Introduction

Since the invention of the semiconductor bipolar transistor by William Shockley, Walter Brattain, and John Bardeen in 1947, followed by that of the unipolar Metal-Oxide-Semiconductor Field-Eect Transistor (MOSFET) by Atalla and Kahng in 1959, the microelectronic industry has revolutionized our society thanks to the continuous production of innovative and evermore performant applications with a direct impact on our everyday's life. Indeed, thanks to its much lower power consumption and higher density than bipolar junction transistors, MOSFET has made it possible to build high-density integrated circuits (ICs) allowing the integration of a continuously increasing number of transistors in a single PC (from few thousands in the 1970s to more than one billion in today's microprocessors). Today, semiconductor-based Integrated Circuits (ICs) are present in a large amount of applications: data processing (personal computers, laptops, servers and tablets), communications (mobile phones, smartphones. . . ), consumer electronics (television sets, music players, gaming consoles. . . ), automotive, industrial (transport. . . ), military in addition to spatial and medical applications. Besides, a further step in the electronics revolution is expected with the development of the Internet of Things (IoT), where a new range of low-power devices will be able to directly interact and communicate through the internet.

In all cases, in order to answer miniaturization and diversication challenges, the physical properties of the materials being used in nowadays devices must be known and controlled at the nanometre scale, calling for the development of more and more sophisticated characterization techniques. For instance, one of the key issues for the miniaturization of 3D semiconductor nanodevices is the precise control of their doping which determines the device electrical properties. Today, the spatial distribution of the doping concentration must be controlled with a precision higher than 1 nm while atomic concentrations below 1 at.% have to be measured. Moreover, the electrical properties can also be improved by modifying the material structure as in the case of SiGe alloys, which combine the advantages of Germanium (high carrier mobility) with those of Silicon (high-quality Si oxide insulator). Again, the precise control of the alloy stoichiometry is crucial for the successful implementation of this material.

Several characterization techniques exist for the measurement of both dopant concentration and alloy stoichiometry. Over the years, Secondary Ion Mass Spectroscopy (SIMS) has established as the most reliable technique. It is based on the sputtering of the sample surface with a focused primary ion beam and the concomitant collection and 10 CONTENTS analysis of the ejected secondary ions. The mass/charge ratios of these secondary ions are measured with a mass spectrometer to determine the elemental, isotopic, or molecular composition. One of the main advantages of SIMS is its particularly low detection limit (which is around 1 x10

14 atoms/cm3 for most impurity elements in Silicon). How- ever, the SIMS quantication technique is relevant only on one-dimensional devices and requires large analysis areas (~30 m x 30 m) to achieve signicant result. For this reason, alternative techniques must be developed to ensure the localized measurements in the most advanced 2D and 3D devices being developed in the semiconductor industry.

The work presented in this thesis is based on the use of Scanning Transmission Electron Microscopy (STEM) associated with Energy Dispersive X-ray spectroscopy (EDX),

which is an excellent candidate due to its versatility (almost all elements of the periodic table can be mapped) and its high spatial resolution. In particular, the availability of probe spherical aberration correction, high brightness sources, and high angle silicon drift detectors (SDD) make this technique even more attractive since a high signal to noise ratio can be obtained with the characteristic resolution of STEM imaging techniques. In practice, the quantication of EDX spectra is achieved thanks to commercial softwares in which the dierent physical phenomena occurring between the X-ray emission in the TEM specimen and the X-ray detection are taken into account, so to avoid any impacts of experimental and theoretical artifacts on the nal result.

Artifacts can be divided in two categories: artifacts related to the X-ray detector (peaks deconvolution, shelf, sum peak, etc. . . ) whose impact is correctly taken into account in most commercial softwares; and artifacts related to the interaction between X-rays emitted in the analyzed specimen and the specimen itself, especially the X-ray absorption eect which depends on the mass absorption coecient of each element in the sample as well as the specimen thickness and density. The most critical issue in the calculation of the absorption correction is that the mass absorption coecient and the specimen density depend both on the specimen composition, which is the parameter to be measured. Consequently, iterative methods must be implemented for this calculation. In addition, when the exact relation between the specimen density and composition is not known, the reliability of the quantication results strongly depends on the approximation, which is used to describe it.

Optimized methods for the calculation of the X-ray absorption eect are nally implemented in the data treatment procedure leading from raw EDX spectra to quantied elemental maps. The two most common quantication procedures are the x -factor method and the Cli-Lorimer method. The x -factor method is based on the principle that the intensity of an X-ray is proportional to the mass-thickness and composition of the specimen. However, the disadvantage of this method is that an in-situ measurement of the beam current is required which is not measurable in the microscope setup used in this work. In contrast, the Cli-Lorimer method that we have used in this thesis is based on the principle that the ratio of the X-ray intensity of two elements is proportional to the ratio of the corresponding mass fractions, with the proportionality factors called the Cli-Lorimer (C-L) factors. In this method, the measurement of the beam current is not needed. However, its reliability relies on the precise knowledge of the C-L factors. Commercial softwares are typically based on theoretical values of the C-L factors, which can produce signicant systematic errors.

Within this context, the objective of this thesis is to develop an improved Cli-Lorimer method for the quantication of EDX elemental maps in semiconductor materials, with the specic aim to allow reliable measurements in the case of low concentrations of dopant impurities (below 1 at. %). In our improved method, the iterative correction of the X-ray absorption eect is based on the use of physical models describing the relation between the specimen density and its composition. For each one of the case studies considered in this work, the best-suited model is therefore specically identied and applied to the quantication procedure. In particular, the amorphous solid approximation (based on the Ideal Amorphous Solid structure) will be used for the quantication of Arsenic dopants in Silicon before thermal annealing, while the same radius approximation (that assumes a negligible variation of the covalent radius between matrix and dopant atoms) will be shown to be valid for the quantication of Arsenic dopants in Silicon after thermal annealing. Finally, the amorphous solid approximation will be shown to be consistent with the Vegard's law and will therefore be used for the measurement of the Ge content in SiGe alloys pseudomorphically grown on a Si substrate.

In addition, the C-L factors used in our improved method are experimentally determined for the materials systems investigated in our work (i.e. As-doped silicon and pseudomorphically grown SiGe/Si stacks). For this, we used reference samples whose chemical composition was determined by the Rutherford Backscattered Spectroscopy (RBS) technique. By a direct comparison of EDX quantied data with RBS results, a reliable calibration of the C-L factors for the investigated elements could therefore be achieved.

Finally, we have developed an error estimation procedure based on linear regression and standard error propagation techniques in order to estimate the total experimental error in the dopant density as well as the detection limit for our experimental setup. This total experimental error includes (i) the statistical error on the measured X-rays intensity, (ii) the error on the calibrated C-L factors and (iii) the experimental error on the specimen thickness.

The background and the results of this work are presented in the rst three chapters of this manuscript.

Chapter 1 contains a general description of the fundamentals of the electron microscopy and its related techniques, such as high resolution, Scanning electron Microscopy (SEM), Transmission Electron microscopy (TEM) and Scanning TEM (STEM).

Then a detailed description of the EDX quantication technique will be given, including the various artifacts that have an impact on the quantied results. Finally, the existing quantication models will be presented, including the advantages and limitations of the related algorithms, with special attention to the Cli-Lorimer method.

The new model that we have developed will be discussed in chapter 2. First, the iterative calculation for the self-consistent absorption correction is presented. Then, we will discuss the experimental calibration of the C-L factors that we have achieved by CONTENTS comparison with RBS measurements. The chapter will end with the presentation of the procedure for the estimation of the total experimental error.

Finally, in chapter 3, our quantication model will be applied to two dierent material systems, which are relevant in semiconductor processing: the quantication of Arsenic dopants in Silicon and the investigation of pseudomorphycally grown SiGe/Si stacks. In both cases, the experimental calibration of the C-L factors will rst be carried out, followed by the application of the method to a technologically relevant case.

In particular, in the case of Arsenic-doped Silicon, a 3D FinFET test structure will be investigated.

In parallel to the research activity focused on the quantication of EDX elemental maps, I have contributed, during my PhD, to some additional research projects being carried out in collaboration with other researchers within the LPCNO and the LAAS laboratories. In these activities, X-ray quantication techniques (in SEM or STEM mode) are needed to support other TEM-based structural characterisations of nanoelectronics materials of interest in future nanometric MOS or IoT devices. The results related to these activities will be summarised in the last chapter of this manuscript (chapter 4).

In the rst part of the chapter, we report a simple and versatile laser-writing procedure to integrate exible micro-supercapacitors and micro-batteries on current-collector free polyimide foils, starting from commercial powders. Such microdevices are good candidates for future IoT applications, to store energy and deliver power pulses while providing non-constant voltage output with time. The EDX analysis was rst used to quantify the elements concentration in the fabricated structures (and therefore conrm the validity of the fabrication technique). Additional HREM and XRD (X-Ray Diraction) investigations were conducted to identify the crystallographic structure of fabricated alloys.

In the second part of the chapter, we investigate the formation of metal silicides in Silicon nanowires, which are expected to replace the current FinFET architecture in future technology nodes. Our work in this project was focused on the determination of the crystalline structure of the fabricated metallic alloys using HREM and FFT (Fast Fourier Transform) method and on the subsequent conrmation of the structural analyses thanks to EDX quantitative analysis.

A summary of the main results obtained during my PhD as well as some perspective of the upcoming work will nally be presented in the General Conclusion of the manuscript.

Chapter 1

Energy Dispersive X-ray spectroscopy

EDX is an analytical technique used for chemical characterization. It relies on the stimulated emission of X-rays by electrons. This technique is normally used in an electron microscope and the emission is stimulated by a high-energy electron beam.

The emitted characteristic X-rays can be used to obtain information on the specimen composition. The energy distribution of the characteristic X-rays is measured using an energy-dispersion spectrometer.

One of the advantages of the EDX technique over other spectroscopy techniques used in electron microscopy is the possibility to map a large range of elements with high spatial resolution. When compared with Electron Energy Loss Spectroscopy (EELS), for example, EDX gives better results on heavy elements (Ru, Au, Rh, Pt ...) and it can still be used for light elements [1].

Recent improvements in electron microscopy instrumentation have opened the way to new of EDX applications by allowing the experimenter to obtain spectral maps with an atomic resolution [START_REF] Kothleitner | Quantitative Elemental Mapping at Atomic Resolution Using X-Ray Spectroscopy[END_REF] and EDX techniques can be thus considered as en excellent candidate for elemental mapping on the nanoscale.

In particular, the good versatility of the technique in terms of observable elements combined with the elevated spatial resolution suggest that EDX could be used to characterize nanodevices (by studying both the doping and the metal distribution in the interconnection) as well as complex magnetic nanoparticles based on heavy metals, like those used in catalytic and magnetic applications.

But in spite of the recent instrumental progress, the quantitative application of EDX to nano-objects and nanodevices is still a challenging task. The low Signal To Noise Ratio (SNR) associated to the measurement of small objects or of elements in small concentration (such as dopants in nanodevices) calls for spectrum quantication models taking into account all the possible measurement artifacts in a rigorous way. In these particular cases, parasitic eects can be a critical issue since they can be of the same order of magnitude as the measured signal. The quantication algorithms thus represent a key step in the elemental mapping of nano-objects. In this chapter, an introduction to EDX techniques will be given. The discussion will focus on the basic concepts and formalism needed to explain the new quantication model developed during my thesis work. The discussion will be organized as follows:

First of all the fundamentals of several electron microscopy techniques (transmission and scanning electron microscopy) will be discussed.

Then, a detailed description of the EDX technique will follow, including some details on the theory of electron-matter interaction , the physics of detection, and the quantication models Finally, a discussion on the available algorithms and computer program for the quantication of EDX spectra will follow with a specic focus on their limits for nanoscale and low concentration applications.

Electron microscopy

In Electron microscopy ([3] ch.1) electrons are used to obtain high resolution imaging.

In general, it can be considered that the spatial resolution d of a wave-optical device, is given by the the Rayleigh formula ([3] p.5):

δ = 0.61 × λ NA (1.1)
where λ is the wavelength of the electron and NA is the numeric aperture of the magnifying lens. The numeric aperture depends on the refractive index of the medium (a refractive index analog can be dened for electron waves [4]) and on the angular aperture of the lens. It can be shown that its value is approximately equal to 1. For an optical microscope using green light, the resolution amounts to 300 nm (NA= 1, and l = 550 nm). This same formula can be applied to electrons since the waveparticle duality principle tells us that electrons behave, statistically, as waves. The idea behind electron microscopy is to obtain a high resolution by exploiting electrons. In 1925, Louis De Broglie [START_REF] De | Recherches sur la théorie des Quanta[END_REF] postulated that the wavelength of an electron depends on its energy according to the following law:

λ = h p (1.2)
where p, is the momentum of the electron and h is the Planck constant.

By using relativistic dynamics the following expression can be obtained for the electron wavelength:

λ = hc (eV ) 2 + 2eV mc 2 (1.3)
where m and e are the electron mass and charge, respectively, c the speed of light and V is the acceleration voltage. With this formula it can be shown that a resolution of the order of the picometer can be obtained with a 200kV acceleration voltage. This is obliviously a considerable improvement when compared with light optics.

In practice, the theoretical resolution is not really achievable because of the presence of aberrations in the imaging system. Nevertheless, at present, a resolution of some Angstroms can be obtained with standard commercial devices and a sub-Angstrom resolution can be obtained by using aberration corrected microscopes. There are two main kind of electron microscopes, the Scanning Electron Microscope (SEM) and the Transmission Electron Microscope (TEM), both exploiting electrons but with a very dierent operation mode.

Scanning Electron Microscopy

In a SEM [START_REF] Seiler | Secondary electron emission in the scanning electron microscope[END_REF], a condensed beam is used to obtain information on a particular area of the specimen and the electrons forming the beam are characterized by an energy ranging between a fraction of a keV and 30 keV. As its name suggests, in a SEM, a specic area can be observed by scanning the specimen surfaces with the electron beam. The electrons from the primary beam are scattered in the material and, before being absorbed, they spread in a region called the volume of primary excitation. The characteristic length of this region depends on the material density and on the primary electron energy but is in the order of some microns. The electron-matter interaction in this volume causes the emission of several dierent signals (see gure 1.1(a) and section 1.2.1 for more details on the interaction mechanisms). Several information can thus be obtained by detecting dierent signals. In particular:

In the primary region Secondary Electrons (SE) are emitted by ionization of atoms in the specimen by the primary beam. The secondary electrons have a small energy thus their absorption length is in the order of some nanometers the exact value depends on specimen density and incident beam energy). As a consequence only the secondary electrons emitted in the upper part of the excitation volume get o the specimen to be collected by the SE detector. The detection of secondary electrons can be used to obtain information on a layer with a thickness of some nanometers located just below the surface of the specimen. SE electrons are generally used to obtain 3D information on the specimen surfaces.

To understand why it must be noted that their emission intensity is proportional to the cosinus of the angle between the surface normal and the SE detector. This is the same dependence followed by the intensity of light reected by a surface, which depends on the angle between the reecting surface and the optical detector (or the eye of the observer). The obtained image can thus be used to visualize the surface reliefs by just looking at the obtained image as we would look at a 3D object illuminated by light. More sophisticated techniques can be used to obtain additional information on the surface (composition, electric potential ...) by using the SE signal.

Back-scattered Electrons (BSE) can also be collected using a specic detector. CHAPTER 1. ENERGY DISPERSIVE X-RAY SPECTROSCOPY Since they have larger energy than SE, BSE are originated from deeper regions of the interaction volumes (characteristic lengths of the order of 10 to 100 nm).

Since they come from regions farther away form primary electron beam, BSE imaging has lesser resolution than SE imaging. On the other hand, since the intensity of back-scattering strongly depends on the local specimen density, it can be used to obtain direct information on the specimen composition. As a rule of thumb it can be considered that high density regions in the specimen correspond to brighter regions in the BSE image. This is only applicable to at specimens since on irregular surfaces some region of specimen can be located between the emission point and the detector. This shadowing eects, together with a careful design of the detector can be used to obtain topographic contrast with BSE.

Electrons from the primary beam also cause the emission of X-rays (see section 1.2.1). Since the absorption length of the emitted X-rays is of the order of some microns, the resolution of EDX techniques in the SEM is micrometric. Still the application of these technique on specimens with a uniform distribution is very interesting since a high SNR can be obtained due to the large size of the interaction volume. For the quantication special techniques (known as ZAF techniques [START_REF] Goldstein | Scanning Electron Microscopy and X-Ray Microanalysis[END_REF]) can be used to quantify the EDX signal by taking into account the fact that X-rays are not only emitted near the beam impinging region but in the whole interaction volume.

It must be noted that the interaction volume can be greatly reduced by observing specimens having a negligible thickness with respect to the size of the volume of interaction.

In this case it can be considered that all X-rays are emitted in the beam focusing point (see gure 1.1(b)) and the EDX resolution equals the beam size. This can be achieved by using specimen preparation techniques (mechanical thinning and grinding, ion beam thinning ...) to the specimen before observation. This is the conguration normally used in a transmission electron microscope in scanning mode (STEM) or in a SEM by using a dedicated specimen holder.

Transmission Electron Microscopy

In a TEM, as the name suggests, the detected signal consists of electrons transmitted trough the specimen. As a consequence only thin specimens can be observed.

The maximum acceptable thickness for the specimen to be transparent to electrons depends on the specimen density and composition but is in the order of 100 nanometers. Specimens typically observed in a TEM are either nanometric objects supported on thin organic grids or bulk material and devices thinned using the combination of mechanical and ion beam techniques. The energy of the electrons of the beam ranges, in a TEM, between 80 keV and 300 keV.

The basic TEM column can be divided in several blocks [START_REF] Williams | Transmission Electron Microscopy: A Textbook for Materials Science[END_REF] (see gure 1.2).

An electron gun located on the top of the microscope generates the electron current.

Several kind of guns are used in commercial microscopes. In the most simple (thermionic) guns, a metallic lament is heated and electrons are emitted by thermal eect. The degree of spatial coherence of this kind of source is low since electrons are emitted from a large region near the gun tip and the emitted electrons have an energy spread (temporal coherence) due to the heating of the gun. In a thermionic gun the source size is of the order of some microns and the energy spread of the order of some electron volts.

Better coherence and an improved beam current can be obtained by using Schottky Field Emission Guns (FEG). In this kind of sources the extraction potential is lowered by the creation of a strong eld on the gun tip. Since the eld strength of the created eld is stronger when the radius of curvature is lower, most electrons are emitted from a narrow region near the tip. Schottky guns are also heated up to obtain electron emission but less heating is needed thanks to the extraction potential lowering. Schottky guns have better coherence than thermionic guns.

They also have a higher brightness, which means that a higher current can be obtained when focusing the beam in a small region. The source size is of the order of 10 nm and the energy spread of approximately 0.7eV . In cold FEG guns, electrons are also extracted thanks to potential lowering near the gun tip but no additional heating is provided. As a consequence better coherence and higher brightness is obtained. The source size is of the order of some nanometers and the energy spread of about 0.3eV . Since surface contamination has a strong eect on the emitted current, cold FEGs have more strict requirements in terms of vacuum than other sources and must be periodically cleaned.

For that a short temperature spike (also called ash) is used. The current in cold FEG guns degrades with time and source ashes must be periodically used to restore it to its maximum value.

The current generated by the gun can be focused or spread on the specimen using the CHAPTER 1. ENERGY DISPERSIVE X-RAY SPECTROSCOPY condenser system. The microscope can thus be used in a TEM mode or a scanning TEM (or STEM) mode (see gure 1.3). In TEM mode the beam covers the observed region and the electron wave, after interacting with the specimen, is used to form an image using the objective lens and the projection system. In the STEM mode the beam is focused on a specic location of the specimen and and an image is created by scanning the beam over the observation region. The main dierence between the SEM and STEM mode is that in STEM, in addition to the EDX detectors, several detectors located after the projection system can be used to obtain transmission images. Since the resolution in STEM mode is dictated by the beam size, in modern microscopes the condenser system is equipped with a spherical aberration corrector. Using aberration correction an high coherence sources such as cold FEG sources, electron probes of sub-Angstrom size can be obtained.

The objective lens is used to provide a rst magnication of the image. To minimize lens aberrations, an immersion lens is normally used. This means that the specimen is inserted, using the specimen holder rod, in the middle of the polar pieces generating the objective lens eld. A magnied image is created on the plane that is optically conjugated to the specimen plane, and a diraction image on the image focal plane of the objective. The distribution of the magnetic eld located before the specimen can be modied using separate coils. The pre-specimen eld and the additional coils are sometimes referred to as the mini-lens even the specic design varies in dierent microscope models. The mini-lens can be used to further modify beam convergence. It can for example be used to provide additional convergence in STEM mode or to obtain a parallel beam in diraction mode.

The projection system is composed by the diraction intermediate and projection lenses. It provides further magnication and can be used to select the plane to be projected on the uorescent screen or the image recording system. In diraction mode, for example, the image focal plane of the objective lens is selected.

Image formation in the TEM and STEM modes

As already explained in the description of the condenser system in a TEM two main operation modes are used to characterize a thin specimen the TEM mode and the STEM mode (see Figure 1.3). The image formation mechanisms are quite dierent in the two cases and complementary information can be used by using the two modes. In this section the advantages and disadvantages of each mode will be discussed together with the fundamentals of image formation.

As previously discussed, in TEM mode the electron beam covers entire observed area and an image is obtained by using a recording device such as a photographic plate or a digital camera. This means, in the case of digital acquisition, that the image recording consists in a parallel digital acquisition of all the pixels composing the picture. On the other hand, in STEM mode, the beam is focused in a small region and the image is obtained by scanning and serial acquisition of the recorded signal. A direct consequence of the dierences between the operating modes is that acquisition of an image in STEM is normally longer than a TEM acquisition. Moreover, since the beam is focused, specimen damage is much more critical in STEM mode.

On the other hand the STEM mode is the most suited to perform elemental map-20 CHAPTER 1. ENERGY DISPERSIVE X-RAY SPECTROSCOPY ping. In the particular case of EDX the only elemental information that can be obtained in the TEM mode is the average composition of specimen area under the electron beam.

For EELS things are a bit dierent but it can be shown that even if elemental maps can be obtained in the TEM mode, the resolution obtained using STEM/EELS is much higher [START_REF] Egerton | Electron Energy-Loss Spectroscopy in the Electron Microscope[END_REF].

To understand what kind of information can be obtained form TEM and STEM images, we must discuss how images are formed in these two modes. Since in this thesis we have studied nanometric objects the discussion will be focused on high resolution imaging.

In the High Resolution TEM (HRTEM) mode [START_REF] Fultz | Transmission Electron Microscopy and Diractometry of Materials[END_REF] the incident wave is modied after interacting with the specimen. We will call this modied wave the object wavefunction. The objective lens modies the object wavefunction by magnifying it and adding distortions due to lens aberrations. A nal image is then formed by the projection lens. It can be shown that in order to obtain a maximum resolution, it is necessary to obtain a defocused image using the projection system. This means that the object plane of the projection systems must be separated by a distance ∆z s = 3C S λ from the objective image plane. In the previous equation, C s corresponds to the spherical aberration coecient of the objective lens. This distance is known as Scherzer defocus while the associated optimal resolution d = 4 Cs 6 λ 3 is called the Scherzer resolution.

Without going into the details of the formed image it can be noted at this point that an HRTEM image provides information on the periodicity of the atomic arrangement of the observed specimen.

HRTEM imaging is often coupled with Fourier analysis This is accomplished by using Fast Fourier Transform (FFT) [START_REF] Cochran | What is the fast Fourier transform[END_REF] algorithms on digitally acquired images. The use of FFT allow the experimenter to easily analyze the spatial frequencies present in the HRTEM image. This information can be used to identify the crystal structure of the specimen since the measured spatial frequencies are related to the crystal reciprocal lattice [START_REF] Fultz | Transmission Electron Microscopy and Diractometry of Materials[END_REF].

In the High Resolution STEM mode, quantum mechanics must again be used

to understand what kind of information can be obtained from the nal image. The analysis of the wave function behavior can be simplied by using a very useful principle called the TEM/STEM reciprocity principle [START_REF] Peter | The Principles of STEM Imaging[END_REF]. One interesting consequence of this principle concerns the images obtained a Bright Field (BF) detector, i.e. a detector collecting only the electrons forming a small angle with the optical axis. It can in fact be shown, using the reciprocity principle, that a Bright Field STEM (BF-STEM) image must show the same kind of contrast of a HRTEM image.

Things are dierent when electrons scattered at high angles are detected using a High Angle Annular Dark Field Detector (HAADF) . In this case it can be shown that the collected signal is approximately proportional to Z 2 where Z is the atomic number of the scattering atoms. In a STEM/HAADF image the contrast is thus directly Both BF-STEM and STEM/HAADF can be combined with STEM/EDX allowing the experimenter to obtain simultaneous information on the specimen composition and structure.

The physics of EDX spectroscopy

In the following discussion, we will explain the basic phenomena involved in EDX spectroscopy including the dierent electron-matter interaction mechanisms, the EDX detector and the various artifacts associated with this technique. In addition, an overview of the existing quantication models and available quantication programs will be provided. The discussion will be focused on the critical issues aecting the mapping of nanodevices and nano-objects and on the limitations of the dierent models. In an electron microscope, the interaction between beam electrons and the specimen can have several consequences (see Figure 1.4).

Electron-Matter Interaction

First of all, characteristic X-rays ([3] p.605) can be generated when a high energy electron interacts with an atom of the sample and ionizes it by releasing an electron from an inner shell. The energies associated to the characteristic peaks can be obtained using the X-ray transition energies database [START_REF] Suplee | X-Ray Transition Energies Database[END_REF]. When the incomplete inner shell is lled by an electron coming from an external one, the excess energy can be released in the form of an X-ray. The energy of the emitted X-ray depends on the energy dierence between the outer and the inner shell. Each element of the periodic table therefore gives rise to the emission of characteristic X-rays with dierent energies. Each peak is iden-tied using the Barkla notation [START_REF] Barkla | Secondary radiation from gases subject to X-rays[END_REF]. In this notation a rst roman uppercase letter is used to indicate the inner shell (K for the n=1 shell, L for the n=2 shell ...). The letter is followed by a Greek letter used to sort the characteristic X-rays (also called lines) in order of increasing energy. For example the lowest characteristic X-ray involving a n=1 inner shell is called Kα, the following one Kβ ...). In fact the notation is more complex and involves the employ of subscripts and superscripts in order to take into account subshells.

In addition bremsstrahlung radiation (braking radiation) [START_REF] Trincavelli | Standardless quantication methods in electron probe microanalysis[END_REF] can be emitted emitted when the electrons are deected by an atomic nucleus. Even if the associated probability is low, inelastic scattering can take place and the electron can loose a signicant amount of energy. This energy is released in the form of an X-ray (see Figure 1.5(a)). Statistically speaking, a Bremsstrahlung x-ray can have an energy ranging from zero to the total incident electron energy. Since the probability varies with the energy loss, Bremsstrahlung radiation has a continuous shape. It can be shown that the noise intensity as a function of the energy follows Kramer's law [START_REF] Manjunatha | Beta Bremsstrahlung dose in concrete shielding[END_REF]. From the point of view of an EDX measurement, this radiation can be interpreted as a background noise.

Finally, the electron can be backscattered ([3] p.611) in a collision with an atom of the specimen. In this case the deection angle is greater than 90°and electrons have an energy higher than 50 eV. It has to be noted that in a TEM backscattering not only takes place in the sample but can also be caused by the presence of components located near to the specimen (such as the specimen holder or the objective aperture, see gure 1.2). In this case the backscattered electron can be responsible for the presence of artifacts due to the emission of X-rays in regions located outside the electron beam (see labels 3 and 4 in Figure 1.4 ). Once emitted and independently of their origin and energy the X-rays can interact in many ways with the sample before reaching the EDX detector. This means that the emitted radiation is modied before being measured by the detector and it is important to take these eects into account in order to obtain quantitative information from the measured spectra. In particular, X-rays can be absorbed before reaching the detector.

Two kind of absorption are possible:

X-ray uorescence absorption [START_REF] Van Espen | A computer analysis of X-ray uorescence spectra[END_REF]: in this case a characteristic X-ray emitted by an atom of the sample is absorbed by another atom of the sample before reaching the detector. A new X-ray, with a potentially dierent energy is created in the process. By the energy conservation principle, X-ray generated by uorescence must have a lower energy than the absorbed X-ray. As in the case of backscattering, uorescence can take place in regions of the specimen or of the microscope located outside the electron beam at it is responsible for the presence of artifacts. (see Figure 1.5(b)).

Non-radiative absorption of X-rays in the sample such as those described in the following section for the absorption of an X-ray in a semiconductor.

In summary we have seen that two kind of x-rays can be measured by an EDX detector: the characteristic X-rays, that contains the composition information and the bremsstrahlung radiation, that can be considered a background noise. The X-rays can be absorbed in the specimen and spurious X-rays can be generated by uorescence or backscattering causing measurement artifacts. Since these artifacts are due to the presence of the specimen we will refer to them as specimen artifacts in the following. Specimen artifacts cannot be eliminated and must be taken into account in the quantication methods. In particular we will show in the following that a rigorous treatment of absorption eects is a key issue for quantitative mapping at the nanoscale.

The EDX detector

To take into account all the physical phenomena involved in an EDX measurement, we need to describe the basics of EDX detection.

The EDX detector is an instrument capable of measuring the energy of incident Xrays and represent them in the form of a spectrum, i.e. an histogram of the X-rays energy distribution measured during a specic time interval (also called exposition time). As already discussed a spectrum contains characteristic peaks superimposed to the Bremsstrahlung background noise. In order to discuss the possible measurement artifacts associated to the detection step the X-ray/matter interactions taking in place in the detector will be reviewed.

The simplest form of EDX detector consists in a reverse-biased p-i-n diode (made up by a succession of a P-doped an intrinsic and an N-doped region). The p and n regions are heavily doped (and will be thus labeled p+ and n+) to be able to create ohmic contacts 1.2. THE PHYSICS OF EDX SPECTROSCOPY 25 at its extremities. In the case of an ideal detector the emitted X-rays are absorbed in the intrinsic zone and they generate electron-hole pairs (see Figure 1.6). The number of generated carriers is proportional to the X-ray energy and the carriers are collected at both ends of the detector due to the applied polarization. The collected charge which is proportional to the energy of the incident X-ray, is then converted into a voltage that represents the measurement output. Even if modern detectors, such as Silicon Drift Detectors (SDD) have a more elaborate design (see section 1.2.3), we will use the simple structure depicted in gure 1.6 to better illustrate the dierent phenomena. The same reasoning can in fact be applied to more complex geometries such has the one used for modern SDD detectors. In real detectors some instrumental limitations must be taken into account.

First of all the detector has a nite extension and it can only capture a fraction of the X-rays emitted by the specimen. The geometric eciency of the detector is expressed by the detection angle, which represents the solid angle covered by the detector active surface. The detector solid angle is obviously a function of the detector active surface and its distance from the specimen.

In addition the detector has a limited energy resolution. We will not detail the physical mechanisms but we will limit ourselves to observe that because of these eects a characteristic peak, which ideally corresponds to a maximum of intensity located at a given energy, will have, in practice, a Gaussian prole [START_REF] Goldstein | Scanning Electron Microscopy and X-Ray Microanalysis[END_REF]. The energy resolution will corresponds to the Full Width at Half Maximum (FWHM) of the peak σ F W HM (E) that can be calculated using the following equation:

σ F W HM (E) = A(E -E ref ) + [σ F W HM (E ref )] 2 (1.4)
where A is a constant depending on the EDX detector, E ref is an arbitrarily chosen reference energy, and all energies are expressed in electron volts.

Moreover, absorption can take place for the less energetic x-rays in the metallic contacts or in the other layers which may possibly cover the surface of the detector.

This eect can be eliminated, at least in principle, by calibrating the response of the detector to dierent energies with X-rays of known energy and intensity [START_REF] Trincavelli | Model for the bremsstrahlung spectrum in EPMA. Application to standardless quantication[END_REF].

To conclude the discussion on the detector limitations, two kind of artifacts must be discussed into detail.

1. Collection artifacts arising when the X-ray is absorbed in the detector but the generated charges are not fully collected.

2. Temporal superposition artifacts which can take place when the intensity of the incident photons is very high. In this case there is a non negligible probability that a photon can arrive in the active zone before the charges generated by a previous photon are collected. In this case the contribution of the two photons cannot be distinguished.

Collection artifacts

To understand collection artifacts it must be considered that the conversion of an X-ray into electron-hole pairs can be done through intermediate conversion phenomena (see Figure 1.7(a)). More specically two intermediate phenomena can be responsible of the formation of an electron-hole pair by absorption of an X-ray:

1. Auger electron emission ([3] p.60), which occurs when X-rays ionize the electron from an atomic inner shell. The additional energy created when the inner shell is lled by an electron coming from an outer shell can cause the ionization of weakly bound outer shell electron (see Figure1.7(b)).

2. The emission of an X-ray by uorescence, which has already been detailed previously.

In addition it must be considered that the detector has inactive zones in the p+ and n+ regions (often called "dead layers" in literature). This is due to the presence of defects introduced during the creation of highly doped regions. This is mainly due to the presence of crystalline defects (precipitates, interstitial, residual ionic implantation defects, ...) in the doped zones. The defects acts like traps for the carriers and can cause pair recombination.

The combination of intermediate conversion phenomena and the presence of dead layers is at the origin of several artifacts (see Figure 1.8): Absorption in the dead layer -The X-ray is absorbed in a dead zone and no electron-hole pair is generated. This is additional absorption eect to take into account when evaluating X-ray absorption in the detector.

Incomplete charge collection -Some of the intermediate electrons generated by the X-ray are absorbed into the dead zones of the detector ([3] p.595). This means, in practice, that a fraction of the X-ray energy is not detected and the value of the measured energy is therefore lower than the real one. Moreover, because of incomplete charge collection, the energy of certain photons is underestimated by the detector. The energy of some photons are thus shifted and this causes a deformation of the theoretically predicted Gaussian shape of the characteristic peaks. Due to the form of the deformed peaks the incomplete charge collection mechanisms are normally addressed as tail and shelf ([3] p.610) artifacts. It is important to consider shelves and tails as a part of the signal to be measured and not be associated to background noise. Escape Peak -A uorescence X-ray can be emitted in the detector but not detected since it leaves the detector or is collected in the dead layer ([3] p.606). The missing X-ray which must necessarily be a characteristic X-ray of the material that composes the diode. As in the previous case, the measured energy for the incident X-ray will be lower than the real energy. In the specic case of detectors based on Si diodes, for example, the detected energy is reduced by 1.74 keV which corresponds to the Kα line, the strongest line of the characteristic silicon peaks.

Temporal collection artifacts

These artifacts are due to the temporal superposition of X-rays. To simplify the explanation we will not give details on the electronic conversion of the detector and we will limit the description on its general operation mechanisms. These artifacts take place when an energy X-ray E B arrives on the detector while carriers previously excited by an energy X-ray E A are still present in the detector. The detector electronics is equipped with a rejection function to deal with the temporal events overlap. If a temporal overlap is detected both measurements are rejected. But the detector rejection function obviously is not perfect and if the arrival times of the two X-rays is below a the rejection function sensibility the two X-rays are measured as a single X-ray and the measured energy is equal to (E A + E B ). It is the possible to observe, in a spectrum obtained with an elevated x-ray yield, a sum peak located in E A + E B in addition to the characteristic peaks located in E A and E B ([3] p.607).

SDD detectors

SDD EDX detectors are new generation devices exploiting a circular geometry to drive electrons, generated between the back contact and the front contacts, in the central region of the detector (see image 1.9 ). The collection is made possible by the Without going into the full details of its operation (see [START_REF] Gernert | Comparing the Si(Li)-detector and the silicon drift detector (SDD) using EDX in SEM[END_REF]for more details), it must be noted that one of the main advantages of an SDD detector is that, thanks to its specic design, the detector has a faster response than standard devices, and can be safely used for high counts measurements. Moreover, it has negligible dead layer regions, and this reduces the inuence of charge collection artifacts.

The vast majority of new generation EDX detectors (including those used in our experiments) are equipped with and SDD device.

Summary of EDX artifacts and limitations

In this section we have reviewed all the instrumental limitations and possible artifacts aecting EDX spectra. To better understand how dierent artifacts aect the nal measurement we can summarize the previous discussion as follows:

The non ideality of the EDX detector can be taken into account by considering that the detector has a limited energy resolution, a nite detection solid angle, and because of absorption eects in the detector surface and dead layers, the intensity response of the detector at various energies must be calibrated. This is normally taken into account by the manufacturer, and for commercial EDX detectors, information on the solid angle and resolution is provided. The equation CHAPTER 1. ENERGY DISPERSIVE X-RAY SPECTROSCOPY 1.4 is used to evaluate the resolution for a specic energy after calibration of the A constant using a reference peak. The intensity response is taken into account by direct spectrum correction or by providing the intensity response vs energy curve, also called the detector quantum eciency curve. If the dead layers of the detector have a large extension the peak shape can be distorted by incomplete charge collection eects and the manufacturer must also provide some methods to take into account or correct these distortions. It has to be noted that when using modern devices such has SDD detectors, thanks to improved control of doping mechanisms and dead layers, peak distortions can be safely neglected.

Compositional information can be extracted by measuring the characteristic peaks intensities after bremsstrahlung subtraction. For this dedicated computer programs can be used (more details will be provided in the next section).

If the intensity of measured spectrum is high or the extension of the dead layer is non-negligible, spurious peaks such as escape and sum peaks can appear in the spectrum. This must be taken into account by the experimenter. As already discussed since the dead layers thickness is reduced, in modern devices escape peaks can safely be neglected. Moreover, as already discussed, low X-rays current are normally obtained when applying EDX techniques to nano-objects. As a consequence possible complications due to the presence of sum peaks can be safely neglected.

There is a an additional unavoidable category of artifacts due to the presence of the specimen. For obvious reasons specimen artifacts cannot be accounted for by the EDX detector manufacturer. The rigorous physical modeling of these kind of artifacts has been the main focus of my thesis work and more details on the techniques developed and their applications to nanodevices will be given in the following chapters.

EDX Quantication

As already discussed, the compositional information in a spectrum is encoded in the height and position of the dierent peaks. In order to obtain quantitative information on the specimen composition a model correlating characteristic peak heights to the concentration of the dierent elements composing the specimen must be used. The dierent types of quantication methods and the existing computer programs implementing the quantication techniques will be described in this section. The discussion will be restricted on the quantication of EDX data in TEM/STEM.

f M i f M j = K i,j:F i ,F J I i:F i I j:F j (1.5)
where f M i is the mass fraction of the i th element and I i:F i is the associated intensity of the specic peak. It has to be noted that to apply this method, a specic peak family F i =K, L, M, . . . has to be selected for each element. K i,j:F i ,F J is the C-L factor and depends on the peak family chosen for each element. The only information needed to apply this method is the C-L factors associated to the dierent peaks used for the composition quantication. It is common practice in EDX software to provide a full set of simulated values for the C-L coecients. As an alternative they can be experimentally measured by using a specimen of known composition containing the two elements.

Zeta Factor method

The ξ-factor method is based on the principle that the intensity of an X-ray line from an element A in a thin-lm (i.e. ignoring absorption or uorescence eects) is proportional to the mass-thickness and composition as the following [START_REF] Garmannslund | Renement of the ζ-factor Method for Quantitative Energy-Dispersive X-ray Spectroscopy in Scanning Transmission Electron Microscopy[END_REF][START_REF] Watanabe | The quantitative analysis of thin specimens: a review of progress from the Cli-Lorimer to the new ζ-factor methods[END_REF]:

ρt = ξ A I A C A D e (1.6)
where ρ is density, t thickness, I A the intensity of the peak above background, C A the composition and D e the total electron dose and can be expressed as the following:

D e = I p τ e (1.7)
where I p is the in-situ beam current, τ is the acquisition time and e=1.6022*10 -19 is the electron charge.

If the same equation is applied on an element B we obtain:

ρt = ξ B I B C B D e (1.
ρt = ξ A I A + ξ B I B D e , C A = ξ A I A ξ A I A + ξ B I B , C B = ξ B I B ξ A I A + ξ B I B (1.9) CHAPTER 1. ENERGY DISPERSIVE X-RAY SPECTROSCOPY
The advantage of the zeta factor method in comparison with the Cli-Lorimer method is that pure element standards can be used for calibration. [START_REF] Watanabe | The quantitative analysis of thin specimens: a review of progress from the Cli-Lorimer to the new ζ-factor methods[END_REF]. Pure-element standards have the distinct advantages of being easy to fabricate and they don't change composition during the thinning procedure or due to the beam damage. The main disadvantage is that an in-situ measurement of the beam current is necessary to apply this method, and standard TEM/STEM are not normally equipped for this kind of measurement.

Cross Section method

This method is used to calculate an EDX partial cross section in similar way of the ξ-factor method [START_REF] Macarthur | Quantitative Energy-Dispersive X-Ray Analysis of Catalyst Nanoparticles Using a Partial Cross Section Approach[END_REF]. For thin samples, X-ray absorption and uorescence can be neglected. In this situation, the number of X-ray counts is linearly proportional to sample thickness, therefore we can dene an EDX partial cross section, sv , for a single atom of any given element, x, as:

σ x = I x e iτ n x t (1.10)
where I x is the intensity of the X-ray of the element x, e is the unit electronic charge, i is probe current, τ is the total pixel dwell time and n x t is the atom density per unit area.

If the partial cross section of each element for a particular microscope are known, it is possible to calibrate EDX maps into atom counts, N x , per pixel of the spectrum image using the following equation:

N x = n x tA = I x e iτ σ x (1.11)
where A is the illuminated area. As in the case of the ξfactor method, pure calibration specimens can be used an in-situ beam current measurement is needed.

The advantage of this method is its simplicity of implementation but it is not well adapted to specimens providing a low EDX signal.

Quantication software

As already discussed, in order to extract information on the specimen composition from EDX measurements, background noise must be subtracted rst. Then the characteristic peaks intensity (corresponding to the peak area) must be calculated. Finally a quantication model must be applied to obtain information on the composition. This sequence of operation is implemented in some specic computer programs that can be used for the treatment of EDX data. In this section we will review the available computer programs with particular attention to their strength and weakness. The possibility to obtain quantitative information using the existing software will also be discussed.

It has to be noted that both commercial and free software exist. A really interesting feature of existing free computer programs for EDX data treatment is that they are also open-source. This means that a precise knowledge of the algorithm implementation can be obtained by looking at the code, and some of the software features can be reused.

Code re-usability has been used in the development of the model used in this thesis, as explained in the following. Two free computer programs will be reviewed in the following NTSA-II and Hyperspy.

Commercial software is mainly developed by manufacturers to be sold with the EDX detectors. In the following just two commercial computer programs will be reviewed since all the presented EDX data have been obtained using two detectors provided by two dierent manufacturers: Bruker and JEOL. As a consequence I have only used the Esprit program, developed by Bruker, and Analysis Station, developed by JEOL.

It must also be noted that the NTSA-II software is not suited for EDX treatment in TEM since the allowed maximum beam energy is of 80 keV. Still this software can be used for EDX quantication since a large set of routines that can be used for TEM are available using a library, called EPQ library, provided with the computer program and accessible using the Java programming language.

Finally it must be considered that Hyperspy is not, strictly speaking, an EDX analysis software but a framework for the analysis of EDX data using the python programming language. For this reason it does not provide a full-featured graphical user interface, and the experimenter must write his own python programs in order to analyze the data.

To illustrate the dierence between the dierent programs the basic operation mode of an EDX data treatment will be discussed in the following. Some images, obtained with Esprit, will be used for each step to better illustrate the procedure. For each step the dierences between the techniques used by each program will be discussed in order to assess their strengths and weaknesses. The discussion will be divided in two parts, corresponding to the main quantication steps: measurement of peak intensity and quantication of specimen composition.

Measurement of peak intensity

The quantication of a spectrum begins with the identication of the elements, and this is done graphically [START_REF] Eggert | Automated element identication for EDS spectra evaluation using quantication and integrated spectra simulation approaches[END_REF][START_REF] Newbury | Mistakes encountered during automatic peak identication of minor and trace constituents in electron-excited energy dispersive X-ray microanalysis[END_REF]. The mechanism diers slightly from software to software but the general idea is that by selecting a peak on the spectrum (also called a line for historical reasons) all the elements that can be associated to that specic line are listed. It is up to the user to choose the element by inspecting the presence of the other lines associated to each characteristic peak family of the element. He must also pay attention to the possible presence of escape or sum peaks and to avoid identifying them as additional elements. The element can be considered as identied when all the characteristic peaks in the measurable energy window are present and the identied peaks can then be tagged. In the example of the Figure 1.10, Si and Ge can be identied because all the visible characteristic peaks are present in the spectrum. For the measurement of the selected peaks intensity the background noise has to be subtracted, the possible peak distortion due to detector artifacts has to be taken into account and the intensities of eventually overlapped peaks need to be separated.

In all the fore-mentioned computer programs, background (bremsstrahlung) subtraction (see Figure 1.12) is handled by removing the regions where characteristic peaks are present in the spectrum and interpolating the spectrum background obtained after removal of the peak windows. The width of each characteristic peak window is calculated using equation 1.4 and the A coecient is calculated using a reference peak (usually the Kα peak of copper that is also used to assess the spectrometer resolution).

Dierent interpolation functions are provided by the dierent applications.

It has to be noted that the precise algorithm used by commercial software is not always available. As a consequence, for commercial programs, it is not always possible to provide precise information and he following discussion will be based on the information available in the documentation or obtained using a series of comparative tests.

In Esprit two models are available for TEM. The mathematical model (see 1.11(a)), that is a variation of the Kramer's model (the exact mathematical expression is not available) and the physical model that, in addition, takes into account the absorption of bremsstrahlung X-rays in the sample 1.11(b). The second model is therefore more precise and gives better results.1.11.

Several models are provided by the EPQ library, based on papers by Lifshin [25],

Reed [START_REF] Ware | Background corrections for quantitative electron microprobe analysis using a lithium drifted silicon X-ray detector[END_REF] and Small [START_REF] Small | Modeling of the bremsstrahlung radiation produced in pure-element targets by 1040 keV electrons[END_REF]. The model used by default is a slightly modied version of the Lifshin model.

No information is given on the model used by Analysis Station. Since the interpolated background has a continuous shape it can be supposed that an analytical model based on Kramer's law is used and that absorption in the specimen is not taken into account.

In Hyperspy the background shape can be obtained by averaging the intensity in two windows on each side of a X-ray line and using several standard interpolation functions (polynomial, spline ...).

It has to be noted that, even if the most complete approach is provided by Esprit, the bremsstrahlung interpolation procedure is only critical for low energy peaks, since in this region the noise is higher and the presence of several peaks due to contaminants (O, C, N, ....) are present (see again gure 1.11). For the quantication of characteristic peaks at higher energy no substantial dierence is obtained using the dierent models.

The correction of peak distortions due to incomplete charge collection in the detector is only specically provided by Esprit. But as already noted this kind of corrections can be safely avoided for modern SDD detectors. Finally to measure characteristic peak intensities and taking into account possible overlaps a peak integration and deconvolution procedure must be used [START_REF] Newbury | Mistakes encountered during automatic peak identication of minor and trace constituents in electron-excited energy dispersive X-ray microanalysis[END_REF]. It has to be noted that, as already mentioned, in all programs the calibrated information on peak width is available. The procedure of deconvolution consists in tting a background subtracted spectrum with a series of Gaussian peaks of known width. This means that the adjustable parameters of the t procedure are the Gaussian peak heights.

It has to be noted that the intensities of the dierent characteristic peaks forming the spectrum are not completely uncorrelated. For example, if no absorption is present, there is a xed intensity ration between the dierent lines of the same characteristic peaks family that can be retrieved using the various X-ray emission cross sections.

The main dierence between the various deconvolution algorithms consists on how this correlation is handled.

In Esprit for example three approaches can be used:

CHAPTER 1. ENERGY DISPERSIVE X-RAY SPECTROSCOPY
The spectrum is considered as a sum of independent Gaussian peaks. This is the most versatile approach since it can handle absorption eects but can be less precise in the case of small overlapping peaks.

A constraint on the relative heights of lines of the same family is imposed. The relative heights of the peaks of the same series (K series, L series, etc.) are calculated by using the theoretical values of the cross-sections contained in the XCOM photon cross-section database [START_REF] Suplee | XCOM: Photon Cross Sections Database[END_REF]. This technique gives better results for small overlapping peaks.

An approach based on a statistical iterative deconvolution technique called Bayes deconvolution, using no a-priori information [START_REF] Eggert | A rapid deconvolution method based on the Bayes theorem applied to energy dispersive X-ray emission analysis -improvements and error inuences[END_REF]. In our tests this method has never given acceptable results.

The independent peak approximation is the most suited for our case, since specimen absorption can be present, and this algorithm is available in Analysis Station and Hyperspy, that additionally provides more advanced signal separations techniques based on principal component analysis [START_REF] Rossouw | Multicomponent Signal Unmixing from Nanoheterostructures: Overcoming the Traditional Challenges of Nanoscale X-ray Analysis via Machine Learning[END_REF]. DTSA-II and the EPQ libraries are mainly focused on quantication using standard specimens and reference spectra in SEM, and is less usable in our case.

In summary, commercial and free EDX programs provide dierent algorithms for the measurement of peak intensities. In general, the better results for all programs are obtained for high energy peaks (energy above 2 keV, since in this region background subtraction is easier) and neglecting peak intensity correlation (algorithms are more stable in this case, if absorption is present).

Quantication of specimen composition

Once the peak intensities have been measured, one of the techniques discussed in 1.3.1 must be used to extract information on the specimen composition. It has to be noted that all the fore-mentioned programs implement the C-L quantication method while the ζ-factor method is only implemented in Hyperspy.

As already explained, in order to avoid in-situ current measurement the C-L approach has been used for my work. Two main problems must be addressed when using with the C-L approach for quantitative mapping of nano-objects: the problem of absorption in the specimen and the problem of the quantication of the measurement error.

As already explained specimen absorption can modify the measured peak intensity. This modication must be taken into account in order to obtain an accurate value. When the C-L approach is used, all the fore-mentioned programs provide an absorption correction method (described in the next chapter), based on the knowledge of the specimen density. This means that the user is asked for the value of the specimen density in order to apply absorption correction. The problem is that in the vast majority of cases the specimen density is unknown since it depends on the specimen composition, and the specimen composition has to be determined using the EDX measurement. To solve this problem a self-consistent approach, described in the next chapter, has been developed.

In order to obtain a quantitative result, the accuracy of the C-L coecients and the associated experimental error must be obtained. Moreover the dierent sources of experimental error must be combined to obtain the error on the nal measurement. This feature ins not available in the fore-mentioned programs. The approach used in this thesis for the calibration of the C-L coecients and the calculation of the nal error will be described in the next chapter.

EDX and the other Quantication Techniques

It is interesting to compare EDX to the other characterization techniques that can be used to quantify the specimen composition.

As already mentioned, one of the most used techniques for chemical analysis in semiconductors is SIMS [START_REF] Stevie | Secondary Ion Mass Spectrometry: Applications for Depth Proling and Surface Characterization[END_REF]. The SIMS technique is based on the analysis of the secondary ions sputtered when a (primary) ion beam is focused on the specimen. As described in the introduction the main advantage of this technique is its low detection limit (around 1 x10

14 atoms/cm3). On the other hand,the lateral resolution of SIMS is about 0.2 µm and the depth resolution is in the order of 2 nm. For this reason the SIMS quantication technique can be easily used to obtain depth proles on specimens with a multi-layered structures but cannot be used in modern nano-scaled 3D specimens.

Another important technique for composition measurement is Rutherford Backscattering Spectroscopy (RBS) [START_REF] Chu | Backscattering Spectrometry[END_REF]. RBS uses a beam of high energy (E~1MeV) He+ ions to obtain an absolute measurement of the composition of the specimen. Information on specimen composition is obtained by measuring the energy distribution of back-scattered He+ ions at a xed angle. Like for SIMS, the lateral resolution of the technique prevents it to be used for composition proling of modern 3D devices, and it is usually used to obtain depth concentration proles. Nonetheless, it has to be noted that there is an important dierence between RBS ans SIMS. RBS is an absolute technique: no calibration is needed in order to obtain a quantitative measurement.

For this reason this technique has be used in in this work to calibrate the C-L factors needed to obtain quantitative STEM/EDX measurements on calibration specimens ( see chapter 2 and 3).

There is another technique worth mentioning for concentration measurement in nanoscale devices, that has the additional advantage of being able to obtain 3D information. This technique is called Atomic Probe Tomography (APT) [START_REF][END_REF] and is based on the extraction of atoms from a point-shaped specimen using the electric eld (in non conducting specimens such as semiconductors, laser excitation is coupled to eld eect). The analytic information and 3D distribution is obtained by using spatially resolved time of ight mass spectroscopy. The main reasons to prefer RBS to APT in this work are that the precision of the concentration measurement obtained by RBS is higher and that specimens to be measured by APT need a complicated specimen preparation (a pointlike structure must be created using Focused Ion Beam techniques). Since STEM/EDX calibration can be performed on planar specimens, the 3D capability of APT was not needed in this work and RBS has thus been chosen to improve the precision of the calibration and avoid an additional specimen preparation step.

Chapter 2

Development of a new quantication method

As explained in the previous chapter, the available EDX software do not provide a rigorous approach to treat absorption correction using the C-L equation. Moreover, it is necessary, for quantitative measurements, to have an estimation of the measurement error and of the minimum density measurable by the technique (the technique detection limit). For this, a new quantication method based on the Cli-Lorimer equation has been developed. The advantage of this approach when compared with other methods (see section 1.3.1) is that an in-situ current measurement is not needed. This kind of measurement need the presence of a Faraday cage, and this is not a standard equipment in commercially available microscopes. As a consequence, this method can be applied in a standard microscope, without any need for special equipment.

In this chapter, we will describe the absorption correction problem and justify the need of a self-consistent method. Then, we will detail the procedure used for the calibration of C-L factors and for the estimation of the error in the nal measurement.

Mass fractions and density calculation

As already discussed (see section 1.3.1 and equation 1.5) the C-L approach is based on a proportionality relation between peak intensities and the elements mass fractions.

The proportionality constant K i,j:F i ,F J is the Cli-Lorimer factor relative to the i th and j th element, it depends on the peak families F i and F j that have been chosen for each element. In the vast majority of commercial software, the C-L constants values are extracted from a simulated database.

It can be shown that the C-L coecients can be also written as follows:

K i,j:F i ,F J = k i:F i ,F i 0 k j:F j ,F i 0 (2.1)
where k i:F i ,F i 0 is the C-L coecient relative to the element i and a common reference element i 0 . It has to be noted that, by denition, the reference element has a unitary C-L coecient (k i 0 :F i 0 ,F i 0 = 1) and F i 0 denotes the peak family that has been chosen for the reference element. With this convention it can be shown that there is a direct formula relating measured intensities and the mass fractions for a material composed of N A dierent atoms :

f M i = k i:F i ,F i 0 I i:F i N A j=1 k j:F j ,F i 0 I j:F j (2.
2)

The total intensity I i:F i of the family F i for a specic element is obtained by adding the intensity of the N F i lines (K α, Kβ...) belonging to the family chosen for the element i.

I i:F i = N F i k=1 I i:F i ,k (2.3)
Following this notation each characteristic peak line has an intensity given by I i:F i ,k and is associated to the energy E i:F i ,k where i identify the element F i the peak family and k the index identifying the peak line.

It should also be noted that the intensities to be used in the formula correspond to the integral of the characteristic peaks, to which must be added the intensity of the X-rays which are shifted in other regions of the spectrum (in the peaks sum or escape, in the tail, in the shelf ...) because of measurement artifacts. All the previously described eects must therefore be taken into account for the application of this formula. Moreover, for a correct interpretation of the results, the absorption eects in the sample must be taken into account because they are neglected in the C-L model. Finally, it should be noted that the coecients of C-L are normally calculated using simplied models and they must be calibrated in the case where results with a controlled precision are needed. They must therefore be considered as empirical parameters with an associated measurement error.

Even if it is easy to calculate mass fractions from EDX measurements, it can be useful ( and customary, in semiconductor literature on doping ) to use atomic concentrations (in atoms/cm 3 ) instead of mass fractions. The conversion between mass fractions and atomic concentrations c A i can be obtained using the following formula:

c A i = ρ M f M i m i N AV (2.4)
where ρ M is the mass density of the element i, m i its atomic mass and N AV = 6, 02214076 • 10 23 mol -1 is the Avogadro number.

Another quantity conventionally used to express concentration is the mass fractions. This is used, for example, for binary alloys (composed by the elements A and B) whose chemical formula is normally expressed as A 1-x B x where x is the atomic fraction of B. The mass fractions relative to a series of elements (f A i ) can be calculated from the corresponding mass fractions using the following formula:

f A i = f M i /m i N AT j=1 f M j /m j (2.5)

Absorption Correction

To quantify an EDX spectrum, the intensities of the various peaks are measured. But since the measured X-rays pass through the specimen before reaching the detector the measured intensity of X-rays I * can be dierent of the emitted intensity I. This eect need to be taken into account to obtain reliable results. The following formula can be used to calculate the measured intensity I * (E) of X-rays of energy E going through a homogeneous material of thickness z and mass density ρ M ([3] p.654):

I * (E) = I (E) e -µ(E)ρ M z (2.6)
where µ(E) is the mass absorption coecient of the material, I(E) is the initial X-ray intensity and I * (E) is the intensity measured by the detector after absorption in the specimen.

This formula can be obtained under the hypothesis that X-rays emission is uniform in depth. This means that for a specimen of thickness t the emission density is constant and its value is equal to I(E) t . X-rays are all emitted with the same probability, but the distance they go through in the specimen before reaching the detector depends on the emission depth. Using equation 2.6 we obtain:

I * (E) = t 0 I (E) t e -µ(E)ρ M z dz = 1 -e -µ(E)ρ M t µρ M t I (E) (2.7)
Therefore we can dene a transmission coecient which relating the intensity of emitted rays I(E) with that measured by the detectorI * (E) :

T (E, f M i ) = 1 -e -µ(E)ρ M t/ sin(α) µρ M t/ sin (α) (2.8)
The transmission factor depends on the X-ray energy and on the mass fractions f M i of the dierent elements since both µ(E) and ρ M depends on the mass fractions (for this reason they will be noted as µ E, f M i and ρ M f M i in the following). α is the elevation angle of the detector (see gure 2.1)

There is no universal formula connecting the density of a material to the mass fractions of the atoms composing it, since the density obviously depends on the atomic arrangement of the composing atoms. As a consequence the analytical expression of ρ M f M i depends on the specimen.

On the other hand there is a universal formula for µ(E) ([3] p.654, [START_REF] Eibl | New method for absorption correction in high-accuracy, quantitative EDX microanalysis in the TEM including low-energy x-ray lines[END_REF]):

µ E, f M i = N AT i=1 µ i (E) f M i (2.9)
where µ i (E) is the mass absorption coecient associated to the i th atom.

This relation is normally cited in literature without any proof of reference. It is hence hard to ascertain its limit of validity. For this reason a proof of equation and the underlying approximations needed to obtain it will be given here. Using the total interaction cross sectionsσ i (E) for an X-ray of energy E with an atom of an element i we can write [START_REF] Hubbell | Tables of X-Ray Mass Attenuation Coecients and Mass Energy-Absorption Coecients[END_REF]:

µ i = σ i m i (2.10)
Equation 2.9 can then be obtained by making the hypothesis that the cross section of a mixture of atoms is the sum of the cross sections of the individual atoms. This means that the contribution of the bonding between atoms does not aect their interaction with X-rays. This is a good approximation since it is used in many other calculations (characteristic X-ray energies and relative intensities for example). Supposing that the cross sections are additive and that the number of atoms of type i in the alloy is n i , we can write:

µ(E) = N AT i=1 n i σ i N AT j=1 n j m j = N AT i=1 σ i m i n i m i N AT j=1 n j m j (2.11) 2.2. ABSORPTION CORRECTION 43
and this equation is equivalent to equation 2.9 since µ i = σ i /m i and by denition f M i = n i m i / N AT j=1 n j m j . The value of µ i (E) for the various elements can be obtained using one of the several available tabulated values collections (see for example [START_REF] Chantler | Atomic form factors and photoelectric absorption cross-sections near absorption edges in the soft X-ray region[END_REF] and [START_REF] Heinrich | Mass absorption coecients for electron probe microanalysis[END_REF]). In our case they have been calculated using the EPQ library implementation of the model proposed by Heinrich [START_REF] Heinrich | Mass absorption coecients for electron probe microanalysis[END_REF].

For each characteristic peak E i:F i ,k the intensities I i:F i ,k can be calculated from the measured intensities I * i:F i ,k using equation 2.8.

If the sample is homogeneous and µ E, f M i and ρ M f M i are known, equation 2.8 and equation 2.9 can be used to calculate the mass fractions of the values elements. This is the approach normally used by commercial software. The specimen is assumed to be homogeneous, the user is asked for the value of the specimen density and the mass absorption coecients are evaluated applying equation 2.9 to the mass fractions calculated without taking into account the absorption eects. The main improvements of our approach to take into account specimen absorption are:

First of all, we have developed an iterative approach to overcome the limitation of the existing algorithms. More details on the algorithm we have used are given in the section 2.2.

In addition we have taken into account that the specimen is not homogeneous.

This diculty can be solved for specimen with a 1D geometry, such as those observed in my thesis, by carefully choosing the specimen orientation in the microscope. If the specimen is oriented as in Figure 2.2 and we consider that the detector size is negligible, we can consider that the X-rays are going through a homogeneous slice to reach the detector. In this case equation 2.8 holds for each slice but µ and ρ M can't be considered as constant since their value varies from slice to slice (see again Figure 2.2). For this reason, from now on, we will explicitly indicate dependence on the x (depth) coordinate of the various quantities in the

following (f M i (x), µ x, E, f M i (x) , ρ M x, f M i (x) . . . ).
To take into account specimen absorption, an iterative approach is needed since to calculate the mass fraction f M i (x) at each position x of the prole, we need the value of the transmission factor T which depends on f M i (x) (see equation 2.8). For the iterative procedure we use as a starting point the mass fractions f M i (x) 0 calculated by neglecting absorption (i.e. by assuming that

I * i:F i ,k (x) = I i:F i ,k (x)
) for all the characteristic peaks. For each iteration, a new set of mass fractions is obtained by calculating the transmission factor using the values obtained in the previous iteration:

[T i:F i ,k (x)] n = T (x, E i:F i ,k , f M i (x) n (2.12) = 1 -e - N A j=1 µ i( E j:F j ,k )[f M j (x)] n-1 ρ M (x, [f M i (x)] n-1 t/ sin(α) N A j=1 µ j E j:F j ,k f M j ρ M (x, [f M i (x)] n-1 t/ sin (α)
[I i:F i ,k ] n = I * i:F i ,k [T i:F i ,k (x)] n (2.

13)

Since:

[I i:F i ] n = N F i k=1 [I i:F i ,k ] n (2.14)
we can write :

f M i n = k i,F i ,F i 0 * [I i:F i ] n N A j=1 k j,F j i ,F i 0 * I j:F j n (2.15)
The nal values I i:F i ,k (x) and f A i (x) are obtained after N I iterations, the stopping criterion being that the relative variation of the mass fractions between two subsequent iterations is below a xed tolerance (

(f M i ) n -(f M i ) n-1 (f M i ) n < ).
As already discussed, the relation between the specimen local mass density ρ M (x), and the local elements mass fractions f M A (x) and f M B (x) must be known to apply the iterative procedure. This expression depends on the specimen structure and will be discussed, for dierent materials, in the next paragraph 2.3.

Mass density calculation

The relation between the specimen local mass density ρ M (x) and the local mass fractions f M i (x) of the atoms forming the material can be, in general, quite complicated.

In these section we will discuss some simple approximations that can be used in three cases of interest for the study of semiconductors:

The case of doped crystalline specimens

The case of amorphous alloys (that can be obtained for example by implanting a crystalline specimen with a high dose impurity)

The case of pseudomorphic alloys (such as for example the SiGe alloy).

Doped crystalline materials and the same radius approximation.

In an ideal doped crystalline material, we can consider that a small fractions of the substrate atoms (labeled by 1) are replaced by a dopant atom (labeled by 2). This is equivalent to say that all the doping atoms are in substitution. Since the doping atoms are in small concentration the mass density can be easily calculated starting from the mass density of the substrate ρ M 1 (x) and evaluating the mass dierence obtained when a dopant atom replaces a substrate atom.

ρ M (x) = ρ M 1 f A 1 (x) + m 2 m 1 f A 2 (x) = ρ M 1 1 + m 2 m 1 -1 f A 2 (x) (2.

16)

where f A i (x) are the atomic fractions for the two elements, related, in a binary alloy, by the following equation f

A 1 (x) = 1 -f A 2 (x).
The atomic fractions for the element can be calculated from the mass fractions using equation 2.5.

In the case of two elements, this formula simplies to:

f A 1 (x) = 1 -f A 2 (x) (2.17) f A 2 (x) = f M 2 (x) f M 2 (x) + m 2 m 1 f M 1 = f M 2 (x) f M 2 (x) + m 2 m 1 1 -f M 2 (x)
and equation 2.16 can be written as:

ρ M (x) = ρ M 1 1 + (m 1 -m 2 ) f M 2 (x) m 1 -(m 1 -m 2 )f M 2 (x) (2.18)
This is the formula we will looking for since it relates the mass density to the mass fractions. Since in this approximation it is assumed that the dopant atom causes no deformation, we will refer to it as the same radius approximation since it is equivalent to consider that the doping atoms an the substrate atoms have exactly the same radius.

Amorphous solids and the amorphous solid approximation

A better approximation can be obtained by using a formula that has been originally derived for solids that do not present any kind of ordering (also known as ideal amorphous solids (IAS)) [START_REF] Stachurski | Fundamentals of Amorphous Solids: Structure and Properties[END_REF]:

ρ M = N AT i=1 ρ M i f V i (2.19)
where ρ M i is the mass density of an amorphous amorphous solid composed exclusively by atoms of the i element and f V i is the volume fraction of the same elements in the alloy. We will refer to this approximation in the following as the amorphous solid approximation. The main hypothesis behind the derivation of this formula is that the space occupied in the solid by an atom of a specic element is always the same in any alloy, in the sense that it does not depend on the alloy concentration. Another way to express the same approximation is to say that the structure around a specic kind of atom in the alloy does not depend on the alloy concentration. This requirement is obviously fullled for IASs since in this case the atomic distribution is fully random and does not depend on the concentration of the two elements in the alloy. The space around an individual atom just depends on the size of the atom that can be estimated using its covalent radius. Equation 2.19 can also be used in solids that have not an IAS structure provided the fore-mentioned approximations holds. An example of its application to amorphous solid that do not have an IAS structure will be given in section 3.1.3.

To establish a relation between the alloy mass density and the mass fractions the following formula can be used to calculate the volume fractions from the mass fractions (this formula can be obtained under the same assumptions of equation 2.19):

f V i = f M i ρ M i N AT j=1 f M j ρ M j (2.20)
Inserting equation 2.20 into equation 2.19 and considering that the mass fractions are normalized to 1 ( N AT j=1 f M j =1) we obtain the amorphous mass density equation:

ρ M = 1 N AT j=1 f M j ρ M j (2.21)

Pseudomorphic alloys

It is interesting, to better understand the perfect amorphous approximation, to calculate the density of pseudomorphic alloys. Pseudomorphic alloys are binary alloys composed by two elements (Si and Ge for example) that have exactly the same crystalline structure in their pure form (Si an Ge both have a cubic structure of type diamond). The element A will identied by the index 1 and the element B by the index 2. We will also indicate with x the atomic fraction f A 2 of the second element so that f A 1 = 1 -x and the alloy chemical formula will have the formA 1-x B x . In a pseudomorphic alloy the two lattices have a very close lattice parameter ( lattice parameter is 0.227 Angstrom larger in Ge than in Si) and the binary alloy retains the same structure of the pure elements (SiGe has a diamond structure). The lattice parameter of the alloy follow Vegard's law [START_REF] Vegard | Die Konstitution der Mischkristalle und die Raumfüllung der Atome[END_REF][START_REF] Kasper | Test of Vegard's law in thin epitaxial SiGe layers[END_REF]:

a A 1-x Bx = a 1 (1 -x) + a 2 x (2.22)
where a i is the lattice parameters of the crystals formed by a pure element. Using Vegard's law the mass density of a A 1-x B x alloy can be calculated by considering that a unit cell contains, in total n atoms: n(1 -x) of type A and nx of type B:

ρ M = n m 1 (1 -x) + m 2 x [a 1 (1 -x) + a 2 x] 3 (2.23)
The nal expression can then be obtained by transforming the atomic fractions x and (1 -x) into mass fraction. We will not derive this formula because it can be shown that for this case, the amorphous solid approximation can be used. This seems paradoxical since in this case the alloy is not at all amorphous. But in fact it is easy to understand that in this case it is a good approximation to suppose that the volume occupied by a A or B atom in the A 1-x B x alloy does not depend on x. This derives from the fact that in the pure crystal an atom of type A occupies a volume

V A 1 = a 3 1 n (atomic volume)
where n is, again, the number of atoms per unit cell. In the same way V A 2 = a 3 2 n since the two atoms have exactly the same crystalline structure and the number of atom per unit cell n is the same. Since a 1 and a 2 have a close value, the values of V A 1 and V A 2 are also close and we can expect that the atomic volume does not have a strong variation.

As a consequence it is a good approximation to suppose that the atomic volumes of the two atoms are almost constant in the alloy and the amorphous solid approximation can be applied. This can be shown in a rigorous way by comparing the exact result (equation 2.23) with the result that is obtained using the amorphous solid approximation. In this case in the volume occupied by n

(1 -x) atoms is of type A is V 1 = n(1 -x)V A 1 , the volume occupied by atoms of type B is V 2 = nxV A 2 and the total volume is V = V 1 + V 2 .
The density in this approximation is given by equation 2.19.

Taking into the account that we have f V i = V i /V , and that for a crystalline solid with n atoms per unit cell ρ M i = nm i a 3 i we can write:

ρ M = ρ M 1 n(1 -x)V A 1 + ρ M 2 nxV A 2 n(1 -x)V A 1 + nxV A 2 (2.24) = nm 1 a 3 1 n(1 -x) a 3 1 n + nm 2 a 3 2 nx a 3 2 n n(1 -x) a 3 1 n + nx a 3 2 n =n m 1 (1 -x) + m 2 x a 3 1 + x(a 3 2 -a 3 1 )
Expressions 2.23 and 2.24 are obviously dierent,. But it can be shown that they are approximately the same for a pseudomorphic alloy. For that we can express both denominators using the parameter ε = a 2 -a 1 a 2 that can be considered to be small ( = 4.2% in SiGe). For the denominator of equation we can write 2.23:

[a 1 (1 -x) + a 2 x] 3 = (2.25) = a 3 1 [1 + x] 3 = a 3 1 1 + 3 x + 3 ( x) 2 + ( x) 3 = a 3 1 1 + 3 x + O 2
while for the denominator of equation 2.24 it can be shown that : It can be noted that the two expressions are equal up to the rst order and the amorphous solid approximation is indeed a good approximation for pseudomorphic alloys.

a 3 1 + x(a 3 2 -a 3 1 ) = (2.26) = a 3 1 + x a 3 1 (1 + ε) 3 -a 3 1 = a 3 1 1 + 3εx + 3ε 2 x + ε 3 x = a 3 1 1 + 3εx + O 2

Model calibration and error estimation

As already discussed, in order to be able to obtain a reliable measurement, the C-L coecients must be calibrated. In addition, in order to obtain a quantitative measurement the error on the calibrated coecient must be estimated, and the error on the nal composition measurement calculated by combining all the possible source of errors. The procedure and formulas we have used for calibration and error estimation will be discussed in this section. A direct application of these techniques using Rutherford Backscattered Spectroscopy RBS [START_REF] Chu | Backscattering Spectrometry[END_REF] for the calibration will be given in chapter 3.

In the general case of an alloy composed by N AT atoms, all the k i:F i ,F i 0 C-F coecients relative to the reference element i 0 must be calibrated.

The general technique for the calibration and the error estimation of the error of the C-L coecient relative to two elements i and i 0 for the peak families F i and F i 0 will be explained in the following. This procedure must be applied on two peaks at a time until the full set of C-L coecients is obtained. Several calibration specimens can be used, the only requirement is that using the specimen a measurement f M i (x) M eas of the mass fractions can be obtained with an experimental technique dierent from EDX.

For two elements i and i 0 the C-L factors relative to the peak families F i can be obtained by linear regression starting from the following formula (see equation 1.5 and 2.1 ):

k i:F i ,F j I i:F i (x) I i 0 :F i 0 (x) = f M i (x) M eas f M i 0 (x) M eas (2.27)
To estimate the error on ∆k As:F As on the calibrated value of k As:F As ,F Si we have assumed that the statistical distribution of the measured intensities follow Poisson law.

In this case the error on the measurement of the intensity I is given by ∆I(x) = I(x) and the relative error is given by

I = ∆I I = 1 √ I
.The best t value for k i:F i ,F j can be obtained by using the weighted least squares techniques and minimizing the expression

([41] p.75): R = k 1 w(x k ) f M i (x) M eas f M i 0 (x) M eas -k i:F i ,F i 0 I i:F i (x k ) I i 0 :F i 0 (x k ) 2 (2.28)
where {x k } is the set of positions where the densities have been experimentally measured. In order to favor the most precise points the weighting factors w(x i ) have been chosen to be equal to the relative experimental error associated to I As:F As (x i )

I Si:F Si (x i ) .
By using error propagation laws we can then write w(x k ) = N AT i=1 1

I i:F l (x i ) . 50 CHAPTER 2. DEVELOPMENT OF A NEW QUANTIFICATION METHOD
The best t value of k As:F As and the associated statistical error ∆k As:F As have been calculated using the standard formula for parameter error estimation in weighted least square regression ( [START_REF] Wasserman | All of Nonparametric Statistics[END_REF], [START_REF] Taylor | Introduction To Error Analysis: The Study of Uncertainties in Physical Measurements[END_REF]).

When performing a quantitative measurement, it is of course important to evaluate the total measurement error. In the case of a concentration measurement carried out by EDX using the calibrated value of the C-L coecients three kind of errors must be taken into account:

1. The fore-mentioned statistical error on the measured intensities ∆I(x) = I(x).

The contribution of this error to the mass fraction can be calculated, using standard error propagation techniques [START_REF] Taylor | Introduction To Error Analysis: The Study of Uncertainties in Physical Measurements[END_REF], to be:

∆ I f M i (x) = k i:F i ,F i0 N AT j=1 k j:F j ,F i0 I j:F j (x) 2 ×
(2.29)

j =i k j:F j ,F i0 I j:F j (x)∆I i:F i + I i:F i (x)∆I j:F j (x)
where i =j indicates the sum over all atoms except the i atom.

2. The errors ∆k j:F j ,F i0 on the calibrated value of the CL constant. The contribution of this error can be calculated to be :

∆ k f M i = I i:F i ,F i0 N AT j=1 k j:F j ,F i0 I j:F j (x) 2 ×
(2.30)

j =i I j:F j (x) k j:F j ,F i0 ∆k i:F i ,F i0 + k i:F i ,F i0 ∆k j:F j ,F i0
3. The experimental error on the specimen thickness ∆t. In this case it is not possible to apply the error propagation formula to evaluate the error since the dependence of f M j (x) on t is given by the previously discussed iterative calculation. In this case the contribution to the total error has been estimated using the following formula

∆ t f M i (x) = [f M i (x)] t+∆t -[f M i (x)] t-∆t 2
where f M i (x) t is the density calculated for a thickness t.

Assuming that the error on k and the error on the intensities are statistically dependent (the same detector is used for the calibration and the measurements) and that the error on the thickness is statistically independent of the other errors we can obtain the expression of the total error :

∆f M i (x) = [∆ I f M i (x) + ∆ k f M i (x)] 2 + [∆ t f M i (x)]
∆c A i (x) = N AV m i N AT j=1 ∂ ∆ρ M ∂f M j ∆f M j + ρ M ∆f M i (2.32)
The precise formula depends on the formula used for ρ M . We can the estimate the error for the two mass density approximation discussed in section 2.3.

For the constant radius approximation the following formula is obtained:

∆c A 2 (x) = N AV m j ρ M 1 1 + (m 1 -m 2 ) f M 2 (x) m 1 -(m 1 -m 2 )f M 2 (x) × (2.33) × 1 + m 2 (m 2 -m 1 ) [m 2 -(m 2 -m 1 )f M 2 (x)] 2 f M 2 (x) ∆f M 2 (x)
while for the amorphous solid approximation we have:

∆c A i (x) = N AV m As 1 N AT k=1 f M k ρ M k    ∆f M i (x) + 1 N AT k=1 f M k ρ M k N AT j=1 ∆f M j (x) ρ M j    (2.34)
As already discussed, it is conventional, for binary alloys, to use the atomic fractions instead of the mass fractions. In this case the error on the mass fractions can be calculated applying error propagation formula on equation 2.5. The obtained expression for the error on the atomic fractions is:

∆f A i = 1/m i N AT j=1 f M j /m j 2 × (2.35) j =i 1/m j f M j ∆f M i + f M i ∆f M j 2.

Discussion and conclusions

In this chapter we propose a new quantication technique taking into account the absorption in the specimen with a self-consistent method and including model calibration for error estimation. The critical point for the model application is to establish a relation between the mass density and the specimen composition. For, this two approximations have been proposed.

It will be demonstrated in the following chapter that these two approximation can be successfully applied to obtain quantitative concentration measurements in amorphous and crystalline semiconductors.

To conclude this chapter, in this section we will compare the model we have developed with other existing methods and discuss how it can be applied to materials dierent from the ones discussed in the next chapter.

Comparison with other models

As already stated, the main novelties of this new approach are the use of RBS data to calibrate the k-factors and that the specimen local density has been calculated using physical models. The knowledge of the local density is used to correct the absorption eects without any in situ measurement of the electron beam current, which is an advantage with respect to other techniques such as the ζ-factor method or the crosssection method (see section 1.3.1).

This being said, other techniques have been proposed in the past that do not require any current measurement and yet can be used to correct absorption. We can cite, for example, the techniques proposed by Morris et al. [START_REF] Morris | EMAG'79 Brighton[END_REF] and Qiu et al. [START_REF] Qiu | Calibration of thickness-dependentk-factors for germanium X-ray lines to improve energydispersive X-ray spectroscopy of SiGe layers in analytical transmission electron microscopy[END_REF], where the dierence in absorption between dierent peaks is used to deduce the specimen thickness. Another approach uses measurements performed at dierent tilt angles (see again Morris et al. [START_REF] Morris | EMAG'79 Brighton[END_REF]) to estimate the specimen thickness.

Concerning the technique using dierent peaks to obtain information on the thickness, it must be noted that it can only be applied to specic materials. To be applicable, two main peaks of the substrate must be measurable and their characteristic energy must be big enough for the dierence in absorption to be measurable. This technique is, for example, not applicable in pure silicon substrates since at at the energy commonly used in TEM (~100 kV), just one characteristic peak (the SiK peak) can be measured.

Some limitations also apply to the technique using several tilt angles. First of all the specimen must be reasonably at (but this also apply to our model since a xed specimen values is used in the calculations). Moreover the dierence in tilt angles must be high enough for absorption eect to vary, and this is problematic in modern microscopes. In high resolutions microscopes the maximum tilt angle can be limited by the small objective lens pole-pieces gap, and in analytical microscopes equipped with large EDX detectors, shadowing eects can appear at small tilt angles.

To our knowledge, the idea of using RBS to calibrate the C-L coecients and error propagation to estimate the measurement error has never been used before. This is an original feature of the model proposed here.

The main disadvantage of our model when compared with techniques such as ζfactor method, is that a density model to be used depends on the specimen and must be established while in the ζ-factor method, the additional information provided by the in-situ current measurement can be used to measure thickness and density at the same time. The model presented in this chapter can, in principle, also be applied, to alloys composed by more than two elements, or alloys with a known phase-diagram provided that the density in each domain can be calculated using one of the fore-mentioned approximations. This being said, in cases in which the development of a density model becomes too complicated (because of the presence of several elements or several phases in the alloy phase-diagram) the ζ-factor method must be preferred.

Chapter 3

Quantitative EDX mapping in semiconductors

As already discussed in the introduction, Elemental mapping at the nanoscale is a particularly critical issue for semiconductor devices. The emergence of devices based on ultra-shallow junctions or 3D geometries has created a demand on new techniques since standard techniques such as Secondary Ion Mass Spectroscopy cannot be used on new generation devices.

The quantication model described in the previous chapter has specically been developed to obtain quantitative concentration measurements by STEM/EDX and can be used for composition proling in semiconductor nanodevices.

In this chapter two applications will be described:

The measurement of low concentration impurities in a Si nanodevice structure.

The measurement of Ge concentration in a SiGe nanostructure.

The discussion will include, for each application, details on the implementation of the model, observed specimens and experimental setup. The problem of this devices is that their characteristic size is of the order of some nanometers.

Using the developed model, STEM/EDX can actually be used for this kind of measurements. For that, dopant mapping has been carried out applying STEM/EDX to the source drain regions of a FinFET device.

A FinFET device [START_REF] Colinge | FinFETs and Other Multi-Gate Transistors[END_REF] is specic type of 3D transistor that can be obtained by using standard lithography techniques on Silicon On Insulator (SOI) substrates. SOI consists One of the main problems when creating a FinFET is that doping must be symmetrically distributed on all the walls used to control the structure. This means that (again with reference to gure 3.1(b)) the lateral walls in a three walls structure must have the same doping concentration than the upper wall. A doping distribution fullling this requirement is referred to as a conformal distribution.

It is easy to understand that for a transistor to work correctly the source and drain doping must also have a conformal distribution. And since the gate can be lowly doped (and even unopened) in FinFET devices while the source drain region must be highly doped to assure an ohmic contact, the problem of the conformal doping of the source and drain region is the most critical problems for FinFET devices. For this reason we have decided to apply the developed model to the measurement of a high concentration As conformal doping in FinFET test structure.

In the following the experimental details on the observed specimen and the observation conditions will be described. In section 3.1.4, the calibration of the model will be discussed and it will be shown that a quantitative As prole can be obtained on the nanometer scale. Finally the measurement errors and doping detection limit will be estimated, and the possibility to improve the detection limit by changing the observation parameters will be discussed.

Specimen preparation and TEM observations

For TEM observation, a cross sectional lamella has been created using a FEI Helios 600i dual beam focused ion beam (FIB) [START_REF] Ke | TEM sample preparation by FIB for carbon nanotube interconnects[END_REF]. A lift-out procedure has been used to transfer the specimen on a supporting copper grid. In order to minimize curtaining eects a layer of platinum has been deposed on the wafer surface before the creation of the lamella. Platinum deposition has been carried out in two consecutive steps by using Electron Beam-Induced Deposition (EBID) followed by Ion Beam-Induced Deposition (IBID) in order to avoid any kind of ionic damage near the surface, that can be potentially created during IBID. It has to be noted that there is an additional advantage in using FIB specimen preparations for EDX analysis. The use of a lamella reduces the emission of secondary X-rays by uorescence and the backscattering in comparison with specimens prepared by other techniques like mechanical grinding followed by ion thinning (see gure 3.4). FIB preparation has also been used in the case of the FinFets since a precise specimen region has to be selected in order to observe the implanted regions (such as the region shown in gure 3.2). The obtained intensities have been used as a starting point for the calibration, the absorption correction and the error estimation.

Application of the quantication Method

The quantication of the data have been carried out using the methods in the previous chapter. To be able to apply the the self-consistent calibration method the formula to be used to calculate the mass density has been established both for amorphous SiAs alloy (a-SiAs) found in the calibration specimens and for the As-implanted crystalline silicon (c-SiAs) composing the FinFETs.

For a-SiAs, the constant radius and amorphous solid approximations can be considered. The value of the covalent radius for silicon and arsenic have a close value, R Si = (111 ± 2)pm and R As = (119 ± 4)pm, while the value of their atomic masses As of amorphous arsenic [START_REF] Greaves | Amorphous arsenic[END_REF] is shown in the plot.

It could then be tempting to use the amorphous solid approximation (equation 2. [START_REF] Gernert | Comparing the Si(Li)-detector and the silicon drift detector (SDD) using EDX in SEM[END_REF] in this case. But, as already discussed in section 2.3.2, to justify this choice one of this two assumptions must be satised:

The structure of a-SiAs is compatible with that of an IAS structure.

The structure of the alloy is similar to that of the amorphous forms of the pure It is easy to prove that neither a-As nor a-Si are ideal amorphous solids. For a-Si this can be proved by noting that the density predicted by the IAS model ρ M a-Si,IAS = 4.32g/cm 3 [START_REF] Stachurski | Fundamentals of Amorphous Solids: Structure and Properties[END_REF] is higher than the density of crystalline Si (c-Si) ρ M c-Si = 2.32 g/cm 3 for the diamond structure. This is in clear contradiction with the experimentally determined density of amorphous silicon ρ M a-Si = 2.285 g/cm 3 [49] which is slightly lower than ρ M c-Si . The same kind of reasoning can be used to prove that the structure of amorphous As (ρ M a-As = 4.7 g/cm 3 ,ρ M c-As = 5.73 g/cm 3 ) is also not compatible with that of an IAS.

In conclusion there is no reason to think that the a-SiAs alloy (whose structure is a combination of the structure of amorphous Si and amorphous As) has an IAS structure.

On the other hand it is easy to show that a-Si and a-As have a similar structure.

It can then be assumed that this same atomic arrangement is preserved in the SiAs alloy for any value of As concentration which is precisely the assumption used to derive equation 2.19. To show this the fraction s = V S V A between the atomic volume V A and the atom size V S must be evaluated. This parameter is an estimator of the packing ratio of the solid. The volume V A is by denition the inverse of the atomic density ρ A = ρ M /m of the material while the atom size calculated by using using a spherical The comparison between the amorphous solid approximation and the same radius approximation is shown in gure 3.5. It can be noted that the amorphous solid approximation gives better results than the same radius approximation for high arsenic concentrations while the two approximations are almost equivalent for low As concentrations as we can see in gure 3.5.

For c-SiAs, it is much simpler to obtain the mass density formula since this correspond exactly to the case the constant radius approximation has been derived for.

Mass density can then be calculated using 2.16 with

m 1 = m Si , m 2 = m As , ρ M 1 = ρ M Si .

Experimental results and discussion

In this section we will present the experimental results obtained on the FinFET test structure. For that, the calibration results, that have been used to obtained the calibration value of the C-L coecients will be described rst. Then the nal results obtained on dopant proling will be discussed.

Calibration results

In order to apply the calibration techniques the atomic density concentrations c A A/RBS (x) measured by RBS have been converted into mass fractions. Then the C-L coecients have been obtained by weighted linear regression as described in section 2.4. Equations 2.4 and 2.21 can be combined to obtain the mass fractions measured by RBS:

f M As (x) M eas = m As c A As/RBS (x) N AV ρ M a-Si + m As c A As/RBS (x) 1 - ρ M a-Si ρ M a-As
The calibration technique described in section 2.4 have been applied to the a-SiAs calibration specimens described in section 3.1.1. An EDX hyper-map consisting of a series of spectra (one for each pixel of the corresponding STEM image) has been recorded (see gure 3.6 for the 2% specimen and gure 3.7 for the 10% specimen).

For the acquisition the FIB specimen has been oriented as in gure 2.2. Then a line prole has been obtained by integrating the obtained map (see gure 3.6(a) and 3.7

(a)) in a direction perpendicular to the depth direction (dened in gure 2.2). Several characteristic peak families have been used associated: the K peaks for silicon, and either the K or the L peaks for arsenic (see gure 3.8).

The absorption correction procedure described in section 2.2 has then been used to calculate the absorption corrected intensities I Si:K , I As:K and I As:L . The iterative absorption algorithm has shown a rapid convergence and no numerical instability (see gure 3.9(a)). Convergence is basically obtained after two iterations (only few curves are visible since the proles for the iterations from 2 to 10 are superimposed). Moreover it can be observed that the absorption eects are more pronounced in the high density region (see gure 3.9(b)). This can be understood by noting that the mass density of the specimen is higher in this region and, as a consequence, the absorption eects are bigger. This is a clear demonstration of the fact that it is important to establish a density formula in order to take into account the mass density dependence on the local concentration.

Then fore-mentioned linear regression techniques have then been used on the absorption corrected data to calibrate:

the CL factor K As,Si:K,K = k As:K,K associated to the K family of peaks of arsenic when the K family of peaks of silicon is used as reference the CL factor K As,Si:L,K = k As:L,K associated to the L family of peaks of arsenic when the K family of peaks of silicon is used as reference

In order to minimize the error on the C-L coecients we have carried out the calibration on the higher dose specimen. The reason of this choice can be easily understood by observing that under the fore-mentioned hypothesis that the X-ray count has a Poisson The error on the C-L coecients can be used to estimate the nal errors on the density prole as explained in section 2.4 where the possible sources of error have been discussed. Between the dierent errors the error arising from the uncertainty on the specimen thickness deserves a special discussion. It is generally complicated to measure specimen thickness in a reliable and precise way. Even if some techniques exist, measurement by SEM, EELS, convergent beam electron diraction, it is complicated, in general to obtain a value with a low error. As a consequence measurement errors on the thickness can be in the order of 30% or higher. It is then important to assess the inuence of the thickness error on the density measurement. For this, the measurement variation on the peak region of the two specimens (that, as already noted are the most aected by absorption) is reported in gure 3.11. It can be easily observed that the variation of the concentration is small even for large thickness variations. This means that a precise measurement of the doping concentration can be obtained even if the error on the specimen thickness (∆t) is high. The obtained proles as well as the total experimental errors on the EDX As concentration measurement in the 10% specimen are shown in gure 3.12. In order to check the validity of our procedure and the applicability of these coefcients to arsenic distribution with lower concentrations, we have used the previously obtained values of the CL coecients to measure the doping concentration in the 2% a-SiAs specimen. The results are shown in gure 3.13 where the experimental errors on the EDX measurements are also reported.

First of all it must be observed that the experimental error is larger for higher concentrations. This is due to hypothesis that X-ray intensities follow a Poisson distribution. This implies that the absolute error on the measured intensity increases with the intensity while the relative error decreases. The error on doping concentration follow the same trend since it directly depends on the error on the measured intensity. This means that, in the doping distribution peak, the measured value of doping concentration is more precise (i.e. has a lower relative error) than the values in the distribution tail, even if larger error bars appear in the graph. Figure 3.13 clearly shows that the results obtained using the K peaks of arsenic are more accurate than those obtained using the L peaks. This is still more evident for low concentrations (i.e. in the rightmost tail of the doping prole).

The origin of this discrepancy can be sought, in our opinion, in the lack of precision on the measured intensity of the L peaks of arsenic. This is due to the position of the L peaks, located in the low energy region of the spectrum. In this region the subtraction of the bremsstrahlung noise is more dicult since the noise is higher and there are neighboring peaks associated to oxygen and carbon. For low As concentrations, such as those we are measuring, this problem becomes critical since the bremsstrahlung and peak intensities can be of the same order of magnitude. This means that a small error on the background noise interpolation has a strong impact on the measurement of As concentration. We have thus decided to use the arsenic K peaks for all the

Dopant prole measurement

To check the applicability of our method to a real nanodevices we have carried out some measurements on the FinFET device already described in section 3.1.1. For that an EDX hyper-map has been acquired. Some intensity maps and proles extracted from the hyper-map are shown in gure 3.14. In order to be able to apply the same measurement geometry illustrated in chapter 2 and in gure 2.2 we have decided to study the distribution of one of the FinFET vertical wall and the specimen have been oriented for the acquisition as in gure 3.15(a). The newly developed quantication method described in the previous section has then been applied to the silicon and arsenic K peaks proles. The nal result is shown in gure 3.15(b) where the doping prole on the lateral wall and the corresponding errors are shown.

The results clearly show that a doping prole in a nanodevice can be obtained with this technique and that the obtained experimental error is low enough to be able to appreciate the concentration variations in the prole.

Detection limit

It is interesting, in the context of doping prole by STEM/EDX to try to understand what is the detection limit for arsenic concentration. To estimate the detection limit we have used a simple approach based on the experimental results. It is easy to understand that a density measurement c cannot be distinguished by zero if zero is included in the condence interval (i.e. it is between c -∆c and c + ∆c). The obtained result obviously depends on the chosen condence level [START_REF] Taylor | Introduction To Error Analysis: The Study of Uncertainties in Physical Measurements[END_REF] (a Student coecient equal to 1 has been chosen in this paper).

The analysis of the proles (see gure 3.15) clearly show that doping concentration is always above the concentration limit. The concentration limit can be obtained from the calibration data shown in 3.13(b) and can be estimated to be 7.7 • 10 19 cm -3 . This is not an absolute estimation since the value of the detection limits clearly depends on the measured doping peaks intensity (see equation 2.29 and 2.30) and can be improved by increasing the hyper-map exposition time. The previously discussed formulas can be used to calculate the error associated to a specic set of measured intensities but it is impossible to deduce a theoretical lowest limit since the maximum exposition time depends on many experimental factors such as the stability of the specimen to drift and its resistance to electron irradiation.

Yet a target intensity value can be estimated. To give an example of this procedure, we have estimated the exposition times needed to obtain a given detection limit using the specic parameters of our experimental setup. More specically we have used the values of the count per seconds of our setup to calculate the measuring time needed to obtain dierent values of the detection limit. For the calculations we have supposed that the counts-per-seconds are not changing with time.

Using equation 2.2, 2.31 and 2.34 and making the approximations that the error on the thickness is negligible and that the As peak has a negligible intensity when compared to the Si peak we we can obtain the following formula:

I Si:F Si = k As:F As ,F Si * f M As (1-f M As ) f M As (x) + 1              ∆ c A As m As N AV            ρ M a-SiAs + f M As (x) ρ M a-Si η 1 -η f M As 2            f M As 1 -f M As              2 - ∆ f M As k As:F As ,F Si 2 (3.1)
This formula express the intensity I Si:F Si (x) of the Si peak needed to obtain the following value for the As density: f M As ± ∆ f M As . According to our denition of the detection limit, the intensity I D needed to obtain a detection limit D can be obtained by putting ∆ f M As = f M As = D and in equation 3.1 The exposition time needed to obtain a measured intensity I D can be estimated by using the counts per second (cps) as the following (it is obviously assumed that

I Si:F Si = 0 for t exp = 0) : t exp = I Si:F Si (x) cps (3.2)
The measuring time needed to improve the detection limit can then be calculated, and some results are shown in table 3.1. It can clearly be observed that in our case it As density detection limit Measuring time (live time) 3.1: Estimated measuring time (live time) needed to obtain higher detection limits is dicult to considerably improve the detection limit just by increasing the measuring times, since the presence of specimen drift, beam damage and current instability (in our case a cold Field Emission Gun gun is used) limit the measuring time to a few tens of minutes.

Composition proling of SiGe nanostructures

SiGe layers are of high interest in modern microelectronics due to their improved hole mobility compared to silicon (p-type Metal Oxide Semiconductor transistors channel engineering) and their compatibility with current devices. In modern devices very low resistivity boron-doped SiGe sources and drains are used to compressively stress the channel of short gate length devices, resulting in better electrical performances [START_REF] Parton | Strained silicon the key to sub-45 nm CMOS[END_REF].

To obtain highly-doped and ultra-shallow source and drain regions in SiGe devices some specic annealing processes are needed, such as (RTP).

In particular, Ultra Violet Nanosecond Laser Annealing (UV-NLA) receives much attention because of several promising results obtained on pure silicon and pure germanium [START_REF] Shayesteh | Optimized Laser Thermal Annealing on Germanium for High Dopant Activation and Low Leakage Current[END_REF]. However, there are few studies dealing with the behavior of silicon/germanium layers upon UV-NLA.

On these samples SIMS studies can be performed in order to measure the Ge concentration, but the lack of reliability in nanoscale devices brings us to apply the STEM/EDX technique we have developed. The results obtained in SiGe devices will be discussed in this section.

As in the section 3.1, the experimental details on the observed specimen and the observation conditions will be described, the calibration of the model will be examined and the measurement errors will be calculated.

Description of the samples

As in the case of the SiAs specimen, in order to have an accurate measurement of the Ge concentration two calibration samples with two dierent thicknesses (28nm and 48 nm) of the SiGe layer have been used as we can see in the gure 3.16. The growth of the SiGe layer was performed in a 300 mm reduced Pressure Chemical Vapor Deposition (PCVD) chamber after surface preparation.

To obtain a calibrated value of the C-L coecients, Rutherford backscattering spectroscopy (RBS) measurements have been carried out using He + ions accelerated at 1.5

MeV. The detector resolution and solid angle have been obtained on a SiO 2 reference specimen using a detector collection angle of 10°. For the Ge concentration measurements a collection angle of 65°has been used. Using RBS the atomic atomic fraction of Ge in the two calibration specimens have been obtained and these results were used to calibrate the C-L coecients.

To check the applicability of our technique at the nanoscale, a germanium prole has been measured by STEM/EDX on a nanodevice test structure consisting in an undoped pseudo-morphic Si 1-x Ge x layer grown at 550°C on n-type Si (100) substrates.

The samples were then subjected to UV-NLA in a SCREEN-LT3100 system, operating at 308 nm wavelength with a 145 ns FWHM pulse. Energy densities were chosen to study regimes from sub-melt to full SiGe layer melting. Melt threshold was detected using in-situ Time-Resolved Reectivity (TRR) at 635 nm, and surface quality was studied by means of atomic force microscopy and SP2 Haze.

FIB preparation has been performed on all the specimen, in the same conditions described in section 3.1.2.

For the self-absorption correction, the mass density function contained in equation 2.21 and described in section 2.3.3 has been used.

STEM/EDX Results

Calibration results

In order to apply the calibration techniques the atomic fraction f A A/RBS (x) measured by RBS have been used in this case. Then the C-L coecients have been obtained by weighted linear regression as described in section 2.4 after conversion from the mass fraction to atomic fraction by equation 2.5:

EDX hyper-maps have been also recorded for the 28 nm specimen and the 48 nm specimen (see gure 3.17 and 3.18). For the acquisition the FIB specimen has been oriented as in gure 2.2. Then a line prole has been obtained by lateral integration.

As discussed in section 3.1.4 for the case of Arsenic, the results obtained using the K peaks are more accurate than those obtained using the L peaks since background subtraction is harder to perform on L lines. This is also true for germanium, and in the following discussion we will focus on the results obtained using silicon and germanium K peaks.

To obtain a precise value of the C-L coecients, the calibration technique described in section 2.4 have been applied to SiGe specimens.

This has allowed us to obtain the calibrated value of the C-L coecient: K Ge,Si:K,K = 3.83 ± 0.02.

The atomic fraction measured using the Ge K lines before and after calibration are reported, for the two calibration specimens, in gure 3.19 and 3.20.

As in the case of SiAs specimens, the iterative absorption algorithm has shown a rapid convergence and no numerical instability (see gure 3.21). As in the previous case, it can be observed that the absorption eects are more pronounced in the high density region.

Following the same procedure as in the SiAs specimen, the error has been estimated on the two calibration specimen and the results are reported in gure 3.22.

Nanostructure test

To check the applicability of our method to a real nanodevices we have carried out some measurements on the nanodevice test structure described in section 3.2.1. For that, an EDX hyper-map has been acquired. Some intensity maps and proles extracted from the hyper-map are shown in gure 3.23.

As in the SiAs specimen, the newly developed quantication method has then been applied to the silicon and germanium K peaks proles. The nal result is shown in gure 3.24 where the germanium and silicon prole and the corresponding errors are shown.

The results show, that this technique is relevant for nanodevices and that the obtained experimental error is low enough to be able to appreciate the concentration variations. IoT is a generic term used to describe the exchange of data between portable, smart and connected devices [START_REF] Biljana | A review of Internet of Things for smart home: Challenges and solutions[END_REF][START_REF] Lethien | Challenges and prospects of 3d micro-supercapacitors for powering the internet of things[END_REF] creating a new range of devices able to interact and communicate through the internet. Applications relying on wearable devices such as healthcare, smart clothes or electronic papers have motivated many research works to achieve for instance individual management and continuous monitoring of patients [START_REF] Chana | mart wearable systems : Current status and future challenges[END_REF].

CHAPTER 4. ADDITIONAL RESEARCH ACTIVITY

Reliable micro-sized energy storage sources (ESS) such as batteries and Electrochemical Capacitors (ECs) are needed to power these devices [START_REF] Miller | Valuing Reversible Energy Storage[END_REF][START_REF] Maeng | Wafer-scale integrated microsupercapacitors on an ultrathin and highly exible biomedical platform[END_REF][START_REF] Simon | Materials for electrochemical capacitors[END_REF]. ECs and batteries are complementary ESS covering a broad range of applications. While batteries can deliver low power for long time (tens of hours), ECs are suitable for delivering high peak current or power for short times (up to tens of seconds), making them promising candidates to power smart wearable systems [START_REF] Jost | Textile energy storage in perspective[END_REF][START_REF] Liu | Flexible and Stretchable Energy Storage: Recent Advances and Future Perspectives[END_REF]. Screen-printing of carbon [START_REF] Jost | Carbon coated textiles for exible energy storage[END_REF] and pseudocapacitive materials [START_REF] Zhang | Fully Printed Ultraexible Supercapacitor Supported by a Single-Textile Substrate[END_REF] or stamping processes [START_REF] Chuanfang (john) Zhang | Stamping of Flexible, Coplanar Micro-Supercapacitors Using MXene Inks[END_REF] were reported with limited areal capacitance (< 20 mF.cm -2 ) or resolution. Similarly, micro-devices prepared from Atomic Layer Deposition (ALD) , reactive sputtering and electron beam evaporation [START_REF] Si | On chip, all solid-state and exible micro-supercapacitors with high performance based on MnOx/Au multilayers[END_REF] techniques suer from a poor areal capacitance (few mF.cm -2 ) due to the small electrode thicknesses (<100 nm).

Wet processing routes, such as electrodeposition [START_REF] Lang | Nanoporous metal/oxide hybrid electrodes for electrochemical supercapacitors[END_REF][START_REF] Sieben | Flexible ruthenium oxideactivated carbon cloth composites prepared by simple electrodeposition methods[END_REF] or electropolymerization [START_REF] Meng | Ultrasmall Integrated 3d Micro-Supercapacitors Solve Energy Storage for Miniature Devices[END_REF], are suitable for preparing thick electrodes (> 50 µm) on exible substrates with capacitance exceeding 10 mF.cm -2 . However, these multi-step wet processing routes are not fully compatible with the techniques used in the semi-conductor industry and are therefore dicult to upscale.

Laser-writing process focused a lot of interest these past years for preparing electrodes for energy storage applications. Following pioneer work from Arnold et al [START_REF] Arnold | Direct-Write Planar Microultracapacitors by Laser Engineering[END_REF], laser irradiation was used to reduce graphite oxide (GO) into laser-scribed graphene (LSG) [START_REF] Maher | Laser scribing of high-performance and exible graphene-based electrochemical capacitors[END_REF] or to achieve direct conversion of polyimide (PI) substrates into laser-induced graphene (LIG) [START_REF] Lin | Laser-induced porous graphene lms from commercial polymers[END_REF][START_REF] Peng | Flexible Boron-Doped Laser-Induced Graphene Microsupercapacitors[END_REF][START_REF] Peng | Flexible and Stackable Laser-Induced Graphene Supercapacitors[END_REF] with a limited areal capacitance (< 20 mF.cm -2 ).

Ruthenium oxide exhibits one of the highest gravimetric capacitance (1450F.g -1 )

together with decent electrical conductivity (from 4 to 46S.cm -1 depending on the hy- dration rate of RuO 2 ) [START_REF] Barbieri | Eect of Electronic Resistance and Water Content on the Performance of RuO2 for Supercapacitors[END_REF]. Since the realization of micro-supercapacitors requires only a limited amount of RuO 2 , typically less than 1mg.cm -2 , the cost is not that much an issue [START_REF] Mai Dinh | Hydrous RuO 2 /carbon nanowalls hierarchical structures for all-solid-state ultrahigh-energy-density microsupercapacitors[END_REF]. Therefore, Kaner's group recently adapted their GO laser-scribing method to prepareLSG/RuO 2 based exible micro-supercapacitors fromGO/RuCl 3 mixture [74].

Laser irradiation of the mixture coated onto exible substrate led to simultaneous reduction of GO into LSG and Ru(+III) oxidation into Ru(+IV). However, the small RuO 2 weight loading that could be achieved by this method about 90µg.cm -2 limited the areal and volumetric capacitance of the device to 10mF.cm -2 and 11 F.cm -3 , respectively, in 1M H 2 SO 4 (estimated from Ragone plot). In addition, such oxidation dependent process is not convenient for the preparation of every carbon-(microporous carbon), metal oxide-(Nb 2 O 5 , MnO 2 for example, for which a specic morphology has to be obtained) or phosphates-based electrodes (LiFePO 4 ), as the nal active material is obtained from the transformation of a precursor under the laser beam. Dierently, the present work reports about the preparation of RuO 2 -based micro-supercapacitors from the direct laser-writing of a commercial RuO 2 .xH 2 O powder. For this, powdercontaining ink spin-coated onto an electrically insulating exible polyimide lm, which does not need the design of masks or controlled atmospheres. This process, which results in the preparation of stable electrodes with improved mechanical properties, can be extended to other materials such as metal oxides, carbons or phosphates to design a large variety of micro-devices for various applications. In the following, we will give a description of the RuO 2 specimen preparation, then we will detail the electron microscopy study and its importance in this work.

Description of the samples

Electron microscopy characterization

The facile laser-writing procedure developed in this work was used to make RuO 2based electrodes on a polyimide substrate. To understand the role of the HAuCl 4 .3H 2 O transition layer, the cross-section of an electrode was prepared by cross-polishing 1 . Indeed, capacitance was very limited without the Au precursor, i.e. the Ru amount deposited on polyimide was too small. SEM images (obtained using secondary electrons) representative of the as-prepared laser-scribed RuO 2 electrodes reveal that the lm covers entirely the exible substrate (gure 4.2A). The deposit grows as small crystals and some popcorn-like structures can also be locally observed, accounting for about 20% of the total surface (gure 4.2B).

These crystals form a homogeneous and dense structure onto the substrate as observed in gure 4.2C (top view) and gure 4.2D (cross section). In gure 4.2E is shown a cross sectional view obtained using SEM in backscattered electrons mode, representative of the deposit, which reveals the bi-layered aspect of the deposit (the tilt angle between the observation axis and the specimen surface is 54°). The popcorns start to grow from the RuO 2 crystals; such structure might be due to a local over-heating during the laser-writing process. EDX mapping allowed us to show the presence of oxygen and ruthenium in both layers, but it can be clearly observed that the oxygen concentration is much higher in the 500 nm-thick layer containing the pillars (gure 4.2F), in comparison with the thin interface layer at the PI lm / RuO 2 pillars. This suggests that the upper layer is composed by RuO 2 while the lower one could be mainly composed by metallic Ru. In addition, gure 4.3A shows that several Au nanoparticles are visible in the RuO 2 layers, which are likely to enhance the electrical conductivity of the RuO 2 . XRD measurements conrmed the presence of crystalline RuO 2 (gure 4.3B) as well as Au A surrounding gate allowing optimal electrostatic control of the channel.

An increase in the density of transistors per unit area due to the vertical structure of NWFETs.

However, like any technological breakthrough, an optimization phase is essential in order to rst identify the axes of improvement of the new architecture and, secondly, to establish strategies to understand and solve them. In the case of NWFETs, there is still a lack of understanding regarding the formation of metal contacts, still with a view to reducing access resistance. Indeed, the usual procedure for forming these contacts is based on a rapid annealing (RTA) at around 300°C after deposition of a platinum lm on silicon of a few tens of nanometers until the silicide PtSi is obtained.

PtSi is not the only phase to come into play during the silicon-platinum [START_REF] Gas | Formation of silicide thin lms by solid state reaction[END_REF][START_REF] Chen | Kinetic Manipulation of Silicide Phase Formation in Si Nanowire Templates[END_REF] inter-diusion.

It is possible to obtain mainly 3 phases: PtSi, Pt 2 Si and Pt 3 Si. The parametrization and the understanding of the mechanisms that take place in the silicidation reaction are In the case of a bulk substrate, the parameters and mechanisms of platinum silicidation are known and, in the case of horizontal nanowire structures, have been recently investigated [START_REF] Chen | Kinetic Manipulation of Silicide Phase Formation in Si Nanowire Templates[END_REF][START_REF] Lin | Single Crystalline PtSi Nanowires, PtSi/Si/PtSi Nanowire Heterostructures, and Nanodevices[END_REF][START_REF] Liu | Silicidation of Silicon Nanowires by Platinum[END_REF]. However, these studies do not make it possible to apprehend the impact of the "ultimate" size reduction on the silicidation reaction.

The observations made by [START_REF] Han | Réalisation et caractérisation de dispositifs MOSFET nanométriques à base de réseaux denses de nanols verticaux en silicium[END_REF] revealed a non-negligible variation in the kinetics of reaction of the platinum-silicon system when the dimensions involved are reduced to the extreme. Indeed, the idea of this work was to achieve and characterize the metal contacts of vertical nanowires transistors with a progressively decreasing diameter (from 60 down to 12 nm). In particular, we wanted to achieve a better phenomenological understanding of the silicidation reaction. For that, electron microscopy has been used

to investigate the existence of several phases in the nanowires. In order to validate the results, TEM-EDX analysis has been coupled with HRTEM crystallographic analysis. The crystallographic analysis was carried out by using Digital Micrograph and Carine Crystallography. The observed Fourier patterns have been compared with the theoretical patterns of PtSi, Pt 2 Si and Si structure obtained using the kinematic approximation [START_REF] Williams | Transmission Electron Microscopy: A Textbook for Materials Science[END_REF].

Electron microscopy characterization

The crystalline structure of the the PtSi phases are reported in The Fourier transform analysis allowed us to conrm that this nanowire structure is compatible with that of a PtSi crystal imaged through a triclinic <0 1 0> zone axis.

It can also be shown that PtSi this is the only structure compatible with the observed pattern. We can see that, within the experimental errors, the ratio between the Pt ans the Si

Specimen annealed at 350°C

To study the impact of nanowire size on the silicidation kinetics, the structure of the silicide regions has been studied in wires of dierent sizes. The results obtained for a 50 nm wire are shown in gure 4.13. The analysis show that the whole contact region has a PtSi structure.

The results obtained on a thinner nanowire with a 13 nm diameter are shown in gure 4.14 and gure 4.15).

In this case a PtSi region (see gure 4.14) as well as a Pt 2 Si region through a tetragonal <1 1 0> zone axis (see gure 4.15) can be identied by HRTEM and STEM/EDX. The combined analysis has allowed us to obtain the dimensions of the regions associated to each dierent phase.

A summary of the results obtained on nanowires of dierent diameter is reported in gure 4.16. It can be observed that in larger nanowires, PtSi regions are larger than Pt 2 Si regions while the opposite situation is found in smaller nanowires. Moreover, the Pt 2 Si phase is always found in the region closer to the nanowire tip. This results can be interpreted assuming that Pt diuses suciently fast in large nanowires to allow the formation of a uniform PtSi phase in the whole silicide region. In contrast, if Pt diusion is reduced in smaller nanowires the diusion is slower, only the region close to the interface with the Si substrate can go through the entire reaction sequence and form the PtSi phase, while the region close to the nanowire the tip (where the initial Pt reservoir is located) remains richer in Pt, resulting in the formation of the intermediate Pt-rich P t 2 Si phase. The reason beyond this variation is currently unclear and will be the object of further studies.

Conclusion

The main objective of this thesis was to develop an improved Cli-Lorimer method for the quantication of EDX elemental maps in semiconductor materials, with the specic aim to allow reliable measurements in the case of low concentrations of dopant impurities (below 1 at. %).

For that, dierent electron microscopy techniques have been presented in the rst chapter including their main peculiarities and advantages. In addition, the STEM/EDX technique and the related quantication methods have been discussed in detail. Finally, the quantication procedures available in commercial softwares, and specically the Xray absorption correction and the calibration of the C-L factors have been presented, allowing us to identify their main weaknesses.

Our improved model was then presented in the second chapter. The study started with the iterative calculation of the X-ray absorption in the specimen, which depends on the density, the mass absorption coecient and the specimen thickness. In particular, we showed that there is no universal formula for the expression of the density as a function of the specimen composition. For this reason, we developed an appropriate approximation of the relation between the density and the specimen composition for each investigated materials system. In particular, the cases of amorphous solids and of pseudomorphic alloys were studied. We then detailed the procedure we used for the experimental calibration of C-L factors by comparison with absolute concentration measurements obtained by the RBS technique. Finally, in order to know the precision of our method, the last step was estimation of the total experimental error on the measured specimen concentration using the error propagation procedure.

The new algorithm, with all its modications compared to the existing commercial softwares, has been tested in the third chapter, rst on reference specimens and then applied to samples with unknown composition. A Silicon substrate doped with As by ion implantation has been used as a reference for the quantication of As dopant concentration (based on the amorphous solid approximation model), while a SiGe/Si stack was used for the quantication of Germanium (based on the pseudomorphic alloy approximation). The Gaussian shape of the Arsenic concentration prole allowed us to demonstrate that the absorption eect is more important on the high concentration region due to the high mass density of the specimen in this region. The calibration of the C-L factors was then performed by comparison with the RBS measurements carried out on the reference samples. The results show that the calibration on the K peak is 99 100 CHAPTER 4. ADDITIONAL RESEARCH ACTIVITY more accurate than on the L peak. This is due to the L peaks being located in the low energy region of the X-ray spectrum where the intensity of the bremsstrahlung noise is comparable (or even stronger) than the characteristic peak to be analysed. The noise subtraction is therefore more dicult to achieve. Finally, we estimated the total experimental error aecting the EDX measurements. As expected, we found that this error is less important in the high As and Ge concentration regions, where the x-ray count is higher.

In the case of As doping, once the algorithm has been conrmed, we applied the calculation to a 3D FinFET test structure. We could demonstrate the validity of this technique for nanometric scale specimens, where the error is low enough to trust the measurement obtained with the new algorithm, down to concentrations below 1 at. % (the actual detection limit was estimated at 7.7x1019 cm-3). Finally, the exposition times needed to improve the detection limit have been calculated using the measured count rates. The results clearly suggest that, due to instabilities of the experimental setup, the resolution limit cannot be drastically improved by just increasing the measuring time.

In the fourth chapter, I reported the results obtained within some additional research projects carried out at the LPCNO and the LAAS laboratories. In these activities, X-ray quantication techniques (in SEM or STEM mode) are needed to support other TEM-based structural characterisations of nanoelectronics materials of interest in future nanometric MOS or IoT devices.

The rst project focused on the fabrication of supercapacitors obtained by laser writing. The EDX measurements in particular the distribution mapping of the dierent elements in the layer allowed us to identify the phase and the structure of the specimen. That makes the EDX technique one of the important elements to understand the originality of the laser-writing fabrication method.

The second project concerned the fabrication of GAA (gate-all-around) nanowirebased transistors, where the fabrication process includes the formation of low resistivity PtSi silicide contacts obtained by Pt deposition and subsequent annealing. Our results, obtained by HREM and FFT method, evidenced an impact of the nanowire diameter (and the annealing temperature) on the silicide formation mechanism. In particular, for high temperature annealing and large nanowire diameters, the expected PtSi crystallographic structure was found. In contrast, when the nanowire diameter (or the annealing temperature) are decreased, a Pt2Si structure is obtained on the top surface of the nanowire, while the low resistance PtSi phase is obtained at the interface with the Si substrate. These results were further conrmed by EDX analysis carried out on the same structures.

Finally, the following issues should be addressed in future work to achieve further improvements of our quantication method:

In our model, we considered that the detector size is very small and far from the specimen so that all the X-rays emitted from the specimen are collected in a single point. In fact, a more rigorous description of the X-ray collection mechanism angles and hence a modication of the total intensity of the collected X-rays.

We showed that the calibration of the C-L factors based on the K lines is more accurate than the one based on L lines, because of the strong impact of the Bremsstrahlung radiation in the low energy region of the X-ray spectrum. Further eorts should therefore be devoted to the improvement of the Bremsstrahlung subtraction procedure in the low energy region of the spectrum.

Finally, it would be useful to demonstrate the validity of our quantication method by using the zeta factor technique. For this, the microscope must be equipped by in-situ current measurement device [START_REF] Nicholson | Electron beam current measurement in the electron microscope[END_REF]. Dans ce chapitre, nous décrirons le problème de la correction d'absorption et justierons le besoin d'une méthode itérative. Ensuite, nous détaillerons la procédure utilisée pour la calibration des facteurs C-L et pour l'estimation de l'erreur de la mesure nale.

Fractions massiques et calcul de la densité

Comme déjà discuté (voir la section 1.3.1 et l'équation 1.5), l'approche C-L est basée sur une relation de proportionnalité entre les intensités des pics et les fractions massiques des éléments. La constante de proportionnalité K i,j:F i ,F J est le facteur de Cli-Lorimer par rapport au i-ème et au j-ième élément. Elle dépend des familles de pics F i et F j qui ont été choisies pour chaque élément et, dans la grande majorité des logiciels commerciaux, est extraite d'une base de données de valeurs simulées. On peut montrer que les coecients C-L peuvent également être écrits de la forme suivante : où k i:F i ,F i 0 est le coecient C-L relatif à l'élément i et à un élément de référence commun i 0 . Il convient de noter que, par dénition, l'élément de référence a un coecient C-L unitaire (k i 0 :F i 0 ,F i 0 = 1) et F i 0 indique la famille de pics choisie pour l'élément de référence. Avec cette convention, on peut montrer qu'il existe une formule directe reliant les intensités mesurées et les fractions de masse pour un matériau composé de N A atomes diérentes :

K i,j:F i ,F J = k i:F i ,F i 0 k j:F j ,F i 0 (2.
f M i = k i:F i ,F i 0 I i:F i N A j=1 k j:F j ,F i 0 I j:F j (2.2)
La densité totale I i:F i de la familleF i pour un élément spécique est obtenue en additionnant les intensités des raies de N F i (K α, Kβ...) appartenant à la famille choisie pour l'élément i.

I i:F i = N F i k=1 I i:F i ,k (2.3)
Suivant cette notation, chaque pic caractéristique a une intensité I i:F i ,k et est associé à l'énergieE i:F i ,k où i identie l'élément F i de la famille des pics et k est l'indice identiant la raie.

Il faut aussi noter que les intensités à utiliser dans la formule correspondent à l'intégrale des pics caractéristiques, à laquelle il faut ajouter l'intensité des rayons X qui sont décalés dans d'autres régions du spectre à cause des artefacts de mesure qui doivent être pris en compte pour l'application de cette formule. De plus, pour une interprétation correcte des résultats, les eets d'absorption dans l'échantillon doivent être pris en compte car ils sont négligés dans le modèle C-L. Enn, il convient de noter que les coecients de C-L sont normalement calculés à l'aide de modèles simpliés et qu'ils doivent être étalonnés si on souhaite connaître la précision des résultats. Ils doivent donc être considérés comme des paramètres empiriques, qui sont aussi aectés par une erreur de mesure. Même s'il est facile de calculer les fractions massiques à partir de mesures EDX, il peut être utile (c'est le cas dans la plupart de la littérature sur les dopants) d'utiliser des concentrations atomiques (en atomes/cm 3 ) au lieu des fractions massiques. La conversion des fractions massiques en concentrations atomiques c A i peut être obtenue à l'aide de la formule suivante :

c A i = ρ M f M i m i N AV (2.4)
où ρ M est la masse volumique de l'élément i, m i sa masse atomique et N AV = 6, 02214076 • 10 23 mol -1 est le nombre d'Avogadro.

La concentration des éléments dans un matériaux peuvent aussi être indiquées en utilisant les fractions atomiques. La formule chimique d'un alliage binaire composé par les 2.2. CORRECTION DE L'ABSORPTION 107 éléments A et B, par exemple, est normalement exprimée comme A 1-x B x où x est la fraction atomique de l'élément B dans l'alliage. La fraction atomique relative à une série d'éléments (f A i ) peut être calculée à partir des fractions massiques correspondantes à l'aide de la formule suivante :

f A i = f M i /m i N AT j=1 f M j /m j (2.5)

Correction de l'absorption

Pour quantier un spectre EDX, il est nécessaire de mesurer les intensités des différents pics. Mais puisque les rayons X traversent l'échantillon avant d'atteindre le détecteur, l'intensité mesurée des rayons X I * peut être diérente de l'intensité émise I.

Cet eet doit être pris en compte pour obtenir des résultats ables. La formule suivante permet de calculer l'intensité mesurée I * (E) des rayons X de l'énergie E traversant un matériau homogène d'épaisseur z et de masse volumique ρ M :([3] p.654) :

I * (E) = I (E) e -µ(E)ρ M z (2.6)
I * (E) = t 0 I (E) t e -µ(E)ρ M z dz = 1 -e -µ(E)ρ M t µρ M t I (E) (2.7)
On peut donc dénir un coecient de transmission liant l'intensité des rayons émis I (E) à la densité mesurée I * (E) :

T (E, f M i ) = 1 -e -µ(E)ρ M t/ sin(α) µρ M t/ sin (α) (2.8)
Le coecient de transmission dépend de l'énergie des rayons X et des fractions de

µ E, f M i = N AT i=1 µ i (E) f M i (2.
µ(E) = N AT i=1 n i σ i N AT j=1 n j m j = N AT i=1 σ i m i n i m i N AT j=1 n j m j (2.11)
Cette équation est équivalente à l'équation 2.9 puisque µ i = σ i /m i et par dénitionf M i = n i m i / N AT j=1 n j m j . La valeur de µ i (E) pour les diérents éléments peut être obtenue en utilisant l'un des diérents modèles disponibles (voir, par exemple, [START_REF] Chantler | Atomic form factors and photoelectric absorption cross-sections near absorption edges in the soft X-ray region[END_REF] et [START_REF] Heinrich | Mass absorption coecients for electron probe microanalysis[END_REF]). Dans notre cas, elles ont été calculées en utilisant la bibliothèque EPQ qui implémente le modèle proposé par Heinrich [START_REF] Heinrich | Mass absorption coecients for electron probe microanalysis[END_REF]. L'accès direct à la bibliothèque EPQ a été eectuée avec Octave en utilisant ses fonctionnalités d'interfaçage avec le code java. Pour chaque pic caractéristique E i:F i ,k , les intensités I i:F i ,k peuvent être calculées à partir des intensités mesurées I * i:F i ,k en utilisant l'équation 2.8. T qui dépend également de f M i (x) (voir l'équation 2.8). Pour la procédure itérative, nous utilisons comme point de départ les fractions de masse f M i (x) 0 calculées en négligeant l'absorption (c'est-à-dire en supposant que I * i:F i ,k (x) = I i:F i ,k (x)) pour tous les pics caractéristiques. A chaque itération, un nouvel ensemble de fractions massiques est obtenu en calculant le facteur de transmission à l'aide des fractions massiques obtenues lors de l'itération précédente :

[T i:F i ,k (x)] n = T (x, E i:F i ,k , f M i (x) n (2.12) = 1 -e - N A j=1 µ i( E j:F j ,k )[f M j (x)] n-1 ρ M (x, [f M i (x)] n-1 t/ sin(α) N A j=1 µ j E j:F j ,k f M j ρ M (x, [f M i (x)] n-1 t/ sin (α)
Les nouvelles valeurs des intensités absorbées et des fractions de masse corrigées sont évaluées comme suit :

[I i:F i ,k ] n = I * i:F i ,k [T i:F i ,k (x)] n (2.13)
puisque :

[I i:F i ] n = N F i k=1 [I i:F i ,k ] n (2.14)
on peut écrire :

f A i n = k i N F k=1 [I i:F i ,k ] n (
(f M i ) n -(f M i ) n-1 (f M i ) n < ).
Comme indiqué plus haut, pour appliquer la procédure itérative, il est nécessaire d'établir, pour chaque matériau, la relation appropriée entre la masse volumique locale de l'échantillon ρ M (x) et les fractions massiques des éléments locaux f M A (x) et f M B (x).

Ce calcul dépend de la structure de l'échantillon et varie selon les cas (voir section 2.3).

Calcul de la densité massique

Pour appliquer la procédure itérative, il est nécessaire, pour chaque matériau, d'établir la relation appropriée entre la densité de masse locale du spécimen ρ M (x) et les fractions de masse locales de silicium et d'arsenic f M i (x). Dans le cas idéal on peut considérer que dans un cristal dopé, des petites fractions d'atomes du substrat (marquées par 1) sont remplacées par des atomes dopants (marqué QUANTIFICATION par 2). Cela revient à dire que tous les atomes dopants sont en substitution. Comme les atomes de dopage sont en faible concentration, la masse volumique peut être facilement calculée à partir de la masse volumique du substrat ρ M 1 (x) en évaluant la diérence de masse obtenue lorsque un atome de dopant remplace un atome de substrat.

ρ M (x) = ρ M 1 f A 1 (x) + m 2 m 1 f A 2 (x) = ρ M 1 1 + m 2 m 1 -1 f A 2 (x)
(2.16) où f A i (x) sont les fractions atomiques des deux éléments, liées, dans un alliage binaire, par l'équation suivante f

A 1 (x) = 1 -f A 2 (x).
Les fractions atomiques peuvent être calculées à partir des fractions massique en utilisant la formule 2.5 :

Dans le cas de deux éléments, cette formule se simplie et on obtient :

f A 1 (x) = 1 -f A 2 (x)
(2.17)

f A 2 (x) = f M 2 (x) f M 2 (x) + m 2 m 1 f M 1 = f M 2 (x) f M 2 (x) + m 2 m 1 1 -f M 2 (x)
L'équation2.16 peut donc être ré-écrite :

ρ M (x) = ρ M 1 1 + (m 1 -m 2 ) f M 2 (x) m 1 -(m 1 -m 2 )f M 2 (x) (2.18)
C'est la formule recherchée car elle relie la masse volumique aux fractions massiques.

Puisque dans cette approximation, on suppose que l'atome dopant ne provoque aucune déformation, nous l'appellerons approximation du rayon équivalent car elle est équivalente à considérer que les atomes du dopage et les atomes du substrat ont exactement le même rayon.

Approximation des solides amorphes

Une meilleure approximation peut être obtenue en utilisant une formule qui a été originalement dérivée pour les solides qui ne présentent aucun type d'ordre (également connu sous le nom de solides amorphes idéaux ou Ideal Amorphous Solids (IAS)) [START_REF] Stachurski | Fundamentals of Amorphous Solids: Structure and Properties[END_REF] :

ρ M = N AT i=1 ρ M i f V i (2.
f V i = f M i ρ M i N AT j=1 f M j ρ M j (2.20)
En insérant l'équation 2.20 dans l'équation 2.19 et en considérant que les fractions massiques sont normalisées à 1 ( N AT j=1 f M j =1) on obtient l'expression de la densité de masse d'un solide amorphe idéal : Si on associe l'indice 1 à l'élément A, l'indice 2 à l'élément B et on indique avec x la fraction atomique f A 2 de B on peut écrire la formule chimique de l'alliage dans la forme A 1-x B x . Dans un alliage pseudo-morphique, les deux réseaux ont des paramètres de maille très proches (la diérence de paramétre de maille entre Si et Ge est de 0,227 Angström) et l'alliage binaire conserve la même structure des éléments purs (le SiGe a une structure de type diamant). Le paramètre de maille de l'alliage suit la loi de Vegard [START_REF] Vegard | Die Konstitution der Mischkristalle und die Raumfüllung der Atome[END_REF] :

ρ M = 1 N AT j=1 f M j ρ M j ( 2
a A 1-x Bx = a 1 (1 -x) + a 2 x (2.22)
où les coecients a i sont les paramètres de réseau des cristaux formés par un élément pur.

QUANTIFICATION

En utilisant la loi de Vegard, la masse volumique d'un alliage A 1-x B x peut être calculée en considérant que dans une maille élémentaire , il y a n atomes, dont n (1-x) de type A et nx de type B :

ρ M = n m 1 (1 -x) + m 2 x [a 1 (1 -x) + a 2 x]
1 et V A
2 sont pratiquement identiques. La variation du volume ato- mique avec la composition de l'alliage est donc négligeable car dans le cas de l'alliage le volume atomique est compris entre V A 1 et V A 2 qui représentent le deux cas extrêmes. Par conséquent, on peut supposer que les volumes atomiques des deux atomes ne varient pas avec x et que l'approximation du solide amorphe peut être appliquée.

Ceci peut être démontré de manière rigoureuse en comparant le résultat exact (équation 2.23) avec le résultat obtenu avec l'approximation du solide amorphe. Dans ce cas, le volume occupé par n(1 -x) atomes de type A est

V 1 = n(1 -x)V A 1 , le volume occupé par nx atomes de type B est V 2 = nxV A 2 et le volume total est V = V 1 + V 2 .
ρ M = ρ M 1 n(1 -x)V A 1 + ρ M 2 nxV A 2 n(1 -x)V A 1 + nxV A 2 (2.24) = nm 1 a 3 1 n(1 -x) a 3 1 n + nm 2 a 3 2 nx a 3 2 n n(1 -x) a 3 1 n + nx a 3 2 n =n m 1 (1 -x) + m 2 x a 3 1 + x(a 3 2 -a 3
[a 1 (1 -x) + a 2 x] 3 = (2.25) = a 3 1 [1 + x] 3 = a 3 1 1 + 3 x + 3 ( x) 2 + ( x) 3 = a 3 1 1 + 3 x + O 2
alors que pour le dénominateur de l'équation 2.24, on peut montrer que :

a 3 1 + x(a 3 2 -a 3 1 ) = (2.26) = a 3 1 + x a 3 1 (1 + ε) 3 -a 3 1 = a 3 1 1 + 3εx + 3ε 2 x + ε 3 x = a 3 1 1 + 3εx + O 2
On peut noter que les deux expressions sont égales jusqu'au premier ordre en et que l'approximation du solide amorphe est en eet une bonne approximation pour les alliages pseudo-morphiques. L'équation 2.21 peut donc être utilisée pour calculer la densité massique à partir des fractions massiques. k i:F i ,F j I i:F i (x)

I i 0 :F i 0 (x) = f M i (x) M eas f M i 0 (x) M eas
f M i (x) M eas f M i 0 (x) M eas -k i:F i ,F i 0 I i:F i (x k ) I i 0 :F i 0 (x k ) 2 (2.28)
où {x k } est l'ensemble des positions où les densités ont été mesurées expérimentalement. An de privilégier les points les plus précis, on a choisi comme facteur de pondération w(x i ) l'erreur relative associée

I As:F As (x i ) I Si:F Si (x i ) .
En utilisant les lois de propagation des erreurs, nous pouvons alors écrire w(x k ) =

N AT i=1 1
I i:F l (x i )
.

La valeur de k As:F As et l'erreur statistique associée ∆k As:F As ont été calculées à l'aide de la formule standard pour l'estimation de l'erreur dans un régression pondérée aux moindres carrés ( [START_REF] Wasserman | All of Nonparametric Statistics[END_REF], [START_REF] Taylor | Introduction To Error Analysis: The Study of Uncertainties in Physical Measurements[END_REF]).

Lorsque vous eectuez une mesure quantitative, il est bien sûr important de pouvoir évaluer l'erreur de mesure totale. Dans le cas d'une mesure de concentration eectuée par EDX en utilisant la valeur calibrée des coecients C-L, trois types d'erreur doivent être pris en compte :

1. L'erreur statistique sur les intensités mesurées ∆I(x) = I(x). La contribution de cette erreur à la fraction de masse peut être calculée, à l'aide des techniques de propagation de l'erreur [START_REF] Taylor | Introduction To Error Analysis: The Study of Uncertainties in Physical Measurements[END_REF] :

∆ I f M i (x) = k i:F i ,F i0 N AT j=1 k j:F j ,F i0 I j:F j (x) 2 ×
(2.29) j =i k j:F j ,F i0 I j:F j (x)∆I i:F i + I i:F i (x)∆I j:F j (x) où i =j indique la somme sur tous les atomes sauf l'atome i.

∆ k f M i = I i:F i ,F i0 N AT j=1 k j:F j ,F i0 I j:F j (x) 2 ×
(2.30)

j =i I j:F j (x) k j:F j ,F i0 ∆k i:F i ,F i0 + k i:F i ,F i0 ∆k j:F j ,F i0
∆ t f M i (x) = [f M i (x)] t+∆t -[f M i (x)] t-∆t 2 où f M i (x)
∆f M i (x) = [∆ I f M i (x) + ∆ k f M i (x)] 2 + [∆ t f M i (x)] 2 (2.31)
Comme on l'a déjà noté, même si les fractions de masse sont plus faciles à calculer à partir de mesures EDX, il est parfois préférable d'utiliser la concentration atomique.

Pour évaluer l'erreur sur les concentrations atomiques c A i (x), l'équation 2.4 peut être utilisée. Le résultat obtenu est :

∆c A i (x) = N AV m i N AT j=1 ∂ ∆ρ M ∂f M j ∆f M j + ρ M ∆f M i (2.32)
La formule précise dépend de la formule utilisée pour ρ M (voir section 2.3).

Pour l'approximation du rayon constant, la formule suivante est obtenue pour l'erreur sur la concentration atomique du dopant :

∆c A 2 (x) = N AV m j ρ M 1 1 + (m 1 -m 2 ) f M 2 (x) m 1 -(m 1 -m 2 )f M 2 (x) × (2.33) × 1 + m 2 (m 2 -m 1 ) [m 2 -(m 2 -m 1 )f M 2 (x)] 2 f M 2 (x) ∆f M 2 (x)
Alors que pour l'approximation du solide amorphe, la formule suivante est obtenue :

QUANTIFICATION ∆c A i (x) = N AV m As 1 N AT k=1 f M k ρ M k    ∆f M i (x) + 1 N AT k=1 f M k ρ M k N AT j=1 ∆f M j (x) ρ M j    (2.34)
Comme indiqué précédemment pour les alliages binaires on utilise souvent les fractions atomiques au lieu des fractions massiques. Dans ce cas, l'erreur sur les fractions massiques peut être calculée en appliquant la formule de propagation d'erreur sur l'équation 2.5. L'expression obtenue pour l'erreur sur les fractions atomiques est :

∆f A i = 1/m i N AT j=1 f M j /m j 2 × (2.35) j =i 1/m j f M j ∆f M i + f M i ∆f M j Chapitre 3
Cartographie EDX quantitative dans les semi-conducteurs Mais dans la pratique, cette approximation surestime la densité de masse de SiAs amorphe (a-SiAs) pour des concentrations élevées en arsenic par rapport à la valeur de l'arsenic amorphe indiqué dans la littérature (ρ M a-As = 4.7 g cm 3 [48]) comme le montre la gure 3.5.

Il pourrait alors être tentant d'utiliser l'approximation du solide amorphe (l'équation 2.19) dans ce cas. Mais, comme déjà discuté dans la section 2. As de l'arsenic amorphe [START_REF] Greaves | Amorphous arsenic[END_REF] est aussi indiquée dans le graphique.

2.32g/cm 3 . Cette valeur de densité théorique ne correspond pas à la valeur la densité de silicium amorphe déterminée expérimentalement ρ M a-Si = 2.285 g/cm 3 [49] qui est légèrement inférieure à ρ M c-Si . Le même type de raisonnement peut être utilisé pour prouver que la structure de l'amorphe As (ρ M a-As = 4.7 g/cm Il est intéressant, dans le contexte du prol de dopage de STEM / EDX, d'essayer de comprendre quelle est la limite de détection de la concentration en arsenic dans un nano-dispositif. Pour estimer la plus faible concentration en arsenic mesurable, nous avons utilisé une approche simple basée sur les résultats expérimentaux. Il est facile de comprendre qu'une mesure de densité ne peut pas être distinguée de zéro si l'intervalle de conance (c'est-à-dire l'intervalle qui se situe entre c -∆c et c + ∆c) inclut le zéro.

f M As (x) M eas = m As c A As/RBS (x) N AV ρ M a-Si + m As c A As/RBS (x) 1 - ρ M a-Si
Le résultat obtenu dépend évidemment de intervalle de conance choisi [START_REF] Taylor | Introduction To Error Analysis: The Study of Uncertainties in Physical Measurements[END_REF].

L'analyse des prols montrés sur la gure 3.15 montre clairement que la concentration de dopage est toujours supérieure à la limite de détection. Pour obtenir la limite de détection on s'est donc servi des résultats obtenus sur les échantillons de calibration (voir gure3.13 (b)) qui nous ont permis de mesurer une limite de détection de 7.7 10 19 cm -3 .

Bien évidemment, la valeur obtenue ne représente pas une estimation absolue, car la valeur de la limite de détection dépend de l'intensité des pics (voir l'équation 2.29 En partant de l'équation 2.2, en utilisant l'équation2.31 et 2.34 et en considérant que l'erreur sur l'épaisseur est négligeable et que le pic As a une intensité négligeable par rapport au pic Si nous nous pouvons obtenir la formule suivante : As of amorphous arsenic [48] is shown in the plot. . . . . . . . . . . . . .

I Si:F Si = k As:F As ,F Si * f M As (1-f M As ) f M As (x) + 1       ∆ c A As m As N AV      ρ M a-SiAs + f M As (x) ρ M a-Si η [ 1-η f M As ] 2      f M As (1-f M As )      

3.6

Intensity maps obtained on the calibration specimen with a 2% atomic density peak.

14 CHAPTER 1 .

 141 ENERGY DISPERSIVE X-RAY SPECTROSCOPY

Figure 1 . 1 :

 11 Figure 1.1: Schematic drawing of electron-matter interaction in bulk (a) and thin (b) specimens.

Figure 1 . 2 :

 12 Figure 1.2: Schematic drawings of the main elements of a transmission electron microscope.

Figure 1 . 3 :

 13 Figure 1.3: Schematic drawing illustrating the TEM and STEM modes.

Figure 1 . 4 :

 14 Figure 1.4: Diagram explaining the three physical phenomena resulting from the interaction of the electron beam with the sample: characteristic X-rays, Bremsstrahlung radiation and electron backscattering.

Figure 1 . 5 :

 15 Figure 1.5: Schematic drawing illustrating the origin of Bremsstrahlung radiation (a) and Fluorescence (b).

Figure 1 . 6 :

 16 Figure 1.6: Diagram showing the structure and operation of a simple p-i-n EDX detector.

Figure 1 . 7 :

 17 Figure 1.7: Diagram representing the three intermediate mechanisms for carrier ionisation in the detector (a) and the emission process of an Auger electron (b).

Figure 1 . 8 :

 18 Figure 1.8: Diagram representing the detection artefacts.

1. 2 . 29 Figure 1 . 9 :

 22919 Figure 1.9: Schematic drawing of an SDD detectors, taken from [18].The image is available under the license Creative Commons Attribution International 4.0 (https://creativecommons.org/licenses/by/4.0/).

8 )

 8 And since C A + C B = 1 then C A ,C B and ρt can be expressed as the following:

Figure 1 . 10 :

 110 Figure 1.10: EDX spectrum obtained on a SiGe sample using the 2100F microscope installed at UMS Castaing.

Figure 1 . 11 :

 111 Figure 1.11: Diagram showing the two Bremsstrahlung elimination models: (a), Mathematical and (b), Physical TEM.

Figure 1 . 12 :

 112 Figure 1.12: Example of subtraction of background noise (a) and deconvolution (b).

1. 4 .

 4 EDX AND THE OTHER QUANTIFICATION TECHNIQUES

Figure 2 . 1 :

 21 Figure 2.1: Schematic diagram of the absorption geometry in a homogeneous material.

Figure 2 . 2 :

 22 Figure 2.2: Schematic illustration of the detector position and of X-ray path, when the detector is placed on a plane perpendicular to the variation direction of the doping prole.

2. 4 .

 4 MODEL CALIBRATION AND ERROR ESTIMATION 49

3. 1

 1 Dopant proling in Si nano-devices Dopant proling in modern semiconductor devices is a key issue. The problem arises from the development of transistors based on new substrates (such as silicon on insulator) or based on new 3D architectures (3D transistors on substrate, nano-wires ...).

56 CHAPTER 3 .

 563 QUANTITATIVE EDX MAPPING IN SEMICONDUCTORS in a wafer composed by a thin crystalline Si layer of uniform thickness on top of an oxide layer. The top layer can be used to create horizontal nano-rods, called ns, using standard lithography techniques (see gure 3.1(a)). A MOSFET transistor, called aFinFET, can then be build using the obtained ns. Several types of FinFETs exist, depending on the specic design of the several oxide and metal layers composing the transistor. However, the general idea behind all possible geometries is to obtain an enhanced control of the transistor by improving the gate geometry. For this, small ns are normally used (some tens of nanometers), and the gate has a 3D structure composed by several walls. A schematic drawing of a FinFET with a three walls gate is shown in gure 3.1(b).

Figure 3 . 1 :

 31 Figure 3.1: Schematic illustrations of a SOI substrate with ns (a) and a FinFet (b).

3. 1 . 57 3. 1 . 1

 15711 DOPANT PROFILING IN SI NANO-DEVICES Description of the samplesDopant distribution A FinFet test structure with a channel width of 80 nm a height of 100 nm having received a high dose implant (see Figure3.2(b)). The test structure did not present any gate oxide or metallic contacts since we wanted the observation to be focused on dopant distribution. For the creation of the doped region an arsenic plasma implant has been used to dope the specimen in order to obtain a conformal distribution of the doping prole.To be able to measure arsenic doping concentration in silicon nanodevices, we have designed some calibration specimen obtained by ionic implantation of arsenic in monocrystalline silicon wafers. In order to check the precision of the calibration on dierent concentrations, two specimens have been used. Both were implanted using an acceleration voltage of 21 keV but with a dierent dose amounting, respectively, to 9.4x 10 15 cm -2 and 1.8x10 15 cm -2 . It has to be noted that due to the high dose, the implanted wafers are amorphized in the implantation region (see gure 3.3). The implantation parameters have been chosen by simulating the implantation prole using the SRIM (Stopping and Range of Ions in Matter ) Monte Carlo simulation software ([46] ch.3). The simulation results corresponding to the two dierent doses are shown in 3.2(a). It can be noted that the value of the concentration peak amounts to 10 21 cm -3 (equivalent to 10% atomic As concentration) for the higher dose implant and to 5x10 21 cm -3 (2% atomic As concentration) for the lower dose implant.

Figure 3 . 2 :

 32 Figure 3.2: (a) Implanted doping distribution simulated with SRIM for As implant in silicon with an energy of 21keV; (b)TEM image of the cross section of the FinFET sample.

Figure 3 . 3 :

 33 Figure 3.3: RBS measurement of As atomic density superimposed to the TEM image obtained in the same specimen. The contrast variation in the TEM images indicate that the high concentration implanted region is amorphized.

3. 1 . 59 Figure 3 . 4 :

 15934 Figure 3.4: Schematic illustration of the inuence of the specimen geometry on EDX artifacts (secondary X-rays created by uorescence and backscattered electrons) in specimens prepared using: a) FIB. b) mechanical thinning.

60 CHAPTER 3 .

 603 QUANTITATIVE EDX MAPPING IN SEMICONDUCTORS greatly dier m Si = 28.0855 g mol and m As = 74.9216 g mol . Hence the constant radius approximation could in principle be a good approximation and 2.16 could be used for density calculation.But in practice this approximation overestimates the mass density of amorphous SiAs (a-SiAs) for high arsenic concentrations comparing to the value of the amorphous arsenic found in the litterature (ρ M a-As = 4.7 g cm 3[48]) as shown in gure 3.5.

Figure 3 . 5 :

 35 Figure 3.5: Mass density a function of the As mass fraction calculated with two dierent approximations. The experimentally determined value of the mass density ρ M

3. 1 .

 1 DOPANT PROFILING IN SI NANO-DEVICES 61 elements. In this case the same reasoning applied in section 2.3.3 to the pseudomorphic alloys can be applied in the case of a-SiAs.

πr 3 )

 3 and the covalent radius r. The values of s = 4 3 πr 3 ρ A for amorphous Si and amorphous As can be calculated by using the previously dened data for the density, the radius and the mass of the two elements. The obtained values are quite close (s Si = 0.28 ,s As = 0.26 ) conrming that the packing of the two structure is almost the same.

62 CHAPTER 3 .

 623 QUANTITATIVE EDX MAPPING IN SEMICONDUCTORS

Figure 3 . 6 :

 36 Figure 3.6: Intensity maps obtained on the calibration specimen with a 2% atomic density peak. In inset (a) the BF-STEM image of the mapped region is shown together with the geometry of the linescan used for the extraction of a prole. The intensity maps after peak integration and bremsstrahlung subtractions are show in the other insets for the following peak families: SiK(b), AsL(c), Pt(M) (d).

Figure 3 . 7 :

 37 Figure 3.7: Intensity maps obtained on the calibration specimen with a 10% atomic density peak. In inset (a) the BF-STEM image of the mapped region is shown together with the geometry of the line scan used for the extraction of a prole. The intensity maps after peak integration and bremsstrahlung subtractions are show in the other insets for the following peak families: SiK(b), AsL(c), Pt(M) (d).

Figure 3 . 8 :

 38 Figure 3.8: EDX spectrum obtained from the 10% specimen and showing the position of the Si and As peak families.

Figure 3 . 9 :

 39 Figure 3.9: The result of the correction absorption for the rst 10 iterations. In (a) the full prole is shown while in (b) a zoom on the high density region is displayed.

Figure 3 . 10 :

 310 Figure 3.10: Plot of the measures As concentration c A As (x) before (a) and after (b) calibration of the CL coecients. The values obtained by using the I As:K (AsK) and I As:L peaks are reported and compared with the values obtained by RBS.

Figure 3 . 11 :

 311 Figure 3.11: Variation of the absorption corrected peak density in the 2% (a) and 10% (b) specimen. The concentration has been obtained by using the AsK and SiK families.

Figure 3 . 12 :

 312 Figure 3.12: Plot of the measures As concentration c A As (x) of the 10% specimen using (a) K peaks for Si and As and (b) K peak for Si and L peaks for As. In both cases the obtained values are compared with the values obtained by RBS.

Figure 3 . 13 :

 313 Figure 3.13: Plot of the measures As concentration c A As (x) of the 10% specimen using (a) K peaks for Si and As and (b) K peak for Si and L peaks for As. In both cases the obtained values are compared with the values obtained by RBS.

3. 1 .

 1 DOPANT PROFILING IN SI NANO-DEVICES 69 measurements reported in the following.

Figure 3 . 14 :

 314 Figure 3.14: Bright eld STEM images(a) and EDX intensity maps (b-c) obtained on a FinFET device. The K peaks have been used both for oxygen and arsenic to obtain the elemental maps shown in (b) and (c) from the EDX hyper-map. Intensities proles obtained using a laterally averaged line-scan are superimposed to maps ((b) and (c)). The direction and position for the line-scan as well as the lateral integration width are shown in gure (a), (b) and (c).

Figure 3 .

 3 Figure 3.15: (a) Schematic image illustrating the geometry used to obtain the doping prole and (b) doping prole with experimental error obtained by applying the quantication method described in chapter 2.

Figure 3 . 16 :

 316 Figure 3.16: Cross-sectional TEM micrographs showing the calibrated specimens structure.

Figure 3 . 17 :

 317 Figure 3.17: Intensity maps obtained on the calibration specimen with a 28 nm of SiGe layer. In inset (a) the STEM of the mapped region is shown together with the geometry of the line scan used for the extraction of a prole. The intensity maps after peak integration and bremsstrahlung subtractions are show in the other insets for the following peak families: SiK(b), GeK(c), CK (d).

Figure 3 . 18 :

 318 Figure 3.18: Intensity maps obtained on the calibration specimen with a 48 nm of SiGe layer. In inset (a) the STEM of the mapped region is shown together with the geometry of the line scan used for the extraction of a prole. The intensity maps after peak integration and bremsstrahlung subtractions are show in the other insets for the following peak families: SiK(b), GeK(c), CK (d).

3. 2 . 77 Figure 3 . 19 :

 277319 Figure 3.19: Plot of the measures Ge atomic fractionf A Ge (x) before (a) and after (b) calibration of the CL coecients for the 28 nm specimen. The values obtained by using the I Ge:K (AsK) and I Ge:L peaks are reported.

Figure 3 . 20 :

 320 Figure 3.20: Plot of the measures Ge atomic fractionf A Ge (x) before (a) and after (b) calibration of the CL coecients for the 48 nm specimen. The values obtained by using the I Ge:K (AsK) and I Ge:L peaks are reported.

Figure 3 . 21 :

 321 Figure 3.21: The result of the correction absorption for the rst 10 iterations of the two calibration specimens.

Figure 3 . 22 :

 322 Figure 3.22: Plot of the measures Ge atomic fraction f A As (x)using K peaks for Si and Ge in the 48 nm (a) and 28 nm (b) specimen.

Figure 3 . 23 :

 323 Figure 3.23: Bright eld STEM images(a) and EDX intensity maps (b-c-d) obtained on a nanodevice. The K peaks have been used both for germanium and carbon to obtain the elemental maps shown in (c) and (d) from the EDX hyper-map. Intensities proles obtained using a laterally averaged line-scan are superimposed to maps ((b), (c) and (d)). The direction and position for the line-scan as well as the lateral integration width are shown in gure (a), (b) and (c).

Figure 3 . 24 : 4

 3244 Figure 3.24: SiGe prole with experimental error obtained by applying the quantication method described in chapter 2.

4. 1 .

 1 RESULTS OBTAINED IN THE MICRO-SUPERCAPACITORS STUDY 83

Figure 4 .

 4 Figure 4.1 shows the fabrication process of the RuO 2 -based micro-supercapacitors onto a polyimide (PI) exible substrate. An ink containing a mixture of a tetrachloroauric acid ( HAuCl 4 .3H 2 O) gold precursor and cellulose acetate dissolved in tetrahydrofurane (THF) is spin-coated onto the exible substrate (layer 1).In a second step, a

Figure 4 . 1 :

 41 Figure 4.1: Fabrication process of RuO 2 -based exible micro-supercapacitors. HAuCl 4 .3H 2 O / cellulose acetate mixture (1) and RuO 2 (2) are dispersed in THF and successively spin-coated on polyimide. Laser-writing is performed on the as-deposited bilayered lm (3) and the non-exposed areas are removed by washing with ethanol and acetone (4) to reveal the interdigitated electrodes (5).

Figure 4 .Figure 4 . 2 :

 442 Figure 4.2: Structure of the Au/RuO2 growing under laser beam. (A) Tilted and (B) top view of the Au/RuO 2 deposited on KaptonTM and (C) zoom on the RuO 2 crystals. (D) Cross-sectional view of the Kapton TM /laser-scribed Au/RuO2 with (E) tilted SEM image obtained using backscattered electrons and the corresponding EDX map (F) evidencing the presence of Ru at the interface. The large 50°tilt angle used in Figure E and F to highlight the porous layer prevent from using reliable scale bar.

Figure 4 . 3 :

 43 Figure 4.3: Characterization of the RuO 2 -based electrodes. (A) Cross sectional view of the Kapton TM /Ru/RuO2 prepared by Cross Polisher technique. (B) XRD pattern of the laser-scribed Au/RuO2 deposited on PI exible substrate without Ti/Au current collector.

Figure 4 . 4 :

 44 Figure 4.4: Observation of the Kapton TM / RuO 2 interface. TEM image of (A) a popcorn like structure grown (B) on the Ru/ RuO 2 at the interface with the exible substrate. High resolution TEM image of (C) the RuO2/KaptonTM interface; the inset shows the Fourier transform of the image in the dotted rectangle. (D) Morphology of the RuO 2 , with a Fourier transform of the full image as an inset and (E) Ru contained in the pillars (the inset shows the superposition of the hexagonal ruthenium structure in a <0001> orientation and a region of the image). (F) Bright-eld TEM observation of the top of the pillars.

Figure 4 . 5 :

 45 Figure 4.5: EDX analysis of the exible substrate / laser scribed electrode interface of 4.4 Analysis shows that there is a higher O concentration in the upper region compared with the lower region. This is compatible with a Ru (lower) /RuO 2 (upper) stacking.Au is present in the layers in nanoparticle form.

4. 2

 2 Results obtained in the vertical nanowire transistors study 4.2.1 Project motivation and description One of the most limiting factors to follow Moore's law is the short channel eect that aects planar devices and can be reduced by adopting new architectures. In this project a specic architecture capable of reducing this kind of eects has been studied: the NanoWire Field Eect Transistors NWFETs [75]( see gure 4.6).

Figure 4 . 6 :

 46 Figure 4.6: Diagram of a silicon nanowire transistor (NWFET).

4. 2 .Figure 4 . 7 :

 247 Figure 4.7: Diagram of a NWFET and network architecture with grid, insulators and common contacts taken from the "Nanowire Gate All Around Transistors" article of the nextBIGFuture site and inspired by [75].

  Vertical silicided nanowires are manufactured in several stages (see gure 4.8). Firstly, the silicon nanowires are created by Reactive Ion-etching (a), then an anisotropic plat-inum layer of 15nm is deposited by electron beam evaporation (PLASSYS MEB 550S) (b) and nally, the silicidation reaction is thermally activated by rapid annealing under hydrogenated nitrogen N2: H2 (95: 5) (c).

Figure 4 . 8 :

 48 Figure 4.8: Process for producing platinum silicides on a network of vertical silicon nanowires with a diameter of 32 nm and a height of 215 nm. (a) RIE etching, (b) deposition of a platinum layer, (c) after RTA at 350 °C for 3 minutes under H 2 N 2 from [80].

Figure 4 . 9 :

 49 Figure 4.9: Schematic showing dierent thicknesses of the nanowires.

Figure 4 .

 4 Figure 4.10 shows an example of the comparison between the experimental Fourier pattern obtained from a HRTEM image and the theoretical diraction pattern obtained by the software Carine. The image was obtained on a nanowire annealed at 500°C for 3 minutes.

Figure 4 . 10 :

 410 Figure 4.10: HRTEM of the 53nm Nanowire annealed at 500°C (a), Fourier transform obtained using Digital Micrograph in the selected region and theoretical diraction pattern obtained with the software Carine (c). The spot represented by a cross, corresponds to double diraction spots.

  atomic concentrations is close to unity, conrming the formation of the PtSi structure in several regions of the nanowire (see gure 4.12).

Figure 4 .

 4 Figure 4.11: EDX spectrum obtained in the zone shown in gure 4.10(a) and corresponding quantication results.

4. 2 .Figure 4 . 12 :

 2412 Figure 4.12: Diagram showing the quantication results for dierent zones on the 53 nm Nanowire annealed at 500°C.

Figure 4 . 13 :

 413 Figure 4.13: HRTEM of a 50nm Nanowire annealed at 350°C (a), Fourier transform of the selected region (b), specimen composition obtained by STEM/EDX in the same region (c) and theoretical diraction pattern corresponding to the Fourier transform (d). The spot represented by a cross, corresponds to double diraction spots.

4. 2 .Figure 4 . 14 :

 2414 Figure 4.14: HRTEM of a 13 nm Nanowire annealed at 350°C (a), Fourier transform of the selected region (b), specimen composition obtained by STEM/EDX in the same region (c) and theoretical diraction pattern corresponding to the Fourier transform (d). The spot represented by a cross, corresponds to double diraction spots.

Figure 4 . 15 :

 415 Figure 4.15: HRTEM of a 13 nm Nanowire annealed at 350°C (a), Fourier transform of the selected region (b), specimen composition obtained by STEM/EDX in the same region (c) and theoretical diraction pattern corresponding to the Fourier transform (d).

4. 2 .Figure 4 . 16 :

 2416 Figure 4.16: Diagram resuming the height of silicide phases for dierent diameters of nanowire annealed at 350°C.

4. 2 .

 2 RESULTS OBTAINED IN THE VERTICAL NANOWIRE TRANSISTORS STUDY 101 should include the actual geometry of the detector, leading to dierent collection

Figure 2 . 1 :

 21 Figure 2.1: Illustration schématique de la géométrie d'absorption dans un matériau homogène.

Figure 2 . 2 :

 22 Figure 2.2: Illustration schématique de la position du détecteur et du trajet des rayons X, lorsque le détecteur est placé dans un plan perpendiculaire à la direction de variation du prol de dopage.

.21) 2 . 3 . 3

 233 Alliages Pseudo-morphiques Le calcul de la densité pour les alliages pseudo-morphes constitue également un cas d'intérêt. Les alliages pseudomorphiques sont des alliages binaires composés de deux éléments A et B (Si et Ge par exemple) qui ont exactement la même structure cristalline dans leur forme pure (le Si et le Ge purs ont une structure cubique de type diamant).

2. 4 3 .

 43 Calibration du modèle et estimation d'erreur Comme indiqué précédemment, pour pouvoir obtenir une mesure able, les coecients C-L doivent être calibrés. De plus, pour obtenir une mesure quantitative, il faut estimer l'erreur sur ces coecients et calculer l'erreur sur la mesure de la composition nale en combinant toutes les sources d'erreur possibles. La procédure et les formules que nous avons utilisées pour la calibration et l'estimation des erreurs seront abordées dans cette section. Une application directe de ces techniques utilisant la Spectroscopie de rétrodiusion de Rutherford RBS [32] pour la calibration sera discutée dans le chapitre Dans le cas général d'un alliage composé de N AT atomes, tous les coecients k i:F i ,F i 0 C-L relatifs à l'élément de référencei 0 doivent être étalonnés. Nous nous concentrerons, dans la suite, sur la technique générale pour la calibration et l'estimation de l'erreur expérimentale sur le coecient C-L relatif à deux éléments i et i 0 appartenant aux familles F i et F i 0 . Cette procédure peut être appliquée à diérents couples de pics de l'ensemble jusqu'à ce que la totalité des coecients C-L soient obtenus. Plusieurs échantillons peuvent être utilisés pour l'étalonnage à condition de disposer, pour chaque échantillon d'une mesure f M i (x) M eas des fractions de masse obtenue avec une autre technique expérimentale. QUANTIFICATION Pour deux éléments i et i 0 , les facteurs C-L relatifs aux familles des pics F i peuvent être déterminés par régression linéaire à partir de la formule suivante (voir l'équation 1.5 et 2.1 ) :

(2. 27 ) 1 √.

 271 Pour estimer l'erreur ∆k As:F As sur la valeur calibrée de k As:F As ,F Si , nous avons supposé que la distribution statistique des intensités mesurées obéissait à la loi de Poisson. Dans ce cas, l'erreur sur la mesure de l'intensité I est donnée par ∆I(x) = I(x)et l'erreur relative s'écrit I = ∆I I = I La valeur de k i:F i ,F j peut être obtenue en utilisant les techniques des moindres carrés pondérées et en minimisant l'expression ([41] p.75) :

  de lithographie standard (voir gure 3.1(a)). Un transistor MOSFET, appelé FinFET, peut ensuite être obtenu en utilisant les FIN. Il existe plusieurs types de FinFET. L'idée générale derrière toutes les géométries est d'obtenir un contrôle amélioré du transistor en améliorant la géométrie de la grille. Pour cela, on utilise normalement de petits FIN (quelques dizaines de nanomètres) et la grille présente une structure 3D composée de plusieurs parois. Un dessin schématique d'un FinFET avec une grille à trois parois est montré dans la gure 3.1(b).

Figure 3 . 1 :

 31 Figure 3.1: Illustrations schématique de FIN obtenus à partir de un substrat SOI (a) et d'un FinFet (b).

Figure 3 . 2 :Figure 3 . 3 :

 3233 Figure 3.2: (a) Simulation de la distribution de dopage implanté obtenue avec SRIM pour une énergie de 21keV ; (b) Image MET de la section transversale de l'échantillon FinFET.

3. 2 Figure 3 . 4 :

 234 Figure 3.4: Illustration schématique de l'inuence de la géométrie des échantillons sur les artefacts EDX (rayons X secondaires créés par la uorescence et les électrons rétro-diusés) dans les échantillons préparés : a) par FIB. b) par amincissement méca-nique+ionique.

3 . 2 ,Figure 3 . 5 :

 3235 Figure 3.5: Densité de masse en fonction de la concentration obtenue avec les diérents modelés. La valeur déterminée expérimentalement de la masse volumique ρ M

IFigure 3 . 6 :

 36 Figure 3.6: Distributions d'intensité obtenus dans l'échantillon de calibration avec un pic de densité atomique de 2% pour le pic SiK (b) pour AsK(c) et PtM(d). L'image STEM de la région cartographiée ainsi que la géométrie utilisée pour l'extraction du prol est indiquée en (a).

Figure 3 . 7 :

 37 Figure 3.7: Distributions d'intensité obtenus dans l'échantillon de calibration avec un pic de densité atomique de 10% pour le pic SiK (b) pour AsK(c) et PtM(d). L'image STEM de la région cartographiée ainsi que la géométrie utilisée pour l'extraction du prol est indiquée en (a).

Figure 3 . 9 :

 39 Figure 3.9: Résultat de la correction d'absorption pour les 10 premières itérations. Le prol complet est aché en (a) et la région a haute densité en (b).

Figure 3 . 10 :

 310 Figure 3.10: Prol de la concentration de dopage c A As (x) avant (a) et après (b), calibration pour les pics I As:K (AsK) et I As:L . Les prols RBS sont aussi montrés sur la gure.

Figure 3 . 12 :

 312 Figure 3.12: Concentration atomique de As ( c A As (x)) mesurée en utilisant les pics K pour les Si et le as As (a) et les pics K le Si et les pics L pour le As (b). Dans les deux cas, les valeurs obtenues sont comparées aux valeurs obtenues par RBS.

3. 3 .Figure 3 . 13 :

 3313 Figure 3.13: Concentration atomique de As ( c A As (x)) mesurée en utilisant les pics K pour les Si et le as As (a) et les pics K le Si et les pics L pour le As (b). Dans les deux cas, les valeurs obtenues sont comparées aux valeurs obtenues par RBS.

3. 3 . 2

 32 Figure 3.14: Images STEM à champ clair (a) et cartographie d'intensité EDX (b-c) obtenues sur la structure de test FinFET. Les pics K ont été utilisés à la fois pour l'oxygène et pour l'arsenic an d'obtenir les proles d' intensité montrés en (b) et (c). Les prols d'intensités obtenus par intégration latérale sont superposés aux cartes ((b) et (c)). Les èches en (a), (b) et (c) indiquent la direction et la position de la direction d'intégration ainsi que la largeur d'intégration latérale.

Figure 3 .

 3 Figure 3.15: (a) Image schématique illustrant la géométrie utilisée pour obtenir le prol de dopage et (b) prol de dopage avec erreur expérimentale obtenue en appliquant la méthode de quantication décrite dans le chapitre 2.

et 2 .

 2 30) et peut être améliorée en augmentant le temps d'exposition utilisé pour la mesure. Les formules dérivée dans cette thèse, peuvent, d'ailleurs, être utilisées pour calculer l'erreur associée à un temps d'exposition spécique, mais il est impossible de déduire une limite théorique minimale car le temps d'exposition maximal dépend de nombreux facteurs expérimentaux tels que la stabilité de l'échantillon à la dérive et de sa résistance à l'irradiation électronique. Pourtant, le temps nécessaire pour obtenir une valeur d'intensité cible peut être estimé. Pour donner un exemple de cette procédure, nous avons estimé les temps d'exposition nécessaires pour obtenir une limite de détection donnée à l'aide des paramètres spéciques de notre conguration expérimentale. Plus précisément, nous avons utilisé les valeurs du nombre de coups par seconde mesurées dans notre conguration pour calculer le temps de mesure nécessaire pour obtenir diérentes valeurs de la limite de détection. Pour les calculs, nous avons supposé que la valeur des comptes par seconde ne changeait pas avec le temps.

2 - 2 ( 3 . 1 )( 3 . 2 ) 1 :

 2231321 Cette formule exprime l'intensitéI Si:F Si (x) du pic de Si à obtenir si nous voulons obtenir la valeur suivante pour la densité As f M As ± ∆ f M As . Selon notre dénition de la 136 CHAPITRE 3. CARTOGRAPHIE EDX QUANTITATIVE DANS LES SEMI-CONDUCTEURS limite de détection, on peut alors obtenir l'intensité I D nécessaire pour obtenir une limite de détection D en imposant que ∆ f M As = f M As = D dans l'équation 3.1 Le temps d'exposition nécessaire pour obtenir une intensité mesurée I D peut être estimé en utilisant les coups par seconde (cps) de la manière suivante (on a aussi supposé que I Si:F Si = 0 pour t exp = 0) : t exp = I Si:F Si (x) cps Le temps de mesure nécessaire pour obtenir une limite de détection spécique, peut ainsi être calculé. Les résultats sont présentés dans le tableau 3.1. On peut clairement Limite de détection de la densité d'As Temps d'exposition (temps réel) Temps de mesure estimé nécessaire pour obtenir des limites de détection plus élevées constater que, dans notre cas, il est dicile d'améliorer considérablement la limite de détection en augmentant simplement les temps de mesure, puisque la présence de la dérive de l'échantillon, des dommages causés par le faisceau et de l'instabilité du courant (dans notre cas, le microscope est équipé avec un canon à émission de champ à froid) limitent le temps de mesure à quelques dizaines de minutes. 1.4 Diagram explaining the three physical phenomena resulting from the interaction of the electron beam with the sample: characteristic X-rays, Bremsstrahlung radiation and electron backscattering. . . . . . . . . . . . . . . . . . . . . . 1.5 Schematic drawing illustrating the origin of Bremsstrahlung radiation (a) and Fluorescence (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Diagram showing the structure and operation of a simple p-i-n EDX detector. 1.7 Diagram representing the three intermediate mechanisms for carrier ionisation in the detector (a) and the emission process of an Auger electron (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Diagram representing the detection artefacts. . . . . . . . . . . . . . . . . 1.9 Schematic drawing of an SDD detectors, taken from [18].The image is available under the license Creative Commons Attribution International 4.0 (https://creativecommons.or 1.10 EDX spectrum obtained on a SiGe sample using the 2100F microscope installed at UMS Castaing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Diagram showing the two Bremsstrahlung elimination models: (a), Mathematical and (b), Physical TEM. . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Example of subtraction of background noise (a) and deconvolution (b). . . 2.1 Schematic diagram of the absorption geometry in a homogeneous material. 2.2 Schematic illustration of the detector position and of X-ray path, when the detector is placed on a plane perpendicular to the variation direction of the doping prole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Schematic illustrations of a SOI substrate with ns (a) and a FinFet (b). . 3.2 (a) Implanted doping distribution simulated with SRIM for As implant in silicon with an energy of 21keV; (b)TEM image of the cross section of the FinFET sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 RBS measurement of As atomic density superimposed to the TEM image obtained in the same specimen. The contrast variation in the TEM images indicate that the high concentration implanted region is amorphized. . . . 3.4 Schematic illustration of the inuence of the specimen geometry on EDX artifacts (secondary X-rays created by uorescence and backscattered electrons) in specimens prepared using: a) FIB. b) mechanical thinning. . . . . 3.5 Mass density a function of the As mass fraction calculated with two dierent approximations. The experimentally determined value of the mass density ρ M

  In inset (a) the BF-STEM image of the mapped region is shown together with the geometry of the linescan used for the extraction of a prole. The intensity maps after peak integration and bremsstrahlung subtractions are show in the other insets for the following peak families: SiK(b), AsL(c), Pt(M) (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . on the calibration specimen with a 10% atomic density peak. In inset (a) the BF-STEM image of the mapped region is shown together with the geometry of the line scan used for the extraction of a prole. The intensity maps after peak integration and bremsstrahlung subtractions are show in the other insets for the following peak families: SiK(b), AsL(c), Pt(M) (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 EDX spectrum obtained from the 10% specimen and showing the position of the Si and As peak families. . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 The result of the correction absorption for the rst 10 iterations. In (a) the full prole is shown while in (b) a zoom on the high density region is displayed. 3.10 Plot of the measures As concentration c A As (x) before (a) and after (b) calibration of the CL coecients. The values obtained by using the I As:K (AsK) and I As:L peaks are reported and compared with the values obtained by RBS. 3.11 Variation of the absorption corrected peak density in the 2% (a) and 10% (b) specimen. The concentration has been obtained by using the AsK and SiK families. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Plot of the measures As concentration c A As (x) of the 10% specimen using (a) K peaks for Si and As and (b) K peak for Si and L peaks for As. In both cases the obtained values are compared with the values obtained by RBS. . 3.13 Plot of the measures As concentration c A As (x) of the 10% specimen using (a) K peaks for Si and As and (b) K peak for Si and L peaks for As. In both cases the obtained values are compared with the values obtained by RBS. . 3.14 Bright eld STEM images(a) and EDX intensity maps (b-c) obtained on a FinFET device. The K peaks have been used both for oxygen and arsenic to obtain the elemental maps shown in (b) and (c) from the EDX hypermap. Intensities proles obtained using a laterally averaged line-scan are superimposed to maps ((b) and (c)). The direction and position for the line-scan as well as the lateral integration width are shown in gure (a), (b) and (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 (a) Schematic image illustrating the geometry used to obtain the doping prole and (b) doping prole with experimental error obtained by applying the quantication method described in chapter 2. . . . . . . . . . . . . . . 3.16 Cross-sectional TEM micrographs showing the calibrated specimens structure.

3. 17

 17 Intensity maps obtained on the calibration specimen with a 28 nm of SiGe layer. In inset (a) the STEM of the mapped region is shown together with the geometry of the line scan used for the extraction of a prole. The intensity maps after peak integration and bremsstrahlung subtractions are show in the other insets for the following peak families: SiK(b), GeK(c), CK (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  As already discussed, the establishment of a density formula is the main drawback of our model and it is the important to understand in which cases the approximations discussed in 2.3 can be used.If we focus on the case in which only two elements are present (binary alloys), there are three categories of materials for which it is simple to establish a density model: Alloys composed by atoms of comparable size (covalent radius) in which one of the two atoms is in small concentration. In this case the same-radius approximation can be used. Interesting cases included in this category are some doped semiconductors, Si:B, Si:P, Si:Al, Si:As but also metallic alloys provided the two fore-mentioned conditions are satised (low concentration iron alloys such as FeCo, FeNi, FeCu ...).

	2.5. DISCUSSION AND CONCLUSIONS	53
	2.5.2 Model applicability

Pseudo-morphic alloys (see 2.3.3) like SiGe, CuPd, AgAu... Alloys presenting no local ordering (ideal amorphous solid see section 2.3.2). It will be shown in section 3.1.3 that binary alloys composed by elements having the same packing ratio fall in this category.

  ).In a second step, a RuO 2 .1.8H 2 O commercial powder dispersed in THF is deposited by spin-coating (layer 2) onto the layer 1.Then, laser-writing was performed on the bilayered HAuCl 4 .3H 2 O / RuO 2 lm with a UV laser (wave length of 405 nm) with a spot size of 5µm.When the bilayered lm was deposited, laser-writing was performed at 116 mW with a writing speed of 3 ± 1 mm.s -1 . After washing in acetone and ethanol, mechanically stable patterned electrodes adhering to the exible PI lm were obtained.

table 4

 4

	.1.

Table 4 .

 4

1: Crystalline structure of PtSi phases

  En supposant que les sections ecaces sont additives et que le nombre d'atomes i dans l'alliage est n i , nous pouvons écrire :

	2.2. CORRECTION DE L'ABSORPTION		109
	par exemple).
			9)
	où µ i (E) est le coecient d'absorption massique associé à l'atome i.
	Cette relation est normalement citée dans la littérature sans démonstration ou
	référence bibliographique. Il est donc dicile de déterminer sa limite de validité. Pour
	cette raison, il est nécessaire de donner une démonstration de cette formule et
	d'indiquer les approximations nécessaires à son obtention. En utilisant les sections
	ecaces totales d'interaction σ i (E) pour un rayon X d'énergie E avec un atome d'un
	élément i, nous pouvons écrire [35] :
	µ i =	σ i m i	(2.10)
	La formule 2.9 peut alors être obtenue en émettant l'hypothèse que la section e-
	cace d'un mélange d'atomes est la somme des sections ecaces des atomes individuels.
	Cela signie que les liaisons entre les atomes n'ont pas d'inuence sur l'interaction des
	atomes avec les rayons X. C'est une bonne approximation, car elle est utilisée dans de
	nombreux autres calculs (énergies caractéristiques des rayons X et intensités relatives,

  2.15) Les valeurs nales I i:F i ,k (x) et f A i (x) sont obtenues après N I itérations, le critère d'arrêt étant que la variation relative des fractions de masse entre deux itérations ultérieures est inférieure à une tolérance relative xée (

  i dans l'alliage. L'hypothèse principale à la base de cette formule est que l'espace occupé dans le solide par un atome d'un élément spécique est toujours le même, indépendamment de la concentration de l'alliage. Une autre façon de formuler la même approximation consiste à armer que la structure autour d'un type d'atome spécique dans l'alliage est toujours la même et qu'elle ne dépend donc pas de la concentration.Cette approximation est évidemment correcte pour les IAS car dans ce cas, la distribution atomique est totalement aléatoire et ne dépend pas de la concentration des deux

	éléments dans l'alliage. L'espace autour d'un atome individuel dépend simplement de
	la taille de l'atome qui peut être estimée à l'aide de son rayon covalent.
	L'équation 2.19 peut également être utilisée pour des solides qui ne possèdent pas
	une structure IAS à condition que les approximations susmentionnées soient respec-
	tées. Il n'est pas toujours facile de vérier la validité des approximations. Un exemple
	d'application à un solide amorphe non idéal sera donné dans la section 3.1.3.
	Pour établir une relation entre la masse volumique de l'alliage et les fractions mas-
	siques, on peut utiliser la formule suivante qui lie les fractions volumiques et les fractions
	massiques (cette formule peut être obtenue en utilisant les mêmes hypothèses de l'équa-
	tion 2.19) :
	19)
	où ρ M i est la densité de masse du solide amorphe et f V i est la fraction volumique de l'

élément

  -x) en fraction massiques. Nous ne déduirons pas cette formule car il peut être démontré que, dans ce cas, l'approximation du solide amorphe peut être utilisée.Cela semble paradoxal puisque dans ce cas l'alliage n'est pas du tout amorphe. Mais en réalité, il est facile de comprendre que, dans ce cas, on peut supposer que le volume occupé par un atome A ou B dans l'alliage A 1-x B x ne dépend pas de x. Cela dérive du fait que dans le cristal pur, un atome de type A occupe un volume V A le nombre d'atomes par unité de cellule n est le même dans les deux expressions. Étant donné que l'écart entre a 1 et a 2 est négligeable, on peut considérer que les valeurs deV A

			3	(2.23)
	L'expression nale peut ensuite être obtenue en transformant les fractions atomiques
	x et (1 1 = atomique) où n est, encore une fois, le nombre d'atomes par cellule unitaire. De la a 3 1 (volume n
	même manière, V A 2 =	a 3 2 n	, puisque les deux atomes ont exactement la même structure
	cristalline et que

  Ladensité dans cette approximation est donnée par l'équation 2.19.Tenant compte du fait que nous avons f V i = V i /V , et que pour un solide cristallin avec n atomes par cellule élémentaire ρ M i = nm i

	i a 3	, nous pouvons écrire :

1 )

 1 Les expressions 2.23 et 2.24sont évidemment diérentes, mais on peut montrer que les dénominateurs sont égaux pour un alliage pseudo-morphique. Pour cela, on peut 2.4. CALIBRATION DU MODÈLE ET ESTIMATION D'ERREUR 115 exprimer les deux dénominateurs en utilisant le paramètre ε = a 2 -a 1

	qui peut être considéré comme petit ( = 4.2% en SiGe). Pour le dénominateur de l'équation 2.23, a 2
	nous pouvons écrire :

  Les erreurs ∆k j:F j ,F i0 sur la valeur calibrée de la constante C-L. La contribution de cette erreur peut être calculée de la manière suivante :

	2.4. CALIBRATION DU MODÈLE ET ESTIMATION D'ERREUR	117
	2.

  3. L'erreur expérimentale sur l'épaisseur de l'échantillon ∆t. Dans ce cas, il n'est pas possible d'appliquer la formule de propagation d'erreur pour évaluer l'erreur car la dépendance de f M j (x) sur t est liée au calcul itératif présenté précédemment.

	Dans ce cas, la contribution à l'erreur totale a été estimée en utilisant la formule
	suivante

  Les deux ont été implantés avec une tension d'accélération de 21KeV avec une dose diérente s'élevant respectivement à 9, 4 10 15 cm -2 et 1, 8 10 15 cm -2 . En raison de la

	3.1. CARTOGRAPHIE DU DOPAGE DANS LES NANODISPOSITIFS SI	121
	gure 3.1(b)) les trois parois latérales doivent avoir la même concentration de dopage
	que la paroi supérieure. Une distribution de dopage répondant à cette exigence est
	appelée distribution conforme.
	Il est facile de comprendre que pour qu'un transistor fonctionne correctement, le
	dopage source et drain doit également présenter une distribution conforme. Et comme
	le canal est faiblement dopée dans les dispositifs FinFET (et même intrinsèque dans
	certains cas) et que les régions source et drain doivent être fortement dopées pour
	assurer un contact ohmique, le fait d'obtenir une distribution conforme dans les régions
	source/drain est d'importance capitale. Pour cette raison, nous avons décidé d'appliquer
	le modèle développé à la mesure d'un dopage conforme d'arsenic à haute concentration
	dans une structure de test de type FinFET.
	Dans la suite, les détails expérimentaux sur l'échantillon observé et les conditions
	d'observation seront décrits. Dans la section 3.3, la calibration du modèle sera discutée
	et on montrera que un prol As quantitatif peut être obtenu à l'échelle nanométrique.
	Enn, les erreurs de mesure et la limite de détection de dopage seront estimées et la
	possibilité d'améliorer la limite de détection en modiant les paramètres d'observation
	sera discutée.
	dose élevée utilisée pour l'implantation, les plaquettes implantées sont métamorphisées
	en surface, après implantation (voir gure 3.3). Les paramètres d'implantation ont été
	choisis en simulant le prol d'implantation à l'aide du logiciel de simulation Monte
	Carlo SRIM (Stopping and Range of Ions in Matter) ([46] ch.3). Les résultats de la
	simulation correspondant aux deux doses diérentes sont présentés dans3.2 (a). On
	peut noter que la valeur du pic de concentration s'élève à 10 21 cm -3 (équivalent à une
	concentration atomique de As de 10%) pour l'implant à plus forte dose et à 5 10 21 cm -3
	(soit une concentration atomique de As de 2% ) pour l'implant à plus faible dose.

3.1.1 Description des échantillons

Une structure de test de type FinFET avec un canal de 80 nm de largeur et 100 nm de hauteur ayant reçu une implantation à dose élevée (voir la gure 3.2 (b)) a été utilisée pour les mesures. La structure de test ne présentait aucun oxyde de grille ni contact métallique, car nous souhaitions que l'observation soit focalisée sur la distribution des dopants. Pour la création de la région dopée, un implantation plasma d'arsenic a été utilisé an d'obtenir une distribution conforme du prol de dopage.

Pour pouvoir mesurer la concentration de dopage en arsenic dans le nanodispositif, nous avons conçu deux échantillon de calibration obtenus par implantation ionique dans un wafer de silicium monocristallin. An de pouvoir vérier la précision de la calibration des cocients de C-L pour diérentes concentrations, deux types d'échantillons ont été utilisés.

An d'obtenir une mesure absolue de la concentration en arsenic, des mesures de spectroscopie par rétrodiusion de Rutherford (RBS) ont été eectuées sur les échantillons de calibration. Pour les mesures RBS des ions He + accélérés à 1,5 MeV ont été SEMI-CONDUCTEURS

  -CONDUCTEURS estimation du taux de remplissage du solide. Le volume V A est par dénition l'inverse de la densité atomique ρ A = ρ M /m du matériau tandis que la taille de l'atome est calculée en utilisant une approximation sphérique (V S = 4 3 πr 3 ) et le rayon covalent r. Les valeurs de s = 4 3 πr 3 ρ A pour le Si amorphe et le As amorphe peuvent être calculées en utilisant les données précédemment dénies pour la densité, le rayon et la masse des deux éléments. Les valeurs obtenues sont assez proches (s Si = 0.28 ,s As = 0.26 ) , ce qui conrme que la structure des deux matériaux purs est presque identique. La comparaison entre l'approximation du solide amorphe et l'approximation de même rayon est illustrée à la gure 3.5. On peut noter que l'approximation du solide amorphe donne de meilleurs résultats pour des concentrations élevées en arsenic, alors que les deux approximations sont presque équivalentes pour des concentrations en As faibles, comme on peut le voir sur la gure 3.5. Pour les c-SiAs, il est beaucoup plus simple d'obtenir la formule de la masse volumique puisque celle-ci correspond exactement au cas où l'approximation de rayon constant a été déduite. La densité de masse peut simplement être calculée en utilisant 2.16 avec m 1 = m Si , m 2 = m As , ρ M 1 = ρ M

	Si .
	3.3 Résultats expérimentaux et discussion
	Dans cette section, nous présenterons les résultats expérimentaux obtenus sur la struc-
	ture de test FinFET. Pour cela, les résultats de la calibration des coecients de C-L
	seront tout d'abord décrits. Ensuite, les résultats naux obtenus sur le prolage des
	dopants seront discutés.
	3.3.1 Résultats de Calibration
	An d'appliquer les techniques de calibration, les concentrations de densité atomique
	c A A/RBS (x) mesurées par RBS ont été converties en fractions masiques. Ensuite, les
	coecients de C-L ont été obtenus par régression linéaire pondérée comme décrit dans
	la section 2.4. Les équations 2.4 et 2.21 peuvent être combinées pour obtenir les fractions
	de masse mesurées par RBS :
	3 ,ρ M c-As = 5.73 g/cm 3 ) n'est
	pas non plus compatible avec celui d'un IAS. En conclusion, il n'y a aucune raison de
	penser que l'alliage a-SiAs (dont la structure est une combinaison de la structure de Si
	amorphe et de As amorphe) présente une structure IAS.
	D'autre part, il est facile de montrer que a-Si et a-As ont une structure similaire. On
	peut alors supposer que cette structure est conservée dans l'alliage SiAs indépendam-

ment de la concentration d'As dans l'alliage, ce qui est précisément l'hypothèse utilisée pour dériver l'équation 2.19. Pour montrer ceci, on peut se servir des fraction s = V S V A entre le volume atomique V A et la taille de l'atome V S . Ces fractions sont un bonne SEMI

  'obtenir les distribution d'intensité des pics de Si et As dans la zone observée.

	3.3. RÉSULTATS EXPÉRIMENTAUX ET DISCUSSION	127
	a permit dEnsuite, un prol d'intensité a été obtenu pour chaque pic en intégrant les distributions
	(voir g. 3.6(a) et 3.7(a)) dans une direction perpendiculaire à la direction z (dénie
	à la gure 2.2). Les distributions de composition ont été obtenues en appliquant notre
	modèle à diérentes familles de pics caractéristiques : les pics K pour le silicium et les
	pics K ou L pour l'arsenic (voir gure3.8).
	La procédure de correction d'absorption décrite dans la section 2.2 a ainsi été utilisée
	pour calculer les intensités corrigées par absorption
	ρ M a-As
	La technique d'étalonnage décrite dans la section 2.4 a été appliquée aux échantillons
	de calibration a-SiAs décrits dans la section 3.1.1. Plusieurs spectres EDX (un pour

chaque pixel de l'image STEM correspondante) ont été enregistrées (voir la gure 3.6 pour l'échantillon à 2% et la gure 3.7 pour l'échantillon à 10%). Pour l'acquisition, l'échantillon FIB a été orienté comme sur la gure2.2. Les traitements des spectres nous

1.1. ELECTRON MICROSCOPY

1.3. EDX QUANTIFICATION

See http://www.gatan.com/products/sem-specimen-preparation/pecs-ii-system

K As,Si:K,K = 2.18 ± 0.03 , K As,Si:L,K = 1.89 ± 0.03.L'erreur sur les coecients C-L peut être utilisée pour estimer les erreurs nales sur le prol de densité, comme expliqué à la section 2.4, où les sources d'erreur possibles ont été examinées. Entre les diérentes erreurs, l'erreur résultant de l'incertitude sur l'épaisseur de l'éprouvette mérite une discussion particulière. Il est généralement compliqué de mesurer l'épaisseur des échantillons de manière able et précise. Même si certaines techniques existent (mesure par SEM, EELS, diraction électronique en faisceau convergent etc...), il est généralement compliqué d'obtenir une valeur able avec une faible erreur.En conséquence, les erreurs de mesure sur l'épaisseur peuvent être de l'ordre de 30%

Remerciements

Je tiens en premier lieu à exprimer ma profonde reconnaissance aux membres du jury pour l'honneur qu'ils m'ont fait en acceptant de juger ce travail : La Présidente, Madame le Docteur Caroline BONAFOS, les rapporteurs, Madame le Docteur Marie-France BEAUFORT, Monsieur le Docteur Alain PORTAVOCE et l'examinateur Monsieur le Docteur Etienne TALBOT.

C'est un véritable plaisir pour moi de pouvoir remercier ici toutes les personnes dont la présence m'a été si précieuse durant ces 5 années d'études en France, même si de simples mots ne suront jamais pour exprimer mon immense reconnaissance. Je tiens donc à remercier Monsieur le Professeur Marc RESPAUD. Marc je te remercie pour m'avoir accueilli dans ton équipe, pour le réel intérêt que tu as manifesté pour ce projet au cours de ces trois années, pour tes conseils scientiques et surtout de m'avoir gentiment pousser pour mieux avancer quand il a fallu le faire. J'adresse mes plus profonds REMERCIEMENTS à Monsieur le Docteur Pier-Francesco FAZZINI pour m'avoir plus-que parfaitement encadré pendant ces 3 ans. Pier je pense que j'ai eu une énorme chance de t'avoir comme encadrant. Je te remercie pour ton optimisme, pour ta disponibilité, pour ton accompagnement au cours de chaque étape de la thèse. Sans toi je n'aurais jamais pu arriver jusqu'au bout ! Je te souhaite une excellente suite autant au plan professionnel qu'au plan familial. J'adresse aussi mes profonds remerciements à Monsieur le Docteur Fuccio CRIS-TIANO pour m'avoir encadré pendant ces 3 ans. Je te remercie pour la grande base de connaissance scientique que tu étais toujours prêt de me la transmettre. Comme Pier, tu m'as appris ce que je sais faire aujourd'hui. Tu étais toujours présent pour me donner autant de la connaissance scientique que de la motivation et des conseils pour tous les jours. J'ai eu vraiment de la chance de t'avoir comme encadrant. Je te souhaite aussi une excellente suite autant au plan professionnel qu'au plan familial.

Je voudrais exprimer toute ma gratitude et surtout toute mon estime pour les différents membres de l'équipe pour le soutien morale, surtout lors de la rédaction de ce manuscrit. Abstract One of the key issues for the miniaturization of semiconductor nanodevices is the precise control of their doping. In new generation devices, the doping spatial distribution must be controlled with a precision higher than 1nm while atomic concentrations below 1% have to be measured. This calls for the use of high resolution techniques. Scanning transmission electron microscopy (STEM) associated with Energy Dispersive X-ray spectroscopy (EDX) is an excellent candidate due to its versatility (almost all elements of the periodic table canbe mapped) and its high spatial resolution. On the other hand, quantitative analysis of doping by STEM/EDX is complicated by the presence of measurement artifacts that can safely be ignored for high concentrations impurities but become critical for low concentration impurities.

Je tiens à remercier aussi l'ensemble des membres des plateformes

In this thesis a modied Cli-Lorimer (C-L) method has been developed to obtain quantitative dopant proling in a nanoscale device structure. The method has been applied on specimens prepared by Focused Ion Beam, in order to reduce the inuence of secondary X-rays produced by uorescence or back scattered electrons, and is based on the iterative correction of X-rays absorption eects in the specimen. In order to obtain reliable results the C-L coecients have been calibrated using Rutherford Back Scattering (RBS) measurement and the total experimental error has been calculated using standard error propagation techniques. The results obtained on a FinFET test structure and on a laser annealed SiGe substrate show the applicability of this technique to devices on the nanometer scale and with low concentration impurities.

Résumé L'un des problèmes clés pour la miniaturisation des nanodispositifs à semi -conducteurs est le contrôle précis de leur dopage. Dans les dispositifs de nouvelle génération, la distribution spatiale du dopage doit être contrôlée avec une précision supérieure à 1 nm, tandis que les concentrations atomiques inférieures à 1% doivent être mesurées. Cela nécessite l'utilisation de techniques de haute résolution. La microscopie électronique à balayage en transmission (STEM) associée à la spectroscopie par rayons X à dispersion d'énergie (EDX) est un excellent candidat en raison de sa polyvalence (presque tous les éléments du tableau périodique peuvent être cartographiés) et de sa haute résolution spatiale. D'autre part, l'analyse quantitative du dopage par STEM/EDX est compliquée par la présence d'artefacts de mesure qui peuvent être ignorés sans risque pour les impuretés à haute concentration, mais deviennent critiques pour les impuretés à faible concentration.Dans cette thèse, une nouvelle méthode basée sur la méthode de de Cli-Lorimer(C-L) a été développée pour la mesure quantitative de la distribution de dopant dans un dispositif à l'échelle nanométrique. La méthode a été appliquée sur des échantillons préparés par faisceau ionique focalisé, an de réduire l'inuence des rayons X secondaires produits par uorescence ou par électrons rétrodiusés, et est basée sur la correction itérative des eets d'absorption des rayons X dans l'échantillon. An d'obtenir des résultats ables, les coecients de C-L ont été étalonnés à l'aide de la mesure de Rutherford Back Scattering (RBS) et l'erreur expérimentale totale a été calculée à l'aide de techniques de propagation d'erreur standard. Les résultats obtenus sur une structure de test FinFET et sur un substrat de SiGe ayant subi des recuits laser montrent l'applicabilité de cette technique aux dispositifs à l'échelle nanométrique et avec des impuretés à faible concentration.