surface is lower at 2 keV than at 500 eV and 100 eV, and so is the TEEY.

Conclusion of Chapter 2

In this chapter, we have detailed the numerous electron-matter interactions we need to model, in order to simulate the transport of low energy electrons in dielectrics. First, we have seen that the transport of electrons and the electron emission properties of all material types are the result of a competition between several interactions: the elastic interactions, which can strongly scatter the electrons, and the inelastic interactions, where primary electrons lose energy and secondary electrons are generated. The source of secondary electrons are quite varied: core shell electrons, weakly bound electrons in the conduction or valence band, Auger relaxation processes, or plasmon excitation and decay. The surface of the material has also been evidenced to be a strongly limiting factor for the emission of very low energy electrons into vacuum.

However, to model the TEEY and the charge buildup of insulators, additional interactions must be considered. Electron interactions with optical phonons are the main source of energy loss below the band gap, and acoustic phonon interactions must be taken into account to correctly describe the random motion walk of electrons below 100 eV in insulators. The charge buildup cannot be correctly modeled without simulating the transport of thermalized electrons, implanted in the material, and holes created during inelastic interactions. This drift transport, enabled by thermal agitation and the electric field in the material, can be interrupted by several trapping processes. Indeed, polarons, localized states and deep level traps may capture a charge carrier and immobilize it for a given period. However, under the effect of thermal excitation, a charge carrier can exit a shallow trap, and move in the material by hopping between traps. For deep traps however, the escape of the particle must be aided by the electric field, with the Poole-Frenkel and Phonon Assisted Tunneling enhancements. Finally, as the density of trapped charges increase, mobile holes and electrons can be captured by a trap filled by an oppositely charged particle, which becomes more attractive. Charge carriers may thus recombine, through trap-assisted recombination of electron-hole pairs.

Since the elastic, inelastic and surface processes are common to all material types, we can start designing our Monte-Carlo simulation by integrating these models, and simulating the transport of low energy electrons in metals and semiconductors. Indeed, given the lack of publicly available Monte-Carlo transport models for low energy electrons, developing our own tool is mandatory and will allow us to study the secondary electron emission of metals and semiconductors. As we have seen in section 2.4, such a Monte-Carlo code can also yield additional information on the transport of electrons, like extrapolated ranges and ionizing doses, for which experimental data in the literature is lacking below 1 keV. The development of this transport model will be detailed in Chapter 3. While this model cannot simulate the effects of charging on the TEEY of insulators since it does not consider the charge buildup, such a simulation can still be used as a base and extended to the simulation of the TEEY of insulators. The transport model for insulators including the drift transport, trapping, detrapping and recombination processes will be developed in Chapter 5.

Chapter 3: Development of a Monte-Carlo low energy electron transport model to simulate the secondary electron emission of metals and semiconductors

Presentation of MicroElec and Geant4

In this section, we shall present the models used to simulate the transport of electrons in metals and semi-conductors. In this work, we have chosen to develop our Monte-Carlo model using Geant4 [1][2][3], a free open-source toolkit coded in C++. Geant4 is made of many different packages that allow the simulation of the transport of various types of radiation (electrons, protons, photons, neutron…) through matter. The main advantage of Geant4 is its flexibility: the user can freely design their geometry, and choose which physical models should be used for each particle type. Multiple models can be chosen for a given particle, which can be specific to an energy domain, a material or a region of the geometry.

To enable such a modular structure, the architecture of Geant4 relies on an object oriented approach. Several well-defined classes need to be constructed by the user, which will indicate to the Geant4 kernel the key data of the simulation, such as the geometry or particles. In the scope of this flexibility, 'Messenger' objects can be attributed to almost any class of the Geant4 application. Through text files containing macro commands, the user can modify the parameters of the simulation outside of the code. At the start of the simulation, the macro files will then be read, and the data will be imported in the classes by their messengers. Some of the key classes of Geant4 we have used in this work are listed below.

 The geometry is created by the user in the class DetectorConstruction. Here, several geometrical shapes can be used to construct complex 3D structures. The user also creates the sensitive detectors that will count the particles going through them and retrieve their data from the simulation. For instance, we can create a spherical detector around a cubic material sample that can count the electrons emitted by the sample and record their energy. This allows the simulation to reproduce the data obtained from the spherical electron collector found in experimental TEEY measurement facilities. The user can also separate the geometry in different regions, where different physical models can be activated depending on the region.

 The particle source is defined in the class PrimaryGeneratorAction. The user can pick different source types, such as a G4ParticleGun or a GeneralParticleSource, and set the properties of the incident particles (single energy or energy spectrum, type…) and of the source (angle of emission, position…).

 A Sensitive Detector class must be created separately to properly record the information when particles go through the detector created in the geometry. In practice, we attribute to a given shape the sensitive detector object, so that Geant4 knows when to call the detector's functions and record information. In the sensitive detector class, we can define the data to be recorded and save it in a hit collection.

 The data can be extracted from the simulation with Geant4 AnalysisManager. It is first created at the beginning of the simulation, where the user can set the type of file to export the data into (ROOT, CSV…). The analysis manager can then be called at various steps of the simulation to save data such as strings, doubles, or vectors.

 The run is the fundamental unit of the Geant4 simulation. The user can choose to simulate as many runs as they want. During a run, several events are simulated depending on the number of incident particles. We can choose to have only one event per incident particle, or group multiple incident particles in a single event. An individual particle is followed in a track until it is killed. The secondary particles generated along the track are put in a stack by order of creation. When the primary particle is killed, the secondaries of the stack are then transported one by one until they are killed. If the secondary particle creates tertiary particles, they are added on top of the stack and are transported after the secondary particle is killed, and before the other secondaries. Finally, the smallest simulation unit is a step. A step begins right after an interaction, and ends when another interaction happens. However, Geant4 can also force a step to end when the particle is crossing a geometry boundary. Finally, in Geant4, the runs, events, and tracks of a simulation are completely independent.

 User classes are provided to allow the user to perform various action on the simulation units we have just presented. These are the RunAction, EventAction, TrackingAction, StackingAction and SteppingAction classes. For example, we can use the SteppingAction while debugging to get information on the particles after each interaction, or we can retrieve the values of the electron counters at the end of a run in RunAction and save the TEEY in a file with the AnalysisManager.

 The Physics List is where the user will choose which models to use in the simulation. For each simulated particle, they can define a model to be used in a specific region of the geometry, over a given energy domain. For instance, we can choose to use a very precise model for low energy electrons and switch to a less precise but faster model above a few tens of keVs. We can also choose to only model the full electron cascade generated by an incident proton in a section of the geometry made of silicon, and just compute the energy lost by the proton with a condensed history model in another section made of tungsten.

 In practice, the physics list is filled by Processes, for instance the inelastic interaction of electrons is a process. Depending on the process, different interaction Models can be set to a single process, to recreate the examples we have just mentioned. The models handle the computation of the interaction cross sections, which are fed to the Geant4 kernel for the Monte-Carlo random draw of the physical interaction length. If the process is selected, the model class will then compute what happens during the interaction, such as the amount of energy lost or the angular deviation. For some processes, no models can be set and the process class itself handles the computation of cross sections and pre/post step actions. The processes we will use for the simulation of secondary electron emission are only Post-Step processes, which perform an action at the end of a step. Some other Geant4 processes can modify the particle in between two interactions, and are called Along-Step processes.

 The electric, magnetic, or electromagnetic field is defined in DetectorConstruction. Two classes need to be created separately to handle the field. The FieldSetup object contains the parameters for the resolution of the field equation and the interpolation of the trajectory, which is mostly done with Runge-Kutta methods. The value of the field at a certain position and time is given by the object Field. It can be tweaked to have an uniform field or a field that can vary in space and time. This object is first called by the transportation manager of Geant4 to get the value of the field at several points. Then, the transportation manager integrates the trajectory during the step and computes the modification in energy and direction.

The models found in Geant4 can cover a very wide range of energies, up to several GeV, for various applications. For instance, ion track structures simulated in Geant4 have been used for the study of Single-Event Effects in electronic components [4,5]. We can also cite GRAS, a Geant4 based simulation tool for the study of space environment effects [6].

However, some studies have underlined the limits of the Geant4 ionization models for very integrated technologies applications [4,7,8], due to the recommended production threshold of 250 eV for secondary electrons. The MicroElec project is one of the physics model packages found in Geant4. It is developed by CEA DAM in collaboration with ONERA. As a Geant4 extension for incident electrons, protons and heavy ions in silicon, MicroElec aims to implement lower energy ionization models in Geant4 to improve track structures simulation. MicroElec has been used for microdosimetry and Single-Event Effects applications [9,10] and is based on the already existing framework of the Geant4-DNA extension [11,12].

Geant4 offers a wide selection of physical models. For electromagnetic physics (EM), there are standard EM processes, above 1 keV, and low energy EM processes, valid down to ~250 eV for electrons and gamma rays. In Geant4, we can distinguish for electrons, four different electromagnetic processes applicable in different energy ranges:

-G4eIonization (STANDARD) -G4LivermoreIonisationModel (LIVERMORE) -G4PenelopeIonisationModel (PENELOPE) G4eIonization (STANDARD) physics is a continuous process valid above 1 keV. Similarly, according to the Geant4 documentation, the Livermore continuous ionization model can be used down to few tens of eV. The PENELOPE (PENetration and Energy LOss of Positrons and Electrons) model is also a continuous ionization process which is valid down to ~50 eV [13]. These continuous inelastic processes must be combined with some elastic interaction models to account for the scattering of particles on nuclei. These later processes can be a discrete (Single Scattering : SS) or a Multiple SCattering (Msc) process.

Multiple scattering is mainly used for high energy particles. In this approach, the deviations and energy losses of the incident particle's trajectory caused by many individual interactions are condensed in one single mean trajectory after one single interaction (hence the name "Condensed history"). On the other side, discrete processes simulate each single interaction and compute the energy losses and direction changes step by step. This approach is obviously much slower than multiple scattering, but is also much more precise. This is especially true when low energy particles need to be simulated, as the distances between two interactions can be very small and the deviation can be very significant. Hence, the multiple scattering approach can be too approximate in this case. According to the nature and the energy of the incident particles, Geant4 provides some scenarii combining inelastic and elastic processes. For instance, the G4EmStandardPhysics_option4 uses the PENELOPE continuous ionization process with the multiple scattering. The accuracy is increased by replacing the multiple scattering by the single scattering process for large deviation angles. Similar options are proposed based on the G4LivermoreIonisationModel. The best accuracy can thus be reached by using discrete processes for both ionization and elastic scattering, as provided by MicroElec.

The first version of the module (Geant4.9.6) was presented in [14,15] and validated for electrons of 16.7 eV up to 50 keV and incident protons and ions of 50 keV/nucleon -23 MeV/nucleon in silicon only. Later improvements (Geant4.10.0) [16] have been added to extend the highenergy range of the models up to 100 MeV for electrons and 10 GeV/nucleon for protons and ions, with the inclusion of relativistic corrections. This version of MicroElec, which has been designed for microdosimetry simulation in silicon, is the version that was publicly available in Geant4 at the start of this PhD thesis. On the other hand, other improvements of MicroElec were made during two previous PhD theses at ONERA, but were not released in Geant4. First, during the thesis of J. Pierron [17][18][19][20][21], MicroElec and the OSMOSEE code were extended and validated for the simulation of the secondary emission of aluminum, silver and silicon. In this regard, improvements were brought to the modelling of the elastic and inelastic interactions, and a model for the crossing of the surface was also added for material/vacuum interfaces. In later work during the thesis of P. Caron [22], the treatment of the dielectric function for the computation of the inelastic cross sections was modified, to improve the accuracy of the stopping powers of electrons below 1 keV, and protons/ions below 100 keV/nucleon. He also developed computer programs for the fitting procedure of the energy loss function and the computation of the inelastic cross sections, which we have used in this work.

First, these improvements to the interaction models of MicroElec for very low energy particles will be presented. An extensive description of the elastic and inelastic interaction models can be found in the PhD works of J. Pierron [17][18][19][20][21] and P. Caron [9,22], hence we shall only give here a brief description of these models. This section will rather focus on the additional improvements that were brought during this PhD thesis, that is to say the development of acoustic phonon and optical phonon interaction models for insulators, and the extension of the surface crossing model to material/material interfaces, allowing MicroElec to be used with multilayered materials.

We have also computed the inelastic cross sections for 16 materials (Be, C, Al, Si, Ti, Fe, Ni, Cu, Ge, Ag, W, Au, SiO2, Al2O3, Kapton, BN). The new interaction models allow the Geant4 extension to be used for the tracking of electrons, protons and heavy ions in several new materials compared to its last publicly available version, for microdosimetry and secondary electron emission applications. We will then present some simulation results of the TEEY obtained from MicroElec.

3.2 Implementation of interaction models for low energy electrons

Modeling of the elastic interaction

Elastic processes for electrons are handled in MicroElec by the classes G4MicroElecElastic and G4MicroElecElasticModel. In the previous version of MicroElec (Geant4 10.0), the total and differential elastic cross sections for Si in the 50 eV -100 MeV range were extracted from the ICRU database [23]. These cross sections were completed by ab-initio cross sections from Bettega et al. [24] from 16.7 eV to 50 eV. However these calculations were originally made for CS2, and had to be adapted for Si following the approach of Akkerman et al. [25].

In the new version, the elastic cross sections are calculated with the Partial Wave Analysis (PWA) method and the code ELSEPA [26], an approach also followed by the Geant4 DNA team [27].

Since ELSEPA can be used down to 10 eV for compounds and monoatomic materials, this new approach allows us to use a single model on the whole energy range to compute the elastic cross sections for all materials. This improves the consistency of the simulations and avoids the use of individual adjustments when required for a given material. The elastic cross sections were computed in this work for all elements up to uranium and down to 0.1 eV, covering all corresponding monoatomic materials. Although the model validity is limited to 10 eV, computing the cross sections to lower energies gives us some extrapolation points that can be used to track electrons down to the energy of the surface barrier, which is about a few eVs.

The elastic cross sections for compounds can be computed with the ELSEPA module for molecular cross sections, which are then converted to cross sections per atom, or by computing the average of the monoatomic cross sections, weighted by the stoichiometry. These two approaches are compared for SiO2 in Figure 3-1. For energies higher than 1 keV, the difference between the Total Cross Sections (TXS) given by the two approaches is negligible. Below 1 keV however, the silicon atoms seem to have a stronger interaction probability than the oxygen atoms, as the molecular TXS follows the Si TXS. This would indicate that the free-atom approximation used with the weighted average becomes less valid, as low energy electrons may be subject to a stronger coupling with the lattice and/or aggregation effects from the molecule, as suggested in [26]. Due to the wide availability of material parameters, the molecular cross section approach has been chosen for SiO2. Indeed, the computation of the molecular cross sections requires additional parameters that may not be easily available for some materials, such as the molecular polarizability. The user also needs to enter the exact coordinates of the atoms in the molecule, which can quickly become overwhelming for polymers such as Kapton. For this reason, the mean atom approach was followed for the computation of the elastic cross sections of aluminium oxide, kapton, and boron nitride. Even though we know that the accuracy of this approach is reduced at low energies, this should be offset by the fact that we will be modelling the elastic interactions in insulators with an acoustic phonon-electron interaction model at 100 eV and below. In MicroElec, the inelastic interactions for incident electrons, protons and heavy ions are handled by the classes G4MicroElecInelastic and G4MicroElecInelasticModel. The cross sections (XS) calculations are based on the complex dielectric function theory of Lindhard and Ritchie [28] and the modeling of the energy loss function (ELF), following the work of Akkerman et al. [25] on silicon and the Geant4-DNA package. This approach is widely used to model the radiation transport at low energy, and has also been used to compute inelastic mean free paths by Flores-Mancera et al. [29], de Vera and Garcia-Molina [30], and Montanari et al [31].

The ELF defines the response of a material to an electronic perturbation. It is expressed as

𝐸𝐿𝐹(ℏ𝜔, 𝑞 ) = 𝐼𝑚 [- 1 𝜀(𝜔, 𝑞 ) ] Equation 3-1
with 𝜀(𝜔, 𝑞 ) the complex dielectric function, and ℏ𝜔, ℏ𝑞 the energy and momentum transferred by the primary particle to an electron of the material. In the dielectric function framework, the differential cross section

𝑑 2 𝜎 𝑑(ℏ𝜔)𝑑(ℏ𝑞)
for an electron of incident energy 𝑇 can be expressed from ELF following the equation

𝑑 2 𝜎 𝑑(ℏ𝜔)𝑑(ℏ𝑞) = 1 𝑁𝜋𝑎 0 𝑇 𝐼𝑚 [- 1 𝜀(ℏ𝜔, ℏ𝑞 ) ] 1 ℏ𝑞 Equation 3-2
Where 𝑁 is the atomic density, and 𝑎 0 = ℏ² 𝑚𝑒² ⁄ is Bohr's radius. The differential cross section in energy can then be obtained by integrating Equation 3-2 over the transferred momentums:

𝑑𝜎 𝑑(ℏ𝜔) = 1 𝑁𝜋𝑎 0 𝑇 ∫ 𝐼𝑚 [- 1 𝜀(ℏ𝜔, ℏ𝑞 ) ] 𝑑|𝑞 | |𝑞 | 𝑞 𝑚𝑎𝑥 𝑞 𝑚𝑖𝑛 Equation 3-3
Finally, the total cross section is obtained by integrating the differential cross section over the transferable energies

𝜎 = 1 𝑁𝜋𝑎 0 𝑇 ∫ ∫ 𝐼𝑚 [- 1 𝜀(ℏ𝜔, ℏ𝑞 ) ] 𝑑|𝑞 | |𝑞 | 𝑞 𝑚𝑎𝑥 𝑞 𝑚𝑖𝑛 ℏ𝜔 𝑚𝑎𝑥 ℏ𝜔 𝑚𝑖𝑛 Equation 3-4
Two steps are involved in the Monte-Carlo simulation of the inelastic interaction for a particle of energy 𝐸. First, the inelastic mean free path 𝜆 is computed from the total cross section, using the relation

𝜆(𝐸) = 1 𝑁𝜎(𝐸)
with 𝜎 from Equation 3-4. If an inelastic interaction happens, a random sampling of the the differential cross section is done in a second step, to determine the amount of energy transferred. Indeed, as we will now see, some values of energy transfers may be more probable depending on the value of the energy loss function.

At the limit 𝑞 = 0 ⃗ , the ELF is called the Optical Energy Loss Function 𝑂𝐸𝐿𝐹(ℏ𝜔, 0

⃗ ) = 𝐼𝑚 [- 1 𝜀(𝜔,0 ⃗ ⃗ ) ]
. This quantity is accessible through experimental measurements of either the OELF itself, or the optical indices 𝑛 and 𝑘. Indeed, the OELF can be obtained from these indices by the relation

𝐼𝑚 [- 1 𝜀(𝜔, 0 ⃗ ) ] = 2𝑛𝑘 (𝑛 2 + 𝑘 2 ) 2 Equation 3-5
Several databases of experimental OELF or n,k measurements are available in the literature, such as Palik's handbook [32] or the OELF database of Sun et al. [33]. A few examples of OELFs from the latter are shown on Figure 3-2. We can see several peaks and regions on the OELFs, which correspond to the various sources of energy loss we have identified in Chapter 2. The value of the OELF at a given energy is an indication of the probability of an energy transfer having this value. This means that energy transfers with a higher value of OELF will be more probable. For instance, the most probable energy loss for particles in Al is around 15 eV. First, at low energies around 10 -20eV, can be found the plasmon peak, which is a marked peak of highest intensity for many materials. This means that the main source of secondary electrons for these materials (Al and Al2O3 in Figure 3-2) will be the excitation of a plasmon, followed by its relaxation by the emission of a secondary electron. While the plasmon peak is very narrow for lighter elements, for heavier elements (Ni, Au), the peak becomes much more widened. This is due to the fact that interband transitions become more probable, that is to say the excitation of an electron from a lower level of the valence/conduction band to the conduction band above the Fermi level. As we go up towards heavier elements, the region of interband transitions significantly widens, and transition metals such as Ag, Cu, W or Au have a plateau region on the OELF that can span up to several hundreds of eVs. In the case of Au here, the region of interband transition ranges up to 2 keV. The effect of this plateau region is that the energy transfers are susceptible to take a wider range of values compared to the very marked plasmon peak found in lighter elements. Finally, for the highest energy transfers, a series of peaks can be observed, which correspond to the excitation of an electron from a core shell of an atom (K, L) to the conduction band. The binding energy of a given shell is a hard threshold for the excitation of an electron from this shell. As the core electrons have a stable bond with the nucleus, the energy required to set one of these electrons into motion can be quite high. This is especially true for heavier Z atoms, which are larger and have core electrons that are very strongly bound. For Al, the first accessible core shell is the L shell at around 100 eV, and the deepest shell is the K shell at 1.5 keV. For Ni, the core shells cover a wider range, spanning from the M shell at 100 eV to the K shell at 8 keV. Comparatively, for Au, the first core shell that can be excited is the shell at 2 keV up to the shell at 14 keV. Finally, for a compound material such as Al2O3, the particles can excite the core electrons of either atom of the compound, in this case the K shell of either the Al atom (1.5 keV) or the O atom (500 eV). The OELF of Al2O3 is quite interesting, as it is an insulating material with a gap of around 9 eV. Therefore, no ionization is possible for energy transfers below 9 eV, which is observable on the OELF with the sharp decrease of the plasmon peak at 9 eV. On the other hand, some energy transfers around a few eV are still possible, since the OELF is not null. These are attributed to the energy losses by creation of optical phonons.

Since the OELF only gives us information on the energy transfers at 𝑞 = 0, it needs to be fitted with dielectric function models and extended to 𝑞 ≠ 0 using dispersion relations. In the previous version of MicroElec, the OELF was fitted with a sum of extended Drude functions [28]:

𝐼𝑚 [- 1 𝜀(𝜔, 0 ⃗ ) ] = ∑ 𝐷 𝑗 (ℏ𝜔) 𝑗 Equation 3-6
With 𝐷 𝑗 the expression of the Drude function for the 𝑗 𝑡ℎ peak

𝐷 𝑗 (ℏ𝜔) = 𝐸 𝑝 2 𝐴 𝑗 𝑤 𝑗 ℏ𝜔 (ℏ 2 𝜔 2 -𝐸 𝑗 2 ) 2 + (𝑤 𝑗 ℏ𝜔) 2 Equation 3-7
The parameters 𝐴 𝑗 , 𝑤 𝑗 and 𝐸 𝑗 are respectively the amplitude, width, and energy of the peak, and 𝐸 𝑝 is the plasmon energy. Finally, the 𝐸𝐿𝐹(ℏ𝜔, 𝑞 ) was obtained with the dispersion relation

𝐸 𝑗 (𝑞) = 𝐸 𝑗 + ℏ 2 𝑞 2 2𝑚 0
In the new version of MicroElec, the OELF is fitted with a sum of Mermin dielectric functions following the approach given by Abril et al. [34]:

𝐼𝑚 [- 1 𝜀(𝜔, 0 ⃗ ) ] = ∑ 𝐹(𝜔)𝐴 𝑗 𝐼𝑚 [- 1 𝜀 𝑀 (𝜔, 0 ⃗ , 𝐸 𝑗 , 𝛾 𝑗 ) ] 𝑗 Equation 3-8
with the fitting parameters 𝐴 𝑗 , 𝛾 𝑗 = 𝑤 𝑗 and 𝐸 𝑗 . A simple step function 𝐹(𝜔) cuts the peak below a threshold energy 𝐸 𝑡ℎ . The Mermin function [35] 𝜀 𝑀 (𝜔, 𝑞 )

= 1 + (1 + 𝑖 𝛾 𝜔 ⁄ )[𝜀 𝐿 (𝑞 , 𝜔 + 𝑖𝛾) -1] 1 + (𝑖 𝛾 𝜔 ⁄ ) [𝜀 𝐿 (𝑞 , 𝜔 + 𝑖𝛾) -1] [𝜀 𝐿 (𝑞 , 0) -1] ⁄ Equation 3-9
is expressed in terms of the Lindhard dielectric function [34,36]

𝜀 𝐿 (𝑞 , 𝜔) = 1 + 𝜒 2 𝑧 2 [𝑓 1 (𝑢 ⃗ , 𝑧 ) + 𝑖𝑓 2 (𝑢 ⃗ , 𝑧 )]
𝑓 1 ⃗⃗⃗ (𝑢 ⃗ , 𝑧 ) = 1 2 + 1 8𝑧 [𝑔 (𝑧 -𝑢 ⃗ ) + 𝑔 (𝑧 + 𝑢 ⃗ ) 𝑓 2 ⃗⃗⃗ (𝑢 ⃗ , 𝑧 ) = { 𝜋 2 𝑢 ⃗ , 𝑧 + 𝑢 ⃗ < 1 𝜋 8𝑧 [1 -(𝑧 -𝑢 ⃗ ) 2 ], |𝑧 -𝑢 ⃗ | < 1 < |𝑧 + 𝑢 ⃗ | 0, |𝑧 -𝑢 ⃗ | > 1 𝑔(𝑥) = (1 -𝑥 2 ) ln | 1 + 𝑥 1 -𝑥 |
An interesting feature is that the dependence in both energy and wave vector is included in the Mermin dielectric function. As a result, no separate dispersion relation needs to be applied as a post treatment, contrary to the extended Drude model. The improvement of the Mermin approach over the Drude approach, especially for low energy particles, can be seen on the stopping powers computed in section 3.3.1.

The peaks can be fitted to the different transitions observed on the OELF by tweaking the three fitting parameters, as shown in Figure 3-3, where the peaks for different shells and transitions are shown. This method has been used by Denton et al. [37] and Da et al. [38] to fit the OELF for silicon and some metals. In the case of silicon, our OELF fit is similar to the fit from the previous version of MicroElec [15], while our Kapton fit is based on the work of de Vera et al. [39]. Our fit parameters for all materials shown in this work are given in Appendix I. We have fitted the OELF of 16 materials using Mermin's model: Be, C, Al, Si, Ti, Fe, Ni, Cu, Ge, Ag, W, Au, SiO2, Al2O3, Kapton, BN. The accuracy of our fit is verified using the average ionization potential I, and P-and Z-sumrules as used by Tanuma et al. [40]. The P-sum rule is given by the condition

𝑃 𝑒𝑓𝑓 = 2 𝜋 ∫ 𝐼𝑚 [- 1 𝜀(𝜔, 0 ⃗ ) ] 𝑑𝜔 𝜔 +∞ 0 + 𝑅𝑒 [ 1 𝜀(0) ] = 1
Where 𝑅𝑒 [ ] = 𝑛 -2 (0) for insulators. The Z-sum rule follows

𝑍 𝑒𝑓𝑓 = 2 𝜋Ω 𝑝 2 ∫ 𝜔𝐼𝑚 [- 1 𝜀(𝜔, 0 ⃗ ) ] 𝑑𝜔 +∞ 0 = 𝑍
Where Ω 𝑝 2 = (4𝜋𝑛 𝑎 𝑒² 𝑚 ⁄ ) 1 2 ⁄ and, 𝑛 𝑎 is the density of atoms or molecules.

The values obtained for the sum rules are shown in Table 1, and compared with data from Mukherji et al. [41], SRIM [42], ICRU [43] and Vera et al. [39]. The target value (Z) for Zeff is the atomic number for a monoatomic material, or the total number of electrons per molecule for a compound. There are some non-negligible errors in the sum rules for some materials, as we have defined our fits to be the closest to the experimental OELF. We have also chosen to validate their correctness by favoring the agreement with stopping power and SEY data over the sumrule values. For some materials, we have chosen to degrade the OELF fit to improve the stopping powers at low energies. In the case of Ni, Cu, Ge, W, SiO2, the stopping powers and SEY are still satisfying despite the errors in the sum rules, as shown in 3.3.1.

The differential cross section (DXS) for an incident particle of energy E [MeV] and mass 𝑀 [MeV/c²] is then calculated from the fitted ELF and using the following relationship, by summing the partial DXS for each j th shell:

𝑑𝜎 𝑑(ℏ𝜔) (𝐸, ℏ𝜔) = 𝑍 𝑒𝑓𝑓 2 𝜋𝑁𝑎 0 𝐸 ′ ∫ ∑ 𝐹(𝜔)𝐴 𝑗 𝐼𝑚 [- 1 𝜀 𝑀 (𝜔, 0 ⃗ , 𝐸 𝑗 , 𝛾 𝑗 ) ] 𝑗 𝑑|𝑞 | |𝑞 | 𝑞+ 𝑞- Equation 3-11
with the atomic density N [#/cm 3 ], the Bohr radius 𝑎 0 , the electron mass 𝑚 𝑒 , 𝐸 ′ = 𝑚 𝑒 𝑀 𝐸, and 𝑍 𝑒𝑓𝑓 the effective charge, equal to 1 for electrons and protons. For other particles, the effective charge was previously calculated using the Barkas formula [44]:

𝑍 𝑒𝑓𝑓 = 𝑍[1 -exp(-125𝛽𝑍 -2 3 ⁄ )] Equation 3-12
where

𝛽 = √1 -(1 + 𝐸 𝑀𝑐 2 ⁄ ) -2
. Nevertheless, at low incident ion velocity, the target electron velocities cannot be neglected. Brandt and Kitagawa (BK) theory provides an expression for describing the effective charge, connecting the charge state of the projectile and the Fermi velocity of the target. Some suggestions can be found in [42] to improve the BK formulation compared to data analysis. The original expression of the effective charge given by BK is (in atomic units):

𝑍 eff,BK (𝑍, 𝑞, 𝑣 𝐹 , 𝜁) = 𝑍[𝑞 + 𝜁(1 -𝑞) ln(1 + (2Λ 0 𝑣 𝐹 ) 2 )] Equation 3-13
Where Λ 0 is the screening radius, treated as a variational parameter in the BK theory and expressed as: 152.4 [41] 166 [42] 160.3 [41] 171 [42] 235 [42] 321 [42] 255 [41] 332 [42] 335 [42] 349.3 [41] 488 [42] 618 [41] 735 [42] 139 [43] 82.4

Material (Z) C (6)
[39]

𝛬 0 = 0.48 (1 -𝑞) 2 3 𝑍 1 3 (1 - 1 -𝑞 7 ) Equation 3-14
𝜁 is generally assumed to be ∼ 0.5 [45,46] and 𝑞 = (𝑍 -𝑁)/𝑍 is the fraction of the electrons that have been stripped from the moving ion (𝑁 being the number of electrons still bound to the projectile). An improved formulation of the screening radius is given by Kaneko [47]:

𝛬 = 𝛬 0 1 + 𝑓𝛬 0 Equation 3-15
Where the relation giving 𝑓 can be found in [47]. To obtain the best fit of stopping powers, Ziegler et al. [42] have proposed the following relation for describing 𝑞:

𝑞 𝑍𝑖𝑒𝑔𝑙𝑒𝑟 (𝑍, 𝑣 𝑟 ) = 1 -exp (-𝑐 ( 𝑣 𝑟 𝑍 2 3 -0.07)) Equation 3-16
with 𝑐 a constant close to 1 (𝑐 = 0.95 in our calculations) and 𝑣 𝑟 the relative velocity of the incident particle and a target electron. A complete expression of 𝑣 𝑟 is given by Kreussler et al. [48]. In addition, Ziegler et al. have suggested a modified value of ζ to adjust the effective charge results to extensive data analysis [42]:

𝑍 𝑒𝑓𝑓,𝑡ℎ𝑖𝑠 𝑤𝑜𝑟𝑘 (𝑍, 𝑞, 𝑣 𝐹 ) = 𝑍 𝑒𝑓𝑓,𝐵𝐾 (𝑍, 𝑞, 𝑣 𝐹 , 𝜁 𝑍𝑖𝑒𝑔𝑙𝑒𝑟 ∼ 1 2𝑣 𝐹 2 ) Equation 3-17
Finally, our expression of the effective charge depends on the atomic number, the Fermi velocity of the target and the incident velocity, through the relative velocity. This description is used in the new version of MicroElec. It does not however account for the charge modifications of the incident projectile, unlike the charge fraction approach used in CasP [49] and by Moreno-Marin et al. [50], Behar et al. [51] or Heredia-Avalos et al. [52]. Although the charge fraction description is more accurate, it is much less convenient to use in the code as the modification of the inelastic MFP needs to be handled for each charge state, which could significantly slow down the simulations. We have chosen to not use charge fractions in the new version of MicroElec as a compromise between accuracy and ease of use. Moreover, the effective charge description is much more complete than the previously used Barkas formula.

The relativistic corrections detailed in [16] by Raine et al. are also applied to the expression of the DXS and its integration limits. The effect of these corrections is directly visible on the stopping powers in section 3.3.1 computed with the Mermin approach. The comparison of these stopping powers with literature data is used as a validation metric for the inelastic cross sections. This has been done for seven metals (C, Al, Ag, Ti, Ni, Cu and W), two semi-conductors (Si and Ge) and two insulators (SiO2, Kapton), considerably extending the simulation capabilities of MicroElec. Low energy electron transportation in insulators does however require additional energy loss models, among which the interactions of electrons with phonons.

There are other corrections to the dielectric function theory at low energy as detailed by Salvat & Fernandez-Varea in [53], such as exchange effects or plasmon decay, that have not been included in this work. We have assumed that the dampening of the plasmon always leads to the production of a secondary electron, which may decrease the accuracy of our calculations at very low energies. Although the comparison with low energy experimental data is still satisfying, exchange effects are considered for future versions of MicroElec, following the method of Ochkur [54] and its implementation in other works, such as Fernandez-Varea et al. [55] or de Vera et al. [39].

Procedure of computation of the energy transfers

In the last version of MicroElec (Geant4 10.0) [17], the energy ℏ𝜔 and moment 𝑞 transferred by a primary particle of energy 𝐸 was calculated via rejection sampling from the differential cross sections per shell

𝑑𝜎 𝑗 𝑑(ℏ𝜔)
, which are given by

𝑑𝜎 𝑗 𝑑(ℏ𝜔) (𝐸, ℏ𝜔) = 𝑍 𝑒𝑓𝑓 2 𝜋𝑁𝑎 0 𝐸 ′ ∫ 𝐹(𝜔)𝐴 𝑗 𝐼𝑚 [- 1 𝜀 𝑀 (𝜔, 0 ⃗ , 𝐸 𝑗 , 𝛾 𝑗 ) ] 𝑑|𝑞 | |𝑞 | 𝑞+ 𝑞-

Equation 3-18

In this subsection, the energy ℏ𝜔 lost by the incident particle will be noted as 𝑄, and the differential cross section for a given energy transfer 𝑄 as 𝑑𝜎 𝑑ℏ𝜔 ⁄ (𝑄). In rejection sampling, we use a normalized distribution function of the energy transfers obtained from the differential cross section:

( 𝑑𝜎 𝑑ℏ𝜔 ) 𝑛𝑜𝑟𝑚 = (𝑑𝜎 𝑑ℏ𝜔 ⁄ ) max (𝑑𝜎 𝑑ℏ𝜔 ⁄ ) Equation 3-19
The rejection sampling procedure is shown in Figure 3 If the normalized differential cross section for 𝑄 𝑡𝑟𝑖𝑎𝑙 verifies the condition

( 𝑑𝜎 𝑑ℏ𝜔 ) 𝑛𝑜𝑟𝑚 (𝑄 𝑡𝑟𝑖𝑎𝑙 ) ≥ 𝑅 2 Equation 3-21
The value of the energy transfer 𝑄 is then taken equal to 𝑄 𝑡𝑟𝑖𝑎𝑙 . If this condition is not verified, the procedure is repeated with new random numbers 𝑅 1 and 𝑅 2 until we find a 𝑄 𝑡𝑟𝑖𝑎𝑙 that verifies Equation 3-21. The issue with rejection sampling is that we need to draw random numbers and interpolate the distribution until the condition is verified. This step can thus be repeated many times for each occurrence of an inelastic interaction, which considerably hinders the simulation time. For instance, in the example given in Figure 3-4, the condition of Equation 3-21 is not verified, so we need to start the procedure again.

To address this issue, MicroElec has been extended to handle direct sampling from cumulated differential cross sections. While the functions for direct sampling were already coded in MicroElec at the beginning of this thesis, the cumulated DXS were not computed and this option was not enabled. The use of cumulated DXS per shell and direct sampling has been validated for the new version and set as the default sampling method, considerably improving computation time.

In any case, the value of 𝑄 we have obtained is removed from the initial energy of the incident particle, which has a final energy 𝐸 𝑓 = 𝐸 -𝑄 after the inelastic interaction. Still, 𝑄 is not entirely transferred to the secondary electron generated. Indeed, for electrons generated from core shells with a binding energy 𝐸 𝑏 , a part of 𝑄 equal to 𝐸 𝑏 is spent to excite the electron from the core level above the Fermi level. In this case, the secondary electron is generated with an energy

𝐸 𝑠𝑒𝑐 = 𝑄 -𝐸 𝑏 Equation 3-23
For semiconductors and insulators, the secondary electron also needs to be brought above the energy gap, so that the energy of the secondary electron in these materials is

𝐸 𝑠𝑒𝑐 = 𝑄 -𝐸 𝑏 -𝐸 𝑔𝑎𝑝 Equation 3-24
The case of plasmon excitation and interband transitions is more complicated. In this case, the secondary electron is excited from the valence band into the conduction band, or within the conduction band from below to above of the Fermi level in a metal. We will call these electrons "weakly bound", in opposition to the strongly bound core electrons. The main difference between these two populations is that the core electrons are localized on a discrete level with a well defined binding energy, whereas the weakly bound electrons are located within a continuum of energies below the Fermi level 𝐸 𝐹 . In a first approximation, we can consider that all weakly bound electrons come from the Fermi level, which was the approximation used in the version of MicroElec in Geant4 at the beginning of the thesis. In such a case, the binding energy of the weakly bound electron in Equation 3-23 or Equation 3-24 is 0 eV. However, if we look at the densities of states (DOS) for a few materials, we can see that the Fermi level is not necessarily the most populated level, even in the simplest approximation of a free electron gas DOS. Since the Fermi-Dirac statistic at room temperature is a staircase function with a hard cut at 𝐸 𝐹 , we can consider that the density of states is a direct indication of the population of potentially excitable electrons at a given depth below 𝐸 𝐹 .

Consequently, for interactions with weakly bound electrons, the initial energy of the weakly bound electrons has to be taken into account in the energy transfer. In fact, during an interaction with a weakly bound electron, we can consider that it is excited from the conduction band below the Fermi level for metals, or the top of the valence band for semi-conductors and insulators.

This electron has a potential energy 𝐸 𝑖𝑛𝑖𝑡 compared to the Fermi level, which we will treat as similar to the binding energy of core shells. In result, the secondary electron is generated with an energy

𝐸 𝑠𝑒𝑐 = 𝑄 -𝐸 𝑔𝑎𝑝 -𝐸 𝑖𝑛𝑖𝑡 Equation 3-25
where 𝐸 𝑔𝑎𝑝 = 0 for metals. Following the example of the Monte-Carlo code OSMOSEE for low energy electrons [17,56], we have reintroduced the initial energy of weakly bound electrons. This energy applies to all plasmon and interband transitions. The references of energy in MicroElec correspond to the highest occupied state at 0K. Hence, for a metal, the reference is the Fermi level, for a semi-conductor or insulator it is the bottom of the conduction band.

In OSMOSEE, 𝐸 𝑖𝑛𝑖𝑡 was selected from the density of states. This selection was done for each individual interaction with weakly bound electrons by rejection sampling. The DOS 𝑔(𝐸) used for sampling was an approximation of the real DOS with the free electron gas theory, in the form of 𝑔(𝐸) ∝ √𝐸. However, as we have seen for the sampling of the differential cross sections, rejection sampling is quite expensive in computation time.

To address this issue, we have used instead a unique value for 𝐸 𝑖𝑛𝑖𝑡 , chosen for each material. This value has been selected as either the median value of the DOS, or the energy of the most populated state in the DOS. The objective is to have a value that will be representative of an average value obtained over a great number of random samplings. In this sense, the median or the most populated state are values that have the highest probability of being drawn, and are more statistically consistant. Nevertheless, the choice of a unique value has quite a few limits that are due to the simplicity of the approach. Indeed, when computing the median or choosing the most populated state, we assume that all states of the DOS are accessible. However, this will not be the case if the incident electron has a very low energy, below the Fermi energy (ex: 11.6 eV in Al). In this situation, the deepest states of the DOS are not accessible anymore. In consequence, the sampling interval is also reduced, and a random sampling over the accessible states may not yield the same average value. This case was handled in OSMOSEE by reducing the interval [𝑢 𝑎 ; 𝑢 𝑏 ] for rejection sampling and excluding the inaccessible states. In the worst case scenario where the incident electron has an energy of only a couple of eVs, its energy 𝐸 𝑖 may even be below 𝐸 𝑔𝑎𝑝 + 𝐸 𝑖𝑛𝑖𝑡 which should prevent any kind of energy transfer. This exception is handled in our code, by verifying that the energy lost by the electron and taking into account 𝐸 𝑖𝑛𝑖𝑡 does not exceed its energy 𝐸 𝑖 . If it is the case, we assume that the secondary electron comes from the Fermi level and has an initial energy 𝐸 𝑖𝑛𝑖𝑡 = 0.

As we will see in the computation of the TEEY in section 3.3.3, the addition of the initial energy is necessary to avoid an overestimation of the TEEY values. Despite the simple approximation made here, its impact on the TEEY is clearly visible and allows the simulation to gives values are in good agreement with the reference data. The values chosen for the initial energy of weakly bound electrons are given in Table 3 Finally, Auger transitions are included for core shells using the class G4AtomDeexitation. This class is called when a core electron has been ejected during an inelastic interaction, and Auger reorganization processes are susceptible to happen. For this, we need to know which shell the secondary electron is coming from. Hence, there is first a random selection based on the differential cross sections per shell, to determine if the secondary electron will be produced from a given shell or a plasmon/interband transition. Then, the corresponding table of cumulated cross section per shell is retrieved and used in the direct sampling procedure. If a core shell has been selected, the info is passed to G4AtomDeexitation along with the atomic number of the atom where the deexcitation process takes place. This class will then compute the Auger processes and add the secondary electrons generated by the Auger cascade to the tracking stack if necessary.

3.2.3

Improvements for the simulation of multilayer materials

Interaction model for the crossing of electrons through a surface or material interface

At the vacuum/material boundary, the periodic potential of the crystal is disturbed by the discontinuity induced by the interface, which can be expressed as a potential barrier [57]. The energy reference for an electron in vacuum is set as the vacuum level. In MicroElec, the energy reference for an electron inside of the material was chosen as the lowest unoccupied state, namely the Fermi level for a metal, or the bottom of the conduction band for an insulator. In other works, such as the OSMOSEE code, the bottom of the conduction band is generally taken as the energy reference for metals instead of the Fermi level, and the surface potential barrier is the sum of the Fermi energy and the workfunction. For consistency with the previous versions of MicroElec, we have chosen to retain the Fermi level as the energy reference in metals. Due to this change of reference when crossing the surface potential barrier, illustrated in Figure 3-6, an electron penetrating in the material will lose potential energy; this loss is transformed into a gain of kinetic energy. An electron exiting the material will, on the contrary, gain potential energy and lose kinetic energy. In both cases, the gain or loss is equal to the value of the potential barrier, taken as the work function 𝑊 for metals, or the electron affinity 𝜒 for insulators and 3.2 -Implementation of interaction models for low energy electrons semi-conductors. The energy modifications for an electron going through the surface are shown in Figure 3-6. In both cases, a free electron has a positive energy. Captured or bound electrons of the material have a negative energy, such as the weakly bound electron energy used for plasmon and interband transitions in Equation 3-25. The electron needs to have a greater energy than the surface barrier to be emitted outside of the material.

In MicroElec, the surface has been added as an exponential potential barrier, based on the model of the Monte-Carlo code OSMOSEE [56]. This gives the following expression of the transmission probability for an electron of energy 𝐸 and incident angle 𝜃:

𝑇(𝜃, 𝐸) = 1 - sinh 2 (𝜋𝑎(𝑘 𝑖 -𝑘 𝑓 ))
sinh 2 (𝜋𝑎(𝑘 𝑖 + 𝑘 𝑓 ))

Equation 3-26
with the pre-and post-transmission wave vectors [nm -1 ]: For an interface between two materials, 𝐸 𝑡ℎ is taken as the difference between the thresholds of the two materials as a first approximation.

𝑘 𝑖 = √(2𝑚 0 𝑒) ℏ √(𝐸 cos 2 𝜃), 𝑘 𝑓 = √(2𝑚 0 𝑒) ℏ √(𝐸 + 𝐸 𝑡ℎ ) cos 𝜃 𝐸
If not transmitted, the electron is reflected by the surface. The post reflexion angle, with respect to the normal to the surface, is then taken equal to the pre reflexion angle, as it is the case with optical reflexion, and the energy is unchanged. Finally, an electron traveling near the surface may interact with a surface plasmon, as opposed to the volume plasmons treated by the inelastic process, the interaction probability depending on the depth of the electron. Although the interaction has not been implemented, the secondary emission yields given by the code in section 3.3.3 are still in good agreement with experimental data.

The surface processes for electrons are included in Geant4 in the new class G4MicroElecSurface, a discrete process called at each interface and based on the other Geant4 class G4OpBoundaryProcess that handles the surface processes for optical photons. G4MicroElecSurface handles Vacuum/Material and Material/Material interfaces. This allows the simulation of simple multi-layer structures using the supported materials. In such a case, the difference in work function/electron affinity between the materials needs to be taken into account. Hence, the threshold energy 𝐸 𝑡ℎ takes the value of this difference. The modelling of the interfaces of a multilayer structure is shown on Figure 3-7, in the example of a SiO2 layer on a Si layer. We can notice that electrons coming from the Si layer need to overcome a potential barrier ∆𝜒 = 𝜒 𝑆𝑖 -𝜒 𝑆𝑖𝑂 2 . The surface processes allow the electron tracking limit to be extended down to the work function or electron affinity of the material, which corresponds to energies of a few eVs. In the simulation, all electrons are followed until their energy falls below the height of the surface potential barrier.

In this case, they are unable to escape and are killed from the simulation.

Modifications of MicroElec to allow the simulation of multiple materials

In the original version of MicroElec, the transport of electrons, protons and heavy ions could only be simulated in silicon. The module could also model the transport of particles in the 16 new materials with a few fixes in the code, but with only a single material per simulation. As a result, some more extensive modifications had to be made to handle the simulation of multilayered materials, using the interaction models we have presented so far.

The first addition brought during this thesis work is the creation of a material structure data file. Indeed, there are several parameters that we need to know for each individual material, such as the binding energies for each shell or the work function. For each material, a structure file Data_ [Material_Name].dat is created to store these relevant quantities. Two examples of the files are given for a monoatomic material (silver) and a compound (SiO2) in Figure 3-8. The first line of the file is a header, giving information on the material's name and its atomic number. For a compound, the keyword "Compound" is used instead of the atomic number. All the other lines (except the comments marked by a #) contain data on a given variable or table of values, and follow the structure:

[Number of values] [Name of variable] [Variable unit, or "noUnit"] [Values of the variable]

For some variables, a single value is needed. However, for other variables such as the low energy limits of the Mermin functions used for fitting (treated here as the binding energy for a core shell), we need to store a vector since we have one value per shell. The values stored in the structure file and their significance are listed in Table 3-3. Low and high energy limit of the inelastic interaction models for electrons (_e) or protons and ions (_p) ElasticModelLowEnergyLimit ElasticModelHighEnergyLimit Low and high energy limit of the elastic interaction model for electrons The new class G4MicroElecMaterialStructure reads the data files and stores the data in an object that can be created by the different processes. They can then retrieve the data for a given material by calling the object G4MicroElecMaterialStructure for a material. In this thesis work, we have also implemented map structures in MicroElecElasticModel and MicroElecInelasticModel to store the cross section tables for each material of the geometry. In these maps, each cross section table is identified by a key (the name of the material). At each simulation step, the current material can then be used to search the storage maps and retrieve the corresponding cross section tables or material structure data. Using both storage maps and material structure objects, the new MicroElec is then able to handle the transport of particles in multiple materials in a single simulation.

Implementation of electron-phonon interaction models for insulators

Due to the wide band gap in insulators (8-9 eV in SiO2 or Al2O3), the inelastic mean free paths (MFP) shown in Figure 3-9 become divergent as the electron energy approaches the band gap, and ionization becomes impossible below the band gap. The transport of electrons at this energy level (<~ 10 eV) is thus dominated by other processes. This is especially true for insulators that have large band gaps. New physical processes for the interaction of low-energy electrons with optical and acoustic phonons have been added to MicroElec for the insulator SiO2, following the work of Schreiber and Fitting [58]. The models have also been extended to other insulators, namely Al2O3 and BN.

Inelastic interactions with optical phonons

An electron can interact with a Longitudinal-Optical (LO) Phonon, of vibration mode 𝜔 𝐿𝑂 , and either absorb or create a phonon. Phonon absorptions or emissions are respectively a gain or a loss of energy for the primary electron, the value of which corresponds to the phonon vibration mode ∆𝐸 = ±ℏ𝜔 𝐿𝑂 . This interaction has been described by Fröhlich [59] [60], and used by Fitting et al. [61] in SiO2, Akkerman et al. [62] in alkali halides, and Ganachaud et al. [63] in Al2O3. Optical phonons have also been implemented in PENELOPE, though with a different formalism based on an integration of the ELF in the infrared range [64]. In the Fröhlich formalism, the scattering rates 𝑓 [s -1 ] for the absorption (-) or emission (+) of a LO phonon by an electron of energy 𝐸 are given by [65]

𝑓 ∓ (𝐸) = 𝑒 2 4𝜋𝜖 0 ℏ 2 • (𝑁 𝐿𝑂 + 1 2 ∓ 1 2 ) • ( 1 𝜖(∞) - 1 𝜖(0) ) • √ 𝑚 * 2𝐸 • ℏ𝜔 𝐿𝑂 • ln [ 1 + 𝛿 ∓1 ± 𝛿 ] Equation 3-27 With 𝛿 = √1 ± ℏ𝜔 𝐿𝑂 𝐸 , 𝑁 𝐿𝑂 = 1 𝑒 ℏ𝜔 𝐿𝑂 /𝑘 𝑏 𝑇 -1
the Bose-Einstein distribution of the population of phonons for the mode 𝜔 𝐿𝑂 and the temperature 𝑇 [K], 𝜖(∞) = 2.25 and 𝜖(0) = 3.84 the optical and static dielectric constants, with the values here given for SiO2. In this material, two phonon modes ℏ𝜔 𝐿𝑂 = 63 and 153 meV are considered. The emission and absorption rates for the two phonon energies of SiO2 are given in Figure 3-10 for electrons below 10 eV. We can see that the emission of a phonon of a given mode ℏ𝜔 𝐿𝑂 becomes impossible when the electron energy falls below ℏ𝜔 𝐿𝑂 . Moreover, phonon emission is much more probable than phonon absorption in SiO2, so the latter has been neglected. However, the case of SiO2 is particular compared to Al2O3 and BN, since for these materials only one LO phonon mode is considered.

Hence, for SiO2, we would need to create two LO phonon interaction processes in Geant4, whereas for other insulators we would have to only create one. To ensure the flexibility of our Geant4 module, we have combined the 0.063 eV and 0.153 eV emission processes into a single emission process for SiO2, with the emission frequency plotted in black in Figure 3-10. In fact, the 0.153 eV emission process is more probable than the 0.063 eV emission for ballistic electrons, so they have been weighted by a factor of respectively 75% and 25% by Schreiber [58].

We have used these factors in our LO phonon emission process for SiO2, where we use a single weighted energy ℏ𝜔 𝐿𝑂 = (0.75 * 0.153 eV + 0.25 * 0.063 eV) = 0.131 eV in Equation 3-27. Now that the scattering rate corresponds to the emission of two phonon modes, it also has to be multiplied by 2.

Figure 3-10: Emission and absorption rates of LO phonons in SiO2

The effective mass 𝑚 * is assumed to be equal to the free electron mass 𝑚 0 . The angular deflection 𝜃 of the primary electron is calculated with a random number 𝑅 ∈ [0; 1] following the angular distribution from [61]:

cos(𝜃) = 𝐸 + 𝐸 ′ 2√𝐸𝐸 ′ (1 -𝐵 𝑅 ) + 𝐵 𝑅 Equation 3-28
Where 𝐵 =

𝐸+𝐸 ′ +2√𝐸𝐸 ′ 𝐸+𝐸 ′ -2√𝐸𝐸 ′ ; 𝐸 ′ = 𝐸 -ℏ𝜔 𝐿𝑂
The mean free path can be obtained as 𝑀𝐹𝑃 = √2𝐸/𝑚 0 𝑓 ± ⁄ . As can be seen in Figure 3-9, the interactions with LO phonons will be the main energy loss process for electrons under 15 eV in SiO2. Indeed, the LO phonon MFP becomes lower than the inelastic MFP below 15eV, as the latter becomes divergent. Below the energy gap, the energy loss by LO phonon is also the only possible energy loss interaction for electrons. The parameters used for the LO phonon model are given in

Elastic interactions with acoustic phonons

Electrons are also able to interact with acoustic phonons, in this case the interaction is analog to the elastic interaction with nuclei. In fact, the energy lost by an electron interacting with an acoustic phonon is about a few meV, which will be neglected here. However, as pointed out by Akkerman [25] or Valentin [14], the validity of the Partial Wave Analysis (PWA) cross sections for the elastic interaction of electrons becomes questionable below 50 eV. In SiO2, this phenomenon is especially noticeable as the elastic MFP, given by the molecular cross sections, is inferior to the inter-atomic distance below 50 eV (Figure 3456789). It is thus recommended to switch to an acoustic phonon-electron interaction model at lower energies. In such a model, the interaction between the electrons and the lattice and their behaviour as Bloch electrons should be more realistic [58].

In order to simplify the expressions, we will use the approximation of a parabolic band, which gives the density of states of electrons [58] :

𝐷(𝐸) = 2 √2𝜋 2 ℏ 3 𝑚 0 3/2 √𝐸 Equation 3-29
The dispersion relationship is then given by:

𝐸(𝑘) = ℏ 2 𝑘 2 2𝑚 0 ⁄ Equation 3-30
and the effective mass is assumed equal to the free electron mass as in the LO phonon model.

Due to the lattice's periodicity, the unique values of the acoustic phonons' wave vectors 𝑘 are contained in the first Brillouin zone, that is to say in [-𝑘 𝐵𝑍 ; 𝑘 𝐵𝑍 ]. Hence, a wave vector outside of this domain can be expressed as a multiple of a wave vector belonging to it. In the reciprocal lattice, the limit of the Brillouin zone is given by 𝑘 𝐵𝑍 = 𝜋 𝑎 with the lattice parameter 𝑎.

In our simulations, we have implemented the expression of the collision frequency given by Schreiber & Fitting [58]. It is based on the integral relationship given by Bradford and Woolf [66]. The collision frequency is then given by:

𝑓 𝑎𝑐 = 𝜋𝑘 𝑏 𝑇 ℏ𝑐 𝑠 2 𝜌 ℰ 𝑎𝑐 𝐷(𝐸) 1 + 𝐸 𝐴 ⁄ if E < E BZ 4 𝑓 𝑎𝑐 = 2𝜋𝑚 * * (2𝑁 𝐵𝑍 + 1) 𝜌ℏℏ𝜔 𝐵𝑍 ℰ 𝑎𝑐 2 𝐷(𝐸)𝐸 2 ( 𝐴 𝐸 ) 2 * [- 𝐸 𝐴 ⁄ 1 + 𝐸 𝐴 ⁄ + ln (1 + 𝐸 𝐴 )] if E > E BZ Equation 3-31
The parameters ℰ 𝑎𝑐 (also noted 𝐶) and 𝐴 are specific to the material. We also have 𝑘 This model was first conceived by Sparks et al. [67] for alkali halides (NaCl, HCl…), and used by Fischetti [68] [69] for SiO2 according to the equation :

1 𝑓 ± (𝑘) = ∑ 𝑚 * 4𝜋𝑀𝑁 𝑐 ℏ²𝑘 ∫ 𝑑𝑞 𝑞 3 𝜔 𝛼 (𝑞) 𝑞 𝑚𝑎𝑥 0 |𝑆 𝛼 (𝑞)|² 𝛼 [𝑛 𝛼 (𝑞) + 1 2 ± 1 2 ] Equation 3-32
We have here a sum that extends over the acoustic phonon branches 𝛼, since the probability given by Sparks is only given for the scattering of an electron of wavevector 𝑘 with a phonon of a given wavevector 𝑞. Hence, we need to sum over all possible phonon wavevectors 𝑞 to get the scattering frequency. 𝑁 𝑐 Is the density of primitive cells, 𝑀 is taken as the mass of a primitive cell for small 𝑞, or or the mass of the heaviest constituent of the cell when 𝑞 approaches 𝑘 𝐵𝑍 . 𝑛 𝛼 (𝑞) are the Bose factors of the phonon branches, and 𝑞 𝑚𝑎𝑥 = 2𝑘 ± 2𝑚 * 𝐶 𝑠 /ℏ. |𝑆 𝛼 (𝑞)|² is the electron-phonon matrix element.

Bradford and Woolf [66] note that the expressions proposed by Sparks et al. [67] and Fischetti [68] [69] use approximations that become invalid at higher energies. This is notably true for electrons with wave vectors beyond the Brillouin zone edge (𝐸 > 𝐸 𝐵𝑍 ), around a few eV. The interaction frequency extrapolated by Sparks from Equation 3-32 for electrons with an energy greater than 𝐸 𝐵𝑍 /2 gives a dependence in energy in 𝐸 3/2 , which makes the inverse mean free paths diverge. To fix this problem, Bradford and Woolf have proposed to take into account the screening of the electrons belonging to the atoms of the lattice. This is made by adding a correction factor 𝛼 in the integral relationship of Equation 3-32. They obtain the formula for the matrix element |𝑆 𝛼 (𝑞)| 2 = 𝐶 2 𝑞 2 /(1 + 𝑞 2 /𝛼²)², and for the scattering rate

𝑓 ± = 3𝐶 2 4𝜋𝑀 𝑝 𝑁 𝑐 ℏ𝑣 ∫ 𝑑𝑞 𝑞 3 𝜔(𝑞) * [𝑛(𝑞) + 1 2 ± 1 2 ] * 𝑓(𝑞) (1 + 𝑞 2 𝛼 2 ⁄ ) 2 𝑞 𝑚𝑎𝑥 ± 0 Equation 3-33
Where 𝑀 𝑝 and 𝑁 𝑐 are the mass and density of the unit cell, 𝑛(𝑞) is the Bose function at room temperature,

𝑞 𝑚𝑎𝑥 ± = { 2𝑘 ∓ 2𝑚 * 𝑐 𝑠 ℏ ⁄ for q < k BZ 𝑘[1 + (1 ∓ 𝑐 𝑠 𝑘 𝐵𝑍 ℏ 𝐸 ⁄ ) 0.5 ] for q ≥ k BZ 𝑓(𝑞)
is a function that alters the mass of the unit cell to the mass of the heaviest constituent depending on 𝑞, 𝜔(𝑞) is the phonon frequency. The screening parameter has the dimension of a wavevector in this expression, whereas it is expressed as an energy in the expression of Schreiber & Fitting [58] (the parameter 𝐴). This correction allows the convergence of the acoustic MFP towards the elastic MFP at higher energies.

Nevertheless, a major drawback of this model is that it is very difficult to determine its parameters. Arnold et al. [70] underline the difficulty in finding parameters that work at both lower and higher energies. They attribute this difficulty to the lack of dependence of the scattering rates on the density of states of electrons. Akkerman et al. [62] also highlight this omission, as an explanation to the fact that their simulated energy spectrum of the secondary electrons emitted by alkali halides derive towards higher energies compared to the experimental measurements. However, the density of states is taken into account in the expression of Schreiber & Fitting [58], which we have used here. In fact, the authors have proposed three models for the band structure of SiO2, including the 1-band free electron approximation we have retained here. Two other models were proposed, namely a 3-band and a 5-band structure. They both include the free electron band, along with one or two hole and electron bands, and yield a different density of states from the free electron approximation. As they mentioned that the best agreement with the experimental data was obtained with a single free-electron band structure, we have chosen this structure for the density of states in Equation 3-29.

The value of the deformation potential 𝐶, or ℰ 𝑎𝑐 , is unknown. It is the limit of the matrix element |𝑆 𝛼 (𝑞)| from Equation 3-32 when 𝑞 converges to 0 (|𝑆 𝛼 (𝑞)| ≅ 𝐶). For higher 𝑞, the matrix element can be approximated from the total elastic scattering cross section 𝜎 evaluated at the energy of the exciton as [69]:

|𝑆 𝛼 (𝑞)| 2 ≅ ( 𝜋ℏ 4 𝑁 2 𝑚 * 2 ) 𝜎 Equation 3-34
Where 𝑁 is the atomic density. In the absence of values for 𝐶, Fischetti [69] uses Equation 3-34

for the whole energy range, that is to say 𝐶 2 = ( 𝜋ℏ 4 𝑁 2 𝑚 * 2 ) 𝜎. For SiO2, Fischetti and Schreiber & Fitting use C = 3.5 eV. The cross section can be exactly computed for an electron at the energy of the exciton, as done by Wang [71]. An alternate method is proposed by Sparks et al [67]. They have used for alkali halides the integrated cross section 𝑄 = 4𝜋𝜎 for elastic scattering with the negative ion at the exciton energy, as it is supposed to be much larger than for the positive ion. The cross section is estimated from molecular scattering data. For HCl, which has an exciton energy of 8 eV, they obtain a cross section 𝑄 𝐻𝐶𝑙 = 0.35 nm 2 . The cross section for other materials is computed from 𝑄 𝐻𝐶𝑙 using a scaling relationship depending on the radius of the halide ion 𝑟 ℎ : 𝑄 = ( 𝑟 ℎ 𝑟 𝑁𝑎𝐶𝑙 )²𝑄 𝐻𝐶𝑙 . Fischetti have followed the same approach in SiO2, assuming that the cross section is dominated by the larger oxygen ions. They have obtained an integrated cross section Q = 3.5 x 10 -15 cm² which they have rescaled by the effective charge of oxygen (≅ 1.1 𝑒) in SiO2. For Q in cm² and the atomic density N in at/cm 3 , we can then obtain the deformation potential via the relation 𝐶 = √𝜋ℏ 4 (𝑁 * 10 6 ) 2 (𝑄 * 10 -4 )/𝑚 0 ², which yields the value of 3.5 eV. In contrast, we have seen on Figure 3-1 that the molecular elastic cross sections in SiO2 from ELSEPA converge towards the atomic elastic cross section of silicon at lower energies. This could indicate that silicon is the dominating atom in SiO2 for the elastic scattering at low energies rather than oxygen, in contradiction with the assumption made by Fischetti.

The screening parameter can be chosen to make the acoustic MFP converge to the elastic MFP at a certain energy [72]. Schreiber and Fitting [58] have chosen to completely replace the elastic MFP from the PWA method by the acoustic MFP for all elastic interactions. Using their parameters, the convergence to our elastic MFPs happens at 3 keV. For the alkali halide CsI, Boutboul et al. [72] have fixed the convergence between the two models at 20 eV. This parameter can also be determined from the deformation potential following the equation [66]:

𝑙𝑖𝑚 𝑞→0 ( 4𝜋 𝑉 𝑐 * 1 4𝜋𝜀 0 𝑍 1 𝑍 2 𝑒 2 * 1 𝛼 2 + 𝑞 2 ) = 𝐶 ⇒ 𝛼 = √𝑍 1 𝑍 2 𝑒 2 𝑉 𝑐 𝐶𝜀 0
For SiO2, we have 𝑍 1 = 14 and 𝑍 2 = 8 the atomic numbers of Si and O, and 𝑉 𝑐 = 113 Å is the volume of the unit cell.

The speed of sound in the material 𝐶 𝑠 can be computed using the elastic theory from the longitudinal and transverse velocities 𝐶 𝑙 and 𝐶 𝑡 [71]:

3 𝐶 𝑠 = 2 𝐶 𝑡 + 1 𝐶 𝑙 Equation 3-35
𝐶 𝑙 and 𝐶 𝑡 can be obtained from the constants 𝐶 11 , 𝐶 12 and 𝐶 44 from the elasticity tensor by:

𝐶 𝑙 = √ 1 3 (𝐶 11 + 2𝐶 12 + 4𝐶 44 ) 𝜌 , 𝐶 𝑡 = √ 1 3 (𝐶 11 -𝐶 12 + 𝐶 44 ) 𝜌 Equation 3-36
Finally, the value of 𝑘 𝐵𝑍 can also be defined as the radius of a sphere with the volume of the first Brillouin zone [69], which gives 𝑘 𝐵𝑍 = (6𝜋 2 𝑉 𝑝 ⁄ )

1 3 [71] with 𝑉 𝑝 the volume of the cell.

The angular distribution is assumed to be isotropic in some works, such as Akkerman et al. [62]. This approximation can be justified by the fact that the transport of very low energy electrons becomes a random walk motion. However, this approximation may not be realistic for energies above 100 eV. In this work, we have also assumed an isotropic angular distribution, and this model replaces the PWA model in SiO2 for energies below 100 eV. Parameters are given in Table 3-5. The value of A has been modified, in comparison with the parameters given by Schreiber [58] (5*EBZ), to improve the acoustic MFP convergence to our PWA elastic MFP at 100 eV. For Al2O3, we have also computed the parameters following the different laws given in this section. The PWA elastic cross section obtained from the average atom approach was used to evaluate the deformation potential, the speed of sound was computed from the elastic constants, and 𝑘 𝐵𝑍 For this material, the switch between the PWA elastic model and the acoustic phonon models also happens at 100 eV. Due to the difficulty in finding the parameters of this model, another approach has been proposed by Ganachaud et al. [63,73,74]. They have proposed the following empirical cut-off function depending on the electron energy 𝐸, which multiplies the elastic cross sections at low energies by:

𝒌

𝑅 𝐶 (𝐸) = tanh (𝛼 𝑐 ( 𝐸 𝐸 𝑔 ) 2 ) Equation 3-37
With 𝐸 𝑔 the gap of the material, and 𝛼 𝑐 an adjustment parameter. The effect of this function is to increase the elastic MFP in a similar way to the acoustic phonon model. This is mandatory to avoid unphysically low values of the MFP, which will also result in particles getting stuck in place. We have used this function in place of the acoustic phonon model for boron nitride, as in the work of Chang et al [75], since we could not find some of the necessary parameters for the acoustic phonon model. For BN with a gap of 𝐸 𝑔 = 5.2 eV, the adjustment parameter has been taken as 𝛼 𝑐 = 0.5.

Empirical modeling of trapping using a polaronic interaction model

A basic model of trapping has also been proposed by Ganachaud et al. [63,73,74,76] for Al2O3.

They have assumed that the trapping of electrons is due to the polaronic effect [77][78][79], which is itself tied to the coupling of electrons with phonons. As we have seen, electrons can selflocalize by forming small polarons in the materials where they have a strong coupling with the lattice.

In this formalism, an electron of energy E has a probability (inverse mean free path) of becoming trapped given by:

𝑃 𝑡𝑟𝑎𝑝 (𝐸) = 𝑆 exp(-𝛾𝐸) = 1 𝜆 𝑡𝑟𝑎𝑝 (𝐸) Equation 3-38
𝑆 defines the amplitude of the trapping, and 𝛾 defines the energy domain where the process applies. If 𝑆 increases, the mean free path is reduced, and if 𝛾 increases, the mean free path starts to diverge at lower energies, reducing the trapping of electrons of a ten of eV. This model has been implemented in our simulations for Al2O3 and extended to SiO2. The parameters for these materials are given in Table 3-6. This gives a capture mean free path of 3.5 nm for electrons at 5 eV in Al2O3. When an electron is captured by the polaronic process, we assume that is has been lost permanently and it is removed from the simulation.

Material

𝑆 (nm -1 ) 𝛾 (eV -1 ) Al2O3 0.5 0.35 SiO2 0.1 0.2

Validation of the low energy transport model with reference data 3.3.1 Computation of stopping powers

The stopping power 𝑆(𝐸) [MeV/mm or MeV. cm 2 /g] for a particle of energy 𝐸 is related to the differential cross section according to the following relation:

𝑆(𝐸) = 𝑁 ∫ ℏ𝜔 𝑑𝜎 𝑑(ℏ𝜔) (𝐸, ℏ𝜔)𝑑(ℏ𝜔) 𝐸 𝑚𝑎𝑥 0 Equation 3-39
It can be used to check the validity of our modelling and fit of the ELF. The stopping powers given by the new version of MicroElec for silicon are plotted in Figure 3-11 The improvement of the Kaneko (B-K) [47] approach over the Barkas [83] formula for 𝑍 𝑒𝑓𝑓 is clearly visible below 30 keV/nucleon.

Ionizing dose calculations

The calculation of the ionizing dose-depth profile for an incident particle gives information on the energy deposited in the material and its uniformity. It can thus be used as another verification for the inelastic process. The ionizing dose given by the new version of MicroElec is compared in Figure 3-13 with data from SRIM and the standard physics processes of Geant4 which include continuous and multiple scattering processes. Both rejection sampling and direct sampling methods give the same results for MicroElec, but with significantly improved computation time for direct sampling. The good agreement between MicroElec, SRIM and the Standard physics models in the case of 10 keV protons is another validation of the inelastic processes. However, due to the exclusive use of discrete processes, MicroElec is about 4 times slower than the standard physics in this case, with a time difference that may vary depending on the configuration used. In the case of low energy electrons (the example of 200 eV electrons is shown here), the standard physics underestimate the range of electrons. Indeed, the range given by the stopping powers of Luo et al. [81], using the continuous slowing-down approximation for 200 eV electrons (8.9 x 10 -6 mm) is greater than the range given by the standard physics (5.5 x 10 -6 mm) and closer to MicroElec (1.1 x 10 -5 mm). There is in fact a plateauing behaviour of the extrapolated range under 1 keV. It has been illustrated with the code OSMOSEE in [21], and we will study the extrapolated range of electrons in more details in Chapter 4. Finally, the introduction of the initial energy for weakly bound electrons does not modify the dose profile. A more detailed comparison between the new version of MicroElec and the different Geant4 ionization models regarding dosimetry applications for microelectronics has been published in [START_REF] Inguimbert | Surface ionizing dose for space applications estimated with low energy spectra going down to hundreds of eV[END_REF].

Computation of the TEEY of metals and semi-conductors and validation with experimental data

The emission of secondary electrons from materials under electron irradiation is driven by several parameters: First, the TEY of silicon obtained with MicroElec is compared with simulated data from OSMOSEE [18] and experimental data from Bronstein [START_REF] Bronshtein | VTORICHNAYA ELEKTRONNAYA EMISSIYA[END_REF] in Figure 3 Option (b) shows how the initial energy of weakly bound electrons acts as a slight reduction of the TEEY while it did not have a significant effect on the dose profile. As a result, this energy can be used as a tweakable parameter to fit the TEEY to experimental data. It also shows that we need to take this energy into account in order to not over-estimate the energy of the secondary electron generated. The importance of surface processes is clearly visible in (c), with the total yield being even much higher than the experimental data (about a factor of 2). Finally, version (a) with all processes shows a good agreement with our experimental reference data [START_REF] Bronshtein | VTORICHNAYA ELEKTRONNAYA EMISSIYA[END_REF].


The yields are compared for all other metals and semi-conductors with experimental data from the SEY and BEY database of Joy [START_REF] Joy | A database on electron-solid interactions[END_REF], data from Bronstein [START_REF] Bronshtein | VTORICHNAYA ELEKTRONNAYA EMISSIYA[END_REF], and EEY measurements from ONERA [START_REF] Gineste | Investigation of the electron emission properties of silver: From exposed to ambient atmosphere Ag surface to ion-cleaned Ag surface[END_REF][START_REF] Balcon | Secondary Electron Emission on Space Materials: Evaluation of the Total Secondary Electron Yield From Surface Potential Measurements[END_REF] in Figure 3-15. Simulated data from other M-C codes, such as OSMOSEE [18] are also added as references. The TEEYs for metals and semi-conductors show good agreement with literature and experimental data. The TEEY of Kapton is also correctly modelled, even if we have not taken into account the phonon process as in the other insulators. This could be because the work function (4.7 eV) is greater than the energy gap (2.05 eV), so we do not have electrons that can go below the gap and still be able to escape. Hence, we do not have to model the energy losses below the band gap to compute the TEEY for Kapton.

Material

Computation of the TEEY of insulating materials without charging effects

First, the SEY of Al2O3 obtained from MicroElec is shown in Figure 3-16 with the phonon and polaron processes deactivated. We can see that the yield is completely overestimated and increases linearly with the incident energy. Indeed, more energetic incident electrons create more secondary electrons. However, due to the energy gap and the absence of models for the energy losses by phonons, any electron falling below the energy gap (8.8 eV) cannot lose energy anymore. Given the value of the electron affinity (0.5 eV), practically all low energy electrons can move in the material without losing any energy, and are able to freely escape from the surface. This gives the unrealistic yields seen in Figure 3-16. Then, we show in Figure 3-17 the SEY and BEY of SiO2 computed with the acoustic and LO phonon interaction models enabled, but without the polaronic capture model. The BEY for SiO2 is consistent with the data from Schreiber and Fitting [65]. However, a significant discrepancy can be seen for the SEY of SiO2 between MicroElec, the Monte-Carlo codes of Ohya et al. [START_REF] Ohya | Monte Carlo study of secondary electron emission from SiO2 induced by focused gallium ion beams[END_REF], Schreiber and Fitting [58], and the experimental data of Bronstein [START_REF] Bronshtein | VTORICHNAYA ELEKTRONNAYA EMISSIYA[END_REF]. An important point is that all the simulated SEY and BEY shown in this section do not include charge or recombination effects. The trapping of particles is only modelled empirically in our code and the code of Ganachaud & Mokrani [63], and has no impact on the successive electron cascades. This is equivalent to a fresh unirradiated and uncharged sample, which is not the case for experimental data related to insulators. The electron generation cascade can be disrupted by the trapping of low energy electrons, reducing the escape probabilities and the SEY. The creation and motion of positive and negative charges in the insulator can also generate internal and external electric fields which can dynamically evolve during the irradiation. These fields modify the trajectories of low energy electrons inside and outside of the material, and their penetration depths. As these effects significantly alter the electron cascade in insulators, this explains the discrepancy between simulated and experimental data.

Conclusion of Chapter 3

Several additions and improvements have been brought to MicroElec regarding the transportation of low energy electrons, protons, and ions in 16 materials (Be, C, Al, Si, Ti, Fe, Ni, Cu, Ge, Ag, W, Au, SiO2, Al2O3, Kapton, BN). The Energy Loss Function fitting model has been switched from the extended Drude model to the more accurate Mermin approach. The inelastic model for ions and protons is now valid down to 1 keV/nucleon, and the inelastic and elastic models for electrons are now provided down to a few eVs. Computation time is improved using cumulated DXS.

A model for the vacuum/material and material/material interfaces has been added, handling the interaction of electrons with the interfaces and the transition between the different materials of the simulation. In MicroElec, the Fermi level was initially chosen as a reference for the electron energies in the previous version of the code, which is why this reference was also retained for the computation of the crossing of the potential barrier. This will be corrected in a future version of MicroElec, since the formulas used in this work for the crossing of the surface followed the implementation in OSMOSEE, where the reference for metals was at the bottom of the conduction band. All the TEEY computations shown in this work for metals were made using an energy reference at the Fermi level. If the reference of energy in metals is changed to the bottom of the conduction band, a small modification of the energy of weakly bound electrons (for instance from 4.2 to 3.6 eV in Al) allows us to practically obtain the same TEEY curves as shown in this work. Therefore, taking a reference at the Fermi level does not invalidate the TEEY results for metals shown in this chapter, but this will be modified in a future version of MicroElec for physical consistency.

Although the module is slower than the standard physics of Geant4, it has an increased accuracy for low energy electrons under 1 keV. As the secondary electron production threshold and the low energy tracking limit for electrons are set to the value of the surface barrier, MicroElec can be used for secondary electron emission applications. The processes have been modified to allow the use of nine new supported materials. Although a simple model is used for material/material interfaces, it already allows the simulation of basic multi-layers. The refinement of the model and the addition of more oxides to the supported materials would open many possibilities for MicroElec within the study of surface or layer effects.

The new processes have been validated using multiple data. The inelastic process has been validated with the electron and proton stopping powers, which have shown good agreement with literature data. The low secondary electron production threshold in MicroElec also gives improved dose profiles. The secondary electron yield has been used to verify the complete transportation model for electrons, underlying the importance of the surface potential barrier for electron emission modeling. Satisfying agreement with experimental TEY data is observed, despite surface plasmons and exchange effects not being supported, which could decrease the accuracy of our calculations at low energies.

Finally, electron-phonon interactions have been added for insulators and used to simulate electron transportation in SiO2. The inelastic stopping powers and the BEY have been validated, but the SEY calculations do not match experimental data since charge and recombination effects are not modelled. Consequently, further developments are needed to extend MicroElec to the simulation of insulators by including the effects of charging. These developments will be presented in Chapter 5, where we have developed a new Monte-Carlo code for the simulation of charging effects in SiO2, based on MicroElec.

On the other hand, the Monte-Carlo code we have shown here is capable to compute the TEEY of several metals and semi-conductors. However, for some applications, an analytical expression of the TEEY can be more convenient to use instead of a full Monte-Carlo code. In Chapter 4, we will present an analytical SEY model we have developed, based on the data acquired by the Monte-Carlo model we have developed in this chapter.

Chapter 4: Development of an analytical electron emission model for metals and semi-conductors

As mentioned in the introduction, the TEEY is the driving parameter of many effects of the space environment on spacecraft systems. To quantify these risks, systems simulation packages need the SEY as an input parameter, for instance to compute the power thresholds to prevent multipacting in a certain geometry, or the evolution of surface potentials in a certain radiation environment. In the example of SPIS, some trials were made to combine the higher-level simulation packages with a Monte-Carlo module for the computation of the SEY. Nevertheless, this approach was very inefficient in computation time and resources. For this reason, most simulation packages use analytical expressions for the SEY, such as Dionne's model [1], Vaughan's model [2], or Furman & Pivi's model [3]. Even if these models can be freely fitted to an experimental SEY data set, their flexibility relies on the use of several arbitrary parameters and approximations that become unphysical for low energy electrons below a few keV.

The objective of this section is to derive an analytical secondary electron emission yield model that is based on the physics of low energy electron transport. In this regard, our Monte-Carlo code has been used to extract the electron transport parameters that will be studied in sections 4.1 and 4.2, that is to say the penetration depth, the ionizing dose-depth profile, and the transmission rate of electrons. The TEEY, which was our validation metric for MicroElec in the previous section, is dependent on all these parameters. Hence, the results obtained from the Monte-Carlo code should form a solid reference for the development of our analytical models.

We have then combined these parameters into a single analytical expression for the secondary electron emission yield, which will be presented in section 4.3. The analytical models have been validated with Monte-Carlo data (and experimental data for the SEY model) for 11 monoatomic metals and semiconductors: Be, C, Al, Si, Ti, Fe, Ni, Cu, Ge, Ag and W. These particular materials were selected because we have already modeled the transport of electrons in them with MicroElec, therefore we will be able to easily get reference data on these materials for our models. As we will see in this chapter, some essential quantities of our models are based on the atomic number, which is why we have not focused on compound materials such as Al2O3.

Gathering data on the range, transmission rate and dose of low energy electrons is especially critical, due to the low availability of experimental data for electrons below 10 keV. There is also practically no experimental data for electrons below 1 keV. This is because the penetration depths of these particles are about a few tens to a few hundreds of nanometers. Measuring the range of particles within a nanometric accuracy is particularly difficult, hence the interest of developing an analytical model for the range of electrons down to a few tens of eV.

The analytical models for the extrapolated range and transmission rate of low energy electrons were published during the thesis in ref. [4], and the ionizing dose model in ref. [5] in Applied Surface Science. A third paper on the SEY model has been submitted. The content of these papers has been reused in this section with the agreement of the publisher. Many empirical range-energy expressions have been proposed by several authors [6][7][8][9][10][11][12][13][14][15], describing the electron extrapolated ranges in various materials, aluminum being the most extensively studied material. Most of these relationships are in the form of a power law with the energy of the electron 𝐸 in MeV:

𝑟(𝐸) = 𝑘 * 𝐸 𝑛 Equation 4-1
Katz and Penfold [6] have made a very thorough compilation of experimental results for aluminum, and have proposed an empirical formula for the extrapolated range of electrons between 10 keV and 3 MeV. In this formula, widely used in the past, the n factor in Equation 4-1 is a function of the energy of the electrons 𝐸 in MeV:

𝑛 = 1.265 -0.0954 ln (𝐸) Equation 4-2
With 𝑟(𝐸) in mg/cm², 𝑘 = 412. Sometimes 𝑛 also depends on the material [12]. Weber [7] proposed a different expression, valid for aluminum in the energy region [3 keV, 3 MeV]. Kobetich and Katz [8] extended this model to the 0.3 keV-20 MeV range by adjusting the constants: The same authors [9] proposed further improvements for this formula, by introducing a dependence on the atomic number Z of the material for the three parameters: Most of the other models found in the literature suppose that the extrapolated range tends to zero when the electron energy decreases towards zero, by following a power law as in Equation 4-1 [16][17][18][19][20][21]. However, contrary to the Weber expression extended by Kobetich and Katz (Equation 4-3), they are not relevant on the whole energy range. This expression has also the advantage of being able to either express the extrapolated range 𝑟 as a function of the energy E (𝑟(𝐸)) or inversely E as a function of the range (𝐸(𝑟)). For these reasons, we have chosen to use this expression as a basis for our extrapolated range model, with the objective of improving this formula for very low energy electrons down to a few tens of eV. For this, we need to take into account the strong diffusion that low energy electrons undergo when traveling in the material, which also creates a significant dispersion in the penetration depths. Given that the penetration depths of sub-keV electrons is about a few tens of nanometers, we cannot neglect this dispersion and use the same assumptions as for higher energies. As we will show in the next section, the high-energy formula fail to reproduce the behaviour of the extrapolated range below 1 keV.

𝑟(E) = 𝐴𝐸 [1 - 𝐵 (1 + 𝐶𝐸) ]
𝐴 = (

Study of the extrapolated range and transmission rate of low energy electrons with Monte-Carlo simulations from MicroElec

To understand why the expressions valid for high energy electrons fail to model the behaviour of low energy electrons especially below 1 keV, we propose in this section to study the transmission rate and penetration depth using our Monte-Carlo model.

First, the transmission rates should be computed as they will be needed to deduce the extrapolated range, following the method described in section 1.5.4 of Chapter 1. We remind here that the implantation depths of electrons are first extracted from the Monte-Carlo simulations and sampled as a depth distribution. The distribution is then integrated and normalized to get the repartition function, which is the transmission rate. Finally, the tangent at 0.5 is taken and followed until its intersection with the X axis, the value obtained at the intersection being the extrapolated range.

In the following, the implantation depths for incident electrons between 25 eV and 5 keV have been sampled from MicroElec simulations and used to calculate the transmission probability as a function of the incident electron energy. They are shown in Figure 4-1 for beryllium, aluminum, iron and silver, which have respective densities of 1.85, 2.7, 7.87 and 10.5 g/cm 3 . At low energies, the penetration depths are comparable for all four materials, around 1-2 nanometers. However, as the energy increases, the penetration depths, expressed in nm in Figure 4-1, increase more quickly for low density materials. This effect is visible in Figure 4-1 as the transmission curves become less widespread when the material density increases. Indeed, for 2 keV electrons the penetration depth can reach hundreds of nanometers in aluminum and beryllium, but only 40 nm in iron and silver. Moreover, the intervals between the transmission rate curves of all four materials become very small for electron energies below 100 eV. This implies that the extrapolated ranges extracted from these transmission curves should decrease more slowly below 100 eV, as will be studied next. In Figure 4-2, the extrapolated ranges 𝑟 computed with MicroElec, which uses discrete processes, are compared with extrapolated range computations made with the continuous processes of Geant4's standard physics list (Opt4) for aluminum and silicon. This physics list is designed for precise transportation of electrons on a wide range of energies. For energies under 10 keV, the PENELOPE continuous energy loss ionization model is used with a multiple scattering model for elastic interactions. Monte-Carlo simulated data from Colladant et al. [16], Akkerman et al. [17] and experimental data from Kanter & Sternglass [18] are also displayed, but the available experimental data for the extrapolated range of low energy electrons below 1 keV is very scarce. Finally, the range given by equations ( 5) and ( 7) (Weber Formula), and the CSDA range (𝑆 ̅ ) obtained from MicroElec's stopping powers (in green, labeled CSDA) for both The extrapolated range curves show a typical behavior below a few hundreds of eVs. They are no more proportional to the incident energy, and the range/energy function stabilizes as the range reaches a plateau region whose height depends on the material. This effect is more visible on the MicroElec curve and can also be seen on the CSDA curves computed from the MicroElec stopping powers, which become parallel to each other at lower energies. The level of this plateau region changes from a material to another as a function of the relative values of the inelastic and elastic mean free paths.

One can notice that, for both materials, the CSDA ranges (𝑆 ̅ ) obtained from Equation 4-4 with MicroElec's stopping powers converge to about 2 times the extrapolated range 𝑟 𝐸 obtained from MicroElec simulations. However, the CSDA gives values which are much higher than the simulated data sets below a few keV. This can be explained by the fact that the CSDA is a maximum range which neglects the deflection induced by the elastic and inelastic interactions, and would only be attained by a hypothetical electron with a strictly linear trajectory. As this is never the case for electrons, this parameter is an unreachable limit for the actual range 𝑟. Indeed, below 100 eV, the elastic scattering becomes prevalent over the inelastic scattering for electrons. This can be shown qualitatively by calculating the probability 𝑃 𝐸𝑙 that the interaction made by the electron is an elastic interaction. It is obtained from the total cross sections (𝜎) as:

𝑃 𝐸𝑙 = 𝜎 𝐸𝑙𝑎𝑠𝑡𝑖𝑐 𝜎 𝐸𝑙𝑎𝑠𝑡𝑖𝑐 + 𝜎 𝐼𝑛𝑒𝑙𝑎𝑠𝑡𝑖𝑐
As can be seen in Figure 4-3, 𝑃 𝐸𝑙 increases strongly below 50 eV, where energy losses become rarer, and approaches 1 below 20 eV. This behavior can also be seen for other Monte-Carlo codes, as mentioned by Pierron et al. [19]. But, as mentioned previously, this range is a maximum depth that will not be reached by most electrons. Indeed, at these energies the angular distribution of the elastic scattering becomes quasi-isotropic (Figure 1). Consequently, the transportation of low energy electrons below 50 eV becomes a case of random diffusion: they are highly scattered without any energy loss until an inelastic interaction occurs, then leading the particle to come at rest. Hence, the electron can travel a longer total path before being stopped, while remaining close to the surface. However, we can see that the CSDA range, which does not take into account the elastic diffusion, becomes parallel with the extrapolated range at lower energies. This shows that the elastic interaction does not change the dynamic of the range. It only reduces the values of the projected range by diffusing the trajectories of the particles, and the intensity of this reduction becomes larger as the energy and elastic MFP decrease. Hence, the flattening dynamic could rather be caused by the divergence of the inelastic MFP.

The plateauing effect at very low energies does not seem to be reproduced by the Geant4 continuous processes (Std Phys Opt4). Indeed, this physics package uses multiple scattering instead of discrete elastic models, which can also generate more differences. Above 1 keV however, the Geant4 continuous processes give similar values to MicroElec. This behaviour can be linked with the transmission probabilities in Figure 4-5, in the example of silicon. The intervals between the transmission curves computed with MicroElec become narrower as the energy decreases, as seen for the other materials in Figure 3. However, the transmission curves obtained with the standard physics keep a spread of about the same order at lower and higher energies on the lin-log plot. This implies a linear (on a log-log plot) decrease of the extrapolated range for the whole energy range of 25 eV-10 keV seen in Figure 4-2. The comparison is given here as an indication of the differences which can occur at very low energies, when using different interaction models with different approaches (continuous vs discrete) and different mean free paths. We have already conducted a more extensive comparison between the different physics models of Geant4 for low energy dosimetry applications, which can be found in ref. [20].

Analytical expressions for the extrapolated range and transmission rate

Above some keV, Equation 4-1 reproduces faithfully the Monte-Carlo simulations and experimental data. However, the commonly used power law expressions are no longer applicable below 1 keV. As shown in Figure 4-2, Equation 4-1 does not reproduce the dynamic of the extrapolated range below 1 keV, with a linear (log/log scale) evolution in place of the flattening phenomenon observed in the simulations. Thanks to the Monte Carlo simulations shown in the previous section 4, this expression has been modified in order to be relevant down to 10 eV. In our new expression, the model of ref. [9] in Equation 4-3 is maintained for electron energies over 14.5 keV, as it is able to correctly model the dynamic of the range over this energy. This energy correspond to the points where our new expression best fit the formula of ref. [9]. Below this threshold, a power law function replaces the expression of the extrapolated range 𝑟(𝐸):

𝑟(𝐸) = { 𝐷(𝐸 + 𝐸 𝑟 ) 𝐹 ∶ 10𝑒𝑉 < 𝐸 ≤ 14.5𝑘𝑒𝑉 𝐴𝐸 [1 - 𝐵 (1 + 𝐶𝐸) ] : 𝐸 ≥ 14.5𝑘𝑒𝑉 Equation 4-5
With the following parameters:

𝑟 𝐴𝑙 = 3 * 10 -7 𝑔 𝑐𝑚 2 ⁄
is the extrapolated range of 50 eV electrons in aluminum obtained from the MicroElec simulations, 𝐸 0 = 14.5 𝑘𝑒𝑉 𝐴 = (1.06 𝑍 -0.38 + 0.18) 10 -3 𝑔 𝑐𝑚 2 ⁄ . 𝑘𝑒𝑉 𝐵 = 0.22 𝑍 -0.055 + 0.78 𝐶 = (1.1 𝑍 0.29 + 0.21) 10 -3 𝑔 𝑐𝑚 2 ⁄ . 𝑘𝑒𝑉 Although the range below 50 eV is over estimated and the agreement with the simulations is decreased for low Z (Be, C) and high Z (W) materials, overall a satisfying agreement with the simulations is observed. Indeed, below 100 eV, the average difference between the model and the simulation is between 3% and 12% for all materials, except for Be with 18%. With 𝑟 𝑍 (50 𝑒𝑉) the extrapolated range of 50 eV electrons in the material Z. The values for 𝑟 𝑍 can be extracted from the Monte Carlo simulations. They are more representative of the actual differences between the ranges of the materials and give a better estimation of G. The ranges shown in Figure 4-7 use the G values from the simulations. Alternatively, G can also be extracted from the ranges available in the literature. However, for energies this low, most available data are CSDA ranges which are not representative of the random walk of electrons. In spite of this, the data is available for many materials, like in the case of the stopping powers of Shinotsuka [21] that can be used to get the CSDA range and G parameter for 41 materials. Moreover, the dynamic of the CSDA range is similar to the one of the extrapolated range, as shown in Figure 4-2, due to the divergence of the inelastic mean free paths which strongly increase below a few tens of eVs. The 𝐹(𝑍) factor defines the slope of the extrapolated range 𝑑𝑟 𝐸 /𝑑𝐸 between 1 and 10 keV, so that the curve reproduces the dynamic of the reference ranges. The values for F and G in both cases are shown in Figure 4-8 shows the correlation between the values of G(Z) and F(Z) for each material with the atomic number of the material 𝑍. The strong correlation on Figure 4-8a shows that the height of the plateau region for a material is strongly dependent on the atomic properties of the material. Indeed, the range of low energy electrons below 100 eV tends to be higher in high Z materials, as shown in Figure 4-7. However, it is not trivial to provide a definite explanation for this correlation. At low energies, many properties depending on the Z or the material may influence the height of the plateau region, such as the number of conduction electrons per atom. This number is lower in Cu and Ag than for neighboring metals which could explain why their G values are slightly lower than expected. We can also observe that both Si and Ge have lower plateau heights than other materials with a close Z. We could then suppose that this discrepancy is due to the fact that these materials have an energy gap, which plays an important role in the energy loss process. Thus it is difficult to conclude on the origins of this dependence on Z and the observed discrepancies for some materials, and a more extensive study would be required.

𝐸 𝑟 = 𝐸 0 ( 𝐴𝐸 0 [1 - 𝐵 (1 + 𝐶𝐸 0 ) ] 𝐺 𝑟 𝐴𝑙 -1) 1 𝐹 𝐷 = 𝐴𝐸 0 [1 - 𝐵 (1 + 𝐶𝐸 0 ) ] (𝐸 0 + 𝐸 𝑟 ) 𝐹
Figure 4-8b shows a more limited correlation between the slope of the range curve (F) and the atomic number, as F varies in a narrower range than G. This correlation is rather given as a starting point to extrapolate the parameter F and extend the model to other materials than the 11 shown here. This can also be done for G using the correlation relationship between Z and G. However, some fitting work may still be needed to improve the agreement with the reference data. The comparison between the analytical model (dotted lines) and MicroElec (solid lines) for all 11 materials can be found below. As in the case of the extrapolated range, a satisfying agreement is seen between the model and the simulations, although the agreement is degraded for very low and high Z materials (in the case of Be and W), and very low energies below 20 eV.

Development of an analytical model for the ionizing dose deposited by low energy electrons

It is also possible to derive a model for the ionizing dose deposed by primary electrons of a certain incident energy arriving at the surface of the material, following an analytical approach.

The ionizing dose is defined as the energy transferred by these electrons to the material, which can then generate secondary electrons. This is in opposition to the non-ionizing dose, where the energy deposed does not generate additional particles, but can instead result in the displacement of an atom from its position in the lattice. Therefore, in insulators, electrons with an incident energy below the bandgap are not able to depose ionizing dose in the material. In the following, we will first derive a general formula for the dose deposited in a layer of thickness 𝑑ℎ, located at the depth ℎ of a material irradiated with a flux 𝜑 of incident electrons having an energy 𝐸. The amount of energy 𝑑𝐸 deposited in this layer of thickness 𝑑ℎ is given by the subtraction from the amount of energy arriving at the depth ℎ of the amount of energy leaving the volume at the depth ℎ + 𝑑ℎ. The amount of energy transported at the depth ℎ is also proportional to the transmitted fluence 𝜑 * 𝜂(ℎ, 𝐸), 𝜂(ℎ, 𝐸) being the transmission probability of the electrons of energy E through a material of thickness ℎ. It is also proportional to the energy of the particle at the depth ℎ, which is 𝐸(r -ℎ), where the extrapolated range 𝑟 is the maximal distance traveled by the particle. Hence, 𝑟 -ℎ is the remaining distance that the particle has to travel after having previously travelled the distance ℎ in the medium. This gives the expression for the energy deposited at ℎ: 𝜑 ⋅ 𝜂(ℎ, 𝐸) ⋅ 𝐸(𝑟 -ℎ)

Equation 4-9
Similarly, the amount of remaining energy at the depth ℎ + 𝑑ℎ is given by:

𝜑 ⋅ 𝜂(ℎ + 𝑑ℎ, 𝐸) ⋅ 𝐸(𝑟 -(ℎ + 𝑑ℎ)) Equation 4-10
The substraction of these two terms leads to the general expression of the dose deposited at depth ℎ, which is proportional to the flux of electrons:

𝑑𝑜𝑠𝑒(ℎ) = 𝜑 𝑑[𝜂(ℎ, 𝐸) ⋅ 𝐸(𝑟 -ℎ)] 𝑑ℎ Equation 4-11
What is noticeable here is that the dose is, for electrons of energy E, a simple function of both the transmission probability 𝜂(ℎ, 𝐸), and the extrapolated range vs. energy expressions 𝑟(𝐸) (or its inverse function 𝐸(𝑟 -ℎ) in Equation 4-11) of these electrons. We must also note that this expression only describes the energy transferred to the material by a primary electron traveling deeper into the material. It does not take into account the retrodiffusion of primaries, or the diffusion of this energy through the subsequently created secondaries. The objective of the model is to give the energy deposed by an incident electron before the generation of secondary electrons, hence we will not be taking into account the diffusion of this energy induced by secondaries. However, the retrodiffusion of the primary electron needs to be taken into account to have a good estimation of the dose, which is why we will introduce some corrections later in this section.

We have already shown in section 4.1 that most range and transmission rate models become invalid below 1 keV, due to the omission of elastic scattering. For these reasons, we will use the analytical models we have developed in section 4.1, to propose an analytical formula for the dose profile of low energy electrons.

4.2.1 Analytical expressions of the ionizing dose from the low energy range and transmission rate models 4.2.1.1 Derivation of an initial expression for the ionizing dose For higher energies, above 14.5 keV, the extrapolated range expression of Kobetich & Katz (Equation 4-3) can be used to calculate the ionizing dose-depth profile, leading to a different formula from the one we will develop here based on our models. This high energy relationship will be shown in the comparison between the models in section 0 and compared to the low energy model presented below. In the following, the ionizing dose will be expressed as a function of the depth, and given for a single incident energy 𝐸. Therefore, this expression can be used over the thickness of the material to give the dose-depth profile. Some of the quantities of the model are also dependent on the incident energy, which will be displayed as a subscript [ ] 𝐸 below the quantity. For instance, the extrapolated range 𝑟(𝐸), which depends on the incident electron energy, will be noted 𝑟 𝐸 .

Under some keV, the domain of energy of interest for secondary electron emission and surface analysis, the dose-depth profile for a given fluence can be directly derived from Equation 4-11 our models for the range-energy relationship in Equation 4-5 and the transmission rate in Equation 4-8. This gives the expression for the dose per fluence as:

𝐷𝑜𝑠𝑒 𝐸 (ℎ) = 𝑑 𝑑ℎ (𝜂 𝐸 ⋅ 𝐸(𝑟 𝐸 -ℎ)) = 𝐼′ 𝐸 (ℎ)𝜂 𝐸 (ℎ) -𝐼 𝐸 (ℎ)𝜂 𝐸 ′ (ℎ) Equation 4-12
Where the transmission rate 𝜂 𝐸 (ℎ) is given by Equation 4-8, and 𝐼 𝐸 (ℎ) = 𝐸(𝑟 𝐸 -ℎ) is the inverse function of the range expression of Equation 4-5. Knowing that the incident electron has an extrapolated range of 𝑟 𝐸 , it gives the remaining energy of this electron after having travelled the distance ℎ in the material. 𝐼 𝐸 (ℎ) can be deduced from Equation 4-5 as:

𝐼 𝐸 (ℎ) = 𝐸(𝑟 𝐸 -ℎ) = [( 𝑟 𝐸 -ℎ 𝐷 ) 1 𝐹 -𝐸 𝑟 ] Equation 4-13
The expressions of the derivatives 𝐼′ 𝐸 (ℎ) and 𝜂 𝐸 ′ (ℎ) can be written as follows:

𝐼′ 𝐸 (ℎ) = 𝑑𝐸(𝑟 𝐸 -ℎ) 𝑑ℎ = [ (𝑟 𝐸 -ℎ) 1 𝐹 -1 𝐹𝐷 1 𝐹 ⁄ ] Equation 4-14 𝜂 𝐸 ′ (ℎ) = 𝑑𝜂 𝐸 (ℎ) 𝑑ℎ = - 𝑝𝑞 𝑟 𝐸 ( 𝑞ℎ 𝑟 𝐸 ) 𝑝-1 𝑒 (- 𝑞ℎ 𝑟 𝐸 ) 𝑝 Equation 4-15
Combining all these expressions leads to a pure analytical expression for the dose depth profile.

Corrections to the initial dose model expression: modeling of the retrodiffusion of primaries and improvements for electrons below a few hundred eV

The initial expression we have derived (Equation 4-12 combined with eqs. 13-15) is found to slightly underestimate the dose near the surface. Indeed, we have assumed that all primary electrons can only travel forward into the material. However, a part of the primary electrons can be inelastically backscattered, and leave the material. These electrons will only transfer a part of their energy to the material before exiting it, this energy being deposited near the surface. Another part of the incident fluence can also be elastically backscattered, in which case the electrons will not lose energy in the material. As a result, some corrections have been brought to Equation 4-12 to take into account these effects and better fit the dose-depth profiles.

For a given depth ℎ, a retrodiffusion factor 𝜁 𝑅𝑒𝑡𝑟𝑜 has been added to reduce the energy deposited by the electrons moving forward into the material. The removed part is compensated by the energy deposited at the depth ℎ by the electrons reflected from deeper into the material, which are traveling back to the surface. The effect of this factor is to redistribute the dose towards the surface and simulate the electrons which have escaped the material but deposited a significant part of their initial energy. The Equation 4-12 becomes:

𝐷𝑜𝑠𝑒(ℎ) = 𝐼 ′ 𝐸 (ℎ)𝜂 𝐸 (ℎ) -[(1 -𝜁 𝑅𝑒𝑡𝑟𝑜 )𝐼 𝐸 (ℎ)𝜂 𝐸 ′ (ℎ) + 𝜁 𝑅𝑒𝑡𝑟𝑜 ∫ 𝐼 𝐸 (𝑧)𝜂 𝐸 ′ (𝑧) 𝑑𝑧 𝑟 𝐸 ℎ ] Equation 4-16
The retrodiffusion factor has been chosen as 𝜁 𝑅𝑒𝑡𝑟𝑜 = 0.1. We can interpret the two parts of this equation as follows: 𝐼 ′ 𝐸 (ℎ)𝜂 𝐸 (ℎ) is the remaining energy of the electrons that are transmitted through h, and 𝐼 𝐸 (ℎ)𝜂 𝐸 ′ (ℎ) is the energy deposited by the electrons that have stopped at h. The dose deposited at h is therefore the difference between the energy deposited by the electrons stopped at h, and the energy kept by the electrons transmitted through h. The aim of the retrodiffusion factor is to reduce the energy deposited by electrons deeper in the material, so in this case we want to reduce 𝐼 𝐸 (ℎ). Then, the energy deposited by the backscattered electrons when they stop is redistributed through the integral.

At very low energy, around some tens of eV, the approach of Equation 4-11 does not apply anymore because the transport regime changes to a kind of random walk motion governed by the elastic process that becomes dominant. Indeed, the inelastic mean free path starts to increase very significantly at energies close to the plasmon energy, as we have already shown in Figure 4-3 of the previous section. Equation 4-11 overestimates the deposited energy, which can become higher than the incident energy. To correct that, below a cutoff energy chosen equal to the plasmon energy of the target material, the ionizing dose have been simplified, following

𝑑𝐸 = 𝜑 ⋅ 𝐸. 𝑑𝜂⋅ 𝑑ℎ
. And the dose-depth profile is simplified to:

𝐷𝑜𝑠𝑒 𝐸 (ℎ) = -𝐸 • 𝜂 𝐸 ′ (ℎ) if 𝐸 ≤ ℏ𝜔 𝑝 Equation 4-17
This formula avoids any overestimation of the deposited dose at very low energy (E< ħp). In the intermediate energy range [ħp, ~300eV], in order to connect smoothly both Equation 4-12 and Equation 4-17, a linear combination of these two formula is proposed:

𝐷𝑜𝑠𝑒 𝐸 (ℎ) = [ 𝐸 -ℏ𝜔 𝑝 𝐸 𝐿𝑜𝑤 -ℏ𝜔 𝑝 ] 𝐼 ′ 𝐸 (ℎ)𝜂 𝐸 (ℎ) -[𝐸 𝐸 𝐿𝑜𝑤 -𝐸 𝐸 𝐿𝑜𝑤 -ℏ𝜔 𝑝 + 𝐼 𝐸 (ℎ) 𝐸 -ℏ𝜔 𝑝 𝐸 𝐿𝑜𝑤 -ℏ𝜔 𝑝 ] 𝜂 𝐸 ′ (ℎ) Equation 4-18
Where 𝐸 𝐿𝑜𝑤 = 𝐸 𝑟 /2, with Er from the extrapolated range model of Equation 4-5. This expression can be used in the energy range (ħp <E < 𝐸 𝑟 /2). Depending on target material, Er/2 corresponds to energies of a few hundreds of eV.

Modeling the contribution of the backscattered electron yield to the ionizing dose

The deposited energy expressions obtained so far are defined for any electrons entering a target solid. But as we have shown when studying the energy distribution of electrons exiting the material, a part of the incident fluence is elastically backscattered and does not deposit any energy in the material. The deposited dose, which is equal to the product of the incident fluence by the energy loss, must be reduced from the amount of reflected electrons. The proportion of incident electrons entering the irradiated material is equal to 1 -𝐵𝐸𝑌 𝐸 , 𝐵𝐸𝑌 𝐸 being the Backscattered Electron Yield for an incident energy 𝐸. To summarize, the contribution of the elastically backscattered electrons for a given energy must be removed from the three relationships for 𝐷𝑜𝑠𝑒 𝐸 (ℎ) (Eqs. [16][17][18] by removing the BEY. The dose-depth profile is thus obtained by the final expression:

𝐷𝑜𝑠𝑒 = 𝐷𝑜𝑠𝑒 𝐸 (ℎ) * (1 -𝐵𝐸𝑌 𝐸 ) Equation 4-19
In this work, we propose an expression for the BEY depending on the material and the incident electrons' energy. It is based on data from MicroElec Monte-Carlo simulations, where we have computed the proportion of backscattered electrons. The BEY is modeled by two single values for low and high energies, which are linearly connected in the intermediate region.

For materials with Z > 22, we have to reduce the BEY value used in the model at higher energies to get a better agreement with the Monte-Carlo simulations. In this case, we have used the elastic yield, which is the fraction of the BEY that only contains the incident electrons that are elastically backscattered. This choice was made to improve the agreement of the dose peofiles with the reference data. We can suppose that for higher Z materials, the backscattered electrons are able to travel deeper into the material and lose more energy. This effect is only partially included in the model, by the use of the factor 𝜁 𝑅𝑒𝑡𝑟𝑜 in Equation 4-16. Consequently, this modification of the BEY is required to reproduce the Monte-Carlo simulations, which compute the full backscattering process for the incident electrons.

The BEY values with their domain of validity are given in Table 4-2. Between the low energy and high energy values, a linear fit is applied. For W, a unique value has been used. The dose-depth profiles, calculated with this model, are first compared with Monte Carlo simulation results from MicroElec, in an aluminum target for energies ranging from 10 eV up to 2000 eV. To get these dose profiles, only the primary electrons have been simulated. The secondary electrons, which would have been created from the energy deposited by the incident electrons, have not been generated in the simulation. As the resulting electronic cascade is not simulated, the dose is independent on the SEY. This allows us to get the dose-depth profile per incident electron fluence. Indeed, the elastically and inelastically backscattered electrons are also simulated, and their contribution is thus included in the dose. 50 000 incident electrons have been simulated for each energy, with a computation time of about 1 min per energy.

As can be seen in Figure 4-10 the analytical model is also in quite good agreement with the Monte Carlo simulation of Walker [22] (9%) at 2 keV. The dispersion between the data of Walker, OSMOSEE [23], MicroElec and the analytical model is higher at 500 eV (25%) than 2 keV (11%), which can be attributed to a difference in the mean free paths used by the different Monte Carlo codes. Indeed, the authors of ref. [22] indicate that the dose is given per primary electron without modeling the secondary electron cascade, in the same way as MicroElec. The dose profiles given by this work's model for the rest of the materials are provided below in Figure 4-11. The analytical formulation of the deposited energy is in relatively good agreement with the Monte Carlo simulations for most materials. The agreement is less satisfactory for lower energies, which can be linked to the limitations of the models used. Indeed, the error increases on average up to 50% at 50 eV and below, up to a factor 2 in worst cases. This is due to the fact that the transport of electrons at very low energies becomes a random-motion walk due to the predominance of the elastic interactions. When an inelastic interaction happens, the electron deposits all its energy on a single point and comes to rest. As a result, the amount of energy deposited between h and h+dh cannot be approximated by a continuous energy loss anymore, which can explain the discrepancies between the model and the simulation at 50 eV and below. For the same reason, the dose-depth profiles have been validated for depths above a few angströms only. Indeed, a depth of a few angströms becomes very close to the atomic distances and the notion of a deposited dose in a volume becomes questionable in both the analytical model and the Monte-Carlo simulations. The quantity of backscattered electrons and the energy they can deposit near the surface may also vary with the incident energy.

Nevertheless, the average error decreases down to 30% at 100 eV. At 250 eV and above, the error for all materials is about 15% on average, and always less than 30%. Consequently, despite the approximations of the model, we can still consider that the agreement with the simulations is satisfactory above 50 eV, and the shape of the dose-depth profiles are generally well reproduced. At 50 eV and below, the model reaches its limits, but still gives a correct reproduction of the depth reached by the very low energy electrons and the dose is estimated within the right order of magnitude, as can be seen in the figures above.

Comparison between Geant4

Monte-Carlo models (MicroElec and GRAS(em_lowenergy))

Our Monte-Carlo reference code (MicroElec) can be compared with the other electromagnetic physics modules of GEANT4. In Figure 4-12, the dose profiles of electrons in Si given by MicroElec and GRAS [38] are plotted. In this study, GRAS [24] is used with the Geant4 em_lowenergy electromagnetic physics, which are continuous processes with a condensed history and multiple scattering approach. MicroElec however uses discrete processes, which are slower but more precise at low energies where the step lengths become nanometric. At 2 keV and above, both models give similar dose profiles. Below 1 keV, the electron energy gets closer to the low energy limit of GRAS, and the dose below a depth of 2E-6 g/cm² given by GRAS is much higher than MicroElec. MicroElec is able to give the dose profiles of electrons down to a few eVs, which is an improvement over the low energy limit of GRAS (250 eV). The comparison between MicroElec (dots) and em_lowenergy GRAS (lines) for the other materials studied in this work can be found in Figure 4-13, and a more detailed comparison between MicroElec and other Geant4 models can be found in ref. [20]. MicroElec has been chosen as the reference for the analytical dose model due to its ability to transport electrons below 250 eV down to a few eVs.

Comparison between the low and high energy models

The high energy model proposed in ref. [11], which includes the range expression given by Equation 4-3 , can be used to calculate the ionizing dose leading to a formula different from our model of Equation 4-16 based on the relationships of 4.1. In the same way as in the low energy model (section 3.2), the ionizing dose in the high energy model can be expressed as:

𝐷𝑜𝑠𝑒 𝐸 (ℎ) = 𝑑 𝑑ℎ (𝜂 𝐸 ⋅ 𝐸(𝑟 𝐸 -ℎ)) = 𝐼′ 𝐸 (ℎ)𝜂 𝐸 (ℎ) -𝐼 𝐸 (ℎ)𝜂 𝐸 ′ (ℎ) Equation 4-20
Where

𝑟 𝐸 = 𝐴𝐸 [1 - 𝐵 (1+𝐶𝐸)
] is the high energy range expression from Equation 4-3. This gives the inverse range function 𝐼 𝐸 (ℎ):

𝐼 𝐸 (ℎ) = 𝐸(𝑟 𝐸 -ℎ) = 1 2𝐴𝐶 [(𝑟 𝐸 -ℎ)𝐶 -𝐴(1 -𝐵) + √∆(𝑟 𝐸 -ℎ)]
And its derivative The expressions of the transmission probability and its derivative remain the same as Equation 4-8 and Equation 4-15 respectively. However, the values of p and q are changed, following the expressions given by Kobetich & Katz [9]: At 10 keV, the maximum difference between the high energy analytical model and MicroElec is 40%, and the shapes of the dose profiles are similar. However, the maximum error for the low energy dose model is 20%, thus it can still be considered acceptable. At 14.5keV, which corresponds to the limit of validity of the low energy model, it underestimates the peak of the dose with an error of 40%. However, the dose deposited near the surface is still more accurate with the low energy model. Indeed, the high energy model is based on the first assumption of Equation 4-12 which does not consider the backscattering process. For the energies below 10 keV, the dose given by MicroElec is significantly underestimated by the high energy model, and the improvements brought by the low energy model are clearly visible.

𝐼′ 𝐸 (ℎ) = 𝑑𝐸(𝑟 𝐸 -ℎ) 𝑑ℎ = 1 2𝐴𝐶 [-𝐶 + 1 2√∆(𝑟 𝐸 -ℎ) (2𝐶(𝐴(1 -𝐵) -(𝑟 𝐸 -ℎ)𝐶) -4𝐴𝐶)]
𝑞 =

Development of an analytical model for the secondary electron emission yield

As the ionizing energy released by incident electrons in irradiated materials is dissipated in the form of secondary electrons, the analytical model presented in the previous section can be used to develop simple expressions of secondary emission yield models. The proportion of electrons produced close to the surface of the solid and escaping from the material can also be estimated by combining the dose analytical expression with the extrapolated range and the transmission rate models described previously. This approach will be followed in this section. Here, we propose a semi-empirical model based on a similar approach as Dionne's model [1,25]. The simplifying assumptions used by Dionne have been left out to obtain more accurate calculations. This leads to a more complex formulation which cannot be reduced to a simple analytical formula. But in counterpart, the proposed model has an angular dependency and is a function of a limited number of parameters having a physical meaning: the work function (Wf), the average energy of the inelastic recoil electrons <Es> and the mean energy I lost by the primary particle to create a secondary electron.

Principle of calculation of the secondary electron emission yield

The secondary electron emission yield can be separated in two different processes [26][27][28]. In a first step, during the slowing down of the incident electrons, secondary electrons are produced as the result of the interaction between the primary beam and the lattice electrons. The second step addresses the transport of the secondary electrons to the surface as well as its crossing.

If one calls 𝐺(ℎ, 𝐸) the number of secondary electrons produced within a thickness dh at a distance h from the surface and p(h) the probability that these secondary electrons reach the surface and are emitted into the vacuum, The amount of secondary electrons produced at a given depth that escapes from the solid is given by:

𝐺(ℎ, 𝐸) • 𝑝(ℎ) • 𝑑ℎ Equation 4-21
The secondary emission yield is then simply the sum of the produced electrons from the surface down to the penetration depth of the incident electrons, ℎ 𝑒 (𝐸). The SEY can be estimated by the following integral expression:

𝑌 𝑆𝐸 (𝐸) = ∫ 𝐺(ℎ, 𝐸) • 𝑝(ℎ) • 𝑑ℎ h e (𝐸) 0 Equation 4-22
ℎ 𝑒 (𝐸) can be evaluated with the extrapolated range (practical range) of the incident electrons [4,29]. The generation term 𝐺(ℎ, 𝐸) is a function of the number of inelastic interaction induced by the incident electron with the electrons of the medium. It is proportional to the ionizing dose and then to the average energy loss per unit path length of the incident electrons (dE/dh). The escape probability of secondary electrons p(h) can be approximated with a quite good accuracy with an exponential function [1,25,29]: 𝑒 -𝐶 2 .ℎ (C2 being a fitting parameter). In order to integrate the Equation 4-22 and get an analytical expression of the SEY, Dionne [1,25] assumed a constant energy loss along the path of the incident electrons. According to that approximation, the generation term dE/dh is simply given by the following affine function:

    E h E h dh dE e  Equation 4-23
If we assume that the range follows Equation 4-1 [30], the Equation 4-22 can be integrated and leads to the following expression for the SEY:

𝑌 𝑠𝑒 (𝐸) = 𝑐 1 𝑐 2 𝐸 ℎ 𝑒 (𝐸) (1 -𝑒 -𝑐 2 ℎ 𝑒 (𝐸) ) Equation 4-24
𝐶 1 is a constant used to express the deposited dose, which is a deposited energy by unit of mass of target material. But 𝐶 1 and 𝐶 2 are arbitrary parameters used to fit the model to experimental SEY data. The maximum of SEY (𝐸 𝑚𝑎𝑥 , 𝑌 𝑠𝑒,𝑚𝑎𝑥 ) is commonly used to define 𝐶 1 and 𝐶 2 . These models are valid only for the calculation of the "true" secondary electron emission yield. It does not take into account the contribution of backscattered electrons (elastic & inelastic), even if it is used to fit the experimental total emission yield. However, the amount of backscattered electrons can be implicitly taken into account by adjusting the C1 parameter. The main assumption at the basis of this formula, i.e. a constant energy loss function, prevents this model to reproduce faithfully the shape of the experimental SEY.

In this work, we propose to improve the approach used by Dionne [1,25], in order to get a more accurate model that will depend on physical parameters that could be measured experimentally or deduced from Monte Carlo modelling. It aims at providing not simply a fitting formula but an expression based on physical parameters. In addition the approach proposed here have a relevant angular dependency which is not the case of former expressions. For that we can rewrite the Equation 4-22 as the following general expression:

       dh E w p h I E h Dose E Y s f w E E h se s e , , ) ( 0      Equation 4-25
In that expression, the generation term is given by the integral of the ionizing dose over the path of the primary electron divided by the mean ionizing potential I, to get the amount of secondary electrons produced between h and h+dh. The escape probability of the secondary electrons is separated in two terms: <Es>(h) is the probability for the secondaries produced at depth h to reach the surface. As we saw in section 4.1, the transmission probability depends on h and the energy of the secondaries <Es>. The electrons that reach the surface have then a probability to cross the potential barrier given by pw(wf, <Es>). It depends also on the energy of the secondaries, and on the work function wf. We will assume in the rest of the paper, that the average energy of secondaries that reach the surface of the material is identical to the average energy of all secondaries. This hypothesis is supported by the fact that the secondary emission is driven by the secondary electrons produced near the surface. A large part of these electrons will be able to leave the material without producing any additional interaction and therefore without energy loss.

The goal is then to elaborate analytical expressions combining the dose depth profile and the two different transmission probability <Es>(h) and pw(wf, <Es>). It is possible to formulate such kind of analytical functions for Dose(h, E), <Es>(h) and pw(wf, <Es>), but it will be very difficult to get an expression, product of these three functions that could be integrated. The final Yse function will be expressed as an integral. Is the transmission rate of the secondary electrons created at the depth x with a mean energy <Es>, from our model in Equation 4-8. In Equation 4-26, we assume that the generation of secondary electrons is isotropic, hence the factor ½ applied to 𝜂(〈𝐸 𝑠 〉, 𝑥) since only half of the electrons are moving towards the surface.

Analytical expression of the SEY based on our low energy range, transmission rate and ionizing dose models

Dose(x) is obtained from our expressions in section 4.2.1, that is to say Equation 4-16, Equation 4-17 or Equation 4-18 depending on the incident energy. Assuming that the secondary electrons are produced isotropically, the probability of these electrons to reach the surface is simply given by the transmission probability through a thickness x divided by 2, to take into account the fact that only half of the isotropically produced electrons move in the direction of the surface. No factor is applied to the dose as it is already given per incident electron, while the elastically backscattered electrons have already been removed from the expressions of the dose profile.

pw(wf, <Es>) is the crossing probability of electrons through the surface potential barrier wf, for the mean secondary electron energy <Es> inside of the material. The crossing probability is limited by two phenomena: the quantum reflection by the potential barrier, and the limit angle θlim above which the electron is reflected by the surface. As only electrons with an energy greater than the work function are considered here, the reflection remains negligible and the crossing probability is mostly affected by the limit angle:

          E w E f arcsin lim  Equation 4-28
Assuming that the secondary electrons crossing the surface are isotropically distributed, by integrating the flux of secondary electrons from 0 up to lim the crossing probability becomes:

                                s f s s f w E w E E w p arcsin 2 cos 1 2 1 ,

Equation 4-29

Using all our models, Equation 4-26 is only valid for energies below 14.5 keV. It is also defined for normal incident electrons. But the dependence on the incident angle is implicit. It can be taken into account by replacing simply the depth x by x/cos() in the integral. That leads to the following expression for the SEY, which should be used in place of Equation 4 This model depends on a number of different physical parameters that can be defined with more or less ease. The work function wf can be found in literature for many materials. The average energy of recoil electrons <Es> and the mean ionizing energy I required to produce a secondary electron are more difficult to estimate. MicroElec Monte Carlo simulations show that <Es> can vary, for the tested materials, from around 10 eV up to 35 eV. Moreover, this energy varies as a function of the incident energy, and depending on the generation of secondary electrons that are considered. For the first generation of SEs, the median energy can be linked to the energy loss function of the material. The optical energy loss functions (OELF) from the database of Sun et al. [32] for Al (small Z metal), Si (small Z semi-conductor), Ge (higher Z semiconductor) and Ag (higher Z transition metal) are shown in Figure 4-15. The median energy of 1 st generation SEs is generally close to the plasmon energy when a unique plasmon peak is present in the OELF, as in the case of Si or Al. For other materials however, generally transition metals, the OELF is characterized by a plateau region where energy losses to plasmon and valence band transitions are superposed, as in the example of Ag. In this case, the median energy is close to the center of the plateau, though its value increases with the energy of the primary electrons. Indeed, a wider range of the OELF becomes accessible to the primary electrons when their energy increases, the limit being the energy of the primary electron.

In our case, in order to limit the number of parameters and to get an amount of ionized electrons representative of the population of emitted secondary electrons, the energy 𝐼 lost by the primary electrons when creating a secondary electron has been assumed equal to the mean energy of secondary electrons 〈𝐸 𝑠 〉. Both parameters have been set to the median energy of the 1 st generation SEs, which corresponds to the most probable transferable energy for the incident electrons, so that the model remains consistent with the OELF of the material. Nevertheless, our approach has some limitations and approximations which need to be compensated by the introduction of the parameter 𝜅 in the SEY. First of all, the angular distribution of the secondary electrons generated is not considered in the expression of the transmission probability. The model assumes that all electrons put into motion towards the surface follow a normal direction to this surface, which can lead to an overestimation of the SEY.

Moreover, we have used single values for 〈𝐸 𝑠 〉 and 𝐼, whereas the energy of the 1 st generation SEs generated by the primary particle should vary according to the depth. Indeed the primary electron is slowed down by the previous interactions, so the secondary electrons produced deeper into the material may not have the same energy as the ones produced near the surface. What's more, these 1 st generation SE can create other subsequent generations of secondary electrons. As a result, the median energy of all escapable SEs when all generations are considered becomes lower. This is shown in Figure 4-16 where the median energies of the 1 st generation SEs are compared with the median energy of all subsequent generations (2 nd to n th gen). What is noticeable here is that the median energies of the 1 st gen SE for Si and Al reach a limit value after 200 eV, while the energies for Ag and Ge continue to increase. This is linked to the shape of the OELF, where a plateau is present in the OELF of Ag, while the OELF of Al and Si have a single plasmon peak, and the OELF of Ge has a wider peak. Finally, the simulations show that the proportion of 1 st generation SEs that have escaped from the surface varies between 40 and 60% of all escaped SE, depending on the material and the energy of the primary electrons. Consequently, the energy of the secondary electrons in the escape probabilities and the energy used to divide the dose are very complex to model as their dynamic can be different for each material. Moreover, the energy distribution of the SEs that reach the surface may also be different from the mean SE production energy we have just focused on. A single value for 〈𝐸 𝑠 〉

and 𝐼 is limiting but much simpler to use, hence the introduction of a factor 𝜅 to compensate for all the approximations. The values for this factor have been chosen so that the model is in better agreement with the SEY data from Monte-Carlo calculations with MicroElec. The values for all parameters used in the model can be found in Table 4-3. The values of the work functions 𝑤 𝑓 have been chosen from the values commonly found in the literature, such as Lin & Joy [33].

Validation of the SEY model with Monte-Carlo and experimental data

In this section, the SEYs given by the analytical model will be compared with MicroElec simulations [4,34], and experimental data from Bronstein & Fraiman [35] and from Joy's database [33,36]. The SEY increases with the angle of incidence because the ionization is produced closer to the surface, increasing the probability for the secondary electrons to escape from the material. We can also notice that the maximum of yield also shifts to higher energies as the incident angle increases. This behavior can be linked to the location of the maximum of deposited dose, i.e. the depth at which the production of secondary electrons is maximum. This can be explained by the fact that, at normal incidence, the depth at which the maximum of dose is deposited is large enough, to prevent most of the electrons coming from this depth to escape the material. By tilting the incident beam, the peak of dose is brought closer to the surface, increasing the amount of secondary electrons able to escape the surface. The consequence is a shift of the energy of the electrons at which the maximum of SEY is reached.

In Figure 4-17 the comparison is made for Al for angles between 0° and 75° between Monte Carlo simulations and our analytical approach. The agreement is quite good from normal to 60° incidence, and the shape of the secondary emission yield is reproduced faithfully. The SEY is overestimated at 75° however, which shows the limitations of the model. The comparison of the analytical SEY with Monte Carlo simulations and experimental data of Bronstein and Joy are shown in Figure 4-18 for all materials in the case of a normal incidence. The secondary emission yield of "true secondary" electrons given by the analytical model and MicroElec [5,34] is shown in this set of figures for Be, C, Al, Si, Ti, Fe,Ni, Cu, Ge, Ag, W. In experimental references [35,36] however, the SEY is evaluated by counting electrons having an energy lower than 50 eV. Within this assumption inelastic backscattered and secondary electrons are mixed. This can be a source of small deviation with the model. Moreover, due to the surface state of the sample and the measurement conditions which can both be variable, there is an important spread between the experimental SEY data for some materials. In this case, multiple data sets have been used as a reference, to verify the overall agreement of the analytical model. Some small discontinuities can be seen on the model curves, which correspond to the transitions between the different dose (0.5Er) and transmission probability (2 keV) expressions.

The maximum of the curve is predicted at the good energy for most materials. A shift can be observed on the SEYs of Be and C, while the SEY is overestimated for W after the maximum of the SEY. This can be linked to the limitations of the other analytical expressions used in the SEY model, as they hit their limitations when used for low-Z and high-Z materials. The same observation can be made for the values of . The factor is close to 1 for transition metals but approaches 2 for the lower and higher Z materials.

Overall, with regard to the assumptions of the model, a fairly good agreement is found with the Monte-Carlo data, which was expected as it was used as the reference for the calibration of the other models used in the SEY analytical model and the SEY model itself. The agreement is also satisfying with the experimental SEY from Bronstein and Fraiman and Joy's data. Even if the data of Bronstein and Fraiman have been measured on samples evaporated under vacuum, and thus are considered free from oxidation and contamination, the surface state of the samples may be relatively different from the "ideal" flat surface simulated numerically. The surface state is known to impact significantly the SEY. Both Monte Carlo and analytical models ignore the roughness of the surface which could be very important to consider. In addition, it should be recalled that the model depends also on the backscattered emission yields from MicroElec. The BEYs have been chosen as a couple of values, each value covering a given energy domain. The dependence of the BEY on the incidence angle [5] is also not simulated, which is obviously not realistic. Finally, while the inelastically backscattered electrons are simulated in the expression of the dose, it is done by a very simple model which can be refined. The development of a backscattering emission yield model is thus necessary to get a more relevant model for the total emission yield. Nevertheless, the simulations and the experimental data of Bronstein and Joy can be considered in relatively good agreement for these materials, according to these considerations.

Comparison of key SEY parameters with Monte-Carlo simulations

The maximum SEYmax and its energy value (Emax) have been extracted from both Monte Carlo and model SEY vs. energy curves. The factor 𝜅, whose values are shown in Figure 4-19, was applied to improve the amplitude of the SEY curves and compensate for the approximations of the model. Nevertheless, one can see from the values of 𝜅 that the error in amplitude between the analytical model and the Monte-Carlo simulations without this correction is between 5% and 30% for most materials, which is satisfying given the approximations and simplifications of the analytical model. The reason of such good correlation can be related to the range of low energy electrons, and in particular to the plateau region. This region and the knee of the range/energy curve (Figure 4567) are particularly important to accurately model the maximum of SEY, which is located around some hundreds of eV. The accuracy of the range/energy curve is closely connected to both the accuracy of the deposited ionizing dose and the electron transmission probability. The height of the plateau of the range/energy curve, which is given by the G(Z) factor of our extrapolated range model (Equation 4-5), closely defines the location of the knee of this curve and the shape of its derivative (dr/dE). Indeed, G appears in the parameters D and Er of the range expression of Equation 4-5, and thus in the derivative in Equation 4-14. The quantity F is also tied to the height of the plateau. In fact, when modifying F the height of the plateau is changed, which then changes the slope of the range between the plateau region below 100 eV and the linear region above 1 keV (Figure 4567). Since the slope is changed, the derivative is also changed, and so is the dE/dr. The higher the plateau, the greater the dE/dr, this latest being proportional to the ionizing dose. The amount of secondary electrons is itself proportional to the ionizing dose. Analogously a high range at low energy pushes the knee of the range energy curve toward higher energies. This knee is clearly correlated to the location of the maximum of SEY.

To show this correlation, the G(Z) factor has been plotted as a function of SEY data from Monte-Carlo simulations: the maximum SEY and the energy of the maximum of SEY. These figures depict a linear behavior. In the case of the max SEY value, carbon is an outlying point since its Z is low but the max SEY is about 1.3 (Figure 4-18). This is an exception in the evolution of the SEY with Z, as the next material to have a SEY>1 is Fe with a Z=26. The correlation is improved from 0.47 to 0.85 when C is not considered. Likewise, W is an outlying point for the position of the max SEY, which is due to the fact that the uncertainties and discrepancies of the range and dose models are more important for high Z materials. Although in this case, the correlation is still good without removing W, at 0.77. Despite these two points being exceptions and showing the limits of the model, we can still consider for the remaining materials that there is a strong correlation between the SEY and the range level of low energy electrons (<1 keV). This parameter depends closely on the range/energy relationship that shall be known with a good accuracy. This latest parameter can be defined only knowing with a good accuracy the electron transmission probability. But knowing these parameters seems to be sufficient to make a rough estimation of the secondary electron emission for different materials. The approach of the dose model used in this work (based on the transport of low energy electrons) can be confronted to the hypothesis used by Dionne [1,25] of a constant energy loss along the penetration depth of incident electrons, where the dose is uniformly deposited on the whole range of the primary particle. In this hypothesis, the energy loss function is given as:

𝑑𝐸 𝑑ℎ (ℎ) = 𝐸 𝑟(𝐸) Equation 4-31
Where the range follows a power law based on the Continuous Slowing Down Approximation:

𝑟(𝐸) = 𝐸 𝑛 /𝛽𝑛 Equation 4-32
and 𝛽 and n are individual fitting constants which need to be determined for each material. Hence, the dose-depth profile in this approximation is given by: In Figure 4-23, the dose given by the constant loss model is compared with the analytical dose model used in this work, in the example of Cu. The parameters 𝛽 and n for the constant energy loss have been chosen respectively as 5.43 and 1, following the work of Plaçais et al. [30] who have fitted Dionne's model to the SEY of experimental Cu samples. As can be seen in Figure 4-23 the assumption of the constant energy loss is not able to precisely evaluate the dose depth profiles, despite it being able to give accurate SEYs. As a result, the surface dose is significantly underestimated. The constant energy loss dose profile is flat and has the same value for all electron energies, as a result of the choice of parameters by the authors of ref. [30]. They have chosen a value of 𝑛 = 1, which removes the dependence in energy of Equation 4-33. While the maximum ranges given by the continuous energy loss model are in good agreement with the analytical model and Monte-Carlo data at 500 eV and 1000 eV, they are underestimated at 100 and 50 eV. Indeed, below 500 eV the range does not follow a power-law dynamic anymore (Figure 4-6), so the power law in Equation 4-33 becomes invalid. Consequently, the constant energy loss approximation is less realistic than the approach proposed in the analytical dose model used here.

We can also study how the SEY can be correlated to the total amount of dose deposited within the escape depth of electrons, which will be referred to as "surface energy deposit". Indeed, the quantity of energy deposited in the first few nanometers of the material is the result of a compromise between the incident energy available to generate the secondaries and the capability of the medium to stop incident particles near the surface (density of the material). At very low energy (<~100 eV) the penetration distance of electrons of increasing energy remains constant.

In the plateau region of the range/energy function (Figure 4-7), electrons of increasing incident energy deposit more and more energy in a constant thickness. Consequently, the energy deposit increases rapidly as a function of the incident energy. The maximum of surface energy deposit is reached when the practical range function starts to become steeper (~ hundreds of eV, ). After this stage, the deposition is spread deeper in the target material, and as a consequence, starts to decrease near the surface.

Considering this, the energy of the maximum of SEY should be close to the energy where the dose deposited within the escape depth of electrons is also maximal. From the transmission rates of However, there is a much weaker correlation (0.56) between the amplitudes, as shown in Figure 4-24(b). Indeed, there is a significant dispersion between the maximum SEY and the maximum surface energy deposit on the same figure. Plotting 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑒𝑛𝑒𝑟𝑔𝑦 𝑑𝑒𝑝𝑜𝑠𝑖𝑡 /𝐼, which is the generation term for the first 2 nanometers, also does not give a correlation with the max SEY.

If we plot the analytical SEY and compare it to the surface energy deposit in Figure 4-25 in the example of a few materials, we can see that the amount of deposited energy closely follows the evolution of the SEY. A similar observation has also been made by Pierron et al. [23] for Si and Al where they showed that the percentage of deposited energy remaining at the end of the electron cascade is inversely proportional to the SEY, since the other part has been dissipated in the form of secondary electrons. This also highlights the importance of having a precise evaluation of the dose in the first few nanometers for SEY calculations, which as we have seen on Figure 4-23, is not made possible by the constant energy loss approximation. The same observation regarding the lack of correlation between the amplitudes can be made in Figure 4-25, where the difference in amplitude between the dose and the SEY varies for each material (the SEYs and doses have been plotted using the same axis scales).

In conclusion, this shows that the other models for the transportation of electrons, namely the transmission and surface crossing probabilities, are of crucial importance in order to get an accurate amplitude of the SEY. Indeed, the secondary electrons generated from the surface dose will have different energies and different probabilities to reach and cross the surface, which could significantly vary between the different materials. This also shows that generation of electrons also needs to be correctly estimated in the integral by the upper limit he(E), hence the knowledge of the surface energy deposit alone is not sufficient for the amplitude of the SEY. 4.4 -Discussion of the approach: Limits at very low energies

Discussion of the approach: Limits at very low energies

At very low energies and for low projected path lengths, the notions of an average range, transmission probability or average dose become debatable. Indeed, at higher energies, many inelastic interactions can be made by the electrons in a unit path length dx. The ranges of electrons follow a Gaussian distribution with a well-defined average range and a limited spread.

A single value of the range can thus be extrapolated and be used as a representative parameter for an individual electron. When the electron energies become very low, their projected path lengths reach the interatomic distances, they are highly scattered by the elastic interactions and they are only able to make a couple of inelastic interactions before coming at rest. Subsequently, the number of interactions made per unit path length dx is very small and the projected path length distribution has a significant spread. Crucially, the depths reached by very low energy electrons become close to the interatomic distances (a few angströms), which is another limitation for the number of interactions. The notion of a continuous function for the transmission probability through a few atomic layers may also become debatable. Finally, the Monte-Carlo code itself reaches its limits for very low depths and energies. Indeed, the material cannot be treated as a bulk material anymore and the use of the dielectric function theory as in the case of a bulk material is questionable.

To sum up, while the ranges and transmission rates evaluated at high energy are representative of the average path for an individual electron through a unit path length dx, the same extrapolation cannot be made at low energies where these quantities become statistical. Instead of an average electron, these parameters are applicable to a flux of a large number of electrons, where the global range and transmission rate of the flux should follow the models detailed in this work. An analogy can be made with the case of photons going through a certain thickness.

A photon is fully absorbed by the material at the first interaction it makes, as in the case of very low energy electrons. Consequently, the transmission probability derived for photons is not representative of the path of each individual photon but of the flux of photons as a whole.

Conclusion of Chapter 4

In this chapter, we have proposed an analytical model for the computation of the secondary electron emission yield. The model is based on a physical approach, which required us to also develop analytical models for the different processes of electron transport that are involved in the secondary electron emission.

First, Monte-Carlo simulations of the penetration depths of electrons have been performed for 11 materials and used to compute the transmission rates and extrapolated ranges of electrons.

An analytic formula for each of these two quantities is proposed, depending on the atomic number and two parameters that are specific for each material. The simulation results have been used to calibrate these expressions. A correlation for the material-dependent parameters F and G can be established with the atomic number Z. In the case of the plateau height G, a fairly strong correlation has been found, indicating that the behavior of very low energy electrons (below 100 eV) is strongly dependent on the atomic properties of the material. These correlation laws can be used to extend the model to new materials.

We have then proposed a dose model based on the energy loss process of electrons in matter.

Since the model is given per incident electron, corrections have been introduced to simulate the elastically and inelastically backscattered electrons. These are respectively the backscattered electron yield of the material, and the computation of the energy deposited by the electrons traveling back to the surface corrected by a retrodiffusion factor. This also allows the model to better reproduce the dose-depth profiles and give a better estimation of the surface dose induced by the inelastically backscattered electrons. The accuracy of the model has been checked with MicroElec and other Monte-Carlo simulations for 11 materials. The model is less accurate at 50 eV and below, and some discrepancies can be observed depending on the energy or the material, but on the whole the dose-depth profiles are faithfully reproduced by the model. The agreement with the reference data from MicroElec is acceptable (error less than 30% above 100 eV), as well as the one with the other Monte-Carlo codes (Walker, OSMOSEE, GRAS) above 1 keV. This low energy analytical model is also a definite improvement over the high energy model below 10 keV.

Finally, these analytical models were combined in an expression for the SEY. The comparison of the predicted SEY with Monte Carlo and experimental data is in satisfactory agreement for the 11 studied materials (Be, C, Al, Si, Ti, Fe Ni, Cu, Ge, Ag, W). This work has also demonstrated the very strong correlation of the SEY (SEYmax and Emax) with the range of very low energy electrons (plateau region of the range/energy curve). This work highlights the fact that the SEY needs a more accurate description of the transportation of low energy electrons, than the hypothesis of a constant energy loss used in some analytical SEY models.

However, this analytical approach presents some strong limitations. In principle the models can be applied on any monatomic target material. But, the parameterized functions they depend on shall be validated for each new material. The SEY model depends also on the knowledge of the average energy of secondary electrons, which is a quite difficult parameter to determine. This parameter has been defined in this work by the use of Monte Carlo simulations, though this allowed us to show that the average energy of secondaries <Es> can significantly vary according to the primary electron energy. However, we have also shown that <Es> is strongly dependent on the shape of the Energy Loss Function (ELF), which dictates the energy loss distribution. As a result, <Es> can be chosen close to the statistical median of the ELF.

To get a better prediction of the total emission yield, a more accurate expression for the backscattering of electrons must also be added to the model presented here, in order to get all the different contributions and the variation in energy. Finally, the surface state of the material is known to strongly affect the SEY, which is neglected by this model that is only valid for ideal materials having no contamination, no oxidation and no roughness. In principle, the model can be directly extended to stacks of different materials. It could be used to account for oxidized or contaminated surfaces. These are various possible improvements of the model, which are considered for future developments. Nevertheless, the analytical models presented here can be used as input data in other simulation codes, and provide an improved computation time compared to the Monte-Carlo models.

Finally, this model is a good basis for an extension to compound or layered materials, as its current version has been validated in this paper for monoatomic and monolayer materials only.

The effect of surface roughness can also be included in the form of geometrical models [37,38]. These improvements will allow the model to be adapted to the SEY of technical materials.

Chapter 5: Development of a low energy charge transport model to simulate the secondary electron emission and charge buildup in silicon dioxide

Introduction

With the analytical model presented in Chapter 4, we now have another tool that can be used for the study of secondary electron emission. The parameters of this model can be modified to model the emission yield of insulators. However, it cannot simulate the evolution of the TEEY caused by the charge buildup in the material. The Monte-Carlo model we have developed in Chapter 3 can also simulate the TEEY of a fresh insulator sample in the static case, but it is also unable to simulate the evolution of the TEEY according to charging. As the goal of this PhD thesis is to model the experimental TEEY measurements on insulators that are affected by the charge buildup, we will have to extend either of these models with new charge transport models.

Various works have followed either of these approaches, with silicon dioxide being by far the most studied material. This means that several of the parameters that may be needed by our model (trap density, capture cross section…) should be more available for this material. Moreover, we already have a modeling of the transport of low energy electrons in SiO2 down to the electron affinity of 0.9 eV, and the simulated TEEY in the static case is coherent with other simulations. For these reasons, we have chosen to develop our charging simulation model for SiO2 only, and we will focus on this material only in this chapter and Chapter 6. However, we shall create a model that is as general as possible, so that it can be extended to other insulators if the simulation parameters can be found. Where 𝐹(𝑥, 𝑡) is the electric field, 𝜌(𝑥, 𝑡) is the charge density, 𝐽(𝑥, 𝑡) is the current density of charges, 𝜎(𝑥, 𝑡) is the conductivity, and 𝑆(𝑥, 𝑡) is a source term.

These models are faster than Monte-Carlo codes, however their description of the charge transport is more macroscopic, since they simulate the charges densities globally. The interactions of the charges are also not modeled explicitly. They are contained in macroscopic terms of the equations that describe the variation of charge density with time, since this variation depends on the individual interactions such as the trapping of the secondary electrons, or the generation of holes by the inelastic interactions. On the other hand, our objective is to understand the effect of the charge buildup on the electronic cascades, the transport of electrons and the secondary electron emission. Consequently, we have rather chosen to use a Monte-Carlo code, which allows us to follow the particles individually and precisely model each single interaction made by the charges in the material. As for the analytical approach, many Monte-Carlo codes can be found in the literature for the simulation of secondary electron emission and charge transport. Nevertheless, as we mentioned before, they focus on the external and global charge effects, whereas we are concerned in this study by the internal charging effects and their influence on the transport of electrons.

In this regard, we will have to model the transport of the thermalized electrons, contrary to our current Monte-Carlo model which stops the transport of electrons when they cannot overcome the surface potential barrier. We will also have to model the transport of holes created during inelastic interactions, which were completely neglected before. The electric field generated by the charge density influences the drift of these very low energy particles, so it needs to be computed and taken into account in the simulation. For this, we can take advantage of the builtin field classes of Geant4 to apply an electric field to particles, but we will still have to compute the field. Finally, for all particles, the trapping, detrapping and recombination processes need to be implemented. In essence, by simulating the transport of the charge carriers, we want to model the radiation induced conductivity created by an incident electron cascade and its influence on the subsequent cascades.

In this chapter, we will present a new Monte-Carlo code based on the Geant4 Monte-Carlo model developed in Chapter 3 for SiO2. This code follows an iterative approach according to time, since the computation of the field and the charge densities is done dynamically. Indeed, we need to simulate the evolution of the charge buildup if we want to see the evolution of the TEEY with time. Hence, instead of a single simulation of the TEEY over several energies, we have to model the transport of incident electrons of a single energy over a given time, by simulating successive electron cascades. Most importantly, the result of the previous electronic cascade N-1 are reused and will have an impact on the next electron cascade N, so the different iterations of the simulation are not independent. This is in strong contrast with the approach of Geant4, where simulation runs are completely independent. Nevertheless, this iterative approach is mandatory in order to correctly model the effect of internal charging on the TEEY and its evolution in time. During this PhD thesis, we have also made experimental measurements of the TEEY on SiO2 thin film samples, including time-resolved measurements for the validation of our model. We will compare the results of the model developed in this chapter with the experiments in Chapter 6, and use our model to explain the experimental observations.

In this chapter and Chapter 6, we now make the distinction between ballistic electrons and drift electrons. The incident electrons and the secondary electrons generated in the material will be referred to as ballistic electrons. They are tracked until they escape the material and the electron detector surrounding the sample collects them. The transport of ballistic electrons is also stopped if their energy falls below the surface potential barrier (0.9 eV). Hence, ballistic electrons always have an energy above 0.9 eV. They follow all the interaction models shown in Chapter 3 and are subjected to the elastic, inelastic, surface interaction, optical phonon and acoustic phonon processes. If a ballistic electron becomes unable to escape, we consider that it is thermalized and becomes a drift electron. These electrons follow a different regime of transport known as drift, which is dominated by phonon collisions and trapping effects. Their energy comes from the thermal agitation, and they can be accelerated by an electric field. The energy of the drift electrons is related to the Boltzmann distribution of speeds, but we can consider that they have an energy of 3/2 𝑘𝑇 on average. This gives an energy of 40 meV at room temperature (300 K) in the absence of an electric field. Due to electrostatic conservation, we also simulate the transport of positively charged holes in the material. They are assumed to follow the same drift regime than the drift electrons. In the following, we will also be mentioning many charge densities, defined in Table 5-1 𝑛 will be defined as the number of charges stored in the counters of the simulation. 𝑁 is the number of trapped charges per cm 3 , which can be multiplied to a cross section 𝜎 (cm 2 ) to get a mean free path for trapping or recombination processes. Lastly, the charge densities 𝜌 used in Poisson's equation are expressed in C/cm 3 The simulation configuration is presented in Figure 5-1. A flat rectangular sample of SiO2 on a Si substrate is placed in a spherical electron collector, set to the ground. The thickness 𝐿 of the SiO2 layer can be freely adjusted. For practically all simulations, we have used a thickness of 20 nm, which is the thickness of the experimental samples. However, bulk materials of a few µm can also be simulated. An electron gun set to the ground is sending electrons with a normal incidence on the sample, with a distance ℎ between the gun and the surface. The reference for the 𝑧 axis is the surface of the sample. Positive 𝑧 values are in the material, while negative values are in vacuum above the surface.

The dielectric sample is formed by a 1D mesh in depth, which is used for the computation of the electric field and the sampling of the charge densities. The 1D approximation can be considered valid, as the surface irradiated by the gun (cm²) is much larger than the thickness of the samples (20 nm) studied here, so that the radial field will be negligible compared to the field in depth. The mesh is made of a series of nodes 𝑧 𝑖 spaced by an interval ∆𝑧 𝑖 = 𝑧 𝑖+1 -𝑧 𝑖 , from the surface of the material (𝑧 0 ) down to the contact between the SiO2 and the Si layer (𝑧 𝑛 ). The nodes define the cells of the mesh as 𝐶 𝑖 [𝑧 𝑖 ; 𝑧 𝑖+1 [. We store the densities of charge carriers 𝜌 𝑖 trapped between 𝑧 𝑖 up to 𝑧 𝑖+1 in the i th cell of the mesh 𝐶 𝑖 .

For the computation of the electric field, we assume that all charges in the cell 𝐶 𝑖 are located on the i th node 𝑧 𝑖 of the mesh. The density of charges 𝜌 𝑖 is then plugged into Poisson's equation to get the potential at a given node 𝑉 𝑖 :

( 𝜕𝐹(𝑧) 𝜕𝑧 ) i = ∆𝑉 𝑖 (𝑧 𝑖 ) = - 𝜌 𝑖 (𝑧 𝑖 ) 𝜖 0 𝜖 𝑟 Equation 5-2
The mesh also has a surface 𝑆 𝑚𝑒𝑠ℎ that needs to be defined accordingly, since it will be used for normalization of the charges in the computation of volumetric densities. We want to model an electron gun that has current densities from 10 -8 to 10 -5 A/cm², which can irradiate a surface from 0.1 cm² up to a few cm², and our samples have a diameter of a few cm. Consequently, it seems pertinent to use an elementary surface 𝑆 𝑚𝑒𝑠ℎ = 1 cm² for the normalization.

The sample holder can be biased to a set potential in the experiment, generally +27 V or -9 V in DEESSE. This potential, along with the potential of the electron gun (0 V) and the behavior of the electric field at the surface (Gauss' law), need to be taken into consideration as initial conditions for the resolution of Poisson's equation. The discretization of this equation will be presented in section 5.2.2.

The mesh used in the simulation is made of about 200 nodes for a 20 nm sample, which gives an average step of 1 Angstrom per node, and the first node of the mesh is set to be at 0.5 Angstroms from the surface. The rest of the nodes are logarithmically spaced. This allows us to have a very fine mesh close to the surface, which is where most secondary electrons are produced. Indeed, we need cells that have less than a nm of thickness in the first few nm of the surface since most holes are created there, but we do not need this amount of refinement past the escape zone of the secondary electrons, which is around 10-15 nm in our simulations of SiO2. Hence, we can use a much broader mesh for the rest of the material, particularly beyond the region of implantation of the primary electrons. With this logarithmic distribution, the mesh broadens as we move deeper into the material. This is especially useful in the case of bulk materials, where we can have several hundreds of nm without any deposed charge, or for electrons of a few keV, which have an implantation region that is much deeper (several tens or hundreds of nm) than the production region of the secondary electrons (10 nm). The general procedure of the iterative Monte-Carlo simulation follows these 4 phases:

1. The incident electrons are emitted by the electron gun with a normal incidence to the surface. The transport of the ballistic electrons is simulated including all interaction models from Chapter 3. A ballistic electron is stopped when it escapes from the material and hits the detector, or if its energy falls below the surface potential barrier, which means that it is unable to escape. In the latter case, we save the position of the ballistic electron; it is assumed to be thermalized and will become a drift electron in the following steps. Ballistic electrons can be captured by free traps, or recombine with trapped holes. The trajectories and energies of the electrons are modified by the electric field following the classical equation of the electrostatic force (𝑎(𝑧) ⃗⃗⃗⃗⃗⃗⃗⃗ = 𝑞 𝑚 * 𝐹(𝑧) ⃗⃗⃗⃗⃗⃗⃗⃗⃗ ), and using the built-in tools of Geant4 for field handling. Holes are also created when an inelastic interaction happens and a secondary electron is generated. The thermalized electrons and new holes are added to two stacks of "new drift holes" and "new drift electrons". They will be used for the computation of the charge densities, and as a list of charges to be transported in the next steps.

2. When the tracking of all of the ballistic electrons is finished, the positions of the created holes and thermalized electrons resulting from the electron cascade are sampled in depth along the 1D mesh. The densities of trapped holes and electrons from the previous cascades are also sampled. These densities are updated for the computation of the capture probabilities, which depend on the density of free and occupied traps. The total charge density obtained is used to update the external and internal electric field, using Poisson's equation. This equation is discretized using a resolution scheme along the 1D mesh.

3. The detrapping of the trapped electrons and holes is computed according to the detrapping probability. By multiplying the probability of detrapping for a given trap in a given cell of the mesh by the number of trapped charges in this cell, we can obtain the number of charges detrapped from the cell.

4. The transport of the drift holes and electrons is simulated. The positions of creation of the holes and of thermalization of the electrons during the electron cascade of phase 1 were saved to compute the charge distribution, so these charges are generated here at their exact position of thermalization (for electrons) or creation (for holes). In the first version of the simulation, the exact position of the trapped charges was also saved, and they were generated from their exact position of capture. However, this created an excessive consumption of memory, which made the program crash if too many trapped particle positions were stored (more than 20 million). This would systematically happen when irradiation times of several tens of ms were simulated. To address this issue, the current version of the program only stores a counter of the number of charges trapped in a given cell of the mesh. As we know which cell a given particle was detrapped from, its position of generation is randomly drawn in the cell using an uniform law. For a cell 𝐶 𝑖 [𝑧 𝑖 ; 𝑧 𝑖+1 [ and a random number 𝑅 ∈ [0; 1], a detrapped particle from this cell will be generated at a depth 𝑧 = 𝑧 𝑖 + 𝑅(𝑧 𝑖+1 -𝑧 𝑖 ). The distribution of depths of trapped charges in a given cell is probably not uniform and we are making an approximation here. But this error should be reduced if the mesh is precise enough. The detrapped charges are also added to a stack of either detrapped electrons or detrapped holes. The drift particles from the electron cascade and the detrapping are both generated with a thermal energy 𝐸 = 3 2 𝑘𝑇 in a random direction to take into account the thermal agitation. Their trajectories can also be distorted by the electric field. The charges are followed until they are captured by a trap. Depending on the filling status of the trap (empty, or filled by a trapped particle of the opposite sign), the drift particle is either saved as a trapped particle, or recombines with the particle that is already in the trap. In the case of capture by a free trap, the trapped charge density of the corresponding cell is increased.

In the case of recombination, both charges are deleted from the simulation, and the trap is freed. The drift particles are generated in the following arbitrary order in 4 distinct runs: 1) Drift holes from the new electron cascade 2) Drift electrons from the new electron cascade 3) Drift holes from the detrapping 4) Drift electrons from the detrapping.

Obviously, if the detrapping computation returns that no charges were detrapped during the simulation step, only the transport of the drift particles created during the electron cascade will be simulated.

When all drift particles have been transported, a new simulation step begins, ballistic electrons are sent again by the gun and we go through the 4 phases again. The distribution of the implanted charges and the electric field generated by this distribution are also computed after each step, summing all the distribution of charges. Hence, the volume charge densities used in Poisson's equation for the electric field follow the sum 𝜌 = 𝜌 𝑛𝑒𝑤 + 𝜌 𝑠ℎ𝑎𝑙𝑙𝑜𝑤 𝑡𝑟𝑎𝑝𝑠 + 𝜌 𝑑𝑒𝑒𝑝 𝑡𝑟𝑎𝑝𝑠 , where we add the density of charges generated in the electron cascade (𝜌 𝑛𝑒𝑤 ) with the densities of trapped charges. Given the large time intervals to be simulated (a few ms), a time step of 𝜏 = 1µs has been attributed to each iteration of the simulation. The number N of incident electrons to be sent during each iteration of the simulation is then obtained from the incident current 𝐼 using the relation

𝑁 𝑖𝑛𝑐 (𝜏) = 𝐼𝜏 𝑒 Equation 5-3
However, for an incident current of 1 µA as in ONERA's experimental setup DEESSE, this means that 6 250 000 electrons would have to be sent for each simulation step, which would lead to an insane computation time. Consequently, we have to make another approximation to get a reasonable simulation time. Instead of sending 𝑁 𝑖𝑛𝑐 incident electrons per simulation step, we choose to send a more reasonable number of incident electrons 𝑁 𝑒𝑓𝑓 per step that is related to the true number of electrons by a charge bias factor 𝛽 = 𝑁 𝑖𝑛𝑐 (𝜏) 𝑁 𝑒𝑓𝑓

Equation 5-4

In this approximation, we suppose that a single particle of the simulation is actually representative of the transport of 𝛽 real particles. So, for instance, when a hole is detrapped or an electron recombines in the simulation, this means that 𝛽 holes have detrapped or 𝛽 electrons have recombined during the time step 𝜏. There is a compromise to be made in the number of incident electrons per step 𝑁 𝑒𝑓𝑓 , between computation time and statistical consistency. While a low number of incident electrons will greatly improve the computation time, the statistical dispersion will also be very important and the consistency of the simulation will be severely degraded. For these reasons, a 𝑁 𝑒𝑓𝑓 of 500 to 1000 electrons per step has been chosen, to speed up the simulations while aiming to keep a good statistic. This gives a charge bias factor of 𝛽 = 12500 to 6250 for an incident current of 1µA. The TEEY results are degraded for a 𝑁 𝑒𝑓𝑓 lower than a few hundred, but increasing 𝑁 𝑒𝑓𝑓 past 1000 did not seem to change the TEEY results.

The trapped charge densities stored in the simulation also need to be multiplied by the charge bias factor when computing the capture mean free path, to get the real number of particles trapped in a given cell and not the effective number of trapped particles that is actually stored in the simulation. So, for a given number of trapped particles 𝑛 𝑖 (unitless) stored in a simulation, the density of trapped charges 𝑁 𝑖 (cm -3 ) is given by 𝑁 𝑖 (cm -3 ) = 𝑛 𝑖 𝛽 ∆𝑧 𝑖 𝑆 𝑚𝑒𝑠ℎ

Equation 5-5

With ∆𝑧 𝑖 the thickness of the cell, and 𝑆 𝑚𝑒𝑠ℎ = 1 cm 2 is the surface of the mesh.

During the experiment, the TEEY is sampled with two current measurements over the duration a pulse, with a relaxation period between two pulses. So, a single TEEY point is obtained per pulse, as the average of TEEY over the duration of the pulse. To reproduce this measurement procedure, the TEEY returned by the simulation at a given time is the average of the TEEY of 50 simulation steps, that amount to a duration of 50 µs. This also limits the number of points in the output data and improves the statistical noise. The simulation phases 1 to 4 are followed during the duration of the pulse. At the end of the pulse, the relaxation period is also simulated. This is done by cutting the generation of incident electrons and removing phase 1 from the simulation step. At each simulation step, no incident electrons are sent on the target, but the program will still compute the evolution of the charge densities and the electric field. It will then simulate the transport of the drift particles that are detrapped during the relaxation process, before refreshing the electric field after the charges have drifted.

Computation of the electric field with a 1D Poisson solver

The mesh presented in section 5.2.1 is used for the discretization of Poisson's equation shown here. This discretization has been done using the explicit method, following the PhD thesis work of R. Pacaud [1][2][3] who has proposed a 1D resolution scheme for Poisson's equation in the THEMIS code. This gives the Poisson equation at a given node i:

( 𝜕 2 𝑉 𝜕𝑧 2 ) 𝑖 = 𝑉 𝑖+1 -𝑉 𝑖 ∆𝑧 𝑖 - 𝑉 𝑖 -𝑉 𝑖-1 ∆𝑧 𝑖-1 ∆𝑧 𝑖 + ∆𝑧 𝑖-1 2 = - 𝜌 𝑖 𝜀 Equation 5-6
With 𝜀 = 𝜀 0 𝜀 𝑟 . From there we can isolate the coefficients of the potential levels 𝑉 𝑖 as

2𝜀 ∆𝑧 𝑖-1 (∆𝑧 𝑖 + ∆𝑧 𝑖-1 ) 𝑉 𝑖-1 - 2𝜀 ∆𝑧 𝑖 ∆𝑧 𝑖-1 𝑉 𝑖 + 2𝜀 ∆𝑧 𝑖 (∆𝑧 𝑖 + ∆𝑧 𝑖-1 ) 𝑉 𝑖+1 = -𝜌 𝑖 Equation 5-7
This equation can be rewritten under a more general form :

𝑎 𝑖 𝑥 𝑖-1 + 𝑏 𝑖 𝑥 𝑖 + 𝑐 𝑖 𝑥 𝑖+1 = 𝑑 𝑖 Equation 5-8
For a given node of the mesh 𝑖, we can identify from Equation 5-7 the 𝑎 𝑖 , 𝑏 𝑖 , 𝑐 𝑖 coefficients, which depend on the mesh size and the permittivity, the unknown variables 𝑥 𝑖 , which are the potential levels, and the coefficients 𝑑 𝑖 = 𝜌 𝑖 are the charge density on the nodes of the mesh. However, the charge densities in Poisson's equation have the dimension of a volumetric density, while the charge densities sampled in 1D along the mesh are lineic densities. The number of charges in a given cell needs to be normalized by the volume of the cell to get a volumetric density, which gives

𝜌 𝑖 (C cm 3 ⁄ ) = 𝑛 𝑖 𝑒𝛽 ∆𝑧 𝑖 𝑆 𝑚𝑒𝑠ℎ = 𝑁 𝑖 𝑒 Equation 5-9
This equation should be valid for all cells of the mesh, hence it can be rewritten in a matrix form:

[ 𝑏 1 𝑐 1 0 𝑎 2 𝑏 2 𝑐 2 0 𝑎 3 𝑏 3 ⋱ ⋱ ⋱ 𝑐 𝑛-1 𝑎 𝑛 𝑏 𝑛 ] [ 𝑥 1 𝑥 2 ⋮ 𝑥 𝑛 ] = [ 𝑑 1 𝑑 2 ⋮ 𝑑 𝑛 ] Equation 5-10
This matrix equation is solvable through Thomas's algorithm [1][2][3], which is an iterative algorithm that is able to solve equations including a tri-diagonal matrix. It allows us to determine the unknown variables 𝑥 𝑖 from the coefficients 𝑎 𝑖 , 𝑏 𝑖 and 𝑐 𝑖 . From Equation 5-7, we can see that 𝑎 𝑖 , 𝑏 𝑖 and 𝑐 𝑖 only depend on the discretization steps of the mesh, which are known for any point i. 𝑑 𝑖 is also known since we have sampled the charge distribution in depth before solving Poisson's equation. However, there are some boundary conditions that need to be verified. From Figure 5-1 we can see that the potential needs to follow three conditions at the position of the electron gun, at the surface, and at the SiO2/Si interface, which will be our boundary conditions.

The electron gun, which is placed at a height -ℎ away from the material's surface, is set to the ground. This gives the first boundary condition

𝑉 𝑔𝑢𝑛 = 0 𝑉 Equation 5-11
However, we need to rewrite our boundary conditions in the form of Equation 5-8 in order to identify the coefficients a, b, c and d. Hence, we get the coefficients:

{ 𝑎 0 = 0 𝑏 0 = 1 𝑐 0 = 0 𝑑 0 = 𝑉 𝑔𝑢𝑛 = 0 Equation 5-12
At 𝑧 = 𝐿, we have the interface between the SiO2 layer and the Si substrate. In the experiment, the sample holder is biased to a potential 𝑉 𝑝𝑜𝑙 = -9 𝑉 when measuring the emitted current, so this is the potential we will use here. We also assume that the silicon layer is conductive enough, that the potential at the interface with the SiO2 layer is equal to the bias set at the metallic sample holder. This also means that we assume there is no electric field in the Si layer. This gives the boundary condition: Finally, at the Vacuum/SiO2 interface, the potential is not forced to a given value, but the electric field needs to follow Gauss' law. This law tells us that the scalar product between the surface normal 𝑛 ⃗ and the difference of the displacement vectors in the material 𝐷 𝑀 ⃗⃗⃗⃗⃗ and in vacuum 𝐷 𝑉 ⃗⃗⃗⃗⃗ is equal to the surface charge density 𝜎 𝑆 , which is equal to 0 here. So we have the following relationship:

𝑉 𝑛 = 𝑉 𝑝𝑜𝑙 = -9 𝑉
𝑛 ⃗ • (𝐷 𝑀 ⃗⃗⃗⃗⃗ -𝐷 𝑉 ⃗⃗⃗⃗⃗ ) = 𝜎 𝑆 = 0 ⇒ 𝐷 𝑉 = 𝐷 𝑀 Equation 5-15
From there we can convert the displacement vectors into the electric field as

𝐹 𝑉 = 𝜀 𝑟 𝐹 𝑀 Equation 5-16
To find an equation in the form of Equation 5-8, the potential levels need to appear. We can discretize Equation 5-16 as a potential gradient along the mesh and make the potentials at the nodes of the mesh appear as:

𝑉 𝑠 -𝑉 𝑔𝑢𝑛 ℎ -𝜀 𝑟 𝑉 1 -𝑉 𝑠 ∆𝑧 0 = 0 Equation 5-17
From Figure 5-1, the electron gun is not included in the mesh. The mesh is beginning at the surface of the material with the surface potential 𝑉 0 = 𝑉 𝑠 being the first node of the mesh. The next node of the mesh is thus the potential 𝑉 1 that is at a distance ∆𝑧 0 = 𝑧 1 -𝑧 0 from the surface, and we have 𝑉 -1 = 𝑉 𝑔𝑢𝑛 . We then transform Equation 5-17 to get an equation in the form of Equation 5-8:

(𝑉 𝑠 -𝑉 𝑔𝑢𝑛 ) -∆𝑧 0 𝜀 0 ℎ𝜀 + (𝑉 1 -𝑉 𝑠 ) = 0 ⇒ 𝑉 𝑔𝑢𝑛 ∆𝑧 0 𝜀 0 ℎ𝜀 -𝑉 𝑠 (1 + ∆𝑧 0 𝜀 0 ℎ𝜀 ) + 𝑉 1 = 0 Equation 5-18
By identification, we can get the coefficients:

{ 𝑎 1 = ∆𝑧 0 𝜀 0 ℎ𝜀 𝑏 1 = -(1 + ∆𝑧 0 𝜀 0 ℎ𝜀 ) 𝑐 1 = 1 𝑑 1 = 0 Equation 5-19
Now that we know the values of the coefficients 𝑎 𝑖 , 𝑏 𝑖 , 𝑐 𝑖 and 𝑑 𝑖 at any point of the mesh, we can determine the potential levels with Thomas' algorithm. The principle of the algorithm is to first compute new values of c and d, namely 𝑐 𝑖 ′ and 𝑑 𝑖 ′ , from the previous values. For this, we have

𝑐 𝑖 ′ = { 𝑐 𝑖 𝑏 𝑖 if 𝑖 = 0 𝑐 𝑖 𝑏 𝑖 -𝑎 𝑖 𝑐 𝑖-1 ′ if 𝑖 ∈ [1; 𝑛 -1] 𝑑 𝑖 ′ = { 𝑑 𝑖 𝑏 𝑖 if 𝑖 = 0 𝑑 𝑖 -𝑎 𝑖 𝑑 𝑖-1 ′ 𝑏 𝑖 -𝑎 𝑖 𝑐 𝑖-1 ′ if 𝑖 ∈ [1; 𝑛 -1] Equation 5-20
In a second phase of back substitution, we obtain the potential for 𝑖 ∈ [1; 𝑛 -1] as

𝑉 𝑖 = 𝑑 𝑖 ′ -𝑐 𝑖 ′ 𝑉 𝑖+1 Equation 5-21
Finally, the electric field is obtained from the potential gradient as

𝐹 𝑖+1 2 ⁄ = - 𝑉 𝑖+1 -𝑉 𝑖 ∆𝑧 𝑖 Equation 5-22
Since this is a gradient, we actually obtain the field on the inter-node points 𝑧 𝑖+1 2 ⁄ . To get the electric field on the nodes of the mesh, we simply do the mean of the fields on the closest intermesh nodes:

𝐹 𝑖 = 𝐹 𝑖+1 2 ⁄ -𝐹 𝑖-1 2 ⁄ 2 Equation 5-23
This operation is then repeated for each simulation step. The sampling of the charge densities is done using a dichotomy method. Due to the efficiency of this method and of Thomas' algorithm, the computation of the electric field during the simulation is practically instantaneous.

Transport of the drift particles

At each time step, holes and drift electrons generated in the material move according to a drift diffusion process. Due to their low energy and strong coupling with the lattice, they are strongly scattered by the collisions with the phonons and undergo a random walk motion. Between these collisions, the particles can be accelerated by the electric field. From a macroscopic point of view, the density of holes (electrons) can drift in the (opposite) direction of the field. This motion can be evaluated with the drift velocity of the density, which is generally expressed as

𝑣 𝐷 ⃗⃗⃗⃗ = 𝑑 𝑑𝑡 〈𝑟 〉 = ±µ𝐹 Equation 5-24
Where 〈𝑟 〉 is the average movement of the distribution, 𝐹 is the electric field and µ is the electron/hole mobility. In essence, the drift velocity is the displacement of the position of the centroid of the distribution 〈𝑅〉 over a given time 𝑑𝑡. It is not representative of the movement of the individual particles however. Indeed, each particle has a random trajectory due to the scattering with phonons. In the absence of an electric field and in pure Gaussian transport, the distribution of particles is spreading with time but the centroid is not moving. Here the drift velocity will be null, whereas the particles are definitely not immobile. Hence, we cannot apply the drift velocity directly to the individual particles, but we have to generate their individual trajectories.

The drift particles are characterized by their thermal velocity. In steady state, the simplest approximation is to consider that the particles have a thermal energy 𝐸 = 𝑚𝑣² we can get an average velocity 𝑣 𝑡ℎ = √3𝑘𝑇/𝑚 * which is about 10 7 cm/s at room temperature. However, in all rigor, the particles have a unique velocity defined from the Maxwell-Boltzmann distribution of velocities. In this distribution, the probability of a particle having a velocity 𝑣 is given by:

𝑃(𝑣) = ( 𝑚 2𝜋𝑘𝑇 ) 1/2 4𝜋𝑣 2 𝑒 -𝑚𝑣 2 2𝑘𝑇
Equation 5-25

For consistency, the velocity of each particle should be randomly drawn from the Maxwell-Boltzmann distribution. To avoid the computational cost induced by a random sampling before generating each drift particle, the holes and drift electrons are simply generated with an unique energy 𝐸 = 3 2 𝑘𝑇. The drift particles are also generated in a random direction, to take into account the thermal agitation. The trajectory and energy of the particle are modified by the electric field following the classical equation of dynamics:

𝑎 = 𝑞 𝑚 * 𝐹
For this step, we take advantage of Geant4's built in tools for the transport of particles through an electric field. The equation solved by Geant4's tools is Newton's equation, which is why it is used in this simulation. The differential equation of the trajectory is integrated using a Runge-Kutta method. In Geant4, the user can define the interpolation step and the precision of the integration. A high precision is necessary to obtain realistic curved trajectories and not simply a sum of large segments. But an excessive precision will hinder computation time. In this work, the integration step has been set to 1 nm. The electric field is also applied to the ballistic particles, which can be accelerated and deviated between two interactions.

To have a physically accurate simulation, all collisions with phonons should be simulated. However, as we have seen in section 3.2.4.1 of Chapter 3, the interaction frequency of very low energy electrons (< 0.1 eV) with LO phonons is on the order of 10 -13 s to 10 -14 s .This is very short compared to the time of flight before trapping in SiO2 (10 -9 to 10 -7 s) [4]. This is especially true for holes, which have a strong coupling with Si and O atoms and are able to form small polarons [5]. When a hole becomes a polaron, it is self-trapped and immobilized at a given interatomic trapping site for very short times, comparable to the vibration frequency of the lattice (10 -12 s) [6]. As a result, it is not possible to simulate all interactions with phonons in a reasonable computation time, since the time step attributed to each iteration of the simulation is much larger (10 -6 s).

We have adopted here an approach that is comparable to the condensed history approach used for high energy particles. The drift particle is assumed to travel in a single trajectory driven by the electric field. It would correspond to the sum of all trajectories between the collisions with the phonons. This drift motion is modeled thanks to the mobility of the particle. In practice, we attribute an effective mass to the drift particle, given by 𝑚 * = 𝑞𝜏 𝑝ℎ µ

Equation 5-26

where τ ph = 10 -13 s is the time between two collisions with LO phonons [7]. The mobilities for electrons and holes have been set to common values found in the literature for SiO2, respectively µ e = 20 cm 2 V -1 s -1 [8] and µ h = 10 -5 cm 2 V -1 s -1 [4]. While these values are only valid at a given temperature and electron density, this approximation does not seem to hinder the TEEY results. Using this method, we are able to take into account the effect of the electric field on the drift of the particles, which can force the populations of electrons and holes in different directions. We are also able to model the thermal spread of the distribution in the absence of an electric field.

Modeling of the trapping of holes and electrons

In the simulation, the drift holes and electrons are followed until they are trapped. Ballistic electrons of a few eVs are also susceptible to being trapped, although with a much lower probability given their energy. In this work, we have chosen to model the effect of both deep traps and shallow traps. The trap distributions used in our simulations are shown in Figure 5-2.

We have considered two populations of traps for each particle type, which are split into deep traps due to impurities and shallow traps due to localized states.

In this section, we will also detail the trap energy depths chosen in our model, which will be noted 𝐸 𝑖 . For electron traps, the energy depth is defined as the difference between the bottom of the conduction band and the trap energy level, following the convention shown in Figure 5-2.

For hole traps, their energy depth is defined as the difference between the top of the valence band and the energy level of the trap. Hence, when the energy depth 𝐸 𝑖 of an electron (hole) trap increases, its energy level is located further away from the conduction (valence) band edge, and closer to the middle of the band gap.

Deep traps

Many types of defects and impurities are able to capture electrons or holes in SiO2. The nature and concentration of these impurities is highly variable depending on the material and the fabrication process. Indeed, in silicon dioxide, several types of electron and hole traps have been identified [9]. These traps are induced by the presence of impurities and defects in the material, or by the electron irradiation itself. They create energy levels in the band gap below the conduction band or above the valence band, in which electrons or holes can fall. The cross sections of these traps are highly variable [9], depending on whether the trap is coulombic attractive (10 -13 -10 -15 cm²), neutral (10 -15 -10 -18 cm²) or coulombic repulsive (< 10 -18 cm²), and their activation energy is about 1 to 3 eV. Finally, the concentration of the traps is also dependent on the fabrication process. For instance the water related trap concentration can vary from 10 15 cm -3 for a dry oxide to 10 19 cm -3 for a wet oxide [10]. Here, we aim at modelling the charging of plasma grown oxides, however these oxides also have specific traps that do not appear in thermal grown oxides, with cross sections of 10 -15 cm² [11] (value used for our simulations). Finally, the interface between the SiO2 and Si layers comprises a zone of a few nanometers of thickness where the concentration of defects is very high (10 20 cm -3 ). As a result, we can see that there are several possible values used in the literature for both parameters of trapping (density and cross section), and the choice of representative parameters is not trivial. The situation is also complicated by the fact that we have to model traps for electron and for holes, but the nature of the most attractive electron traps may not be the same as hole traps.

In this work, the deep trapping of electrons and holes has been modeled using a unique cross section 𝜎 𝐷 = 10 -15 cm² for drift electrons and holes. Therefore, we assume that the main capture mechanism for deep traps is due to coulombic attractive traps. The density of deep level traps is taken as N D = 10 18 cm -3 [12]. This density of traps is able to capture either electrons or holes. The mean free path is obtained by:

𝜆 𝐷 = 1 𝜎 𝐷 𝑁 𝐷 ⁄ Equation 5-27
The energy depth of the deep traps is taken as 𝐸 𝑖 = 2 𝑒𝑉 from the value proposed by Cornet et al. [13] This value is close to the energy depth of the most commonly encountered electron traps in SiO2, namely the oxygen vacancy or the Na impurities.

Shallow traps

The deep level traps shown above can be considered as "extrinsic" traps, since they are not a property of the material itself but are the result of defects created during the fabrication process.

Other types of traps exist in amorphous materials, which are known as intrinsic traps since they are linked to the nature of the material itself. Here, the disorder and ruptures in the atomic bonds create a band of localized states below the conduction and valence bands where electrons and holes may be trapped [14]. While the energy depth of these traps is very shallow (0.1 eV or less), their density is very high (10 21 cm -3 ) due to the significant disorder in amorphous materials.

In other simulations of the TEEY of SiO2, the energy depth of the shallow traps has been modeled by either a single energy level [12] or a Gaussian distribution centered on the deeper level traps [15]. However, an accurate model of this distribution of traps is especially important for the transport of holes in amorphous SiO2. Indeed, electrons follow a Gaussian transport, which means that their distribution of positions is moving in a global direction with a welldefined drift velocity [16]. On the contrary, we have seen in Chapter 2 that the transport of holes is dispersive [4] and follows the Continuous Time Random Walk theory. The cause for this transport is that the time of immobilization of the holes in the traps is not constant. Silver et al. [17,18] have shown that the dispersive transport of holes can be modeled in a Monte-Carlo simulation by an exponential distribution of trap energy depths, which induces a distribution of trap residence times.

Consequently, the density of localized states for holes is modeled in this work by an exponential law [10,17,19], so that the density of localized states with an energy 𝐸 𝑇 above the valence band edge between 𝐸 𝑖 and 𝐸 𝑖 + 𝑑𝐸 is given by:

𝑁(𝐸 𝑖 ≤ 𝐸 𝑇 < 𝐸 𝑖 + 𝑑𝐸) = 𝑁 𝑆 𝐸 𝑐 exp (- 𝐸 𝑖 𝐸 𝑐 ) 𝑑𝐸 Equation 5-28
The distribution of trap energy depths has an expected value 𝐸 𝑐 , taken as 𝐸 𝑐 = 0.07 𝑒𝑉 [10].

Since electrons follow a Gaussian transport and are much more mobile, the electron shallow traps are modeled in our work by a single level of energy depth E c = 0.02 eV, following the value proposed by Wager [10]. Here, the single energy level leads to constant release times. In fact, Mady et al. [20] have shown that the hopping transport of electrons through traps with identical energy depths follows the characteristic of a Gaussian transport. For both hole and electron traps, the total density of shallow traps is 𝑁 𝑆 = 10 21 cm -3 [10].

The capture cross sections used are 𝜎 𝑆 = 1 × 10 -14 𝑐𝑚 2 for drift electrons, 2.5 × 10 -14 𝑐𝑚² for holes. The capture mean free path is obtained in the same way as the deep level traps, using the total density of traps:

𝜆 𝑆 = 1 𝜎 𝑆 𝑁 𝑆 ⁄ Equation 5-29
The exponential distribution of energy depths is simulated using a discrete distribution of 20 trap levels 𝐸 𝑖 regularly spaced between 0 and 0.4 eV. The density of each level 𝐸 𝑖 is obtained from Equation 5-28, where 𝑑𝐸 is the width of the level, given by the discretization step as 𝑑𝐸 = 0.02 eV. Hence, shallower level have a higher density than deeper levels. This discretization is illustrated in Figure 5-3. The capture mean free path is computed globally for the exponential distribution, using the total density of traps 𝑁 𝑆 in Equation 5-29.

𝑊(𝐸 𝑖 ) = 𝑊 0 exp (-𝐸 𝑖 𝑘𝑇 )

Equation 5-31

The frequency factor 𝑊 0 (s -1 ) is a fundamental parameter of the detrapping phenomenon, since it describes the intrinsic mobility of the charge carrier in the trap. It can significantly vary according to the energy depth of the trap 𝐸 𝑖 (eV), as shown by Cornet et al. [13]. They have proposed a law linking the energy level of the trap in eV and the frequency factor in s -1 as:

log 10 (𝑊 0 ) = 4 + 5𝐸 𝑖 (eV) Equation 5-32
This expression is assumed to be valid for both electron and hole traps. Notably, from this law, one can see that shallow traps with 𝐸 𝑖 around 0.1 eV will have a low frequency factor of around 10 4 s -1 . On the other hand, deeper traps with 𝐸 𝑖 of about a few eVs will have a very high frequency factor of around 10 14 s -1 , even though it is very difficult for the charge carriers to escape from these traps. We have chosen a value for the shallow traps of 𝑊 0 = 10 3 s -1 , close to the law proposed in Equation 5-32. Since the activation energies of the shallow hole traps follow a distribution, each trap level has a distinct escape frequency, which leads to a dispersive transport for holes. For deep traps, the frequency factor is chosen as 𝑊 0 = 10 14 s -1 , from ref. [13].

Detrapping enhancements for deep traps

The detrapping of deep traps is not only enabled by thermal activation. Indeed, as we have seen in Chapter 2, some effects related to the electric field or the tunneling of charge carriers can increase the detrapping probability. Since we have assumed that our deep traps are coulombic attractive, these enhancements apply to them and should be taken into account in the modelling of the detrapping.

The potential barrier of the trap can first be lowered by an electric field 𝐹 due to the Poole-Frenkel effect [23,24]. The Equation 5-31 has to be modified by introducing the Poole-Frenkel (PF) enhancing factor 𝑒 𝑃𝐹 :

𝑒 𝑃𝐹 = exp (- ∆𝐸 𝑖 𝑘𝑇 ) , ∆𝐸 𝑖 = √ 𝑒 3 𝐹 𝜋𝜖 0 𝜖 𝑟 ⁄ Equation 5-33
We can see here that the Poole-Frenkel lowering is only dependent on the value of the electric field and the permittivity of the material. It is completely independent on the energy depth of the trap or the nature of the trapped particle.

Detrapping can also be enhanced by the Phonon-Assisted Tunelling (PAT) effect. In this case, the trapped charge can absorb a phonon and get excited to a higher level in the trap, where the probability of tunneling through the trap barrier is more favorable. The probability is dependent on the transparency of a potential barrier, given by 𝑇 = exp (-4 3

(2𝑚 * ) 1 2 ⁄ 𝐸 𝑖 3 2 ⁄ 𝑞ℏ𝐹 ) Equation 5-34
The PAT enhancement factor is obtained by an integral over the energy depth of the trap, of the probability of excitement at a given energy level 𝑧 times the transparency of the barrier at 𝑧. The potential barrier is assumed to be triangular, which gives the PAT enhancement factor as [24,25]:

𝑒 𝑃𝐴𝑇 = ∫ 𝑒𝑥𝑝 (𝑧 -𝑧 3 2 ⁄ ( 4 3 (2𝑚 * ) 1 2 ⁄ (k𝑇) 3 2 ⁄ 𝑞ħ𝐹 )) 𝑑𝑧 𝐸 𝑖 /𝑘𝑇 0 Equation 5-35
In this expression, 𝑚 * is the effective mass of the particle, 𝑞 is its charge, and we integrate the probability of tunneling from a level 𝑧 over the possible levels that the charge carrier can be excited to. The barrier is assumed to be triangular in this case. The final emission rate for the deep coulombic trap use in the Monte-Carlo model is written by including both PAT and PF enhancements [26]: In the integration of the transparency factor for the tunneling probability, the term

𝑊(𝐸 𝑖 ) =
(1 -( ∆𝐸 𝑖 𝑧k𝑇 ) 5 3 ⁄
) appears. This is because the potential barrier is deformed by the Poole-Frenkel lowering effect. In this situation, Hill [23] and Vincent et al. [24] mention that the triangular barrier model from Equation 5-35 is invalid. Consequently, the potential barrier of the trap should rather be modelled as an hyperbolic potential barrier, which modifies the tunnelling probability by a factor (1 -(

∆𝐸 𝑖 𝑧k𝑇 ) 5 3 ⁄
).

-Modeling of the recombination of electron -hole pairs

This equation has no analytical solution and has to be solved numerically. To do so, we integrate the tunneling probability over 10 energy levels z in the trap, for each time the detrapping probability for deep traps has to be computed.

Modeling of the recombination of electron-hole pairs

From a general point of view, a trap in a material is associated with a defect in the lattice which carriers can scatter on, regardless of the state of this trap (empty, filled, charged, neutral). The scattering process will occur for a carrier with a given probability to occur related to the state of the trap. Depending on the state, the scattering process can lead to the recombination of the carrier. As the material is irradiated, holes or electrons fill more and more traps, and the drift carriers have a higher probability of being captured by a trapping site which is already filled. If a hole or an electron falls into a trap which is already occupied by the opposite particle, the two particles recombine and disappear, and the trap is freed. The cross-section for the recombination of a drift particle by a trap occupied by the opposite particle is set as σ e-h = 2 × 10 -12 cm².

This value is high but is coherent with the values found in the literature, which range from 10 -13

[15] to 10 -11 cm² [27,28]. It also takes into account the fact that the coulombian interaction caused by the charge of the trapped particle adds itself to the native attraction of the trap and makes it even more attractive. On contrary, a trap filled by a particle of the same sign than a drifting particle will be much less attractive due to the coulombian repulsion caused by the trapped charge. While some kind of traps may be able to capture multiple charges of the same sign, the capture cross section for neutral or coulombic repulsive traps is much lower, so this phenomenon has not been modeled here.

The recombination mean free path for electrons and holes is given respectively as:

{ 𝜆 𝑒 = 1 𝜎 𝑒-ℎ 𝑁 ℎ ⁄ 𝜆 ℎ = 1 𝜎 𝑒-ℎ 𝑁 𝑒 ⁄ Equation 5-37
Where 𝑁 ℎ and 𝑁 𝑒 are the densities of trapped holes and electrons. However, as the new drift particles fill the traps, the densities of hole-occupied and electron-occupied traps increase after each simulation step. Since the total density of traps 𝑁 𝑇 is fixed, the density of free traps is also reduced as the traps get filled. Hence, the trapped charge and free trap densities for either shallow or deep traps follow the relation

𝑁 𝐹𝑟𝑒𝑒 = 𝑁 𝑇 -(𝑁 ℎ + 𝑁 𝑒 ) Equation 5-38
As the charge densities are sampled in depth, 𝑁 ℎ and 𝑁 𝑒 will also vary according to the position of the particle. This evolution needs to be taken into account in the capture mean free path of charge carriers by the empty traps. As a result, Equation 5-27 and Equation 5-29 have been modified and combined with Equation 5-37 to derive a unique mean free path for the capture by the shallow (𝑆) or deep (𝐷) traps including their occupation status, following the approach proposed by Li et al [15]. This combined MFP for both capture by empty traps and trap-assisted recombination is used in the simulation, in place of the separate recombination MFP of Equation 5-37 and capture by empty traps MFP of Equation 5-27 and Equation 5-29.

This gives the combined mean free paths for holes (ℎ) at a given depth z: For the exponential distribution, the density of free traps used in the mean free paths is computed globally over the 20 levels of the distribution as the sum of the available traps for each level.

When the particle falls into a trap, a random number 𝑟 1 is sampled between zero and one to determine if the particle is captured by an empty trap, or if it falls in a trap that is occupied by an opposite charge. Capture of an electron or a hole by a free trap happens if 𝑟 1 < 𝑃 𝐹𝑟𝑒𝑒 where 𝑃 𝐹𝑟𝑒𝑒 (𝑒/ℎ) = 𝜎 𝐹𝑟𝑒𝑒 𝑁 𝐹𝑟𝑒𝑒 𝜎 𝐹𝑟𝑒𝑒 𝑁 𝐹𝑟𝑒𝑒 + 𝜎 𝑒-ℎ 𝑁 ℎ/𝑒 Equation 5-42 𝑃 𝐹𝑟𝑒𝑒 is the percentage of free traps modified by the capture cross section to include the fact that free traps are less attractive than traps filled by a particle of the opposite charge. In the opposite case, the particle is captured by a trap filled with a particle of the opposite charge, and the two particles recombine.

In the case of trapping of holes in the shallow trap distribution, the occupied and free trap densities are saved individually for each trap level. Consequently, a random number is first sampled in the exponential law, to select a trap level and retrieve its charge densities. For this, we compute a trial value for the energy level 𝐸 𝑡 of the trap where the particle has fallen into, from a random number 𝑟 2 ∈ [0; 1] and the mean energy of the distribution 𝐸 𝑐 as [19]:

𝐸 𝑡 = -𝐸 𝑐 log(1 -𝑟 2 )
Equation 5- 43 We then find the energy level 𝐸 𝑖 such that 𝐸 𝑖 < 𝐸 𝑡 < 𝐸 𝑖+1 using a dichotomy method, and retrieve the density of trapped charges (𝑁 ℎ ) 𝑖 and available traps (𝑁 𝑓𝑟𝑒𝑒 ) 𝑖 for the level 𝐸 𝑖 to be used in Equation 5-42.

Modeling the effect of the incident beam surface and the current density

So far, the trapped charge and free trap densities have been defined globally in a 1D manner. That is to say, we consider in the simulation that the charge densities 𝑁 ℎ , 𝑁 𝑒 and 𝑁 𝑓𝑟𝑒𝑒 are uniform on the whole surface, and the capture+recombination mean free paths will always depend on the same charge density, regardless of the point of impact of the electron. However, this is only true if the current density is high enough, so that the electron cascades are uniformly distributed and an incident electron is guaranteed to arrive in an area filled with charges. Indeed, in TEEY experiments, the electron beam is not a single point but it covers a surface that ranges from a few mm² up to several cm². Depending on the current density and the transport properties of the trapped charges, there is a possibility that the charge densities may not be uniform on the surface at all. If the current density is low, the electron impacts on the surface will be quite widespread and the surface will not be irradiated uniformly. This will make the charge densities have a very poor uniformity, and there is a high probability that some areas will be free of charges after a given time, while other areas that have received an incident electron will have many trapped charges. This can also happen if the charges are able to quickly escape the traps and diffuse between two electron impacts at a given zone. In these cases, if a particle is created in a region where all traps are free, the charge densities used in the computation of the mean free paths should be empty. Nevertheless, in the present state, our Monte-Carlo model is unable to simulate the lack of uniformity in the charge densities that could be caused by a low incident current density.

In the following, we will present the method used to model the effect of the current density on the uniformity of the charge densities. As we sample the charges in 1D in depth and the objective is to understand a 2D surface effect, an empirical approach was used. This empirical approach can be justified by comparing the radius of an electron cascade (roughly 10 nm, Figure 5-4) with the beam area (up to several cm²). Indeed, if we want to have an accurate 3D mesh for the charge densities, it would need to have cells with an area that is not larger than the size of a cascade. Consequently, we would have to use cells that have an area not greater than 10 nm², to model a surface on the order of the cm². This would lead to a mesh with 10 12 cells, which would require excessive computation time and resources.

From the incident current 𝐼 0 , we can get the number of electron impacts 𝑁(𝜏) after a given time 𝜏 from Equation 5-3. The total area of the material where electron cascades have been created

after 𝜏 should then be the product of the number of electrons 𝑁(𝜏), with the area of an electron cascade 𝑆 𝐶 . Hence, we need to determine the area of a single electron cascade. Assuming that the electron cascades are spherical, which gives us a circle on a 2D projection on the surface, this area is given by:

𝑆 𝐶 = 𝜋𝑟 𝐶 ² Equation 5-44
Where 𝑟 𝐶 is the radius of an electron cascade taken from the incident electron's point of impact. However, given the Brownian motion of very low energy electrons, the distance between the point of capture of a secondary electron and the incident electron's point of impact can vary quite significantly, and should follow a statistical distribution. Nevertheless, we can still extract this radius from the Monte-Carlo simulations, by first sampling the final positions of the secondary electrons created in the cascade. In essence, we want to measure here the same quantity as the extrapolated range, but laterally rather than in depth. We can then follow the same approach than for the computation of the extrapolated range, by computing the normalized integral of the distribution of radii and obtaining an extrapolated radius from the tangent at P = 0.5. The extrapolated radius obtained following this method from the simulated data is given in Figure 5-4 for SiO2. In the simulations, the cascade radius 𝑟 𝐶 ranges from a few nm at 100 eV to a few tens of nm above 1 keV. Interestingly, the extrapolated radius follows a behavior that is identical to the extrapolated range of low energy electrons [29], with a plateau region below 100 eV. However, the radius was measured without simulating the drift of the charge carriers generated in the electron cascade, which will increase its spread. To take the drift into account, the value of 𝑟 𝐶 that was used in the computation of the surface of the cascade was increased by 5 nm, which is an estimation of the length that can be traveled by a hole through a few hopping events. In the simulation, we use a unique value for the cascade radius 𝑟 𝐶 = 15 nm regardless of the incident electron energy. We consider that a cascade radius of 10 nm before drift should be representative for most incident energies of interest (100 eV to 2 keV) for the TEEY. This factor, illustrated in Figure 5-5, corresponds to the macroscopic probability that an incident electron arrives in an area where another electron cascade was previously generated. In this case, the two cascades overlap, and the secondary electrons created in the new electron cascade can interact with the trapped particles created by the previous cascade. However, if the electron arrives in an area that is still free of charges after 𝜏, the electron cascade will not be affected by recombination with the charges created by the previous electrons. As we mentioned earlier, the charge densities must be modified in order to take into account the uniformity (or lack thereof) of the electron impacts on the surface. To do so, the overlap factor is used to compute an effective charge density 𝑁 𝑒𝑓𝑓 = % 𝑜𝑣𝑒𝑟𝑙𝑎𝑝 (𝜏) * 𝑁 which is then used in the computation of the mean free paths, and in Equation 5-42 for the probability of recombining. The relation between the densities given by Equation 5-38 thus becomes

𝑁 𝑓𝑟𝑒𝑒 = 𝑁 𝑇 -% 𝑜𝑣𝑒𝑟𝑙𝑎𝑝 (𝑁 𝑒 + 𝑁 ℎ ) Equation 5-46
Indeed, in the simulation, we store the total number of trapped charges created at a given depth by the incident electrons. Nevertheless, we need to model the fact that the electron has a probability of hitting a region of the material that is still free of charges, and the average density of available traps also needs to be modified in consequence. The overlap factor is computed after each simulation time step 𝜏, knowing the number of electrons sent during this step. This factor is capped at 100%, which is the limit condition where the whole surface is covered by electron cascades.

5.7 -Description of the architecture of the Geant4 simulation 207

Description of the architecture of the Geant4 simulation

To integrate all the physical models for the transport of drift electrons and holes, and to develop an iterative simulation, many new classes were conceived in the Geant4 application we have developed. Here, we will only list these various classes and provide illustrations showing how they articulate with the classes of Geant4. The full description of the architecture can be found in Appendix I.

 The most important class of the simulation is the DriftManager. It is a singleton with functions related to the charging effects, and the center of the charging simulation. It stores the charge densities, computes and interpolates the electric field, launches the different phases of the simulation, computes the capture mean free paths and detrapping probabilities, and computes what happens when a particle is captured by a trap (free capture or recombination). As a singleton, DriftManager is called by the physical interaction processes, field handling classes and particle sources throughout the simulation.

 The DriftManager is accompanied by a DriftMessenger. It is tasked with loading the different parameters of the model through macro commands stored in the file ChargingParameters.mac. Other macro commands are passed to the DriftManager by the DriftMessenger to signal it to begin the different phases of the simulation, and tell it whether the electron cascade, the drift phase or a relaxation phase should be simulated.

 New particle types DriftHole and DriftElectron were also added. These particles inherit from the class G4ParticleDefinition and use the same particle definition parameters as G4Electron, except for the mass which is set to the effective mass from Equation 5-26, and the charge which is +e for a hole. DriftHole and DriftElectron particles are generated by the DriftManager during the drift phases.

 A new physical process Trapping has been created, following the structure of the other physical processes of Geant4. It returns a mean free path from DriftManager for the capture of drift electrons and holes by shallow or deep traps, and calls DriftManager to handle the trapping or recombination of the particle. By creating a new interaction process following the nomenclature of Geant4, we can let the toolkit handle the transport of the drift particles as any other particle.

 The process G4ElectronCapture tasked with killing electrons below a preset energy threshold has been overhauled. It is now handling the capture and recombination of ballistic electrons, in a similar way to the G4Trapping process.

 The PrimaryGeneratorAction (PGA) class is driven by the DriftManager, which tells it whether the current run is a ballistic or drift run. The PGA also retrieves the stack of particles to be tracked in a drift run from the DriftManager. It then communicates to the Geant4 transportation manager the positions and directions of the drift particles generated.

 The process DriftTimeStepMax stops the drift of the particles if their drifting time exceeds the simulation step. Here, a definite interaction value is returned instead of a mean free path. The particles that have not finished drifting at the end of the simulation time step are added to a postponed stack, and the simulation of their transport is resumed after the next ballistic run.

 A set of enum variables is stored in the file driftEnums.h. They are used to set and check conditions regarding the type of the current run, the current particle type, or the type of trap.

The figures By simulating the charge buildup in the insulator and its effect on the successive electron cascades, our new Monte-Carlo model should be able to simulate the evolution of the TEEY of SiO2 with time, depending on the internal and external charge effects. We have seen that some experimental studies were made in conditions of positive charging with the external charging effects removed, either by sample holder biasing or electron collector biasing to prevent the recollection of secondary electrons. These studies had shown a decrease of the TEEY that could only be attributed to internal charging effects. In this section, we want to study whether our Monte-Carlo code is able to simulate this decrease of the TEEY in the case of SiO2 samples.

The simulation results will be validated quantitatively in Chapter 6, where we will present the experimental TEEY results obtained during this thesis and confront them to the simulation results. In the present section, we shall rather focus on a qualitative analysis. The objective is thus to determine whether the code can reproduce a decrease of the TEEY on SiO2 that would be similar to what was observed experimentally on other insulators. All simulated TEEY were obtained with an incident current of 1µA, and a beam area of 0.1 cm². This gives a current density of 10 µA/cm², to ensure that the surface is irradiated uniformly and that we can observe the interactions with internal charges. 500 incident electrons were sent at each simulation step, which gives a charge bias factor 𝛽 = 12500.

Time-resolved simulation of the TEEY of a bulk SiO 2 sample: study of the external charging effects

Before studying the influence of internal charging effects, we must ensure that our program is correctly modeling the evolution of the TEEY due to the macroscopic external charging effects, in a qualitative way. To do so, we must place ourselves in conditions where the external charging effects are not removed. Consequently, in this section, the sample holder voltage will be set at 0 V instead of the -9 V bias. Simulations were made on a bulk sample of SiO2 with a thickness of 100µm and a surface of 1 cm², instead of the 20 nm thin films we have mentioned so far. Indeed, the surface potential 𝑉 𝑠 is linked with the capacitance of the material 𝐶 and its thickness 𝐷 by the relation

𝑉 𝑠 = 𝑄 𝐶 = 𝑄 𝐷 𝜖 0 𝜖 𝑟 𝑆 Equation 5-47
Therefore, using a thick sample with a small capacitance will make the surface potential evolve much faster than with a thin film sample, and we can easily observe the effect of the external electric field. For the same reasons, the TEEY simulated in this section were also obtained under continuous irradiation instead of pulsed measurements. We have previously identified three situations regarding the global charge buildup and the evolution of the TEEY, depending on the incident electron energy 𝐸 0 with regard to the crossover points 𝐸 𝐶1 and 𝐸 𝐶2 . In the following, we will place ourselves in these conditions and verify if the time-resolved TEEY returned by the Monte-Carlo code follows the expected behavior. a) 𝐸 0 < 𝐸 𝐶1 , 𝑇𝐸𝐸𝑌 < 1

In this case, the sample is charging negatively and the TEEY is decreasing to 0 until 𝐸 = -𝑒𝑉 𝑠 . When this limit condition is reached, the incident electrons are unable to overcome the external electric field and hit the surface. They get electrostatically reflected and the TEEY increases sharply back to 1. In Figure 5678, electrons with an incident energy of 20 eV are sent on the sample. Since the TEEY is lower than 1, the surface potential is decreasing, which slows down the incident electrons. At -𝑒𝑉 𝑠 = 20 eV, the incident electrons cannot hit the surface, the surface potential does not evolve anymore, and the TEEY increases to one, as expected. In conclusion, the global charge buildup is correctly modeled for very low energy electrons below 𝐸 𝐶1 . However, in the experimental TEEY measurements made in the DEESSE facility, the TEEY curve is sampled from 50 eV up to 2 keV. For many materials, 50 eV will be below the first crossover point, but this is not the case for the SiO2 samples we have used in this work. Hence, we should not be in this situation when comparing experiments and simulations of the TEEY in Chapter 6.

b) 𝐸 𝐶1 < 𝐸 0 < 𝐸 𝐶2 , 𝑇𝐸𝐸𝑌 > 1

In this situation, the sample is charging positively, which increases the landing energy of the incident electrons, and the TEEY converges to 1 where it stabilizes. In Figure 56789, electrons with an energy of 500 eV are sent on the SiO2 sample. We can see in Figure 5-9 that the TEEY is decreasing rapidly due to the recollection of secondary electrons, caused by the increase of the surface potential. The equilibrium of charges is reached when the TEEY stabilizes at 1, where the surface potential also stops evolving, as expected. What is interesting here though is that the surface potential has stabilized at 14 V. Indeed, if we follow the conventional theory for the evolution of the TEEY due to external charging, the TEEY should only stabilize at one when the landing energy of the incident electrons has reached the second point of crossover, which is about a few keV. This means that the surface potential should stabilize at around a few thousand volts, instead of 14 V only. Here, the incident electrons have a final landing energy of 514 eV but the TEEY has still stabilized at 1. What the conventional theory does not take into account however, is the recollection of the secondary electrons. Indeed, the electrons escaping the surface must be energetic enough to overcome the external electric field generated by the positive charge. If the energy of a secondary electron is lower than 𝑒𝑉 𝑠 , it will not be able to overcome the electric field and will be recollected by the surface. Macroscopically, we can consider that a potential barrier of height 𝑒𝑉 𝑠 must be overcome by the secondary electrons. However, if we look in Figure 5-10 at the energy spectrum of electrons produced by a sample of SiO2 irradiated by 500 eV electrons, we can see that the energy of the secondary electrons is centered around 5-6 eV. Interestingly, there is a discontinuity of the energy distribution around 7 eV. This is roughly equal to the value of the energy gap used in the simulation (8 eV) minus the 0.9 eV lost by an electron when crossing the surface potential barrier into vacuum. The energy spectrum in the simulation is quite dependent on the fit of the energy loss function used, but it is unknown if this could be the reason for this discontinuity. Practically all secondary electrons have an energy that is lower than 20 eV. Therefore, as the surface potential increases, the recollection of low energy electrons compensates the positive charges that were left in the material when the secondary electrons escaped the surface in the first place. At the final value of 𝑒𝑉 𝑠 = 14 eV, the majority of secondary electrons are unable to escape and are captured by the surface. Since the TEEY has reached 1, the global charge is not evolving anymore. Hence, the value of 14 eV is an equilibrium, where a sufficient quantity of secondary electrons are recollected and compensate all the new holes created in the material, without introducing additional negative charges. This is why the surface potential also stabilizes, but at a value that is much lower than was is expected from the conventional theory.

From Figure 5-9, we can also note that the surface potential has already increased past 10 V after 1 ms only. If we were to use such a thick sample in TEEY experiments, a stronger negative bias than the -9V used in DEESSE would need to be applied to the sample holder, for instance -20 V. There is also the possibility that the simulation underestimates the final value of the surface potential, which may be even higher with real samples. Therefore, if we were to study the internal charging effects on the bulk samples, there is a significant probability that the samples would charge too quickly. The TEEY we would measure could then be falsified by the apparition of an uncompensated external electric field. When doing experimental measurements on the 20 nm thin films however, we have observed that the material does not charge past a couple of volts under continuous irradiation for a few tens of seconds. For that reason, we can safely use a negative bias of -9V, which is always stronger than the increase of the surface potential caused by the positive charge. This is the reason why we have chosen to use thin film samples to study the internal charging effects, instead of the bulk samples simulated in this section.

In summary, the Monte-Carlo model is able to simulate the external charging effects and the recollection of secondary electrons in conditions of positive charging. This is especially important since all experimental measurements will be made between the two points of crossover.

c) 𝐸 0 > 𝐸 𝐶2 , 𝑇𝐸𝐸𝑌 < 1

In this last case, the sample is charging negatively and the TEEY increases to 1, where it also stabilizes. However, the maximal incident energy used in the experiment is 2 keV, which is below the second crossover point of our SiO2 samples. Consequently, we will never be in this situation when confronting the simulation with the experiment. Moreover, the internal charging effects shown in Chapter 1, which we need to explain, were observed for incident energies between the two crossover points, therefore the situation of negative charging is out of the scope of this study. In any case, it is still necessary for consistency to study the behavior of the Monte-Carlo code for energies beyond the second crossover point.

Here, we will be using 2.5 keV incident electrons. This energy is close to the second point of crossover in the simulation, which is roughly 2.1 keV. According to the conventional theory, the sample needs to charge negatively up to -400 V, so that the energy of the incident electrons reaches the second point of crossover. Using our current simulation parameters on a 1 mm thick sample to improve computation time, the stabilization of the TEEY is not reached after 50 ms in Figure 5-11a. The TEEY actually starts to diminish before increasing and converging to 1 as what would be expected, and the negative surface potential is overestimated. When the recombination cross section is reduced to 10 -13 cm² (Figure 5-11b), the TEEY does stabilize at 1. This modification of the recombination cross section is a purely unphysical tweak, since we should expect the recombination cross section to remain the same for a given material, regardless of the incident electron energy. Despite this, the surface potential has stabilized at -780 V, which is a factor two above the expected value. This could be due to a dependence in energy that becomes relevant for electrons of 2 keV and above and/or for bulk samples that is missing in the model, or an issue that prevents the incident electrons from being slowed down by the negative surface potential. Another major drawback of the Monte-Carlo model in this situation is the computation time. Indeed, more energetic electrons create a lot of secondary electrons and drift particles that need to be transported individually. For instance, the 500 incident electrons of 2.5 keV used here generate 120 000 particles for each simulation step, if no particles are lost by recombination. In comparison, 500 incident electrons of 500 eV (as in the positive charging case) generate 20 000 drift particles per step. Hence, the computation time strongly increases for electrons of a few keV and above. Moreover, the sample needs to charge up to several hundreds of volts, but the initial TEEY is relatively close to 1. This means that the net negative charge implanted per electron is quite small compared to the case of positive charging with a TEEY of 2.5 or more. Contrary to the latter, we also do not have the recollection of secondary electrons accelerating the decrease of the TEEY. As a result, the TEEY and the surface potential can take a much longer time to reach an equilibrium, as we can see in In conclusion, the Monte-Carlo model can simulate the negative charge buildup created by electrons beyond the second crossover point, but with significant limitations. The initial parameters of the model had to be modified, since the effect of recombination was overestimated in this case, but this is an unphysical tweak as the recombination cross section should be the same for a given material. There is also probably a dependence in energy that could missing in some of the parameters of the model, which would create a significant error for electrons of a few keV. As a result, the simulation of the negative charge buildup caused by energetic electrons can only be validated in a strictly qualitative manner. The computation time also becomes excessive for incident electrons of a few keV. Admittedly, we have highlighted here a shortcoming of the Monte-Carlo code. In this work, we have chosen to not focus on the accuracy of the model beyond the second crossover point, given that the code already works in the 50 eV -2 keV range, which is the most relevant to our experimental measurements. Nevertheless, this is still a physical problem that needs to be fixed, in order to extend the simulation to electrons of a few keV and thicker materials.

Time-resolved simulation of the TEEY of SiO 2 thin films in the case of positive charging with external charging effects removed

In this section, we shall go back to simulating our experimental samples of 20 nm thin films, in positive charging conditions under a negative bias of -9V. This bias should prevent the global positive charge from generating a recollecting electric field in vacuum. As in the experimental studies, we will used a pulsed measurement procedure instead of the continuous measurement simulated in the previous section. To get an accurate time resolution, pulses of 100 µs with a spacing of 50 ms were used. The simulation results for the TEEY of 300 eV (a), 500 eV (b) and 1 keV (c) electrons are shown in Figure 5-12, obtained during 80 pulses of 100 µs. We can observe for the three energies a decrease of the TEEY. The TEEY decreases faster if its initial value is higher, as we can see by comparing the difference between the initial and final TEEY for the three energies. At 1 keV, the TEEY has been reduced by 13% after 6 ms, while at 300 eV the reduction is about 20% after 6 ms. At the same time, positive charges are created in the material and the surface potential is increasing, since the TEEY is greater than 1. However, due to the large capacitance of the thin film samples, the surface potential is evolving very slowly compared to the bulk samples of 5.8.1, and remains negative. Consequently, the decrease we observe here for the three energies cannot be caused by the recollection of secondary electrons, as in 5.8.1b. The phenomenon we observe here is strictly due to internal interactions of the electrons with charges. The assessment we make here was also observed on MgO [27], polycrystalline diamond [30], or Al2O3 [31], in conditions where positive external charging effects were compensated. For these materials, a decrease of the TEEY of electrons of a few hundred of eV was also observed as more and more incident electron pulses were sent on the target. Therefore, we can qualitatively validate the Monte-Carlo model for the simulation of the decrease of the TEEY initiated by internal charging effects, in the case of SiO2 samples.

Study of the effect of the parameters of the model on the TEEY

In this section, we compare the influence of the simulation parameters on the charge-less TEEY curve and on the decrease of the TEEY observed in the previous section. The key quantities for our simulation are the capture mean free path for empty traps, the recombination mean free path, and the detrapping frequency. These parameters will be modified compared to the reference values used so far.

First, we shall look at the TEEY spectrum of a charge-less sample, which is the static case at the beginning of the simulation. The material is free of charges so that no recombination can happen. We will not simulate the drift of charge carriers, hence the only parameter that can have an influence here is the capture mean free path of secondary electrons by empty traps. The combined capture+recombination mean free path of Equation 5-41 is used in all simulations. However, when the density of charges is null, it is equal to the capture mean free path by empty traps of Equation 5-29. This mean free path can be modified by changing either the capture cross section 𝜎 𝑆 or the trap density 𝑁 𝑆 .

In the examples shown in Figure 5-13, the capture cross section of the free shallow electron traps has been modified, whereas the electron shallow trap density remains constant at its default value of 10 21 cm -3 . We can see that the TEEY curve increases significantly when the capture cross section is increased, which creates a reduction of the capture mean free path. A capture cross section of 10 -16 cm² for the shallow traps makes the yield increase back to the value from Chapter 3 which was obtained without any trapping model. By modulating the capture mean free path, we are effectively doing the same thing as in Chapter 3, where we have changed the parameter S from the empirical polaronic capture model to decrease the TEEY from 8 to 4. The choice of a capture mean free path for secondary electrons is quite difficult, due to the significant dispersion in the TEEY measurements for a single material. It is also not trivial to choose between the various values of trap cross section and densities, and determine which kind of traps will be prevalent in a given sample. Indeed, in Figure 5-14, the TEEY obtained on our experimental samples used as a reference is plotted with the simulated charge-less TEEY and other experimental TEEY obtained on SiO2 samples. The TEEY from this work is higher than the data of Bronstein [32], but lower than the TEEY of Yong et al. [33] obtained on wet SiO2 samples. However, we have mentioned in section 5.3.1 that plasma grown samples had a different structure than wet oxides, which led to additional traps with a cross section of 10 -15 cm² in the plasma grown samples compared to the wet oxides. Wet oxides are also reported to have a trap density of 10 19 cm -3 [10] which is lower than the density used in the simulations for shallow traps (10 21 cm -3 ). Hence, the different nature of the samples can be a source of discrepancy between the various TEEY measurements. To further emphasize this point, data from Rigoudy et al. [34] obtained on thermally grown SiO2 samples has been included. Notably, this data was obtained with the same TEEY measurement facility used in this work, so the difference in TEEY between their samples and ours should only be due to the nature of the samples, if we assume that the effect of charging was limited. Indeed, there is also the possibility that some experimental measurements found in the literature could have been affected by charging effects that were not entirely compensated. It is also difficult to know whether the various points of the TEEY curves are representative of the start of the decrease of the TEEY due to internal charging, or if the TEEY has already been reduced before the final value was obtained. In fact, in the example of pulsed measurements, the value obtained for a given energy is the average TEEY over the pulse. Yet, if the TEEY has decreased during the pulse as in Figure 5-12, it is unclear what the TEEY value obtained over the pulse actually represents. Finally, we can study the effect of the charging parameters on the decrease of the TEEY observed in 5.8.2, in the example of 500 eV electrons, as in Figure 5-15. In this case, we add back all physical models we have detailed in this chapter: trapping, detrapping, drift transport of charges, dynamic computation of the field… First, the capture mean free path by free traps for secondary electrons is modified in Figure 5-15a, using the same values as Figure 5-13. This modification increases the starting point of the TEEY, since it increases the charge-less TEEY.

From the observations of Figure 5-12, we can also see that the decrease of the TEEY is much faster from a higher initial value. However, after an initial sharp decrease phase, the TEEY obtained with 𝜎 = 10 -15 cm 2 and 𝜎 = 10 -16 cm² appear to converge to the same final value at 8 ms. It is also possible that, if we increase the time period simulated, the TEEY for the three cross sections would converge to the same point. Therefore, there seems to be a second phase of decrease of the TEEY appearing after a few ms, which would not depend on the initial value of the TEEY.

In Figure 5-15b, the recombination cross section has been lowered from its initial value to 10 -13 cm². The starting point of the TEEY is not modified, since no charges are in the material at the start of the simulation. However the TEEY converges to a higher value at 8 ms with a lower recombination cross section. The decrease of the TEEY between these two points is also much sharper with a higher recombination cross section. This could indicate that the recombination of the drift charge carriers or even the secondary electrons would have a major influence on the decrease of the TEEY, which was notably suggested empirically in the experimental study of Belhaj et al [27].

Lastly, in Figure 5-15c the detrapping frequency constant 𝑊 0 for shallow traps has been reduced, which increases the time of residence of the charge carriers in the traps, and effectively disabling detrapping. By doing so, the evacuation of charges through the Si layer between two electron impacts is prevented and the TEEY in this case is decreasing more than with the default parameters. Since the charge density should be higher than with detrapping activated, the interactions of the electron cascade with the trapped charges should also be more probable, which joins the hypothesis made in the previous paragraph. Furthermore, the electric field should also be higher in the sample if more charges are trapped, and the field has also been shown to cause a reduction of the TEEY if it becomes large enough [7].

Conclusion of Chapter 5

In this chapter, we have conceived a Monte-Carlo model for the effect of positive internal charging on the TEEY of insulating materials, with a focus on SiO2 thin films due to the wide availability of reference data on this material. In this model, the experimental configuration of the DEESSE facility has been reproduced, so that our simulation results can be later compared to experimental measurements made with this facility. The code is able to simulate the drift transport of electrons and holes in 3D, until they are trapped and stored in a 1D mesh, so the effects of charging are effectively modeled in 1D. We have also modeled the trapping, detrapping, and recombination of holes, drift electrons and low energy ballistic electrons. To build our simulation, numerous new classes had to be created in a Geant4 application, and designed to interact with themselves and the Geant4 kernel to ensure proper transportation of the particles. We also had to add a 1D Poisson solver to Geant4, to compute the electric field after each simulation step.

In this chapter, the Monte-Carlo code has first been validated qualitatively, by verifying if it could reproduce the dynamic of the TEEY induced by external charging effects. The negative charging for electrons below the first crossover point and the positive charging for electrons between the crossover points is correctly modeled, with coherent values for the steady-state TEEY and surface potential. However, the simulation of the negative charging and TEEY for electrons above the second crossover point needs to be improved. In this case, the simulation parameters need to be modified, and the surface potential is overestimated. Given that the computation time also becomes disproportionate for electrons above a few keV (a few tens of hours up to a few days), we reach here the limits of the Monte-Carlo model. The perspectives of this PhD work regarding this limit could be to improve the modeling of the charging induced by electrons above a few keV, and improve the computation time by converting the code from single-thread to multithreading.

The architecture of the simulation and its processes are designed to be as general as possible. Therefore, the present Monte-Carlo code can be extended to any insulator. Simulating the TEEY of another insulator with charging effects can be done, if the transport of ballistic electrons can be simulated in this material with the interaction models of Chapter 3, and if the various parameters involved in the simulation of drift transport can be found. The choice of these parameters is especially capital, since each parameter can have a significant effect on the simulation of the TEEY. However, many different values for the charging parameters can be found in the literature in the case of SiO2. It can be difficult to choose a single value for the interaction cross sections or the trap density, for instance, since these values are highly variable.

For other less studied insulators, there can be no reference values in the literature and the charging parameters may have to be chosen arbitrarily.

Nevertheless, this also highlights the complexity in modeling the charge dependent TEEY of insulators, since many physical phenomena are involved, and these phenomena may have a stronger or weaker influence on the TEEY, depending on the material and the structure and purity of the reference sample. Despite all these uncertainties, the Monte-Carlo model can simulate a decrease of the TEEY in conditions of positive charging with the external field compensated, as the charge buildup increases in the material. This reduction of the TEEY strictly caused by internal charging effects has been properly simulated for SiO2. It is also coherent with the TEEY decrease observed experimentally on other insulators. However, we do not know yet what exactly creates this decrease, or how the positive charge buildup can influence the production or escape of secondary electrons. The parametric study made in 5.8.3 hints towards interactions between the electron cascade and the trapped charges, or between the drift particles themselves. It could also be an effect of the internal electric field on the transport of particles, but further proofs are required to validate any of these hypotheses. We also know that the simulation results are similar qualitatively to the decrease of the TEEY observed on other insulators, but we still need to quantitatively validate our results.

This will be the focus of Chapter 6, where we will present the experimental measurements made during this PhD thesis on SiO2 thin film samples in the DEESSE facility. We will then use the experimental results to validate our Monte-Carlo model. Finally, we will attempt to explain the experimental observations using the wide range of data we can extract from the simulations.

Chapter 6: Study of the effect of internal charging on the electron emission yield of silicon dioxide samples

The Monte-Carlo model developed in Chapter 5 should allow us to understand the physical processes behind insulator charging and its effect on the TEEY. Nevertheless, we have only compared the observations made with the simulation with experimental studies made on other insulators, in a strictly qualitative assessment. To ensure the coherence of our simulation, experimental measurements on SiO2 samples are needed for a quantitative validation.

In this chapter, we will use both simulation and experimental TEEY results that were obtained during this thesis work. The DEESSE and ALCHIMIE experimental measurement facilities of ONERA have been used to measure the TEEY of insulator samples. The validity of the ballistic electron transport models has already been shown quantitatively in Chapter 3, where we have demonstrated that the MicroElec Monte-Carlo model can accurately simulate the TEEY curve of non-insulator materials without charging effects. Therefore, the validation will focus on of the charging models introduced in Chapter 5. In this scope, we will present measurements of the TEEY depending on the incident energy, like the TEEY curves we have seen so far. Indeed, the incident energy/TEEY curve is the standard data used to evaluate the electron emission of a material, which is why the majority of TEEY data for insulators is only given in the form of an energy/TEEY curve. However, the effect of positive charging and the evolution of the TEEY can be assessed much more accurately with time-resolved measurements showing the temporal evolution of the TEEY at a single incident energy, as the time-resolved simulated results shown in section 5.7. Consequently, most of the results presented in this chapter will be time-resolved TEEY measurements. Several time-resolved simulations of the TEEY of dielectric materials can be found in the literature, but experimental data of the sort is much scarcer, and the simulated evolution of the TEEY was mainly driven by global charge effects. Hence, getting our own timeresolved experimental data on the decrease of the TEEY depending on internal charging is a focal point of this study, and will allow us to validate the Monte-Carlo code.

First, we will detail the measurement protocol used for the experimental measurements presented throughout this chapter. Then, the time-resolved experimental TEEY results obtained at room temperature on SiO2 thin film samples will be presented, and compared with the simulated results in order to validate the Monte-Carlo model. Once the model is validated, we can use it to retrieve data on the physics of charge transport, and provide explanations for the experimental observations. The code will first allow us to explain why the TEEY is decreasing with the positive charge buildup. From the conclusions gathered in this phase, we will highlight several experimental artifacts and bias that can appear when measuring the TEEY of dielectric materials. We will then explain how the charge buildup can falsify the TEEY obtained on an insulator, and what steps can be made during the experiment to avoid this falsification. We will also study the evolution of the TEEY until its stabilization, and the differences in the two TEEY measurement facilities used in this work. Finally, we will move away from TEEY studies in a controlled environment, and explore how the electron emission properties of a dielectric can vary in the conditions of the space environment. In this scope, we will study the variation of the TEEY with temperature from -180°C to 200°C, which is the typical temperature interval that space used materials are subjected to. We will also briefly probe what happens when the samples are not bombarded by a single energy electron gun, but by an energy spectrum and a current density that are representative of the incident radiation received from the space environment.

The values of the parameters used throughout this chapter for the simulation of charge transport in SiO2 are summarized in Table 6-1. Charge carrier mobility 20 cm 2 V -1 s -1 (drift electrons)

10 -5 cm 2 V -1 s -1 (holes)[1][1]
Electron cascade radius

Experimental measurement protocol

In this work, we have made experimental TEEY measurements on samples of amorphous SiO2 thin films, which we have already presented when developing the Monte-Carlo model. The samples were obtained from NEYCO, with the SiO2 layer grown on a Si substrate using plasma growth. The SiO2 layer has a thickness of 20 nm, and the samples are 5 cm wide.

All experimental measurements were conducted using the two TEEY measurement facilities at ONERA: DEESSE (Dispositif d'étude de l'Emission Electronique Secondaire Sous Electrons, Facility for emission of secondary electron under electron bombardment) and ALCHIMIE (AnaLyse CHImique et Mesure de l'émIssion Electronique, Chemical analysis and measurement of electronic emission). The equipments available in DEESSE (Figure 6-1) and ALCHIMIE (Figure 6-2) are listed below. An extensive description of the facilities can be found in the thesis of T. Gineste [2,3].

Most of the measurements presented in this chapter were made with the DEESSE facility [4]. It is made of an analysis chamber under ultrahigh vacuum (10 -9 mbar). In this chamber, a 1 eV -2 keV and a 1 keV -22 keV electron gun (Kimball Physics) can be used to measure the TEEY. These can be used in either a continuous or pulsed configuration. The sample can be tilted by 80° to study the TEEY at a given incidence angle. The TEEY can be measured with the hemispherical electron collector, or through the sample holder, following the method shown in Chapter 1. The latter technique is the one used in this work.

The Omicron hemispherical analyzer can also be used to study the energy spectrum of emitted electrons from 1 eV to 2 keV, and to perform in-situ Auger Electron Spectroscopy for surface analysis or Electron Energy Loss Spectroscopy. In this work, Auger spectrum analysis has been used to verify the chemical composition of the surface and ensure that the samples have been properly decontaminated. When sending electrons on the sample, we can measure the energetic spectrum of electrons emitted by the material with a hemispherical analyzer placed above the sample. Auger electrons are especially interesting, since each element has its own characteristic Auger transition energies. The Auger electrons can appear in the energy spectrum as very distinctive peaks around the transition energy. The spectrum can then be compared with a database of Auger transition energies per element, which gives us information about the percentages of each element in the surface. This analysis method has been used on the SiO2 samples after the decontamination phase. Only a small amount of hydrocarbon contamination remains (CKLL Auger peak, about 7% of contamination).

The samples can be decontaminated before the measurement in a separate transfer chamber, also under vacuum (10 -8 mbar). This chamber is equipped with a 10 eV -5 keV ion gun, which can be used with Ar or Xe ions for sample sputtering and erosion. The temperature of the sample holders can be regulated from room temperature (23°C) up to 200°C. This is the method we have chosen to decontaminate the samples before each campaign of measurements, by heating the sample during 48h at 200°C. Finally, the facility is equipped with a Faraday cup for electron beam characterization. The ALCHIMIE facility is also made of an ultrahigh vacuum analysis chamber (10 -10 mbar)

and a transfer chamber (10 -8 mbar). The analysis chamber is equipped with three electron guns (1 eV-2keV, 5eV-1keV, and 1keV-30keV) usable in pulsed or continuous configuration. The sample can be tilted up to 80°, and a 10 eV -5 keV ion gun can be used for erosion with Ar or Xe ions. The facility is equipped with a Sigma hemispherical electron analyzer which can record the energy spectrum of electrons from 1 eV to 3.5 keV. This analyzer can be used to perform in-situ surface analysis with X-ray Photoelectron Spectroscopy (XPS) using the built-in X-ray source. In this method, X-rays are sent on the target to generate photo electrons that can escape the material and be detected by the analyzer. Due to the low escape depth of the electrons (less than 10 nm), the energy spectrum (and therefore the TEEY obtained in the subsequent measurements) is strongly dependent on the chemical composition of the surface. Since photoelectrons have a distinctive energy that is proper to the chemical element they were generated from, the spectrum can then be used to determine the proportion of each chemical element of the surface. The facility is also equipped with a Faraday cup and a Pfeiffer mass spectroscopy gas analyzer. Finally, the sample holders can be cooled with liquid nitrogen down to -180 ± 10°C, and heated up to 200 ± 10°C. In this work, TEEY measurements were made with ALCHIMIE from -180 ± 10°C to 100 ± 10°C, following the same protocol as with the DEESSE installation. During the secondary electron emission measurements in both installations, the sample holder is biased to a -9 V potential, so that the surface potential remains negative and the secondary electrons that escape the material are not recollected by the surface. The sample is irradiated by a 2 keV Kimball Physics electron gun with pulsing capabilities in a defocused beam configuration (> 25 cm² down to 0.1 cm²), with an incident current of 0.1 to 1 µA, giving an incident current density ranging from less than 25 nA/cm² up to 20 µA/cm².

All measurements and simulation results were obtained under normal incidence. Indeed, sending electrons with a given angle of incidence may introduce 3D effects that we may not be able to reproduce with our 1D sampling of the charge distributions. Moreover, we are focusing on interactions of the electrons inside of the material, which should modify the TEEY regardless of the incidence angle.

The TEEY measurement procedure used in this work is the one presented in Chapter 1, which is based on two measurements of the current flowing through the sample. First the sample holder is biased to a potential of +27V, to force the recollection of all low energy secondaries. The current 𝐼 𝑆 + measured during this step is very close to the incident current (𝐼 0 ≅ 𝐼 𝑆 + ). Then, the sample holder is biased to a potential of -9V, to prevent the recollection of secondary electrons that can be induced by the positive charging of the sample. The current 𝐼 𝑆 -measured in this case can be used to deduce the emitted current, using the value of 𝐼 0 from the previous step. Finally, the TEEY is obtained from the ratio of emitted current over incident current:

𝑇𝐸𝐸𝑌 = 𝐼 𝐸 𝐼 0 = 𝐼 0 -𝐼 𝑆 𝐼 0 = 𝐼 𝑆 + -𝐼 𝑆 - 𝐼 𝑆 + Equation 6-1
Since we are working with dielectric samples, all measurements were made in a pulsed configuration instead of continuous incident current. This means that the sample is only irradiated with incident electrons for a certain duration (a few ms typically), then the electron gun is cut for a relaxation period (a few tens to hundreds of ms). This procedure greatly limits the influence of charging on the TEEY. In pulsed measurements, a first series of pulses is sent to measure the incident current, then a second series is sent to get the emitted current, from the procedure shown above. In TEEY experiments made on dielectric materials, we do not directly measure the current flowing through the sample, since they are of course insulators. Instead, we measure the opposite current generated by the creation of an image charge in the metallic sample holder.

In the simulation, the number of incident electrons is already known, and stays constant for each simulation step. Therefore, only a measurement of the number of emitted electrons is needed, and the sample holder is permanently biased to -9 V in the simulation. We also count the number of electrons exiting the material with an electron collector surrounding the sample, instead of a measurement through the sample holder.

The energy/TEEY curve is obtained through an automated program. For each incident energy, two series of 10 pulses are sent, with a duration of 6 ms per pulse and a relaxation period of 200 ms between each pulse. The first series gives the value of 𝐼 0 , and the second series gives 𝐼 𝐸 . The current measured in both cases is the sum of the current acquired during each pulse. Hence, the TEEY for each incident energy is the averaged TEEY over 10 pulses. During time-resolved measurements of the TEEY at a single energy, 80 to 100 pulses of incident current of 1 µA and 100 µs duration are sent, with a 50 ms relaxation period between each pulse. The pulses were made shorter in order to improve the resolution of the measurement. Indeed, a single point is acquired at the end of each pulse, which is the current measured during the pulse. By using 100 µs pulses instead of the standard 6 ms pulses, the resolution of the measurement is greatly improved, and we can more accurately observe the decrease of the TEEY. The relaxation period is necessary for the instruments to record the current through an oscilloscope. If the relaxation period is too short (below 10 ms), parasitic capacitances can perturb the recording of the sample current. The standard pulse parameters used in both experimental measurements and numerical simulations of this chapter are summarized in Table 6-2. In this section, all measurements and simulations were made with an incident current of roughly 1 µA, and a beam spot size of 0.1 cm² was used to ensure that we could observe a modification of the TEEY due to the internal charge interactions. Therefore, the current density J0 of all simulations and measurements of this section is 10 µA/cm², unless specified otherwise.

Using the DEESSE facility, the TEEY has been measured at room temperatures for incident energies of 300 eV and 1 keV. In these conditions, the material is charging positively: more holes than electrons are created in the material. For most of this study, we will interest ourselves in measuring and studying the evolution of the TEEY during 80 or 100 pulses of incident electrons, for a total irradiation time of 8 to 10 ms. This duration should allow us to observe a marked decrease during the experiments, while keeping the computation time reasonable. Indeed, in Chapter 5, we observed in such conditions a decrease of the TEEY with time. We noted a reduction of the TEEY by 13% to 20% after only 6 ms, which is the length of a single pulse in conventional TEEY measurements. Therefore, a total irradiation time of 8 ms should already be enough for us to validate the code, and try to provide physical explanations to this decrease.

From Figure 6-3, we can see that a very similar decrease to the simulation is measured experimentally for these two incident energies. The TEEY starts at a value greater than one, and is reduced at variable speeds depending on the initial value of the TEEY and the electron energy. This decrease is observed even though the positive external charging effects are removed. For 1 keV electrons the decrease of the TEEY is very sharp during the first ms, and a steady state is reached quite rapidly after 2 ms. The simulation results can then be compared with the experimental measurements we have obtained, with the comparison shown in Figure 6-4. In this figure, the experimental results are adjusted by 20 % so that the starting point of the decrease of the experimental TEEY matches with the simulations. That is to say, the experimental TEEY curve at 300 eV is multiplied by 1.2, and the experimental curve at 1 keV by 0.8. This allows us to directly compare the temporal decrease of the TEEY. As in the experiment, a series of 100 µs incident electron pulses are sent on the material, with a relaxation time of 50 ms between each pulse. For both 300 eV and 1 keV incident electrons, the simulation is able to reproduce the evolution of the TEEY due to the positive internal charging. The difference between the initial TEEY of the experiment and the simulation is due to the fact that the simulations start from a perfectly flat and decontaminated target, with no trapped charges inside. This is not the case for the experimental samples, which may have some surface roughness, a small amount of residual hydrocarbon and/or hydroxide contamination, or residual charges that are already deeply trapped at the start of the experiment. The starting point of the TEEY in the simulations is also strongly dependent on the density of traps, which limits the mean free paths of the low energy secondary electrons, and the final point of the decrease strongly depends on the recombination cross section. However, these parameters are difficult to choose accurately, as already stated.

From Figure 6-5, we can see that there is a difference of about 10 to 15% in amplitude between the charge-less TEEY from the simulations, and the experimental TEEY obtained with a broad beam configuration ( > 25 cm²). We will show in 6.3 why the TEEY measured with a broader beam is the closest we can get to the charge-less TEEY. This difference in the charge-less TEEY can also be a source of discrepancy for the start of the decrease in the time resolved results.

Finally, there is also the possibility of a space charge close to the surface which can force the recollection of secondary electrons during the measurement. However, previous studies on this experimental setup have shown that these effects are negligible [5]. Another possibility is that the parameters chosen for the simulation of charge transport are only representative for low electric fields and short times. Indeed, at 8 ms, the error between the simulation and the experiment increases up to 15 % and the curves start to diverge. As the time of irradiation increases, the electric field inside of the material is also increasing. As we will investigate in 6.2.2.1, if the electric field becomes higher than a certain threshold, the transport of charge carriers enters a different regime, which is governed by the electric field instead of the scattering mechanisms. To model the transport of charges in this high-field regime, we could have to change of our current charging parameters.

In section 6.5.1, we will present additional measurements that were made on ALCHIMIE to sample the TEEY over several seconds until it reaches its final steady-state. However, for the remainder of this current study, we shall interest ourselves in the 8 first ms of the decrease. We consider an error of 20% at the start of the decrease and a divergence of 15% after 80 pulses to be very satisfying, given the approximations of our model, the numerous processes involved in insulator charging which can be a source of errors, the wide spectrum of possible values found in the literature for the parameters of our model, and the strong dispersion of the experimental TEEY data obtained on insulating samples. What's more, the model is able to reproduce the decrease of the TEEY over multiple energies which have a quite distinct behavior. As a result, we consider that the simulations are accurate enough to explain how the internal charging leads to a decrease of the TEEY.

Explanation of the experimental observations through the simulation:

Study of the internal charge transport

Study of the internal electric field

Even if the recollection of the low energy secondary electrons is impossible due to the negative applied surface potential, other internal mechanisms could be the cause of the reduction of the TEEY. First, the electric field generated in the material by the trapped charges can modify the trajectories of the secondary electrons by accelerating them between interactions. According to Fitting et al. [6], electric fields above 0.5 MV/cm are strong enough to increase or reduce the escape depths by a few nanometers. Such high electric fields can also strongly accelerate the drift electrons and force them to move in the direction of the field [7]. The drift electrons can then be accelerated up to a few eVs [8], which can modify the phonon collision and trapping mean free paths. In such a case, the mobilities of the charge carriers may also be significantly affected. There are in fact three regimes of transport that the drift electrons can follow in silicon dioxide, depending on the value of the electric field and the energy they gain between two collisions [9,10]. When the field is lower than 0.5 MV/cm, the electrons cannot gain enough energy between two collisions with LO phonons and their mobility is steady. Above 0.5 MV/cm, the LO phonon collisions cannot prevent the electrons from heating up and gaining energy, which is the optical runaway phase. Here, the acoustic phonon collisions contain the electron energies up until fields of 3-4 MV/cm [9], when acoustic runaway occurs. If the field increases past this value, the electrons may gain enough energy to create secondary electrons through impact ionization, for fields close to the breakdown value in SiO2 (10 MV/cm). The trapping cross sections for coulombic attractive traps should also be lowered if the field and the electron energies increase [11].

Due to the approximations used for the drift particles, the transport of these particles may not be accurately modeled at very strong electric fields. Indeed, we approximate the collision with phonons as a single trajectory which would be the sum of all collisions, and we use a single value of mobility for the particles. With this approach, we should be able to model the fact that the drift particles are all forced to travel in the direction of the field, if it is high enough. But as we saw, the mobility should evolve for strong electric fields, along with the phonon collision frequency and mean free path. We also do not consider the evolution of the capture cross section depending on the drift electron energy. Nevertheless, such fields are not reached in our case, as shown in Figure 6-6 where the internal electric fields in a SiO2 sample after 80 pulses of 500 eV and 1 keV electrons are plotted. The maximum field value at 500 eV is about 0.04 MV/cm, and 0.34 MV/cm at 1 keV. These values are much below the 0.5 MV/cm threshold above which the transport of electrons will be impacted. Since 500 eV is around the maximal value of TEEY, this is the field obtained when the net quantity of charges created in the material is maximal, and should be the highest value of electric field attainable. Indeed, if we sample the electric field in the material after 100 pulses of 1 keV electrons, the value of the field is lower, as is the TEEY. Since the electric fields remain low enough, the approximations used for the transport of drift particles are still valid. Indeed, while the value of the electric field could be strong enough to force the charge distribution to move in its direction, the electric field is not high enough to create a regime of optical runaway.

On the other hand, the dependencies of the phonon collision MFPs with energy are all included in the transport of ballistic electrons, and we should be able to simulate the chargeless TEEY/energy curve in the presence of a strong internal field. Therefore, probing the transport of very low energy electrons at high fields could be another validation of the Monte-Carlo code.

In Figure 6-7, the internal electric field has been set to given values from 0 MV/cm up to 4 MV/cm. The field is uniform in the whole thickness of the SiO2 sample. The sign of the field is its orientation along the z axis. For positive values (a), the field is oriented down to the Si layer, and the electrons are accelerated towards the surface. For negative field values (b), the electrons are accelerated deeper into the material. We can see that there is no modification of the TEEY at 0.04 MV/cm, which is the maximal value reached in the simulations. From ±0.5 MV/cm however, the escape depth of the secondary electrons starts to be modified, and the TEEY is increased or reduced depending on the orientation of the field. This modification of the TEEY is becoming more visible This is coherent with other Monte-Carlo simulation works, who have observed a modification of the TEEY in SiO2 from 0.5 MV/cm and above [6,7,12]. In conclusion, our simulation code can qualitatively model the effect of a high electric field on the transport of ballistic electrons. However, we do not have data on the actual field reached in the thin film samples and its evolution during the experiments, and we cannot affirm whether the electric field reached in the time-resolved simulations is correctly estimated. We know that the final potential difference between the sample holder and the surface is less than 9 V over 20 nm, which gives a global electric field of at most 4.5 MV/cm. In such a case, we could have strong field effects modifying the TEEY. However, in both simulations and experiments, we do not reach the final stabilization of the TEEY and potential. Therefore, we do not know the actual value of the field at the end of our measurements, and we do not have access on the value of the field depending on the depth, such as in Figure 6-6. Moreover, the final surface potential is observed in continuous irradiation, whereas we have used pulsed measurements throughout this chapter, so that the charges can be evacuated between two pulses. Hence, the 4.5 MV/cm is a worst case value, which is never reached during pulsed measurements. Given that the decrease of the TEEY is consistent with the experiment, we will assume that the field computed in the simulation is within the right order of magnitude, despite the lack of experimental data on the internal field during the measurement.

Taking this into account, we did not simulate the drift transport in high fields since we know that the approximations used in this work become invalid. However, this regime is never reached in the simulations, as the field always remains lower than 0.1 MV/cm. One of our hypotheses to explain the decrease of the TEEY was that the charge buildup was creating an electric field, which could interfere with the transport of secondary electrons and reduce their escape probability. However, the actual field is not strong enough to have a significant impact on both the ballistic and drift transport. Therefore, we can rule out this hypothesis.

Study of the internal charge density

We have found that the decrease of the TEEY is not created by the internal electric field. Consequently, it is very probable that some interactions between the drift charge carriers and the secondary electrons could perturb the electron cascade and prevent the secondary electrons from escaping, or even being produced. While the transport of low energy electrons is already disturbed by the capture by free traps, the presence of charges in the material could amplify this phenomenon. To verify this, it is necessary to study the internal charge density, and its evolution with depth and time. First, we will interest ourselves in the shape of the charge/depth profile at the end of the decrease. In Figure 6-8, the total simulated charge densities at the end of 100 pulses (10 ms) of 300 eV and 500 eV electrons are plotted (a). They are compared with the charge density sampled by sending 10000 electrons on the target without simulating the drift of charge carriers or using a charge bias factor (b). This plot gives us information on what the charge density would be if there was no charge carrier transport or recombination, which would be the result of a single electron cascade before the drift transport is simulated. In this specific case, we deviate from the default 500 electrons sent for each simulation time step (1µs). Indeed, increasing the number of electrons allows us to reduce the simulation noise. We do not send 10000 electrons in the charging simulation since this needs to be repeated every step, and the computation time would be excessive. Here, the simulation of Figure 6-8 (b) only needs electrons to be sent once, which is why we can afford to send 10000 electrons at once. The 300 eV and 500 eV curves have a similar profile. Browsing the charge density profile from the surface, a positive region appears in the first few nanometers of the surface. In this region, more holes than electrons are captured, with the peak density of holes at 2.5 nm below the surface for 300 eV and at 5 nm for 500 eV electrons. The material has a strong positive charge up until 7.5 nm at 300 eV, and 14 nm at 500 eV. Indeed, electron-hole pairs are created as the incident electrons go through the sample and depose their ionizing dose. However, only the electrons that are close to the surface are able to successfully escape the material. When a secondary electron crosses the surface, the hole, which is much less mobile, is left in the material as a net positive charge. The total charge density becomes negative at a higher depth for 500 eV electrons, since these electrons are more energetic. Indeed, their ionizing dose is deposed

(a) (b)
through a greater thickness than 300 eV electrons, which leads to electron-hole pairs being created deeper. From the drift-less charge profiles, we can extrapolate the escape depths of the secondary electrons produced by electrons of a given incident energy. Going by the position of the peak hole density, we can also estimate that most secondary electrons are created around 3 nm below the surface. The escape depth should then be 4 nm for 300 eV and 6 nm for 500 eV incident electrons. Below the escape depth, the secondary electrons are not able to escape and will become thermalized. Since no negative charges are lost, the sum of charges in this region should be equal to 0. However, there is still a net negative charge. Indeed, we are now reaching the implantation region of the incident electrons. The extrapolated range of electrons in SiO2 has been extracted from simulations on a charge-less sample, and plotted in Figure 6789. We can see that the peak of the negative charge density is spread around the value of the extrapolated range, when the transport of the particles is not simulated. This is also why the negative region is placed deeper for 500 eV electrons, which have a greater range than 300 eV electrons. The charge density profile of 1 keV electrons is quite interesting, as shown in Figure 6-10, since the whole thickness of the material is positively charged. There is also a larger net positive charge than for the other electron energies, especially close to the interface with the silicon substrate. It is known that more energetic electrons will create more electron-hole pairs, but the TEEY of 1 keV electrons is lower than 300 eV or 500 eV electrons, which contradicts the fact that the positive net charge is larger. The extrapolated range obtained for 1 keV electrons in SiO2 (30 nm) is greater than the thickness of the sample (20 nm). This means that the implantation region of the primary electrons is mainly in the Si substrate, which should explain the lack of a distinct negative charge region in the SiO2 layer. The electrons may also be implanted deep enough that a significant part of them can escape through the substrate, resulting in the loss of negative charges and a net positive charge appearing close to the SiO2/Si interface. This leads to a large net positive charge close to the SiO2/Si interface when we sum the quantity of positive and negative charges to get the total charge density plotted here, hence the peculiar shape of the charge distribution.

In Figure 6-11, we can see the charge profiles obtained by sending 100 000 electrons and disabling the drift transport, which gives us the result of a single electron cascade as in Figure 6-8b. We find for the 20 nm sample a conventional shape for the charge density instead of the strictly positive curve after 100 pulses. We can also see that most electrons in the 20 nm sample are created within the 5 closest nanometers to the SiO2/Si interface, which is why they are able to escape the dielectric layer very easily. Finally, the peak of negative charges when the drift transport is disabled is located at 25 nm in Figure 6-11. This gives the mean implantation depth of 1 keV electrons, which is also greater than the 20 nm thickness of our samples. As we want to understand a time-dependent decrease, we should now look at the evolution of the charge density through the irradiation time. In Figure 6-13, the charge density profiles measured through the 100 pulses of the simulation are shown, with one curve sampled each 50 µs. The elapsed time increases as we move from blue to red. In Figure 6-12, the position of the positive and negative charge peaks is also plotted as a function of time, which gives a clearer estimation of the migration of charges. From these figures, we can see that the motion of holes is very limited, with at most 1 nm of difference between the positive charge peak of 500 eV electrons at 0 pulses and 100 pulses. For electrons, which are much more mobile, the peak of negative charges is moving towards the SiO2/Si interface at 300 eV and 500 eV. At 300 eV, it moves from 7 to 12nm, and at 500 eV from 10 to 17 nm. However, given the orientation of the field, we would be expecting the electrons to travel towards the surface instead.

It is also notable that the positive charge peak is increasing, but the negative charge peak starts decreasing after a given time. It is possible that the electrons are actually moving towards the surface and meeting the hole population. When these electrons reach the hole population, they get lost by recombination, which decreases the negative charge. This phenomenon can be seen for 500 eV electrons, where the charge density becomes negative at 8 nm at the start of the simulation, but is only negative above 15 nm after 100 pulses. However, another phenomenon is occurring in the opposite direction: the drift electrons are escaping through the silicon substrate, resulting in the loss of these negative charges and a net positive charge. The leak of electrons in the silicon layer should then be more pronounced for 500 eV electrons than 300 eV. This is evidenced by the migration and reduction of the negative charge peak at 500 eV, hinting towards a leak of electrons in the later stages of the simulation.

In Figure 6-6, we saw that the peak value of the field was on the order of 10 4 𝑉/𝑐𝑚. According to Fitting & Friemann [7], such a field does not have any effect on the trajectories of electrons of 0.1 eV. Our drift electrons are generated with an energy of 3/2 kT = 0.04 eV. It is therefore possible that the electric field in the sample is not strong enough to force most of the electrons to follow its opposite direction, and that electrons are traveling in all directions due to thermal agitation. Hence, the electrons implanted close to the Si layer still have a significant probability of leaking into the substrate, even though the direction of the field tends to accelerate them away from the interface. Another part of the electrons can be traveling towards the surface and recombining with the holes, which is why we have two opposite phenomena occurring at the same time. This would also explain why the holes are practically not moving, as they have an even lower mobility.

Finally, we can compare the charge density profiles of 1 keV electrons during the first pulse and the second pulse, following the 50 ms relaxation period. This comparison is shown in Figure 6-14. We can see that during the first pulse, we still have a conventional charge profile with a negative region after 12.5 nm. Due to the neutralization of positive and negative charges, the total charge profile is very close to 0. However, after the relaxation period, the charges have separated. Most electrons have escaped in the silicon layer, but the holes have remained in the SiO2 layer. This results in a net positive charge near the interface, which is due to the escape of electrons in the silicon layer and confirms our previous explanation. The evolution of the charges in the first 5 nanometers of the surface should be critical, as it is the region where most secondary electrons are created and escaping from. In the following, we will study the evolution of the total density of charges in the first 5 nanometers below the surface, which will be referred to as the surface charge density. From the charge profiles we have seen so far, we can expect this charge density to be strictly positive. It should almost be composed of trapped holes only, although a small part of electrons may get trapped very close to the surface. This density will be expressed in charges/cm², and given by an integral of the number of charges per cm 3 𝑁 that is analogue to the formula of the surface energy deposit from Chapter 4:

𝑁 𝑆𝑢𝑟𝑓 (cm -2 ) = ∫ 𝑁(𝑧) 𝑑𝑧 5 𝑛𝑚 0 Equation 6-2
In Figure 6-15, the evolution of the surface charge density is plotted with the evolution of the TEEY for our three energies of interest. We can see a definite correlation between the timeresolved increase of the surface hole density, and the decrease of the TEEY. Indeed, when the surface hole density increases at a stronger rate, the TEEY also decreases at a stronger rate. For instance, at 1 keV, the decrease of the TEEY is weaker than for the other energies, and the increase in the surface hole density is also weaker. From the interactions of the charge carriers shown in Chapter 5, we can recall that the probability of recombination of the drift electrons and ballistic electrons increases as the density of trapped holes also increases. From the very similar evolution of the TEEY and the charge density, it is very probable that the recombination of electrons with holes is the driving mechanism of the decrease of the TEEY. Indeed, a higher hole density increases the recombination probability, which should lower the overall capture mean free path of secondary electrons escaping the material. If an increasing number of secondary electrons is lost due to recombination before they can reach the surface, this should cause a reduction of the TEEY. The incident electrons are also susceptible to recombine, which could prevent them from creating secondary electrons in the first place. However, this recombination is only possible at the end of their path, when their energy has fallen down to a few eV and they have already created an electron cascade above their implantation depth.

Since the value of surface charge density is known through the whole simulation, we can try to prove this correlation by quantifying the decrease of the TEEY and confronting these two quantities. This can be done by computing the relative variation of the TEEY (

∆𝑇𝐸𝐸𝑌 𝑇𝐸𝐸𝑌

), compared to its value at t = 0. It is expressed as:

∆𝑇𝐸𝐸𝑌 𝑇𝐸𝐸𝑌 (𝑡) = 𝑇𝐸𝐸𝑌(0) -𝑇𝐸𝐸𝑌(𝑡) 𝑇𝐸𝐸𝑌(𝑡) Equation 6-3
Using this formula, we aim to follow the method given in the experimental study of Belhaj et al. [13]. They have calculated the surface density of positive charges from the surface potential ∆𝑉 𝑠 , the capacitance 𝐶, and the surface of the sample 𝑆 with the formula

𝑁 𝑆𝑢𝑟𝑓 (cm -2 ) = 𝐶 𝑞𝑆 ∆𝑉 𝑠 Equation 6-4
They have then plotted the relative variation of the TEEY of MgO at 200 eV with the surface hole density, and observed a linear relation between these two quantities. They have deduced an empirical correlation relationship that links the relative decrease of the TEEY to the surface charge density 𝑁 𝑠𝑢𝑟𝑓 as:

- ∆𝑇𝐸𝐸𝑌 𝑇𝐸𝐸𝑌 ≈ 𝑆 𝑒-ℎ 𝑁 𝑠𝑢𝑟𝑓 Equation 6-5
Where 𝑆 𝑒-ℎ is defined as an effective recombination cross section. In Figure 6-16., we have plotted the same correlation, using the TEEY/TEEY and the 𝑁 𝑆𝑢𝑟𝑓 extracted from the Monte-Carlo simulation for energies ranging from 100 eV up to 2 keV. There is in fact, for all energies, a strong linear correlation between the hole density and the relative variation of TEEY, with a R² of 0.99 except for the higher energies. For all energies except 100 eV, the curves are superimposed, and the values of the effective recombination cross section are almost identical with an average of 𝑆 𝑒-ℎ = 5.7 × 10 -12 cm 2 . This cross section is in the same order of magnitude as the recombination cross section used in the Monte Carlo simulations, which is 𝜎 𝑒-ℎ = 2 × 10 -12 cm². This result definitely shows that the recombination of electrons with holes is the internal mechanism at the origin of the decrease of the TEEY at all energies. When the hole density increases, more secondary electrons recombine and are lost before they can escape the material. In consequence, the TEEY is reduced. With our simulation results, we have also confirmed the hypothesis formulated experimentally by Belhaj et al. Noticeably, the linear correlation does not fit the very start of the decrease of the TEEY, as shown in Figure 6-17 in the example of 300 eV and 1 keV electrons. In this phase, the material is not uniformly irradiated yet, and the overlap factor is lower than one. When the material is uniformly filled by holes, the recombination takes over as the driving mechanism of the evolution of the TEEY. Hence, the evolution of the TEEY we measure after a given time becomes only dependent on the properties of the material regarding charge creation, transport, and trapping. This is coherent with the fact that this evolution of the TEEY was not only observed by ourselves on SiO2, but also by several experimental work on other insulators.

We can now also explain why the various parameters of the model had a strong impact on the evolution of the TEEY shown in the parametric study of section 5.7.3. Lowering the recombination cross section meant the TEEY had a very small reduction compared to the default case. By lowering this cross section, the trapped holes are less attractive to the secondary electrons, which have a longer global capture mean free path. Their probability of recombining before escaping is lowered, and the TEEY is increased. When the detrapping frequency factor was lessened, the decrease of the TEEY was sharper. In this case, the migration of trapped charges by hopping was prevented. From Figure 6-8, we have shown that the holes are created in the first 5 nanometers of the surface, but they can migrate a couple of nm deeper in the material, which will reduce the surface hole density. By deactivating the detrapping, the density of holes close to the surface is higher, and so is the recombination probability. The recombination with holes had already been proposed by other experimental works as an explanation of the decrease of the TEEY, in the case of negatively biased samples under a defocused beam [13][14][15][16][17]. This hypothesis is confirmed here both qualitatively and quantitatively by our numerical simulations, and the experimental measurements we have made on SiO2 samples. In conclusion, we have successfully provided an explanation for the misunderstood decrease of the TEEY, with our Monte-Carlo simulation code and the study of the charge transport.

6.2.3 Highlighting the effect of the presence of residual holes in the sample at the start of a measurement

So far, we have assumed in our simulation that the samples are perfectly charge-less at the start of the measurement. However, this may not be the case for the real samples used in the experimental study. Indeed, the charges created in the material during irradiation may not be completely evacuated in between two TEEY measurements. If the residual surface hole density is high enough, the secondary electrons produced will have a significant probability of recombining with these holes, as we have shown in the previous subsection. In simulation codes, we have complete control and knowledge of the charge density inside of the material. In TEEY experiments however, it is not possible to know the evolution of the charge in the same time as the evolution of the TEEY. We can only use a Kelvin probe to measure the surface potential after the sample has been bombarded by electrons, which only gives us information on the total charge remaining in the sample after a relaxation period. Consequently, at the start of the TEEY measurement, we do not know the charge state of the material. This can be problematic if we perform an experimental measurement on a sample that is still filled with residual holes. In fact, the presence of charges can lead to an error in the measurement of the TEEY, which can be lowered due to the recombination.

This effect is shown in Figure 6-18, where we have made two experimental measurements of the TEEY of 1 keV electrons in SiO2, in the DEESSE facility. The first TEEY measurement was made on a sample that was left at rest for two days since the last measurement, so we can consider that practically all charges should have been evacuated, except the charges captured by very deep traps (> 1 eV). The second measurement was made in the middle of a measurement campaign on the same sample. One can see that the second measurement is shifted compared to the first data set. It is possible that the electron cascades created during the previous TEEY measurements may have left residual charges that have perturbed this measurement. While we do not precisely know the quantity of charges injected between the two measurements, this first plot already shows in a qualitative way that two measurements following an identical protocol may not yield the same value of TEEY, depending on the charge state of the sample. A more quantitative study on this phenomenon has been made with the ALCHIMIE facility. This time, we have made two TEEY measurements using 300 pulses of 500 eV electrons. The second measurement was made right after the first one, so we have a better estimation of the quantity of charges left in the material at the start of the second measurement. From the experimental results in Figure 6-19, we also observe a downwards shift of the TEEY obtained in the second measurement. To prove that it is indeed the presence of residual holes in the sample that is falsifying the measurements, this phenomenon has been reproduced in the Monte-Carlo code. We will attempt to simulate the same situation as in the experiments made with ALCHIMIE, where we have made a second measurement of the TEEY immediately after the first measurement. However, we cannot know precisely the quantity of charges inside of the material at the end of an experimental measurement. Consequently, this study can only be qualitative. Instead of having the simulation start from a perfectly virgin sample, we can introduce a density of holes that are already deeply trapped at the beginning of the simulation. We can show that only the deeper traps can retain holes between two measurements, by calculating the time of residence of holes 𝜏(𝐸 𝑖 ) in a shallow trap of depth 𝐸 𝑖 . It is given by the standard activation law as:

𝜏(𝐸 𝑖 ) = 1 𝑊(𝐸 𝑖 ) = 1 𝑊 0 exp (- 𝐸 𝑖 𝑘𝑇 ) Equation 6-6
Where 𝑊 0 = 10 3 s -1 for shallow traps, and 𝑇 = 300 K. If we take a shallow trap with an energy depth of 0.1 eV, this gives a time of residence of 10 -3 s. Let's compare this to the most frequent deep trap in SiO2, which is the oxygen vacancy with an energy depth of 2.4 eV. According to Equation 6-6, the immobilization time of a charge carrier captured by such a trap is 10 20 years. While the actual immobilization time of the charge carrier in this trap will be reduced thanks to the Poole-Frenkel and Phonon Assisted Tunneling enhancements, it should still remain much larger than the time scale of a TEEY measurement. Accordingly, in the ten of seconds that separate two measurements, the particles in shallow traps should have had ample time to either exit the sample, or get fixed in deep traps.

The distribution of residual holes we introduced is assumed to follow the distribution of positive charges shown in Figure 6-8 at the end of 100 pulses. To take into account the fact that not all charges will end up in deep traps, we will assume that 50% of the charges at the end of the 100 pulses have been captured by deep traps, and the rest have escaped or recombined. Therefore, the actual charge density used in the simulation will be half of what is given below. The charge densities of Figure 6-8 for 300 and 500 eV electrons at a depth z (in nm) are approximated by a Gaussian law as:

𝑛 ℎ (𝑧) = 𝑛 0 exp (- (𝑧 -µ) 2 2𝜎 2 ) Equation 6-7
With 𝑛 0 = 1.38 * 10 9 , µ = 2.7 nm, 𝜎 = 1.8 nm at 300 eV, and 𝑛 0 = 1.5 * 10 9 , µ = 4.7 nm, 𝜎 = 3.5 nm at 500 eV. Due to the singular shape of the charge distribution after 100 pulses of 1 keV electrons, a better fit was achieved for this energy with an exponential law in the form of:

𝑛 ℎ (𝑧) = 𝑛 0 exp(𝛼(𝑧 -19 nm))

Equation 6-8

Where 𝑛 0 = 5.5 * 10 9 and 𝛼 = 0.1. For this energy, there was an excessive lowering of the TEEY (below 1) when using 50% of the density of Equation 6-8, so we have used 25% of the density instead. The approximated densities given by these relationships are plotted in Figure 6-20 in dotted lines, and compared with the reference charge density after 100 pulses in solid lines. In the simulation, the 𝑛 ℎ (𝑧) used is 50% or 25% of the values given by Equation 6-7 or Equation 6-8, which is then divided by the charge bias factor to get the actual number of trapped particles to add in each cell of the mesh. The simulated TEEY of 300 eV, 500 eV and 1 keV electrons for a sample including residual holes is compared in Figure 6-21 with the simulation results obtained on a sample that is initially charge-less. The shift of the TEEY curve in the charged sample is clearly visible, and the phenomenon we have observed experimentally is accurately modeled by the simulation code. This confirms that surface trapped holes are at the origin of the phenomenon shown in Figure 6-18, by increasing the recombination probability of the secondary electrons. For all curves, the residual holes TEEY starts roughly at the value reached at the end of the charge-less TEEY. This shows that our estimation of the proportion of holes remaining in the sample immediately after the first simulation is within the right order of magnitude. When doing successive experiments, the difference in the TEEY is not as pronounced as in the simulation, since there is always a rest time of several seconds when changing the measurement parameters. A part of the holes may also be removed during the measurement of the incident current, which is made before measuring the sample current. In section 6.2.1, a scaling factor of 20 % had to be applied to the experimental data, so that it could be compared to the simulation. While this is due to the differences in the charge-less TEEY and the presence of surface contamination on the experimental samples, this difference could also be due to these residual holes. It is also possible that the starting point of the TEEY in our time-resolved measurements does not correspond to a fresh sample, and that the actual charge-less TEEY is higher than our reference data.

In conclusion, the presence of residual holes from a previous irradiation can influence the TEEYs measured afterwards if the holes have not evacuated or been compensated. Since the transport of holes is heavily dependent on the material, the situation shown in this section may only occur in the less conductive insulators. Indeed, this phenomenon can only occur if the hole density remains high enough after a given relaxation period. This can be due to a high density of deep traps, or a very long time of residence in traps, which can prevent the holes from being evacuated.

Implementation of a charge compensation procedure to avoid falsification of TEEY measurements

We have established that the charge state of the sample is a source of errors for the TEEY. In order to avoid falsification of the experimental data, we should then find a way to remove these remaining charges before starting a new measurement. It has actually been shown experimentally by Belhaj et al. [13,18] that the residual holes can be suppressed by sending very low energy electrons in the material. They have studied this phenomenon over several pulses, using a charge compensation procedure proposed by Hoffman et al. [19]. This procedure consists in sending very low energy electrons to discharge the sample. When no charge compensation method was used, they measured a lowering of the TEEY after each incident pulse, similarly to what we have simulated in Figure 6-21. When the sample was discharged between two pulses however, the TEEY measured after each pulse was identical and the lowering was eliminated. From section 6.2.2.2, we have understood that most holes are created within 5 to 10 nm of the surface, and the density of holes in the first 5 nanometers was directly tied to the relative variation of the TEEY. The principle of this charge compensation method would then be to send very low energy electrons, which have a penetration depth of a few nanometers. Since these electrons will be implanted close to the capture depth of the residual holes, this allows us to specifically target the surface holes, and remove them of the material by recombination.

We have followed the experimental protocol from ref. [13] based on this phenomenon, to check whether the suppression of the holes close to the surface can increase the TEEY in our simulations. In the first phase, the sample is bombarded by 10 to 20 pulses of incident electrons, using the same beam parameters as previously. The sample is then bombarded by several pulses of 3 eV electrons and biased to +27 V. In this phase, we can also use a continuous low energy beam instead of incident pulses. Since the energy of most secondary electrons is lower than 27 eV, this bias allows the recollection of the secondary electrons that may escape. The incident electrons arrive at the surface with an effective energy of 30 eV. From Figure 6-9, the extrapolated range of these electrons is 3 nm, which is right at the peak of the hole density of Figure 6-20. Therefore, the injected electrons can progressively discharge the sample by recombining with the holes. This phase is run until the positive charge density has been eliminated. Finally, the sample holder is biased back to -9 V and the energy of the incident electrons is set back to its initial value. The sample is bombarded again by 10 pulses to see if the TEEY has evolved compared to its value before the charge compensation phase.

Several examples of simulation results using this procedure are shown in the figure below. First, in Figure 6-22, the sample is irradiated by 10 pulses of 300 eV electrons. At the end of this phase, 80 pulses of 3 eV electrons are sent, which progressively discharge the sample. We can see this discharge in Figure 6-22b, where the evolution of the surface charge density is shown to decrease during the whole compensation phase until it becomes negative. From Figure 6-22c, we can indeed see from the charge density profiles that the whole thickness of the sample is negatively charged at the end of the compensation. After these 80 pulses, 10 pulses of 300 eV electrons are sent again. We can see in Figure 6-22a that the TEEY has increased from 2.2 before the charge compensation up to 2.4 after this phase.

Another example is given for 500 eV electrons in Figure 6-23 following the same procedure.

Here, the TEEY has increased from 2.2 up to 2.34 after the discharge. Despite the fact that the holes are created deeper than at 300 eV, the low energy electrons are still able to completely compensate these holes. In both Figure 6-22c and Figure 6-23c, the 30 eV electrons are able to eliminate the peaks of holes, as we can see from the plots of the charge density before and after the discharge. While the TEEY of both energies has significantly increased after the discharge, it still has not reached the charge-less value at the beginning of the simulation, despite the fact that the whole thickness of the material is negatively charged at the end of the compensation phase. It may thus be necessary to change our simulation parameters, in order to enhance the recombination and ensure that practically every trapped hole has been removed. Our goal is then to bring the TEEY back to its initial value, which would guarantee reproducible measurements. In Figure 6-24, the discharge phase was run with a continuous beam of 3 eV electrons bombarding the sample for 6 ms, followed by a relaxation period of 1 second. During this period, the implanted electrons and holes are able to detrap and recombine with each other. Since we have an excess of electrons at the end of the low energy electron beam phase, the trapped holes should all be removed by recombination while these electrons move by hopping. In fact, when we resend 300 eV electrons on the target, the TEEY has increased back to its charge-less value. This simulation shows that the removal of the surface holes is done in two steps. First, the secondary electrons can recombine with the trapped holes and decrease the surface hole density, as shown in Figure 6-22 and Figure 6-23. But the relaxation period of 50 ms between each pulse was not enough to let the new electrons remove all holes, as the TEEY did not increase to its initial value. By using a much longer relaxation period of 1 second after the injection of electrons, these new drift electrons are able to get trapped and detrapped multiple times, which increases their probability of recombining with a hole. In Figure 6-25, no low energy electrons were sent after the 10 pulses of 300 eV electrons. The sample was left at rest for 10 seconds instead, before sending 300 eV electrons again. The objective of this simulation is to verify if the sample can discharge itself in a short time between two TEEY measurements, without the need to inject electrons. We can see in (a) that the TEEY has only had an increase of 0.03 at the rest period, which could be due to simulation noise. The TEEY effectively stays at the same value as before the relaxation. The positive surface charge density in (c) and the surface potential in (b) have also not evolved. From (d), the peak of positive charges in the first 5 nanometers only had a very small decrease during the relaxation. However, the negative charges trapped after 5 nm have spread in the material and were able to escape in the silicon substrate. Indeed, the density of negative charges is significantly lowered between 8 nm and 20 nm after the relaxation period. This is due to the fact that the electrons are more mobile than holes. Therefore, during the rest period, most of the charges lost by the sample are electrons that were implanted close enough to the SiO2/Si interface. Since we have not removed the surface holes, we find ourselves in the situation of section 6.2.3, and we can see that the second TEEY measurement after the relaxation period is lowered by the presence of the remaining positive charge. These results show that leaving the sample at rest for a few seconds is not enough to avoid a remanence in the TEEY, and we still need to inject electrons in the sample to effectively remove the charges. On the other hand, when the sample was left at rest for a few hours (or two days in the case of between two measurements in our experiments, the TEEY obtained after this rest period was much higher than the measurements made before. In result, the sample can be discharged by leaving it at rest for a given period of In conclusion, it is possible to remove the residual positive charge that can falsify the TEEY between two measurements, by sending very low energy electrons into the first nanometers of the surface and letting them recombine with the holes. This procedure will allow us to make more reproductible TEEY measurements without remanence. The lowering of the TEEY by the positive charging and its removal are dependent on the density of holes close to the surface, which further demonstrates the importance of recombination on the secondary electron emission process. 10 seconds 10 seconds 10 seconds

Low energy electron beam

Study of multiple-hump TEEY curves 6.3.1 Experimental measurements of multiple-hump TEEY curves

A TEEY curve with a standard shape is not always what we obtain when measuring the TEEY of an insulator sample. Indeed, several studies have reported experimental TEEY measurements on SiO2 thin films that exhibit a double-hump shape, with the apparition of a TEEY local minimum between the two humps. [20][21][22]. This behavior was also observed on other spaceused dielectric materials. In this section, we will use the DEESSE installation to study the conditions of appearance of the double-hump TEEY curves on SiO2 thin films. The most recent study on this subject was made by Rigoudy et al. [20]. The authors have used the DEESSE facility to measure the TEEY of thermal grown SiO2 thin films. They have observed doublehump TEEY curves similar to what was reported elsewhere, with a minimum of TEEY appearing around 1 keV. They have proposed an explanation based on an analytical model taking into account the radiation induced conductivity. This a phenomenon we should be able to model with our simulation code, in order to verify their theory. Notably, they were able to remove the local minimum of TEEY and obtain a conventional shape by changing the measurement parameters, such as the collector voltage. The beam spot area can also be adjusted, by changing the focus voltage F of the electron gun, from 0 V to 1000 V. A focus of 0 V (F0) is expected to produce a very broad beam spot (a few cm²), while a focus of 300 V (F300) should give a narrower beam spot (a few mm²). In all experimental results shown in this section, we have measured the TEEY over several pulses of 6ms spaced of 200ms. The energy/TEEY curves were obtained with the automated TEEY measurement program. A first series of 10 pulses is sent with the sample biased to +27V to measure the incident current 𝐼 0 , then a second series of 10 pulses is sent with the sample biased to -9V to measure the sample current 𝐼 𝑆 . In both cases, the current is averaged over the 10 pulses, and the TEEY is obtained with Equation 6-1.

In Figure 6-26, we have measured the TEEY of our plasma-grown SiO2 samples using a focus voltage of 300V. This is the value of the focus voltage that was used by Rigoudy et al. [20] when they observed a local minimum of TEEY. It is indeed the value of focus used during the standard TEEY measurement procedure in DEESSE and ALCHIMIE. One can see the appearance of a local minimum of the TEEY around 1 keV in our experimental data, as in the study of ref. [20]. The maximal TEEY (2.35) is also significantly lowered compared to the value of Figure 6-5 (2.7), where the experimental data was obtained with a focus voltage of 0V. It is important to note that changing the focus voltage from 300 V to 0 V has eliminated the local minimum at 1 keV, which allowed us to obtain the TEEY curve of Figure 6-5 that follows the standard behavior. Since the focus voltage seems to enable or disable the apparition of the local minimum, we have also made other measurements of the TEEY curve using different values of focus voltage. The results are shown in Figure 6-27. As seen before, the local minimum of the TEEY disappears if a broad beam is used (F = 0V). Moreover, changing the focus voltage from 300V to 100V or 500V changes the position of the local minimum of the TEEY and its relative amplitude compared to the maximum TEEY. When a 100 V focus voltage is used, the local minimum is moved to 500 eV. In this case, the dip in the TEEY curve is very marked, since the local minimum has a higher relative variation compared to the max TEEY (54 % of the maximum TEEY). In comparison, the local minimum at 300V is equal to 64% of the maximum. In the case of a 500V focus voltage, there is even a second local minimum that appears at 250 eV, with the local minimum around 1 keV that has been previously observed. Finally, two TEEY measurements were made in DEESSE with a thin film sample of MgO using a focus voltage of 250 V and 0 V. The results are given in Figure 6-28. Here again, we find at F 250V a double-hump TEEY curve with a local minimum at 900 eV, which is only slightly shifted compared to the minimum observed at 1 keV at a focus of 300 V. This local minimum is also eliminated if the focus is changed to 0V, as was the case for SiO2. By changing the focus voltage, we were able to make appear or disappear a local minimum of TEEY on a different insulator, which demonstrates that this effect is not a property of SiO2 alone. Given that we observe TEEY curves that are very similar to Figure 6-27, this points to physical processes which would be common to insulator thin films. On the other hand, this could also be linked to the protocol or parameters we use to measure the TEEY, because we obtain the same results on two different insulators but with the same measurement parameters. Another set of interesting data we have gathered on this subject, is the following TEEY measurement made on the SiO2 sample with a focus of 250 V, before the sample was decontaminated. It is compared in Figure 6-29 with the data of Figure 6-26 after decontamination. First, we can see that the TEEY of the contaminated sample is higher by a factor of 40% compared to the decontaminated sample. As a result, it is crucial to properly decontaminate the sample before performing any TEEY experiment, so that the data obtained is representative of the material and not of the surface contamination layer. Second, there is also a local minimum of TEEY that appears at 900 eV with a focus voltage of 250 V, like for MgO. The surface contamination layer is composed of many chemical species such as organic compounds or water. As a result, we cannot say if it is conductive or insulating. On the other hand, the SiO2 underneath the contamination layer retains its insulating properties, hence there could be some charging effects or recombination inside the SiO2 that would affect the secondary electrons before they reach the surface contamination layer. In summary, by changing the focus voltage, we can make local TEEY minimums appear at 300 eV, 500 eV and 1 keV. Nevertheless, it is not possible at this stage to accurately investigate the apparition of the local minimums with an explanation based on physics only, since the minimums can appear at various energies. The explanation proposed by Rigoudy et al. was that the local minimum was appearing at 1 keV, since the penetration depth of electrons was comparable to the thickness of the SiO2 sample. This would create a conductive channel which would force the secondary electrons into the Si layer. We have seen indeed in section 6.2.2.2 for 1 keV incident electrons that the drift electrons could easily be evacuated into the Si layer, which is coherent with their hypothesis. However, when we use different values of the focus voltage, the TEEY local minimum still appears at lower electron energies. These electrons have an extrapolated range that is lower than the sample thickness, so they are not able to create a conductive channel. Since the shape of the TEEY drastically changes according to the focus voltage, it is very possible that the double-hump curves are also a result of the measurement parameters.

Measurement of the variation of the beam area with the incident electron energy

To understand the effect of these different beam parameters, we have first studied the evolution of the incident current, to see whether there was a dependence on the focus voltage and/or energy. Indeed, if the incident current increases, more electron-hole pairs will be created in the material in a given time period. Therefore, it could be possible that the TEEY minimums are generated when the incident current is the strongest. In this regard, the incident current 𝐼 0 for the focus voltages F = 100V, 300V and 500V have been measured, and are shown in Figure 6-30. One can see an increase of the incident current with energy, from a few tens of nA at 50 eV to 0.3 µA at 2 keV. However, this variation of the incident current is independent on the focus voltage, so that the three plots are superimposed. Given the significant variation of the position and amplitude of the TEEY local minimum, the variation of the incident current alone cannot explain the dependence of the TEEY on the focus voltage. The incident beam surface 𝑆 𝐵 has then been measured for these 4 different values of the focus voltage in the ALCHIMIE facility. We have observed the same kind of multiple-hump TEEY curves in DEESSE and ALCHIMIE, and the two facilities use the same model of electron gun, a Kimball Physics ELG2. Consequently, we will assume that the beam area of DEESSE follows the same variation as measured in ALCHIMIE. The measurements were made by using the electron gun on a 5 cm wide aluminum square plaque covered by sodium salicylate powder. This powder is phosphorescent when hit by electrons above 200 eV. By using a camera filming the plaque, we can save an image of the surface irradiated by the beam, which can then be measured to obtain its area. The measurements are given in Figure 6-31. For the three focus voltages F = 100V, 300V and 500 V, there is a significant variation of the beam surface with the electron energy. For instance, the area goes from 1.8 cm² at 300eV to 0.08 cm² at 1 keV for F = 300V.

In the case of F = 100 V, the beam area goes from 0.1 cm² at 500 eV up to 13 cm² at 2 keV, as shown in the secondary plot. No data is provided for a focus voltage of 0V, because the beam surface was wider than the area covered by the phosphorescent powder at all energies. Therefore, we can only estimate the beam area at F = 0 V to be greater than 25 cm². of beam surface appear at the same energy, which is about 0.1 cm² for F = 100V at 550 eV, F = 300V and F = 500V at 1 keV, and 1mm² for F = 500V at 200eV. One can also see that the presence of two minimal area points at F = 500 V (250 eV and 1 keV) creates a triple-hump TEEY curve with two local minimums at the same energies. The beam area is also equal to or lower than 0.2 cm² at F = 500V on a large energy range (200 eV to 1.2 keV). This leads to a TEEY that is constantly lower than with the other focus voltages, except at the local minimum of beam area at 500 eV for F = 100V. Such variations of beam surface with electron energy were also reported in other works [23], but they have not made any measurements on insulators.

We have demonstrated that the incident current 𝐼 0 remains constant for various focus voltages, but the surface irradiated by the beam 𝑆 𝐵 varies with the electron energy. Therefore, the key parameter tied to the TEEY variations should be the incident current density 𝐽 0 , as:

𝐽 0 = 𝐼 0 𝑆 𝐵 Equation 6-9
Rather than sampling the whole TEEY curve averaged over several pulses, it is also possible to sample the evolution of the TEEY for a given energy after each pulse, and see if there is also a correlation with the current density. In the following, we have used the same method for timeresolved TEEY measurements as in section 6.2, where the currents were not averaged over the pulses but sampled for each individual pulse. Instead of using 100 µs pulses with a spacing of 50 ms, we have chosen here to stay with 6 ms pulses with a spacing of 200 ms, to use the same parameters as when measuring the whole TEEY curve. The resolution will be much lower, but this will allow us to observe how the TEEY is evolving during a standard measurement.

In Figure 6-32, we compare the time evolution of the TEEY for the energies and focus voltages where a TEEY minimum appears, with the evolution of the TEEY at the same energy but using a broader beam For the energies and focus voltages combinations where the beam area is equal to or below 0.2cm² (300 eV at F500, 500eV at F100 and F500, and 1 keV at F 500V and F 300V), the current density is on the order of 1 to 10 µA/cm². In this case, the TEEY is immediately lowered after one pulse compared to the broader beams, which have an area of a few cm² and a current density lower than 1 µA/cm². For 300 eV and 500eV electrons, this lowering stabilizes after 10 to 15 pulses, while for 1 keV electrons the TEEY does not seem to evolve after the first pulse. This is why our study of the decrease of the TEEY has focused on these three energies. Indeed, at these energies and focus voltages, we could easily get time-resolved experimental data on the decrease of the TEEY, and this decrease was more pronounced than for other energies which were not located at a TEEY minimum. We know that the decrease of the TEEY we observe at a local minimum is due to the recombination of secondary electrons with holes, but this could indicate that the recombination mechanisms are enhanced when a current density of 1-10 µA/cm² is used. Indeed, when the current density becomes lower than 1 µA/cm², the TEEY is much higher and has either a much smaller decrease (300 eV) or no decrease at all (1 keV). This significant variation of the TEEY depending on the current density is coherent with the apparition of the local TEEY minimums in Figure 6-27. If we were to compute the average TEEY over 10 pulses for a given focus voltage and from the data of Figure 6-32, we can clearly see how a local minimum can appear at the energies where the current density is increased.

There is a systematic shift between the TEEY curves that were obtained at the same energy and beam area, for instance at 500 eV between F 500V and F 100V, which have a beam area of 0.2 cm² and 0.1 cm². An identical type of shift is found for the wide beam curves, for instance at 1 keV between F 100V (4 cm²) and F 0V (> 25cm²). In all cases, the TEEY curve that is lower corresponds to the measurement that was made later. In the previous section, it has been shown that the presence of residual deeply trapped holes in the sample may cause a shift of the measurements. Since we did not use the charge remove procedure between each measurements, this could also be the source of the error observed here.

Explanation of the experimental observations by the simulation: Study of the effect of the current density on the TEEY

To understand why the TEEY is decreasing much faster with a focused beam than with a broad beam, we can do the same exercise as in 6.2.2.2 and study the evolution of the internal charge buildup. Using an incident current of 1 µA as previously, the TEEY was simulated with a narrow and a wider beam surface in an attempt to qualitatively reproduce the experimental results of Figure 6-32. The simulations of section 6.2, which were made with a beam area of 0.1 cm² and a current density of 10 µA/cm², have been plotted along with simulations made with a beam area of 1 cm² and a current density of 1 µA/cm², in Figure 6-33 (300 eV), Figure 6-34 (500 eV) and Figure 6-35 (800 eV). At 800 eV, the charge density has a very similar profile to the one obtained with 1 keV electrons, with a strong net positive charge near the interface. The extrapolated range of 800 eV electrons is 15 nm, so this positive charge should also be due to the leak of the trapped holes in the silicon substrate, as for 1 keV electrons. When the beam area is small and the current density is large (0.1cm², 10 µA/cm²), we observe a decrease of the TEEY due to recombination, which is coherent with the experimental measurements made with a narrow beam. However, when the current density is lowered (1 µA/cm²), the TEEY has a higher value and a slower decrease. Notably, we can observe for all energies a contradictory behavior between the evolution of the charge buildup and the TEEY at 1 µA/cm², compared to the observations we made at 10 µA/cm². In Figure 6-15, we had shown that a higher surface density of positive charges led to a stronger decrease of the TEEY. At 1 µA/cm², the positive surface charge density is much higher than at 10 µA/cm² for all three energies. As we can see from the comparison between the charge density profiles, the peak of positive charges is increased at 1 µA/cm². However, the TEEY is also higher at 1 µA/cm² than at 10 µA/cm², and the decrease is much weaker.

It is also possible to highlight this change of behavior of the TEEY by plotting the correlation between the relative variation of TEEY and the surface charge density, like we did in Figure 6-16 and Figure 6-17 at 10 µA/cm². This correlation is given in Figure 6-36. For all energies, the correlation between the two quantities is weaker at 1 µA/cm² than at 10 µA/cm², where the R² was 0.99. The slope of the linear fit is also decreased on average from 𝑆 𝑒-ℎ = 5.7 × 10 -12 cm² in Figure 6-17 down to 𝑆 𝑒-ℎ = 5.5 × 10 -13 cm². There is finally a stronger dispersion in the values of the effective cross section, and the curves are not superimposed like in Figure 6-16 and Figure 6-17. This is due to the apparition of a plateau region for the lower surface charge densities, where the relative variation of TEEY is null.

Since the effective recombination cross section is decreased, this indicates that the interactions between the electrons and the trapped holes are less probable. There is indeed a reduction of the recombination probability, but this is not due to a change of the actual recombination cross section, which remains at 2 × 10 -12 cm². The total charge density in the material is higher than at 10 µA/cm², but due to the lack of uniformity the average recombination mean free path is also higher. Indeed, with a 1 µA/cm² current density, the incident electrons do not hit the sample as uniformly as with a 10 µA/cm² density. Therefore, the electrons have a much higher chance of arriving in a region of the material that has not been irradiated before. Thanks to the overlap factor, the simulation can consider this phenomenon. We are able to simulate the fact that most electrons arrive in a region where the trapped charge density is empty, and that the density of charges seen by an individual electron is thus reduced. Indeed, after 10 pulses of 6ms, the overlap factor is 26% at 1 µA/cm², so the electrons will only see 26% of the total charge densities of Figures Figure 6- In Figure 6-33, the simulation results for 300 eV electrons at 1 µA/cm² have also been plotted on a time scale reduced by a factor x0.1 (in green) to get an equivalent time scale compared to the TEEY at 10 µA/cm². The aim of this plot is to show that the two curves are not superimposed. Dividing the current density by 10 does not simply make the TEEY decrease 10 times slower. We can see that the slope of the decrease of the TEEY is still sharper at 10 µA/cm², but on the other hand the surface potential at 1 µA/cm² still has a much shaper increase.

Therefore, the effect of the current density is not simply to change the time scale at which the TEEY is decreasing, as we have seen that the charge buildup is also modified by the current density. In fact, there is a competition between the time between two electron impacts at the same place, and the time needed for the holes to migrate by hopping. If the hole density at a given place is emptying faster than the time between two electron impacts, the decrease of the TEEY due to recombination should be less pronounced, which could be what happens at 1 µA/cm². At 10 µA/cm², the time between two electron impacts is short enough that a significant part of holes still remains and can capture the secondary electrons, which leads to the decrease of the TEEY. Depending on the material, this reduction of the TEEY may be more pronounced if the hole traps are deeper or more concentrated, or less pronounced if the holes are very mobile and can be dissipated in the sample.

Using the data from Figure 6-31, the simulation can then be used to reproduce the experimental TEEY curves for the three focus voltages. To do so, the TEEY has been simulated for 10 pulses of 6 ms with a 200 ms relaxation period. Each point of the TEEY curve is averaged over the 10 pulses, as in the experiment. The beam surfaces from the measurements in Figure 6-31 were used to compute the overlap factor, in order to simulate the effect of the focus voltage. In Figure 6-37, the simulation of the TEEY for focus voltages of 100, 300 and 500 V is displayed. By modifying the recombination probability according to the variation of current density, we can successfully simulate the apparition of the one or two local TEEY minimums that we had observed experimentally. Consequently, we are able to prove that the multiple hump TEEY curves are indeed linked to physical interactions of the electrons with the trapped charges. We have shown however that the physical interaction involved is the loss of secondary electrons by recombination, instead of RIC and the creation of a conductive channel. We have also demonstrated that these TEEY minimums are created by the variations of the current density with energy, which are themselves created by the variations of the beam surface with energy. These variations can be quite significant over the energy range. Indeed, for a focus of 300V and an incident current of 1µA, the current density ranges from 0.5 µA/cm² at 2 keV to 20 µA/cm² at 1 keV. In Figure 6-38, simulations of the TEEY curve were made with a constant current density of 1 µA/cm² and 10 µA/cm². The TEEY of each energy was averaged over 100 pulses of 100 µs. In this case, the TEEY obtained at 10 µA/ cm² is lower than 1 µA/cm², as expected due to the decrease induced by the recombination. On the other hand, the TEEY at 1 µA/cm² is very close to the charge-less TEEY, with a maximum TEEY of 2.4 instead of 2.5 for the chargeless case.

For both TEEY curves, there is no local minimum of TEEY appearing, since the current density is constant. With a narrower beam, the TEEY is lowered globally but not at select energies like in Figure 6-37. This demonstrates that the local TEEY minimums can be eliminated by working with a constant current density during TEEY experimental measurements. However, this current density should be low enough to avoid a global lowering of the TEEY and a falsification of the data by the recombination effects. For SiO2, this threshold appears to be below 1 µA/cm², but it may be different for other dielectrics with other charge transport properties. In conclusion, the multiple-hump TEEY curves of thin dielectric layers are due to internal charging effects. Nevertheless, these can also be created depending on the measurement parameters, and can in fact be a measurement artefact. A careful choice of experimental parameters can eliminate this artefact, by using a constant current density that is also low enough to limit recombination effects. Therefore, the experimental data obtained with a focus of 0V and a beam surface larger than 25 cm² should be the closest to the charge-less TEEY of the sample. With a current density on the order of the nA/cm², the great majority of incident electrons should hit regions of the sample that are free of charges. We were not able to measure the variation of the beam surface at 0 V, so it is possible that the current density has similar variations to what we measured for the other values of focus. Yet, given the absence of a local TEEY minimum in all measurements made with a focus of 0V, we can assume that the current density remains low enough at all energies to prevent the lowering of the TEEY, even if it is not uniform. An experimental study by Belhaj et al. [16] had also shown that a higher current density could globally lower the TEEY curve. In the present study, we were able to confirm this phenomenon, and explain it by the proportion of overlapping electron cascades on the surface.

Study of the effect of temperature on the electron emission yield

So far, we have interested ourselves in the electron emission of insulating samples in standard measurement conditions, that is to say in a controlled environment. Indeed, the temperature (25°C) and incident current (0.1 to 1 µA) remain constant throughout the experimental TEEY measurements. On the other hand, spacecraft materials are subjected to a large gradient of temperatures, typically ranging from -200°C to +200°C. It is known that the surface chemistry of a contaminated sample can strongly evolve depending on the ambient temperature, which should modify the TEEY. At higher temperatures, the chemical compounds can evaporate from the surface, which is the phenomenon we have used to decontaminate the samples by heating them at 200°C. At lower temperatures below 0°C, water molecules may condensate and freeze on the surface, forming a dielectric layer of ice that can charge during the experiment.

In this section, we will only focus on the variation of the dielectric properties of a SiO2 sample and its influence on the TEEY, depending on the temperature of the sample. The effect of the surface chemistry should be limited, since the samples are decontaminated and the measurements are made in an ultra-high vacuum chamber, preventing the condensation of contamination or water when the samples are cooled. Measurements at 27°C and 200°C were first made in DEESSE. We have then used the ALCHIMIE facility for TEEY measurements from -180°C to 100°C, since it is equipped with a supply of liquid nitrogen. The temperature reached when cooling or heating the sample holder is fluctuating around the target value in an interval of ±10°C. Consequently, all the temperatures given in this section (except room temperature, 27°C) are given with a precision of ±10°C.

Experimental measurements of the TEEY at -180°C and 200°C

First, we will focus on the temporal evolution of the TEEY, using time-resolved measurements with the same protocol as before (Pulses of 100µs separated by 50ms). In Figure 6-39, we have measured the TEEY of 300 eV electrons in DEESSE, at room temperature (27°C) and at 200°C, to study the effect of sample heating. The focus voltage used was F 500V, which gives a current density on the order of 10 µA/cm². The data at 200°C is starting higher than at 27°C, hence it has been reduced by 10% to match the starting point of the data at 27°C. This allows us to directly compare the evolution of the TEEY after its initial point. We find the same kind of recombination-induced decrease at 200°C. However, the slope of this decrease is lessened compared to the TEEY at 27°C, and the TEEY is already diverging at 1 ms. In Figure 6-40, the TEEY has been measured in ALCHIMIE for 500 eV electrons at F 100V, which also gives a current density on the order of 10 µA/cm². The measurements at room temperature and -180°C are compared. This time, the starting point of the TEEY is immediately lowered by 20% when the sample is cooled compared to 27°C. On the other hand, if we scale the TEEY to match the starting point of the TEEY at 27°C, we do not observe a stronger decrease of the TEEY. Instead, the TEEY first seems to have a weaker decrease at -180°C than at 27°C during 10 ms. The authors of ref. [16] have also observed an increase of the TEEY curve of polycrystalline diamond when the temperature was raised from 25°C to 90°C. On the other hand, when the current density was increased, the TEEY was lowered at all energies by recombination and the effect of temperature was much less visible. To verify if a similar effect happens in amorphous SiO2, we can compare the previous results obtained at 10 µA/cm² with the TEEY at 100°C, 27°C and -180°C measured with a current density of a few nA/cm² (Focus 0V). This comparison is shown in Figure 6-41. At -180°C, we do observe a lowering of the TEEY compared to 27°C, albeit only by 0.10. At 100°C however, the TEEY has the same value than at -180°C. Therefore, in our case, the effects of temperature on the decrease of the TEEY are removed when the current density is lowered. Finally, we can also compare the TEEY/energy curve from -180°C to 100°C measured in ALCHIMIE for a given focus voltage. The variation in current density will allow us to see if the behavior of the TEEY is coherent with what we have highlighted with the time resolved measurements. In Figure 6-42, the TEEY has been measured with a focus voltage of 100V, which creates a local minimum of TEEY at 500 eV. We can see that this local minimum still appears regardless of the temperature. The reduction of the TEEY at 500eV is the strongest when the sample is cooled to -180°C, but we do not observe any meaningful difference between the TEEY at -100°C and at 23°C. On the other hand, the TEEY is increased at 100°C, and the dip of the TEEY at 500 eV is also less important than at the other temperatures. Indeed, at -180°C, the ratio between the maximum TEEY at 200 eV and the minimum TEEY at 500 eV is 1.79. At 100°C, this ratio is reduced to 1.56. In Figure 6-43, a focus of 0V is used. We obtain identical TEEY curves regardless of the temperature, with only very slight variations akin to what we have observed in the time-resolved measurements of Figure 6-41. We know that with a focus of 0V, the current density of less than 25 nA/cm² is low enough to limit the interactions of the electrons with the trapped charges created by the previous cascades. Since we do not observe any variation of the TEEY from -180°C to 100°C in such a case, it is very probable that the transport of ballistic electrons is not affected by temperature.

In conclusion, we have observed a dependence of the TEEY on temperature, when a current density of 10 µA/cm² is used. However, when the current density decreases down to a few tens of nA/cm², the effect of temperature on the TEEY is removed. To understand why the temperature is modifying the TEEY, we can make the same study as in 6.2 and compare the evolution of the TEEY with the charge density. Indeed, this change of slope could indicate that the temperature is also acting on the transport of the drift charge carriers. It could be that the charges are migrating deeper in the material, reducing the surface charge density, or spreading laterally, modifying the overlap between electron cascades. To verify these hypotheses however, we have to make simulations of the TEEY from -180°C to 200°C and see whether we find the same behavior as in the experiment. In this subsection, we will present Monte-Carlo simulations which were all made with an incident current density of 10 µA/cm² as in section 6.2.

First, in Figure 6-44, the TEEY of electrons at 300 eV has been simulated at 27°C, -180°C and 200°C. The energy depths of the shallow hole and electron traps were increased respectively to 0.1 and 0.05 eV, instead of 0.07 and 0.02 eV previously. This change was made to get a better reproduction of the modification of the TEEY with temperature. Indeed, the measurements at 27°C, -180°C and 200°C were all made during different measurement campaigns, and it is unsure whether the exact same sample was used each time. Hence, the trapping parameters of each samples may vary, but it is impossible to obtain the actual trap distribution using experimental means. These new values remain within the right order of magnitude expected for shallow trap energy depths, but we must note that this modification of the energy depths is a purely arbitrarily fit to improve the agreement with the experimental data.

We can immediately observe that the starting point of the TEEY is increased when the sample is cooled, and decreased when the sample is heated. This is in opposition with the experimental results, where an increase of the temperature also increased the starting point of the TEEY, except with a broad beam at focus 0V. On the other hand, when the sample is charged at the end of 100 pulses, the TEEYs have reached the same value. Given the different slopes of the decreases, the TEEY at 200°C should become higher than at 27°C and the TEEY at -180°C should go below the TEEY at 27°C over a longer period of time. Thus, it is possible that we would have a better match with the experimental data if we were to start the simulation on a precharged sample, as when studying the effect of the residual holes. We must also remember that the starting point of the TEEY in the simulation is the perfectly charge-less TEEY, which is not the case of the experimental sample. The LO phonon scattering frequency is reminded below. We can see that it varies linearly with the population of optical phonons. Consequently, the interaction mean free path of ballistic electrons with phonons is reduced at higher temperature. The energy of the LO phonons does not change but the frequency of the collisions is greater at higher temperatures, which accelerates the thermalization of electrons below the band gap.

𝑓 ∓ (𝐸) = 𝑒 2 4𝜋𝜖 0 ℏ 2 • (𝑁 𝐿𝑂 + 1 2 ∓ 1 2 ) • ( 1 𝜖(∞) - 1 𝜖(0) ) • √ 𝑚 * 2𝐸 • ℏ𝜔 𝐿𝑂 • ln [ 1 + 𝛿 ∓1 ± 𝛿 ] Equation 3-27 of Chapter 3
The acoustic phonon scattering frequency is also reminded below. This time, the temperature directly appears in the collision probability for electrons with an energy below As a result, when the temperature increases, the ballistic electrons are even more scattered in random directions by the acoustic phonons, and they also lose more energy through the increased collisions with optical phonons. These two phenomena reduce the escape probability of the secondary electrons, which is why the charge-less TEEY increases when the temperature decreases in the simulation, since we compute the phonon population in the simulation depending on the temperature. In the experiment however, there was no variation of the TEEY at a focus of 0V. Hence, it is possible that the electron-phonon interaction models we have used can overestimate the effect of the phonon population on the scattering rates. On the other hand, this effect was only seen in our simulations for a perfectly charge-less sample, and it is very improbable that our experimental samples were perfectly charge-less at the start of the measurements.

In Figure 6-45, the data at -180°C and 200°C have been scaled by 5% to match the starting point of the TEEY at 27°C. This allows us to compare more accurately the relative variation of the TEEY. We can notice the same change of slope between 27°C and 200°C than in the DEESSE measurements of Figure 6-39, with the TEEY at 200°C decreasing more slowly. On the other hand, the TEEY at -180°C is decreasing faster than at 27°C. From this comparison, we can deduce that the ballistic electron-phonon interactions are not the only physical processes modified by the temperature. Indeed, we have demonstrated at room temperature that the decrease of the TEEY was directly linked to the increase of the quantity of surface positive charges. Hence, it is very probable that the change of slope of the TEEY we observe here is linked to the transport of holes in the material. In fact, several mechanisms of the drift regime are dependent on the temperature. In Equation 5-31 of Chapter 5, reminded below, we have defined the detrapping probability for a shallow trap of given depth 𝐸 𝑖 as a classical thermally activated law. In consequence, the increase in temperature also increases the detrapping probability. Therefore, when the sample is heated, the holes and electrons can easily migrate through the material, by hopping between traps.

𝑊(𝐸 𝑖 ) = 𝑊 0 exp (-𝐸 𝑖 𝑘𝑇 )

Equation 5-31 of Chapter 5

The detrapping enhancement of deep traps by Phonon-Assisted Tunneling (PAT), given in Equation 5-35 of Chapter 5, is obviously dependent on the temperature. Indeed, the energy gained by the trapped charge carrier through the collisions with phonons is what allows it to jump to a higher energy level and tunnel through the trap. When the phonon population is increased, the charge carrier is subjected to more collisions from the thermal agitation, and can more easily be excited to a higher level of energy within the trap. Thus, the formula of the enhancement factor by PAT given below is also dependent on the temperature. Finally, the energy of the thermalized charge carriers is directly dependent on the temperature. Whether we take the Boltzmann distribution of velocities as in

𝑒 𝑃𝐴𝑇 = ∫ 𝑒𝑥𝑝 (𝑧 -𝑧 3 2 ⁄ ( 4
𝑃(𝑣) = ( 𝑚 2𝜋𝑘𝑇 ) 1 2 4𝜋𝑣 2 𝑒 -𝑚𝑣 2 2𝑘𝑇
Equation 5-25 of Chapter 5

or an average energy of

3 2
𝑘𝑇, the drift charge carriers have an increased energy when the temperature is raised. The mobility of the charge carriers is also modified by the temperature, as shown by Hughes [24] who has determined that the hole mobility increases exponentially with the temperature. Mady et al. [25] have also shown that the polaron mobility increases with temperatures for internal electric fields below 1 MV/cm. In consequence, two phenomena can lead to a reduction of the surface hole density at high temperature, which would increase the TEEY. First, the migration of holes by hopping deeper into the material is favored by the reduction of the trap immobilization time. It is also possible that the capture cross section of the traps is lowered, when the particles gain more energy due to the thermal agitation. Second, the electrons can also be detrapped easily and either recombine with the surface holes, or escape in the Si layer.

We can see this charge migration in Figure 6-46, where the charge density profiles at the end of 100 pulses of 300 eV electrons are plotted for our three temperatures of interest. First, at -180°C, there is a very strong negative charge after 5 nm, which is more important than at the other temperatures. The charge density profile also resembles the profile of Figure 6-8b, where the drift transport was disabled. Indeed, the maximum of negative charges at -180°C is at 7.5 nm, compared to 6 nm in the drift-less simulation. The positive charge at -180°C is also more important than at the other temperatures, and there are no negative charges at the SiO2/Si interface. This shows that when the sample is cooled, the transport of the drift particles is severely limited. In result, more holes can remain at the surface of the material for a longer period, which can strongly reduce the TEEY. This explains why the experimental TEEY is strongly lowered at -180°C. With a current density of 10 µA/cm², the interval between two electron impacts at a given zone is shortened. However, the holes were not able to evacuate during this interval because their transport is limited by the cold temperature. Consequently, more electrons are lost by recombination and the TEEY is decreased. In this situation, it is even more important to discharge the sample as much as possible with the procedure of 6.2.4 before starting a TEEY measurement. Indeed, the charges are not mobile enough to evacuate themselves when the sample is left at rest, and it is very probable that these charges will cause an error in the next TEEY measurement. When studying the charge profile at 200°C however, we can notice a migration of the positive and negative charges deeper into the material. The surface hole density has moved of 1.5 nanometers in depth, and the negative charge density is also reduced. Here, we have evidence of the thermally activated transport that enables the migration of charges, and can increase the experimental TEEY. Indeed, despite using a current density of 10 µA/cm², the holes are still able to be evacuated between two electron impacts. This limits the loss of electrons by recombination and we have a higher TEEY when the temperature is increased. Notably, the hopping transport of electrons is also activated by the temperature, and they are able to reach the SiO2/Si interface more easily. Hence, we could also be measuring a bigger leakage current at 200°C, which could artificially increase the experimental TEEY.

Finally, in Figure 6-47, the evolution of the surface charge density is plotted. The positive surface charge increases very quickly at -180°C, which is coherent with our analysis of the charge density profile. The lack of evacuation of holes creates a strong reduction of the TEEY, as we have observed experimentally. On the other hand, the positive surface charge evolves more slowly at 200°C due to the enhanced evacuation of holes. We can thus confirm the link between the surface charge density and the decrease of the TEEY we have found at room temperature. The dependence of the TEEY on this density is also prevalent at low and high temperature, and the modification of the drift transport with temperature is what causes a modification of the TEEY. The activation of the transport of holes by raising the temperature is actually a phenomenon that can be used to discharge the sample after a measurement. However, this discharge is not instantaneous, since we have to wait until the charge carriers are detrapped and evacuated. In the case of deep traps, the time of residence of the charge carrier in the trap can be much longer than for shallow traps even when the temperature is raised. Thus, several hours of discharge may be needed to completely remove the deeply trapped particles.

In conclusion, the Monte-Carlo model is able to simulate the effect of temperature on the drift transport and the ballistic transport. It is also able to model the effect of temperature on the TEEY, albeit with a change of parameters to improve the fit to experimental data. After studying the trapped charge densities, we can explain the increase of the TEEY with temperature by the activation of the hopping transport. This phenomenon predominates over the increase of energy losses of the secondary electrons by phonon collisions, which tends to reduce the TEEY. In fact, the phonon scattering models we have implemented in this work were only used by Schreiber & Fitting [26] in simulations of a sample at room temperature. It is therefore possible that the effect of the temperature on the electron-phonon scattering rates is overestimated. Finally, we have shown that the TEEY can decrease or increase of 20% at -180°C or 200°C compared to the measurements made at room temperature with a current density of 1 µA/cm². Therefore, it is possible that space insulators subjected to temperatures ranging from -200°C to 200°C exhibit such variations of TEEY, which we cannot observe through standard qualification at room temperature. However, such variations of the TEEY with temperature are not observed when a current density of a few nA/cm² is used.

Further discussions on internal charging effects and the TEEY of dielectrics

In this section, we give additional results gathered during this thesis, some of which are preliminary. They are given as a basis for discussion and tracks to explore, in the scope of further perspectives on the effects of charging on the TEEY of insulators.

6.5.1 Probing the decrease of the TEEY towards the equilibrium state

The Monte-Carlo code is able to simulate the first 8 ms of the decrease of the TEEY, and we have been able to explain this decrease by the recombination of secondary electrons with holes. However, in standard TEEY measurements, the sample is receiving 10 pulses of 6 ms, for a total irradiation duration of 60ms, and it is possible that space-used dielectrics may be irradiated continuously for even longer time periods.

When studying the multiple-hump TEEY curves, we have made TEEY simulations that followed exactly the same protocol as the experiment, with 10 pulses of 6ms. Nevertheless, the comparison was only qualitative, and we were only able to compare the TEEY averaged over these 10 pulses. Therefore, we can go further and interest ourselves in the simulation of the TEEY past the 80 or 100 pulses of 100 µs. Since we know that the code can simulate the start of the decrease, additional experimental data will be helpful to probe the capabilities of the code in simulating the TEEY over a greater period. In this regard, measurements were made in the ALCHIMIE facility using the same experimental protocol as in DEESSE, with pulses of 100 µs spaced by 50 ms. This time however, special care was taken by using the charge removal procedure of 6.2.4 before each measurement, to limit remanence in the TEEY.

First, the decrease of the TEEY was measured over 300 pulses with the standard procedure (measurement of 𝐼 0 at + 27 V and 𝐼 𝐸 at -9 V), at the energies of the TEEY minimums of Figure 6-27. In this case, the beam area is about 0.05 to 0.1cm². The decrease of the TEEY measured in DEESSE was much stronger than the decrease measured in ALCHIMIE, due to the difference in calibration of the electron guns. In Figure 6-48, the incident current used during the standard measurement procedure for the energy/TEEY curve is plotted for the DEESSE and ALCHIMIE installations. The parameters of the electron guns are identical in both installations. However, we can see that for the same gun parameters, the incident current in DEESSE is higher (1.5 µA) than ALCHIMIE (0.7 µA). Assuming that the beam area variations are identical, this results in a stronger current density in DEESSE (15 µA/cm²) than ALCHIMIE (7 µA/cm²), which, as we have seen before, creates a stronger reduction of the TEEY. The TEEY measurements of ALCHIMIE are compared with the simulations from section 6.2 in Figure 6-49 for 300 eV, 500 and 1 keV. As for the comparison of Figure 6-4, the experimental results have been scaled to match the starting point of the simulated data. In Figure 6-49, the experimental data from DEESSE is also plotted for 300 eV electrons. The simulations have a good agreement with the data from DEESSE over 80 pulses. In the case of the ALCHIMIE measurements however, we have measured a weaker decrease of the TEEY due to the reduced current density. In consequence, the simulations severely overestimate the reduction of TEEY compared to the ALCHIMIE experimental data at the three energies. It was possible to improve the agreement of the simulations with the data from ALCHIMIE, by using an incident current of 0.6µA in the simulation. However, the agreement was only improved for the start of the measurement.

It was possible to improve the agreement at the end of the 300 pulses with the experimental data by reducing the recombination cross section to σ Recomb = 6 × 10 -13 cm², instead of its default value σ Recomb = 2 × 10 -12 cm² used in all other simulations. This modification is a purely arbitrary fit to improve the agreement to the ALCHIMIE data. It is possible that the sample used in DEESSE has different physical characteristics than in ALCHIMIE, but this isn't something we can confirm or use as a justification for this change. We have seen in the previous section that the overlap factor approach could qualitatively simulate the fact that the decrease of the TEEY is slower when the current density is lower. However, the need for such a tweak seems to highlight that more improvements are needed to this approach, in order to have a 0,0E+00 quantitative modeling of the time-resolved decrease of the TEEY at various current densities.

Although there is an inflexion around 7 ms that is not as visible in the simulations, the overall dynamic of the decrease is well simulated with this change of parameter. Therefore, the Monte Carlo model can also simulate the decrease of the TEEY past 10 ms, albeit with a change of parameters, and we have validated it for the simulation of the TEEY during 300 pulses or 30 ms in total. We can observe for both energies a behavior that is identical to the measurements of Figure 6-32 made in DEESSE. For a wide beam surface (F0), the TEEY is practically constant, with only a very small variation at 300 eV from 2.36 to 2.33. This confirms that the effect of charging and recombination on the TEEY is very limited, if a low current density is used (nA/cm² in this case). On the other hand, with a narrow beam surface (F500 or F100) the TEEY immediately decreases and stabilized after 100 ms. What is especially interesting here is that the final value reached by the TEEY is not 1. At 300 eV, both data from ALCHIMIE (Figure 6-50) and DEESSE (Figure 6-32) reach a final value of 1.6. At 500 eV, the TEEY measured in ALCHIMIE stabilizes at 1.9, while the TEEY from DEESSE reaches 1.6 after 30 pulses of 6 ms. This would mean that the sample would be charging indefinitely since it is always emitting more electrons than received. This is impossible however, because we have measured that the surface potential is stabilizing around a few volts under a continuous incident current. This measurement is made by scanning the spectrum of secondary electrons, using the hemispherical analyzer above the sample. It is known that the energy spectrum of electrons should start at 9 eV, since the sample bias at -9V accelerates the secondary electrons through a potential of 9V towards the analyzer.

However, a positive change of the surface potential of 𝑛 volts will reduce the energy gained by the electrons of 𝑛 eV. In consequence, we can measure the energy value at which the distribution of secondary electron starts, and evaluate the positive change of surface potential from the shift of this distribution.

Consequently, it is possible that the actual emitted current 𝐼 𝐸 stabilizes at the value of the incident current 𝐼 0 , which makes the TEEY stabilize at 1 and prevents the surface potential from increasing infinitely. This would mean that we are somehow measuring a current 𝐼 𝑆 that is greater than 𝐼 0 . If we go back to the very first equations of this chapter, the fundamental current conservation law used for the TEEY measurement procedure is given by 𝐼 0 = 𝐼 𝐸 + 𝐼 𝑆 , and the TEEY is computed with 𝑇𝐸𝐸𝑌 = 𝐼 𝐸 𝐼 0 ⁄ = 𝐼 0 -𝐼 𝑆 𝐼 0 ⁄ 𝐼 𝑆 is the current flowing through the sample holder, measured when the sample is negatively biased. From a particle based point of view, these equations can only describe the TEEY accurately if 𝐼 0 are the incident electrons, 𝐼 𝐸 are all electrons exiting the sample, and 𝐼 𝑆 is only made of the electrons remaining in the sample. In metals and semi-conductors, the implanted electrons can easily reach the sample holder and be recorded as a current, therefore this condition is verified. For insulators, we have assumed that the current measured at the metallic sample holder is only made of the current created by the image charge, which will be noted 𝐼 𝐶 . This is true for bulk samples, where the implanted electrons are unable to reach the sample holder, so the leakage current 𝐼 𝐿 created by these electrons is negligible. In section 6.2.2.2 however, we have shown that the drift electrons can escape from the SiO2 thin film through the silicon layer, which should create a leakage current. This leakage current should also be higher for incident electrons with a higher energy. Indeed, we have also shown that, as the incident energy increases, the drift electrons are implanted deeper in the SiO2 layer and can leak more easily in the silicon substrate. As a result, the expression of the TEEY based on the sample current becomes

𝑇𝐸𝐸𝑌 = 𝐼 0 -(𝐼 𝐶 + 𝐼 𝐿 ) 𝐼 0 Equation 6-11
Indeed, if there exists a leakage current that is able to flow through the insulator, the total current 𝐼 𝑆 measured at the sample holder will be equal to the sum of the leakage current 𝐼 𝐿 and the image charge current 𝐼 𝐶 . In the case of our thin film samples, the leakage current is not negligible. It will add itself to 𝐼 𝐶 and increase 𝐼 𝑆 . In consequence, the measured TEEY is falsely increased, even if the actual quantity of emitted electrons does not increase. The TEEY becomes erroneous, because our measurement procedure interprets the leakage of electrons through the sample as secondary electrons emitted into vacuum. This could be why the experimental TEEY does not stabilize at 1 but at 1.6 at 300 eV. At 500 eV, the leakage of electrons is higher, and so is the final value of TEEY (1.9).

With this knowledge in mind, we can attempt to explain the TEEY of 1 keV electrons measured over 5000 pulses, which is very peculiar. In Figure 6-51a, the TEEY is practically not evolving at F0, which was expected. At F300, the TEEY starts decreasing rapidly, but then immediately increases after a couple of ms and reaches its initial value. From section 6.2.2.2, we have seen that a part of the 1 keV incident electrons can actually travel through the whole SiO2 layer, and be implanted in the Si layer. The drift electrons are also created very close to the SiO2/Si interface, which enables them to leak very easily. This resulted in a strong positive charge in the SiO2 layer after 100 pulses, as shown in Figure 6-10. As a result, the thermalized secondary electrons can easily flow to the sample holder and be measured as leakage current, and we can also record a part of the incident current that goes directly through the insulating layer. This should create a strong leakage current that can significantly increase the measured TEEY. In Figure 6-13, the positive charge near the SiO2/Si interface increases with time. This could indicate that the leakage current also increases with time, which would be why the measured TEEY is also increasing. Indeed, if we zoom in the first 300 pulses of the measurement (Figure 6-51b), we only notice a quick stabilization of the TEEY after 5ms. In every experimental measurement made at 1 keV (Figures Figure 6-3, Figure 6-18, Figure 6-32, Figure 6-49), the TEEY also stabilizes very quickly, after a couple of ms only. In the simulation however, we were not able to reproduce this quick stabilization, and the simulated TEEY at 1 keV continues to decrease. Indeed, in the simulation, we count the secondary and backscattered electrons with a spherical collector. With this method, the leakage current is not included in the computation of the TEEY, and we always measure the true emitted current. This explains why the simulation struggles to fit the TEEY of 1 keV electrons after a few ms, where the leakage current starts to modify the experimental TEEY. The effect of the leakage current on the TEEY is not immediately visible over 300 pulses, and it was necessary to measure the TEEY over a larger period to highlight a possible measurement bias.

Another possible explanation for this phenomenon, is that we could be seeing the effect of the electric field, due to the amount of charges created in the material on longer times of irradiation (a few tens of seconds). It could be possible that the electric field becomes greater than the 0.1 MV/cm threshold necessary to force the electrons to drift in the opposite direction of the field. In such a case, the positive charge should force the electrons towards the surface and oppose itself to the leakage current, so when the field increases over time, the leakage should be lowered. The electrons could then be neutralizing the holes and creating an equilibrium of charges. As we have seen before, a reduction of the surface hole density creates an increase of the TEEY. It is finally possible that the leakage of drift electrons does occur, but only at shorter times when the internal field is below the 0.1 MV/cm threshold shown before. However, further investigations on the electric field effects are needed, especially to explain why the TEEY is increasing at 1 keV. Hence, another possibility is that both electric field and leakage effects could be occurring at the same time.

To validate any of these hypotheses, it would be interesting to perform TEEY measurements using the electron collector instead of the sample current method, and compare the two measurements over a few hundred of ms. Indeed, when measuring the TEEY with the electron collector, the leakage current is not taken into account. If there is still a stabilization of the TEEY over 1, then this could be due to the electric field and not the leakage current. On the other hand, if we don't have a stabilization of the TEEY over 1, then the leakage current would be the culprit.

Extending the simulation to measure this leakage current should be more difficult. Indeed, simulating the effect of the leakage current is not as simple as adding the number of drift electrons reaching the Si layer to the TEEY. The current is measured at the sample holder, and the transport of electrons through a bulk silicon layer is not instantaneous. As a result, we would also have to simulate the drift transport of electrons in silicon until they reach the sample holder, which would necessitate the use of a drift-diffusion code instead of a Monte-Carlo code.

6.5.2 Discussion on the charge buildup of a SiO 2 sample under a low energy GEO incident electron spectrum (200 eV -10 keV)

To place ourselves in the context of space used dielectrics, we can also try to simulate the evolution of TEEY of an energy spectrum, instead of the TEEY of a monoenergetic beam. Indeed, spacecraft materials are hit by a spectrum of electron energies ranging from a few hundreds of eV up to several MeV. In this study, the Monte Carlo code has been used to compute the SEY of 20nm SiO2 samples under an energy spectrum. We will reuse the -9V negative bias, so the external charging effects should be limited. This is to simulate the negative bias that can be applied to the solar panels on board of satellites, which can be covered by a layer of SiO2.

The spectrum we have used in this study is obtained from the GREEN model and based on the GEO space environment. The GREEN spectrum ranges from 200eV to several MeV. In this work, we focus on the contribution of low energy electrons of 10 keV and below only, for several reasons. From a physical point of view, electrons above 10 keV only depose a very small part of their dose near the surface, so their TEEY is very small. From a technical point of view, the cross sections for the inelastic interactions of electrons were only computed up to 10 keV. This is because the computation times above 10 keV becomes excessive compared to condensed history models such as the Geant4 standard physics, by a factor of 4 to 5. We have also shown in Chapter 4 that the dose depth profiles of 10 keV and 14.5 keV given by MicroElec and GRAS had a good agreement, so it is more pertinent to switch from MicroElec to a condensed history model above 10 keV.

In Chapter 5 however, we have shown that the Monte-Carlo model reaches its limits when modelling the external charging effects on the TEEY of electrons past the second crossover point, and that a change of parameters was needed to accurately model these effects. The TEEY results shown in this section should only be taken as preliminary results, which need to be improved. On the other hand, we should still be able to use the charge density profiles to study the charge buildup induced by an energy spectrum.

The differential incident spectrum from GREEN in the GEO orbit is given in Figure 6-52. One can see that the majority of electrons have an energy below 1 keV, which have a TEEY greater than 1 and will induce positive charging. However, a significant part of the spectrum is made of electrons that have an energy beyond the 2 nd crossover point of the material used in the simulations, which is around 2 keV. As a result, these higher energy electrons should create negative charging. The simulated TEEY obtained from this spectrum is shown in Figure 6-53, for a total simulated time of 30 ms in continuous irradiation and a current density of 10 µA/cm². We can see the apparition of a temporal evolution of the electron emission yield, which decreases with time. However, given that the surface potential remains negative and around -9V, there is no recollection of the secondary electrons by the surface. The computation was made with all the default parameters from Table 6-1, but we had shown in Chapter 5 that these parameters overestimated the recombination effect for electrons of 2.5 keV and above. Therefore, the decrease of the TEEY shown in Figure 6-53 is overestimated. Due to the computation time needed to produce these results (1 week), it was not possible to redo this computation. However, we can compute the average TEEY of the electron spectrum without charging effects from the TEEY at a given energy and the differential flux Still, we can use the charging simulation results of Figure 6-53 to study the charge density in the 20 nm SiO2 layer after 30ms of irradiation, which is given in Figure 6-54. Three distinct regions can be seen. In the first 5 nanometers, the material is positively charged due to the creation of holes and the escape of secondary electrons. According to the range of electrons of 1 keV and below in SiO2, which can be deduced from the dose depth profiles shown in Figure 6-55, the negative region after 5 nm corresponds to the implantation region of these electrons. Finally, we have shown in 6.2.2.2 that the positive region close to the interface is due to the escape of drift electrons in the silicon substrate, an effect that appeared for electrons above 300 eV. Overall, we can deduct from the surface potential in Figure 6-53 that the global charge of the material is positive. The ionizing dose-depth profiles for the full 10 keV GEO spectrum and for the GEO spectrum cut at 1 keV are shown in Figure 6-55. In the case of the 1 keV spectrum, the whole dose is deposed in the SiO2 layer. The average TEEY of the spectrum cut at 1 keV is 2.25, so they will generate a positive charge. However, when adding the contribution of higher energy electrons to get the 10 keV spectrum, we can see that the dose deposed close to the surface is reduced, which reduces the average TEEY of the spectrum down to 1.19. More importantly, a significant part of the dose is deposed in the Si layer. Indeed, the extrapolated range of 10 keV electrons in SiO2 is 700 nm, and we have already shown that 1 keV electrons are able to go through the 20 nm dielectric layer. Consequently, the electrons above 1 keV are implanted in the Si layer, and they do not contribute to the charging of the SiO2 layer. Due to the lack of their negative charge, the sample can thus get charged positively. From the dose depth profile of the 10 keV spectrum, we can deduce that the maximal range of 10 keV electrons is 1µm. In Figure 6-56, simulations were made by sending the 10 keV GEO incident spectrum on a 2 µm thick layer of SiO2. The same TEEY as for the 20 nm sample is obtained, however the 2 µm sample is charging negatively. This is because the thickness of the insulating layer is greater than the penetration depth of all electrons. As a result, the higher energy electrons beyond the 2 nd crossover point get implanted in the SiO2 layer and can generate a negative charge, which was lost in the Si layer in the case of the 20 nm SiO2 sample. We can highlight this effect by plotting the charge density profile in the 2 µm sample after 30 ms, which is given in Figure 6-57. In this case, there is a positive region in the first ten nanometers of the surface, as for the 20 nm sample. On the other hand, a large negative charge appears from 10 nm down to 1 µm. Indeed, the drift electrons contained in this negative region are unable to travel 1 µm and escape in the Si substrate. The dielectric sample is also thick enough that no electrons can go through it. Therefore, the drift electrons remain in the SiO2 layer, and generate a global negative charge. These preliminary results were obtained with an incident current density of 10 µA/cm², which is the current density in DEESSE in the cases where the beam area is minimal (0.1 cm²). In the standard measurement protocol for the TEEY qualification of space materials in DEESSE, a focus of 300 V is used, which gives a current density varying between 0.7 and 28 µA/cm² for an incident current of 1.4 µA on average. However, the incident current density in the space environment is much lower. We can easily compute the incident current densities received by a spacecraft in the LEO and GEO orbits, from the integrated flux of electrons per cm² in the GREEN model. These current densities are given in Figure 6-58. In the worst case scenario of the GEO orbit, the spacecraft will receive an incident electron current density of 0.1 nA/cm², which is far below the standard TEEY test conditions. Notably, the incident current density used in the measurements at focus 0 V in ALCHIMIE is at most 24 nA/cm². With such a current density, we did no observe any reduction of the TEEY after 0.5 seconds in section 6.5.1. Since the current densities in the GEO and LEO space environments are even lower, we also should not observe a variation of the TEEY in these situations, if we were to perform a similar experiment. Indeed, in Figure 6-59, a simulation of the TEEY of the 1 keV spectrum was made under a continuous current, with a current density of 1 nA/cm². Even after 130 ms, the TEEY has not evolved and remains at the average value of the TEEY of the spectrum, 2.25. Consequently, in the case of dielectric materials directly exposed to the space environment, it is very possible that their TEEY remains stationary at a high value for an extended period. Thus, we may never observe a lowering of the TEEY due to recombination in this situation. Due to the very low current density of 1 nA/cm², there is a large time interval between two electron impacts in the same area, which can allow the trapped charges to be evacuated and prevent the loss of electrons by recombination. This obviously depends on the transport properties of the drift charge carriers in the insulator. For very poorly conductive insulators, the charges may remain trapped long enough to recombine with the incoming electrons, or create a high electric field in the material. However, to verify if the TEEY is indeed stationary in the conditions of the space environment, we would need to make TEEY measurements at a continuous current density of 0.1 nA/cm² over a much longer time (a few tens of seconds, or even a few minutes). We also know that very low energy electrons of a few tens of eVs have a penetration depth that corresponds to the positive charge region. As a result, we have shown in 6.2.4 that these electrons can immediately recombine and remove the positive surface charge. Therefore, it would be very interesting to have access to an energy spectrum that goes below 200 eV, and study the TEEY of SiO2 samples including the contribution of these very low energy electrons. We would be able to determine whether these low energy electrons are able to remove the surface charge, or if the electrons of a few hundreds of eVs can still create an overall positive charge. Finally, the energy spectrum we have used may only be applicable to the study of the TEEY of dielectric materials that are directly exposed to the space environment, such as solar panels. In the case of RF devices placed inside of the spacecraft, the shielding could modify the energy spectrum and flux intensity of the electrons arriving in the device. We also have to take into account the electronic cascades generated by other types of incident radiation, or more energetic particles.

Conclusion of Chapter 6

In this chapter, we have gathered a significant amount of knowledge on the effects of charging on the TEEY of SiO2 thin films, by combining our simulation code with experimental TEEY measurements we have made with the ALCHIMIE and DEESSE measurement facilities of ONERA. These experiments were performed on SiO2 thin films of 20 nm of thickness, plasma grown on a Si substrate. We have notably performed time-resolved TEEY measurements, which are not commonly found in the literature but are especially pertinent for the study of the effects of charging on the TEEY. Indeed, these measurements have allowed us to observe how the TEEY is diminishing with time due to the internal charge buildup. With this data, we have first validated our simulation of the reduction of the TEEY induced by the internal charge buildup.

The Monte-Carlo code we have developed is able to accurately model the decrease of the TEEY observed in our experimental measurements. The slope of this decrease and the starting point of the TEEY agree with the simulation within an error of 20%. Given the numerous processes involved in insulator charging, the accuracy of this simulation model is very satisfying.

By studying the evolution of the charge density with time in the material, we have highlighted a direct correlation between the increase of the surface density and the decrease of the TEEY. This allowed us to demonstrate that the reduction of the TEEY, which was observed experimentally and in the simulations, is due to the recombination of the secondary electrons with the holes. As the surface hole density increases due to the bombardment of incident electrons, the probability of losing the secondary electrons by recombination also increases, which makes the TEEY diminish with time. We have thus confirmed the hypotheses that were proposed in other works [13,15,27], who had suggested that the recombination of electrons with holes was lowering the TEEY.

With this knowledge, we have also demonstrated that the presence of residual holes in the sample at the start of a measurement could cause some significant errors in the TEEY. Due to enhanced recombination compared to a virgin sample, the sample must be properly discharged to avoid a remanence effect in TEEY measurements. From this, we have tried in our simulation a discharging method proposed by Belhaj et al. [13] and Hoffman et al. [19], by sending several pulses of very low energy electrons on the target followed by a relaxation period of several seconds. By studying the internal charge density and the TEEY before and after the discharge, we have confirmed that this procedure can effectively remove the residual surface holes, and can be used during experimental TEEY measurements to improve their reproducibility.

We have made other measurements of the TEEY of SiO2 thin films, this time focusing on the apparition of multiple humps in the energy/TEEY curves. We have shown a direct correlation between the variation of the beam spot size with energy, which depends on the electron gun parameters, and the apparition of a local minimum of TEEY at select energies. The effect of the current density on the TEEY has been investigated with the Monte-Carlo simulation. By modeling in our simulations the effect of the proportion of overlapping electron cascades after a given time, we have proved that the reduction of the TEEY by recombination was enhanced by a higher current density. Experimental works on other insulators than SiO2 had also highlighted a stronger reduction of the TEEY with a stronger current density [16,28,29]. However, to our knowledge, the effect of the proportion of overlapping electron cascades and their uniformity on the surface was not taken into account in Monte-Carlo models. The novel approach of our work has allowed us to prove that the multiple-hump TEEY curves are due to a variation of the current density. These are most probably a measurement artefact, which can be removed when working with a low and constant current density.

Finally, we have moved away from the controlled lab environment and the standard measurement parameters, towards the conditions of the space environment that the space used dielectrics will be subjected to. First, we have measured and simulated the effect of temperature on the TEEY, in the standard range of temperature felt by spacecraft materials (-180°C to 200°C). We have demonstrated that the thermally activated transport of the electrons and holes could significantly modify the TEEY. If the temperature is increased, the drift charge carriers are evacuated more easily which increases the TEEY, and inversely if the temperature is lowered. This is due to the dissipation of the trapped holes in the material, which increases the escape probability of electrons. Lastly, we have also made a preliminary experimental study on the decrease of the TEEY until its stabilization. These measurements hint towards an error in the TEEY that could be created by the leakage of electrons through the substrate. This could be due to our measurement procedure through the sample holder current. Electric field effects may also become dominant over longer irradiation times, and be the cause behind the increase of the TEEY following the decrease we have observed for 1 keV electrons.

We have demonstrated in this chapter a significant influence of the current density and the temperature on the TEEY of insulators. However, the current densities used in the literature and in standard TEEY qualification for space-used dielectrics (1 µA/cm²) are much higher than the maximal current densities received by the materials on board of spacecrafts (0.1 nA/cm² in a GEO orbit). Moreover, if the insulator is subjected to a temperature gradient, it is also possible that the hotter part of the material becomes more emissive than the colder part, which can create a charge gradient inside of the material. Nevertheless, these phenomena are not taken into account in the standard TEEY measurement protocols, which were first designed for metals. Hence, the TEEY gathered during standard qualification in a controlled laboratory environment should be quite different from the TEEY of the dielectric in its real conditions of usage. In consequence, it should be necessary to develop new experimental protocols, to obtain TEEY data that are more representative of the effective TEEY of dielectrics subjected to the space radiative environment.

Conclusion and perspectives

The aims of this study were to develop a low energy electron transport model in dielectric materials for space applications, to simulate the electron emission properties of dielectric samples, and to study the effect of the internal charge on the TEEY. By conducting experimental measurements and using our model to explain them, we have clarified the misunderstood experimental observations made on dielectrics, and highlighted the physical processes involved. Several steps were involved in our approach, which have yielded significant results.

First, we have developed a low energy electron transport models in metals and semiconductors. This was done by bringing additions and improvements to the MicroElec physics module of the Monte-Carlo toolkit GEANT4. The toolkit has been extended to the transportation of low energy electrons, protons, and ions in 16 materials (Be, C, Al, Si, Ti, Fe, Ni, Cu, Ge, Ag, W, Au, SiO2, Al2O3, Kapton, BN), for the simulation of the secondary electron emission. The transport can also be made in multi-layered materials and complex geometries. The secondary electron yield has been used to verify the complete transportation model for electrons, underlying the importance of the surface potential barrier for electron emission modeling. Satisfying agreement with experimental TEEY data has been observed, which has allowed us to publish these new interaction models in Geant4. Most Monte-Carlo models publicly available for the use of the scientific community have a low energy limit of a few hundred eV at most for the transport of electrons, which includes the physical interactions models of GEANT4 except GEANT4-DNA. Hence, by releasing these models in Geant4, we have provided an accessible tool for low energy electron transport, secondary electron emission and microdosimetry studies in 16 materials. The updated MicroElec models are now available in Geant4 since the 10.6 release.

Using the results from this Monte-Carlo model for metals and semiconductors, we have investigated several key quantities of the low energy electron transport, namely the extrapolated range, transmission rate, ionizing dose and secondary electron emission yield. In this study, we have developed analytical models for all these quantities, which are valid down to a few tens of eV. The comparison of the predicted SEY with Monte Carlo and experimental data is in satisfactory agreement for the 11 studied materials (Be, C, Al, Si, Ti, Fe Ni, Cu, Ge, Ag, W). This work highlights the fact that the SEY needs a more accurate description of the transportation of low energy electrons than the hypothesis of a constant energy loss used in some analytical SEY models.

However, this analytical approach presents some limitations which could be improved in future work. Notably, to get a better prediction of the total emission yield, a more accurate expression for the backscattering of electrons could be added to the model, in order to get all the different contributions and the variation in energy. Nevertheless, this model is a good basis for an extension to compound or layered materials. The effect of surface roughness can also be included in the form of geometrical models. These improvements will allow the model to be adapted to the SEY of technical materials.

One of the main results of this thesis is the development of a Monte-Carlo model able to simulate the effect of positive internal charging on the TEEY of insulating materials, with a focus on SiO2 thin films due to the wide availability of reference data on this material. The code is able to simulate the drift transport of electrons and holes in 3D, until they are trapped and stored in a 1D mesh. We have modeled the trapping, detrapping, and recombination of holes, drift electrons and low energy ballistic electrons. To build our simulation, numerous new classes were created in a Geant4 application, and designed to interact with themselves and the Geant4 kernel to ensure proper transportation of the particles. We also added a 1D Poisson solver to Geant4, to compute the electric field after each simulation step. This simulation code can properly model the reduction of the TEEY of SiO2 in conditions of positive charging with the external field compensated, which is strictly caused by internal charging effects. This modelling is coherent with the TEEY decrease observed experimentally on other insulators in such conditions.

Conversely, the simulation of the negative charging and of the TEEY for electrons above the second crossover point needs to be improved. Given that the computation time also becomes disproportionate for electrons above a few keV, we reach the limits of the Monte-Carlo model. Perspectives of this PhD work regarding the Monte-Carlo model could be to improve the modeling of the negative charging induced by electrons above a few keV, and improve the computation time by converting the code from single-thread to multithreading. Still, the architecture of the simulation and its processes have been designed to be as general as possible. Therefore, this Monte-Carlo code can be extended in future works to other insulators used in space applications.

Most importantly, we have gathered a significant amount of knowledge on the effects of charging on the TEEY of SiO2 thin films, by combining our simulation code with experimental TEEY measurements we have made during this thesis with the ALCHIMIE and DEESSE measurement facilities of ONERA. These experiments were performed on SiO2 thin films of 20 nm of thickness, plasma grown on a Si substrate. We have notably performed several timeresolved TEEY measurements, which are not commonly found in the literature, but are especially pertinent for the study of the effects of charging on the TEEY. With this data, we have first validated our simulation of the reduction of the TEEY induced by the internal charge buildup, with an error of 20%. Given the numerous processes involved in insulator charging, the accuracy of this simulation model is very satisfying.

By studying the evolution of the charge density with time in the material, we have highlighted a direct correlation between the increase of the surface density and the decrease of the TEEY. We have therefore demonstrated that the reduction of the TEEY, observed experimentally and in the simulations, is due to the recombination of the secondary electrons with the holes. We have confirmed the hypotheses that were proposed in other experimental works, who had suggested that the recombination of electrons with holes was lowering the TEEY.

From this knowledge, we were able to explain why the TEEY was modified by internal charging in several observations. Firstly, we have shown that the presence of residual holes in the sample at the start of a measurement could cause some significant errors in the TEEY, and that the sample must be properly discharged to avoid a remanence effect in TEEY measurements. We have then proposed a discharging method usable during TEEY characterization campaigns. The error created by the trapped holes can be removed by sending several pulses of very low energy electrons on the target, followed by a relaxation period of several seconds, which allows the electrons to eliminate the holes, and bring the target closer to its charge-less state at the start of the measurements. With this method, the reproducibility of TEEY data is improved.

Secondly, we have explained the origin of the multiple humps in the energy/TEEY curves. We have demonstrated that the variation of the beam spot size with energy, which depends on the electron gun parameters, was correlated with the apparition of a local minimum of TEEY at select energies. This correlation was explained with the simulation, by proving that the reduction of the TEEY by recombination was enhanced with a higher current density, and linked to the stronger overlap of the electron cascades. To our knowledge, the effect of the proportion of overlapping electron cascades and their uniformity on the surface was not taken into account in any other Monte-Carlo models. The novel approach of our work has allowed us to demonstrate that the current density had a strong effect on the TEEY. The multiple-hump TEEY curves are therefore most probably a measurement artefact, and the effects of internal charging can be mitigated when working with a low and constant current density. In an attempt to go further, we have also studied the decrease of the TEEY until its stabilization. These measurements hint towards an error in the TEEY, which could be created by the leakage of electrons through the substrate. This could be due to our measurement procedure through the sample holder current, but this hypothesis must be confirmed with additional measurements.

Lastly, we have moved away from the controlled lab environment used in standard TEEY studies, towards the conditions of the space environment which the dielectrics will be subjected to. We have measured and simulated the effect of temperature on the TEEY, in the standard range of temperature felt by spacecraft materials (-180°C to 200°C). It has been demonstrated that the thermally activated transport of the electrons and holes could significantly modify the TEEY. This is due to the dissipation of the trapped holes in the material, which increases the escape probability of electrons, and depends on the transport properties of the insulator. We have finally simulated the TEEY of a material under a GEO electron energy spectrum. These results show that the modelling of negative charging induced high energy electrons should be improved. On the other hand, we have demonstrated a lack of variation of the TEEY when an incident current density of 1 nA/cm² was used, which is the incident current densities received by spacecrafts in a GEO orbit.

Multiple articles have been written and published in the literature, based on the results of this PhD thesis. First, the improvements brought to MicroElec have been published in [1] and released in GEANT v10.7. The new version of MicroElec developed in this thesis has been used in two other published studies, which have focused on surface ionizing dose calculations [2] and the simulation of the TEEY of multilayered materials [3]. The analytical models developed for the extrapolated range and transmission rate have been published in [4], and the ionizing dose analytical model in [5]. A third paper on the secondary electron emission yield is currently under review. Lastly, regarding the results gathered on insulator charging, the Monte-Carlo charging model for SiO2 and the study of the effect of recombination have been published in [6]. An article on the double-hump TEEY curves and the effect of current density, entitled "Experimental and Monte-Carlo study of double hump electron emission yield curves of SiO2 thin films" has been submitted, and another article on the effect of the temperature on the TEEY, entitled "Monte-Carlo simulation and experimental study of the effect of temperature on the electron emission yield of SiO2 thin films", is under progress.

As final points, we can draw several perspectives, which stem from the large range of results gathered during this PhD thesis on the low energy electron transport in insulators. When studying the effect of recombination, we have shown a direct correlation between the hole density and the variation of the TEEY. Therefore, it could be possible to simplify our Monte-Carlo model, by treating the drift charge carriers implicitly in global charge densities, instead of explicitly simulating the transport of each particle. An explicit simulation was required at the time of this study, since a lot of physical phenomena are involved in the transport of charges, and it was unknown which of these processes was modifying the TEEY. Now that we have identified recombination as the main mechanism, we could simply simulate the drift, trapping and recombination of charges in a more macroscopic approach.

For this, drift-diffusion codes are especially suited, since they directly compute the spatial and temporal variations of the global charge densities, instead of computing the microscopic transport. Due to this lack of microscopic transport, drift-diffusion codes trade the increased accuracy of Monte-Carlo codes for much faster computation times. This would be especially interesting for the simulation of higher energy electrons or extended time periods, where we have shown that the computation time of our Monte-Carlo model is much too long to be applicable. Moreover, we already have an analytical model of secondary electron emission for the charge-less transport of electrons, in metals and semiconductors. Hence, if this model was extended to insulators, the effect of charging on the TEEY could be treated with a purely analytical approach. We could then study whether a drift-diffusion resolution scheme is needed, or if the effect of recombination can simply be modelled with a purely analytical formalism.

We have used a 1D mesh for the computation of the electric field and the storage of charge densities, given that we had to simulate a nanoscale transport but study the electron emission of samples with an area of several cm². Hence, using a full nanoscale 3D mesh for centimetric dimensions would have been extremely expensive in computational resources. The effects we have studied in this work were mostly 1D effects, so our 1D approach was suited. Nevertheless, it could be useful to extend our simulation model with a 3D mesh for other situations. For instance, we could simulate the electron emission of a target hit with a very high incident current density, and attempt to study the variation of image contrast caused by target charging in scanning electron microscopes. Computing the electric field in 3D would also allow to study the effects of charging on rough samples, where the electric field may be locally intense due to peak effects.

Last of all, from an experimental point of view, we have shown that the standard TEEY measurement protocols, which were designed for metals, are not necessarily adapted for spaceused dielectrics. Indeed, the current densities used (1 µA/cm²) are not representative of the maximal current densities received by the materials on board of spacecrafts (0.1 nA/cm² in a GEO orbit). The controlled tests at room temperature also do not take into account the thermal cycling and temperature gradients that spacecraft materials can undergo (from -180°C to +200°C). However, we have shown that both current density and temperature can significantly change the TEEY measured on an insulator. Hence, such modifications could happen with the dielectrics on board of satellites, but are not evaluated during standard qualification. In result, it is possible that the data so obtained introduces an error when evaluating the discharge or multipactor risks associated with dielectrics in the space environment. In consequence, it is necessary to conceive new TEEY measurement standards that are specific to dielectrics, with a current density that is constant and low enough to avoid charging artefacts, and that takes into account the variation of the TEEY with temperature. This will yield data that is more realistic, and representative of the TEEY of dielectrics in space applications.

Appendix I -Description of the architecture of the GEANT4 simulation

To integrate all the physical models for the transport of drift electrons and holes, and to develop an iterative simulation, many new classes were conceived in the Geant4 application we have developed. Here, we shall list these various classes and then explain how they articulate with the standard architecture of Geant4.

 The most important class of the simulation is the DriftManager. It is a singleton with functions related to the charging effects, and the center of the charging simulation. It stores the charge densities, computes and interpolates the electric field, launches the different phases of the simulation, computes the capture mean free paths and detrapping probabilities, and computes what happens when a particle is captured by a trap (free capture or recombination). As a singleton, DriftManager is called by the physical interaction processes, field handling classes and particle sources throughout the simulation.

 The DriftManager is accompanied by a DriftMessenger. It is tasked with loading the different parameters of the model through macro commands stored in the file ChargingParameters.mac. Other macro commands are passed to the DriftManager by the DriftMessenger to signal it to begin the different phases of the simulation, and tell it whether the electron cascade, the drift phase or a relaxation phase should be simulated.

 New particle types DriftHole and DriftElectron were also added. These particles inherit from the class G4ParticleDefinition and use the same particle definition parameters as G4Electron, except for the mass which is set to the effective mass from Equation 5-26, and the charge which is +e for a hole. DriftHole and DriftElectron particles are generated by the DriftManager during the drift phases.

 A new physical process Trapping has been created, following the structure of the other physical processes of Geant4. It returns a mean free path from DriftManager for the capture of drift electrons and holes by shallow or deep traps, and calls DriftManager to handle the trapping or recombination of the particle. By creating a new interaction process following the nomenclature of Geant4, we can let the toolkit handle the transport of the drift particles as any other particle.

 The process G4ElectronCapture tasked with killing electrons below a preset energy threshold has been overhauled. It is now handling the capture and recombination of ballistic electrons, in a similar way to the G4Trapping process.

 The PrimaryGeneratorAction (PGA) class is driven by the DriftManager, which tells it whether the current run is a ballistic or drift run. The PGA also retrieves the stack of particles to be tracked in a drift run from the DriftManager. It then communicates to the Geant4 transportation manager the positions and directions of the drift particles generated.

 The process DriftTimeStepMax stops the drift of the particles if their drifting time exceeds the simulation step. Here, a definite interaction value is returned instead of a mean free path. The particles that have not finished drifting at the end of the simulation time step are added to a postponed stack, and the simulation of their transport is resumed after the next ballistic run.

 A set of enum variables is stored in the file driftEnums.h. They are used to set and check conditions regarding the type of the current run, the current particle type, or the type of trap.

Creation and initialization of the DriftManager at the start of a simulation

The DriftManager is made of many commands that will be called by most of the other classes throughout the simulation. But it first needs to be created and initialized in the main at the very start of the simulation, using the command DriftManager* DriftManager::GetDriftManager(). This central command is used to call the pointer to the DriftManager from any class of the simulation. To ensure that the same object is retrieved every time and a new manager isn't created instead, we use the concept of the singleton in object-oriented programming. The principle of the singleton is to restrict the instantiation of a class to one single instance, and to provide a global access to this instance. Since the DriftManager stores a lot of data that needs to be common to the whole simulation (field, traps…), it needs to be defined as a singleton. We also follow the example of the Geant4 class RunManager, which is defined as singleton. Likewise, there needs to be a unique instance of the RunManager that is called by many other classes of Geant4. This concept works by defining the GetDriftManager() function as a static public method and defining a pointer DriftManager* fDriftManager as a static private object of the class. This means there is a unique copy of the pointer fDriftManager shared by every instance of the DriftManager class. It belongs to the class DriftManager itself, but not to a specific object of the class unlike standard variables. This pointer is created as a null pointer. The function is then defined as follows. At every call of GetDriftManager(), the function verifies if fDriftManager is a null pointer, which it is necessarily at the first call of the function. If it is a null pointer, a new DriftManager object is created and the pointer to this manager is attributed to fDriftManager. If fDriftManager is not null, which is true for every call of the function except the first one, the function simply returns fDriftManager. Since fDriftManager is a static pointer, the address stored in this pointer is also common to every instance of the class. Moreover, a static public method such as GetDriftManager() is independent of any object of the class. This means that the function can be called from anywhere using the class name, even if there are no objects of the DriftManager class. This is how we can ensure that only one object of the DriftManager class will be created once, and the same object will be returned every time we call GetDriftManager(). What's more, the constructor of the class DriftManager is only called once through GetDriftManager() and it is never called directly.

The DriftManager is created at the very start of the simulation along with the DriftMessenger, before the physics are initialized and the first run starts. Right after they are created, the The following commands are called after these parameters are loaded, and are used to initialize the mesh, field, and traps.

 void DriftManager::CreateMesh() Here, we create the 1D mesh in depth for the charge densities, field computation through Thomas's algorithm and field interpolation for the whole geometry. There are actually 2 meshes to be defined in this function. The global mesh std::vector <G4double> z_mesh is defined from the electron gun to the SiO2/Si interface. However, we only need to sample the charge densities inside of the material for the computation of the mean free paths and recombination. For these functions, we use an auxiliary mesh std::vector <G4double> z_histogram that is defined with a first point at 0.5 Angstrom from the surface down to the SiO2/Si interface. This condition has been chosen because 0.5 Angstrom is roughly the thickness of an atomic layer. Therefore, it would not make physical sense to have a particle trapped between the surface and 0.5 Ansgtrom. This is also necessary to circumvent a Geant4 bug where some particles may be captured in infinitesimal thicknesses of 10 -19 m from the surface. Hence, when sampling the charge distributions, any particle that has a capture position between 0 and 0.5 Angstroms from the surface is attributed to the first cell of this inner mesh, starting from 0.5 Angstrom. In essence, the main mesh z_mesh is simply the inner mesh z_histogram with two points added: the surface of the SiO2 layer and the position of the electron gun. The points of z_histogram are logarithmically spaced between 0.5 Angstrom and the thickness of the sample. We also compute the interval std::vector <G4double> delta_z between the points of the main mesh, and the inter-node points std::vector <G4double> z_field on which the electric field is defined.

 void DriftManager::InitializeThomasCoefficients()

In this function, we initialize the coefficients for Thomas's algorithm. This is done by following the initial conditions and the expressions for the coefficients a, b and c detailed in 5.2.2, with the latter depending only on the permittivity and the spacing of the mesh.

 void DriftManager::PreFillDeepTraps() At the start of the simulation, the sample is normally free of charges. In Chapter 6 however, we will study what happens when we start a TEEY measurement on a sample that was already filled by deeply trapped charges that have not been able to escape. This function can be called to prefill the deep trap densities used in the simulation following a given charge profile. Hence, if this function is called, the sample will be already filled with deeply trapped holes at the start of the simulation.

 void DriftManager::CreateTrapLevels()

In this method, the exponential distribution of shallow hole traps is created, following Figure 5-3. 4 quantities are generated in this method and stored in vectors of doubles. The depths of the trap levels are stored in std::vector <G4double> trapEnergyLevels, here we use 20 levels between 0 and 0.4 eV that are uniformly spaced. The probability density for a given trap level 𝐸 𝑖 is obtained from the exponential law as

𝑃(𝐸 𝑖 ) = 𝑃(𝐸 𝑖 ≤ 𝐸 < 𝐸 𝑖 + 𝑑𝐸) = 𝑑𝐸 𝐸 𝑐 exp (- 𝐸 𝑖 𝐸 𝑐 ) Equation A-1
Where 𝑑𝐸 = 𝐸 𝑖+1 -𝐸 𝑖 , and it is stored in std::vector <G4double> probabilityDensity. The density of states for a given trap level from Equation 5-28 is then obtained by multiplying Equation A-1 by the total density of traps, and is stored in std::vector <G4double> trapDensityOfStates. Finally, we also compute the detrapping probability for all levels as the product between the detrapping frequency of a level (Equation 5-31) and the simulation time step. The probabilities are stored in std::vector <G4double> detrappingProbability.

Simulation procedure of the electron cascade

Now that the charging parameters are initialized, we can start the first phase of the simulation defined in section 5.2.1, by sending incident electrons using the macro commands. In an external macro file, the command /Charging/StartElectronCascadeRun is called, which enables the command void DriftManager::StartElectronCascadeRun(). The current type of run currentRunType is defined in the DriftManager through the enum runType. It can take the values electronCascade (sending incident electrons on the target from the electron gun), regularDriftRun (simulating the drift of new and detrapped holes and electrons), or postponedDriftRun (simulating the drift of the holes and electrons from the previous simulation step, whose drifting time exceeded the simulation time step). When calling DriftManager::StartElectronCascadeRun(), the method first sets the run type as a electronCascade run. It then calls the G4RunManager of Geant4, and sends the number of incident electrons that was defined in the ChargingParameters file through the G4RunManager.

The generation of incident particles in Geant4 actually goes through the PrimaryGeneratorAction (PGA) class, a fundamental class of Geant4 that can be edited by the user. As a result, we have to introduce the interactions between the DriftManager and the PGA. This class allows the user to define the type of particles to be generated, along with their position, direction, and energy, though the command void PrimaryGeneratorAction::GeneratePrimaries(G4Event* anEvent) which is called at the start of each event. Consequently, this method of the PGA will retrieve data from the DriftManager, which will tell it to generate ballistic or drift particles depending on the phase of the simulation. At the start of each event, the method runType DriftManager::GetCurrentRunType() from the drift manager is called by the PGA to get the current run type. There is a switch in GeneratePrimaries depending on the runtype returned that will define the particles to be generated. Since we are in an electronCascade run here, the PGA will then send incident electrons from the electron gun using the GeneralParticleSource (GPS) source type of Geant4. Through macro commands set in an external file, the GPS has been placed at the height of the electron gun and configured to send incident electrons of a given energy on the sample at normal incidence. Consequently, no further settings need to be set internally through the PGA, and we can simply generate an event with the standard command associated to the GPS.

The The particle transportation tools of Geant4 (G4Transportation) are tasked with interpolating the trajectory of the ballistic electrons as they are subjected to the electric field generated by the density of charges. As a result, it requires the value of the electric field at several points of the particle's trajectory to correctly interpolate it. In this case, the transportation tools of Geant4 will call the method void EMField::GetFieldValue(const G4double point [4], G4double *field) const of the EMField object. In this function, the Geant4 kernel passes a given point (x,y,z,t) to the EmField object, which is supposed to return the value of the electric field at this point.

The value of the field in the x and y directions 𝐹(𝑥) and 𝐹(𝑦) are set to 0, and the field is also static thus 𝐹(𝑡) = 0. The value of the field in depth 𝐹(𝑧) is already saved in the DriftManager in electricFieldOnNodes, which stores the values of the electric field on the nodes of the mesh.

To return a value to the Geant4 kernel, EmField calls the method G4double DriftManager::GetFieldValue(G4double depthInMeters, G4double x, G4double y) in the drift manager. Three cases are set depending of the depth of the particle. If the input depth 𝑧 is located within the dielectric layer, the function first finds the nodes 𝑧 𝑖 and 𝑧 𝑖+1 that verify 𝑧 𝑖 ≤ 𝑧 < 𝑧 𝑖+1 using a dichotomy method. The values of the field at 𝑧 𝑖 and 𝑧 𝑖+1 are then used in a linear interpolation to return the field at 𝑧. If 𝑧 is located in vacuum between the electron gun and the surface of the dielectric layer, the electric field returned is simply the surface potential divided by the distance between the gun and the surface as the gun is set to the ground. If the inverse mean free path obtained is used in Equation 5-41, to get the final capture mean free path depending on the electron energy.

In G4ElectronCapture::PostStepDoIt, another function of the drift manager is called to handle the trapping of the ballistic electron: void DriftManager::ComputeRecombinationForDriftParticle(const G4Track& aTrack, particleType part, trapType t). Similar to the computation of the mean free path, the type of particle and the type of trap need to be specified. This function retrieves the trapped charge densities in the cell of the mesh, and determines if the ballistic electron recombines with a trapped hole. The charge densities are computed in the same way as in the computation of the mean free path detailed just above. For shallow traps, a level in the energy distribution is first selected using Equation 5-43 to draw a random level from the exponential law, which is then attributed to one of the 20 discrete levels of the distribution following the procedure of 5.5. The density of holes of this level is then used in Equation 5-42, which gives the probability of recombination with a random sampling. If there is no recombination, the ballistic electron is thermalized and will be able to do an additional step as a drift electron (it joins the population of new electrons from the cascade). However, if the electron recombines, its position will not be saved and we also remove a hole from the trapped hole density. The electron will be automatically thermalized if the density of trapped holes is empty, since recombination is impossible in this case. In any event, the transport of the electron is stopped.

At the end of the run, the RunAction user class of Geant4 retrieves the TEEY measured by the sensitive detectors, and feeds it to the DriftManager with the command void DriftManager::AddSEYtoAverage(G4double s). Indeed, the TEEY returned by the simulation is averaged over several runs, to reproduce the experimental configuration where we get a single integrated point per pulse. The averaging procedure is done in the DriftManager after a given number of runs, using the values fed after each ballistic run by the RunAction class.

Computation of the electric field

Once the transport of the ballistic electrons is finished, the second step of the simulation is the computation of the field, handled by the function void DriftManager::ComputeField(). The first step of this computation is the sampling of the charge densities in depth, which will give us the 𝑑 𝑖 coefficients of Thomas's algorithm. This step is handled by calling the method void DriftManager::SampleChargeDistribution(). To do so, we must retrieve the total number of charges in each cell of the mesh. The trapped holes and electrons are already saved as a number of trapped charges per cell in the vectors std::vector< G4double >electronTrapped, std::vector< G4double > electronDeepTrapped and std::vector< G4double > holeDeepTrapped, which are shown in Figure A-2 (a). Consequently, we can simply take the number of particles saved in a given cell, sum the number of particles in deep and shallow traps, and save the total number of trapped particles per cell in the temporary vectors std::vector< G4double > rhoElectrons and std::vector< G4double > rhoHoles. For holes trapped in shallow traps, the numbers of trapped charges are stored per energy level and per depth, so we have a 2D table in the form of std::vector< std::vector< G4double >> holeTrapped, which is shown in Figure A-2 (b). There is simply an additional step which consists in summing the number of holes trapped in every energy level for a given cell of the mesh, to get the total number of trapped holes in this cell, which is then added to rhoHoles. We now know the total number of trapped charges for every cell, but the new holes and thermalized electrons from the electron cascade we have just simulated also need to be taken into account for the computation of the field. If some charges were postponed from the previous drift step, they also need to be considered when sampling the charge densities. Hence the total number 𝑁 of electrons or holes in a cell should be given by 𝑁 = 𝑁 𝑛𝑒𝑤 + 𝑁 𝑠ℎ𝑎𝑙𝑙𝑜𝑤 𝑡𝑟𝑎𝑝𝑝𝑒𝑑 + 𝑁 𝑑𝑒𝑒𝑝 𝑡𝑟𝑎𝑝𝑝𝑒𝑑 + 𝑁 𝑝𝑜𝑠𝑡𝑝𝑜𝑛𝑒𝑑 Equation A-2

However, for both new and postponed particles, the vectors std::vector<G4ThreeVector> holeNew, std::vector<G4ThreeVector> electronNew, std::vector<G4ThreeVector> holepostponed and std::vector<G4ThreeVector> electronPostponed only contain the current coordinates (x,y,z) of the particles, which were recorded when they were thermalized (for new electrons), created (for new holes), or postponed. For these particles, we need to sample the vectors and attribute each stored position to its corresponding cell of the mesh. Here, the method std::vector<G4double> Histogram(std::vector<G4double> data, std::vector<G4double> bins) is called for each stack of positions. In this function, with the depth 𝑧 of a given new or postponed particle, we find the corresponding cell of the mesh 𝑧 𝑖 using a dichotomy method, as for every other case where we need to find the correct cell with the function GetMeshCell. When the cell of a particle is found, the number of electrons or holes in the cell saved in std::vector< G4double > rhoElectrons or std::vector< G4double > rhoHoles is increased by 1. When the new and postponed charge densities have been sampled, the total number of charges in a cell 𝑛 𝑖 is finally obtained. We can then, for each cell of the mesh, obtain the volumetric charge density 𝜌 𝑖 from Equation 5-9 with the number of electrons (𝑛 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠 ) 𝑖 and holes (𝑛 ℎ𝑜𝑙𝑒𝑠 ) 𝑖 as: Where the charge bias factor 𝛽 appears to get the real number of trapped particles.

After the charge densities have been computed, the rest of the method void DriftManager::ComputeField() is used to obtain the electric field on the nodes of the mesh. Now that all the coefficients 𝑑 𝑖 are known, the computation procedure of the electric field is rather straightforward and follows the steps of Thomas's algorithm detailed in 5.2.2.

Computation of the detrapping

The third phase of the simulation is the computation of the detrapping of trapped holes and electrons, which will be transported along with the new drift particles in the fourth and last phase. This computation is handled by the function void DriftManager::ComputeDetrapping().Since the detrapping frequency 𝑊 can vary depending on the cell of the spatial mesh (especially for deep traps where 𝑊 depends on the field 𝐹(𝑧)), the number of detrapped particles is computed separately for each cell of the mesh. The principle used for the detrapping of any population of trapped particles is as follows. First, the detrapping frequency is computed from the formulae detailed in section 5.4 for shallow and deep traps. The frequency 𝑊 𝑖 obtained for the cell is then multiplied by the simulation time step 𝜏, which gives the detrapping probability 𝑃 𝑖 (𝜏) = 𝑊 𝑖 𝜏. This probability is applied on average to every charge trapped in the cell, which gives the number of detrapped charges in a given cell as 𝑛 𝑑𝑒𝑡𝑟𝑎𝑝𝑝𝑒𝑑 (𝜏) = 𝑃 𝑖 (𝜏) 𝑛 𝑖 . If 𝑃(𝜏) ≥ 1 for a given cell, all particles are detrapped. Once the number of detrapped particles is known, the particles are added to the stacks std::vector<G4ThreeVector> electronDetrapped or std::vector<G4ThreeVector> holeDetrapped, and removed from the trapped charge densities. They contain the coordinates of the detrapped particles, where they will be generated from in the drift transport phase. However, as mentioned in 5.2.1, we only store the number of trapped particles in a given cell and not their exact position. Consequently, the position of each detrapped particle has to be randomly generated according to the coordinates of the cell. For this, the function G4double DriftManager::RandomizeDepthforDetrapping(G4double cellID) is called. Since we know that the particle has been released from the cell 𝐶 cellID [𝑧 cellID ; 𝑧 cellID+1 [, it is thus randomly 5 -Simulation of the transport of drift electrons and holes 325 generated at a depth 𝑧 = 𝑧 cellID + 𝑅(𝑧 cellID+1 -𝑧 cellID ), where 𝑅 is a random number between 0 and 1.

The detrapping probability is calculated separately for electrons in shallow traps, holes in shallow traps, electrons in deep traps and holes in deep traps. For shallow traps, the detrapping frequency from Equation 5-31 only depends on the depth of the trap and the temperature, so it is invariant throughout the simulation and does not need to be computed after each electron cascade. For shallow hole traps, this probability is stored in the vector detrappingProbability for each of the 20 energy levels of the distribution. Hence, for each level, the number of detrapped particles is calculated separately. For deep traps however, the computation needs to be made at each simulation step, since the PF barrier lowering depends on the electric field, which evolves with time and space. Therefore, after each simulation step, the method G4double DriftManager::GetDetrappingProbabilityForDeepTrap (G4double cellID) is called in ComputeDetrapping to compute the detrapping probability at each node of the mesh, depending on the PF and PAT effects.

Simulation of the transport of drift electrons and holes

After the detrapping computation is completed, we have four stacks of drift particles that need to be transported until they get trapped: the holes and electrons from the cascade in holeNew and electronNew, and the detrapped electrons and holes in electronDetrapped and holeDetrapped. If some particles from the previous drift phase were postponed, we also have two additional stacks as holePostponed and electronPostponed. The master command void DriftManager::GenerateDriftParticles() is called through the external macro command file with the command /Charging/StartDriftRun, which immediately follows the command /Charging/StartElectronCascadeRun we have used to send the ballistic electrons. The commands for the computation of the field and the detrapping are actually called successively in GenerateDriftParticles. This function does the following actions:

1. The electric field is computed by calling ComputeField(), which itself calls SampleChargeDistribution() as we have seen in 3. 2. If 50µs have elapsed, the analysis manager is called to fill the .csv output file with the TEEY averaged over 50µs and the current value of the surface potential. Another sampling class created during this thesis work is called at this step to save the charge density profiles in depth after every 50 µs. 3. The overlap factor is refreshed, knowing the number of electrons that were sent from the start of the simulation. 4. The detrapping of the charges is computed by calling ComputeDetrapping(), following the procedure in 4. 5. If there are postponed particles, the current run type is set as a postponedDriftRun. The transport of postponed electrons is simulated, then the transport of postponed holes is simulated. 6. The current run type is set as a regularDriftRun.

7. The transport of the new drift electrons from the latest electron cascade is simulated. 8. The transport of the new holes from the latest electron cascade is simulated. 9. The transport of the detrapped drift electrons is simulated. 10. The transport of the detrapped holes is simulated.

The Figure A-3 shows the interactions between the DriftManager and the other classes of Geant4 during a drift particle cascade run. At the start of each event, the PGA gets the current type of run. If it is a regular drift run (for new and detrapped particles), the PGA calls G4ThreeVector DriftManager::GetEventFromDriftParticleStack() to get the coordinates of origin of the particle. The direction of the particle's momentum is then randomly generated in an isotropic distribution, and the particle's energy is set to

3 2
𝑘𝑇. The definition of the particle is also retrieved from the drift manager with the command GetDriftParticleDefinitionForRun(), so that the PGA knows whether to generate a drift electron or a hole. If it is a postponed run, the PGA retrieves the particle's position and momentum direction, which were saved at the end of the run in the drift manager. As for the transport of ballistic electrons, the field handling classes of Geant4 interact with the drift manager to get the value of the electric field at a given position. Through the function GetFieldValue of the drift manager, the field is interpolated and returned to the EmField object, which then transmits the value to the transportation tools.

Three processes are involved during the transport of the drift particles. In the physics list, the processes need to be defined separately for each type of particle. Consequently, the three physical processes we will present are each created in a copy for drift holes and another copy for drift electrons. The main source of interactions is the trapping process. It behaves in the same way as the G4ElectronCapture process for ballistic electrons, in that two objects are created and tasked with handling the trapping in deep traps or shallow traps. Using DriftManager::GetTrapMFPForDriftParticle(), the trapping process retrieves the capture MFP from the drift manager, depending on the trapping type and the particle that is tracked. Indeed, this process follows the procedure from 5.5, therefore we need to use the density of trapped electrons when transporting a drift hole and the density of trapped holes when transporting a drift electron. Contrary to ballistic electrons, the capture mean free path of drift particles is independent on the energy, so the value returned by the drift manager is directly used in Geant4 for the computation of the interaction length. If the trapping process is selected for the interaction, the PostStepDoIt method follows the exact same procedure as in the G4ElectronCapture class. The method DriftManager::ComputeRecombinationForDriftParticle() is called, which follows the same behavior as before. It finds which cell of the mesh the current particle is located in, computes the trapped charge and free trap densities, and does a random sampling to determine if the particle is trapped in a free trap or if it recombines. For shallow traps, a first random sampling is made to find a trap energy level in the exponential distribution of shallow hole traps with the function G4double DriftManager::GetEnergyLevel(). If a free trap captures the drift particles, the number of trapped charges in the cell (or the level for shallow hole traps) is increased. If the particle recombines, a particle of the opposite sign is removed from the number of trapped particles.

The process DriftTimeStepMax stops the particles if their drifting time exceeds the simulation step time 𝜏. However, the Geant4 kernel does not work in elapsed time between two interactions but in length traveled between two interactions. Hence, we need to convert our time limit constraint into a maximal distance constraint. Since we can retrieve the velocity 𝑣 of the particle when the process is called, we can easily get the maximal distance travelled by the particle during the simulation time step using 𝑣 = 𝑙/𝜏 . This distance will be the constraint used to stop the particles. This physical process is different from all the other processes we have seen so far, because it does not return an interaction mean free path but the physical interaction length itself. As a reminder, the physical interaction length 𝑙 is the true distance traveled by the particle between two interactions. It is determined from the MFP 𝜆 using the standard Monte-Carlo random sampling procedure with the equation 𝑙(𝐸) = -𝜆(𝐸) ln(𝑅) where R is a random number drawn from 0 to 1. Since we need to stop the particles if they exceed the maximum distance, the process DriftTimeStepMax bypasses the Monte-Carlo random sampling phase and always returns the same interaction length as 𝑙 = 𝑣𝜏. If the capture physical interaction length is greater than the limit length 𝑙, the DriftTimeStepMax will be selected for the interaction and its method PostStepDoIt() is activated. In this case, the particle is stopped, and its position and momentum are saved in the electronPostponed or holePostponed stacks of the drift manager, using the commands DriftManager::AddPostponedElectron() or DriftManager::AddPostponedHole(). The postponed particles will be generated in the next drift transport phase, after a new electron cascade has been simulated.

The final process that can act on the drift particles is the G4MicroElecSurface class, which handles the passage of the particles through any interface of the simulation. For ballistic electrons, it first computes the transmission probability. If the electron is transmitted, its transmission angle is computed, along with the energy gained or lost by the electron when going through the interface. Drift particles, however, are not able to go through every interface. At the surface of the SiO2 layer, the surface potential barrier prevents the electrons from escaping the material, unless they get accelerated from 3/2 𝑘𝑇 to at least 1 eV without getting trapped. This would require very high electric fields on the order of a few MV/cm, which are not reached in the simulation. Subsequently, any drift particle that reaches the surface of the insulating layer is always reflected back into the material. At the SiO2/Si interface however, the electrons and holes can flow though the Schottky barrier and into the Si layer due to the narrower band gap in silicon. If a drift particle reaches this interface, it is assumed to have escaped in the silicon layer and is removed from the simulation.

When the transport of all stacks of drift particles is finished, we can go to the first phase of the simulation in 2 and simulate a new electron cascade. Nevertheless, since we want to simulate a pulsed measurement procedure and not a continuous flow of incident electrons, there is a relaxation phase between two pulses that also needs to be simulated. Indeed, even in the absence of incident electrons, the drift particles can move in the material and this transport must be simulated. In this phase, no incident electrons are sent, and we only use the method DriftManager::GenerateDriftParticles() through the macro command /Charging/StartDriftRun. This way, the field is still refreshed after each simulation step, and we can simulate the detrapping, transport and trapping of the drift particles even when no incident electrons are sent. [22] and OSMOSEE [23]. Small circles of different colors are for MicroElec simulations, big circles of different colors are for Walker [36] reference data and different squares are our former simulations performed with OSMOSEE code [37]

Table of figures

  Introduction .............................................................................................................................................. Chapter 1: Context and aim of the study ...................................................................................... 1.1

Figure 1 - 1 : 1 -

 111 Figure 1-1: Integral fluxes of protons and electrons received by a spacecraft in LEO and GEO orbits, from the radiation belt model GREEN

Figure 1 - 2 :

 12 Figure 1-2: Illustration of an electron avalanche between two parallel plates

Figure 1 - 3 :

 13 Figure 1-3: Illustration of a typical TEEY curve

Figure 1 - 4 :

 14 Figure 1-4: Illustration of a typical energy distribution of emitted electrons

Figure 1 - 5 :

 15 Figure 1-5: TEEY measurements for an etched sample of Cu under various angles of incidence. data from [29]

Figure 1 - 6 :

 16 Figure 1-6: TEEY measurements of a copper sample, as received and after decontamination by baking and erosion. Data from [30]

Figure 1 - 7 :

 17 Figure 1-7: Comparison of the TEEY of silver on a flat surface and with roughness patterns with height = 100µm and base length L=80µm. Simulated results from [40]

1 2𝑚𝑣 2

 12 is a helix with a gyration radius of

Figure 1 - 8 :

 18 Figure 1-8: Evolution of the global charge of the insulator according to the TEEY curve

Figure 1 - 9 :

 19 Figure 1-9: Charge buildup and evolution of the TEEY for incident electrons below the first crossover point

Figure 1 - 10 :

 110 Figure 1-10: Charge buildup and evolution of the TEEY for incident electrons between the two crossover points

Figure 1 - 11 :

 111 Figure 1-11: Charge buildup and evolution of the TEEY for incident electrons after the second crossover point

Figure 1 - 12 :

 112 Figure 1-12: Illustration of atypical TEEY curves observed on SiO2 thin films

Figure 2 - 1 :

 21 Figure 2-1: Illustration of the diffusion of particles by a scattering center

Figure 2 - 2 :

 22 Figure 2-2: Illustration of the electron-matter interactions that are common to metals, semiconductors and insulators

Figure 2 - 3 :

 23 Figure 2-3: Illustration of the electronic structure of the isolated atom and of a solid

Figure 2 - 4 :

 24 Figure 2-4: Illustration of the band structure depending on material type

Figure 2 - 6 :

 26 Figure 2-6: Illustration of the electron and charge carrier interactions in insulators

Figure 2 - 7 :

 27 Figure 2-7: Dispersion relations of optical and acoustic phonons Phonons are split into two first categories, acoustic and optical phonons. This separation is made according to the shape of the dispersion relation branches 𝜔(𝑘 ⃗ ) of their associated vibration mode, which are shown in Figure 2-7. If the dispersion relation is not equal to zero for a null wave vector (𝜔(𝑘 ⃗ = 0 ⃗ ) ≠ 0), the vibration mode can be excited by a particle having a null wave

Figure 2 - 8 :

 28 Figure 2-8: Definition of the drift and thermal velocities from the distributions of particle positions, in the case of gaussian transport

Figure 2 - 9 :

 29 Figure 2-9: Shallow traps and associated transport mechanisms

Figure 2 - 10 :

 210 Figure 2-10: Deep traps in the band gap of an insulator

Figure 2 - 11 :

 211 Figure 2-11: Detrapping enhancements factors

Figure 2 - 13 :Figure 2 - 14 :

 213214 Figure 2-13: Definition of the true range and projected range from an electron's trajectory

Figure 2 - 15 :

 215 Figure 2-15: Monte-Carlo and analytical computations of the dose deposed in Al by low energy electrons

Figure 3 - 1 :

 31 Figure 3-1: Comparison of molecular and averaged elastic Total Cross Sections for SiO2 3.2.2 Modeling of the inelastic interaction 3.2.2.1 Modeling of the energy losses using the dielectric function theory

Figure 3 - 2 :

 32 Figure 3-2: Examples of OELFs from Sun et al.'s database

Equation 3- 10 With

 10 𝑢 ⃗ = 𝜔 𝑞 𝜐 𝐹 ⁄ , 𝑧 = 𝑞 2𝑞 𝐹 , ⁄ the density parameter 𝜒 2 = 𝑒² (𝜋ℏ𝜐 𝐹 ), ⁄ the Fermi velocity of the target valence electrons 𝜐 𝐹 , 𝑞 𝐹 = 𝑚 𝑒 𝜐 𝐹 ℏ ⁄ , and the functions

Figure 3 - 3 :

 33 Figure 3-3: Fitted OELFs for a metal (Ag), a semi-conductor (Si), and an insulator (SiO2) supported by MicroElec. Each colored smooth curve is attributed to a core shell/plasmon/interband transition. The reference OELFs are taken from the database of Sun et al.

1 𝜀

 1

- 4 .

 4 The possible energy transfers 𝑄 are contained in an interval 𝑢 = [𝑢 𝑎 ; 𝑢 𝑏 ].

Figure 3 - 4 : 1 Equation 3 - 20

 341320 Figure 3-4: Illustration of the rejection sampling procedure First, two random numbers 𝑅 1 and 𝑅 2 are drawn between 0 and 1. A trial value for the energy transfer 𝑄 𝑡𝑟𝑖𝑎𝑙 in the interval 𝑢 is then computed from the random number 𝑅 1 via the relation 𝑄 𝑡𝑟𝑖𝑎𝑙 = 𝑢 𝑎 + (𝑢 𝑏 -𝑢 𝑎 )𝑅 1 Equation 3-20

  transfer 𝑄 made by an incident electron of energy 𝐸 is given by the following normalized integral: sampling procedure, we can draw a random number 𝑅 between 0 and 1. The reverse function of ( 𝑑𝜎 𝑑ℏ𝜔 ) 𝑖𝑛𝑡 then gives us a value for the energy transfer 𝑄 in a straightforward manner, shown in Figure3-5. While we still need to interpolate the tables of cumulated cross sections to get a single value of 𝑄, this procedure is much faster than rejection sampling.

Figure 3 - 5 :

 35 Figure 3-5: Illustration of direct sampling through cumulated DXS

Figure 3 - 6 :

 36 Figure 3-6: Energy changes for an electron going through the surface barrier in the case of a metal or an insulator. The energy reference in the material used in MicroElec is highlighted in red.

  the post-transmission refraction angle 𝜃 𝐸 = asin (√ 𝐸 𝐸 + 𝐸 𝑡ℎ sin 𝜃) 𝑎 = 0.5 × 10 -10 m, and 𝐸 𝑡ℎ [eV] = 𝑊 or 𝜒.

Figure 3 - 7 :

 37 Figure 3-7: Example of the modeling of a multilayer structure (SiO2 on Si) in MicroElec

Figure 3 - 8 :

 38 Figure 3-8: Material structure files of silver (a) and SiO2 (b)

Figure 3 - 9 :

 39 Figure 3-9: Calculated Mean Free Paths for the simulated processes in SiO2, including interactions with acoustic phonons. Interactions with Longitudinal-Optical (LO) phonons are simulated for the two dominant vibration modes (63 and 153 meV).

4 and

 4 𝜌 and the speed of sound in the material 𝐶 𝑠 . 𝑁 𝐵𝑍 = 1 exp (ℏ𝜔 𝐵𝑍 /𝑘 𝑏 𝑇)-1and 𝜔 𝐵𝑍 = 𝐶 𝑠 𝑘 𝐵𝑍 are respectively the population and energy of phonons at the Brilloun zone edge[62]. A linear fit is applied to connect both expressions between 𝐸 𝐵𝑍 𝐸 𝐵𝑍 , where 𝐸 𝐵𝑍 = ℏ 2 𝑘 𝐵𝑍 2 /2𝑚 0 from the dispersion relationship of Equation 3-30.

Figure 3 - 11 .Figure 3 -

 3113  They are compared with previous versions of MicroElec, namely Geant4 9.6 for electrons and Geant4 10.0 for protons. The effect of the relativistic corrections introduced in[16] is visible on the stopping powers, with a better agreement observed at high electron energy with the ESTAR data from NIST[80]. In the low energy range, the Mermin model (This work) leads to a better agreement with the experimental data of Luo et al.[81] for the stopping powers of electrons (Figure5(a)) than the extended Drude model (MuElec), thanks to its better estimation of the plasmon lifetime. This phenomenon is even more noticeable on the stopping powers of protons (Figure5(b)), where the Drude model fails to reproduce the SRIM[42] data below 30 keV. The shopping powers computed for 18Ar ions (Figure5(c)) with the Mermin approach are also compared with data from SRIM and MSTAR[82] in Figure3-11. The improvement of the Kaneko (B-K)[47] approach over the Barkas[83] formula for 𝑍 𝑒𝑓𝑓 is clearly visible below 30 keV/nucleon.

Figure 3 - 11 :

 311 Figure 3-11: Stopping powers of electrons (a), protons (b) and 18 Ar ions (c) in Si The validation of the new version of MicroElec for all other materials is shown for the stopping powers of electrons in Figure 3-12. They are compared with data from ESTAR [80] , Shinotsuka et al. [84], Ashley et al. [85], de Vera et al.[30], and experimental data from Joy's database[START_REF] Joy | A database on electron-solid interactions[END_REF].

Figure 3 - 12 :

 312 Figure 3-12: Electron stopping powers for the materials modeled in MicroElec. The stopping powers of BN are provided for information purposes only, as no reference data could be found for this material.

Figure 3 - 13 :

 313 Figure 3-13 : Dose-depth profile for 10 keV protons (a) and 200 eV electrons (b) with normal incidence in Si

  The inelastic MFP controls the number of secondary electrons set in motion in the irradiated material and their energy distribution,  The elastic MFP and angular deviation have an influence on the quantity of backscattered electrons and on the random walk at very low energies,  The surface potential barrier is a limitation of the quantity of low energy electrons escaping from the material.Thus, the Backscattered (BEY), Secondary (SEY), or Total (TEEY) electron Emission Yields are quantities of interest that can be used to validate the transport model of low energy electrons.In this section, TEEYs, BEYs and SEYs computed with MicroElec are compared with data from other Monte-Carlo simulation codes ([MC]) and experimental measurements ([EXP]). All data correspond to the emission yields of a flat target irradiated with an electron beam under normal incidence. The processes used in MicroElec include elastic and inelastic interactions, surface processes and phonon interactions in SiO2, Al2O3 and BN.

Figure 3 - 14 :

 314 Figure 3-14: Effects of the corrections implemented in MicroElec for low energy electrons on the TEEY of silicon

  -14. Three options of MicroElec are plotted to show the effect of the different corrections: a) includes all processes and corrections previously mentioned, b) is without an initial energy for weakly bound electrons, c) is without the initial energy of secondaries and without the surface potential barrier.

Figure 3 - 15 :

 315 Figure 3-15 : Comparison of the TEYs calculated for all metals and semi-conductors in MicroElec

Figure 3 -

 3 Figure 3-16: SEY of Al2O3 computed without the phonon and polaron processes

Figure 3 - 17 :

 317 Figure 3-17: Comparison of the SEY and BEY of SiO2 simulated in MicroElec with phonon interactions, without polaronic capture All MC simulations overestimate the SEY compared to the experimental data. One can notice in Figure 3-17 that our simulations are in relatively good agreement with those of Ohya et al [91]. There is a factor of two between our calculation and Schreiber and Fitting's one [58]. The Monte-Carlo codes of Ohya et al. and Schreiber & Fitting also do not include the polaronic capture model. Consequently, the difference between the results obtained with the M-C codes could be attributed to the different approaches used for the other processes. Indeed, the inelastic cross sections have been computed by Schreiber and Fitting with an impact ionization model instead of the dielectric function theory, and the M-C code from Ohya et al. does not use the acoustic phonon model at low energies. However, as we have shown, the parameters of the acoustic phonon model are quite difficult to successfully evaluate. As a result, many adjustments can be made to the model and its energy application domain to improve the SEYs. This was done by Schreiber et al. [58] by introducing a scaling parameter into the acoustic scattering rate to modify its influence on the SEY. Our modification of the screening parameter to improve the transition between the elastic models also lowered our SEY curves. Schreiber & Fitting also include the contribution of transverse optical phonons, contrary to our M-C code and the code of Ohya et al.

Figure 3 - 18 :

 318 Figure 3-18: Comparison of the BEY and SEY from MicroElec for SiO2 including all modelsFinally, the SEY and BEY of SiO2 and Al2O3 obtained from MicroElec are given in Figure3-18 for SiO2 and Figure3-19 for Al2O3, with the polaronic capture enabled. The data obtained for Al2O3 is compared with experimental data from Dawson measured on two samples of sapphire, including one highly polished[START_REF] Dawson | Secondary Electron Emission Yields of some Ceramics[END_REF]. We also compare the data to the simulated data from the M-C code of Ganachaud & Mokrani[63]. The effect of the polaron model is clearly visible in SiO2 compared to Figure3-17, where we have a SEY that is now consistent with the data of Schreiber et al. If we do not use the polaronic capture model in our simulations for Al2O3, we obtain a SEY that follows the standard curve but goes up to 20. The simulated SEY for Al2O3 including the capture model is consistent with the measurements made on a highly polished surface of

Figure 3 - 19 :

 319 Figure 3-19: Comparison of the BEY and SEY from MicroElec for Al2O3 including all models

4. 1

 1 Development of an analytical model for the extrapolated range and transmission rate of low energy electrons (10 eV -10 keV)The definition of the extrapolated range 𝑟(𝐸) can be found in section 2.4.1 of Chapter 2, where the Figures Figure2-13 and Figure 2-14 illustrate how the extrapolated range is computed and how it differs from the true range or the average range of a particle in a material.

Equation 4- 3 where

 3 𝑟 𝐸 is the extrapolated range, E the kinetic energy of the electrons in keV, 𝐴 = 5.37 • 10 -4 𝑔/𝑐𝑚²/𝑘𝑒𝑉, 𝐵 = 0.9815, 𝐶 = 3.123 • 10 -3 𝑘𝑒𝑉 -1 . 4.1 -Development of an analytical model for the extrapolate d range and transmission rate of low energy electrons (10 eV -10 keV)

Figure 4 - 1 :

 41 Figure 4-1: Electron transmission rate for Be, Al, Fe and Ag materials. The energy of electrons range from 25 eV up to 5 keV

Electron energy 4 . 1 -

 41 Development of an analytical model for the extrapolated range and transmission rate of low energy electrons (10 eV -10 keV) 133 materials are displayed on both figures. As a reminder, the CSDA range for an electron of energy E is obtained from the stopping power dQ/dx with the relation:

Figure 4 - 2 :

 42 Figure 4-2: Extrapolated ranges of low energy electrons in Al and Si

Figure 4 - 3 :

 43 Figure 4-3:Elastic scattering ratio for 11 materialsQuantitatively, the elastic MFP becomes an order of magnitude lower than the inelastic MFP, with values of a few nanometers, as shown in Figure4-4. Due to the divergence of the inelastic mean free path, the CSDA range also displays a flattening effect below 100eV.

Figure 4 - 4 :

 44 Figure 4-4: Mean free paths of electrons in MicroElec for Si and Al

Figure 4 - 5 :

 45 Figure 4-5: Comparison of the transmission rates in Si from MicroElec and the standard continuous processes

Figure 4 - 6 :

 46 Figure 4-6: Comparison between the range models of Equation 4-3 (ref. [7]) and Equation 4-5 (This work) As shown in Figure 4-6, above ~10 keV, the proposed new expression converges to the classical formula of Equation 4-3. Below this limit, the different extrapolated range expressions diverge. The new formulation reproduces the plateau region appearing in MC simulations while the former formula cannot mimic this behavior. This formula is compared with the Monte-Carlo simulations for all simulated materials in Figure 4-7(a) and (b).Although the range below 50 eV is over estimated and the agreement with the simulations is decreased for low Z (Be, C) and high Z (W) materials, overall a satisfying agreement with the simulations is observed. Indeed, below 100 eV, the average difference between the model and the simulation is between 3% and 12% for all materials, except for Be with 18%.

Figure 4 - 7 :

 47 Figure 4-7: Comparison of the extrapolated ranges given by eq. 13 and MicroElec for all 11 materials (Be, Al, Ti, Ni, Ge and W in fig (a) and C, Si, Fe, Cu and Ag in fig (b))The quantities 𝐹 and 𝐺 are specific for each material and are determined with the following calibration process. The values of F and G are provided in Table4-1. Aluminum, which is by far the more documented material, will serve as reference in the rest of the work. 𝐺(𝑍) sets the height of the plateau region of the range curve 𝑟 𝑍 for the material Z, relatively to the extrapolated range of 50 eV electrons in aluminum. It is defined as

Figure 4 - 8 :

 48 Figure 4-8: Correlation of the atomic number (y-axis) with the height of the range plateau (a) and the slope of the range curve (b) Analytical expressions have also been proposed for the transmission rate of electrons through a given thickness. But they are generally valid only for energies down to a few keV. As in the case of the extrapolated range, the probabilities calculated with MicroElec can be used to calibrate a new expression that is valid down to a few eV and suitable for SEY modelling. In this work, the model from Kobetich & Katz [9] has been extended to lower energies (~10 eV). Their formula of the transmission probability 𝜂(𝐸, ℎ) for electrons of energy 𝐸 [keV] through a thickness ℎ [g/cm²], is initially given as a function of the extrapolated range 𝑟(𝐸) [g/cm²]. In this case, the extrapolated range is obtained with the analytical expression from Kobetich & Katz shown in Equation 4-3:

Figure 4 - 9 :

 49 Figure 4-9: Transmission rate model (grey) compared with MicroElec (colors)

Figure 4 - 10 .

 410 Figure 4-10. Dose-depth curve given by the analytical model (Model) proposed in this work for incident electrons in aluminum. The model is compared with Monte Carlo simulations of MicroElec, Walker [22] and OSMOSEE [23]. Small circles of different colors are for MicroElec simulations, big circles of different colors are for Walker [36] reference data and different squares are our former simulations performed with OSMOSEE code [37].

Figure 4 - 11 :

 411 Figure 4-11: Validation of the analytical dose model with MicroElec data for 11 materials

Figure 4 - 12 :

 412 Figure 4-12: Comparison of the dose profiles in Si from MicroElec and GRAS

Figure 4 - 13 :

 413 Figure 4-13: Comparion between MicroElec and GRAS

4. 2 -

 2 Development of an analytical model for the ionizing dose deposited by low energy electronsWhere ∆(𝑟 𝐸 -ℎ) = [𝐴(1 -𝐵) -(𝑟 𝐸 -ℎ)𝐶 2 ] + 4𝐴𝐶(𝑟 𝐸 -ℎ) is the discriminant of the high energy range expression.

1 𝑝 = 1 . 8 (

 118 0.0059 𝑍 0.98 + 1.log 10 𝑍) -1 + 0.31 The dose profiles resulting from the high energy model of Equation 4-20 are shown in Figure 4-14. They are compared with MicroElec and our low energy model based on the extrapolated range expression of Equation 4-5 in the example of Cu. Equation4-20 should only be considered valid above 10 keV, as it is the domain of validity of the range energy relationship of Equation4-3, which is used in this case. Nevertheless, both models have also been plotted for electron energies below 10 keV, to demonstrate that the high energy model of Equation4-20 is indeed invalid below 10 keV and to show the improvement of our low energy model compared to the high energy model.

Figure 4 - 14 :

 414 Figure 4-14: Comparison of the low energy and high energy analytical models

For electrons having energies 26 4. 3 -

 263 greater than some keV, one can find in the literature analytical functions capable of adequately representing <Es>(h)[4,29,31]. The model presented in this section is based on both the practical range v.s. energy and the transmission probability functions developed in the previous sections. Based on this, a simple Yse model valid in the [~eV, ~keV] range can be derived as: Development of an analytical model for the secondary electron emission yield Where r(E) is our model for the practical range of the incident electrons of energy E defined by Equation 4-5, and 𝜂(〈𝐸 𝑠 〉, 𝑥)

Figure 4 - 15 :

 415 Figure 4-15: OELFs of Ge, Al, Ag, Si

Figure 4 - 16 :

 416 Figure 4-16: Median SE energy in Al, Ag, Ge and Si from MicroElec simulations

Figure 4 - 17 :Figure 4 -

 4174 Figure 4-17: Secondary emission yield calculated with the model and MicroElec for an aluminum target and for different angles of incidence Figure 4-17 shows the SEY calculated with the analytical model for an aluminum target and various incidence angles going from 0° (normal incidence) up to 75°.The SEY increases with the angle of incidence because the ionization is produced closer to the surface, increasing the probability for the secondary electrons to escape from the material. We can also notice that the maximum of yield also shifts to higher energies as the incident angle increases. This behavior can be linked to the location of the maximum of deposited dose, i.e. the depth at which the production of secondary electrons is maximum. This can be explained by the fact that, at normal incidence, the depth at which the maximum of dose is deposited is large enough, to prevent most of the electrons coming from this depth to escape the material. By tilting the incident beam, the peak of dose is brought closer to the surface, increasing the amount of secondary electrons able to escape the surface. The consequence is a shift of the energy of the electrons at which the maximum of SEY is reached.

Figure 4 - 18 :

 418 Figure 4-18 : Comparison of the analytical SEY model with Monte-Carlo and experimental data

Figure 4 -

 4 Figure 4-19: κ as a function of ZHere, the correlation gives a law in Z² for the values of 𝜅 which can be used as a starting point to get an estimation of this parameter for any new materials. The position of the max SEY Emax has been plotted in Figure4-20 as a function of the Emax extracted from MicroElec. In this case, no compensation factor has been applied. The correlation factor is also satisfying (0.94).

Figure 4 - 20 :

 420 Figure 4-20: Comparison of the energy of the maximal SEY from the analytical model and Monte Carlo data

Figure 4 - 21 :Figure 4 - 22 :

 421422 Figure 4-21: Correlation of G with the max value of the SEY

33 Figure 4 - 23 :

 33423 Figure 4-23: Comparisons of the dose depth curve given by the analytical model of this work for incident electrons in copper with MicroElec and a constant energy loss model.

Figure 4 - 34 Where

 434 Figure 4-9 for 25 eV electrons, we can suppose that the escape depth of secondaries is around 2 nm. Consequently, we can show this correlation by calculating the total energy deposited in the first 2 nanometers of the surface, i.e.

Figure 4 - 24 :

 424 Figure 4-24: Correlation between the surface energy deposit and the SEY from the analytical models: position (a) and value (b) of the max

Figure 4 - 25 :

 425 Figure 4-25: Comparison of the surface energy deposit and SEY curves for C, Ti, Ni, W.

  Several works have developed analytical models for charge transport in insulators. In most cases, the model solves the drift-diffusion equations to compute the evolution of the charge densities with space (𝑥) and time (𝑡). A standard 1D equation scheme found in these codes includes Poisson's equation, Ohm's law and the conservation law respectively in the form of: { ∆𝑉(𝑥, 𝑡) = 𝜌(𝑥, 𝑡) 𝜀 𝐽(𝑥, 𝑡) = 𝜎(𝑥, 𝑡) 𝐹(𝑥, 𝑡) 𝜕𝜌(𝑥, 𝑡) 𝜕𝑡 + ∇𝐽(𝑥, 𝑡) = 𝑆(𝑥, 𝑡) Equation 5-1

Figure 5 - 1 :

 51 Figure 5-1: Simulation configuration

Equation 5 - 14 5. 2 -

 5142 𝑛 = 𝑉 𝑝𝑜𝑙 = -9 V Developing an iterative Monte-Carlo simulation of ch arging and secondary electron emission

3 2𝑘𝑇

 3 , where 𝑘 is Boltzmann's constant and 𝑇 the temperature. Using 𝐸 = 1 2

Figure 5 - 2 :

 52 Figure 5-2: Trap levels modeled in the simulations

Equation 5 - 39 For

 539 ℎ (𝑧) = 𝜎 𝑆 𝑁 𝑆,𝐹𝑟𝑒𝑒 (𝑧) + 𝜎 𝑒-ℎ 𝑁 𝑒 (𝑧) 1 𝜆 𝐷,ℎ (𝑧) = 𝜎 𝐷 𝑁 𝐷,𝐹𝑟𝑒𝑒 (𝑧) + 𝜎 𝑒-ℎ 𝑁 𝑒 (𝑧) drift electrons (𝑒): { 1 𝜆 𝑆,𝑒 (𝑧) = 𝜎 𝑆 𝑁 𝑆,𝐹𝑟𝑒𝑒 (𝑧) + 𝜎 𝑒-ℎ 𝑁 ℎ (𝑧) 1 𝜆 𝐷,𝑒 (𝑧) = 𝜎 𝐷 𝑁 𝐷,𝐹𝑟𝑒𝑒 (𝑧) + 𝜎 𝑒-ℎ 𝑁 ℎ (𝑧) 𝑒 (𝑧, 𝐸) = [𝜎 𝑆 𝑁 𝑆,𝐹𝑟𝑒𝑒 (𝑧) + 𝜎 𝑒-ℎ 𝑁 ℎ (𝑧)] exp (-𝛾𝐸) 1 𝜆 𝐷,𝑒 (𝑧, 𝐸) = [𝜎 𝐷 𝑁 𝐷,𝐹𝑟𝑒𝑒 (𝑧) + 𝜎 𝑒-ℎ 𝑁 ℎ (𝑧)] exp (-𝛾𝐸) Equation 5-41

Figure 5 - 4 :

 54 Figure 5-4: Simulation results of the electron cascade radius in SiO2 for different incident energies Now that we know the size of a single electronic cascade, we can compute the proportion of the material surface filled with electrons and holes resulting from the electron cascades after a given time. It is simply expressed by the number of incident electrons 𝑁 𝑖𝑛𝑐 (𝜏) sent from the start of the measurement, multiplied by the surface of a cascade to get the total surface covered by electron cascades, and divided by the area irradiated by the electron beam. This gives the following overlap factor, where we can make the incident current density 𝐽 0 appear using Equation 5-3: % 𝑂𝑣𝑒𝑟𝑙𝑎𝑝 (𝜏) = 𝑆 𝐶 𝑁 𝑖𝑛𝑐 (𝜏) 𝑆 𝐵 = 𝑆 𝐶 𝑒 𝐽 0 𝜏 Equation 5-45

Figure 5 - 5 :

 55 Figure 5-5: Illustration of the evolution of the overlap factor with time

Figure 5 -

 5 6 and Figure 5-7 illustrate how these classes interact with the Geant4 classes to perform the transport of ballistic electrons and drift charge carriers.

Figure 5 - 6 :Figure 5 - 7 :

 5657 Figure 5-6: Architecture of the simulation for the transport of ballistic electrons

Figure 5 - 8 :

 58 Figure 5-8: Simulation of the TEEY of a 100 µm SiO2 sample hit by 20 eV electrons. Current density of 10 µA/cm²

Figure 5 - 9 :

 59 Figure 5-9: Simulation of the TEEY of a 100 µm SiO2 sample hit by 500 eV electrons. Current density of 10 µA/cm²

Figure 5 - 10 :

 510 Figure 5-10: Energy spectrum of electron emitted by SiO2 under 500 eV incident electrons

Figure 5 - 11 :

 511 Figure 5-11: Simulation of the TEEY of a 1 mm SiO2 sample hit by 2500 eV electrons. Current density of 10 µA/cm². (a): Default parameters (b): Recombination cross section decreased to 10 -13 cm²

Figure 5 - 11 . 8 -

 5118 The sample used here is 10 times thicker (1 mm) than in Figure5-9 (0.1 mm), so according to Equation 5-47 the surface potential should be evolving ten times faster. Yet, the equilibrium is only reached from roughly 30 ms, instead of 2 ms in Figure5-9, so we can suppose that for a 0.1 mm sample we would have to compute a hundred times more simulation steps compared to Figure5-9 and the (Simulation of the TEEY of SiO2 simulation would be ten times longer than the present case. All in all, the individual simulation steps are longer, and we need to compute a lot more of them. As a result, the simulation results from Figure5-11 were obtained after 40 hours of computation, while the simulation of Figure5-8 took less than an hour.

Figure 5 - 12 :

 512 Figure 5-12: TEEY of a 20 nm SiO2 sample for 300 eV (a), 500 eV (b) and 1 keV (c) incident electrons. Current density of 10 µA/cm²

Figure 5 - 13 :

 513 Figure 5-13: Effect of the capture cross section of secondary electrons by free shallow electron traps on the TEEY

Figure 5 - 14 :

 514 Figure 5-14: Comparison of the TEEY with experimental data

Figure 5 - 15 :

 515 Figure 5-15: Effect of the model parameters on the TEEY of 500 eV electrons, with a current density of 10 µA/cm². (a): Effect of the capture cross sections of secondary electrons by free traps. (b): Effect of the recombination cross section. (c): effect of the detrapping frequency factor

Beam area and current density 0. 1

 1 cm² to a few cm², for a current density of 0.1 µA/cm² to 10 µA/cm² Number of incident electrons per time step and bias factor 500 electrons, bias factor of 12500 Density of shallow traps per cm 3 (N S ) 10 21 cm -3 Density of deep traps per cm 3 (N D ) 10 18 cm -3 Capture cross section by a shallow trap (σ S ) 10 -14 cm² (holes and drift electrons) 6 × 10 -15 cm² (ballistic electrons) Capture cross section by a deep trap (σ D ) 10 21 cm -3 Mean value for the exponential distribution of hole shallow trap depths (20 levels ranging from 0 to 0.4 eV) 0.07 eV Modified to 0.1 eV in section 6.5 to improve the fit to experimental data Depth of electron shallow traps 0.02 eV Modified to 0.05 eV in section 6.5 to improve the fit to experimental data Depth of electron and hole deep traps 2 eV Electron-hole recombination cross section (σ e-h ) 2 × 10 -12 cm² Detrapping frequency factor (W 0 ) 10 3 s -1 for shallow traps 10 14 s -1 for deep traps

Figure 6 - 1 :

 61 Figure 6-1: Illustration of the DEESSE facility. Image by S. Dadouch

Figure 6 - 2 :

 62 Figure 6-2: Illustration of the ALCHIMIE facility. Image by S. Dadouch

Figure 6 - 3 :

 63 Figure 6-3: Time-resolved experimental measurement of the decrease of the TEEY of SiO2 thin film for 300 eV and 1 keV electrons.

Figure 6 - 4 :

 64 Figure 6-4: Comparison of the simulated and experimental data of the decrease of the TEEY.

Figure 6 - 5 :

 65 Figure 6-5: Comparison of the simulated charge-less TEEY and experimental TEEY (J0 < 25 nA/cm²)

Figure 6 - 6 :

 66 Figure 6-6: Electric field in SiO2 after 80 pulses of 500 eV and 1 keV electrons

Figure 6 - 7 :

 67 Figure 6-7: Energy/TEEY curve for positive (a) and negative (b) uniform values of the internal electric field

Figure 6 - 8 :

 68 Figure 6-8: Charge distributions after 100 pulses of 300 eV and 500 eV electrons (a) and after electron irradiation without drift transport (b)

Figure 6 - 9 :

 69 Figure 6-9: Extrapolated range of electrons in SiO2 from the Monte-Carlo simulation

Figure 6 - 10 :

 610 Figure 6-10: Charge profiles after 100 pulses of 1 keV electrons in 20 nm and 50 nm thick samples This effect can be shown if we compare the charge distribution after 100 pulses of 1 keV electrons in a 50 nm sample with our 20 nm sample. The comparison is shown in Figure 6-10. The 50 nm sample is thicker than the range of 1 keV electrons. For this sample, we observe a charge profile with very distinctive positive and negative regions, instead of the strictly positive charge profile of 1 keV electrons in 20 nm. The 50 nm sample is positively charged until 24 nm, which can explain why the 20 nm sample has no negative charge region. As a result, what we see on the 20 nm sample is only a part of the positive charge peak, which extends into the silicon layer. Most of the electrons can easily escape in the silicon layer due to their implantation depth.This leads to a large net positive charge close to the SiO2/Si interface when we sum the quantity of positive and negative charges to get the total charge density plotted here, hence the peculiar shape of the charge distribution.

Figure 6 - 11 :

 611 Figure 6-11: Charge profiles of 100 000 1 keV electrons in 20 nm and 50 nm thick samples, with drift transport disabled

Figure 6 - 12 :Figure 6 - 13 :

 612613 Figure 6-12: Evolution of the peaks of positive and negative charges during 100 pulses of 300 eV, 500 eV and 1 keV electrons

Figure 6 - 14 :

 614 Figure 6-14: Evolution of the charge density of 1 keV electrons after 1 and 2 pulses

Figure 6 - 15 :

 615 Figure 6-15: Comparison of the evolutions of the TEEY (a) and the surface hole density (b)

Figure 6 - 16 :

 616 Figure 6-16: Linear fit between the TEEY and the surface charge density from 100 eV to 2 keV

Figure 6 - 17 :

 617 Figure 6-17: Correlation between the TEEY and the surface charge density at 300 eV and 1 keV

Figure 6 - 18 :

 618 Figure 6-18: Comparison of the experimental TEEY of 1 keV electrons for a virgin and a charged sample

Figure 6 - 19 :

 619 Figure 6-19: Comparison of the experimental TEEY of 500 eV electrons for a discharged and a charged sample

Figure 6 - 20 :

 620 Figure 6-20: Approximated residual hole densities used in the simulations

Figure 6 - 21 :

 621 Figure 6-21: Simulation of the effect of residual holes on the TEEY of 300 eV, 500 eV and 1 keV electrons.

Figure 6 - 22 :

 622 Figure 6-22: Simulation of the compensation of the surface holes by very low energy electrons. Case of 300 eV electrons (a) Evolution of the TEEY (b) Evolution of the total surface charge density (c) Charge density profiles after 10 pulses of 300eV and 80 pulses of 3 eV electrons.

Figure 6

 6 Figure 6-23: Simulation of the compensation of the surface holes by very low energy electrons. Case of 500 eV electrons (a) Evolution of the TEEY (b) Evolution of the total surface charge density (c) Charge density profiles after 10 pulses of 500eV and 80 pulses of 3 eV electrons.

Figure 6

 6 Figure 6-24: Simulation of the removal of the surface positive charge by a continuous 3 eV beam for 6 ms and a rest period of 1 second (a) Evolution of the TEEY (b) Evolution of the total surface charge density (c) Charge density profiles before and after the discharge

Figure 6 - 25 :

 625 Figure 6-25: Simulation of the TEEY evolution through two periods of 10 pulses of 300 eV electrons, spaced by a 1 second rest period. (a) Evolution of the TEEY (b) Evolution of the surface potential (c) Evolution of the total surface charge density (d) Charge density profiles before and after the 1 second relaxation period

Figure 6 - 26 :

 626 Figure 6-26: Double hump TEEY of SiO2 thin film sample measured at F = 300 V

Figure 6 - 27 :

 627 Figure 6-27: Comparison of the TEEY curves obtained with different focus voltages

Figure 6 - 28 :

 628 Figure 6-28: Double hump TEEY of a MgO thin film sample measured at F = 250 V and F = 0V

Figure 6 - 29 :

 629 Figure 6-29: Comparison of the TEEY of a contaminated sample at F 250V and a clean sample at F 300V

Figure 6 - 30 :

 630 Figure 6-30: Variation of the incident current with the focus voltage

Figure 6 - 31 :

 631 Figure 6-31: Variation of the beam area with the focus voltage and electron energy From Figure 6-31, we can see a direct correlation between the area irradiated by the beam and the shape of the TEEY curves of Figure 6-27. The local minimum of TEEY and the minimum

Figure 6 - 32 :

 632 Figure 6-32: Evolution of the TEEY in time with the number of pulses, depending on the incident energy and focus voltage

Figure 6 - 33 :Figure 6 - 34 :Figure 6 - 35 :

 633634635 Figure 6-33: Simulation of the TEEY of 300 eV electrons for a current density of 10 µA/cm² and 1 µA/cm² (a) Evolution of the TEEY (b) Evolution of the surface potential (c) Evolution of the total surface charge density (d) Charge density profiles at the end of 100 pulses

  33 to Figure6-35. In comparison, the overlap factor reaches 100% after 4 pulses for a current density of 10 µA/cm². In consequence, for a wider beam or lower current density, the proportion of secondary electrons lost by recombination is lessened, which leads to a higher TEEY.

Figure 6 - 36 :

 636 Figure 6-36: Correlation between the TEEY and the surface charge density for a current density of 1 µA/cm²

Figure 6 - 37 :

 637 Figure 6-37: Simulation of the TEEY curves obtained with different focus voltages. I0= 1µA

Figure 6 - 38 :

 638 Figure 6-38: Simulation of the TEEY averaged over 100 pulses of 100 µs with a constant beam surface. I0= 1µA

Figure 6 - 39 :

 639 Figure 6-39: Measurement of the TEEY of 300 eV electrons at F 500V in SiO2 at 27°C and 200°C in DEESSE. J0 = 10 µA/cm²

Figure 6 - 40 :

 640 Figure 6-40: Measurement of the TEEY of 500 eV electrons at F 100V in SiO2 at 27°C and -180°C in ALCHIMIE. J0 = 10 µA/cm²

Figure 6 - 41 :

 641 Figure 6-41: Measurement of the TEEY of 500 eV electrons at F 0V in SiO2 at 100°C, 27°C and -180°C in ALCHIMIE. J0 ~ nA/cm²

Figure 6 - 42 :

 642 Figure 6-42: Experimental measurement of the TEEY at F 100V in ALCHIMIE from -180°C to 100°C

Figure 6 - 43 :

 643 Figure 6-43: Experimental measurement of the TEEY at F 0V in ALCHIMIE from -180°C to 100°C 6.4.2 Explanation of the experimental observations by the simulation: Study of the thermally activated hopping transport

Figure 6 - 44 :

 644 Figure 6-44: Simulation of the TEEY of 300 eV electrons in SiO2 at -180°C, 27°C and 200°C. J0 = 10 µA/cm² The change of charge-less TEEY in the simulation depending on the temperature can be explained by the interactions of electrons with phonons. Indeed, we have established in Chapter 3 that the transport of the ballistic electrons can be modified by the phonon interactions even in the absence of charges. Notably, the population of phonons 𝑁(𝑇) of a given mode ℏ𝜔 is dependent on the temperature 𝑇, as it follows the Bose-Einstein distribution: 𝑁(𝑇) = 1 𝑒 ℏ𝜔/𝑘 𝑏 𝑇 -1 Equation 6-10

Figure 6 - 45 :

 645 Figure 6-45: Simulation of the TEEY of 300 eV electrons in SiO2 at -180°C, 27°C and 200°C, normalized to match the starting point of the TEEY at 27°C. J0 = 10 µA/cm²

3 (

 3

Figure 6 - 46 :

 646 Figure 6-46: Charge density profiles at the end of 100 pulses of 300 eV electrons at 27°C, 200°C and -180°C. J0 = 10 µA/cm²

Figure 6 - 47 :

 647 Figure 6-47: Evolution of the surface charge density of 300 eV electrons at 180°C, 27°C and 200°C. J0 = 10 µA/cm²

Figure 6 - 48 :

 648 Figure 6-48: Comparison of the incident current used in standard TEEY curve measurements in DEESSE and ALCHIMIE

Figure 6 - 49 :

 649 Figure 6-49: Measurement and simulation of the TEEY over 300 pulses, for 300eV, 500 eV and 1 keV electronsLastly, we have made time-resolved measurements of the TEEY in ALCHIMIE over several seconds, by sending 5000 pulses of 100 µs with an interval of 50 ms, for a total irradiation time of 0.5 s. The objective here is to sample the decrease of the TEEY until it reaches its equilibrium, and define the value of this equilibrium. It is technically possible to simulate such a duration with the Monte-Carlo code, however the computation time would be enormous. Indeed, it took 17 hours on our desktop computer to simulate the TEEY given in Figure6-49 for 500 eV electrons over 300 pulses. In comparison, the time required for the acquisition of the TEEY over 5000 pulses in ALCHIMIE or DEESSE is roughly 4 minutes, regardless of the incident energy. In consequence, we will only use the experimental data acquired with ALCHIMIE for this part of the study, without comparing it to simulated results. Since the incident current remains practically invariable through the time-resolved measurements, only the emitted current was measured. The TEEY was then computed with the average value of the incident current from the TEEY measurements made over 300 pulses. By proceeding this way, we avoid creating a large number of residual holes when measuring I0 before sampling the emitted current, which could happen if we sent 5000 pulses first to measure I0. The TEEY measured during 5000 pulses are plotted for 300 eV and 500 eV in Figure6-50, in a minimal beam area configuration (Focus voltage F500 or F100, 0.1 cm²) and a broad beam configuration (F0, > 25 cm²).

Figure 6 - 50 :

 650 Figure 6-50: Measurement of the TEEY of 300 eV and 500 eV electrons in SiO2 over 5000 pulses

Figure 6 - 51 :

 651 Figure 6-51: Measurement of the TEEY of 1 keV electrons in SiO2 over 5000 pulses (a). In (b), the data of (a) is zoomed in the first 30 ms (300 pulses) of the measurement.

Figure 6 - 52 :

 652 Figure 6-52: GREEN differential GEO electron spectrum

Figure 6 - 53 :

 653 Figure 6-53: Simulated TEEY and surface potential of a 20 nm SiO2 target. J0 = 10 µA/cm².

Figure 6 - 54 :

 654 Figure 6-54: Charge density profile after 30ms of the 20 nm SiO2 target

Figure 6 - 55 :

 655 Figure 6-55: Dose-depth profiles for 1 keV and 10 keV incident spectrums

Figure 6 - 56 :

 656 Figure 6-56: TEEY and surface potential of a 2µm SiO2 target

Figure 6 - 57 :

 657 Figure 6-57: Charge density profile after 30ms in the 2 µm SiO2 target

Figure 6 - 58 :

 658 Figure 6-58: Integrated current densities from GREEN for low energy electrons in the LEO and GEO space environment

Figure 6 - 59 :

 659 Figure 6-59: Simulation of the TEEY of the 1 keV GEO spectrum, J0 = 1 nA/cm²

1 -

 1 Creation and initialization of the DriftManager at the start of a simulation DriftMessenger will load the simulation parameters from the ChargingParameters.mac file, which is shown below. The macro commands setting the parameters for the mesh and the physics are set in the categories "Mesh" and "Phys". The commands in this file allow to respectively set:  The sample holder voltage and electron gun voltage, as boundary conditions for the computation of the electric field  The number of incident electrons (saved in nbOfElecPerRun in the drift manager) and the associated charge bias factor. The number of incident electrons is set as a balance between computation time and statistical consistency. Using an external table file, the charge bias factor is computed from the incident current we want to model with Equation 5-4. The electron and hole mobilities, the relative permittivity  The radius of an electron cascade for the overlap factor  The cross sections for empty trap, respectively for ballistic electrons, drift electrons and holes  The cross sections for recombination, in the same order  The detrapping rate 𝑊 0 and the temperature

  Figure A-1 shows the interactions between the DriftManager and the other classes of Geant4 during an electron cascade run. The physical processes G4MicroElecInelasticModel and G4MicroElecElastic model can respectively register new holes and drift electrons to the DriftManager, using void AddHoleFromCascade(G4ThreeVector pos)and void AddElectronFromCascade(G4ThreeVector pos). When one of these functions is called, the position of creation of the particle is saved in either of the vectors std::vector<G4ThreeVector> holeNew or std::vector<G4ThreeVector> electronNew of the drift manager.

Figure A- 1 :

 1 Figure A-1: Architecture of the simulation for the transport of ballistic electrons

Figure A- 2 :

 2 Figure A-2: Illustration of the storage vectors for deep electron traps, deep hole traps and shallow electron traps (a), and for shallow hole traps (b)

  ℎ𝑜𝑙𝑒𝑠 ) 𝑖 -(𝑛 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠 ) 𝑖 ] 𝑒𝛽 ∆𝑧 𝑖 𝑆 𝑚𝑒𝑠ℎ Equation A-3

Figure A- 3 :

 3 Figure A-3: Architecture of the simulation for the transport of drift particles

Figure 1 - 1 : 12 Figure 1 - 2 : 15 Figure 1 - 3 :

 1112121513 Figure 1-1: Integral fluxes of protons and electrons received by a spacecraft in LEO and GEO orbits, from the radiation belt model GREEN ................................................................................. 12 Figure 1-2: Illustration of an electron avalanche between two parallel plates ............................... 15 Figure 1-3: Illustration of a typical TEEY curve ............................................................................... 17Figure 1-4: Illustration of a typical energy distribution of emitted electrons ................................ 18 Figure 1-5: TEEY measurements for an etched sample of Cu under various angles of incidence. data from [29] ......................................................................................................................................... 19 Figure 1-6: TEEY measurements of a copper sample, as received and after decontamination by baking and erosion. Data from [30] ..................................................................................................... 21 Figure 1-7: Comparison of the TEEY of silver on a flat surface and with roughness patterns with height = 100µm and base length L=80µm. Simulated results from [40] .............................. 22 Figure 1-8: Evolution of the global charge of the insulator according to the TEEY curve ....... 26 Figure 1-9: Charge buildup and evolution of the TEEY for incident electrons below the first crossover point ....................................................................................................................................... 27 Figure 1-10: Charge buildup and evolution of the TEEY for incident electrons between the two crossover points ..................................................................................................................................... 28 Figure 1-11: Charge buildup and evolution of the TEEY for incident electrons after the second crossover point ....................................................................................................................................... 29 Figure 1-12: Illustration of atypical TEEY curves observed on SiO2 thin films .......................... 32 Figure 2-1: Illustration of the diffusion of particles by a scattering center .................................... 42 Figure 2-2: Illustration of the electron-matter interactions that are common to metals, semiconductors and insulators ............................................................................................................. 44 Figure 2-3: Illustration of the electronic structure of the isolated atom and of a solid ............... 47 Figure 2-4: Illustration of the band structure depending on material type .................................... 48 Figure 2-5: Energy changes for an electron penetrating in a material ............................................ 51 Figure 2-6: Illustration of the electron and charge carrier interactions in insulators ................... 53 Figure 2-7: Dispersion relations of optical and acoustic phonons ................................................. 54 Figure 2-8: Definition of the drift and thermal velocities from the distributions of particle positions, in the case of gaussian transport ........................................................................................ 56 Figure 2-9: Shallow traps and associated transport mechanisms .................................................... 58 Figure 2-10: Deep traps in the band gap of an insulator .................................................................. 60 Figure 2-11: Detrapping enhancements factors ................................................................................. 61 Figure 2-12: Recombination mechanisms in insulators -trap assisted recombination and geminate recombination ........................................................................................................................ 62 Figure 2-13: Definition of the true range and projected range from an electron's trajectory ..... 64 Figure 2-14: Definition of the true range and the projected range of electrons. .......................... 65

Figure 1 - 4 : 18 Figure 1 - 5 : 19 Figure 1 - 6 : 27 Figure 1 - 10 : 28 Figure 1 - 11 : 29 Figure 1 - 12 : 32 Figure 2 - 1 : 42 Figure 2 - 2 : 44 Figure 2 - 3 : 47 Figure 2 - 4 : 48 Figure 2 - 5 : 51 Figure 2 - 6 : 53 Figure 2 - 7 : 54 Figure 2 - 8 : 56 Figure 2 - 9 : 58 Figure 2 - 10 : 60 Figure 2 - 11 : 61 Figure 2 - 12 :

 1418151916271102811129112322142224423472448255126532754285629582106021161212 Figure 1-1: Integral fluxes of protons and electrons received by a spacecraft in LEO and GEO orbits, from the radiation belt model GREEN ................................................................................. 12 Figure 1-2: Illustration of an electron avalanche between two parallel plates ............................... 15 Figure 1-3: Illustration of a typical TEEY curve ............................................................................... 17Figure 1-4: Illustration of a typical energy distribution of emitted electrons ................................ 18 Figure 1-5: TEEY measurements for an etched sample of Cu under various angles of incidence. data from [29] ......................................................................................................................................... 19 Figure 1-6: TEEY measurements of a copper sample, as received and after decontamination by baking and erosion. Data from [30] ..................................................................................................... 21 Figure 1-7: Comparison of the TEEY of silver on a flat surface and with roughness patterns with height = 100µm and base length L=80µm. Simulated results from [40] .............................. 22 Figure 1-8: Evolution of the global charge of the insulator according to the TEEY curve ....... 26 Figure 1-9: Charge buildup and evolution of the TEEY for incident electrons below the first crossover point ....................................................................................................................................... 27 Figure 1-10: Charge buildup and evolution of the TEEY for incident electrons between the two crossover points ..................................................................................................................................... 28 Figure 1-11: Charge buildup and evolution of the TEEY for incident electrons after the second crossover point ....................................................................................................................................... 29 Figure 1-12: Illustration of atypical TEEY curves observed on SiO2 thin films .......................... 32 Figure 2-1: Illustration of the diffusion of particles by a scattering center .................................... 42 Figure 2-2: Illustration of the electron-matter interactions that are common to metals, semiconductors and insulators ............................................................................................................. 44 Figure 2-3: Illustration of the electronic structure of the isolated atom and of a solid ............... 47 Figure 2-4: Illustration of the band structure depending on material type .................................... 48 Figure 2-5: Energy changes for an electron penetrating in a material ............................................ 51 Figure 2-6: Illustration of the electron and charge carrier interactions in insulators ................... 53 Figure 2-7: Dispersion relations of optical and acoustic phonons ................................................. 54 Figure 2-8: Definition of the drift and thermal velocities from the distributions of particle positions, in the case of gaussian transport ........................................................................................ 56 Figure 2-9: Shallow traps and associated transport mechanisms .................................................... 58 Figure 2-10: Deep traps in the band gap of an insulator .................................................................. 60 Figure 2-11: Detrapping enhancements factors ................................................................................. 61 Figure 2-12: Recombination mechanisms in insulators -trap assisted recombination and geminate recombination ........................................................................................................................ 62 Figure 2-13: Definition of the true range and projected range from an electron's trajectory ..... 64 Figure 2-14: Definition of the true range and the projected range of electrons. .......................... 65

Figure 2 - 68 Figure 3 - 1 : 78 Figure 3 - 2 : 80 Figure 3 - 3 : 84 Figure 3 - 4 : 88 Figure 3 - 5 : 89 Figure 3 - 6 :Figure 4 - 1 :Figure 4 - 5 :

 26831783280338434883589364145 Figure 2-15: Monte-Carlo and analytical computations of the dose deposed in Al by low energy electrons ................................................................................................................................................... 68 Figure 3-1: Comparison of molecular and averaged elastic Total Cross Sections for SiO2 ....... 78 Figure 3-2: Examples of OELFs from Sun et al.'s database ............................................................ 80 Figure 3-3: Fitted OELFs for a metal (Ag), a semi-conductor (Si), and an insulator (SiO2) supported by MicroElec. Each colored smooth curve is attributed to a core shell/plasmon/interband transition. The reference OELFs are taken from the database of Sun et al. .......................................................................................................................................................... 84 Figure 3-4: Illustration of the rejection sampling procedure ........................................................... 88 Figure 3-5: Illustration of direct sampling through cumulated DXS .............................................. 89 Figure 3-6: Energy changes for an electron going through the surface barrier in the case of a metal or an insulator. The energy reference in the material used in MicroElec is highlighted in red. ............................................................................................................................................................ 92 Figure 3-7: Example of the modeling of a multilayer structure (SiO2 on Si) in MicroElec ........ 94 Figure 3-8: Material structure files of silver (a) and SiO2 (b) ........................................................... 95 Figure 3-9: Calculated Mean Free Paths for the simulated processes in SiO2, including interactions with acoustic phonons. Interactions with Longitudinal-Optical (LO) phonons are simulated for the two dominant vibration modes (63 and 153 meV). ........................................... 97 Figure 3-10: Emission and absorption rates of LO phonons in SiO2 ............................................ 98 Figure 3-11: Stopping powers of electrons (a), protons (b) and 18 Ar ions (c) in Si ................... 106 Figure 3-12: Electron stopping powers for the materials modeled in MicroElec. The stopping powers of BN are provided for information purposes only, as no reference data could be found for this material. ................................................................................................................................... 108 Figure 3-13 : Dose-depth profile for 10 keV protons (a) and 200 eV electrons (b) with normal incidence in Si ...................................................................................................................................... 109 Figure 3-14: Effects of the corrections implemented in MicroElec for low energy electrons on the TEEY of silicon ............................................................................................................................ 111 Figure 3-15 : Comparison of the TEYs calculated for all metals and semi-conductors in MicroElec ............................................................................................................................................. 114 Figure 3-16: SEY of Al2O3 computed without the phonon and polaron processes ................. 115 Figure 3-17: Comparison of the SEY and BEY of SiO2 simulated in MicroElec with phonon interactions, without polaronic capture ........................................................................................... 116 Figure 3-18: Comparison of the BEY and SEY from MicroElec for SiO2 including all models ................................................................................................................................................................ 117 Figure 3-19: Comparison of the BEY and SEY from MicroElec for Al2O3 including all models ................................................................................................................................................................ 118 Figure 4-1: Electron transmission rate for Be, Al, Fe and Ag materials. The energy of electrons range from 25 eV up to 5 keV........................................................................................................... 132 Figure 4-2: Extrapolated ranges of low energy electrons in Al and Si ........................................ 134 Figure 4-3:Elastic scattering ratio for 11 materials ......................................................................... 135 Figure 4-4: Mean free paths of electrons in MicroElec for Si and Al ......................................... 135

Figure 4 - 19 :Figure 5 - 1 :Figure 5 - 2 :Figure 5 - 3 :Figure 5 - 4 :Figure 5 - 7 :Figure 5 - 12 :Figure 6 - 9 :Figure 6 - 15 :Figure 6 - 34 :Figure 6 - 54 :

 419515253545751269615634654 Figure 4-5: Comparison of the transmission rates in Si from MicroElec and the standard continuous processes .......................................................................................................................... 136 Figure 4-6: Comparison between the range models of Equation4-3 (ref.[7]) and Equation4-5 (This work) ........................................................................................................................................... 138 Figure 4-7: Comparison of the extrapolated ranges given by eq. 13 and MicroElec for all 11 materials (Be, Al, Ti, Ni, Ge and W in fig (a) and C, Si, Fe, Cu and Ag in fig (b)).................... 139 Figure 4-8: Correlation of the atomic number (y-axis) with the height of the range plateau (a) and the slope of the range curve (b) ................................................................................................. 142 Figure 4-9: Transmission rate model (grey) compared with MicroElec (colors) ....................... 144 Figure 4-10. Dose-depth curve given by the analytical model (Model) proposed in this work for incident electrons in aluminum. The model is compared with Monte Carlo simulations of MicroElec, Walker[22] and OSMOSEE[23]. Small circles of different colors are for MicroElec simulations, big circles of different colors are for Walker[36] reference data and different squares are our former simulations performed with OSMOSEE code[37]. ........................................... 150 Figure 4-11: Validation of the analytical dose model with MicroElec data for 11 materials ... 153 Figure 4-12: Comparison of the dose profiles in Si from MicroElec and GRAS...................... 154 Figure 4-13: Comparion between MicroElec and GRAS ............................................................. 156 Figure 4-14: Comparison of the low energy and high energy analytical models ....................... 157 Figure 4-15: OELFs of Ge, Al, Ag, Si.............................................................................................. 163 Figure 4-16: Median SE energy in Al, Ag, Ge and Si from MicroElec simulations .................. 164 Figure 4-17: Secondary emission yield calculated with the model and MicroElec for an aluminum target and for different angles of incidence ..................................................................................... 165 Figure 4-18 : Comparison of the analytical SEY model with Monte-Carlo and experimental data ................................................................................................................................................................ 166 Figure 4-19: κ as a function of Z ...................................................................................................... 168 Figure 4-20: Comparison of the energy of the maximal SEY from the analytical model and Monte Carlo data ............................................................................................................................................. 168 Figure 4-21: Correlation of G with the max value of the SEY .................................................... 170 Figure 4-22: Correlation of G with the position of the max value of the SEY ......................... 170 Figure 4-23: Comparisons of the dose depth curve given by the analytical model of this work for incident electrons in copper with MicroElec and a constant energy loss model. ...................... 171 Figure 4-24: Correlation between the surface energy deposit and the SEY from the analytical models: position (a) and value (b) of the max ................................................................................. 173 Figure 4-25: Comparison of the surface energy deposit and SEY curves for C, Ti, Ni, W. .... 174 Figure 5-1: Simulation configuration ................................................................................................ 185 Figure 5-2: Trap levels modeled in the simulations ....................................................................... 195 Figure 5-3: Modeling of the shallow hole traps .............................................................................. 198 Figure 5-4: Simulation results of the electron cascade radius in SiO2 for different incident energies ................................................................................................................................................. 205 Figure 5-5: Illustration of the evolution of the overlap factor with time .................................... 206 Figure 5-6: Architecture of the simulation for the transport of ballistic electrons .................... 208

Figure 6 - 59 :

 659 Figure 6-54: Charge density profile after 30ms of the 20 nm SiO2 target .................................. Figure 6-55: Dose-depth profiles for 1 keV and 10 keV incident spectrums ............................ Figure 6-56: TEEY and surface potential of a 2µm SiO2 target .................................................. Figure 6-57: Charge density profile after 30ms in the 2 µm SiO2 target ..................................... Figure 6-58: Integrated current densities from GREEN for low energy electrons in the LEO and GEO space environment ...................................................................................................................Figure 6-59: Simulation of the TEEY of the 1 keV GEO spectrum, J0 = 1 nA/cm² ..............Figure A-1: Architecture of the simulation for the transport of ballistic electrons ...................Figure A-2: Illustration of the storage vectors for deep electron traps, deep hole traps and shallow electron traps (a), and for shallow hole traps (b) ............................................................................ Figure A-3: Architecture of the simulation for the transport of drift particles ..........................

Figure A- 1 :

 1 Figure 6-54: Charge density profile after 30ms of the 20 nm SiO2 target .................................. Figure 6-55: Dose-depth profiles for 1 keV and 10 keV incident spectrums ............................ Figure 6-56: TEEY and surface potential of a 2µm SiO2 target .................................................. Figure 6-57: Charge density profile after 30ms in the 2 µm SiO2 target ..................................... Figure 6-58: Integrated current densities from GREEN for low energy electrons in the LEO and GEO space environment ...................................................................................................................Figure 6-59: Simulation of the TEEY of the 1 keV GEO spectrum, J0 = 1 nA/cm² ..............Figure A-1: Architecture of the simulation for the transport of ballistic electrons ...................Figure A-2: Illustration of the storage vectors for deep electron traps, deep hole traps and shallow electron traps (a), and for shallow hole traps (b) ............................................................................ Figure A-3: Architecture of the simulation for the transport of drift particles ..........................

Figure A- 2 :

 2 Figure 6-54: Charge density profile after 30ms of the 20 nm SiO2 target .................................. Figure 6-55: Dose-depth profiles for 1 keV and 10 keV incident spectrums ............................ Figure 6-56: TEEY and surface potential of a 2µm SiO2 target .................................................. Figure 6-57: Charge density profile after 30ms in the 2 µm SiO2 target ..................................... Figure 6-58: Integrated current densities from GREEN for low energy electrons in the LEO and GEO space environment ...................................................................................................................Figure 6-59: Simulation of the TEEY of the 1 keV GEO spectrum, J0 = 1 nA/cm² ..............Figure A-1: Architecture of the simulation for the transport of ballistic electrons ...................Figure A-2: Illustration of the storage vectors for deep electron traps, deep hole traps and shallow electron traps (a), and for shallow hole traps (b) ............................................................................ Figure A-3: Architecture of the simulation for the transport of drift particles ..........................

  1.5 -Computing the TEEY with Monte Carlo simulations

Table 3 -1 : Sum rules values deduced from the OELF fits, for 11 materials, in the new version of MicroElec

 3

  -2.

	Material	SiO2 C	Al	Si	Ti	Ni	Cu Ge	Ag W	Kapton
	Initial energy
	of	weakly
	bound		8	0	4.2	1.73 4	3	2.6 2.5	2.03 4	0.5
	electrons
	(𝐸 𝑖𝑛𝑖𝑡 , eV)

Table 3 -2: Initial energy 𝑬 𝒊𝒏𝒊𝒕 of weakly bound electrons

 3

Table 3 -3: Parameters stored in the Data_Material file

 3

Table 3

 3 -4 for Al2O3, BN and SiO2.

	Material ℏ𝜔 𝐿𝑂 (eV)	𝜖(∞) 𝜖(0)
	SiO2	0.131	2.25	3.84
		(0.75 x 0.153 + 0.25 x 0.063)
		[58]
	Al2O3	0.1 [63]	3	9
	BN	0.17	4.5	7.1

Table 3 -4: Parameters used for the LO phonon model

 3

Table 3 -5 : Table of parameters for the acoustic phonon model in SiO2

 3

Table 3 -6: Parameters of the polaronic capture model

 3

3.3 -Validation of the low energy transport model with reference data

Table 3 -7: Energy parameters for low energy electrons

 3

3.3 -Validation of the low energy transport model with reference data

Table 4 -

 4 1. In the case of the CSDA values, one must use 𝑟 𝐴𝑙 = 9 * 10 -7 𝑔 𝑐𝑚 2 ⁄ .

Table 4 -1: G F parameters from the M-C simulations (bold red) and from CSDA ranges [21]

 4

	Z	G	F
	3	0.06	1.776
	4	0.51	1.8
	6	0.74	1.8
	11	0.09	1.719
	12	0.4	1.714
	13	1	1.73
	14	0.92	1.83
	19	0.2156	1.656
	21	1.386	1.69
	22	1.91	1.8
	23	2.6026	1.657
	24	2.9722	1.64
	26	3.46	1.75
	27	4.774	1.654
	28	3.58	1.8
	29	3.72	1.8
	32	2.71	1.6
	39	1.8018	1.624
	41	5.1128	1.599
	42	5.9444	1.594
	44	6.1138	1.593
	45	5.5748	1.599
	46	5.0974	1.6

Table 4 -2: BEY values

 4 Validation of the model with Monte-Carlo data 4.2.2.1 Comparison of the analytical model with Monte-Carlo data

	Material		C	Be Al Si Ti	Fe Ni Cu Ge Ag (Elastic yield)	W
	Z		6	4 13 14 22	26 28 29 32 47	74
	Low	Value	0.1	0.3	0.35
	energy BEY High	Energy domain Value	E < 21 eV E < 200 eV 0.02 0.225	E < 400 eV 0.14	0.35
	energy BEY	Energy domain	E > 700 eV E > 2 keV	E > 3 keV

Table 4 -

 4 3: Values for the parameters of the SEY model

	Material	(eV)	𝑤 𝑓	⟨𝐸 𝑠 ⟩ (eV)	(eV)	𝐼	𝜅
	Be	4.98		18.4	18.4	1,65
	C	4.81		20	20	1
	Al	4.28		15	15	1,05
	Si	4.05		16.7	16.7	1,3
	Ti	4.33		19.4	19.4	0,85
	Fe	4.5		23.6	23.6	0,85
	Ni	5.15		22	22	0,75
	Cu	4.65		19	19	0,65
	Ge	5		13	13	0,65
	Ag	4.26		22	22	0,75
	W	4.55		35	35	1,15

  .

	Symbol	Definition
	𝑛 𝑒	Number of electrons in the simulation
	𝑛 ℎ	Number of holes in the simulation
	𝑁 𝑒	Density of electrons per cm 3
	𝑁 ℎ	Density of holes per cm 3
	𝜌 𝑒	Volume charge density of electrons in C/cm 3
	𝜌 ℎ	Volume charge density of holes in C/cm 3
	𝑁 𝑇	Density of traps (shallow or deep) per cm 3
	𝑁 𝑆	Density of shallow traps per cm 3
	𝑁 𝐷	Density of deep traps per cm 3
	𝜎 𝑆	Capture cross section by a shallow trap
	𝜎 𝐷	Capture cross section by a deep trap
	𝜎 𝐹𝑟𝑒𝑒	Capture cross section by a free trap
	𝑁 𝐹𝑟𝑒𝑒	Density of free traps
	𝜎 𝑒-ℎ	Electron/hole recombination cross section

Table 5 -1: Definition of the notations for the charge densities, trap densities and cross sections

 5 In this work, we aim to reproduce the experimental setup of the TEEY measurement facility DEESSE at ONERA, which we used to measure the TEEY on amorphous SiO2 samples. It is thus necessary to simulate the experimental setup configuration and the experimental samples, to improve the comparison of our model with the data from this facility.

	5.2 Developing an iterative Monte-Carlo simulation of
	charging and secondary electron emission
	5.2.1 Presentation of the simulation configuration and general procedure

Table 6 -1: Simulation parameters used in the charge transport model

 6

	Parameter	Value
	Simulation time step	1 µs
	Incident current	1 µA

Table 6 -2: Experimental measurement parameters used in this study

 6

	Measurement type	Time-resolved measurements at a single energy	Energy/TEEY curve from 50 eV to 2 keV
	Pulse length	100 µs	6 ms
	Relaxation time between two pulses	50 ms	200 ms
	Number of pulses sent per	80 to 100, up to a few	TEEY averaged over 10
	incident energy	thousand in section 6.4	pulses
	Incident current	0.1 µA to 1 µA
	Area irradiated by the	Minimum: 0.05 cm²
	electron beam	Maximum: > 25 cm²
	Incident current density	Minimum: < 25 nA/cm² Maximum: 20 µA/cm²

  of the edge of the Brillouin zone (a few eVs), the dependence in temperature is contained in the phonon population 𝑁 𝐵𝑍 at the edge of the Brillouin zone, which follows Equation 6-10.

										E BZ 4	. For electrons
	above the energy 𝑓 𝑎𝑐 =	𝜋𝑘 𝑏 𝑇 ℏ𝑐 𝑠 2 𝜌	ℰ 𝑎𝑐 𝐷(𝐸) 1 + 𝐸 𝐴 ⁄	if E <	E BZ 4
	𝑓 𝑎𝑐 =	2𝜋𝑚 * * (2𝑁 𝐵𝑍 + 1) 𝜌ℏℏ𝜔 𝐵𝑍	ℰ 𝑎𝑐 2 𝐷(𝐸)𝐸 2 ( 𝐸 𝐴	) 2	* [-	𝐸 𝐴 ⁄ 1 + 𝐸 𝐴 ⁄	+ ln (1 +	𝐸 𝐴	)] if E > E BZ
			Equation 3-31 of Chapter 3

2.2 -Common electron-matter interactions for all material types

3.3 -Validation of the low energy transport model with reference data

6.3 -Study of multiple-hump TEEY curves

-Simulation of the transport of drift electrons and holes

Remerciements

Ces travaux de thèse de doctorat ont été effectués à lEspace (DPHYCette thèse a bénéficié d'un cofinancement Elle a été

Trapping of ballistic electrons

The deep level traps are also capable of capturing ballistic electrons with a reduced cross section 𝜎 𝐷 = 10 -16 𝑐𝑚² to take into account their higher velocity. For shallow traps, the cross section is also lowered to 6 × 10 -15 𝑐𝑚² for ballistic electrons. However, this trapping is only possible if the electron energy is very low (a few eVs). To take into account this dependency on the energy, the mean free path obtained from Equation 5-27 or Equation 5-29 is modified using Ganachaud and Mokrani's empirical law [21]:

= 𝜎𝑁𝑒𝑥𝑝 (-𝛾𝐸)

Equation 5-30

Where 𝜎 and 𝑁 are the capture cross section and density of the shallow or deep traps, and γ = 0.2 eV -1 , which is the valued used in Chapter 3 and is similar to the value used by Ohya et al. [22] for SiO2 (0.25). Compared to our use of this function for the polaronic empirical capture in Chapter 3, we have substituted the fitting parameter 𝑆 in the interaction probability by the inverse mean free path 𝜎𝑁.

Modeling of the detrapping

Detrapping by thermal activation

The charge carriers immobilized in any kind of trap are able to escape under the effect of thermal agitation. The escape frequency 𝑊(𝐸 𝑖 ) for a trap level of energy depth 𝐸 𝑖 follows a thermally activated law:

5.4 -Modeling of the detrapping particle is located in vacuum but not above the surface of the material (x or/and y are greater than the width of the surface), we assume that the electric field is the same as in the previous case. Lastly, if the particle is in the Si layer, the electric field is equal to zero.

The trapping of low energy electrons is managed in the overhauled process G4ElectronCapture. As a Geant4 process, two phases are involved during the transport of the particle, which each require communication with the drift manager: the computation of the interaction mean free path, and the action to be done at the end of the step if the process is selected. These two phases are respectively handled by G4double G4ElectronCapture::GetMeanFreePath, which returns a mean free path to the Geant4 kernel, and

G4VParticleChange* G4ElectronCapture::PostStepDoIt, which tells Geant4 what to do when the interaction happens at the end of a step.

In G4ElectronCapture::GetMeanFreePath, the function G4double DriftManager::GetTrapMFPForDriftParticle(G4double depthInMeters, particleType part, trapType t) of the drift manager is called. As an input, it receives the depth of the particle, its type, and the type of trap involved. The type of particle is defined by the enum particleType, which can be a driftElectron, balisticElectron or driftHole. Here, we are in the case of a balisticElectron. The type of trap is defined by another enum trapType, it can be a shallowTrap or a deepTrap. The value of the type of trap is stored in the G4ElectronCapture object. In practice, two copies of G4ElectronCapture are created in the physics list. One handles the trapping of low energy electrons by shallow traps, and the other by deep traps. When the two G4ElectronCapture objects are created, we attribute to them the value of either shallowTrap or deepTrap for the type of trap. In the function of the drift manager GetTrapMFPForDriftParticle, the cell of the mesh in which the current particle is located is first determined. This is done by calling the function G4double DriftManager::GetMeshCell(G4double depth), which finds for a given depth 𝑧 the corresponding cell of the mesh 𝑧 𝑖 with a dichotomy method as always. Once the cell is found, we can retrieve the number of holes and electrons trapped in the cell, and compute the density of trapped electrons (e), holes (h) and free traps using Equation 5-9 for the charge densities as:

With 𝑛 𝑒/ℎ the number of trapped electrons/holes saved in the cell. We also use Equation 5-46 for the number of remaining free traps as:

Finally, from Equation 5-39, the function returns the capture inverse mean free path to the class G4ElectronCapture. The density of trapped particles and the cross sections vary depending on the type of trap (deep/shallow), which is why we need to have one capture process for each type of trap. The type of trap also needs to be given as input to the method GetTrapMFPForDriftParticle so that it can retrieve the correct densities and cross sections. The

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