Ce travail de thèse est focalisé sur la détermination, de manière non-destructive, d'interfaces profondément enterrées dans des empilements multi-couches utilisés dans les conditions de technologie réelles au travers d'une méthode innovante basée sur la photoémission avec utilisation de rayons-x de haute énergie (HAXPES) et l'analyse du fond continu inélastique.

Au cours de cette thèse, une procédure numérique à été développée pour quantier la correspondance entre la mesure du fond continu faite par HAXPES et la simulation du fond continu représentative d'une distribution en profondeur donnée. Cette méthode permet de trouver la distribution en profondeur d'un élément grâce à une procédure semi automatisée.

Dans un premier temps cette méthode à été testée en étudiant une couche ultra ne de lanthane enterrée à une profondeur >50 nm dans un dispositif de grille métallique high-κ. L'inuence des paramètres utilisés lors de l'analyse y est étudié et révèle l'importance principale d'un paramètre en particulier, la section ecace de diusion inélastique. La combinaison de mesures HAXPES avec l'analyse du fond continu inélastique utilisant cette nouvelle méthode permet d'augmenter la profondeur de sonde jusqu'à un niveau sans précédent. Ainsi l'échantillon peut être sondé jusqu'à 65 nm sous la surface avec une haute sensibilité à une couche nanométrique.

Dans un second temps, la méthode précédemment validée d'analyse de fond continu inélastique est combinée avec une étude haute résolution des niveaux de c÷ur dans un échantillon servant de source dans un transistor à haute mobilité. Les deux analyses sont complémentaires puisqu'elles permettent d'obtenir la distribution en profondeur des éléments ainsi que leur environnement chimique. Le résultat donne une description complète des diusions élémentaires dans l'échantillon suivant les diérentes conditions de recuit. 11

List of publications:

Probing deeper by hard x-ray photoelectron spectroscopy P. Risterucci, O. Renault, E. Martinez, B. Detlefs, V. Delaye, J. Zegenhagen, C. Gaumer, G. Grenet, S. Tougaard, Applied Physics Letters, 104, 051608 (2014).

Inelastic background analysis of HAXPES spectra: towards enhanced bulk sensitivity in photoemission P. Risterucci, O. Renault, E. Martinez, B. Detlefs, J. Zegenhagen, G. Grenet, S.

Tougaard, Surface and Interface Analysis, 46, 906-910, (2014).

Introduction

Nowadays electronic devices surround our environment. An electronic device is meant to shape and manage electrical signal, to emit, stock or receive an information. This general description shows the multiple purposes of electronic devices, even though the major parts are made for computer components. Those devices are composed dierently upon their application eld and require a large number of complex operations to be made. The increasing demands for optimized device performances come from both consumers (as the life quality increases with technology) and from industries as electronics enables to perform automated accurate tasks with almost innite repeatability. Finally, advanced devices must be ecient in many aspects as power consumption, transportability, strength or execution speed. This is reached only because of the ever increasing integration devices such as Complementary Metal Oxide Semiconductor (CMOS) following Moore's law on the one hand and a diversication of the chip's functionality on the other hand. The key ingredient in this revolution comes from the novel active materials introduced over the past decade to replace silicon derived materials.

The need for new ultra-thin active material is still ongoing and requires a need for innovative characterization techniques. The work materials presented here is related to two of the main challenges expressed by the International Technology Roadmap for Semiconductors (ITRS) in

2013:

Non-destructive characterization method for interface characterization: at extreme miniaturization size where the materials are of nm-thickness, the characterization and metrology methods are needed for control of interfacial layers and require a thorough understanding of nano-scale materials properties and of the physics involved in making the measurement .

Probing the material in integration like conditions. In the real device, the active materials are coated with a nm-thick electrode. The measurement of a technological sample must be performed for the fully deposited structure to account for the real device. Non-destructive characterization techniques must be made to ensure that the derived physical properties are relevant for the real conditions. This thesis tackles the challenge of probing deeply buried interfaces in multilayer stacks used in real devices with a non-destructive technique. Additionally the ultra-thin active layers composing the transistor are dicult to produce and must be controlled between steps of fabrication, implying the use of non-destructive techniques for in-line characterization.

Among the potential non-destructive techniques, x-ray photoelectron spectroscopy (XPS) has an important place as it is a surface sensitive technique perfectly suited for the study of nm-thick materials and interfaces. The limited probing depth of XPS, typically < 10 nm, can be increased with recent use of hard x-rays (HAXPES) delivered by synchrotron radiations as excitation sources.

However HAXPES remains limited so far to the analysis of core-level spectra.

The aim of the thesis is to investigate how inelastic background analysis (IBA) can be applied to HAXPES spectra. Combining HAXPES and IBA would indeed enable much larger probing depth. This work will show that such a method allows to obtain elemental depth distribution up to 65 nm below the surface, making possible to probe a deeply buried active layer in a real device structure.

This thesis was conducted at the Platform For NanoCharacterization (PFNC) of CEA-Minatec (Atomic and Alternative Energy Authority), which is a major actor of French research and innovation. The mission of the PFNC is to provide support for o-line characterization at the nanometric scale for internal or academic purposes and for associated companies of the CEA.

Thanks to the diversity of public and industrial research, the CEA has a great asset for development of characterization techniques, through the large range of technological samples available.

As a result, the CEA is the 3 rd French institute for patents (33 th European) and the 2 nd for publications. The PFNC has state-of-the-art equipment covering the most used characterization techniques and expert researchers to ensure their exploitation.

The rst Chapter of the thesis describes the principles of photoemission spectroscopy. The physical processes involved in photoelectron energy loss are described as they form the inelastic background which will be analyzed in the following. The last paragraphs explain the eects that need to be taken into account when employing hard x-rays and measuring high velocity photoelectrons of HAXPES measurements.

The second Chapter provides a description of the characterization techniques allowing to obtain elemental depth distribution. Among them, photoemission is widely used for its high surface sensitivity in the rst nanometers; depth distribution can be obtained either with varying the photoelectron collection angle or by varying the photon energy. Two inelastic background analysis methods which increase the probing depth of XPS are detailed in this section.

The third Chapter will present the rst results of inelastic background analysis applied to HAXPES spectra of high-k metal gate samples. The investigation of input parameters describing the physical processes in the sample which are inuencing the inelastic background shape are analyzed with extensive analysis of a nanometric La layer buried at depth >50 nm. An original method to obtain the depth distribution with an empirical formula for computing the matching between the measurement and the modeling is shown. A rst application dealing with the annealing of the samples investigated is presented in the last Section.

The last Chapter is dedicated to application cases, based on the results obtained previously.

A series of samples reserved for high Electron Mobility Transistor (HEMT) structures with different thermal treatment is analyzed by combining analysis of high resolution core-level spectra and inelastic background analysis. The results give a global description of the samples allowing to propose a reaction scheme for the formation of an ohmic contact according to the annealing temperature.

1 Relevant aspects of photoelectron spectroscopy Photoelectron spectroscopy is a non-destructive surface chemical characterization technique whose fundamental principles were discovered by Hertz in 1887 [START_REF] Hertz | Ueber einen einuss des ultravioletten lichtes auf die electrische entladung[END_REF]. Theoretical explanations of the photoelectric eect were given in 1905 by Einstein [START_REF] Einstein | Über einen die erzeugung und verwandlung des lichtes betreenden heuristischen gesichtspunkt[END_REF] for which he received the Nobel Prize in 1921. The rst practical results by Siegbahn [START_REF] Siegbahn | Electron spectroscopy for chemical analysis[END_REF] were also rewarded by a Nobel Prize in 1981. This work presents innovative aspects of photoemission with use of a recent advanced high energy light source and analysis of inelastically scattered photoelectron. To ensure a good understanding of such advanced experiments, one must rstly recall the main physical processes involved (section 1.1).

Because this thesis addresses the quantitative treatment of the inelastic loss signal in a photoemission spectrum, a specic emphasis is made in this Chapter on the photoelectron energy loss in section 1.2.

When laboratory X-ray source is used as excitation, only the rst nanometers below the surface are probed. The technique is now commonly referred to X-ray Photoelectron Spectroscopy (XPS).

Photoelectron spectroscopy can also be performed with x-ray synchrotron radiation which oers many alternative experiments exploiting energy and/or polarization tunability of the beam [START_REF] Fadley | X-ray photoelectron spectroscopy: Progress and perspectives[END_REF].

The specicity of HArd X-rays PhotoElectron Spectroscopy (HAXPES) is described in section 1.3.

General aspects 1.Principles of photoemission experiment

A photoemission experiment relies on the photoelectric eect, an interaction between a monochromatic light of energy hν and matter, from which results an electron emission . The so-called photoelectron can be emitted from the atom if the photon energy hν is higher than the electron binding energy E B (> 0). Because of the energy conservation law, the photon energy is equal to the binding energy of the core-level plus the kinetic energy of the photoelectron E K . Practically the binding energy is measured with respect to the Fermi level of the sample grounded to the metallic sample holder and the electron spectrometer as depicted in Fig. 1. In the experiment, the measured quantity is E K,m = E K + (φ s -φ m ), where the analyzer work function Φ m is determined by calibration, usually measuring E F (E B =0) of a clean silver surface. The energy conservation equation is:

hν = E K,m + E B + φ m (1)
The photoelectron spectrum is obtained by plotting the photoelectron intensity as a function of the measured kinetic energy.

Core-level chemical shift

Photoemission is widely used for surface characterization of chemical states [START_REF] Egelho | Core-level binding-energy shifts at surfaces and in solids[END_REF] thanks to core-level photoemission at high energy resolution (∆E < 0.5 eV). The atoms bonded to the element of interest have an eect on the valence and core level electrons, dominated by their dierence in electronegativity χ. If the element A is bonded to an element B of higher χ (i.e. χ A <χ B ), there is a charge transfer of valence electrons from the atoms A to the atom B. The electrons of atom A tighten and decrease their energy of -Δε (with -Δε>0 by convention) which is called chemical shift [START_REF] Fadley | Chemical bonding information from photoelectron spectroscopy[END_REF][START_REF] Sokolowski | Chemical shift eect in inner electronic levels of cu due to oxidation[END_REF]:

∆ε = ke + qV (2)
where k is a constant, e is the charge of the atom and V is the potential of neighboring atoms.

The chemical energy shift is independent of the subshell of the atom. The reported chemical shift in literature for a given compound can be used to assign a chemical environment to the studied chemical element. The energy conservation of eq.1 can be written, considering the chemical shift as:

hν = E K,m + E ε + ∆ε + φ (3)
where E ε represents the binding energy of the free atom, which can be computed by Koopman's equations.

The three step model

The practical description of the photoemission process is usually performed within the so-called three step model [START_REF] Berglund | Photoemission studies of copper and silver: Theory[END_REF]: [START_REF] Hertz | Ueber einen einuss des ultravioletten lichtes auf die electrische entladung[END_REF] the atom is photoionized, (2) the photoelectron is transported in the medium toward the surface, (3) the photoelectron crosses the surface and travels inside vacuum toward the analyzer as shown in Fig. 2. (3) the photoelectron crosses the surface, travels into vacuum and is detected. Adapted from [START_REF] Hüfner | Photoelectron spectroscopy -Principles and applications[END_REF].

The core-level binding energy can be deduced from Eq.1. It enables to determine the elemental composition of the surface, and more importantly the bonding state of the elements from the socalled chemical shift (see section 1.1.3). During the transport to the surface, photoelectrons suer energy losses (see Fig. 2) within a nm-long distance, making XPS very surface sensitive.

The following section addresses in more details some relevant aspects of the rst step, the second and the last steps are detailed in Sec.1.2.

Photoionization cross-section When considering the photoelectron angular distribution within the dipole approximation (see Appendix I) the dierential photoionization cross section of a core level i under a polarized light can be written as:

dσ i dΩ = σ i 4π [1 + βP 2 (cosθ)] (4)
with β the energy-and subshell-dependent asymmetry parameter of the photoelectron angular distribution, σ i the photoionization of the i th atomic subshell and P 2 (cosθ) the second order Legendre Figure 3: Photoionization cross section of La 3s, La 3p and La 3d, calculated from [START_REF] Trzhaskovskaya | Photoelectron angular distribution parameters for elements z=55 to z=100 in the photoelectron energy range 100-5000 ev[END_REF][START_REF] Trzhaskovskaya | Non-dipole second order parameters of the photoelectron angular distribution for elements z=1-100 in the photoelectron energy range 5-10kev[END_REF].

polynomial. In case of unpolarized light the dierential photoionization is:

dσ i dΩ = σ i (1 -[β(3cos 2 ψ -1)] /4) 4π (5)
with ψ the angle between the direction of X-rays and the mean direction toward the analyzer. When renement of this value is needed, two non-dipolar parameters γ and δ can be added to eq.4 to obtain [START_REF] Cooper | Photoelectron-angular-distribution parameters for rare-gas subshells[END_REF]:

dσ i dΩ = σ i 4π [1 + βP 2 (cosθ) + δ + γcos 2 θ sinθcosφ] (6)
The photoionization cross-section decreases with increasing photon energy [START_REF] Trzhaskovskaya | Photoelectron angular distribution parameters for elements z=1 to z=54 in the photoelectron energy range 100-5000 ev[END_REF][START_REF] Trzhaskovskaya | Photoelectron angular distribution parameters for elements z=55 to z=100 in the photoelectron energy range 100-5000 ev[END_REF] as can be seen in Fig. 3 . The asymmetry of the β parameter is of high importance for comparing measurements taken at various angles.

Photoelectron transport inside the material

As x-ray photons get into a few micrometers inside the sample, the probed depth is limited by the distance over which an electron can travel inside the medium. Only electrons that have not suered inelastic scattering have the expected binding energy, they form the photoelectric peak or no-loss peak. The others are detected with less kinetic energy due to inelastic losses. Therefore they appear shifted towards higher binding energies, forming the inelastic background. The various phenomena giving rise to photoelectrons energy loss and their study will be detailed in Chapter II.

Inelastic mean free path The Inelastic Mean Free Path λ (IMFP) is the average distance that an electron with a given energy E (eV) travels between successive inelastic collisions [START_REF]surface chemical analysis -vocabulary, international organisation for standardisation[END_REF]. The IMFP is a calculated quantity and depends on the material. A predictive value can be determined by the Tanuma-Penn-Powell (TPP-2M) formula [START_REF] Tanuma | Calculations of electron inelastic mean free paths. v. data for 14 organic compounds over the 50-2000 ev range[END_REF] as presented in Fig. 4 and which neglects elastic scattering:

λ = E E 2 p [βln(γE) -(C/E) + (D/E 2 )] (7)
with β = -0.10 + 0.944(E 2 p + E 2 g ) -1 /2 + 0.069ρ 0.1

(8) γ = 0.101ρ -1 /2 (9)
C = 1.97 -0.91U

D = 52.4 -20.8U (10)
U = N v ρ/M = E 2 p /829.4 (11)
where E p =28.8(N v ρ/M) 1/2 is the free electron plasmon energy (in eV), N v is the number of valence electrons per atom (for elemental solid) or molecule (for compounds), ρ is the density (in g.cm -3 ), M is the atomic or molecular weight and E g is the bandgap energy (in eV).

Figure 4: Trend curve of inelastic mean free path, taken from [START_REF] Panaccione | Hard x-ray photoemission spectroscopy: Variable depth analysis of bulk, surface and interface electronic properties[END_REF].

The uncertainties on the results of the predictive TPP-2M formula mainly come from the choice of N v . For elemental solids N v is directly the number of valence band electrons but an ambiguity on the choice of N v arises for certain elemental solids in the Periodic Table placed after an element in which an atomic subshell is totally lled. In this case the electrons that were formally in the valence band then occupy a subshell with small binding energy. There are a few recommended N v values given by Tanuma et al. and Seah et al. Although the results vary with material, a change in the value of N v by 50% from the known values will typically lead to a change of <15% in the derived IMFPs results.

For electrons with kinetic energy higher than 70 eV, increasing the kinetic energy of the photoelectron increases the IMFP. Thus the low kinetic energy photoelectrons are representative of the surface whereas high kinetic energy photoelectrons are more representative of the bulk.

Information depth As seen from Monte-Carlo simulations [START_REF] Powell | Eects of elastic and inelastic electron scattering on quantitative surface analyses by {AES} and {XPS}[END_REF] the intensity of photoelectrons elastically scattered (giving rise to the core-level no-loss peak) follow an exponential decay law [START_REF] Seah | Quantitative auger electron spectroscopy: Via the energy spectrum or the dierential?[END_REF][START_REF] Powell | Progress in quantitative surface analysis by x-ray photoelectron spectroscopy: Current status and perspectives[END_REF]. The intensity of the no-loss peak as a function of the probing depth d can thus be written as a Beer-Lambert law:

I(d) = I 0 e (-d λsinθ ) (13)
where I 0 is the peak intensity, λ is the inelastic scattering mean free path and θ is the detection angle from the surface. Therefore 63% of the signal of the no-loss peak arises from λsinθ, 86% from 2λsinθ and 95% of the no-loss peak signal from 3λsinθ as can be seen in Fig. 5. It is possible to experimentally change the angle θ and then obtain information on the depth distribution of elements. This method called Angle Resolved X-rays Photoemission Spectroscopy (ARXPS) will be described later in details.

The eective attenuation length (EAL) is also used in XPS (or AES). If the EAL is introduced instead of the IMFP in an expression derived for XPS (or AES), this expression will be corrected for elastic scattering [START_REF]surface chemical analysis -vocabulary, international organisation for standardisation[END_REF]. Even though the experimental value is very much sample dependent, a large database given by NIST can be used to perform analysis [START_REF] Jablonski | Nist electron eective-attenuation-length datadata[END_REF]. The elastic scattering modulates the exponential attenuation of eq.13, mainly for diraction eects. These eects are anisotropic and depend mainly on the nearest neighbors of the photoionized atom.

Elastic scattering The elastic collisions change the direction of the emitted photoelectron. The intensity measured is aected by this phenomenon.

The eects of elastic scattering of photoelectrons in amorphous and polycrystalline materials can be accounted for by the introduction of two corrections parameters in Eq.6: Q describes the reduction in photoelectron intensity due to elastic scattering and β ef f an eective anisotropy parameter dening the modied angular distribution of photoelectrons. The resulting expression of the photoionization cross section becomes:

dσ i dΩ = σ i Q(1 -[β ef f (3cos 2 ψ -1)] /4) 4π
From extensive series of Monte Carlo simulations, Jablonski found the following equations, tted to the derived values ofQ andβ ef f :

β ef f = a 1 cos 2 α + a 2 cosα + a 3 (14)
Q = b 1 cos 2 α + b 2 cosα + b 3 (15)
where a 1 , a 2 , a 3 , b 1 , b 2 and b 3 are tted coecients listed in an extensive tabulations [START_REF] Jablonski | Database of correction parameters for the elastic scattering eects in xps[END_REF]. The eect of elastic scattering is used in MC simulation to obtain the depth distribution function [START_REF] Powell | Progress in quantitative surface analysis by x-ray photoelectron spectroscopy: Current status and perspectives[END_REF].

Experimental aspects

There are two kinds of x-rays sources, the laboratory sources with xed photon energy and synchrotron radiation sources which deliver photons of tunable energy.

X-ray laboratory sources The specicity of laboratory sources for photoemission are due to the material used as a source. The choice of photon source must be determined by two criteria: the photon ux which should be high to decrease the time measurement and the monochromatic energy of the emitted photons to achieve high resolution. The photon emitted by the source are not monochromatic and must be selected in energy via an optical apparatus called monochromator.

This energy selection decreases the photon ux.

Two sources correspond to these criteria and are usually employed in laboratory experiments, the aluminum with photon emission ray Al K α of 1486.6 eV or magnesium Mg K α which produces photons of 1253.6 eV. This energy regime, called soft x-ray ensures to measure low kinetic energy Synchrotron sources Synchrotron radiation is the electromagnetic eld emitted from a relativistic electron. A synchrotron machine consists of three parts: the linear accelerator which produces the electrons and accelerates them; a booster, where electrons become relativistic and a storage ring where electrons are maintained at constant velocity and orbit as can be seen in Fig. 84. Detailed setup of SOLEIL synchrotron where injection in the storage ring is continuous (top-up mode), on opposite of the ESRF where injection is non-continuous with time can be seen in Appendix II. The main interests of using synchrotron radiation for XPS are the tunability of the photons produced, the high photon ux and the high energy resolution that can be achieved.

The energy range can be between infrared to hard x-rays, allowing to produce low or high kinetic energy photoelectron. Various properties of synchrotron radiation can be used as its coherence for diraction experiments, the polarization can be tuned for absorbance or dichroism experiments.

Electron analyzer Most of the analyzers in XPS are electrostatic hemispherical analyzers (HSA) which scheme is represented in Fig. 7: photoelectrons are retarded with electrostatic lenses, then selected in energy inside the hemispheres and can reach the detector, a table summarizing some of the commercial electron analyzers performances is presented in Appendix III.

The aperture of the analyzer is composed of a thin entrance slit with acceptance angle of a few degrees, which can be changed by the operator. A higher entrance slit increases the number of collected photoelectrons but decreases the resolution, this mode is called high ux.

Figure 7: Schematic of an hemispherical analyzer with retarding lens, Adapted from [START_REF] Thiess | Hard x-ray photoelectron spectroscopy from 5-14.5 kev[END_REF] .

To measure the kinetic energy of the photoelectrons, the analyzer is composed of electrostatic lenses which select the desired kinetic energy by applying a tension reducing the energy to the selected pass energy at which electrons will travel into the analyzer. For example if the selected kinetic energy acquisition is between 100 and 200 eV with a pass energy of 50 eV, the lenses will apply a potential dierence of 50 eV to 150 eV during the measurement. This function is very important as the resolution depends on the pass energy, which must stay constant to operate with constant resolution.

The photoelectron is then selected in energy in by application of a potential dierence ΔV between the two hemispheres of the analyzer. Only photoelectrons of selected energy E can reach the detector, photoelectrons of higher or lower kinetic energy do not reach the detector. The overall resolution ∆E is given by :

∆E E p = s 2R + α 4 (16)
With E p ∝∆V the pass energy of the analyzer, s the entrance slit size, R the mean radius of the hemispheres and α the photoelectron angle from the optical axes of the retarding lenses.

The photoelectrons reach the detector, usually a system based on charge coupled device (CCD) camera and Multi-Channel Plate (MCP) detector. MCP is used to increase the number of photoelectrons with a cascade eect along very-small glass capillaries and convert them into photons.

The photons are detected by the CCD camera. This system is used as it yields high speed recording and very high spacial resolution.

1.2 Energy loss phenomena in photoemission 1.2.1 Two step Vs three step model Section 1.1 described the three step model in photoemission (photoionization; photoelectron traveling inside the medium; photoelectron crossing the surface and traveling into vacuum) but a simplication in a two step model where the third step is neglected can be used. This approximation can be signicant when surface physics is predominant in the experiment but this approximation is valid for HAXPES experiments.

The processes that take place as the photoelectron is traveling inside vacuum toward the analyzer is almost always neglected, as the ultra-high vacuum regime must be maintained in the experimental environment at a pressure lower than 10 -9 mbar, which corresponds to the highest pressure at which the analyzer can work.

Two phenomena can happen at the surface that can change the detected photoelectron kinetic energy:

Band bending can be important. This phenomenon appears when surface atoms are ionized.

To compensate the positive (or negative) charge appearing on the surface atoms, a small negative (or positive) space charge must appear. This zone is usually of a few nanometers [START_REF] Reshchikov | Surface photovoltage in undoped n-type gan[END_REF].

Surface photovoltage, which corresponds to surface potential modication due to radiation on the sample [START_REF] Kronik | Surface photovoltage phenomena: theory, experiment, and applications[END_REF]. The intense beam (especially from synchrotron radiation) can create electron-hole pairs by interband transition or by liberation of carriers trapped inside bands.

In this zone, electric eld leads to diusion toward the bulk for majority carriers whereas minority carriers are trapped on the surface.

Type of excitations inducing photoelectron energy loss

The excitations accompanying photoionization in solids can be roughly classied in four types:

the intrinsic energy losses which are due to the hole created after photoionization, the extrinsic energy losses due to electron transport in the sample, the multiple excitations and interferences between intrinsic and extrinsic contributions. They inuence the spectral shape and decrease the measured photoelectron kinetic energy. It results a tail of the elastic peak (or no-loss peak) toward lower kinetic energy, called inelastic background.

The extrinsic contributions The electric eld created by the photoelectron during its transport in the sample cause excitations, this energy loss processes are referred as extrinsic contributions. They are mainly dominated by plasmons loss. The plasmons are collective excitations of electrons. The plasmon energy ω 0 depends on the energy of the photoelectrons and on the The intrinsic contributions The core-hole created by the photoemission process produces an electric eld which excites the valence electrons and decreases the photoelectron energy. This processes are named intrinsic contributions and account for 20-50% of the measured intensity in an energy range ∼ 15 -25 eV [START_REF] Penn | Role of intrinsic plasmons in conduction-band x-ray photoemission from solids[END_REF] and extend to ∼ 50 eV below the no-loss peak [START_REF] Tougaard | Background removal in x-ray photoelectron spectroscopy: Relative importance of intrinsic and extrinsic processes[END_REF]. They can give rise to satellite peaks as described in Fig. 8:

The shake-up eect is when the photoelectron interacts with the valence band. In this process a valence band electron is promoted to a higher orbital of the valence band. The resulting measured kinetic energy of the photoelectron is then lower and forms a distinguishable satellite peak.

The shake-o eect also arises when the photoelectron interacts with the valence band, but in this process a valence band electron is promoted to the conduction band. The resulting measured kinetic energy of the photoelectron is then lower and contributes to the inelastic background.

Multiple excitations Multiples excitations are possible in photoemission but their inuence is negligible both for energy determination and intensity contribution in the usual experiments. The energy loss due to phonon-electron interaction is also negligible [START_REF] Hedin | Transition from the adiabatic to the sudden limit in core-electron photoemission[END_REF].

Interferences between extrinsic and intrinsic excitations. Interferences between extrinsic and intrinsic contributions are considered to be very weak in energy loss (< 20 eV) and have a small eect on the measured intensity [START_REF] Fujikawa | Theoretical study of plasmon losses in corelevel photoemission spectra[END_REF]. The Begrenzungs eect is the decreasing of the bulk inelastic scattering cross-section close to the surface. This eect which comes from the orthogonal coupling between the volume and surface modes [START_REF] Ritchie | Plasma losses by fast electrons in thin lms[END_REF] is very weak. Fujikawa et al. showed that interference term in bulk plasmon losses is much smaller than that from the surface plasmon losses [START_REF] Fujikawa | Theoretical study of plasmon losses in corelevel photoemission spectra[END_REF] and therefore is of negligible contribution to the spectrum measured.

Hard x-ray photoemission (HAXPES)

The rst x-ray photoemission experiment was performed in the 60's by K. Siegbahn et al. using of hard x-rays [START_REF] Siegbahn | Esca atomic, molecular and solid state structure studied by means of electron spectroscopy[END_REF] produced by Cu K α and Cu K β radiation. Due to very low photoemission intensity, Cu sources were quickly dropped. For 40 years, HAXPES was only performed in synchrotron facility and devolved to fundamental aspects: Lindau et al. [START_REF] Spicer | X-ray photoemission spectroscopy[END_REF] measured the intrinsic linewidth of the Au 4f core-level. At his point HAXPES studies were performed with low resolution. It is only since major technical improvements from the early 2000's that high resolution HAXPES beamlines are available to user experimentation. Since then, HAXPES experiments started to increase and beamlines were created at synchrotron facilities. The number of workshops, congresses and publications is also increasing, K. Siegbahn even went out of retirement to publish an historical article in the special issue of NIMA in 2005 [START_REF] Siegbahn | Preface to hard x-ray photo electron spectroscopy (haxpes)[END_REF]. A table of main HAXPES beamlines and their capabilities are summarized in Appendix II.

The main interest of using hard x-rays is to increase the probing depth which allows studying deeply buried interfaces. The decreasing photoionization cross section and the low eciency of photoelectron analyzer for high energy photoelectrons had been experimental diculties for a long time. Only recent advances in high brilliance X-ray undulator sources at 3rd generation synchrotrons and commercially available high voltage electron analyzers had broadened the community [START_REF] Drube | Preface[END_REF]. The details of these technological advances can be found in ref [START_REF] Kobayashi | Hard x-ray photoemission spectroscopy[END_REF]. The eects of high energy regime (for photons and photoelectrons) on the three steps describing the photoemission are detailed hereafter.

Photoionization in HAXPES

Photoionization cross-section The main dierence compared to soft x-rays photoemission is the drastic decrease in photoionization cross-section, which is in general two orders of magni-tude lower than the one obtained with classical laboratory sources. Theoretical calculations by Trzhaskovskaya et al. [11,[START_REF] Trzhaskovskaya | Photoelectron angular distribution parameters for elements z=55 to z=100 in the photoelectron energy range 100-5000 ev[END_REF][START_REF] Trzhaskovskaya | Non-dipole second order parameters of the photoelectron angular distribution for elements z=1-100 in the photoelectron energy range 5-10kev[END_REF] provide values for photoelectron of 5 and 10 keV. Interpolation between these two values is commonly admitted. Extrapolation for higher kinetic energies is possible with the use of a t made by two exponential decays but without any theoretical justication at present. Furthermore, for the same element, a high principal quantum number and a low angular momentum are promoted (i.e. s shells are easier to be seen than p shells) at high energy as can be seen in Fig. 9.

Figure 9: Photon energy dependence of ratios of subshell photoionization cross-sections for selected atoms from [START_REF] Kövér | X-ray photoelectron spectroscopy using hard x-rays[END_REF].

Recoil energy The photoelectrons produced during HAXPES experiments are emitted with such high energy that a term called recoil energy and noted E R is not negligible anymore [36, 37, 38]:

E R = - Q 2M (17)
where

Q = q -k ≈ 2( ω -E B ) ( 18
)
and M is the mass of the emitter atom, q is the photon momentum and k the momentum of the photoelectron. This term induces a broadening of the spectrum [START_REF] Fujikawa | Theory of recoil eects of elastically scattered electrons and of photoelectrons[END_REF]:

(E R ) 2 = Q 2 k B T M (19)
The recoil energy is neglected in classical XPS analysis where the momentum transfer is very

hν = E K,m + E ε + ∆ε + φ + E R (20)

Photoelectron transport inside the material in HAXPES

It has been shown that usual TPP-2M formula with simple pole approximation leads to good results with Bethe equation when employed for high kinetic energy photoelectrons with an estimated error about 12.3% [START_REF] Tanuma | Calculations of electron inelastic mean free paths. ix. data for 41 elemental solids over the 50 ev to 30 kev range[END_REF]. As seen in eq.7 and in Fig. 4, increasing the kinetic energy of the photoelectron increases the IMFP (thus the EAL) and therefore the probed depth. The dierence between the IMFP and EAL can be studied by dening the single scattering albedo ω as [START_REF] Kövér | X-ray photoelectron spectroscopy using hard x-rays[END_REF]: where λ tr is the transport mean free path dened as:

ω = λ λ + λ tr (21)
λ tr = 1 M σ tr (22)
with M the atomic density and σ tr the transport cross-section dened as:

σ tr = ˆ4π(1 -cos θ)( dσ /dΩ)dΩ (23)
with dσ /dΩ the dierential cross-section for elastic electron scattering. The data are easily accessible via a NIST software [START_REF] Powell | Nist electron elastic-scattering-cross-section database[END_REF][START_REF] Jablonski | Nist electron eective-attenuation-length datadata[END_REF] and calculations of ω allow estimating the importance of the elastic scattering compared to inelastic scattering. This dependence is shown in Fig. 11 for Si, Fe, Ni, Cu, Ge, Ag, W and Au. The linear decreasing after 1.5 keV kinetic energy indicates the decreasing strength of elastic scattering at higher energy [START_REF] Powell | Progress in quantitative surface analysis by x-ray photoelectron spectroscopy: Current status and perspectives[END_REF]. It has been shown by Jablonski and

Powell [START_REF] Jablonski | Practical expressions for the mean escape depth, the information depth, and the eective attenuation length in auger-electron spectroscopy and x-ray photoelectron spectroscopy[END_REF] that average EALs L ave of 16 photoelectron lines and 9 Auger lines of elemental solid Si, Cu, Ag, W, Au and inorganic compounds ZrO 2 , ZrSiO 4 , HfO 2 and HfSiO 4 varied linearly with the single scattering albedo ω as:

L ave = λ(1 -0.735ω) (24)
Eq.24 shows that when elastic scattering tends to 0, which is the case in HAXPES, the average EAL is equal to the IMFP. Thus for analyzing HAXPES spectra, the approximation of replacing the EAL by the IMFP is correct, which simplies the quantication as the IMFP is a computed value whereas the EAL is a measured value.

Crossing the surface and traveling toward the analyzer

Electron transport in materials leads to two extrinsic excitations: surface plasmons which are largely considered in classical theory and bulk plasmons which are neglected for low energy pho-toemission. For high energy photoemission, it's the opposite: a major part of the signal comes from the bulk instead of from the surface [START_REF] Sacchi | Quantifying the eective attenuation length in high-energy photoemission experiments[END_REF] as the probed depth is increasing with increasing photoelectron kinetic energy. Another eects of increasing the photoelectron kinetic energy is the decreasing of surface losses relatively to bulk losses [START_REF] Stern | Surface plasma oscillations of a degenerate electron gas[END_REF].

Photoelectron angular distributions depend strongly on the kinetic energy. For energies higher than a few keV, non-dipolar eects (rst and second order) have a strong inuence on spectra distortion [START_REF] Novák | Polarization and dipole eects in hard x-ray photoelectron spectroscopy[END_REF]. The elastic scattering diusion coecient is also strongly modied by higher order non-dipolar eects and must be amended for use of ARXPS at high energy.

Experimental

There are two major experimental diculties in HAXPES experiments:

Hard x-rays laboratory sources have a much lower photon ux than the soft ones (a few orders of magnitude). Although the technique started in laboratory, acquisition time was very long to obtain a satisfactory signal/noise ratio. Furthermore, it is necessary to have a monochromator in order to achieve high resolution and Bragg reections decrease drastically the photon ux. For this reason, a highly brilliant source in 3rd generation synchrotron must be used. Furthermore, the tunable energy from synchrotron radiation will allow to select the probed depth. Only two HAXPES laboratory machines exist: one can only perform qualitative measurement to our knowledge [START_REF] Kobata | Development of the hard-x-ray angle resolved x-ray photoemission spectrometer for laboratory use[END_REF][START_REF] Kobata | Application of cr k x-ray photoelectron spectroscopy system to overlayer thickness determination[END_REF] and the other was just developed [START_REF] Kobayashi | Development of a laboratory system hard x-ray photoelectron spectroscopy and its applications[END_REF].

High kinetic energy photoelectrons are dicult to measure precisely. In order to achieve high energy resolution, the high kinetic energy photoelectrons must be decelerated [START_REF] Briggs | Surface Analysis[END_REF]. This can be achieved by placing retarding lens between the sample and the analyzer entrance [START_REF] Thiess | Hard x-ray photoelectron spectroscopy from 5-14.5 kev[END_REF].

This was the case for ID-32 beamline at ESRF where experiments were conducted. The diculty relies in the stability of the current delivered in the kV range by the power supply for which the noise must be < 200 μV.

Conclusion of Chapter I

X-ray Photoelectron Spectroscopy is a widely used experiment allowing to obtain chemical information from the surface of a sample by analyzing the photoelectric peak (or no loss peak). In this case the probing depth, which is up to 9 nm in soft x-rays XPS, is limited by the kinetic energy of the photoelectron.

When increasing the photon energy to hard x-rays (HAXPES), the probing depth can increase up to 20 nm but the photoionization cross section decreases. The recoil energy in the energy conservation law becomes non negligible at these high energy and non-dipolar eects aect several parameters, promoting the forward scattering and the bulk predominant information at the expense of the surface.

The energy loss phenomena have been described and the next Chapter will show that analysis of inelastically scattered photoelectrons can increase the probing depth in photoemission and thus allow characterization of deeply buried layer.

2 Inelastic background analysis and other depth proling methods

Here are presented the most commonly used techniques in photoelectron spectroscopy to obtain depth distribution of deeply buried elements. The principal other experiments that allow to obtain such information are presented in Appendix IV. Among them, the photoelectron spectroscopy is used for its non-destructive character and its ability to obtain the elemental composition in the surface and near-surface region.

In classical XPS, the measurement of a high resolution core-level spectrum allows to obtain depth distribution within the 3 rst IMFP below the surface. The maximum probed depth when analyzing core-level is up to 9 nm with use of soft x-rays and 50 nm with hard x-rays (the deepest layer analyzed with hard x-ray published so far is buried at 25 nm [START_REF] Panaccione | Hard x-ray photoemission spectroscopy: Variable depth analysis of bulk, surface and interface electronic properties[END_REF]). This probing depth remains too low to analyze deeply buried interfaces in a stack of technological interest. Therefore the analysis of the inelastically scattered photoelectrons which ensure a probed depth up to 8 IMFP may be a solution to solve this issue [START_REF] Tougaard | Accuracy of the non-destructive surface nanostructure quantication technique based on analysis of the xps or aes peak shape[END_REF].

This Chapter describes the main XPS experiments to obtain depth prole using core-level analysis and inelastic background analysis.

Usual depth-proling methods in core-level photoemission

There are three widely used types of core-level analysis to obtain depth prole: the depth proling with sputtering, the Angle Resolved X-ray Photoemission Spectroscopy (ARXPS) which employs the sinθ dependence of eq.13 to obtain depth prole (i.e. angle of measurement) and the Variable Kinetic Energy X-ray Photoemission Spectroscopy (VKE-XPS) which exploits the energy dependence of eq.13.

They possess the advantage of high sensitivity to chemical states and other characteristics of surfaces but are limited in probed depth to 3 IMFP (i.e. ~9 nm below the surface) as they are based on the study of the no-loss peak, except for the depth proling with sputtering which is explained hereafter.

Depth proling with sputtering

In a sputtering surface sensitive technique, a monoatomic ion source sputters the surface to remove a few layers o the surface and a surface sensitive experiment is done. By cycles of sputtering and/or analysis, depth distribution of elements as a function of the sputtering time is obtained.

The ion bombardment can induce variations of the chemical composition and morphology of the surface, changing the obtained depth prole.

The measurement can be performed either during sputtering (continuous mode) or with cycles of sputtering/measurement (alternate mode). The continuous mode enables fast data acquisition and limits surface contamination if the pressure in the system is not suciently low. The disadvantages are the lower signal-to-noise ratio and the ion beam induced Auger electrons from the crater rim which may distort the prole, especially for low energy peak (Al, Si and Mg for example).

If the study is focused on thin layers or possesses a concentration gradient near detection limit, further complications can arise from continuous mode. To overcome this diculty, the alternate sputtering is performed. This mode can be employed only if the residual background pressure is suciently low to avoid surface contamination.

Sputtering Sputtering is a process where primary ions are accelerated to a few keV onto the sample. As the ions impinge the sample, they transfer energy and momentum to the sample, as a function of the mass of the incident ions and targeted atoms. The collision parameters are described by binary Rutherford scattering. The cross section for elastic energy transfer is high [START_REF] Sigmund | Theory of sputtering. i. sputtering yield of amorphous and polycrystalline targets[END_REF], resulting in a collision cascade. Most of the incident particles are consumed in generating damage and heat within the collision cascade. Sputtering will occur if a struck target atom near the surface is accelerated in a direction toward the vacuum with sucient energy to overcome the surface energy barrier. The process of sputtering can be described by a random ejection of surface atoms during ion bombardment: the roughening of the surface is counteracted by surface diusion [51]. The relocation of the atoms results in interdiusion, resulting in an atomic mixing zone [START_REF] Naundorf | Limits of depth resolution for sputter sectioning: Moments of tracer depth distribution[END_REF][START_REF] King | A model for atomic mixing and preferential sputtering eects in sims depth proling[END_REF]. Also the variation of sputtering combined with variation of crystalline orientation and inhomogeneities of the chemical composition result in a larger-scale roughening. The etching rates are generally given by the manufacturer of the ion gun as it depends on its specicity. For specic samples (with overdensity, voids, exotic alloys etc.) the sputtering rate must be measured with a reference sample.

If the ions impinge the surface at an angle, the sample must rotate during the sputtering to get an uniform crater as seen in Fig. 12(b). The so-called Zalar rotation improves depth resolution [START_REF] Geller | Depth resolution improvements using specimen rotation during depth proling[END_REF]. Recent advance allows to employ ions formed by cluster of hundreds of atoms, decreasing the amount of substance removed and the defects on the sample [START_REF] Yamada | Materials processing by gas cluster ion beams[END_REF]. Depth prole The depth prole is typically obtained by (1) calibration of the intensity scale in unit of instantaneous concentration, (2) converting the sputtering time to a sputter depth,

(3) deconvolution of the measured prole with the aid of a depth resolution function. For many applications, it is sucient to assume that the intensity I measured and the depth prole can be treated separately [START_REF] Hofmann | Sputter depth prole analysis of interfaces[END_REF].

The calibration of the intensity scale in unit of instantaneous concentration can be done when knowing the sputtering yield Y m , from literature or measurement of a reference sample. As the sputtering induces changes on the surface of the sample, a practical way for obtaining depth prole concentration is to measure the total core-level peak area, losing the chemical information. The sputter depth can be written as:

z(t) = ∞ 0 . z(t)dt (25)
with .

z = dz/dt = j p Y m /eN [START_REF] Tougaard | Background removal in x-ray photoelectron spectroscopy: Relative importance of intrinsic and extrinsic processes[END_REF] where j p is the ion-current density of the primary ions, e the elementary charge and N is the atomic density. The error in depth calibration mainly comes from variations in the sputtering rate and surface contamination layer and can change the thicknesses up to 5 nm. Another source of error is the shift of 50% intensity at interfaces toward lower depths, caused by the photoelectron escape depth. The atomic mixing can also gives an error on the estimated thickness up to 5 nm [START_REF] Kupris | Interpretation of sputter depth proles by mixing simulations[END_REF].

Under favorable conditions, these errors can be minimized, but but better results can generally be obtained by using a depth resolution function (DRF).

The experimentally-determined normalized in-depth intensity is given by a convolution of the mole fraction X(z) of an element at depth z, and the DRF g(z) as [START_REF] Ho | Deconvolution method for composition proling by auger sputtering technique[END_REF]:

I(z) I(0) = ˆ+∞ -∞ X(z )g(z -z )dz (27)
where z' is the running depth parameter for which the composition is dened. The shape of g(z)

depends on the material and sputtering process. X(z) can be determined by deconvolution of Eq.27, the diculties are arising if the data precision and the signal to noise ratio are low.

DRF can be approximated by an error function, assuming that the sputter depth prole obtained is representative of a sharp interface. In this case g(z) is a Gaussian function and the depth resolution ∆z is given by 2σ, with σ the standard deviation of the Gaussian function. In a general way, the depth resolution is given by several factors and can be written as [START_REF] Hofmann | Depth resolution in sputter proling[END_REF]:

∆z = (∆z 2 0 + ∆z 2 s + ∆z 2 m + ∆z 2 λ + ∆z 2 r + ...) 1/2 (28)
where 0 stand for the intrinsic surface roughness, s for surface roughening by sputtering statistics, m for the atomic mixing and preferential sputtering, λ for the information depth and r for the ion sputtering induced roughening.

Angle Resolved X-ray photoelectron spectroscopy (ARXPS)

ARXPS is widely used for its ability to retrieve depth proling and quantication of thin layers.

The method is more precise than depth proling with sputtering but is strictly limited to the rst three IMFPs. Despite very good absolute depth resolution, without prior knowledge on the sample, atomic resolution cannot be achieved. It is possible to obtain information of samples with lateral inhomogeneities, roughened surfaces, island covered surfaces with additional assumptions.

For clarity purposes, the next two paragraphs illustrate the simpler cases of a at surface covered with a thin homogeneous layer and the multiple-layeres structure respectively.

The method relies on intensities of no-loss photoelectron peaks. By changing the angular detection, the probed depth is changed as can be seen in the Fig. 13.This gure illustrates the following situation: the sample consists of a top overlayer on a substrate with a broad interface between them. If photoelectrons are collected at low take-o angle (in red), the measured spectrum is representative of the surface overlayer. If the take-o angle increases (in blue), the experiments becomes more bulk sensitive and the measured spectrum is representative of the overlayer, the substrate and the interface. With measurements at various angles, the depth prole can be reconstructed.

Generalities for core-level analysis The analysis of the core-level photoemission spectra possesses several degrees of sophistication for the analysis, and even if cookbooks exist, the methodology is chosen by the operator according to his/her experience [START_REF] Paynter | Comparison of regularization methods for the inversion of {ARXPS} data[END_REF]. In most cases, the core-level high resolution spectrum is corrected for inelastically scattered photoelectron by subtraction of a background. The resulting curve is the tted, generally using a convolution of Gaussian and a Lorentzian. The Lorentzian Full Width at Half Maximum (FWHM) depends on the core-hole lifetime whereas the Gaussian FWHM is dominated by the overall resolution. For metals, the peak can be asymmetric, either by introducing an asymmetry parameter or using a Doniach-Sunjic lineshape [START_REF] Doniach | Many-electron singularity in x-ray photoemission and x-ray line spectra from metals[END_REF]. The center of the corresponding curve is representative of the chemical shift of the element and is used for attribution of the chemical environment of the considered atom.

Single layer analysis of ARXPS measurement Determination of uniform lm thickness is by far the most used system treated by ARXPS.

If S s and S o are the sensitivity factors for the element present in the substrate and the overlayer respectively, and the spectrum is acquired at take-o angle θ, the measured corresponding peak intensities I s and I o are dened by: I s s s = e -λs cos θ (29)

I o s o = e -t/λ 0 cos θ ( 30
)
where t is the overlayer thickness, λ o and λ s are the EALs of the photoelectrons from the overlayer and the substrate when passing through the overlayer material. If the substrate and the overlayer are composed by dierent elements, the EALs are very dierent and resolving the two previous equations involves solving a nonlinear transcendental equation. In the simpler case where the same element is present in the substrate and in a dierent chemical state, i.e. suciently shifted in measured kinetic energy to be resolvable, the two EALS can be approximated and the Eq.29 and 30 can be rearranged to obtain the so-called Hill equation:

t = λ cos θln 1 + I o /s o I s /s s (31)
The curve represented by Eq.31 is plotted in Fig. 14. Two areas are shaded, corresponding to the limit of the model:

When the layer thickness is below the thickness of a single atomic layer, it is more accurate to describe the overlayer with a covering factor.

Figure 14: Layer thickness measurement using the Hill equation (Eq.31). Extracted from [START_REF] Briggs | Surface Analysis[END_REF].

When the overlayer thickness is greater than 3 IMFP (right part of the gure), the substrate peak is too small to be measure accurately.

This method excels at determining surface oxide thickness, typically silicon or aluminum.

Multilayered structures analysis of ARXPS measurement In the case of a multilayered structure, it is possible to obtain the depth distribution of several layers, if they are in the 3 IMFP range from the surface. In the case of a stratied sample (where the chemical states of the layers do not interfere in the XPS spectrum) with n layers, the Eq.29 and 30 can be generalized to: [START_REF] Siegbahn | Preface to hard x-ray photo electron spectroscopy (haxpes)[END_REF] where I n (θ)/I n (0) is the ratio of the peak area at angle θ and normal emission, t n , λ n and d n are the thickness, the EAL and the mean depth of the n th layer. k(θ) is an instrumental function that can be estimated by the intensity ratio of the background I b at the angle θ and at normal emission as:

I n (θ) I n (0) = k(θ) 1 -e tn λn cos θ 1 -e -tn λn e -dn λn cos θ e -dn λn
k(θ) = I b (θ) I b (0) (33)
It is then possible to obtain a value close to the average depth of the layer called characteristic depth parameter Z n , representing a depth approximately 35% of the distance between the top and bottom interface of the n th layer dened as:

Z n = ln I i (θ)I b (0) I i (0)I b (θ) -ln(sec θ) 1 -sec θ (34)
Another useful quantity, the relative amount of material in the nth layer, A n , can be obtained:

A n = I n (0) S n e Zn
The theoretical depth resolution ∆z at depth z is given by [START_REF] Cumpson | Angle-resolved xps and aes: Depth-resolution limits and a general comparison of properties of depth-prole reconstruction methods[END_REF]:

∆z z = 2sinh    π 2 2cosh -1 π(n -1)( I σ i ) 2    (35)
with (σ I /I) the fractional precision of the peak intensities and n the number of layers. For example, if the measurement of the peak intensity is 2% over 5 emission angles for a depth z of 2 nm, the depth resolution ∆z 1.8 nm. From this observation, P.J. Cumpson showed the need of prior knowledge on the sample [START_REF] Cumpson | Angle-resolved xps depth-proling strategies[END_REF].

The stratication method is very powerful for quantication of ARXPS data because of the combination between the depth and relative amount in each layer. However the method cannot distinguish two or more distinct layers having similar composition (where the peak shift is below the resolution of the experiment).

The case of a stratied sample is however not the most common situation encountered. To analyze continuous depth prole (often encountered in diusion process), stepped interface or layers with dierent composition (in this case the approximation of equal EAL cannot be retained), additional assumptions on the sample composition and/or depth model prole must be employed.

The next paragraph illustrates one of the most used numerical method, the Maximum Entropy.

Regularization method -Maximum entropy ARXPS method relies on the determination of the intensity and energy of the no-loss peak. This quantity is not trivial to quantify as the no-loss peak is dependent on the structure of the sample (concentration and location) and on many physical parameters such as the chemical shift, the core-hole lifetime, the photoionization cross-section, the photon beam (size and energy resolution) and the analyzer eciency. To increase the reliability and the consistency of the quantication, the operator can use of a regularization method [START_REF] Cumpson | Angle-resolved xps depth-proling strategies[END_REF].

A regularization procedure quanties a matching between a measurement and a predictive formula by means of two quantities, the residual norm χ and the solution norm α. The residual norm is the standard deviation root mean square χ and is expressed as the formula:

χ = 100 × i (y m i -y b i ) 2 n i (36)
with y i m the measured intensity at energy i, y i b the modelled intensity at energy i and n i the number of points considered. A smaller value of χ indicates a better match between measurement and modeling. The solution norm is a measure of the complexity of the proposed depth prole. It can be of dierent forms, depending on the prior knowledge of the sample structure. Paynter et al. showed that the results obtained for an exponential prole with various regularization schemes depends more on the prior input parameters (concentration, slopes or curvature in depth prole) than on the regulator which calculates the prole entropy [START_REF] Paynter | Comparison of regularization methods for the inversion of {ARXPS} data[END_REF]. However the specic method of Maximum Entropy is widely used in ARXPS and VKE-XPS analysis, which will be explained in the next section.

In both statistical thermodynamics and information theory, the entropy is a measure of disorder (negative quantity). In information theory, the entropy S is the opposite of information content dened as [START_REF] Weaver | The mathematical Theory of Communication[END_REF]:

S = - i p i ln(p i ) (37)
with p i the probability of nding the system in a state i. The Maximum Entropy (MaxEnt) designates a regularization method which minimizes the information content or maximizes the entropy. If extended to include all positive additive terms, then eq.37 is developed as [START_REF] Gull | Maximum entropy method in image processing[END_REF]:

S = i j n j,i -m j,i -n log( n j,i m j,i ) (38)
with n j,i the proportion of element j at depth i and m j,i the initial estimate for the proportion of element j at depth i. The MaxEnt solution is then obtained by maximizing S. It is done by calculation of chi-squared χ 2 (not to be confused with χ of eq.36) mist statistic C dened as:

C = χ 2 = k I calc k -I obs k σ 2 k ≤ N (39)
Q = αS - C 2 ( 40
)
where α is a regularizing parameter. A probability distribution of α can be determined by Bayesian statistical analysis [START_REF] Gull | Maximum Entropy and Bayesian Methods[END_REF], but if the data present sharp peaks, the determination of α can be performed by the operator as a large value of α will result in an over-smooth solution while a small value will lead to overtting the data [START_REF] Smith | Maximum entropy: A new approach to non-destructive deconvolution of depth proles from angle-dependent xps[END_REF].

Active background method The regularization method previously described details the noloss peak tting procedure. However the spectra must be previously corrected from background subtraction of secondary photoelectrons (which will be detailed in the next Section). This static approach implying operator decision upon the background type and energy range (dened as (E left , I left ) and (E right , I right )) can lead to deviations on quantication results [68]. In the active background approach, the shape and intensity of the background is assessed during the peak tting and can be a combination of background types [START_REF] Herrera-Gomez | Practical methods for background subtraction in photoemission spectra[END_REF]. The measurement of intensities depending on the take-o angle is tted with a model for depth distribution of all elements.

The peaks of interest for retrieving the depth prole are treated in parallel, which maximizes the consistency of the procedure. This is illustrated in the Fig. 15(a) showing the intensity versus take o angle for all the elements present in a sample of TiN/Al 2 O 3 /InGaAs [70] and tted with the theoretical behavior of the model in Fig. 15(b). This example also shows the robustness of the active approach which can treat simultaneously two overlapping peaks and the accuracy of the prole obtained [START_REF] Herrera-Gomez | The slope-background for the near-peak regimen of photoemission spectra[END_REF].

This new method has the interest of increasing the independence of the results upon operator background choice. The method is implemented in a commercial software AAnalyzer ® [72] and therefore not widely used, in spite of its high accuracy which can shows sub-nanometric diusion

Variable kinetic energy

The method also relies on intensity determination of no-loss peaks. By increasing the photon energy, the IMFP (or EAL) of a given core-level photoelectron, thus the probed depth, increases.

It allows to probe several interfaces [START_REF] Weiland | Nondestructive compositional depth proling using variable-kinetic energy hard x-ray photoelectron spectroscopy and maximum entropy regularization[END_REF] but requires the use of a reference sample to be quantitative.

To probe the sample with various photon energies, dierent laboratory sources can be employed but the tunability of the photon energy delivered by synchrotron radiation is more adapted to the situation. This paragraph illustrates the simpler case of an overlayer thickness determination with illustration taken from Rubio-Suazo et al. [START_REF] Rubio-Zuazo | Probing buried interfaces on ge-based metal gate/high-k stacks by hard x-ray photoelectron spectroscopy[END_REF] of determination of a Ge overlayer on Si semi-innite substrate.

The peak intensity I, depending on the kinetic energy E kin of the photoelectron of an element located between depth Z A and Z B , referenced to the bulk signal I ∞ is dened as:

I(E kin ) = I ∞ N ∞ A λ(E kin) cos θ ˆZB Z A n A (z) exp - z λ(E kin ) cos θ dz (41)
with λ(E kin ) the EAL depending on the kinetic energy, n A (z) is the atomic density prole as a function of depth, N ∞ A is the density of the bulk and θ is the angle between the surface normal and the emitted photoelectrons.

In the case of an overlayer A of thickness Z A deposited on semi-innite substrate B, the integral and Ge, without the formation of Ge oxide, for both samples. Extracted from [START_REF] Rubio-Zuazo | Probing buried interfaces on ge-based metal gate/high-k stacks by hard x-ray photoelectron spectroscopy[END_REF].

is evaluated between 0 and Z A for the overlayer and between Z A and innite for the substrate, thus:

I A = I ∞ A 1 -exp - Z A λ(E kin ) cos θ (42)
and

I B = I ∞ B exp - Z A λ(E kin ) cos θ (43)
By tting the I n /I ∞ n measurements as a function of the kinetic energy of the photoelectron as presented in Fig. 16, the thickness Z A is obtained, corresponding in this case of to 8.9 nm-thick overlayer. The I ∞ n is the intensity of a pure semi-innite sample measured in the same condition, hence is the requirement of a reference sample.

Many physical parameters such as the photoionization cross-section, the EAL and the plasmons intensity change with variation of the photon energy, complicating the retrieving of the depth prole and appearance of Auger peak can overlap with the measurement. However the method allows to obtain relative depth prole or simple overlayer model [START_REF] Rubio-Zuazo | Information depth determination for hard x-ray photoelectron spectroscopy up to 15 kev photoelectron kinetic energy[END_REF][START_REF] Rubio-Zuazo | Non-destructive compositional depth prole analysis by hard x-ray photoelectron spectroscopy[END_REF][START_REF] Rubio-Zuazo | Non-destructive compositional depth prole in the tens-of-nanometer scale[END_REF][START_REF] Rubio-Zuazo | Probing buried interfaces on ge-based metal gate/high-k stacks by hard x-ray photoelectron spectroscopy[END_REF], the concentrations of the elements depending upon depth prole have not been calculated yet. Similarly to ARXPS, specic data treatment is required in VKE-XPS to retrieve an accurate prole, such as MaxEnt presented previously.

Summary

The main limitation of core-level analysis is the probing depth of 3 IMFP. The choice between the three types of experiments for core-level depth proling is up to the operator. This choice is mainly determined by the location of the point of interest and the in-depth precision needed:

The sputtering depth prole technique can overcome the 3 IMFP probing depth limitation, but the use of the ion gun for sputtering induces surface modication and decreases the depth resolution. However with the use of depth resolution function a precision < 1 nm can be reached.

The ARXPS is strictly limited to 3 IMFP. This experiment, with additional use of a regularization method gives the best accuracy possible, near atomic resolution, but is very sensitive to surface roughness and needs acquisitions of high resolution spectra, which can be rather long.

The VKE-XPS can only be performed with use of several photon sources or in a synchrotron facility. It is dicult to extract the concentration versus depth prole, and to date VKE-XPS has mainly be used for qualitative depth prole or simple overlayer model.

Tougaard method for analysis of inelastic background

The two non-destructive methods to obtain depth prole presented in the previous Section are limited to the rst 3 IMFP in probing depth, because they rely on determination of intensity of the no-loss peak. This Section exposes the inelastic background analysis, which is improving the probing depth up to 8 IMFPs with analysis of a single spectrum.

Principles

This method can be illustrated by spectra of Cu 2p depending on the Cu depth prole in a gold matrix as depicted in Fig. 17. The total amount of Au atoms in the near surface region varies in the four distributions but because of peak attenuation due to inelastic scattering, the observed peak intensities are identical. The inelastic loss shape at lower energy changes from one depth distribution to the other. This method is a modeling of energy loss distribution generated by the photoelectron during its transport in the material. The evaluation of the spectra is based on inelastic scattering cross-sections which may be calculated by a dielectric model. In contrast to the three step model described in sec.1.3.1 the phenomenological description used within Tougaard inelastic background analysis method uses the two step model described in sec.2.1.1.

Modeling of inelastic background is performed by computing the energy loss spectrum and subtracting it from the measured spectrum J(E) to obtain only the spectrum F(E) from photoionization of the element. If F(E 0 ,Ω)d²ΩdE 0 is the photoelectron ux emitted from dE 0 around E 0 from the solid angle Ω, then J(E,Ω) the number of electrons emitted by second and energy unit in

J(E, Ω) = ˆdE 0 F (E, Ω) ˆf (z)G(E 0 , z cos θ ; E)dz (44)
where f(z) is the depth distribution prole from the surface as a function of depth z of the considered element, G(E) is the probability that an electron with initial energy E 0 has an energy in the interval E,E+dE after having traveled a path R = z/cosθ. If the energy loss is small compared to the energy of no-loss peak, then Landau showed [80]:

G(E 0 , R; E) = 1 2π ∞ -∞ e is(E 0 -E) e -R (s) ds (45)
with

(s) = 1 λ i - ˆ∞ 0 K(T )e -isT dT
where s is an integration variable without any physical meaning and K(T)dT is the probability for an electron of energy E 0 to lose energy between T and T+dT per unit path length traveled. To determine the intrinsic spectrum F(E,Ω), a rigorous solution of eq.44 has been found [START_REF] Tougaard | Non-destructive depth proling through quantitative analysis of surface electron spectra[END_REF]:

F (E, Ω) = 1 P 1 J(E, Ω) - 1 2π ˆdE J(E , Ω) ˆdse -is(E-E ) (1 - P 1 P (s) ) (46)
Figure 18: Cross-sections for Fe, Pd, Ti, Cu, Ag and Au determined from REELS analysis and best least-square ts of the Two-and Three-parameters universal cross-sections of eq.49 and eq.50 respectively. Extracted from [START_REF] Tougaard | Universality classes of inelastic electron scattering cross-sections[END_REF].

with

P (s) = ˆf (z)e (-z/ cos θ) (s) (47)
and

P 1 = lim s→±∞ P (s) = ˆ∞ 0 f (z)e (-z/λ cos θ) dz (48)
This function can be evaluated by Fourier transform.

The inelastic scattering cross-section K(T) is the main parameter. These experimental curves which may be obtained by analysis of REELS experiments [START_REF] Tougaard | Dierential inelastic electron scattering cross sections from experimental reection electron-energy-loss spectra: Application to background removal in electron spectroscopy[END_REF][START_REF] Tougaard | Inelastic-electron-scattering cross sections for si, cu, ag, au, ti, fe, and pd[END_REF] depend strongly on the material as can be seen in Fig. 18, but at energy > 30 eV, the individual inelastic scattering cross section have the same trend. The inelastic scattering cross section can also be expressed by mean of an universal curve obtained from least square-tting of several samples, which is a good approximation for transition metals and their oxides [START_REF] Tougaard | Low energy inelastic electron scattering properties of noble and transition metals[END_REF] :

λ(E)K(E, T ) = BT (C + T ²)² (49)
with C = 1643 eV² and B 3000 eV². The cross-section is normalized to unit area with B=2C=3286 eV². For solids with sharp plasmon structure (like silicon, aluminum, germanium ...etc.) [START_REF] Tougaard | Universality classes of inelastic electron scattering cross-sections[END_REF]: The analysis takes into account the eects of inelastic scattering. However in cases where all detected photoelectrons arise from far in depth, the elastic eects can be of importance for the analysis [START_REF] Tougaard | Accuracy of the non-destructive surface nanostructure quantication technique based on analysis of the xps or aes peak shape[END_REF]. From extensive Monte-Carlo simulation, it was found that this correction can be expressed as a simple parameterization factor CF(z,θ,E) accounting for the eects of elastic scattering. For the experimental geometry where θ < 30°(near normal emission) a simple expression was found [START_REF] Lassen | Practical correction procedures for elastic electron scattering eects in arxps[END_REF]:

λ(E)K(E, T ) = BT (C + T ²)² + DT ² (50)
CF (z) = exp(-0.157764τ -1.25132) + exp(-0.0562417τ 2 + 0.00698849τ -0.201962) (51)
where

τ = z λ tr ω = z(λ + λ tr ) λλ tr (52)
with z the burying depth of the element, λ tr the transport mean free path, λ the inelastic mean free path and ω the single scattering albedo from eq.11.

Practical approach

The rigorous and general expression for F(E,Ω) in eq.46 has analytical solutions for some morphologies of depth distribution f(z) [START_REF] Tougaard | Energy loss in xps: Fundamental processes and applications for quantication, non-destructive depth proling and 3d imaging[END_REF] shown in Fig. 20.

f (z) = Ce -z/L (53)
the exact solution of eq.46 becomes [START_REF] Tougaard | Inelastic background intensities in xps spectra[END_REF]:

F (E, Ω) = L + λ cos θ CλL cos θ J(E, Ω) - λL L + λ cos θ ˆ∞ E dE J(E , Ω)K(E -E (54)
and then for an innite sample where L∞ then

F (E, Ω) = 1 Cλ cos θ J(E, Ω) -λ ˆ∞ E dE J(E , Ω)K(E -E) (55)
if one assumes that the sample is homogeneous, i.e. the concentration is constant in the bulk then eq.53 becomes

f (z) = (C s -C b )e -z/L + C b (56)
and analytical expression of 48 can be found

P 1 = C b λ cos θ + (C s -C b ) Lλ cos θ L + λ cos θ (57)
and

P (s) = C b cos θ (s) + (C s -C b ) L cos θ L (s) + cos θ (58)
Layered prole If one consider a delta layer prole where N atoms are located at depth z 0

f (z) = N δ(z -z 0 ) (59)
and the exact expression of eq.48 becomes

P 1 = N e -z 0 /(λ cos θ) (60)
and eq.47 is

P (s) = N e -z 0 (s)/ cos θ (61)
This case is very uncommon in real devices. Instead the atoms are mostly localized between z 0 and z 0 +∆z, thus eq.60 and 61 can be written with

f (z) =          0 f or 0 < z < z 0 C A f or z 0 < z < z 0 + ∆z 0 f or z 0 + ∆z < z (62)
and then

P 1 = C A λ cos θe -z 0 /(λ cos θ) 1 -e -∆z/(λ cos θ) (63)
and

P (s) = C A cos θ (s) e -z 0 (s)/ cos θ 1 -e -∆z (s)/ cos θ (64)
These equations can also be used to represent a surface overlayer with thickness ∆z (where z 0 =0) or for a substrate located at depth z 0 (where ∆z∞).

Equations for modeling island structure have also been found [START_REF] Tougaard | Energy loss in xps: Fundamental processes and applications for quantication, non-destructive depth proling and 3d imaging[END_REF] but as they will not be used in the following they are not be described here.

Equations of 2.4.2 are implemented in a commercial software, QUASES-Analyze ® [START_REF] Tougaard | quases: Software package for quantitative xps/aes of surface nanostructures by peak shape analysis[END_REF]. This software is very user friendly, as can be seen in Fig. 21, only the geometry of the experiment (Angle of emission) and the IMFP value have to be input by the user. There is a possibility to change the inelastic scattering cross section, choosing between the two parameter universal cross section valid for transition metals and their oxides and the computed cross sections for specic elements (polymer, SiO 2 , Si, Ge, Al). The user also has the possibility of performing modeling with its own inelastic scattering cross section (X-Sect, determined for example from a REELS experiment). The software can perform simple manipulation of the data, as noise reduction, correcting it for analyzer transmission and subtracting linear background. The user can choose between several morphology classes to perform modeling. By clicking on Buried Layer, the buttons Start Depth and End Depth opens, and the user can click step by step to increase or decrease the interface locations until a good match between the modeling and the measurement is found. The resulting subtracted spectrum is also plotted (in Fig. 21, the top interface position is set to be 7 nm below the surface and the bottom interface is out of the probed depth).

Comparison with the Shirley background

The so called Shirley background [START_REF] Shirley | High-resolution x-ray photoemission spectrum of the valence bands of gold[END_REF] is used to correct spectra from inelastic scattering in the vicinity of the no-loss peak. The background due to inelastic collision is the exact equivalent of [START_REF] Tougaard | Deconvolution of loss features from electron spectra[END_REF]:

S T (E) = ˆ+∞ E K(E -E)j (E)dE (65)
where j'(E) is the measured spectrum at energy E and K(T) is the inelastic energy loss crosssection function. Assuming that K(T) is constant, the Shirley background correction S S (E) can be expressed as [START_REF] Végh | The shirley background revised[END_REF]: Shirley background for Au 4d 5/2 ; (b) Shirley background for Au 4d 3/2 ; (c) Shirley back- ground for Au 4d 3/2,5/2 ; (d) Tougaard back- ground for Au 4d 5/2 ; (e) Tougaard background for Au 4d 3/2 ; (f ) Tougaard background for Au 4d 3/2,5/2 . where S S,0 (E') is a constant background taken at energy E' dened by the operator. The Shirley background is then iteratively computed as:

S S (E) = k ˆ+∞ E dE (j(E ) -S S,0 (E )) (66)
S S,i (E) = k ˆ+∞ E dE (j(E ) -S S,i-1 (E )) (67)
it was shown [START_REF] Végh | The shirley-equivalent electron inelastic scattering cross-section function[END_REF] that a specic K S (T) applied to 65 can reproduce the behaviour of the iterated Shirley background if K S (T) is expressed as:

K S (T ) = B S C S + T 2 (68)
where T is the energy loss, B and T are parameters.

In practice the Shirley background is set to correct for inelastic scattering of photoelectrons in the vicinity of the no-loss peak and results as a step-like background as can be seen in the left panels of Fig. 22.

The expression of the energy loss depicted in the Shirley background (eq.66) can be seen as the kernel function in eq.68 of the Tougaard background (eq.68 and 49-49). The Tougaard Universal cross-section (eq.49-50) is a functional form t to the average of several materials. It can be widely used but as it uses the energy loss T as a multiplier it enforces a linear behaviour of the cross-section at low energy loss (i.e. in the vicinity of the no-loss peak). In practice the Shirley background is calculated for a precise energy range and not to innity as it should be (see eq.67).

The resulting shape of the Shirley background depends on the selected energy range. This is less important in the case of the Tougaard background as can be seen in Fig. 22 where Au 4d of a pure gold sample is measured and the eects of applying Shirley background or Tougaard background over dierent energy ranges are compared. The Tougaard background shape does not depend on

Comparison with the Partial intensity approach (PIA)

This particular method has the advantage to be potentially the fastest in terms of computing time (it has been developed for simulation program SESSA [START_REF] Smekal | Simulation of electron spectra for surface analysis (sessa): a novel software tool for quantitative auger-electron spectroscopy and x-ray photoelectron spectroscopy[END_REF][START_REF] Smekal | Nist database for the simulation of electron spectra for surface analysis[END_REF]). The method uses a mathematical decomposition of the signal where the sample is referred as a multilayer; and every layer has a contribution to the no-loss peak and inelastic background [START_REF] Werner | Simulation of electron spectra for surface analysis using the partialintensity approach (pia)[END_REF]. In a simple picture depicted in Fig. 23, it is possible to see that the inelastic background is composed of electrons that suer quantied inelastic losses and the sum of these contributions, aected by coecients, forms the overall inelastic background.

Principles The method uses an equivalent of the quasi-elastic (QE) approximation or constant cross-section approximation, where the electron-solid interaction is independent of the value of the energy loss [START_REF] Werner | Electron transport in solids for quantitative surface analysis[END_REF]. It is justied as the energy lost in a scattering event is small compared to the initial energy of the photoelectron. This approximation is strengthened when using hard x-rays.

It also assumes that the energy f 0 (E), the angular distribution g 0 ( Ω) and depth distribution c 0 (z) are uncorrelated. Then the source function S 0 (E, Ω, z) can be written as:

S 0 (E, Ω, z) = f 0 (E) × g 0 ( Ω) × c 0 (z) (69)
The measured spectrum Υ can be expressed with a term G(s,T,μ) representative of the energy loss T [START_REF] Werner | Simulation of electron spectra for surface analysis using the partialintensity approach (pia)[END_REF], assuming a plane symmetry:

Υ(E, µ) = ˆ∞ 0 ˆ∞ 0 ˆ4π G(z, T, µ) × S 0 (z, E + T, µ)dzdT dΩ (70)
with Ω the solid angle and μ the polar deection.

This formula describes the physical dependence of the measured spectrum. The principle of the partial intensity is to express the energy distribution of Υ as the sum of photoelectrons who suer n times an inelastic scattering event. These photoelectrons are aected with a coecient C n corresponding to the number of particles participating to a loss process. The eq.70 can be written in a more convenient form as [START_REF] Werner | Electron transport for spectrum analysis and experiment design[END_REF]:

Υ(E, µ) = ∞ n 1 =0 ∞ n 2 =0 ∞ n 3 =0 C n (µ)Γ n (T ) ⊗ f 0 (E + T ) (71)
with Γ n the distribution of energy loss after n scattering event, f 0 the source energy distribution and ⊗ denotes a convolution over the energy. If the inelastic events are considered as uncorrelated then :

C n 1 ,n 2 ,n 3 ... (µ) C n 1 n 2 n 3 ... (µ) = ∞ 0 Q(s, µ)W n i (s)ds (72)
with Q(s) the pathlength distribution and W n i (s) is the Poisson stochastic process. In the straight line approximation (SLA), the deections are neglected and any two points in space are connected by a unique straight line path, thus Q(s) is expressed as a delta function and:

C n (µ) = ∞ 0 ( z λµ ) n exp(-z/λµ) n! c 0 (z)dz (73)
where z represents the depth of the element, λ is the inelastic mean free path and c0(z) is the compositional depth prole.

The surface excitation can be taken into account using the Tung model where the surface scattering zone is approximated by a straight line [START_REF] Tung | Dierential cross sections for plasmon excitations and reected electron-energy-loss spectra[END_REF], the surface and volume scattering are uncorrelated, then:

C n b ,ns = C n b × C ns (74)
where C n b is the n-fold inelastically scattered bulk coecient and C ns is the n-fold inelastically scattered surface coecient. This approximation is not strictly adequate but has been shown to be eective [START_REF] Werner | Obtaining quantitative information on surface excitations from reection electron energy-loss spectroscopy (reels)[END_REF]. The equation 71 can be written when taking surface eects into account as:

Υ(E) = ∞ n b =0 ∞ ns=0 C n b C ns Γ n b (T ) ⊗ Γ ns (T ) ⊗ f 0 (E + T + T ) (75)
The contribution of the various loss processes can be subtracted to the measured spectrum if the sequence of partial intensities C n and the loss function L n are known using [101]:

Υ k+1 (E) = Υ k (E) -q k ˆΥk (E + T )L k (T )dT (76)
where q k = q k (C n /C 0 ). This operation is applied to the measured spectrum until convergence over the considered energy range. It ultimately yields to f 0 . If the solid is homogeneous with isotropic emitters then

C k = q 1 C k-1 and f 0 (E) = Υ(E) -q 1 ˆΥ(E + T )ω(T )dT (77)
Practical approach The Partial Intensity Approach is implemented in the SESSA software which uses the symmetry property of the linearized Boltzmann equation (the source and the detector are inverted in the algorithm) [START_REF] Case | Linear Transport Theory[END_REF][START_REF] Werner | Trajectory reversal approach for electron backscattering from solid surfaces[END_REF]. The software can compute the distortions of the inelastic background, relevant to depth proling, but is mostly used for analysis of the no-loss peak by resolution of the Eq.76 with n=0. It has the great advantage to use the large NIST database [START_REF] Jablonski | Nist electron eective-attenuation-length datadata[END_REF][START_REF] Powell | Nist electron elastic-scattering-cross-section database[END_REF][START_REF] Jablonski | Nist inelastic-mean-free-path datadata, version 1.1[END_REF] as can be seen in Fig. 24. The software has been recently implemented with possibility of analysis of surface nanostructures and specic HAXPES experiments [START_REF] Werner | Simulation of electron spectra for surface analysis (sessa)for quantitative interpretation of (hard) x-ray photoelectron spectra(haxpes)[END_REF]. During the photoemission process, the photoelectron can suer various physical excitations: the intrinsic excitations due to the core-hole creation; the extrinsic excitations corresponding to the energy loss during the transport of the photoelectron, the multiple excitations and interferences between intrinsic and extrinsic excitations. The multiple excitations in a single scattering event are negligible regarding their intensity contribution to the no-loss peak and the inelastic background.

The interference terms can be evaluated only if the intrinsic and extrinsic contributions are known.

The intrinsic and extrinsic excitations give rise to a tail of lower energetic photoelectrons, the inuence of intrinsic excitation being lower in intensity and in energy loss than the extrinsic contributions.

The main advantage over the no-loss peak analysis which unravels the chemical environment of the element over the rst 3 IMFP is the increasing of the probing depth up to 8 IMFP for the analysis relying on the inelastic background [START_REF] Tougaard | Accuracy of the non-destructive surface nanostructure quantication technique based on analysis of the xps or aes peak shape[END_REF] as illustrated in Tab.1. Three dierent kinds of inelastic background modeling have been shown:

The Shirley background is usually employed when analyzing the no-loss peak, as it corrects the spectrum for inelastically scattered electrons in the vicinity of the no loss peak. for almost every analysis of core-level, but its intensity depends on the selected energy range on which it is applied. As emphasized by Shirley, it is an empirical formula depending on the density of states of the sample.

The PIA has the advantage to treat the no-loss peak and the inelastic background with the same theory frame, but relies on a mathematical deconvolution depending on the sample. The practical application for the PIA is done by SESSA and can properly function with help of the NIST database.

The Tougaard method corrects the measured spectrum from extrinsic excitations, leading to the intrinsic spectrum of the element. By measuring a reference sample (pure and homogeneous), the intrinsic spectrum can be obtained, and this can be used to perform inelastic background modeling with great accuracy. The Tougaard method is very ecient to analyze the observed changes in the inelastic background and from this to determine the corresponding changes in atom depth distributions. The two step model (in which thesurface eects are neglected) is well adapted to HAXPES (in which bulk contributions dominate). The next Chapter will illustrate the implementations made in this work to successfully perform inelastic background analysis in HAXPES spectra with good accuracy.

3 Implementations and limits of background analysis in HAX-PES This Chapter is dedicated to the setting up of background analysis of HAXPES spectra. One of the main advantages of the inelastic background analysis is that the depth distribution of a buried layer can be extracted from a single spectrum recorded at a x experimental geometry. The background analysis was originally developed for soft x-ray experiments.The photon energy has a great inuence on the photoionization, the inelastic mean free path and the signal/noise ratio aecting the measurement.

This Chapter presents the implementations of the method for specic applications to HAXPES spectra. The accuracy and the sensitivity of the method will be discussed by focusing on two dierent situations encountered when studying the same type of MOSFET samples using Hard

X-Ray PhotoEmission:

A deeply buried monolayer of lanthanum will be analyzed to demonstrate the sensitivity and accuracy of the method. This study is of high importance as the lanthanum depth distribution is critical for device performance. The study of La 3d and La 4f are rather complex in XPS due to shakeup and charge transfer processes. The burying depth >50 nm below the surface is limit, even with use of hard x-rays photons (due to the large peak intensity attenuation, analysis of the no-loss peak almost impossible).

Multiple layers of silicon, with various chemical environment and depths, which is a diculty in background analysis to illustrate the accuracy of the depth distribution obtained.

The results obtained by inelastic background analysis will be compared with results from other characterization techniques. Eects of experimental and methodological parameters will be discussed as well as the advantage of using a reference sample in the analysis. The rst MOS device with silicon substrate fully operational was made by D. Kahng and M.

Atalla in 1962 [START_REF] Atalla | Ire solid-state devices research conference[END_REF]. It operates on the transport modulation of charge density in a semi-conductor layer (conduction canal ) by ways of a perpendicularly applied electrical eld. Creation of carriers by eld eect is ruled by the polarization V G of the electrode called grid through an insulating layer called dielectric. Transport of the carriers in the canal is done by pulse of a dierential potential V D between two charge reservoirs, the source and the drain.

The transistor goes from a blocked state where V G = 0 V (the drain current I D is then equal to the leakage current I OFF ) to an operating state where grid polarization is equal to drain polarization (V G = V D ) allowing the drain current I ON to ow. The eective switching from these two states is only possible when the applied grid tension V G is superior to a minimum value V th specic to the device.

From this stage, the architecture of the MOSFET has not changed. The component size reduction down to a few nanometers has made the initial dielectric layer of SiO 2 downscale to 5 nm in MOS devices. At this thickness the leakage current can be signicant [START_REF] Muller | The electronic structure at the atomic scale of ultrathin gate oxides[END_REF]. Therefore to further decrease the dielectric layer thickness, the replacement of the SiO 2 layer by new material is compulsory. Amongst other candidates, Hf and La oxides appeared to be promising as they present a high oxide capacity and a reduced leakage current [START_REF] Wilk | High-k gate dielectrics: Current status and materials properties considerations[END_REF]. They are isolated from the silicon substrate by an interfacial layer (IL) of SiON to reduce the lattice parameter [START_REF] Wong | On the nitridation-induced enhancement and degradation of {MOSFET} characteristics[END_REF]. The gate had also been improved by substituting a metal to the usual silicon because of depletion in silicon and Fermi level pinning [START_REF] De | Impact of gate workfunction on device performance at the 50 nm technology node[END_REF]. TiN metal is used both for its compatibility with CMOS technology and its ability to be easily p-or n-doped (for NMOS or CMOS components).To control the threshold voltage V th precisely, insertion of a thin lanthanum oxide has been integrated in the device in form of a LaO x layer between the TiN and the HfSiON layers. Addition of LaO results in a decreasing of V th and is then used for NMOS components. The obtained improvement is measured as the Equivalent Oxide Thickness (EOT), expressed as a function of the thickness t and dielectric constant K of the considered high-k oxide as:

EOT = t eq = κ SiO 2 κ high-k t high-k (78)
When the dielectric layer is formed by two layers of thickness T and relative permittivity ε then the EOT is expressed as the sum of the contributions:

EOT = ε SiO 2 ( T 1 ε 1 + T 2 ε 2 ) (79)

Samples description

The samples used in this experiment were manufactured by STMicroelectronics. They are made to be integrated in CMOS industrial technology manufacturing as gate part of the device. They were prepared in a gate-rst integration process on 300 nm Si(100) wafers; this gate-rst process is used when dopants are activated through annealing after the deposition. This annealing induces diusion in the already-made stack, which can decrease the device performances if not well controlled. Thicknesses of deposited layers and their depth distribution before and after annealing must be characterized for the nal composition of the stack to assess diusion process. La layer is deposited by Physical Vapor Deposition (PVD), to avoid any carbon contamination.

The thickness of the layer is of 0.4 nm.

The metal gate stack is composed of a 6.5 nm TiN lm deposited by PVD. This specic choice of metal has been motivated by its work function which can be tuned close to the conduction band or the valence band of silicon, depending on the doping element (between 4.1 and 4.3 eV for N-type and between 5 and 5.2 eV for P-type) and its thermal stability compatible with a gate-rst process.

Two samples were then prepared to investigate the eect of burying depth in HAXPES analysis, a 20 nm-thick, a 30 nm-thick and a 50-nm thick amorphous Si (a-Si) lms were deposited using Chemical Vapor Deposition (CVD) with a maximum thermal budget of 600°C, the sample will thereafter be named by their silicon thickness, namely 20 nm sample, 30 nm sample and 50 nm sample.

The same stack was realized with an increasing active layer thickness of 1.5 nm-thick HfSiON and 1 nm-thick La layer. This sample with a 20 nm thick amorphous Si presents the same TiN thickness of 6.5 nm as the 20 nm sample. This sample will be referred as high-k sample. The samples were studied using many techniques. The primordial control step in such devices is to perform very thin active layer to decrease the EOT.

As a result, the active layer of the device is composed (from top to bottom) of a LaO x monolayer, a HfSiON layer of a few nanometres and a very thin SiON layer. Lanthanum and Hafnium have such a close atomic number that dierencing the two layers is not possible in TEM. La and Hf present well separated peaks in photoemission but these active layers are located 50 nm below the surface for the real device. Such a depth is not accessible with classical XPS and this kind of samples has to be studied by XPS at each of the step, but not on the nal device. This Chapter will show how combining HAXPES and inelastic background analysis ensure to probe the entire device. The schemes of the samples presented in Fig. 28 are at scale and illustrate the diculty of such an analysis when the total active layer composed of three nanometric layers with dierent elemental composition is buried under two dierent layers (the poly-Si top electrode and the TiN metal barrier). The actual technological device is composed of a 50 nm top amorphous Si (a-Si) electrode but to investigate the eect of the burying depth for HAXPES analysis, a 30 nm-thick a-Si top electrode will also be investigated.

The deepest buried layer presently published is for a layer located at 20 nm below the surface.

A method is presented here that enables to obtain the depth distribution deeper than this value.

Beamlines description

Three dierent beamlines were used in this work, their experimental stations are shown in Fig. 29.

The ID32 beamline from ESRF [START_REF] Zegenhagen | X-ray standing waves and hard x-ray photoelectron spectroscopy at the insertion device beamline id32[END_REF] was used to measure the sample at very high photon energies up to 18 keV; BM25 from ESRF [START_REF] Rubio-Zuazo | Hard x-ray photoelectron spectroscopy (haxpes) at spline, the spanish beamline at the esrf[END_REF] was used to compare the data; GALAXIES from SOLEIL [START_REF] Céolin | Hard x-ray photoelectron spectroscopy on the galaxies beamline at the soleil synchrotron[END_REF] was used to have the highest resolution and constant ux. Due to high energy storage ring of 6 GeV, the peak ux measured at 25 m was 7.10 14 ph s -1 mm -2 , 0.1% bw, 0.2 A. In order to achieve maximum bulk sensitivity and to optimize for signal intensity, the incident photons impinged on the sample at an angle of 10°and photoelectrons were detected at an angle of 80°both with respect to the surface sample. To reduce data acquisition time, the most relaxed incident beam energy resolution of 1.6 eV available at the given energy with a Si [START_REF] Zegenhagen | X-ray standing waves and hard x-ray photoelectron spectroscopy at the insertion device beamline id32[END_REF] monochromator was used in order to maximize the photon ux. Further signal intensity optimization was achieved by a choice of the largest entrance slit on the electron energy analyzer.

The SPECS PHOIBOS 225 high resolution spectrometer tted with entrance optics enabled the analysis of high kinetic energy photoelectrons up to 15 keV. This experimental setup led to the overall energy resolution of 4.7 eV at 15 keV when tting the Fermi level of gold sample inside the analysis chamber. ID32 was closed for user experiments on 5 December 2011.

Intermediate beamline: BM25, ESRF. BM25 beamline of the European Synchrotron Radiation Facility (ESRF, Grenoble, France) is also designed to deliver high energy photons but has the particularity of having a bending magnet as insertion device. This ensures a wide photon energy accessibility but a lower photon ux, estimated to be 1.10

14 ph s -1 mm -2 0.2 A on the sample. The experimental station is designed to perform high energy photoemission experiments with an electrostatic cylinder-sector FOCUS HV CSA capable of measuring high kinetic energy photoelectrons up to 15 keV. As the method does not imply high energy resolution, the beamline was setup to deliver the maximum photon ux possible by decreasing the resolution. As for ID32 beamline the photon ux is measured through a gold grid and the associated current I 0 is used to normalize the measurements intensity. Unlike other experiments, the geometry of the experiment had to be changed, with photons impinging the surface from 75°from surface normal and emergent photoelectrons were detected with an angle of 15°from the surface normal due to the beam size.

However this slight change doesn't modify strongly the measurement as the signal is only decreased by the cosine of the emission angle from the surface normal.

High ux beamline: GALAXIES, SOLEIL. GALAXIES beamline from SOLEIL synchrotron (Saclay, France) is one of the most recent HAXPES beamlines available for user experiments. This specic synchrotron works in a so-called top-up mode. In this particular mode, electrons are continuously injected from the booster to the storage ring in order to maintain the current constant, and so a constant photon ux in the beamline. For this reason, the photon ux doesn't need to be measured at every acquisition, the spectra can be easily compared in intensity. The beamline is designed to provide a monochromatic focused beam with the highest ux possible in the 2.4-12 keV

(1.5x10

12 ph s -1 ) spectral range and an adaptable photon energy bandwidth between 100 meV and 1 eV. The estimated X-rays spot at the focusing point onto the sample is 30 x 100 um [VxH]

FWHM. The HAXPES chamber is equipped with a EW4000VGScienta hemispherical photoelectron analyzer with a 60°angle. In the experimental geometry used for the study, photons impinge the surface with an angle of 10°and the analyzer is located at 80°with respect to the surface sample. The image shows the 1.4 nm-thick SiON layer, below an intermixed HfSiON and LaO x layer; the no other peaks for La contribution were detected during the experiment, the conclusion is that La is mainly present in the oxidized form in the sample. Hf contribution was also only detected in the oxide ion form at m/n = 98. The dierence in evaporation elds between chemical species can slightly curved interfaces between layers. This is a well-known phenomenon which is intrinsic to the evaporated phases [START_REF] Vurpillot | Pragmatic reconstruction methods in atom probe tomography[END_REF]. To overcome this bias that can be introduced in the concentration proles, a proximity histogram has been used [START_REF] Hellman | Analysis of three-dimensional atom-probe data by the proximity histogram[END_REF]. This has the added advantage of correcting any surface roughness due to the polycrystalline nature of the materials. For the calculation a 37% TiN iso-surface was chosen as the plot reference (0 on the depth scale). The depth prole shows that La and Hf layers are located between the TiN layer and the silicon substrate. Pink and blue dots in TiN layer and Si substrate are due to a small signal at the noise level and so are not dened as La and Hf ions, respectively. Since the intensity of La is non-negligible at Hf maximum intensity, the experiment shows a partial intermixing between the La and the Hf layers induced by the thermal budget during the deposition process. The obtained thicknesses are of 1 nm for the La layer and 1.3 nm for the Hf layer and the resulting layer has an overall thickness of 1.5 nm.

Characterization with other techniques

The TiN layer was found to be stoechiometric (not shown here). the measurement for an element with no prior knowledge on its location can be time consuming.

The second problem arises from the eye control of the matching between the modeling and the measurement. There are possibilities to use computed formula inside the QUASES Analyze software but they are not consistent as the nal result depends on the rst guessed input top and bottom interface positions.

These two diculties have been overcome by computing a bot and by nding an empirical method of computing the matching between modeling and measurement. The resulting error is then plotted as a colour map. All scripts were coded using Python. This specic language has been chosen as it is a free language and thus can be implemented on any computer. Python language is object-oriented and works eciently for treating the large number of les created by the bot.

Python has a math-like syntax with indented structure which makes it easy to read and quick to understand.

bot A bot is a software application that runs automated tasks. Bots perform tasks that are both simple and structurally repetitive, at a much higher rate than would be possible for a human.

In our case, the bot performs a cycle of clicks where the top and bottom interface are changed with a tunable step (which can be dierent for the top interface and the bottom interface); it then saves the obtained background modelled and the subtracted spectrum (dierence between the measurement and the modeling) with a name corresponding to the depth distribution.

Computed error As assumed by theory the matching between the modelled background and the measured background must be taking into account only for energy loss larger than ∼30 eV and up to 120 eV below the no loss peak. For these reasons, the error is computed only for this energy range for background analysis without reference sample. If the analysis applies a reference sample then the shape and intensity can be compared over the full energy of the measurement, χ (see eq.36) is evaluated for all points in the spectrum.

Small variation in top interface location varies strongly the modelled background in the 40-70 eV lower kinetic energy range of the no-loss peak whereas modifying the bottom interface location changes only the far energy loss region (around 100 eV below the no-loss peak). The computed error from eq.36 can lead to erroneous results in which the far from no-loss peak energy region is not well reproduced and the near no-loss peak region is better accounted for. Several empirical calculations for computing an error value have been tried, the best formula (which leads to a good reproduction of the background over the considered energy range) was found when cutting the energy region in three regions of 30 eV each. The rst region α is located between 120 and 90 eV below the no-loss peak, the second region β is located between 90 and 60 eV below the no-loss peak and the third region γ from 60 to 30 eV below the no-loss peak. To each of these regions is applied the error calculation of eq.36 and the total error χ e is computed as the sum of an increasing power law:

χ ef f = α + β 2 + γ 4 (80)
As eq.80 has no physical meaning, it is not possible to extract much information from it. However the matching between the modelled and the measured background is always very good for the minimum χ e value. Assuming that only the matching between the measurement and the modeling accounts for the depth distribution of the element, with this calculation very small variation of χ e corresponding to small depth distribution variation can be emphasized.

In order to have a visualization, χ e values are normalized by the lowest value and is plotted as a colour map with the x-axis the top interface position and the y-axis the bottom interface position. To obtain the nal map, a linear interpolation is done between two points and lines corresponding to 1% changes from the minimum x e value are drawn.

Qualitative results

Even though the background analysis has proven its eciency during the past decade, the application of the method to the hard x-ray regime (and thus high velocity photoelectrons) deserves careful attention. In this section the qualitative eects of high velocity photoelectrons on the background shape will be studied.

Eects of the photon energy

The selected photon energy has two eects on the inelastic background of HAXPES spectra: the photoionization cross section : when the photon energy increases, the IMFP and thereby the escape depth increases but at the same time the photoionization cross-section decreases drastically [START_REF] Trzhaskovskaya | Non-dipole second order parameters of the photoelectron angular distribution for elements z=1-100 in the photoelectron energy range 5-10kev[END_REF]. These two factors inuence the intensity of both the no-loss peak and the inelastic signal to lower kinetic energy. The IMFP depends on the material the photoelectron passes through when traveling inside the medium and on its kinetic energy.

The eective IMFP used in the study is an average of individual IMFP weighted by the nominal thickness of the crossed layer. Tab.3 presents the parameters used to compute the IMFP of La2p 3/2 photoelectrons for the 50 nm sample. Values of the IMFP for high energy electrons have been published for pure elements [START_REF] Tanuma | Calculations of electron inelastic mean free paths. ix. data for 41 elemental solids over the 50 ev to 30 kev range[END_REF] but not for compounds and alloys. Therefore the corresponding IMFPs are computed from the TPP-2M formula [START_REF] Tanuma | Calculations of electron inelastic mean free paths. v. data for 14 organic compounds over the 50-2000 ev range[END_REF]. The weighted average is dominated by the Silicon samples A Silicon wafer substrate was treated by HF for surface cleaning before immediately transferring into an ultra-high vacuum transfer box via a glove box, then transferred into the experimental chamber of the BM25 beamline with less than 2 min air exposure, the resulting spectra measured at 8, 10, 12 and 15 keV are presented in Fig. 34 where no oxide peak is detected. The inelastic background shape is seen for each energy with oscillations which correspond to 1, 2, 3, 4 and 5 multiple excitation plasmons separated by ∼16 eV. The insert of Fig. 34 shows the intensity for the ve rst plasmon losses relative to the no-loss peak intensity. The intensity for each plasmon loss increases when increasing photon energy and their behaviour is similar. This means that in this high energy regime >8 keV, the intensity of plasmon loss relative to the no-loss peak intensity is directly connected with the probed depth inside the sample. oxide peak has a higher intensity than the rst plasmon peak. When increasing photon energy, the rst plasmon becomes more intense than the oxide peak. In the lowest energy experiment, the detected photoelectrons originate mainly from the oxidized silicon layer whereas at the highest energy the detected photoelectrons come from deeper in the layer of pure Si. The resulting energy losses are therefore pure silicon-like at high energy whereas they are a mixture of oxidized silicon and pure silicon at lower energy.

The two samples measured at 10 and 12 keV are shown in brown and purple respectively in Fig. 34 and Fig. 35. The dierence between the two samples can be seen for the oxide peak which is still visible for the oxidized sample but also for maximum intensity of plasmon losses. The pure silicon sample presents higher intensity for plasmon losses and more visible plasmons (the ve rst are plotted in the insert of Fig. 34) as electron scattering is dominated by pure silicon bulk plasmon losses in the sample.

Eects on technological sample

As shown in the previous section, using hard x-rays has an impact on the inelastic background for pure samples. This section will study the behavior of the spectra from the 50 nm sample, 30 nm sample, 20 nm sample and high-k sample using HAXPES at varying photon energy.

Ti 1s

The inuence of varying the photon energy can be seen in Fig. 36 which shows Ti 1s peak for the 50 nm sample at 12, 15 and 18 keV. This peak is much more intense than La 2p 3/2 measured for the same three energies (see Fig. 38). Clear oscillations are seen below the no-loss peak which correspond to the bulk plasmons excitations as the photoelectrons passe through the Si capping layer. The rst oscillation is located 17.5 eV below the no-loss peak and has a higher intensity than the no-loss peak. It can be seen for the three selected photon energies and corresponds to the highest probability of energy loss in silicon. The second plasmon loss is shifted by 16.9 eV toward lower kinetic energy. The energy shift between successive plasmon losses are decreasing with increasing scattering events (16.1 eV for the third and 14.6 eV for the fourth plasmon loss).

Ti 1s photoelectrons suer inelastic losses inside the Si capping layer, dominated by the sharp plasmon losses but also inelastic scattering events inside the TiN layer. In the TiN layer, energy losses are represented by the smooth two-parameters universal cross section valid for metals and their oxides (see eq.49 and Fig. 19). The mixing between these two dierent inelastic scattering events induces a broadening in the inelastic background. Moreover the Ti 1s peak is wider than the Si 1s peak, thus the intensity attenuation of the succesive plasmons makes the number of observed plasmons of Ti 1s inelastic background lower than the one observable from Si 1s. These two reasons give an interpretation of the observed changes in the plasmon peaks seen in Fig. 36.

La 2p

Similar observations can also be made for the ultra thin and deeply buried layer of La.

Fig. 37 shows the La2p 3/2 HAXPES spectra of the 50 nm sample measured at 12, 15 and 18 keV photon energy. The spectra consists in the no-loss peak located at 5523 eV binding energy, and of inelastic loss features extending over typically 120 eV to lower kinetic energy. At 12 keV, the no-loss peak is not detected because, with a calculated IMFP of 10.3 nm, the escape depth is a factor ∼5 smaller than the thickness of the Si overlayer. A closer look at 17 eV below the position of the no-loss La peak reveals another peak due to a bulk plasmon loss in the silicon overlayer [START_REF] Tougaard | Universality classes of inelastic electron scattering cross-sections[END_REF]. Theoretically, with background analysis it could be possible to probe the location of the La buried layer but the poor signal-to-noise ratio makes the inelastic background analysis dicult. In the spectrum at 15 keV, the corresponding IMFP is 15.5 nm and the no-loss peak can be clearly seen since the escape depth now ∼3 smaller than the overlayer thickness. As a result, the inelastic background signal-to-noise ratio is now better although the photoionization cross-section is smaller at 15 keV. At 18 keV, the very low intensity of the no-loss peak is due to the decrease of the photoionization cross-section despite an escape depth of nearly 60 nm (IMFP of 19.6 nm). In conclusion, the best signal/noise and probing depth combination to analyze the ML-thick La buried layer is therefore at the intermediate energy of hν=15 keV. At this energy the probing depth when analyzing the core-level is representative for ∼45 nm below the surface but the inelastic losses can be used to extract the in-depth distribution at larger depths (see sect.2.4.1) under good conditions.

The optimum photon energy must be a compromise between the increasing escape depth and the decreasing photoionization cross section when increasing the photon energy in HAXPES. In conclusion, the 15 keV photon energy has to be selected for the experiments presented in the following since it has the best signal/noise ratio, the photoionization is sucient to see all elements of interest and the IMFP values ensure to probe the entire stack. At this photon energy, the experiment is very low surface sensitive and with a take-o angle of 80°, the bulk sensitivity is enhanced.

3.2.3 La2p 3/2 inelastic background for 20, 30 and 50 nm samples

The 15 keV HAXPES survey spectra of the samples in the spectral regions of the K subshell of Ti (Ti1s peak) and of the L subshells of La and Hf (2s and 2p 3/2,1/2 peaks) are shown in Fig.

38.

The inset shows La 2p 3/2 peak for the three samples. All elements of interest are deeply buried underneath the a-Si overlayer (see Fig. 28). They are detected by their characteristic photoelectron emissions. In particular, the detection of the LaO x deeply buried below the surface in the 50 nm-sample demonstrates the extreme depth sensitivity of HAXPES. The issue is now to retrieve the in-depth distribution of this element. In the inset of Fig. 38, the La 2p 3/2 core level peak is located at 9524 eV kinetic energy. It is separated from the rst energy loss peak by ∼17 eV, which corresponds to the bulk plasmon excitation energy of silicon (measured from Fig.

rst plasmon loss peak has a higher intensity than the main peak. This is true also for the Ti 1s in Fig. 36 and La peaks in Fig. 38 where the plasmon features actually dominate. The inelastic background signal is thus higher than the no-loss peak which is due to the large burying depth of the overlayer.

Quantitative results -Analysis without reference sample

The previous section showed the qualitative impact of high velocity photoelectrons on the inelastic background. These observations were used to determine the best experimental conditions for good background analysis. This section shows the depth distribution obtained without the use of reference sample as it is sometimes dicult to nd a good reference sample.

Input parameters

The calculations for modeling inelastic background need some input parameters: the IMFP, the detection angle and the inelastic scattering cross-section. The IMFP value is calculated from formula (see eq7); the detection angle is an experimental parameter; the inelastic scattering crosssection is both energy and material dependent. Their inuence will be studied in this section.

Inelastic mean free path As previously mentioned, the IMFP for high energy photoelectrons is not perfectly known yet. Tabulations with modied TPP-2M formula are given by Tanuma et al. [START_REF] Tanuma | Calculations of electron inelastic mean free paths. v. data for 14 organic compounds over the 50-2000 ev range[END_REF][START_REF] Tanuma | Calculations of electron inelastic mean free paths. ix. data for 41 elemental solids over the 50 ev to 30 kev range[END_REF][START_REF] Tanuma | Calculation of electron inelastic mean free paths (imfps) vii. reliability of the tpp-2m imfp predictive equation[END_REF] for various elements, but for compounds, the IMFP must be estimated from individual IMFP. The complexity of the presently studied samples increases the diculty of choosing a correct inelastic mean free path. An eective IMFP was used for all elements, which corresponds to a weighted average of individual IMFPs calculated with TPP-2M formula according to the nominal thicknesses of the crossed layers as seen in table 4. Assuming an error on the individual IMFPs of 10% [START_REF] Tanuma | Calculations of electron inelastic mean free paths. ix. data for 41 elemental solids over the 50 ev to 30 kev range[END_REF], the resulting error for the eective IMFP are lower than 20% (see also Tab.4).

hν The eective IMFP is mainly dependent on the IMFP inside the silicon layer because this latter is the thickest one. It was observed that the IMFP choice inuences the background modeling especially in the 60 eV region below the no-loss peak but not that much at the lowest kinetic energy range of the spectrum. To explore more clearly the eect of the IMFP choice, modeling the measurement and the modeling. This is also seen in the inset of Fig. 39 where the normalized χ e varies slowly when increasing the IMFP but decreasing the IMFP value increases drastically χ e and aects the general shape of the background with highly pronounced oscillations. The use of the computed eective IMFP from TPP-2M formula which error is lower than 20% is then a reasonable approximation.

λ
Detection angle The detection angle for photoelectrons is known with an accuracy of around 5°. Fig. 40 shows results of modeling for a detection angle of 0°, 10°(the angle corresponding to our experimental setup) and 40°from surface normal. For angles ±5°from the actual value one leads to a negligible variation of the normalized χ e as can be seen in the inset of Fig. 40. This is due to the sine dependent variation for the probed depth as discussed in 2 since the analyzer acceptance angle is usually ∼ 5°the eect on the angle is negligible. At larger photoelectron collection angle like 40°, strong variations all along the chosen energy range of the inelastic background can be clearly see .

Choice of the inelastic scattering cross-section K(T)

This section considers the critical inuence of the choice of K(T). As previously pointed out, K(T) varies with the energy and the material [START_REF] Tougaard | Universality classes of inelastic electron scattering cross-sections[END_REF][START_REF] Werner | Comparison of hard and soft x-ray photoelectron spectra of silicon[END_REF]. In contrast to most materials, silicon has a sharp plasmon loss feature. In this particular case a three-parameter cross-section is more accurate than the two-parameter universal cross-section. But at the high photoelectron kinetic energies accessible in HAXPES, we found that it is more accurate to apply a cross-section derived from REELS measurements at 10 keV using the QUASES-REELS software.

The bottom panel of Fig. 41 illustrates the result of using three dierent K(T) functions for background analysis of the loss structure in the La 2p 3/2 spectrum. In bottom-left panel of Fig.

41 is presented the use of the two parameter universal cross-section which does not reproduce at all the ne structure in the background. The bottom-middle panel of Fig. 41 shows the result of the analysis performed with the three-parameter universal cross section valid for Si [START_REF] Tougaard | Universality classes of inelastic electron scattering cross-sections[END_REF]. The modeled spectrum reproduces well the rst plasmon loss, but the second one is overestimated and the third loss is underestimated. Furthermore, there is an energy shift of 2.5 eV between modeling and measurement for the rst plasmon energy, whereas the agreement is perfect for the second plasmon. In the bottom-right panel of Fig. 41, is plotted the result of the analysis performed with a silicon inelastic cross section derived from REELS measurements at 10 keV i.e, close to the La2p 3/2 photoelectron kinetic energy at 15 keV. Clearly, such a cross section enables a much better account for the ne structures present in the inelastic background and reproduces quite well the plasmon peaks, including the rst one. This is expected because photoexcitation of the La2p 3/2 peak takes place in the La layer and therefore it does not have contributions from intrinsic Si plasmon excitation. The full intensity at the Si plasmon loss peak is therefore extrinsic, i.e. it is entirely caused by the transport of the La2p 3/2 spectrum through the a-Si overlayer. Theory predicts a matching between measurement and modeling only between 120 and 40 eV below the no-loss peak [START_REF] Tougaard | Energy loss in xps: Fundamental processes and applications for quantication, non-destructive depth proling and 3d imaging[END_REF], therefore only this region of interest (as sketched in Fig. 38) must be taken into account in the modeling.

The top panel of Fig. 41 shows the inelastic scattering cross section used for the modeling.

The two parameter universal cross section has a lower maximum probability than the two others but is much wider. This K(T) is very dierent from the K(T) for silicon. It reasonably accounts for the background ≥ 40 eV from the no-loss peak. As an example of behavior of free electron like materials the inelastic scattering cross section for Si has a sharper distribution corresponding to plasmon excitation. Moreover K(T) at high energy (measured at 10 keV) has sharper distribution centered at 17.5 eV than the one determined for soft x-ray energies centered at 18 eV [START_REF] Tougaard | Universality classes of inelastic electron scattering cross-sections[END_REF]. The interpretation of this behavior is attributed to the probed depth: at high energy, the information is considered as bulk information whereas at the low energy, as surface information.

Eect of the Si overlayer thickness

This Section focuses on the depth at which the La layer is buried on the determination of the depth distribution. The background analysis is performed on the La2p 3/2 HAXPES spectra with photon energy of 15 keV for samples capped with 20, 30 and 50 nm-thick a-Si overlayers.

The results from the bot operation and the χ e calculation from eq.80 are plotted as colormap in Fig. 42 where the lowest χ e value is used for normalization and increasing χ e value of 0.1% are drawn as lines. The center of the rst line (in red) corresponds to the determined depth distribution prole. The best background reproductions are representative of depth distribution of 57.3 nm to 61.8 nm below the surface for the 50 nm sample, 41 to 43 nm for the 30 nm sample Figure 42: Maps of χ e as function of the top interface (x-axis) and bottom interface (y-axis) for (a) 50 nm sample and (b) 30 nm sample . The lines drawn represent an increase of 0.1% in the normalized χ e .

The measured spectra and their associated inelastic backgrounds are shown in Fig. [START_REF] Stern | Surface plasma oscillations of a degenerate electron gas[END_REF]. In comparison to the nominal structure, a strong inuence of the overlayer thickness is observed: the determined La depth distribution is more accurate, when the overlayer is thinner. The top and bottom interface positions are overestimated compared to the expected ones for all samples presented. The discrepancy can be due to uncertainties in the IMFP values used for the analysis, but can also be inuenced by elastic electron scattering eects which are neglected in this analysis.

Moreover, even if inelastic scattering cross section, derived here from REELS measurements of pure Si is suitable, it does not correspond exactly to the real composition of the overlayer which contains a TiN layer of 6.5 nm. When the top a-Si layer is very thick compared to the rest of the sample, the approximation of using inelastic scattering cross-section of pure Si is reasonable.

The results of the inelastic background modeling summarized in Tab.5 show that the error on the absolute interface position decreases when increasing the overlayer thickness.

On the reverse, the error on the thickness decreases with decreasing overlayer thickness.

While performing background analysis, the supposed 0.4 nm-thick lanthanum layer is found to be 4.5, 2 and 1 nm-thick for the 50, 30 and 20 nm samples respectively as summarized in Tab.5. Note rst that the low statistics of the spectra have made the modelling dicult, and second the error on the bottom interface depth determination is not without inuence on the thickness determination.

However for all three samples, a 0.4 nm-thick layer is compatible within the estimated error bars of the analysis.

Sample

Top The Fig. 44 displays HAXPES spectra for two samples, the high-k sample (sketched on the right and in red in Fig. 44) and the 20 nm sample (sketched on the left and in black in Fig. 44). The two spectra are dierent due to the amount of substance (AOS), the two samples dier only by their thicknesses of the LaO x (0.4 and 1.0 nm) and HfSiON layers (1.7 and 11.5 nm). At 15 keV the HfSiON layer is detected by very intense Hf 2s,2p no-loss peaks. The HfSiON layer is more visible, despite its location under the LaO x layer, because Hf 2s,p peaks have higher photoionization cross- section than the La peaks. Moreover, the HfSiON layer has a higher Hf AOS than the LaO x layer has a La AOS. The inset in Fig. 44 shows the La 2p 3/2 region. One can see that increasing the thickness increases the intensity of the no-loss peak and therefore gives a much better signal-tonoise of the inelastic background. Modeling of La depth distribution gives the lowest χ e value for the high-k sample with a top interface located at 43.2 nm and a bottom interface at 45.4 nm below the surface, whereas the 20 nm sample results locate the top interface at 35 nm and the bottom interface at 36 nm as summarized in Tab.6. The overestimation of interfaces positions from the surface is attributed to the uncertainty on the IMFP, on theelastic electron scattering and on the inelastic scattering cross section. The lanthanum thicknesses estimations are now closer to the expected value, for the high-k sample where a 2.2 nm-thick lanthanum layer is found with an expected thickness of 1 nm and for the 20 nm sample where the analysis gives a 1 nm thick La layer (expected value of 0.4 nm) as summarized in Tab.6. The AOS has a strong eect: both the top and bottom interface position error is decreasing with increasing AOS.Background analysis is reliable for measuring the thickness of the layer, but the absolute positions of the interface need to be taken with caution. Silicon Background analysis of silicon is very dicult as silicon is present everywhere in the sample except in the TiN and LaO x layers. Fig. 45 shows a small shoulder located at 5 eV lower kinetic energy from the Si 1s no-loss peak corresponding to a small oxidization of the top part of the silicon capping layer as it was conserved at atmospheric pressure. There is a possibility in QUASES-Analyze ® to model two distinct layers in a sample, but no quantitative information can be extracted from such analysis without reference samples in its depth distribution conguration.

Eect of the amount of substance

Results for all elements

The dierences when modeling inelastic background distortions for various depth distributions close to the nominal values are too weak to be truly reproduced or analyzed. The best modeling obtained by the bot procedure is found for a pure innite homogeneous silicon layer with top interface located at 8 nm below the surface. This result does not agree with results from TEM, Auger depth prole or Atom Probe Tomography. The Fig. 47 shows the background modeling performed on Hf 2p 3/2 spectra measured at 15 keV. The depth distribution presented here is for the nominal values of top interface located at 57 nm and a bottom interface at 58.6 nm below the surface. As for all the other peaks, the inelastic background has a higher intensity than the no-loss peak. The modeled background presents oscillations between 5300 and 5380 eV kinetic energy (as for any other modeling around this depth distribution) which are not present in the measurement. The lowest χ e value is obtained for a depth distribution of 30 to 70 nm below the surface which is not in agreement with previous results of TEM, EELS or Auger depth prole. The error is attributed both to the kinetic energy of the photoelectrons of 5450 eV as the inelastic scattering cross section is taken from 10 keV measurements and to the IMFP value of 9.47 nm which can be dierent from the EAL, due to the calculation method or intrinsic error in the original IMFP calculation. A modeling with too low IMFP leads to a background modeling with too high oscillations (see Fig. 39).

To overcome this diculty, the photon energy was increased to 18 keV which leads to photoelectrons of Hf 2p 3/2 with kinetic energy of 8429 eV. At this photoelectron energy, the IMFP is of

Use of reference samples

The results from the background analysis without reference samples are in good agreement with deposition rate and compared with other characterization techniques. As described in Sec.2.3, the method is good at emphasizing changes in the depth distribution; thus analysis with a reference sample should give more precise results. This section describes background analysis using Tougaard method with use of reference samples.

Experimental conditions

Titanium reference and Hafnium reference were measured at ID32 beamline whereas lanthanum reference was measured at GALAXIES beamline. In theory, background analysis must be done from measurements for all the reference samples using the same apparatus than the one that had been used for the sample of interest. However, in this section, the spectra for Ti and Hf are from another measurement than the one for La.

The The pure silicon sample is the one presented in sec.3.2.1 and in Fig. 34.

Results and discussion

The Sec.3.3.5 showed the diculty of the silicon background analysis as silicon is present in various layers in the stack. This Section will study such analysis with use of a reference sample.

Multilayer modeling of silicon Possibilities are available in QUASES-Analyze ® for performing background analysis of our samples with use of the reference samples. The Fig. 49 presents the dierences between the modeling and the measurement for various options implemented in QUASES-Analyze. The black curve is the result found by our method, it corresponds to a depth distribution of silicon as a single monolayer with a top interface located at 9 nm below the surface and a bottom interface located below detection limit. The two blue curves show results obtained by use of either the Optimized Area (dark blue) or Optimized RMS options in QUASES-Analyze®. If the optimized RMS results are clearly not reproducing the background (oscillations are seen in the subtracted spectrum), the area optimized spectrum can be considered as a good qualitative reproduction of the background (there are no remaining oscillation in the subtracted spectrum) even if the intensity of the subtracted spectrum is always higher than the 0 value. The depth distribution obtained by this method is a single silicon layer located at 17.2 nm below the surface and a bottom interface beyond detection limit. Modeling with a distribution of the expected depth distribution of two layers, the rst as a top layer of 50 nm and the second with top interface located at 55 nm below the surface with a bottom interface located beyond detection limits is plotted in red: it does not reproduce the oscillations in the lower energy loss region. In conclusion, the best background reproduction is found by our error estimation method but the depth distribution is not corresponding to the real depth distribution of silicon. Thin monolayer of lanthanum Lanthanum presents a very weak signal as can be seen in Fig. 38 due to the very small amount of La in the sample and its high burying depth. GALAXIES beamline presents a high photon ux. The measurements at 12 keV gives best analytical conditions. This energy has been chosen as it allows a good resolution and maximum photon ux.

Analysis for depth distribution was performed, resulting in a top interface located at 56 nm and a bottom interface at 61 nm below the surface. The resulting background modeling can be seen in Fig. 50: analysis has been performed with or without the modied inelastic scattering crosssection and with or without use of reference sample. For presentation purposes, spectra have been normalized to maximum no-loss peak and have been shifted in intensity. Comparison of the subtracted spectra without use of reference sample shows the increases in background reproduction, the proper inelastic scattering cross section corresponding to a weighted average of the pure silicon cross-section measured at 10 keV and the universal one gives a atten spectra in the energy loss region (in red).

Additional use of reference sample increases the attening of this part of the spectrum, especially around kinetic energy of 5660 eV. The purple subtracted spectrum corresponding to an analysis with three parameters universal cross-section clearly does not reproduce these oscillations.

The gain in using the modied cross-section is interesting but not spectacular. It is surprising to see the oscillations very well reproduced at this kinetic energy of 5710 eV from the inelastic scattering cross section measured at 10 keV. Oscillations are mainly dominated by bulk plasmon losses. At 5 or 10 keV, the surface information is very poor compared to the bulk information (see Fig. 35).

The various options as optimized RMS and optimized AREA are not presented as they led both to a poor background reproduction and unrealistic depth distributions. The option homogeneous layer also gave unrealistic results and did not even match the specicity of the sample (La is only present in a monolayer). Thick buried layer of Titanium The case of Ti modeling will illustrate how the procedure could be at its best to analyze inelastic background. When treating the measurement with only the use of parameters, the result could be ambiguous as dierent depth distributions can lead to the same background reproduction (see sect. 3.3.5). Use of reference samples reduces the number of possible depth distributions as can be seen in Fig. 52 where the domain of the lowest χ e is narrowed when compared to that of Fig. 46. However two distinct types of depth distributions are equally reproducing the inelastic background and the result remains ambiguous. An additional procedure consists in multiplying the obtained thicknesses by the scale factor of QUASES-software which corrects the signal from the reference sample no-loss peak and the CF factor which takes into account elastic scattering (see sect.2.4.1) as summarized in Tab.8. The two distinct cases for a 10 nm-thick layer (ddp1) and for a 37 nm-thick (ddp2) give thicknesses of 4.5 and 4.1 nm, respectively after correction (close to the 4.6 nm obtained by TEM). The discrepancy between the two obtained thicknesses comes from the uncertainty of the scale factor and of the CF function.

In the case of La, the amount of substance is so low in the sample that the scale factor is 0 and thus this procedure cannot be applied. Thick buried layer of Hafnium The previous case of Hf study is completed by an analysis with reference samples. The reference sample is a 15 nm thick HfSiON layer and not a pure homogeneous sample. As the dierence in the no-loss peak area between the reference and the sample is expected to be small, the total amount of substance are close in the two samples. The result of the bot operation and χ e calculation is presented in Fig. 53. The lowest χ e value is obtained for a layer with top interface located at 51 nm below the surface and the bottom interface at 77 nm. The resulting thickness of 26 nm is absolutely incompatible with nominal thickness or the results obtained by other characterization techniques. However, use of the scale factor of 0.045 (output of the software) and correction function of 0.478 (computed from the Eq. 51), the resulting thickness is of 2.4 nm, which is very close to the 1.7 nm nominal thickness.

Annealing of gate stacks

The background analysis of HAXPES spectra has been successfully applied to the previous samples, which were used to explore depth resolution limits. In technological devices designed in gate last process, the samples undergo annealing for the activation of dopants after deposition of the complete structure. This annealing is typically of a few seconds at high temperature (>1000°C)

and can potentially induce interdiusion of elements in the sample. The diusion can be either required (as for the samples studied in the next Chapter) or undesirable ( because it decreases the device performance in the case of a high-k metal gate sample).

In this Chapter the previous observations are applied to the study of high-k metal gate samples after annealing at 1200°C.

The 50 nm sample, 30 nm sample, 20 nm sample and high-k sample were annealed at 1200°C for 10 s. In device production this step is meant to activate dopants in the sample and force intermixing of the active layers (LaO, HfSiON and SiON layers). The result of the automated procedure is shown in Fig. 54(a) where the lowest χ e is centered around a top interface located at 49.9 nm and bottom interface at 56.2 nm below the surface. The corresponding modeling for inelastic background is shown in Fig. 54(b) where the background is quite well reproduced. The measured spectrum (in black) is very dierent from the unannealed sample where after the rst plasmon loss the inelastic background is smooth (see Fig. 43). The La 2p 3/2 spectrum of the annealed sample presents a second plasmon loss clearly distinguishable (located at 9470 eV kinetic energy) with higher intensity than the rst plasmon loss. Then the inelastic background increases between 9420 and 9460 eV corresponding to a third plasmon loss and nally decreases from 9420 eV. All these qualitative observations are well reproduced in the modeled background. The inelastic background shape dierence is mostly due to the low intensity of the no-loss peak for the annealed sample. This intensity of the La 2p 3/2 peak is too low to perform core-level analysis. The background analysis overcomes this diculty and allows quantication of depth distribution of such a sample. The resulting thickness of the La layer is of 7.4 nm. There is a thickness increase with respect to the unannealed sample (of 4.5 nm without reference sample).

The same procedure with use of reference samples is showed in Fig. 55 the energy region around 5650 eV which is underestimated in the calculation as for the unannealed sample (see Fig. 43). The comparison of the thicknesses of the annealed (6.5nm) and of the unannealed (2nm) 30 nm sample suggests an intermixing of La during the annealing.

The use of reference sample using the same procedure analysis gives very similar results as the obtained top interface is found to be located at 35.5 nm below the surface and the bottom interface at 42.2 nm (instead of 35.8 and 42.3 without reference sample) as can be seen in Fig. 57(a). The dierence in location of the top interface with and without reference sample is of 0.3 nm whereas of the bottom interface of 0.1 nm. For the 50 nm sample, the use of reference sample had only a small inuence on the top interface but a non negligible one in the bottom interface. For the 30 nm sample it is the opposite. In other words, the use of reference sample does not impact the absolute position of interfaces in a systematic way. The resulting thickness obtained is of 6.7 nm and shows an intermixing when compared to the previously obtained 2 nm (without reference sample). The use of a reference sample modies slightly the previous results as showed in Fig. 59(a) where the lowest χ e points to a top interface located at 30.8 nm below the surface and a bottom interface at 35.5 nm (instead of 30.9 and 35.0 nm without reference). The change of the top interface location is very small (0.1 nm) whereasthat of the bottom interface is of 0.5 nm. The resulting thickness of 4.7 nm when compared to the 1 nm found for the unannealed sample (without reference) shows the intermixing.

Summary

The determined depth distribution for the annealed samples are summarized in Tab.9. As was the case for the unannealed sample (see Tab.5), the obtained thicknesses for the 50 nm, 30 nm and 20 nm decrease with the decrease of the capping layer thickness, conrming that the accuracy is better when the studied layer is closer to the surface. The top interface positions of 49. Both annealed 20 nm sample and high-k sample present almost the same depth distribution: the top interface is located at 30.9 nm for the 20 nm sample and at 29.4 nm below the surface for the high-k sample. However the estimated thickness is larger for the high-k sample than for the 20 nm sample. This result is expected because in this latter case, the active layer thickness has been intentionally increased..

In overall, as seen in Tab.9, the use of a reference sample does not change very much the interface positions for both the top and the bottom La interface locations. The layer thickness however, is always found to be thicker with the use of a reference sample. As the inelastic background analysis is expected to be more reliable with the use of reference samples, as conrmed by the better background reproduction, the thus-obtained thickness results are expected to be also more reliable. The dierences between the results with and without use of reference samples are small due to the optimization of the inelastic scattering cross-section used, which is always representative of the sample composition as it is dened as a weighted average of the individual cross sections according to their nominal thicknesses.

Comparison with other techniques

Atom probe Tomography Fig. 61 shows the 3D spatial distribution of atoms for the 50 nm annealed sample, which has been obtained from the atom probe reconstruction. Each dot corresponds to one atom. As for the unannealed sample of Fig. 33, La elemental contribution was detected at m/n = 77 corresponding to LaO 2+ ion compound and Hf contribution was only de- tected in the oxide ion form at m/n = 98. The maximum intensity for La and Hf are located at the same depth and follows the same trend. From comparison with the unannealed sample presented in Fig. 33, the La has diused toward the substrate: the result is a complete intermixing of La and Hf. The oxide layer has a thickness estimated at 1.5 nm, as for the unannealed sample. The small shoulder of La located at 2.3 nm cannot be considered as relevant because its intensity is very low.

Auger depth prole Fig. 62 is obtained from the peak to peak height of the derivative spectra weighted by the corresponding sensitivity factor under the same experimental conditions as for Comparison with the as-received samples The La layer thicknesses obtained before and after annealing are summarized in Tab.10. The atom probe tomography, Auger depth proling and the inelastic background analysis show an increase in thickness of the La layer after annealing.

The APT shows an increas in thickness of 50% from 1 to 1.5 nm during annealing. The Auger depth prole can only show a qualitative increase in thickness because the strong ion implantation prevents to conclude in a more quantitative way.

The inelastic background analysis shows an increase in thickness after annealing for all the samples. Without use of reference sample, the increase is 64%, 225% and 310% for the 50 nm, 30 nm and 20 nm samples respectively.

(a) (b)

Table 11: Layer thicknesses obtained for 20 nm, 30 nm, and 50 nm samples with photon energy of 15 keV.

As seen in this Chapter, the HAXPES experiment is sensitive to deeply buried layer. A deeply buried layer gives an inelastic background intensity higher than the no-loss peak. The shape of this background is determined by the depth distribution of the element, which can be directly analyzed using proper input parameters. The main observations of this Chapter are:

The photon energy must be chosen as a compromise between the photoionisation ratio which decreases with increasing photon energy and the probed depth which increases with increasing photon energy.

Modication of the IMFP parameter in the studied cases does not change a lot the error value (10%). However, if the material is very exotic the dierence between the calculated IMFP and the EAL could be large.

If the sample presents a thick component with plasmon loss as Si, then the inelastic scattering cross section must be carefully chosen to match the physical phenomenon while photoelectrons travel inside the medium. The obtained absolute depth distribution depends on this parameter: the more precise it matches the sample, the more accurate is the absolute position of the modeled interface positions. The obtained thicknesses depend on the location of the layer, a more deeply buried layer gives an overestimated layer thickness whereas a layer located near the surface gives almost perfect match between the expected thickness and the obtained one. The best absolute positions of the layer are then obtained for the 50 nm sample whereas the best thickness modeling is obtained for the 20 nm sample.

Use of reference samples increases the reliability of the background analysis and additional use of corrections for elastic scattering gives a thickness closer to the nominal values.

The original method set up for computing error between the modeling and the measurement gives better reproduction of the background than the methods available in QUASES-Analyze ® , but is more time consuming due to the external treatment of the data.

Application to High Electron Mobility Transistor (HEMT) materials

This Chapter highlights the combination of inelastic background analysis and high resolution core-level analysis by the study of the HEMT sample series (as a function of the annealing temperature). The high-resolution core-level analysis gives the chemical environment of the considered atom in the rst 3 IMFP. From these observations, an inelastic scattering cross-section representative of the scattering events suered by the photoelectrons is determined, as an input of the calculation. This ongoing experiment on technological sample still in development exposes a new methodology to obtain depth distribution and the elemental chemical environment and to depict the structure of the sample and the eects of the annealing.

Among various types of transistors, the need for RF power electronics is increasing. GaN based compounds conveniently correspond to the requirements of wide bandgap materials due to their potential in high output power density, high operation voltage and high input impedance.

GaN also presents a high electron saturation velocity, which allows high frequency operation. The combination of these features results in a high current density and low channel resistance, giving

GaN based transistors good predisposition to be a candidate for high frequency operation and power switching application. To ensure a sustainable high power and high frequency transistors require to be based on semiconductor material with both large breakdown voltage and high electron mobility. The wide bandgap of GaN and its ability to form heterojunctions makes it a good candidate to perform High Electron Mobility Transistor (HEMT).

Transistor are based on GaN/AlGaN heterostructures as the annealing forms a thin AlN layer which conducts electricity between AlGaN and GaN layers like as a Two-dimensional Electron Gas (2DEG) [START_REF] Khan | Observation of a twodimensional electron gas in low pressure metalorganic chemical vapor deposited gan-alxgax-n heterojunctions[END_REF]. At rst the saturation drain current of 40 mA/mm was observed [START_REF] Khan | High electron mobility transistor based on a gan-alxgaxn heterojunction[END_REF] and the rst demonstration of RF power data of 1.1W/mm at 2 GHz for an AlGaN/GaN HEMT was demonstrated [START_REF] Wu | Measured microwave power performance of algan/gan modfet[END_REF]. The increasing performances have been ensured by technological improvements, rstly with use of a SiN x passivation layer [122] to reduce the DC-to-RF dispersion caused by surface trap states, the output power was then measured to 9 and 11 W/mm [START_REF] Shealy | An algan/gan high-electron-mobility transistor with an aln sub-buer layer[END_REF]. Then the addition of a eld plate (FP) in the transistor architecture decreased the peak value of the electric eld, reducing trapping eect and increasing breakdown voltage. As a result a 10 W/mm was demonstrated [START_REF] Ando | 10-w/mm algan-gan hfet with a eld modulating plate[END_REF] and optimization of the FP led to a 30 W/mm GaN based HEMT [START_REF] Wu | 30-w/mm gan hemts by eld plate optimization[END_REF]. It is also possible to connect the source to the FP, then the FP-to-channel capacitance becomes the drain-source capacitance, both possibilities are illustrated in Fig. 63.

The studied samples are designed to be the source electrodes in the global HEMT structure and are based on Ti/Al alloys (labeled S in Fig. 63). As previously stated, a 2DEG of AlN is formed at the interface of the GaN and AlGaN layers. In order to not deteriorate the device performances, the bottom interface of the AlGaN layer must remain unaltered during contact formation in the source, thus the elements forming the source must not diuse toward the substrate down to the GaN layer. The present study is not focused on the 2DEG gas, as this was pinpointed in other studies [START_REF] Gaska | Electron transport in algan/gan heterostructures grown on 6h-sic substrates[END_REF][START_REF] Mishra | Gan-based rf power devices and ampliers[END_REF], but on reactions occurring in the top layers containing Al and Ti which form the source in the device. After the 2DEG is formed, a junction must appear with the rest of the source (i.e. the Ti and the Al layers) to form an ohmic contact (see Fig. 63).

Experimental

Samples description

Samples processed for this study are based on an Al 0.25 Ga 0.75 N/GaN (24/1000 nm) epitaxial stack grown by MOCVD on Si (111) substrate as schemed in Fig. 64.

A 15 nm-thick Ti layer was deposited by electron-beam evaporation; then a 10 nm-thick Al layer was deposited in the same chamber, also by electron beam evaporation; then the sample was cut in three pieces. One is conserved without further treatment whereas the two others were annealed in a Rapid Thermal Annealing (RTA) system under N 2 atmosphere. The result is three dierent samples:

The unannealed sample.

A sample annealed at 600°C for 300 s. This sample is referred to as 600°C sample in the following.

A sample annealed at 600°C for 300 s and then 900°C for 60 s. This sample is referred to as 900°C sample in the following. It presents an ohmic behavior as presented in Appendix V.

The samples are deposited over a large area to avoid any edge eects. They cannot be used to measure DC characteristics, neither I-V curves, but are representative of the physical processes

HAXPES measurements

The measurements were performed on GALAXIES beamline at SOLEIL synchrotron using 8 keV photon energy without any specic surface preparation. The experimental geometry was chosen with x-rays impinging the sample at 10°from the surface and with photoelectrons being detected at 10°from surface normal. High resolution core-level spectra were measured to obtain information on the chemical environment of Ti and Al and high-photon ux measurements were performed for inelastic background analysis of Ti, Al and Ga. At such a high photon energy (8 keV), the corresponding eective IMFPs tabulated in Tab.12 ensure to probe the complete stack with inelastic background analysis. When studying the no-loss peak, 95% of the signal (3 IMFP) comes from 32.4 nm and 14.4 nm below the surface for Al 1s and Ti 1s respectively. The low kinetic energy of Ti 1s photoelectrons of 3011 eV is not sucient to probe the full Ti layer (assuming a bottom interface location at 25 nm below the surface according to deposition tools), but increasing the photon energy (i.e. increasing the IMFP) would lower the energy resolution of the spectrometer and aect the high resolution spectra; for this reason the photon energy was kept at 8 keV. Information given by analysis of the Ti 1s no-loss peak will therefore be representative mainly of the top of the Ti layer, while the interface between Ti and AlGaN layer could be analyzed with background analysis.

Core-level tting procedure

The energy calibration is achieved by measuring the Au 4f 5/2 peak located at 7906.9 eV kinetic energy. The recoil energy shift has been considered using Eq.17, the binding energy has been dened according to Eq.20, thus the photon energy is calculated to be 7994.6 eV. All high resolution spectra have been tted with a Voigt function except for metallic Ti and metallic Al which have been tted with Doniach-Sunjic line shape. The used Voigt functions are products of Gaussian and Lorentzian with ratio of 80% and 20% respectively. These parameter are kept constant during the tting procedure, thus an increase of the Full Width at Half Maximum of the peak is interpreted as an increasing structural and/or chemical disorder in the compound. This Voigt function has been chosen as it is found to be the best shape for the peaks in the spectra under consideration.

Results overview

Fig. 65 shows the high-ux measurements of the Ti 1s used for the inelastic background analysis, corrected to match the baseline, and the insert shows the high-resolution spectra used for the peak tting.

For the unannealed sample, the Ti 1s core-level signal is weak, as Ti is expected to be deeply buried. The strong oscillations seen in the high ux measurements show the metallic character of both the Ti and the Al layers. The inelastic background is of same intensity as the no-loss peak.

The rst annealing at 600°C leads to a new peak located at higher binding energy in the high resolution spectrum and an overall intensity increases. Furthermore only one plasmon loss is seen and slightly shifted toward the lower kinetic energy than the rst plasmon seen for the unannealed sample: it corresponds to a change in the metallic character of Ti and Al. The intensity decrease of the inelastic background indicates a diusion of the Ti toward the surface.

Further annealing at 900°C leads to components located between the two peaks seen at 600°C

and to an intensity increase compared to the one observed for the 600°C sample. These observations indicate the formation of alloys of Ti with dierent compositions. The intensity decreasing of the inelastic background indicates the diusion of Ti toward the surface.

The Fig. 66 shows the high photon ux measurements of the Al 1s used for the inelastic The annealing at 600°C gives rise to a wide peak located at higher binding energy than the oxide peak and an energy shift for the metallic Al peak. The decrease of the intensity near the no-loss peak indicates a change in the bonding of the Al.

The annealing at 900°C gives rise to a large peak located at intermediate position between the two previously observed peaks. The inelastic background is low at the vicinity of the no-loss peak, indicating the loss of metallic character of Al.

The chemical shift toward higher binding energy indicates the bonding with more electronegative elements. With incorporation of high electronegativity element, the energy shift is higher.

For these reasons and as detailed further in the next sections, the highest shift observed is more likely to be due to incorporation of oxygen (for the 600°C sample) whereas the intermediate shift is due to incorporation of nitrogen (for the 900°C sample). For all the samples, the Al 1s inelastic background decreases rapidly, showing a location of Al close to the surface. The Al 1s high-resolution core level spectrum normalized to unit area is shown in Fig. 67(a) along with its peak tting. The majority of the signal, 88%, is attributed to metallic Al located at 1559.8 eV binding energy. The peak is tted with a Doniach-Sunjic line shape with an asymmetry parameter of 0.12 and a Full Width at Half Maximum (FWHM) of 0.70 eV. The binding energy of 1559.8 eV is in perfect agreement with K. Yates et al. [START_REF] Yates | Monochromatized ag l-alpha x-rays as a source for higher energy xps[END_REF] and is very close to the literature value of 1559.6 eV for the element in its natural form [START_REF] Bearden | Reevaluation of x-ray atomic energy levels[END_REF]. The second peak located at 1562.7 eV binding energy is attributed to surface Al oxide due to air exposure [START_REF] Wagner | X-ray photoelectron spectroscopy with x-ray photons of higher energy[END_REF] and corresponds to 12% of the total peak area. The chemical shift of 2.9 eV is in concordance with the Al 1s chemical shift found by J.E. Castle et al. [START_REF] Castle | X-ray photoelectron spectroscopy using si k-alpha radiation[END_REF]. The tting curve is a product between a Gaussian function (80%)

and Lorentzian function (20%) with a FWHM of 1.4 eV. The top layer of aluminum is therefore composed of metallic Al and a layer of oxidized Al on the surface. The peaks characteristics are summarized in Tab.13(a).

Titanium 1s core-level analysis

The Ti 1s high-resolution core level spectrum normalized to unit area is presented in Fig. 67(b) along with its peak tting. This spectrum is tted with a Doniach-Sunjic line shape representative of the metallic Ti at 4964.7 eV binding energy with a smaller asymmetry parameter than for Al 1s of 0.05 and a higher FWHM of 1.1 eV. A second peak located at 4965.6 eV binding energy is representative of TiN states [START_REF] Chourasia | X-ray photoelectron study of tin[END_REF][START_REF] Chourasia | X-ray photoelectron study of tin/sio2 and tin/si interfaces[END_REF][START_REF] Kuznetsov | Xps study of the nitrides, oxides and oxynitrides of titanium[END_REF][START_REF] Burrow | A correlation of auger electron spectroscopy, x-ray photoelectron spectroscopy, and rutherford backscattering spectrometry measurements on sputter-deposited titanium nitride thin lms[END_REF]. This peak is tted with a product between a Gaussian function (80%) and Lorentzian function (20%) with a FWHM of 1.0 eV. The tting with one component characteristic of the metal (97.3% of the signal) and only one other component

Consequences for the inelastic scattering cross-section

The high resolution spectra show the metallic character of Ti and Al for the unannealed sample.

Therefore to describe the inelastic losses for TiAl alloying, the inelastic scattering cross sections could be a pondered average of metallic Al inelastic scattering cross section (dominated by a plasmon loss at 15.1 eV) and metallic Ti inelastic scattering cross section. However, the Ti inelastic scattering cross section is not implemented in QUASES analyze, so the TiO inelastic cross-section, which is very close to the Ti one, has been used. Thus the background analysis was performed with use of a weighted averaging of Al and TiO inelastic cross sections.

As emphasized in the previous Chapter, the eective inelastic cross-section used in the background analysis should take into account the eect of photoelectron high kinetic energy. This energy-dependent cross-section must be measured in REELS from a pure sample. As the photoelectron kinetic energy of Ti 1s is of 3020 eV, such renement is not needed. Thus the inelastic scattering cross-sections used in the following are taken from low energy regime.

For the analysis of the Ga, present in the AlGaN layer and in the thick (micrometric) GaN substrate, the diculty is the deep location of the Ga and its very low photoionization cross section.

The signal was too low to measure a high resolution spectrum for gallium, Ga is located so deep that the energy loss part of the Ga 2s and Ga 2p spectra are more intense than the no-loss peaks themselves as can be seen in Fig. 69(b). The two components are separated by 156 eV (with binding energies of 1299 eV for Ga 2s, 1143.2 eV for Ga 2p 1/2 and 1116.4 eV for Ga 2p 3/2 ). The tting of the gallium spectra was performed on the whole energy range using the two parameters universal scattering cross section.

Inelastic background analysis of Ti 1s and Ga 2s,2p

Fig. 68(a) shows the results of inelastic background analysis for Ti 1s unannealed sample. The lowest χ e obtained by the automated procedure points to a titanium layer with a top interface at 17.9 nm below the surface and a bottom interface at 32.2 nm below the surface. The resulting thickness of 14.3 nm is very close to the 15 nm given by the nominal thickness. The spectrum corresponding to this depth distribution is presented in Fig. 68(b) and gives good account for the inelastic background at high energy loss (2880 eV kinetic energy). However, the low energy loss (<40 eV) region is not so well reproduced in intensity even if the general shape of the measurement, dominated by plasmon loss from the Al layer is well accounted for.

Inelastic background analysis was also performed for the gallium spectrum. The corresponding energy shift compared to the metallic peak is very small, the two peaks are tted with only one component as was the case in Kovàcs et al. [START_REF] Kovacs | In situ surface analytical investigation of the thermal oxidation of ti-al intermetallics up to 1000°c[END_REF]. The small component representative of TiN has increased and represents 11.0% of the signal and the peak is tted with a higher FWHM of 1.3 eV, indicating a broadening of the interface between the metallic Ti and the AlGaN substrate.

A new component appears at 4969.6 eV binding energy, corresponding to 15.4% of the peak area. Its chemical shift is 5.1 eV compared to the metallic peak found previously. Such high chemical shift has been observed by Kovàcs et al. [START_REF] Kovacs | In situ surface analytical investigation of the thermal oxidation of ti-al intermetallics up to 1000°c[END_REF] and Leinen et al. [START_REF] Leinen | An xps study of the mixing eects induced by ion bombardment in composite oxides[END_REF] and attributed to Al 2 TiO 5 . This peak could then be a TiAl alloy with incorporation of an electronegative element.

In Fig. 71(a) and hereafter this component is labeled as TiAl ox . However the only origin of oxygen is the initial surface native oxide but the corresponding amount of O is too small to be responsible for the formation of such species. Another possible source of electronegative element is the N 2 used during RTA at 600°C. However the chemical shift corresponding to TiAl alloy with N atoms is not compatible with the observed chemical shift. The most probable source of oxygen is an unintentional contamination during the RTA, but since we did not look for N or O in our experiment, the exact chemical environment of this particular species cannot be determined and will require further investigation (with use of laboratory source for example). The peaks characteristics are summarized in Tab.14(b).

Aluminum 1s core-level analysis

The high resolution Al 1s spectrum normalized to unit area is presented in Fig. 71(b). In contrast to Fig. 67(b), it presents a very small component of metallic Al of only 8.5%. Two other signicant peaks appear. The one with a chemical shift toward lower binding energy of 1559.2 eV is consistent with the shift observed by Kovàcs et al. [START_REF] Kovacs | In situ surface analytical investigation of the thermal oxidation of ti-al intermetallics up to 1000°c[END_REF] for TiAl. This peak representing 15.4% of the peak area is tted with a product between a Gaussian function (80%) and Lorentzian function (20%) with a FWHM of 0.9 eV.

The other new peak appear toward higher binding energy with an energy shift of 3.4 eV is representative of the formation of TiAl alloy with incorporation of electronegative elements. This peak is tted with a product between a Gaussian function (80%) and Lorentzian function (20%) with a FWHM of 1.7 eV. It cannot be attributed to the Al 2 TiO 5 because it is not consistent with the chemical shift of 3.9 eV assumed by the literature [START_REF] Kovacs | In situ surface analytical investigation of the thermal oxidation of ti-al intermetallics up to 1000°c[END_REF][START_REF] Leinen | An xps study of the mixing eects induced by ion bombardment in composite oxides[END_REF]. The observed chemical shift of 3.4 eV in the measurement have also been observed by Chang et al. for an oxidation state of pure Al in UHV condition at room temperature [START_REF] Chang | Surface oxidation of an al-pd-mn quasicrystal, characterized by x-ray photoelectron spectroscopy[END_REF] and also by Olefjord et al. [START_REF] Olefjord | Intercomparison of surface analysis of thin aluminium oxide lms[END_REF]. As the Ti 1s spectrum shows a high intermixing between Ti and Al, this peak will be considered as representative of the TiAl ox species. The peaks characteristics are summarized in Tab.14(a). Binding energy (eV) Peak shape FWHM (eV) Area (%)

Aluminum 1s core-level analysis

The Al 1s high resolution spectrum normalized to unit area and its decomposition are showed in The inelastic background analysis was performed on Ti 1s and Ga 2s,2p spectra with use of the twoparameter universal cross-section as the high resolution spectra showed an almost total intermixing between Ti and Al. Fig. 76(a) shows the result of the automated procedure for the Ti 1s spectrum.

The lowest χ e value points out two depth distributions reproducing the measurement: one with a top interface located at 1.5 nm below the surface and a bottom interface located at 42 nm and the other with top interface located at 6.8 nm and bottom interface located at 42 nm. In both cases the analysis shows that Ti has not reached the surface. The choice between these two depth proles is discussed in the next section.

The Ga 2s,2p inelastic background analysis in Fig. 77 The sample is considered to be a full intermixing of Ti and Al with incorporation N. A part of the Ti remains in TiN state and the inelastic background analysis shows an overlap between the two interfaces of Ga located at 32 nm below the surface and the bottom interface of Ti 42 nm below the surface. For the same reason as for the 600°C sample, the increase of the TiN component corresponds to the broadening of the interface between Ti and AlGaN. The background analysis has led to two top interface locations at 1.5 nm or 6.8 nm below the surface. However the depth distribution is more likely to be the one with top interface located at 1.5 nm for two reasons. First because the inelastic background analysis of the 600°C sample showed a top interface located at 7.3 nm, the inelastic background of the 900°C sample decreases compared to the 600°C sample, so the Ti in the 900°C is located closer to the surface. Second, the Al 1s has only one component, attributed to a TiAl alloy, and no Al oxide (which should be expected in the case of a Ti top interface at 7.3 nm). For these reasons, the surface is assumed to be covered by a 1.5 nm layer. In the absence of denite information, this surface layer is labelled as a contamination layer.

Binding energy (eV) Peak shape FWHM (eV) Area (%)

Summary and discussion

The components determined by peak tting of the high-resolution spectra are summarized in Tab. [START_REF] Panaccione | Hard x-ray photoemission spectroscopy: Variable depth analysis of bulk, surface and interface electronic properties[END_REF] and their relative evolution after various annealing procedures are shown in Fig. 79.

Results of inelastic background modeling are presented in Tab.17

Comparison with TEM and EDX proles

TEM and EDX proles have been performed on similar samples with thickness increased by a factor 10 (Al layer of 100 nm and Ti layer of 150 nm) for the 600°C sample (Fig. 80) and the 900°C sample (Fig. 81). As the oxygen was not expected in the sample when the experiment was performed, only Ti, Al, Ga and N elements were analysed (and the W used for the sample preparation). The results cannot be directly compared to the samples studied here but can support the peak assignments in the peak tting and the results of the depth distributions. There is no lateral resolution in HAXPES experiments performed here so the measurement is averaged by the size of the irradiated sample area seen by the analyzer, whereas the EDX proles are representative of phenomena at the nm scale. Therefore, care must be taken in the comparison of the two results, as the scale and the sample preparation is very dierent in each case. The top interface location of Ti at 1.5 nm (instead of 6.8 nm) below the surface is interpreted in The structure obtained by combining high resolution spectra and inelastic background analysis is presented in Fig. 82(c

600°C sample

Conclusions of Chapter 4

In this Chapter, how the background analysis of HAXPES spectra could be used at the practical level in device technology has been highlighted.

A new method applied to the study of HEMT samples showing for the rst time the complementarity between high resolution spectra unraveling the chemical information on the rst 3 IMFP and the inelastic background analysis giving depth distribution in the rst 8 IMFP. Results obtained for the chemical environment impacts the choice of the corresponding individual inelastic scattering cross sections because the eective inelastic scattering cross-section must be chosen as the best ones describing the sample. If the inelastic background analysis points to two dierent depth distributions, the observations made from high resolution analysis can support one depth distribution at the expense of the other. The use of averaging multiple individual cross-sections is a good approximation if the exact inelastic scattering cross section is not known.

The respective sensitivity of both high resolution and inelastic background analysis allows to suggest a reaction scheme for the various annealing situations investigated. It showed the diusion of Ti and Al with incorporation of electronegative atoms and a broadening of the interface between Ti and the AlGaN layer. The Ti does not penetrate deeply in the AlGaN layer, which is a requirement for a good device performance.

Conclusion and perspective

This work exposed the rst study of inelastic background analysis of HAXPES spectra and its potential to obtain elemental depth distribution of advanced technological devices.

Characterization of thin buried layers for micro-and nano-electronic is of prime importance for understanding properties of the operating devices. Among the various experimental techniques, X-rays PhotoEmission is widely used as it allows to obtain the chemical environment of an element without modication of the sample. The limiting factor of the XPS measurements is the probing depth which is not beyond 9 nm with laboratory sources and 35 nm with hard x-rays delivered by synchrotron radiation. Increasing the probed depth by analysis of the inelastic background allows to retrieve depth distribution of deeply buried layers up to 20 nm using soft-x-rays source and 35 nm using hard x-rays.

The diculty of the background analysis of HAXPES spectra lies in the wide possibilities of locations for an element if there is no prior knowledge of the depth distribution. Moreover in the case of a deeply buried layer, the variation of the inelastic background for dierent depth distribution is very small, thus determination of the depth distribution by visual inspection is very dicult. A bot and an error estimation procedure can help to explore all the reasonable depth distributions which can lead to the observed spectra. In this direction, a semi-automated procedure has been implemented in order to determine the best matching between the measured and the simulated background.

Sensitivity to deeply buried monolayer

The third Chapter exposed the sensitivity of the HArd X-rays Photoemission Electron Spectroscopy (HAXPES) toward deeply buried layer with detection of a monolayer of lanthanum below electrodes of 20, 30 or 50 nm-thick. Background analysis on the measured spectra was successfully performed in the Tougaard's framework. Extensive analysis on the La 2p 3/2 measurements showed various eects of the parameters:

The photon energy must be chosen as a compromise between the two congurations: (i) A high photon energy where the probed depth is high but the signal/noise ratio is low; (ii) a low photon energy where the probed depth is low and the signal/noise ratio is high.

The inelastic scattering cross-section plays a major role in the eectiveness of the inelastic background analysis. It must be representative of the sample and the photon energy.

The inelastic mean free path (IMFP) has a low incidence on the inelastic back ground simulation. The uncertainty on the individual IMFPs does not aect the inelastic background analysis for more than 30% deviation.

The electron detection angle uncertainty has no inuence on the inelastic background analysis.

The amount of substance (AOS) inuences the inelastic background analusis which is more precise with an increasing AOS.

The use of reference samples increases the reliability of the analysis because the inelastic cross section can be better determined.

Reliability -comparison with other techniques

Previously the methodology for nding the depth distribution corresponding to the inelastic background of the measurement by trial and error and eye inspection. To increase the reliability of the results, a semi-automated procedure has been set up with use of a bot and applying an empirical error formula. Several inelastic backgrounds modeling where showed in this work and they all present a good match between the measurement and the modelled background, ensuring a stable procedure which can be continued to be implemented.

The comparison of the results of inelastic background analysis with destructive techniques (TEM-EELS, APT) and invasive techniques (Auger depth proling) showed the agreement on the results. This is of major importance as the photoemission experiment is a non destructive technique and allows to perform further mesurements on the sample, especially for electric characterization.

Limitations of background analysis of HAXPES spectra

Despite the eort made in this work, the determination of a depth distribution of an element present in two separated layers is still equivocal.

The experiments were performed with analysis of high binding energy core-levels with use of hard x-rays. In this condition the peaks are separated by a few hundred eV. Therefore two overlapping peaks (if they are separated by <120 eV) could not be studied.

Complementarity of core-level and background analysis

The study of the HEMT samples showed the complementarity between the study of the high resolution spectra of core-levels which unravels the chemical composition of the sample within the 3 rst IMFP and the inelastic background analysis which unravels the depth distribution of an element within the 8 rst IMFP. The chemical analysis indicates which inelastic scattering crosssection should be used while the depth distribution of the element obtained with the background analysis can give an insight of the location in the sample of the components determined with the 6 Appendices where P 2 (cos θ) = 1 2 (3 cos 2 θ -1) with θ dened in Fig. 83 as the angle between the momentum vector of the ejected electron and the polarization vector of the incident light, β is a parameter describing the angular distribution of the photoelectron. For unpolarized radiation, the expression of the photoionization cross-section is modied as:

dσ i dΩ = σ i 4π 1 - 1 2 βP 2 (cos θ) (82)
with θ the angle between the momentum vector of the ejected electron and the propagation vector of the incident light.

For an experiment with θ 54.7°, so-called magic angle, one obtain P 2 (cosθ) = 0. It is then possible to determine the relative photoionization cross-section σ i (which is also angular dependent). The parameter β depends on the quantum number l, ranging from -1 to 2 given by [START_REF] Cooper | Angular distribution of photoelectrons[END_REF]:

β = l(l -1)σ 2 l-1 + (l + 1)(l + 2)σ 2 l+1 -6l(l + 1)σ l+1 σ l-1 cos(δ l+1 -δ l-1 ) 3(2l + 1) lσ 2 l-1 + (l + 1)σ 2 l+1

with

σ l±1 = ˆ∞ 0 R n,l rR kl±1 dr (84)
the dipole radical matrix element.

This approximation is widely used in the solid photoemission community. However it was shown that with use of unpolarized Mg and Al K α x-ray, deviation from dipolar angular distribution can be observed and attributed to the inuence of retardation or phonon-momentum transfer . This work have been followed by extensive theoretical eort, mainly using the Independent Particle Approximation (IPA) which showed the dependance of non-dipole eects with atomic number and the orbital angular momentum of the photoemitted electron. For example, experiments on Ne 2s and 2p valence photoionization for 0.25 keV ≤ hν ≤ 1.2 keV showed the limit of the dipole approximation [START_REF] Hemmers | Beyond the dipole approximation: angular-distribution eects in valence photoemission[END_REF] and for specic interference between dierent-parity autoionization resonances, Martin et al. observed non-dipole eects ≤ 0.5% [START_REF] Martin | Electric-dipole quadrupole interference of overlapping autoionizing levels in photoelectron energy spectra[END_REF].

First order corrections

For soft x-rays experiments, the rst order corrections to the dipole approximation is usually sucient for the photon momentum. The correction is based on non-dipole eects in a perturbation theory framework [START_REF] Cooper | Photoelectron-angular-distribution parameters for rare-gas subshells[END_REF] where the truncature of the electromagnetic eld happens in the second term:

exp(ikr) = 1 + ikr. Two new angular parameters δ and γ and the azimuthal angle φ relative to the photoelectron propagation axis (see Fig. 83) are introduced. The dierential cross section of generation sources, the photon beams are produced by insertion devices placed in the straight sections between the bending magnets. A storage ring is then constituted of curved and straight magnetic lenses. Since the circulating electrons must be pumped continuously to balance the radiation losses they must arrive with a well-dened phase at the acceleration electrodes. For this reason they can only circulate in bunches. Every bunch is used to provide synchrotron radiation.

The signicant quantity is the stored current, which can reach several 100 mA, circulating for many hours in the storage ring. Eventually, as more and more electron gets lost, the electron beam dies out and the storage ring must be relled. Typically the widths of bunches are a few picoseconds (13 ps for SOLEIL) and the circulation time is in the order of microseconds (1.181 μs in SOLEIL).

Each time that the electron bunch trajectory is deected in the storage ring by the bending magnet or going through an insertion device, synchrotron radiation is emitted in the forward direction.

SOLEIL operates in the top-up mode providing practically constant beam current in the storage ring: the beam losses are compensated by frequent injections into the storage ring. This procedure overcomes lifetime limitations and keeps a constant photon ux for a thermal equilibrium at the beamline. The beam current is thereby kept constant at 400 mA. We immediately see the impact of insertion device between bending magnet (BM) and undulators (color lines) in g.85 where the brilliance is around 10 15 -10 20 phot/0,1%bw/mm²/mr², for comparison, x-ray tubes have a brightness of 10 9 phot/0,1%bw/mm²/mr² (which is already of the same order of magnitude as the sun).

Insertion devices are the only part of beamline inside the storage ring. The wiggler produce an oscillation in the trajectory of the electrons in the storage ring as can be seen in g.86 and ε = γm 0 c 2 0 and R is the radius of the bending magnet seen in g.87 . We can see that total energy is proportional to the 4th power of the energy particle ε.

Emission pattern for relativistic particle is highly oriented due to Doppler eect and the photons are polarized in the acceleration plane. The half width at half maximum is then:

∆θ = 1 γ = m 0 c 0 ε (89)
The brilliance of the source, dened as the photon ux per solid angle and per area is L = F (∆θ y ∆y)(∆θ x ∆z) [START_REF] Shirley | High-resolution x-ray photoemission spectrum of the valence bands of gold[END_REF] In an undulator, electrons pass through several magnets, resulting in coherent interaction (the undulator is built this way). In its reference frame, electrons see the spatial period of the undulator λp/γ. It is then its undulating period, giving the same period for the emitted radiation. With respect to the Doppler eect in the laboratory frame this period is then divided by a factor 2γ.

The fundamental wavelength λ can be modied by changing the magnetic eld B (in our case this modication is obtained by mechanically changing the gap between the magnetic elements):

λ ∼ = λ p /2γ 2 1 + bB 2 (91)
Where b is a parameter of the undulator. The bandwidth of the emitted beam is then given by:

∆λ λ = N -1 ( 92
)
Where N is the number of magnetic dipoles, then the angular spread is given by: ∆θ = 1 γ √ N

Destructive Methods

Transmission Electron Microscopy -Electron Energy Loss Spectroscopy (TEM-EELS) generally shaped with a FIB. The results are analyzed thanks to a diraction image which can be converted to a real image via Fourier transform [START_REF] Müller | What transmission electron microscopes can visualize now and in the future[END_REF][START_REF] Everts | Transmission electron microscopy of bone[END_REF][START_REF] Midgley | 3d electron microscopy in the physical sciences: the development of z-contrast and eftem tomography[END_REF]. The measurement of electron energy loss (EELS) allowing to obtain depth distribution along the sample surface is proportional to Z, therefore the distinction of two juxtaposed elements of close atomic number can be dicult or even impossible without modeling. These techniques can give results on the chemical state, the depth distribution, the optical and magnetic properties, the structural defects and strain. The detection limit is of 1 10

19 at/cm 3 and the ultimate spatial resolution of a TEM is of 0.2 nm. This resolution is mainly limited by the sample preparation.

Atom Probe Tomography (APT)

Atom Probe Tomography (APT) is an analytical technique for studying materials at the nanoscale.

It enables the generation of a three dimensional elemental map, from which local composition measurement can be extracted. Initially used only for metallic specimens [159], the introduction of a pulsed laser [START_REF] Gault | Design of a femtosecond laser assisted tomographic atom probe[END_REF] in recent years has made possible the analysis of semiconductors and even insulators. APT is based on the eld evaporation eect requiring the sample to be prepared in the form of a needle of radius <50 nm, using a Focused Ion Beam (FIB). This tip is submitted to a high electric voltage (V∼1-15 kV) leading a high electrical eld at the sample surface (E∼V/R, R is the tip radius at the apex) as can be seen in Fig. 89. The atoms at the surface of the tip are ionized, accelerated by the high electric eld and nally detected by a spatial and time resolved detector.

A pulse voltage or laser allows the chemical identication of ions by time of ight (TOF) mass spectrometry. More precisely, the ion TOF from the tip to the detector is proportional to the mass to charge (m/n) ratios. Reconstruction of the evaporated sample is obtained using a projection algorithm taking into account the sequence and nature of detected ions as well as their position [START_REF] Vurpillot | Pragmatic reconstruction methods in atom probe tomography[END_REF]. In ideal conditions, depth resolution can be better than 0.2 nm and lateral resolution better than 0.5 nm [START_REF] Miller | Atom Probe Tomography[END_REF].

Invasive depth proling methods

Elemental depth proling can be done by combining sputtering with surface-sensitive techniques such as X-ray Photoelectron Spectroscopy (XPS) or Auger Electron Spectroscopy (AES). As the photoemission was described in the previous section in detail, the depth proling using sputtering will be illustrated with Auger electron Spectroscopy. The Time of ight Secondary ions Mass Spectroscopy will be explained as it employs two ions beams.

Auger Electron Spectroscopy

Auger Electron Spectroscopy uses an electron focused beam as a source and collects electrons emitted via an Auger process as a function of their kinetic energy. The Auger process is a deexcitation process where a hole in the subshell K allows a transition from the electron of subshell L toward the K subshell. The corresponding energy is given to an electron of subshell M which is ejected from the sample. As the energy of K, L and M subshells are xed the resulting kinetic energy of the emitted Auger electron is xed. The eciency of the Auger deexcitation process decreases with increasing atomic number [START_REF] Krause | Atomic radiative and radiationless yields for k and l shells[END_REF]. For this reason, elements with atomic number > 50 are rarely studied by AES because acquisition time becomes rather long. The ultimate depth resolution is 0.2 nm and the detection limit is of 1 10 19 at/cm 3 .

Time of Flight Secondary Ions Mass spectroscopy (ToF-SIMS)

This technique allows obtaining elemental and molecular composition by analyzing secondary ions emitted when the sample is bombarded with a primary ion beam as shown in Fig. 90. To obtain the depth prole, a second ion beam is used to sputter the surface step by step. The emitted ions are analyzed in a time of ight mass spectrometer, so all elements and their isotopes can be detected.

The molecular ions emitted correspond to molecules in the sample matrix, so that not only depth distribution is obtained but also stereochemistry for organic compounds. The spot size is usually around a few µm 2 and the ToF-SIMS can be used as a microscope with lateral resolution around 100 nm and ultimate depth resolution of 0.5 nm. The primary ion beam has an energy from a few hundred eV to a few keV (typically 25 keV for analysis beam) and can be made of various ions, the most usual being Bi n + , Cs + , O 2 + , Au + and C 60 + . As the kinetic energy of the ionized compounds is xed by the extraction voltage, the time needed for the ions to reach the detector inside the analyzer is proportional to the square root of mass/charge. The primary ion beam has a pulse of a few ns, which allows to measure precisely the time of ight of the ions. The quantication can be performed only with use of reference samples.

Non invasive optical methods

The invasive techniques do not , or very slightly modify the sample. They involve photons as probe source and techniques presented here rely on the photon detection (transmitted or emitted).

Ellipsometry

Ellipsometry is an optical non-destructive characterization technique based on the change in the polarization state of an electromagnetic wave after reexion on the surface of a sample [START_REF] Fujiwara | Spectroscopic Ellipsometry: Principles and Applications[END_REF].

The ellipsometry principle is described in gure 91. The apparatus is composed of a light source (generally between 1 and 10 eV), a polarization state generator (PSG), a polarization state analyzer (PSA) and a detector [START_REF] Mcgahan | Spectroscopic Ellipsometry and reectometry[END_REF]. The PSG determines a specic state polarization to the light impinging the sample, generally linearly polarized. The reected beam after interaction with the sample is analyzed by the PSA. In the case of a monochromatic linearly polarized light, the electrical eld of the incident beam is

- → E = -→ E 0 e i(ω--→ k .
-→ r ) and the same expression is used for , the electrical eld reected by the sample. These two waves can be decomposed in two orthogonal directions, one perpendicular to the incident plane (s wave represented in green in the gure) and the other parallel to this plane (p wave represented in purple in the gure). The p and s vibrations reect on the sample without reciprocal interaction and the corresponding reexion coecients r p and r s are written as:

r p = E p E p r s = E s E s (94)
The p and s polarizations show dierent changes in amplitude and phase upon light reection causing the reected light to become elliptically polarized. By measuring dierent reected intensities, ellipsometry allows the reexion coecients ratio for the p and s waves to be calculated.

This complex number is expressed as:

ρ(θ, λ) ≡ r p r s ≡ tanψe i∆

The parameters tan ψ and Δ are respectively the amplitude and phase variations between the p and s waves of the electric eld. To obtain the measurement of the unknown properties of the analyzed sample (refractive index and thicknesses of the layers), a mathematical model can provide theoretical values for ψ and Δ [START_REF] Azzam | Ellipsometry and polarized light[END_REF]. These values are compared to the experimental data and adjusted through an iterative process. The quality of the obtained values are evaluated by minimization of the mean squared error χ 2 representing the dierence between the experimental data and the calculated values. Nowadays ellipsometry is a widely used technique for measurements of refractive indices n and extinction coecient k of all types of materials due to its non-destructive character and extreme sensitivity to the surface and ultra-thin layers (<1 nm) [START_REF] Cho | Spectroscopic ellipsometry characterization of high-k dielectric hfo2 thin lms and the high-temperature annealing eects on their optical properties[END_REF]. The depth resolution is of 0.5 nm and the probed depth up to a few µm depending on the absorption of the material. equal. This allows one to determine the electron density prole perpendicular to the surface over a range spanning approximately 1 to few hundred of nm below the surface [START_REF] Chason | Thin lm and surface characterization by specular x-ray reectivity[END_REF]. XRR can be equally performed on amorphous, polycrystalline or liquid materials since it is only sensitive to the electronic contrast between materials. The amplitudes of the transmitted and reected waves can be determined by the conditions for continuity of the waves at the interface, resulting in the so-called Fresnel equations. Close to the critical angle, correction from refraction must be also applied according to Snell's law:

n 1 cos(θ 1 ) = n 2 cos(θ 2 ) (96)
as depicted in Fig. 92. For a given θ 1 , called the critical angle, θ 2 is equal to zero, and according to the above equation, the critical angle is directly related to the coecient n 2 of the irradiated material (considering that n 1 is equal to 1 for air). The precise measurement of this critical angle can thus give access to the electronic density of an individual layer [START_REF] Van Der | Grazing incidence specular reectivity: theory, experiment, and applications[END_REF]. Below the critical angle, the beam is almost fully reected. Above the critical angle, the intensity decreases rapidly and this decay can be related to the roughness of the surface. When a lm of dierent electron density to the substrate is present, partial reection occurs at both the top and bottom interfaces. Interference occurs between the waves, resulting in interference fringes as the optical path dierence between them is changed by varying the incidence angle. The fringes are popularly known as Kiessig fringes, after their discoverer. The condition for constructive interference is that the path dierence must be an integral number of wavelengths. These Kiessig fringes periods can thus be used to extract the lm thickness if the thickness is between 1 and 1000 nm [START_REF] Van Der Lee | Mass density determination of thin organosilicon lms by x-ray reectometry[END_REF]. The XRR experiment is not restricted in sort of sample as long as their surface is at. Typical spectrum measured by XRR and the physical information contained within, from [START_REF] Yasaka | X-ray thin lm measurement techniques v. x-ray reectivity measurement[END_REF].

  Complementarity of inelastic background and high resolution core level in HAXPES P. Risterucci, O. Renault, E. Martinez, C. Zborowski, D. Céolin, J.-P. Rue, G. Grenet, S. Tougaard, in preparation. List of abbreviations: AOS: amount of substance APT: atom probe tomography CCD: charge coupled device CMOS: complementary metal oxide semiconductor CVD: chemical vapor deposition DPN: decoupled plasma nitridation DRF: depth resolution function EELS: electron energy loss spectroscopy EOT: equivalent oxide thickness FWHM: full width at half maximum HEMT: high electron mobility transistor HAXPEEM: hard x-ray photoelectron emission microscopy HAXPES: hard x-ray photoelectron spectroscopy ITRS: International technology roadmap for semiconductors ISSG: in-situ steam generation MCP: Multi channel plate MOSFET: metal oxide semiconductor eld eect transistor NMOS: n-doped metal oxide semiconductor PEEM: photoelectron emission microscopy PMOS: p-doped metal oxide semiconductor PNA: post nitridation annealing PVD: physical vapor deposition RMS: root mean square TEM: transmission electron microscope XPEEM: x-ray photoelectron emission microscopy XPS: x-ray photoelectron spectroscopy Parmi les personnes importantes je voudrais également remercier Sven Tougaard pour l'écoute qu'il m'a donné tout au long de la thèse et les échanges que nous avons eus dès que nous pouvions nous voir. Tout ce travail n'aurait pas pus voir le jours si je n'avais pas été formidablement accueilli par mes collègues de la PFNC. Que ce soit l'aide constante d'Eugénie, l'aide à la préparation des échantillons de Jean-Marc, l'aide ponctuelle de Vincent et Dominique, l'aide de dernière minute de Nicolas ou tout simplement pour les bons moments passés à la pause café. Beaucoup de gens m'ont soutenu et encouragé durant mes années d'études; la première personne à l'avoir fait est Mme Assens du collège Sonia Delaunay. Je tiens particulièrement à la remercier pour avoir réussi à éveiller ma curiosité, pour m'avoir montré combien la connaissance était importante. La conance que vous m'avez porté ne m'a jamais quitté et elle m'a souvent servi dans le reste de mon parcours. Bien évidement d'autres personnes m'ont aidés par la suite, notamment M. Rochet qui m'a trouvé mon premier stage au synchrotron SOLEIL. Ce stage à été déterminant dans ma poursuite en thèse. C'est grâce à Fausto que j'ai réellement découvert ce qu'était un scientique, dans l'anticipation mais aussi la réactivité face aux expériences. Je tiens aussi à remercier Matthieu pour sa disponibilité et les échanges que nous avons pus avoir. Je remercie aussi Azzedine pour toute son implication et les nuits passés à travailler sur la ligne. Je remercie également Mario pour m'avoir initié à l'art du peak tting et Christian pour toute son aide. Je souhaite également remercier toute ma famille. En premier lieu mes parents qui m'ont toujours fais ressentir leur amour et leur soutiens. Je remercie mes grands parents de m'avoir accueilli toutes les vacances, pour nous avoir tous aimé et réussit à créer cette joie à chaque fois que l'on se retrouve. Merci à ma s÷ur d'avoir supporté mon attitude durant toute ma jeunesse. Merci à mes tantes et oncles, mes cousins et cousines pour les bons moments passés ensemble. Et merci à tous ceux qui m'ont accueillis à Grenoble. Merci à Stéphanie de m'avoir accepté en collocation du jour au lendemain. Merci à Amandine et Ludovic pour ces presque deux années de vie ensemble. Merci également à toute l'équipe de foot de l'AS, sans vous mon séjour à Grenoble aurait été beaucoup plus calme ! Preface The present thesis was conducted in the Commissariat à l'Energie Atomique et aux Energies Alternatives (CEA) in the Laboratoire d'Electronique et de technologie de l'Information (LETI) from January 2012 to February 2015. The research has evolved around the analysis of inelastic background in photoelectron spectroscopy, mainly at high photon energy. The experiments were performed in synchotron facilities in Grenoble (ESRF) and in Saclay (SOLEIL). The thesis is made in collaboration with the University of Southern Denmark, the Ecole centrale and Institut des Nanotechnologies de Lyon (INL) and the CEA-LETI.

Figure 1 :

 1 Figure 1: Scheme of the relevant energy levels in the sample (left) and measured spectra in a photoemission experiment (right).

Figure 2 :

 2 Figure 2: Three step model of valence-band photoemission. (1) photoionization; (2) the photoelectron is transported toward the surface; (3) the photoelectron crosses the surface, travels into vacuum and is detected. Adapted from [9].

Figure 5 :

 5 Figure 5: Relative intensity contribution of layers with thickness λsinθ from [9].

Figure 6 :

 6 Figure 6: Scheme of a synchrotron facility. (1) LINAC, (2) Booster, (3) storage ring, (4-5) wiggler, (6) insertion device, (7) beamline.

Figure 8 :

 8 Figure 8: Schemes of (1) photoionization; (2) shake up process, (3) shake o process.

Figure 10 :

 10 Figure 10: Inelastic mean free path as a function of the photoelectron kinetic energy for Ti, Sc, K, Si, Al and Mg. The solid circles show calculated IMFPs from the Penn algorithm. The solid lines show ts to these IMFPs with modied Bethe equation and the long-dashed lines indicate the IMFPs calculated from TPP-2M equation. Successive plot have been displaced vertically for clarity. From [39].

Figure 11 :

 11 Figure 11: Single scattering albedo ω as a function of the kinetic energy of the photoelectron for elemental solids Si, Fe, Ni, Cu, Ge, Ag, W and Au. From [35].

Figure 12 :

 12 Figure 12: Scheme of an XPS experimental sputtering (a), optical image of a formed crater (b).

Figure 13 :

 13 Figure 13: Scheme of the principle of ARXPS.

2 k

 2 and observed intensity at the k th measurement (angle for ARXPS or energy for VKE-XPS), σ is the variance of the k th measurement and N is the number of independent observations in the data. The MaxEnt then maximize the joint logarithmic probability function Q dened as:

Figure 15 :

 15 Figure 15: (a) Take o angle dependence of the peak areas for a TiN/Al 2 O 3 /InGaAs sample. The solid lines correspond to theoretical behaviour generated according to the physical model in (b). Extracted from [70].

Figure 16 :

 16 Figure 16: Intensities of the Ge 1s and 2s photoelectron peaks normalized to the corresponding Gebulk signals, for sample A (triangles and sample B (circles). The t (solid line) to the experimental data indicates the presence of a sharp interface between Si

Figure 17 :

 17 Figure 17: First observation of inuence of various depth distribution of copper (orange) inside a gold matrix (white) on inelastic background, from [78].

Figure 19 :

 19 Figure 19: Two parameter Universal cross-sections (red). Three parameters universal cross-section valid for Germanium (black), polymers (green) and silicon (blue) implemented in QUASES-Analyze.

Figure 20 :

 20 Figure 20: Various depth prole morphology that can be analyzed with QUASES-Analyze: (a) exponential prole; (b) single layer; (c) Multilayer; (d) island with active substrate; (e) island with passive substrate; (f ) islands. The structures (d), (e) and (f ) give identical spectra. Adapted from [87].

Figure 21 :

 21 Figure 21: Print screen of the QUASES-Analyze ® software. The measurement is in black, the modelled inelastic background is in purple and the subtracted spectrum is in green.

Figure 22 :

 22 Figure 22:

Figure 23 :

 23 Figure 23: Illustration of the method used. (a) direct simulation approach, (b-e) contribution of scattered electrons. The sum of all these scattered electrons gives (e) and (f ) is the sum aected by coecients. The resulting spectrum (g) is obtained.From[START_REF] Werner | Simulation of electron spectra for surface analysis using the partialintensity approach (pia)[END_REF].

Figure 24 :

 24 Figure 24: Schematic illustration of the internal structure of SESSA. User interacts with the software through CLI (command line Interface) or GUI (Graphical User Interface), from [94].

3. 1

 1 Experimental methods3.1.1 High-k MOSFETMetal Oxide Semiconductor Field Eect Transistor or MOSFET is the principal component of integrated circuits. Its success comes from the ability to be easily made as type P or type N (PMOS or NMOS). The association of two types of transistors is called Complementary MOS (CMOS). It allows to perform complex logical functions and limits the energy consumption of the device.

Figure 25 :

 25 Figure 25: Scheme of a MOS transistor. (S) is the source; (D) is the drain; the red layer corresponds to the conduction canal.

Figure 26 :

 26 Figure 26: Scheme of evolution of the structure of the gate for a MOSFET structure. (a) original design; (b) after introduction of a high-κ; (c) introduction of an Interfacial Layer (IL); (d) integration of a metal gate.

Figure 27 :

 27 Figure 27: Sample production scheme.

Figure 28 :

 28 Figure 28: Sample scheme of the metal gate architecture typically used in CMOS technology. The thicknesses are at scale with respectively 20 nm, 30 nm and 50 nm top electrode composed of polysilicon.

Figure 29 :

 29 Figure 29: Pictures of the used beamlines, (a) GALAXIES from SOLEIL synchrotron; (b) BM25 from ESRF; (c) ID32 from ESRF.

Figure 30 :

 30 Figure 30: TEM cross-section image of the 50 nm sample.

Figure 31 :

 31 Figure 31: EELS prole of the active layer containing HfSiON and LaO layers and the TiN barrier of the 50 nm sample.

Figure 32 :

 32 Figure 32: Auger depth proling of the 50 nm sample, highlighting the thickness of the a-Si cap and the depth distribution of the La layer in between the TiN and HfSiON layers. The broadening of the TiN layer is due to dierent sputter rates for TiN and Si.

Figure 33 :

 33 Figure 33: (a) 3D Atom Probe Tomography of the 50 nm sample. (b) Corresponding proximity histogram of La and Hf calculated from 37% TiN iso-surface. The top panel shows the reconstruction of the La and Hf layers and the depth prole of these elements are presented in the bottom panel.

3. 1 . 5

 15 Implementation for practical use of QUASES Analyze ® There are diculties in the practical use of the software. Firstly the variation of f(z) is made either by typing the interface values or by clicking a button in the software which changes the interface with steps of 0.1, 0.2, 0.5, 1, 2, 5, 10 or 50 Å. Then nding good match between the modeling and

50 (or 30 -Table 3 :

 303 [START_REF] Jablonski | Nist electron eective-attenuation-length datadata[END_REF]) nm a-Si. A source of uncertainty comes from the density of the layers. The samples are technologically advanced and their density may not be corresponding to tabulated values, especially after annealing; this is another source of uncertainty in the evaluated eective IMFP. Eective IMFP for La 2p 3/2 at photon energy of 15 keV in nm for the 50 nm sample.

Figure 34 :

 34 Figure 34: Si 1s spectra measured at 8 (black), 10 (brown), 12 (purple) and 15 keV (blue) photon energy. Spectra are normalized to the same intensity at the no-loss peak maximum.

Figure 35 :

 35 Figure 35: Si 1s spectra of oxidized silicon measured at 2.5 keV (black), 3 keV (blue), 4 keV (red), 6 keV (green), 10 keV (brown) and 12 keV (purple). The inset of the gure shows maximum intensity for the oxide peak located at 4 eV below the Si1s no-loss peak and plasmon losses.

Figure 36 :

 36 Figure 36: Spectra of Ti 1s measured with photons of 12 keV (black), 15 keV (red) and 18 keV (green) with a detection angle of 10°from surface normal.

Figure 37 :

 37 Figure 37: La 2p 3/2 threshold of 50 nm sample measured at 12 keV (green), 15 keV (red) and 18 keV (black) with detection angle of 10°from surface normal. The line points to the no-loss peak binding energy of 5523 eV.

Figure 38 :

 38 Figure 38: HAXPES survey spectra measured at hν = 15 keV on the 50 nm (red) and 30 nm (black) samples. The inset shows the La 2p 3/2 transition along with its energy loss structure. Right: structure of the gate stack samples for the thickest a-Si capping layer (50 nm-sample).

Figure 39 :

 39 Figure39: La 2p 3/2 threshold of 50 nm sample measured at 15 keV photon energy with detection angle of 10°f rom surface normal (black) and modeling for depth distribution of top interface located at 56 nm below the surface and bottom interface at 61 nm. The modeling corresponding to the computed IMFP (15.54 nm) is plotted in red, the green modeling is computed for an increasing value of the IMFP of 30% (20.07 nm) and the blue to a decrease of 30% (10.91 nm). The inset shows the χ e value computed for the energy range between 9400 and 9490 eV.

Figure 40 :

 40 Figure 40: La 2p 3/2 threshold of 50 nm sample measured at 15 keV photon energy with detection angle of 10°fromsurface normal (black) and modeling for depth distribution of top interface located at 56 nm below the surface and bottom interface at 61 nm. The modeling corresponding to the correct detection angle (10°from surface normal) is plotted in red, the green modeling is computed for an angle of 0°from surface normal and the blue to an angle of 40°from surface normal. The inset shows the χ e value computed for the energy range between 9400 and 9490 eV.

Figure 41 :

 41 Figure 41: The top panel presents various inelastic scattering cross-sections: three parameter universal cross section in red, three parameter cross section of Silicon (in green) and cross-section calculated from REELS measurements at 10 keV (brown). The corresponding modeling for nominal depth distribution of lanthanum of the 50 nm sample is plotted on the bottom panel with (a) the three parameter cross section, (b) the three parameters silicon cross section and (c) the cross section derived from REELS measurements at 10 keV.

  and 35.0 nm to 36.0 nm for the 20 nm sample as summarized in Tab.5.

Figure 43 :

 43 Figure 43: Background analysis of the La2p 3/2 HAXPES spectrum of the 50, 30 and 20 nm samples. The blue curves are the calculated backgrounds corresponding to a depth distribution of La extending over 1 nm from 35.0 nm below the surface for the 20 nm sample; 2 nm-thick from 41 nm below the surface for the 30 nm sample; 4 nm from 57 nm below the surface for the 50 nm sample. The hatched part represent the reproducible part of the measurement by inelastic background analysis and the arrow points to the core-level no-loss peak.

Figure 44 :

 44 Figure 44: HAXPES spectra measured for 20 nm samples at 10°from surface normal for 0.4 nm-thin La layer (left and black) and for the 1 nm-thick one (right and red). The inset focuses on La 2p 3/2 peaks where the line corresponds to the kinetic energy of La 2p 3/2 of 12897 eV.

Figure 45 :

 45 Figure 45: Si 1s measurement (in black) with modeling of the expected depth distribution of two layers (in blue) and the best modeling (in green).

Figure 46 :Table 7 :

 467 Figure 46: (a) colormap of χ e for Ti 1s of 50 nm sample; (b)modeling for depth distribution of 49 nm to 60 nm below the surface (blue) and 40 to 77 nm below the surface (in red).

Figure 47 :

 47 Figure 47: Spectrum of Hf 2p 3/2 for the 50 nm sample measured at 15 keV (in black) with background for a layer with top and bottom interface at 57 and 58.6 nm respectively (blue) and the background with lowest χ e value with top interface at 30 nm and bottom interface at 70 nm below the surface (red).

14. 8

 8 nm. Calculation of χ e leads to the result showed in Fig.48(a)where the depth distribution is explored from the thinnest one of 59 to 61 nm below the surface to the thickest one from 50 to 75 nm. These two background modelings are presented in Fig.48(b) and are almost the same.

Figure 48 :

 48 Figure 48: (a) colormap of χ e for Hf 2p 3/2 of 50 nm sample; (b) modeling for depth distribution of 59 nm to 61 nm below the surface (blue) and 50 to 75 nm below the surface (in red).

  TiN reference sample consists of a 150 nm-thick layer of TiN deposited on the same condition as the TiN in the gate stack on silicon. The eective IMFP for Ti 1s in TiN at kinetic energy of 10045 eV is 12.68 nm. Therefore, 8 IMFP overcome the bottom interface of TiN layer located at 56.5 nm below the surface. The hafnium reference is obtained by deposition of a thin lm of HfSiO by MOCVD followed by DPN and PNA as the it has been done for the layer in the studied stack. The resulting thickness of the HfSiON layer in the reference sample is 15 nm.The lanthanum reference consists of a commercial sample LA000200 from goodfellow. It is a 25x25 mm lanthanum foil of 0.025 mm-thick with La concentration of 99.9%. The sample has been directly transferred from the packaging of sample provider to the experimental chamber within a minute of air exposure. Even if this La reference sample can be seen as a pure innite sample from an XPS point of view, its fabrication process does not correspond to the technological sample deposition condition. The studied stack is composed of oxidized lanthanum of unknown oxidation state whereas the reference sample is composed of pure lanthanum.

Figure 49 :

 49 Figure 49: Dierence between the modelled spectrum and the measurement for the Si 1s spectra of the 50 nm samples with various error calculation: the set homogeneous layer from QUASES-Analyze software (red), our error computing method (green), the optimized AREA from QUASES-Analyze software (dark blue), the optimized RMS from QUASES-Analyze software (light blue), results for two homogeneous layers (black), results for set homogeneous layer from QUASES-Analyze software with Universal cross-section (yellow) and an innite layer with top interface located at 9 nm below the surface with the universal cross-section (pink).

Figure 50 :

 50 Figure 50: La 2s subtracted spectra modeling for 50 nm sample corresponding to a depth distribution La of 56 nm to 61 nm, with the pure silicon cross section measured at 10 keV without reference sample (black), with average cross-section without reference sample (red), with the pure silicon cross section measured at 10 keV and reference sample (green), with average cross-section and reference sample (blue) and with universal cross-section and reference sample (pink).

Figure 51 :

 51 Figure 51: Colormap of χ e of La in the 50 nm sample with use of reference s ample.

Figure 52 :

 52 Figure 52: Colormap of χ e of Ti in the 50 nm sample with use of reference sample.

Figure 53 :

 53 Figure 53: plot of χ e calculation for Hf depth distribution prole with help of reference sample in the 50 nm sample.

3. 5 . 1

 51 Results of inelastic background analysis of La 50 nm sample The 50 nm sample annealed was measured at ID 32 beamline with hv = 15 keV Inelastic background analysis was performed with the inelastic scattering cross-section derived from REELS measurement at 10 keV of pure silicon sample and IMFP = 15.4 nm for La 2p 3/2 .

Figure 54 :

 54 Figure 54: (a) Result from automated treatment of the 50 nm sample annealed on La2p 3/2 spectrum, showing a depth distribution with top interface located at 49.5 nm and a bottom interface located at 56.2 nm below the surface. (b) Measured spectrum (in black) and corresponding modeled background (in blue) are presented with the subtracted spectrum (in red).

Fig. 57 (Figure 56 :

 5756 Fig.57(b) shows the subtracted spectra with use of reference sample (in red) and without (in blue). The dierence between the two spectra in the top panel shows the almost perfect superposition of the two spectra in the top panel is representative of two very close depth distributions (the scale is multiplied by 100). It is impossible to decide which one of the two spectra is a better reproduction of the measurement. As the analysis with use of reference sample could be more accurate, the corresponding depth distribution of the top interface located at 35.5 nm and bottom interface at 42.2 nm below the surface will be retained in the following.

Figure 57 :Figure 58 :

 5758 Figure 57: (a) Result from automated treatment of the 30 nm sample annealed on La 2s spectrum with use of reference sample, showing a depth distribution with top interface located at 35.5 nm and a bottom interface located at 42.2 nm below the surface. (b) The two subtracted spectra with (in red) and without (in blue) reference sample.

Fig. 59 ( 2 Figure 60 :

 59260 Fig.59(b) shows the two subtracted spectra with (in red) and without use of reference sample (in blue). The dierence between the two spectra is plotted in the top panel. It shows that the major deviation is located directly below the no-loss peak and extends to 70 eV energy loss (6440 eV kinetic energy). The spectrum with use of reference samples presents a better matching with the baseline. The corresponding depth distribution with top interface at 30.8 and bottom interface at 35.5 nm below the surface is much trustworthy.

Figure 61 :

 61 Figure 61: 3D Atom Probe Tomography of the 50 nm sample unannealed (a) and annealed (b) with corresponding proximity histogram of La and Hf calculated from the 37% TiN iso-surface. The top panels show the reconstruction of the La and Hf layers and the depth prole of these elements are presented in the bottom panels.

Fig. 32 .

 32 Fig. 32. The right panel shows a zoom on the the concentration prole between 60 nm and 70 nm below the surface. It conrmes that the oxided state of both La and Hf as seen in APT. The overall thickness of the layer containing La and Hf is ∼ 6 times wider than the thickness obtained by APT. This is due to ion implantation during sputtering while performing the Auger depth prole (specially for Hf which presents a large tail at >70 nm depth).

Figure 62 :

 62 Figure 62: Auger depth proling of the annealed 50 nm sample annealed (a) and unannealed (b), highlighting the thickness of the a-Si cap and the depth distribution of the La layer in between the TiN and HfSiON layers. The broadening of the TiN layer is due to dierent sputter rates for TiN and Si.

Figure 63 :

 63 Figure 63: Scheme of GaN based HEMT with (a) gate-connected eld plate and (b) source-connected eld plate; (S) is the source and (D) is the drain.

Figure 64 :

 64 Figure 64: Scheme of the sample used as source in HEMT structure. The thicknesses are nominal values given by deposition tools.

Figure 65 :

 65 Figure 65: Measurements of Ti 1s with high photon ux of 8 keV, the insert shows the high-resolution spectra of Ti 1s for the various annealing.

Figure 66 :

 66 Figure 66: Measurements of Al 1s with high photon ux of 8 keV, the insert shows the high-resolution spectra of Ti 1s for the various annealing.

( 2 . 7 %

 27 of the signal) shows the good quality of the deposed Ti layer. There is no presence of N during the deposition of the Ti layer so the TiN found in the measurement is most likely arising from the AlGaN layer on which the Ti is deposited. The peak characteristics are summarized in Tab.13(b).

Figure 67 :

 67 Figure 67: High resolution spectra of (a): Ti 1s; (b): Al 1s measured at 8 keV photon energy for the unannealed sample. The spectra are normalized to unit area.

Fig. 69 (Figure 68 :Figure 69 :

 696869 Figure 68: (a) Result from the automated treatment of the unannealed Ti 1s spectrum, showing a depth distribution with top interface located at 17.9 nm and a bottom interface located at 32.2 nm below the surface. The corresponding measurement (in black) and the corresponding modelled background (in blue) also with the subtracted spectrum (in red) are presented in (b) with the energy range used for calculation of χ e .

4. 3 . 5

 35 Structure of the sample deduced from the analysisThe high resolution spectra unravel the predominantly metallic character of Ti and Al in the unannealed sample in the topmost 3 IMFP and a surface oxidation of the Al layer. These results were used for the choice of the inelastic scattering cross-section used in the background analysis:for Ti 1s an average of pure Al cross section and the universal cross-section for Ti 1s and the two parameters universal cross section for Ga.The top interface of the AlGaN layer is located at 25 nm below the surface whereas the bottom interface of the titanium is located at 32.2 nm. The 7.5 nm dierence between the two interfaces comes from the roughness of the AlGaN layer: as the photoemission experiment is averaging the signal, the step between two AlGaN terraces induces an uppershift of the top interface of AlGaN layer and an lowershift of the bottom interface of the Ti layer. The result of inelastic background analysis is seen as the mean location of the interface. The TiN component seen in the high resolution spectrum is representative of this roughness. The inelastic background analysis dened a depth distribution of Ti with a thickness of 14.3 nm with a top interface located at 17.9 nm below the surface. This value shows that Ti is covered by a larger layer than the expected 10 nm of Al layer. As a surface oxide component has been seen in the Al 1s spectra, the top layer consists of metallic Al and oxidized Al. The depth distribution of Al could not be retrieved as Al is present in the top layer and in the substrate. The real Al thickness may not be of 17.9 nm as the deposition tools forecast a layer of 10 nm. As the sample was kept at air exposure and no surface preparation was made, the discrepancy between the nominal Al layer thickness and the result of inelastic background analysis can be explained by surface contamination. Fig. 70 only shows the results of the present analysis which omits the contamination layer.4.4 Results for the 600°C sample 4.4.1 Titanium 1s core-level analysis For the sample annealed at 600°C for 300 s, new components appear in the Ti 1s high resolution spectrum normalized to unit area presented in Fig. 71(a): the Ti metallic component located at 4964.9 eV has decreased in relative intensity to 73.7% of the total peak area and is shifted

Figure 70 :

 70 Figure 70: Structure of the unannealed HEMT sample. The depths of the layer are extracted from the inelastic background analysis and the dierent layers from the core-level analysis.

Figure 71 :

 71 Figure 71: High resolution spectra of (a) Ti 1s and (b) Al 1s measured at 8 keV photon energy for the 600°C sample. The spectra are normalized to unit area.

Figure 72 :

 72 Figure 72: (a) Result from automated treatment of the 600°C sample Ti 1s spectrum, showing a depth distribution with top interface located at 7.3 nm and a bottom interface located at 20.4 nm below the surface. The corresponding measurement (in black) and the corresponding modelled background (in blue) also with the subtracted spectrum (in red) are presented in (b) with the energy range used for calculation of χ e .

Figure 73 :

 73 Figure 73: (a) Result from automated treatment of the 600°C sample Ga 2s,2p spectrum, showing a depth distribution with top interface located at 22.0 nm and a bottom interface beyond detection limit. The corresponding measurement (in black) and the corresponding modelled background (in blue) also with the subtracted spectrum (in red) are presented in (b) with the energy range used for calculation of χ e .

Figure 74 :

 74 Figure 74: Structure of the HEMT sample annealed at 600°C. The depths of the layer are extracted from the inelastic background analysis and the dierent layers from the core-level analysis.

4. 5

 5 Results for the 900°C sample 4.5.1 Titanium 1s core-level analysis This sample is the 600°C sample which is further annealed at 900°C for 60 s under N 2 atmosphere. The Ti 1s high resolution spectrum normalized to unit area is presented in Fig. 75(b). The main peak representing 78.5% of the peak area is representative of Ti-Al alloy with incorporation of electronegative element. It is tted with the product between a Gaussian function (80%) and a Lorentzian function (20%) with a FWHM of 1.7 eV. Compared to the TiAl ox signal, the peak is shifted of 0.9 eV toward lower binding energy. It indicates a lower oxidation state of the alloy formed, therefore, this peak is attributed to TiAlN species. The tting procedure also quanties the TiN peak located at 4966.1 eV binding energy to represent 21.3% of the signal. The FWHM of the TiN has increased from 1.6 to 2.5 eV. It is the indication of a higher disorder of the TiN. The peaks characteristics are summarized in Tab.15(b).

Fig. 75 (

 75 Fig.75(a). The spectrum is tted with only one component located at 1561.8 eV binding energy with the product between a Gaussian function (80%) and a Lorentzian function (20%) with a FWHM of 1.4 eV. This peak is attributed to Ti-Al alloy with incorporation of electronegative element, but as the binding energy is lower than the one found for the 600°C sample, the alloy is in a lower oxidation state. For the same reason as previously (for Ti 1s), this peak is attributed to TiAlN. The peak characteristics are summarized in Tab.15(a).

Figure 75 :

 75 Figure 75: High resolution spectra of Ti 1s (a) and (b) Al 1s measured at 8 keV photon energy for the 900°C sample. The spectra are normalized to unit area.

134 4. 5 . 3

 13453 Inelastic background analysis of Ti 1s and Ga 2s,2p

  (a) shows a top interface located at 32 nm below the surface. The background reproduction in Fig. 77(b) is almost perfect in the full energy range used for computing the error and even beyond.

Figure 76 :

 76 Figure 76: (a) Result from automated treatment of the 900°C sample Ti 1s spectrum, showing a depth distribution with top interface located at 1.5 nm and a bottom interface located at 42 nm below the surface. The corresponding measurement (in black) and the corresponding modelled background (in blue) also with the subtracted spectrum (in red) are presented in (b) with the energy range used for calculation of χ e .

Figure 77 :

 77 Figure 77: (a) Result from automated treatment of the 900°C sample Ga 2s,2p spectrum, showing a depth distribution with top interface located at 25.0 nm and a bottom interface beyond detection limit. The corresponding measurement (in black) and the corresponding modelled background (in blue) also with the subtracted spectrum (in red) are presented in (b) with the energy range used for calculation of χ e .

Figure 78 :

 78 Figure 78: Structure of the HEMT sample annealed at 600+900°C. The depths of the layer are extracted from the inelastic background analysis and the dierent layers from the core-level analysis.

  . The combination of these results provides an insight on the sample structure and allows to propose the reaction scheme shown in Fig. 82. The results are supported by the conclusion derived from the TEM images and EDX proles shown in the following.

Figure 79 :

 79 Figure 79: Component evolution during annealing; (a) Ti 1s peak; (b) Al 1s peak.

Fig. 80 (

 80 a) shows the TEM image of the 600°C sample with the Al represented in blue, the Ti presented in yellow and the N in red. The sample shows lateral inhomogeneities at the ∼ 50 nm scale. This eect is important as the intermixing of Ti and Al is either total (on the left part) or partial. The encapsulation of the sample needed to prepare the sample for obtaining the TEM image has an impact on the sample surface. For this reason the EDX prole is measured at the arrow position of the TEM image, which corresponds to the mean location of the surface. As the studied sample presents an increased thickness on both Ti and Al layers, the intermixing seen in the EDX prole of Fig. 80(b) at the Ti and AlGaN interface can be qualitatively compared to the results of the 600°C sample studied by photoemission. The Ga and N signals decrease abruptly at the exact location where the Al signal appear. Then theinterface of ∼ 50 nm wide can be decomposed in two parts, one near the AlGaN layer where the Ti and Al concentrations are about 40 and 55% respectively, and a second part where the Al concentration increases and Ti concentration decreases. In both regions, the N atoms are in a non-negligible concentration. The major part of the sample consists in a TiAl 3 alloy, located between the substrate and the surface.

Figure 80 :

 80 Figure 80: TEM image (a) with scale bar of 500 nm. The EDX prole (b) is measured along the arrow in the top image for a sample annealed at 600°C.

Figure 82 :

 82 Figure 82: Scheme of the HEMT structure obtained with high resolution spectra and inelastic background analysis. (a) Unannealed sample, (b) sample annealed at 600°C, (c) sample annealed at 600+900°C.

Figure 86 :

 86 Figure 86: Wiggler scheme

Figure 87 :

 87 Figure 87: Scheme of a bending Magnet

  Photon energy range

Figure 88 :

 88 Figure 88: Scheme of Transmission electron microscope.

Figure 89 :

 89 Figure 89: Schematic diagram of an Atom Probe Tomography experiment.

Figure 90 :

 90 Figure 90: ToF SIMS principles. On the left the pink ions are the sputtering ion beam, whereas the purple are the analysis ion beam. The right panel shows successive sequences of sputtering (green), extracting (blue) and analysis (red). The resulting spectrum (in black) is obtained. Moreover, it is possible to obtain ionic image with a surface scan and combining the depth proling with imaging, permits to visualize a 3D render.

Figure 91 :

 91 Figure 91: Scheme of an ellipsometer experimental apparatus. The elements appear in black whereas the light polarization is in blue.

  Photons(1-10 eV)

Figure 92 :

 92 Figure 92: (a) Reection and transmission at an interface, r is the reected ray and t the transmitted ray. (b)

Table 1 :

 1 Theoretical maximum probed depth for Si 2p of 99.6 eV. The IMFP calculated from TPP-2M formula.

	It is used

Table 2 :

 2 Depth distribution of lanthanum for 50 nm sample.TEM and EELS Because La and Hf have an atomic number too close one from each other ( 57 La and 72 Hf ) and the 1ML La layer is located just next to the Hf-containing layer, TEM experiment cannot give a straightforward result for this type of stack. The result for 50 nm sample from TEM experiment can only give the combined thickness of the La and Hf layers. The a-Si and TiN thicknesses obtained are very accurate so that the burying depth of the active layer can be

	Method	Top interface (nm)	Bottom interface (nm)	Thickness (nm)
		Nominal	Measured	Nominal	Measured	Nominal	Measured
	Background Analysis	56.5	57.3	56.9	61.8	0.4	4.5
	Auger Depth Proling		60.0		70.0		10.0
	TEM (La + Hf )		55.7		57.1		1.4
	Atom Probe Tomography						1.0

Depth distribution proles of lanthanum obtained for TEM measurements, Atom probe Tomography, Auger depth proling and Inelastic Background Analysis are summarized in Tab.2. estimated at 51.1 nm below the surface. Fig 30 shows a high-resolution TEM image measured using a JEOL 2010FEF operated at 200 kV with an overall lateral resolution better than 0.2 nm.

Table 4 :

 4 e Ti1s (nm) λ e La2p 3/2 (nm) λ e Hf2p 3/2 (nm) Table of eective IMFP λ in nm and their associated estimation of uncertainty for 12, 15 and 18 keV for 50 nm sample.

	12 keV	11.9 (12%)	11.4 (11%)
	15 keV	16.2 (16%)	15.4 (15%)	9.5 (9%)
	18 keV	20.3 (21%)	19.6 (19%)	13.8 (14%)

Table 5 :

 5

		interface (nm)	Bottom interface (nm)	Thickness (nm)
		Nominal	Modeled	Nominal	Modeled	Nominal	Modeled
	50 nm sample	56.5	57.3	56.9	61.8	0.4	4.5
	30 nm sample	36.5	41.0	36.9	43.0	0.4	2
	20 nm sample	26.5	35.0	26.9	36.0	0.4	1

Positions of top and bottom interfaces of lanthanum obtained by background analysis.

Table 6 :

 6 of 12897 eV. Positions of top and bottom interfaces obtained by background analysis for the 20 nm sample and the high-k sample.

		top interface (nm)	bottom interface (nm)	Thickness (nm)
		Expected	Modeled	Expected	Modeled	Expected	Modeled
	20 nm sample	26.5	35.0	26.9	36.0	0.4	1.0
	high-k sample	26.5	43.2	27.5	45.4	1.0	2.2

Table 8 :

 8 Parameters inuencing the obtained thickness of a layer when performing inelastic background analysis with reference sample.

Table 9 :

 9 Results of inelastic background analysis for annealed samples.

	107

5, 35.8

and 30.9 nm below the surface (without reference sample) are very close to the nominal values of 56.5, 36.5 and 26.5 nm given by deposition tools.

Table 10 :

 10 Thicknesses of La layer in nm obtained with various methods. IBA stands for inelastic background

	Sample		50 nm	50 nm annealed	30 nm	30 nm annealed	20 nm	20 nm annealed	high-k	high-k annealed
	APT		1.0	1.5
	Auger		10	>10
	IBA		4.5	7.4	2	6.5	1		4.1	2.2	5.6
	IBA with reference	2.3	8.0	$	6.7	$		4.7	$
	analysis.
	3.6 Conclusion of Chapter III
	Thicknesses obtained by background analysis of HAXPES spectra using Tougaard's implemented
	method are summarized in Tab.11.
				20 nm sample	30 nm sample		50 nm sample
	Nominal (nm)	No ref (nm)	No ref (nm)	No ref (nm)	With Ref (nm)
	Ti	6.5							11	4.1-4.5
	La	0.4		1		2			4

Table 12 :

 12 Eective IMFP for Al, Ti and Ga according to Eq.7 with photon energy of 8 keV.

	Element Kinetic energy	Nominal Thickness	Individual IMFP	Eective IMFP	3 x IMFP	8 x IMFP
		(eV)	(nm)	(nm)	(nm)	(nm)	(nm)
	Al 1s	6425	10	10.8	10.8	32.4	86.4
	Ti 1s	3011	15	4.2	4.8	14.4	38.4
	Ga 2s	6750	∞	10.8	9.5	28.5	76

Table 13 :

 13 Components used in the peak tting procedure for Al 1s and Ti 1s of the unannealed sample.

	Assignment Ti metal TiN Al metal Al surface oxide	Binding energy (eV) 4964.7 4965.6 1559.8 1562.7	Peak shape Doniach-Sunjic α=0.05 Gaussian 80% -Lorentzian 20% Doniach-Sunjic α=0.12 Gaussian 80% -Lorentzian 20%	FWHM (eV) Area (%) 1.1 96.5 1.3 3.5 0.7 87.9 1.4 12.1

Table 14 :

 14 Components used in the peak tting procedure for Al 1s (a) and Ti 1s (b) of the sample annealed at 600°C.

Table 15 :

 15 Components used in the peak tting procedure for Al 1s (a) and Ti 1s (b) of the sample annealed at 600+900°C.

Table 16 :

 16 Components area for all samples in percentage; (a) Ti 1s peak; (b) Al 1s peak.*The Ti metal component of the 600°C sample is representative of metallic Ti and stoichiometric Ti-Al alloy.

Table 17 :

 17 ). Table of depth distribution proles obtained.

	Element	Ti

Table 18 :

 18 Table of HAXPES beamline worldwide.*ID32 is no longer in operation.

Table 19 :

 19 Table of commercial HAXPES (a) and XPS (b) apparatus and their principal features.

Table 20 :

 20 Experimental techniques and their principal features.

	6-8 nm

at 17.5 eV). The loss structure is due mostly to inelastic scattering events of photoelectrons traveling through the a-Si overlayer. The contribution to the inelastic background of much thinner layers located below a-Si can be neglected in a rst approximation. It is also noticeable that the

Remerciements

Je tiens à remercier Jean-Claude Royer ainsi que Olivier Renault pour m'avoir permis de venir eectuer ma thèse au seins du service de caractérisation. J'espère leur avoir rendu la conance qu'ils m'ont accordée.

Je tiens également à remercier Geneviève Grenet et la patience qu'elle a eut avec moi. Son aide dans les dicultés scientiques et administratives m'ont été d'un grand secours.

and the bottom interface at 57.4 nm (instead of 49.5 and 56.2 nm without use of reference sample).

If the dierence in top interface of 0.1 nm between the analysis with and without reference sample is too low to be signicant, the bottom interface changes by 1.2 nm. The resulting thickness of the La layer of 8.0 nm shows the diusion of La during annealing when compared to the 2.3 nm obtained (with reference sample) for the unannealed sample.

The Fig. 55(b) shows the two subtracted spectra obtained with (red) and without (blue) use of the reference sample, the dierence between the two spectra is plotted in the top panel. The two spectra correspond to two close depth distributions but the spectrum with use of reference samples is closer to the baseline over the full energy loss region. As this spectrum presents a better match to the measurement, the corresponding depth distribution of top interface located at 49.4 nm below the surface and bottom interface at 57.4 nm is more trustworthy.

Figure 55: The two subtracted spectra with (in red) and without (in blue) reference sample, the top layer shows the dierence between these two spectra. [START_REF] Siegbahn | Esca atomic, molecular and solid state structure studied by means of electron spectroscopy[END_REF] nm sample The inelastic background analysis of the annealed 30 nm sample was performed on the La 2s measurement taken at GALAXIES beamline. The better stability and brilliance of the beam give a better recording the La energy loss signals. The result of the automated procedure is shown in Fig. 56 The drastic decrease of metallic Al component is conrmed by the Ti 1s inelastic background presented in Fig. 72(a): the strong oscillations corresponding to Al plasmon excitations near the no-loss peak seen in the unannealed sample (Fig. 68(b)) have decreased. As the Ti is still in metallic form, these oscillations arise from plasmon loss inside the Ti layer. As a result, the inelastic scattering cross section is chosen as a weighted averaging of the cross-section of TiO and the two parameters universal cross-section.

The result of the automated procedure on Ti 1s is showed in Fig. 72(a). It points to a lowest χ e found for a Ti layer located between 7.3 nm and 20.4 nm for top and bottom interface respectively.

As can be seen in Fig. 72(b), the reproduction of the background for such a depth distribution is rather poor. As a manual trial and error process could not lead to a better matching between the measurement and the modelled background, this discrepancy cannot be imputed to an erroneous estimation made by the automated procedure. The discrepancy is attributed to the choice of the inelastic scattering cross-section: the high resolution spectra show intermixing of Ti and Al in two forms, a TiAl alloy and a TiAl oxidized. Therefore the most accurate inelastic cross-section describing the inelastic losses inside the sample would be an average of the TiAl cross section and the TiAl ox cross section. As these cross sections are not implemented in the software, the use of TiO and the two parameters cross section is an approximation which implies a high uncertainty in the results of inelastic background analysis. It could explain the shrinking of the Ti layer compared to the unannealed sample.

The analysis of Ga 2s,2p inelastic background is shown in Fig. 73(a) and 73(b). The quality of the background analysis is very good over the entire energy range from 6550 to 6900 eV. The top interface is found to be located at 22 nm below the surface and a bottom interface deeper than 75 nm (probing depth limit of this peak). For the same reason as in the case of the unannealed sample, only the top interface location for Ga can be obtained from inelastic background analysis (8xIMFP = 76 nm, see Tab.12). high resolution spectra. This back and forth analysis provides a global description of the sample and allows to suggest a scheme for the various annealing presented.

Future trends

The optimized procedure (bot) proposed in this work allows to discriminate, in the case of the buried layer model, two dierent inelastic background shapes corresponding to depth distributions varying by less than 1 nm at burying depths larger than 20 nm. The procedure can be directly applied to XPS data treatment. However, the reliability of the error calculation needs to be further improved, for example by analyzing HAXPES simpler bilayer systems.

Such an automated procedure could have an important application in real-time processing of XPS data, as required for instance for in-line control in semiconductor industry. Besides in line control, one can also think about in-situ characterization of deposition or annealing processes of materials in a stack. The processing time is however too long for the moment to allow this: in the present version of the program, processing times of 50 min for a depth range of 0 to 300 nm and 0 to 500 nm for top and bottom interfaces (case of Ti 1s of the unannealed HEMT sample) has been achieved. Working on the data processing time is also a prerequirement to be used in laterally resolved background analysis, as can be implemented in a photoelectron emission microscope (PEEM).

In spectroscopic, energy-ltered XPEEM, one would have to consider the analysis of a series of inelastic background spectra generated within the microscope eld-of-view, and ultimately at each image pixel, in order to obtain a map of the elemental depth distribution. The parallel treatment of these spectra would require a drastic reduction of the processing time, and maybe also, to apply noise reduction techniques. Such measurements are becoming possible in HAXPES with the recent HAXPEEM instrument [START_REF] Wiemann | Probing buried layers by photoelectron spectromicroscopy with hard x-ray excitation[END_REF][START_REF] Patt | Bulk sensitive hard x-ray photoemission electron microscopy[END_REF]. During this work, preliminary experiments were performed and the rst results should be available soon, opening the way to microscopic analysis of deeply buried interfaces.

Appendix I -Dipole approximation

In 1927, P. Auger observed that the intensity of the photoelectron emitted with use of unpolarized

x-rays depends only on the angle between the incident photon and the emerging photoelectron [START_REF] Auger | Etude eexperimental des directions d'émission des photoélectrons[END_REF].

The electric-dipole approximation assumes the electromagnetic eld of the photon beam, exp(ikr), expressed as a Taylor-series expansion 1 + ikr + ..., can be truncated to unity [START_REF] Bethe | Quantum Mechanics of One-and Two-electron Atoms[END_REF]. This approximation is valid for UV and far UV photon ranges as the photoelectron velocities following UV photoemission are extremely small compared to the speed of the light (i.e. relativistic eect are negligible) and the wavelength of the UV light is much larger than the orbitals from which electrons are ejected [START_REF] Lindle | Breakdown of the dipole approximation in soft-x-ray photoemission[END_REF].

For high energy regime with hν ≥ 5 keV, the breakdown of the dipole approximation becomes complete and therefore require the use of a full Taylor-series expansion. This phenomenon was predicted and observed decades ago [145,146], only recent description valid for HAXPES experiments will be detailed here. The polar angle θ is measured between the photon polarization vector ε and the momentum vector p of the photoelectron. The azimuthal angle Φ is dened by the photon propagation vector k and the projection of p into the x-z plane.

Eects and limitations of the dipole approximation

The electric-dipole approximation leads to the expression of the dierential photoionization crosssection for linearly polarized light as described by Cooper et al. [START_REF] Cooper | Angular distribution of photoelectrons[END_REF]:

Eq.81 then becomes:

for linearly polarized light. For unpolarized or circularly polarized light the Eq.82 becomes:

These rst order corrections are valid for HAXPES experiments [START_REF] Novák | Polarization and dipole eects in hard x-ray photoelectron spectroscopy[END_REF].

Second order correction

The second order correction is used with exp(ikr) = 1 + ikr -1 2 (kr) 2 . Then the expression of Eq.85 becomes [START_REF] Derevianko | Electric-octupole and pure-electric-quadrupole eects in soft-x-ray photoemission[END_REF]:

where four new angular-distribution parameters ∆β, η, µ and ξ are introduced. However ∆β is not experimentally easily distinguishable from the rst order correction parameter β and the relationship ξ = -η -µ guarantees only two new independent parameters can be experimentally measured.

Appendix II -Synchrotron radiation

Synchrotron radiation is the electrical eld emitted from charged particles (electrons or positrons).

A synchrotron machine is decomposed in three parts: the linear accelerator which produces the electrons and accelerates them; a booster, where electrons become relativistic and a storage ring where electrons are maintained at constant velocity and orbit as can be seen in g.84. The appendix details SOLEIL synchrotron setup, where the storage ring injection is continuous, on opposite of the ESRF where injection is non-continuous.

In order to perform experiments in a synchotron facility, the operator, qualied as a user, must ll a form which is called a proposal. The proposal is unique and follow a template available online on sychrotron's website. It consists in a small text describing the user experience in previous synchotron experiments and a detailed description of the experiment needed.

The proposal is examined by a scientic committee which attributes beamtime. The scientist in charge of the beamline work out the planning with the users.

For every project is attributed a beamline scientist called local contact, whom prepare the beamtime with the user and helps seting up the beamline to delivers specic photons with dened energy.

This appendix presents a table of the HAXPES beamlines used in this thesis and additional beamlines from the German synchrotron DESY, the English synchrotron Diamond and the Japanese SPRing8.

Linear accelerator

The linear accelerator (Linac) produces electrons thanks to an electron gun: a tungsten lament is heated and produces electrons. They are accelerated with a wehnelt at 90 kV. They pass through a grouper in order to have a velocity near c and form bunches, their energy is then of 15 MeV.

They achieve their path in the linac in two linear acceleration devices, with electromagnetic waves of 3000 MHz to reach energy of 100 MeV. They form bunches of 300 ns with a 3 Hz frequency and enter in the booster with a ratio of 0.3. The others are lost.

Booster

The booster is a ring of 157 m perimeter. Due to an electric eld, electrons gain energy from 100 MeV to 2.75 GeV in a period of 166 ms. Magnetic eld of dipoles inside the booster changes from 0,027 T when they electrons enter from the linac and 0.74 T when they reach the energy of 2.75 GeV and are ready to enter in the storage ring. They enter in the storage ring in a transfer section of 42 m with 3 dipoles and 7 quadrupoles with an eciency of 0.7. The storage ring (of perimeter 354 m) is the part where relativistic electrons must be kept at constant velocity and orbit to produce stable synchrotron radiation for the experiments. In 3rd

Storage ring

Beamlines

The Tab.18 shows some beamlines and their features. All beamlines are operational except ID32 which was closed for user operation on 5 December 2011 and I09 which is not open to the users yet.

Appendix III -Commercial XPS and HAXPES apparatus

This appendix presents some commercial apparatus for XPS and HAXPES measurements. The list is not exhaustive and presents only apparatus designed to perform classical experiment, observing chemical shifts. The laboratory apparatus is composed by the buyer with choice in sources and detector, depending on its need to perform several experiments.

ARXPS, where the acceptance angle of the anlyser is very wide, allowing to measure spectrum from dierent take o angle at the same time.

Near Ambiant pressure (NAP) which is a developping eld. in this case, the apparatus allow to have a signicvant presuure in the ordre of mbar near the sample by dierential pumping. The laboratory sources for XPS are generally Al K α , Mg K α .

The HAXPES sources are generally radiation sources but Zr K α , Ag K α can also be used.

SPECS has the possibility to buy laboratory hard x-rays source based on Cr K α and Cu K α which emits photons of 5417 eV and 8055 eV respectively but to our knowledge, no publication were made with these sources.

Appendix IV-State of the art for depth proling methods

In microelectronics, the characterization techniques have to be sensitive to ultra-thin layers, even to monolayers, sometimes buried under rather thick overlayers. Characterization techniques capable of analyzing such deeply buried thin layers can be roughly classied into three major categories depending on their impact on the sample: destructive techniques, invasive and non-invasive techniques.

The destructive techniques require sample preparation. This is not trivial because the sample can be modied during preparation. For example, to perform a TEM experiment, one of the most commonly used techniques, the sample must be made in form of a thin lamina. It must be cut by Focused Ion Beam (FIB) and then polished to reduce roughness [START_REF] Phaneuf | Applications of focused ion beam microscopy to materials science specimens[END_REF].

The invasive techniques are the ones that may modify the sample during the analysis without necessarily requiring complex sample preparation. Most of these techniques employ ion beam as a probe. Cycles of small sputtering before non-destructive surface analysis can also be classied as invasive.

Non-invasive techniques are techniques that do not (or only slightly) modify the sample.

Appendix V: TLM structures

The samples studied in the thesis are full sheet samples; they are deposited in large area to avoid any border eect. They cannot be used to measure DC characteristics, neither I-V curves, but are representative of the physical processes which will occur in the real device. The thicknesses of the metal layers are reduced compared to the thicknesses in the functional device but are the same as the one used to measure DC characteristics.

The electrical characterizations were performed on samples deposited with the same conditions but on patterned Transfer Length Method (TLM) structures as shown in Fig. 93. The sample annealed at 900°C presents a good response as can be seen in Fig. 94(a) where the I-V curves shows an ohmic contact and the corresponding resistivity R C is found to be 3.7 (±0.8) Ω.mm. For these reasons the research was oriented toward this sample. The aim of the study is to nd the chemical composition dierence between the samples which is believed to be responsible for the quality of the contact formation. The study of the unannealed sample will be used as a control step and the 600°C sample will be seen as an intermediate state before the contact formation expecetd in the 900°C samples.